Modeling and Simulation for Material Selection and Mechanical Design

Modeling and Simulation for Material Selection and Mechanical Design edited by

George E. Totten G.E. Totten & Associates, LLC Seattle, Washington, i7.S.A

Lin Xie Solidworks Corporation Concord, Massachusetts, U.S.A

Kiyoshi Funatani IMST Institute Nagoya, Japan

MARCEL

MARCEL DEKKER, INC. DEKKER

.

NEWYORK BASEL

Although great care has been taken to provide accurate and current information, neither the author(s) nor the publisher, nor anyone else associated with this publication, shall be liable for any loss, damage, or liability directly or indirectly caused or alleged to be caused by this book. The material contained herein is not intended to provide specific advice or recommendations for any specific situation. Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress. ISBN: 0-8247-4746-1 This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc., 270 Madison Avenue, New York, NY 10016, U.S.A. tel: 212-696-9000; fax: 212-685-4540 Distribution and Customer Service Marcel Dekker, Inc., Cimarron Road, Monticello, New York 12701, U.S.A. tel: 800-228-1160; fax: 845-796-1772 Eastern Hemisphere Distribution Marcel Dekker AG, Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-260-6300; fax: 41-61-260-6333 World Wide Web http://www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the headquarters address above. Copyright # 2004 by Marcel Dekker, Inc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA

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ENGINEERING

A Series of Textbooks and Reference Books Founding Editor

L. L. Faulkner Columbus Division, Battelle Memorial Institute and Department of Mechanical Engineering The Ohio State University Columbus, Ohio

1. 2. 3. 4. 5.

Spring Designer's Handbook, Harold Carlson Computer-Aided Graphics and Design, Daniel L. Ryan Lubrication Fundamentals, J. George Wills Solar Engineering for Domestic Buildings,William .A. Himmelman Applied Engineering Mechanics: Statics and Dynamics, G. Boothroyd and C. Poli 6. Centrifugal Pump Clinic, lgor J. Karassik 7. Computer-AidedKinetics for Machine Design, Daniel L. Ryan 8. Plastics Products Design Handbook, Patf A: Materials and Components; Patf 6 : Processes and Design for Processes, edited by Edward Miller 9. Turbomachinery:Basic Theory and Applications, Earl Logan, Jr. 10. Vibrations of Shells and Plates, Werner Soedel 1I.Flat and Corrugated Diaphragm Design Handbook, Mario Di Giovanni 12. Practical Stress Analysis in Engineering Design, Alexander Blake 13. An lntroduction to the Design and Behavior of Bolted Joints, John H. Bickford 14. Optimal Engineering Design: Principles and Applications,James N. Siddall 15. Spring Manufacturing Handbook, Harold Carlson 16. Industrial Noise Control: Fundamentals and Applications, edited by Lewis H. Bell 17. Gears and Their Vibration:A Basic Approach to Understanding Gear Noise, J. Derek Smith 18. Chains for Power Transmission and Material Handling: Design and Applications Handbook,American Chain Association 19. Corrosion and Corrosion Protection Handbook, edited by Philip A. Schweitzer 20. Gear Drive Systems: Design and Application, Peter Lynwander 21. Controlling In-Plant Airborne Contaminants: Systems Design and Calculations, John D. Constance 22. CAD/CAM Systems Planning and Implementation, Charles S. Knox 23. Probabilistic Engineering Design: Principles and Applications, James N. Siddall 24. Traction Drives: Selection and Application, Frederick W. Heilich 111 and Eugene E. Shube 25. Finite Element Methods: An Introduction, Ronald L. Huston and Chris E. Passerello Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

, Brayton Lincoln, and 27. Lubrication in Practice: Second Edition, edited by W. S. Robertson 28. Principles of Automated Drafting, Daniel L. Ryan 29. Practical Seal Design, edited by Leonard J. Martini 30. Engineering Documentation for CAD/CAM Applications, Charles S. Knox 31 . Design Dimensioning with Computer Graphics Applications, Jerome C. Lange 32. Mechanism Analysis: Simplified Graphical and Analytical Techniques, Lyndon 0. Barton 33. CAD/CAM Systems: Justification, Implementation, Productivity Measurement, Edward J. Preston, George W. Crawford, and Mark E. Coticchia 34. Steam Plant Calculations Manual, V. Ganapathy 35. Design Assurance for Engineers and Managers, John A. Burgess 36. Heat Transfer Fluids and Systems for Process and Energy Applications, Jasbir Singh 37. Potential Flows: Computer Graphic Solutions, Robert H. Kirchhoff 38. Computer-AidedGraphics and Design: Second Edition, Daniel L. Ryan 39. Electronically Controlled Proportional Valves: Selection and Application, Michael J. Tonyan, edited by Tobi Goldoftas 40. Pressure Gauge Handbook, AMETEK, U.S. Gauge Division, edited by Philip W. Harland 41. Fabric Filtration for Combustion Sources: Fundamentals and Basic Technology, R. P. Donovan 42. Design of MechanicalJoints, Alexander Blake 43. CAD/CAM Dictionary, Edward J. Preston, George W. Crawford, and Mark E. Coticchia 44. Machinery Adhesives for Locking, Retaining, and Sealing, Girard S. Haviland 45. Couplings and Joints: Design, Selection, and Application, Jon R. Mancuso 46. Shaft Alignment Handbook, John Piotrowski 47. BASIC Programs for Steam Plant Engineers: Boilers, Combustion, Fluid Flow, and Heat Transfer,V. Ganapathy 48. Solving Mechanical Design Problems with Computer Graphics, Jerome C. Lange 49. Plastics Gearing: Selection and Application, Clifford E. Adams 50. Clutches and Brakes: Design and Selection,William C. Orthwein 51. Transducersin Mechanical and Electronic Design, Harry L. Trietley 52. Metallurgical Applications of Shock-Wave and High-Strain-Rate Phenomena, edited by Lawrence E. Murr, Karl P. Staudhammer, and Marc A. Meyers 53. Magnesium Products Design, Robert S.Busk 54. How to Integrate CAD/CAM Systems: Management and Technology, William D. Engelke 55. Cam Design and Manufacture: Second Edition; with cam design software for the IBM PC and compatibles, disk included, Preben W. Jensen 56. Solid-state AC Motor Controls: Selection and Application,Sylvester Campbell 57. Fundamentals ofRobotics, David D. Ardayfio 58. Belt Selection and Application for Engineers,edited by Wallace D. Erickson 59. Developing Three-DimensionalCAD Software with the ISM PC, C. Stan Wei 60. Organizing Data for ClM Applications, Charles S. Knox, with contributions by Thomas C. Boos, Ross S. Culverhouse, and Paul F. Muchnicki

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61. Computer-Aided Simulation in Railway Dynamics, by Rao V. Dukkipati and 62. fiber-Reinforced Composites: Materials, Manufacturing, and Design, P. K. Mallick 63. Photoelectric Sensors and Controls: Selection and Application, Scott M. Juds 64. finite Element Analysis with Personal Computers, Edward R. Champion, Jr., and J. Michael Ensminger 65. Ultrasonics: Fundamentals, Technology, Applications: Second Edition, Revised and Expanded, Dale Ensminger 66. Applied finite Element Modeling: Practical Problem Solving for Engineers, Jeffrey M. Steele 67. Measurement and Instrumentation in Engineering: Principles and Basic Laboratory Experiments, Francis S. Tse and Ivan E. Morse 68. Centrifugal Pump Clinic: Second Edition, Revised and Expanded, lgor J. Karassik 69. Practical Stress Analysis in Engineering Design: Second Edition, Revised and Expanded, Alexander Blake 70. An Introduction to the Design and Behavior of Bolted Joints: Second Edition, Revised and Expanded, John H. Bickford 71. High Vacuum Technology:A Practical Guide, Marsbed H. Hablanian 72. Pressure Sensors: Selection and Application, Duane Tandeske 73. Zinc Handbook: Properties, Processing, and Use in Design, Frank Porter 74. Thermal fatigue of Metals, Andrzej Weronski and Tadeusz Hejwowski 75. Classical and Modern Mechanisms for Engineers and Inventors, Preben W. Jensen 76. Handbook of Electronic Package Design, edited by Michael Pecht 77. Shock-Wave and High-Strain-Rate Phenomena in Materials, edited by Marc A. Meyers, Lawrence E. Murr, and Karl P. Staudhammer 78. Industrial Refrigeration: Principles, Design and Applications, P. C. Koelet 79. Applied Combustion, Eugene L. Keating 80. Engine Oils and Automotive Lubrication, edited by Wilfried J. Bartz 8 1. Mechanism Analysis: Simplified and Graphical Techniques, Second Edition, Revised and Expanded, Lyndon 0. Barton 82. fundamental Fluid Mechanics for the Practicing Engineer, James W. Murdock 83. Fiber-Reinforced Composites: Materials, Manufacturing, and Design, Second Edition, Revised and Expanded, P. K. Mallick 84. Numerical Methods for Engineering Applications, Edward R. Champion, Jr. 85. Turbomachinery: Basic Theory and Applications, Second Edition, Revised and Expanded, Earl Logan, Jr. 86. Vibrations of Shells and Plates: Second Edition, Revised and Expanded, Werner Soedel 87. Steam Plant Calculations Manual: Second Edition, Revised and Expanded, V. Ganapathy 88. Industrial Noise Control: Fundamentals and Applications, Second Edition, Revised and Expanded, Lewis H. Bell and Douglas H. Bell 89. finite Elements: Their Design and Performance, Richard H. MacNeal 90. Mechanical Properties of Polymers and Composites: Second Edition, Revised and Expanded, Lawrence E. Nielsen and Robert F. Landel 91. Mechanical Wear Prediction and Prevention, Raymond G. Bayer

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92. Mechanical Power Transmission Components, edited by David W. South and Jon R. Mancuso 94. Engineering Documentation Control Practices and Procedures, Ray E. Monahan 95. Refractory Linings Thermomechanical Design and Applications, Charles A. Schacht 96. Geometric Dimensioning and Tolerancing: Applications and Techniques for Use in Design, Manufacturing, and Inspection, James D. Meadows 97. An lntroduction to the Design and Behavior of Bolted Joints: Third Edition, Revised and Expanded, John H. Bickford 98. Shaft Alignment Handbook: Second Edition, Revised and Expanded, John Piotrowski 99. Computer-Aided Design of Polymer-Matrix Composite Structures, edited by Suong Van Hoa 100. Friction Science and Technology, Peter J. Blau 10 1. lntroduction to Plastics and Composites: Mechanical Properties and Engineering Applications, Edward Miller 102. Practical Fracture Mechanics in Design, Alexander Blake 103. Pump Characteristics and Applications, Michael W. Volk 104. Optical Principles and Technology for Engineers, James E. Stewart 105. Optimizing the Shape of Mechanical Elements and Structures, A. A. Seireg and Jorge Rodriguez 106. Kinematics and Dynamics of Machinery, Vladimir Stejskal and Michael ValaSek 107. Shaft Seals for Dynamic Applications, Les Horve 108. Reliability-Based Mechanical Design, edited by Thomas A. Cruse 109. Mechanical Fastening, Joining, and Assembly, James A. Speck 110. Turbomachinery Fluid Dynamics and Heat Transfer,edited by Chunill Hah 111. High-Vacuum Technology: A Practical Guide, Second Edition, Revised and Expanded, Marsbed H. Hablanian 112. Geometric Dimensioning and Tolerancing: Workbook and Answerbook, James D. Meadows 113. Handbook of Materials Selection for Engineering Applications, edited by G. T. Murray 114. Handbook of Thermoplastic Piping System Design, Thomas Sixsmith and Reinhard Hanselka 115. Practical Guide to Finite Elements: A Solid Mechanics Approach, Steven M. Lepi 116. Applied Computational Fluid Dynamics, edited by Vijay K. Garg 117. Fluid Sealing Technology, Heinz K. Muller and Bernard S. Nau 118. Friction and Lubrication in Mechanical Design, A. A. Seireg 119. lnfluence Functions and Matrices, Yuri A. Melnikov 120. Mechanical Analysis of Electronic Packaging Systems, Stephen A. McKeown 121. Couplings and Joints: Design, Selection, and Application, Second Edition, Revised and Expanded, Jon R. Mancuso 122. Thermodynamics: Processes and Applications, Earl Logan, Jr. 123. Gear Noise and Vibration, J. Derek Smith 124. Practical Fluid Mechanics for Engineering Applications, John J. Bloomer 125. Handbook of Hydraulic Fluid Technology,edited by George E. Totten 126. Heat Exchanger Design Handbook, T. Kuppan

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127.

for Product Sound Quality, Richard H. Lyon in Franklin E. Fisher and Joy R.

Fisher 129. Nickel Alloys, edited by Ulrich Heubner 130. Rotating Machinery Vibration: Problem Analysis and Troubleshooting, Maurice L. Adams, Jr. 131. Formulas for Dynamic Analysis, Ronald L. Huston and C. Q. Liu 132. Handbook of Machinery Dynamics, Lynn L. Faulkner and Earl Logan, Jr. 133. Rapid Prototyping Technology: Selection and Application, Kenneth G. Cooper 134. Reciprocating Machinery Dynamics: Design and Analysis, Abdulla S. Rangwala 135. Maintenance Excellence: Optimizing Equipment Life-Cycle Decisions, edited by John D. Campbell and Andrew K. S. Jardine 136. Practical Guide to Industrial Boiler Systems, Ralph L. Vandagriff 137. Lubrication Fundamentals: Second Edition, Revised and Expanded, D. M. Pirro and A. A. Wessol 138. Mechanical Life Cycle Handbook: Good Environmental Design and Manufacturing, edited by Mahendra S. Hundal 139. Micromachining of Engineering Materials, edited by Joseph McGeough 140. Control Strategies for Dynamic Systems: Design and Implementation, John H. Lumkes, Jr. 141. Practical Guide to Pressure Vessel Manufacturing, Sunil Pullarcot 142. Nondestructive Evaluation: Theory, Techniques, and Applications, edited by Peter J. Shull 143. Diesel Engine Engineering: Thermodynamics, Dynamics, Design, and Control, Andrei Makartchouk 144. Handbook of Machine Tool Analysis, loan D. Marinescu, Constantin Ispas, and Dan Boboc 145. Implementing Concurrent Engineering in Small Companies, Susan Carlson Skalak 146. Practical Guide to the Packaging of Electronics: Thermal and Mechanical Design and Analysis, Ali Jamnia 147. Bearing Design in Machinery: Engineering Tribology and Lubrication, Avraham Harnoy 148. Mechanical Reliability Improvement: Probability and Statistics for Experimental Testing, R. E. Little 149. Industrial Boilers and Heat Recovery Steam Generators: Design, Applications, and Calculations, V. Ganapathy 150. The CAD Guidebook: A Basic Manual for Understanding and Improving Computer-Aided Design, Stephen J. Schoonmaker 151. Industrial Noise Control and Acoustics, Randall F. Barron 152. Mechanical Properties of Engineered Materials, Wole Soboyejo 153. Reliability Verification, Testing, and Analysis in Engineering Design, Gary S. Wasserman 154. Fundamental Mechanics of Fluids: Third Edition, I. G. Currie 155. Intermediate Heat Transfer, Kau-Fui Vincent Wong 156. HVAC Water Chillers and Cooling Towers: Fundamentals, Application, and Operation, Herbert W. Stanford Ill 157. Gear Noise and Vibration: Second Edition, Revised and Expanded, J. Derek Smith

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158. Handbook of Turbomachinery: Second Edition, Revised and Expanded, Earl Logan, Jr., and Ramendra Roy 159. Piping and Pipeline Engineering: Design, Construction, Maintenance, lntegrity, and Repair, George A. Antaki 160. Turbomachinery: Design and Theory, Rama S. R. Gorla and Aijaz Ahmed Khan 161. Target Costing: Market-Driven Product Design, M. Bradford Clifton, Henry M. B. Bird, Robert E. Albano, and Wesley P. Townsend 162. Fluidized Bed Combustion, Simeon N. Oka 163. Theory of Dimensioning: An lntroduction to Parameterizing Geometric Models, Vijay Srinivasan 164. Handbook of Mechanical Alloy Design, George E. Totten, Lin Xie, and Kiyoshi Funatani 165. Structural Analysis of Polymeric Composite Materials, Mark E. Tuttle 166. Modeling and Simulation for Material Selection and Mechanical Design, George E. Totten, Lin Xie, and Kiyoshi Funatani

Additional Volumes in Preparation Handbook of Pneumatic Conveying Engineering, David Mills, Mark G. Jones, and Vijay K. Agarwal Mechanical Wear Fundamentals and Testing: Second Edition, Revised and Expanded, Raymond G. Bayer Engineering Design for Wear: Second Edition, Revised and Expanded, Raymond G. Bayer Clutches and Brakes: Design and Selection, Second Edition, William C. Orthwein Progressing Cavity Pumps, Downhole Pumps, and Mudmotors, Lev Nelik

Mechanical Engineering Sofmare

Spring Design with an IBM PC, Al Dietrich Mechanical Design Failure Analysis: With Failure Analysis System Software for the IBM PC, David G. Ullman

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In Memoriam

During the preparation of this book, one of our most valued authors and mentors passed away on April 29, 2003. Professor George C. Weatherly (1941–2003) graduated from Cambridge University in 1966. He began his career as a research scientist in the Department of Metallurgy at Harwell. In 1968 he moved to Canada where he worked for the University of Toronto for 22 years as a professor in the Department of Metallurgy and Material Science. In 1990 he became a professor of Materials Science and Engineering at McMaster University. He was Director of Brockhouse Institute for Material Research from 1996–2001 and a Chair of the Department of Materials Science and Engineering. Dr. Weatherly has published over 200 papers in different areas of Materials Science. He was Fellow for the Canadian Institute for Mining and Metallurgy and Fellow of ASM International. George was a devoted scientist in the field of electron microscopy and an educator with a distinguished career at McMaster University and the University of Toronto. He will be cherished by his friends, colleagues, and students for the richness of his life, his quiet humor, his humanity and care for others, and above all for his unfailing honesty. His contributions were many and are written clearly in the lives of those with whom he taught and worked. This book is dedicated to his memory.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Preface

In every industry survey, development and use modeling, and simulation technology are cited among the top five critical needs for manufacturing industries to remain viable and competitive in the future. This is particularly true for materials and component design. To address this need, various research programs are currently underway in government, academic, and industry laboratories around the world. This book addresses a number of selected, important areas of computer model development. Effective material and component design procedures are vitally important with increasing pressures to improve quality at lower production costs for all traditional industrial markets. Advanced design procedures typically involve computer modeling and simulation if the necessary algorithms are sufficiently advanced or by using advanced empirical procedures. The objective is to be able to make design decisions based on numerical simulations as an alternative to time-consuming and expensive laboratory or production experimental process development. In fact, advanced engineering processes are becoming increasingly dependent on advanced computer modeling and design procedures. This book addresses various aspects of the utilization of modeling and simulation technology. Some of the topics discussed include hot-rolling of steel, quenching and tempering during heat treatment, modeling of residual stresses and distortion during forging, casting, heat treatment, mechanical property prediction, modeling of tribological processes as it relates to the design of surface engineered materials, and fastener design. These chapters summarize and demonstrate key numerical relationships used in computer

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

model development and their application at various stages in the material production process. In particular, the material covered in this text includes:















Modeling and simulation of microstructural evolution and mechanical properties of steels during the hot-rolling process, calculation of metallurgical phenomena occurring in steel during hot-rolling, and prediction of mechanical properties from microstructure. Heat treatment processes such as quenching and tempering is an active area for process model development. Models used to simulate the kinetics of multicomponent grain boundary segregations that occur in quenched and tempered engineering steels are discussed. These models permit the evaluation of the effect of alloying elements and various tempering parameters on hydrogen embrittlement, stress-corrosion cracking, and other phenomenon. Of all the various problems associated with component design and production, none are more important that residual stress and distortion. Chapter 3 discusses the metallo-thermo-mechanical theory, numerical modeling and simulation technology, coupling of temperature, inelastic behavior and phase transformation and solidification involved with elastic-plastic, viscoplastic and creep deformation as they relate to quenching, forging, and casting processes. Modeling and simulation of mechanical properties, in particular, material behavior during plastic deformation, low-cycle fatigue, creep, and impact strength. This discussion includes the importance of the determination and implementation of adequate material data, consideration of inelastic material behavior, and the formulation of physically founded material models. Chapter 5 discusses the role played by physico-chemical interactions in modifying and controlling friction and wear of critically loaded tribo-couple surfaces during high-speed cutting operations. A comprehensive overview of one of the most important processes in manufacturing is presented in Chapter 6. Threaded fastener selection and design is addressed with many equations and figures included to aid in the design process.

Chapters 1 through 4 describe advanced computer modeling and simulation processes to predict microstructures, material process behavior, and mechanical properties. Chapters 5 and 6 describe more empirical process design procedures for tribological and fastener design.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

This book will be an invaluable resource for the designer, mechanical and materials engineer, and metallurgist. Thorough overviews of these technologies seldom encountered in other handbooks for materials design are provided. The book is an excellent textbook for advanced undergraduate or graduate engineering courses on the role of modeling and simulation in materials and component design. We are indebted to the vital assistance of various international experts. Special thanks to our spouses for their infinite patience with the various time-consuming tasks involved in putting this text together. We extend special thanks to the staff at Marcel Dekker, Inc. including Richard Johnson, Rita Lazazzaro, and Russell Dekker for their invaluable assistance. Without their assistance, this text would not have been possible. George E. Totten Lin Xie Kiyoshi Funatani

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Contents

Preface Contributors 1

A Mathematical Model for Predicting Microstructural Evolution and Mechanical Properties of Hot-Rolled Steels Masayoshi Suehiro

2

Design Simulation of Kinetics of Multicomponent Grain Boundary Segregations in the Engineering Steels Under Quenching and Tempering Anatoli Kovalev and Dmitry L. Wainstein

3

Designing for Control of Residual Stress and Distortion Dong-Ying Ju

4

Modeling and Simulation of Mechanical Behavior Essam El-Magd

5

Tribology and the Design of Surface-Engineered Materials for Cutting Tool Applications German Fox-Rabinovich, George C. Weatherly, and Anatoli Kovalev

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6

Designing Fastening Systems Christoph Friedrich

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Contributors

Essam El-Magd, Dr.-Ing.habil.

Aachen University, Aachen, Germany

German Fox-Rabinovich, Ph.D., D.Sc. Ontario, Canada

RIBE Verbindungstechnik GmbH, Schwa-

Christoph Friedrich, Dr.-Ing. bach, Germany Dong-Ying Ju, Ph.D. Japan Anatoli Kovalev, D.Sc.

Saitama Institute of Technology, Okabe, Saitama,

Physical Metallurgy Institute, Moscow, Russia

Masayoshi Suehiro, Dr.Eng. Chiba, Japan Dmitry L. Wainstein, D.Sc.

Nippon Steel Corporation, Futtsu-City,

Physical Metallurgy Institute, Moscow, Russia

George C. Weatherly, Ph.D.y Canada

y

McMaster University, Hamilton,

McMaster University, Hamilton, Ontario,

Deceased

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1 A Mathematical Model for Predicting Microstructural Evolution and Mechanical Properties of Hot-Rolled Steels Masayoshi Suehiro Nippon Steel Corporation, Futtsu-City, Chiba, Japan

I.

INTRODUCTION

A model for calculating the mechanical properties of hot-rolled steel sheets from their processing condition makes it possible not only to design chemical compositions and processing conditions of steels through off-line simulation but also to guarantee the mechanical properties of hot-rolled steels through on-line simulation. From this point of view, some attempts have been made to develop a mathematical model for calculating the evolution of austenitic microstructure of steels during hot-rolling process and their transformations during cooling subsequent to hot-rolling [1–3]. The mathematical models basically consist of four models for calculating metallurgical phenomena occurring in hot-strip mill and a model for predicting mechanical properties from the microstructure of steel calculated by the metallurgical models. In this chapter, the basic idea and several applications of the mathematical model will be presented.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 1 Schematic illustration of a hot-strip mill.

II.

THE OVERALL MODEL

Since mechanical properties of hot-rolled steels are determined by their microstructure, a model for calculating the mechanical properties of hot-rolled steels is composed of two kinds of models: one for calculating microstructure of steels from their processing conditions, and the other for calculating their mechanical properties from their microstructure. There are several kinds of hot-rolled steel products: sheet and coil, plate, beam, wire, rod, bar, etc. Although the processing conditions are dependent upon each process, each product is produced through the processes such as heating, hot-working, and cooling. Figure 1 shows the schematic illustration of a hot-strip mill. Hot-rolled steel sheets are produced through slab reheating, rough hot-rolling, finish hot-rolling, cooling, and coiling. Table 1 shows the typical thickness and temperature changes in this process and the metallurgical phenomena occurring through this process. In the slab-reheating process, transformation from ferrite and pearlite to austenite and grain growth take place. The

Table 1 The Changes in Thickness and Temperature of Steels and Metallurgical Phenomena in a Hot-Strip Mill Thickness (mm)

Temperature (8C)

Slab reheating

250

1200

Rough rolling Finish rolling Cooling Coiling

!40

1200–1000

!3

1000–850

3 3

— 600–700

Process

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Metallurgical phenomena Transformation, grain growth, dissolution, and precipitation of precipitates Recovery, recrystallization, grain growth, precipitation Recovery, recrystallization, grain growth, precipitation Transformation, precipitation Precipitation

Figure 2 The structure of the model for calculating microstructural evolution and mechanical properties of hot-rolled steels.

recovery and recrystallization, and grain growth of austenitic microstructure occur during and after rough and finish hot-rolling and the transformation from austenite to ferrite, pearlite, bainite, and martensite takes place during cooling and coiling. In the case where steels include alloying elements that form carbides or nitrides, precipitation of such carbides and nitrides takes place and affects recovery, recrystallization, and grain growth in each process. Accordingly, in order to calculate the microstructural evolution of hot-rolled steels, the model used to calculate recovery, recrystallization, grain growth during and after hot deformation, transformation kinetics during cooling and precipitation kinetics in each process is shown in Fig. 2.

III.

BASIC KINETIC EQUATION

In recrystallization and transformation, a new phase forms and grows. These new phases continue to grow until they meet each other and stop growing. This situation is called hard impingement and can be expressed by using the Avrami type equation (4a,4b,4c) X ¼ 1  expðktn Þ

ð1Þ

or the Johnson–Mehl equation (5). In these equations, the concept of extended volume fraction is adopted. By using this concept, the hard impingement can be taken into consideration indirectly. The extended volume

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fraction is the sum of the volume fraction of all new phases without direct consideration of the hard impingement between new particles and is related to the actual volume fraction by X ¼ 1  expðXe Þ

ð2Þ

where X is the actual volume fraction and Xe is the extended volume fraction. The general form of the equation was developed by Cahn [6]. A brief explanation is presented here. The nucleation sites of new phases would be grain boundaries, grain edges, and=or grain corners. In the case of grain boundary nucleation, the volume fraction of a new phase after some time can be expressed as follows. Cahn considered the situation illustrated in Fig. 3 and calculated the volume of the semicircle. In his calculation, firstly, the area at the distance of y from the nucleation site B is calculated. The summation of this area for all nuclei gives the total extended area. From this value, the actual area can be calculated. The extended volume can be obtained by integrating the area for all distances. Finally, the actual volume fraction can be derived.

Figure 3 Schematic illustration of the situation of new phase at time t which nucleates at time t at grain boundary B.

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The area of the section at a plane A for a semicircle nucleated at a plane B is considered. The radius r at time t can be expressed as r ¼ ½G2 ðt  tÞ2  y2 1=2 r¼0

for y < Gðt  tÞ for y  Gðt  tÞ

ð3Þ

where G is the growth rate of new phase and t is the time when the new phase nucleates at plane B. In this calculation, the growth rate is assumed to be constant. From this radius, the extended area fraction dYe for the new phases nucleated at time between t and t þ dt can be obtained as dYe ¼ pIs dt½G2 ðt  tÞ2  y2  for y < Gðt  tÞ for y > Gðt  tÞ dYe ¼ 0

ð4Þ

where Is is the nucleation rate at unit area. By integrating for the time t from 0 to t, the extended area fraction at the plane A at time t can be obtained as Zt Ye ¼

ty=G Z

½G2 ðt  tÞ2  y2  dt

dYe ¼ pIs 0

ð5Þ

0

By exchanging y=Gt for x, this equation leads to
 3 2 3 1x 2  x ð1  xÞ Ye ¼ pIs G t for x < 1 3 Ye ¼ 0

for x > 1

ð6Þ

The actual area fraction of new phases at plane A, Y can be calculated using Ye from Y ¼ 1  expðYe Þ

ð7Þ

The integration of Y for y from 0 to infinity gives the volume of new phases nucleated at unit area of plane B,V0, as     Z1 Z1
3 2 3 1x 2 V0 ¼ 2 Y dy ¼ 2Gt  x ð1  xÞ 1  exp pIs G t dx 3 0

0

ð8Þ Multiplying V0 by the area of nucleation site, the extended volume fraction is obtained as Xe ¼ SV0 ¼ bs1=3 fs ðas Þ

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ð9Þ

where Is Ns ¼ 8S3 G 8S4 G    Z1
3 3 1x 2 1  exp pas dx  x ð1  xÞ fs ðas Þ ¼ as 3 as ¼ ðIs G2 Þ1=3 t;

bs ¼

ð10Þ

0

and Ns the nucleation rate for unit volume. The actual volume fraction X is expressed as X ¼ 1  expðbs1=3 fs ðas ÞÞ

ð11Þ

From this equation, two extreme cases can be considered. One is the case where as is very small and the other is extremely large. For these two cases, the equation becomes X ¼ 1  expðp=3Ns G3 t4 Þ as 51

ð12Þ

X ¼ 1  ð2SGtÞ as 41

ð13Þ

Equation (12) is the same as the one obtained for the case of random nucleation sites by Johnson–Mehl. This equation implies that the increase in the volume of new phases is caused by nucleation and growth. On the other hand, Eq. (13) does not include nucleation rate and it implies that the nucleation sites are covered by new phases and the increase in the volume is dependent only on the growth of new phases. This situation is referred to as site saturation [6]. Cahn did this type of formulation for the cases of grain edge and grain corner nucleations. Table 2 shows all the extreme cases. For all cases, the increase of the volume of new phases for the case of small as conforms to the case of nucleation and growth and site saturation for the case of large as The value of as increases when the nucleation rate is small when compared to the growth rate. The early stage of reaction corresponds to small as and

Table 2 The Kinetic Equations Depending on the Modes and the Nucleation Sites of Reaction in Accordance with Cahn’s Treatment Nucleation site

Nucleation and growth

Site saturation

Grain boundary Grain edge Grain corner

X ¼ 1  expðp=3N_ G 3 t4 Þ

X ¼ 1  expð2SGtÞ X ¼ 1  expðpLG2 t2 Þ X ¼ 1  expðð4p=3ÞCG 3 t3 Þ

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the latter stage corresponds to large as. From Table 2, we can recognize that the exponent of time depends on the mode of reaction and the type of nucleation site for the case of site saturation. A comparison of this information with the experimental results gives useful information on the mode of reaction and the nucleation site. The equations in Table 2 can be used for calculating actual reactions such as transformation and recrystallization by introducing fitting parameters obtained from experiments [7].

IV.

UTILIZATION OF THERMODYNAMICS FOR THE CALCULATION OF TRANSFORMATION AND PRECIPITATION KINETICS

As transformation and precipitation kinetics are closely related to phase equilibrium, thermodynamics can be utilized for their calculation. In this section, the method for utilizing thermodynamics for the calculation will be explained. For the consideration of kinetics, the Gibbs free-energy–composition diagram is much more useful and should be the basis. Figure 4 shows the Gibbs free-energy–composition diagram for austenite and ferrite in steels. Chemical composition at the phase interface between ferrite and austenite is obtained from the common tangent for free-energy curves of ferrite and austenite. The common tangent can be calculated under the condition that chemical potentials of all chemical elements in ferrite are equal to those in austenite. This condition is expressed as mai ¼ mgi

ð14Þ

where m is the chemical potential, the suffix i represents all elements in the system and a and g indicate ferrite and austenite, respectively. In Fig. 4, the driving force for transformation from austenite to ferrite, DGm, is indicated as well. It can be calculated by X  g  ð15Þ xai mi  mai DGm ¼ where x is the fractions of elements. These values are necessary for the calculation of moving rate of the interface during transformation and precipitation. The Zener–Hillert equation [8,9], which represents the growth rate of ferrite into austenite, is expressed as G¼

1 Cga  Cg D 2r Cg  Ca

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ð16Þ

where D is the diffusion coefficient of C in austenite, r the tip-radius of growing phase, Cga, Ca, and Cg is the carbon content in austenite, ferrite at a=g interface and in ferrite apart from interface, respectively. The carbon content at the interface can be calculated from the common tangent between two phases as shown in Fig. 4. There is the other type of expression of moving rate of interface which is expressed as v¼

M DGm Vm

ð17Þ

where M is the mobility of interface, and Vm is the molar volume. The driving force in this equation can be calculated for multicomponent system by Eq. (15). This calculation makes it possible to consider the effect of alloying elements other than the pinning effect and the solute-drag effect. Details of the thermodynamic calculation have been published [10–12]. Recently, some commercial software for the thermodynamic calculation have been used for this type of calculation [13].

Figure 4 Gibbs free energy vs. chemical composition diagram.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

V.

BASIC MODELS

A.

The Concept of the Model

As mentioned above, the overall model for predicting mechanical properties of hot-rolled steels consists of several basic models: the initial state model for austenite grain size before hot-rolling, the hot-deformation model for austenitic microstructural evolution during and after hot-rolling, the transformation model for transformation during cooling subsequent to hot-rolling, and the relation between mechanical properties and microstructure of steels. In the case where steels include alloying elements which form precipitates, the model for precipitation is necessary. Precipitates affect all the models mentioned here. In this section, these basic models will be explained [14,15]. B.

Initial State Model

In this model, austenite grain sizes after slab reheating, namely before hot deformation, are calculated from the slab-reheating condition. In steels consisting of ferrite and pearlite at room temperature, austenite is formed between pearlite and ferrite and it grows into ferrite according to decomposition of pearlite. After all the microstructures become austenite, the grain growth of austenite takes place. We should formulate these metallurgical phenomena to predict austenite grain size after slab reheating. In hot-strip mill, however, the effect of initial austenite grain size on the final austenite grain size after multi-pass hot deformation is small. This can be due to the high total reduction in thickness by several hot-rolling steps in which the recrystallization and grain growth are repeated and the size of austenite grain becomes fine. This means that the high accuracy is not required for the prediction of the initial austenite grain size in a hot-strip mill. From this point of view, the next equation (14) can be applied n

o pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dg ¼ exp 1:61 ln K þ K2 þ 1 þ 5 K ¼ ðT  1413Þ=100 ð18Þ where dg is the austenite grain size after reheating of slab and T is the temperature in K. On the other hand, the initial austenite grain size affects the final austenite grain size in the case of plate rolling because the total thickness reduction is relatively small compared to hot-strip rolling. In this case, the high accuracy of the prediction may be required and the model that is applicable for this case has been reported [16]. Three steps are considered in this model: (1) the growth of austenite between cementite and ferrite according to the dissolution of cementite, (2) the growth of austenite into

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ferrite at a þ g two-phase region, and (3) the growth of austenite in the g single-phase region. The pinning effect by fine precipitates on grain growth and that of Ostwald ripening of precipitates on the grain growth of austenite are taken into consideration. This model is briefly explained in the following paragraphs. The growth of austenite due to the dissolution of cementites can be expressed as dðdg Þ Dgc Cyg  Cga ¼ dg Cga  Ca dt

ð19Þ

where t is the time, Dgc the diffusion constant of C in austenite, and Cg, Cga are the C content in austenite at g=y phase interface and g=a phase interface, respectively. In the a þ g two-phase region, the austenite grain size depends on the volume fraction of austenite, Xg, which changes according to temperature. This situation is expressed as   3Xg 1=3 dg ¼ ð20Þ 4pn0 where n0 is the number of austenite grains at a unit volume when cementites are dissolved. Grain growth occurs in the austenite single-phase region. For grain growth, it is necessary to consider three cases; without precipitates, with precipitates, and with precipitates growing due to the Ostwald ripening. There are equations which are formulated to theoretically correspond to these three cases. They are summarized by Nishizawa [17]. The equation for the normal grain growth is expressed as d2g  d2g0 ¼ k2 t

ð21Þ

where k2 is the factor related to the diffusion coefficient inside the interface, the interfacial energy, and the mobility of the interface. With the pinning effect by precipitates, the growth rate becomes   dðdg Þ 2sV 3sVf ¼M  DGpin ; DGpin ¼ ð22Þ dt R 2r where f is the volume fraction of precipitates and r is the average size of precipitates. When precipitates grow according to the Ostwald ripening, the average size of precipitates used in the Eq. (22) is obtained from r3  r30 ¼ k3 t

ð23Þ

where k3 is the factor related to temperature, interfacial energy and the diffusion coefficient of an alloying element controlling the Ostwald ripening of

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precipitates. By this calculation method, it is possible to predict the growth of austenite grain during heating when precipitates such as AlN, NbC, TiC, and TiN exist in austenite [16].

C.

Hot-Deformation Model

The hot-deformation model is required to predict the austenitic microstructure before transformation through recovery, recrystallization, and grain growth in austenitic phase region during and after multi-pass hot deformation. Sellars and Whiteman [18,19] made the first attempt on this issue and then several researchers [20–27] developed models to calculate recovery, recrystallization, and grain growth. These models are basically similar to each other. In some models, dynamic recovery and dynamic recrystallization are taken into consideration. The dynamic recovery and recrystallization are likely to occur when the reduction is high for single-pass rolling or strain is accumulated due to multi-pass rolling. They should be taken into consideration in finishing rolling stands of a hot-strip mill because, the inter-pass time might be less than 1 sec and the accumulation of strain might take place. Here, the hot-deformation model will be explained based on the model developed by Senuma et al. [20]. In this model, dynamic recovery and recrystallization, static recovery and recrystallization, and grain growth after recrystallization are calculated as shown in Fig. 5. The critical strain, ec, at which dynamic recrystallization occurs is generally dependent upon strain rate, temperature, and the size of austenite grains. The effect of strain rate on ec is remarkable at low strain rate region [28]. One of the controversial issues had been whether the dynamic recrystallization took place or not when the strain rate is high such as that in a hot-strip mill. Senuma et al. [20] showed that it takes place and the effect of strain rate on ec is small at a high strain rate. The fraction dynamically recrystallized, Xdyn, and can be expressed based on the Avrami type equation as   ! e  ec 2 ð24Þ Xdyn ¼ 1  exp 0:693 e0:5 where e0.5 is the strain at which the fraction dynamically recrystallized reaches 50%. On the other hand, the fraction statically recrystallized can be expressed as   ! t  t0 2 Xdyn ¼ 1  exp 0:693 ð25Þ t0:5

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 5 Schematic illustration of microstructural change due to hot deformation.

where t0.5 is the time when the fraction statically recrystallized reaches 50% and t0 is the starting time of static recrystallization. The growth of grains recrystallized dynamically after hot deformation is much faster than normal grain growth in which grains grow according to square of time. This rapid growth was treated with different equations [19–24]. The reason why this rapid growth takes place might be caused by the increase of the driving force for grain growth due to high dislocation density [20], the change in the grain boundary mobility [24] or the annihilation of the small size grains at the initial stage [19]. In the case of multi-pass deformation, the strain might not be reduced completely at the following deformation due to the insufficient time interval and the effect of accumulated strain on the recovery and recrystallization should be taken into consideration. This effect is remarkable for a hot-strip mill because of the short inter-pass time and for steels containing alloying elements which retard the recovery and recrystallization. This effect can be formulated by using the change in the residual strain [22,25] or the dislocation density [20,21,24]. In the modeling process, the accumulated strain is

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

calculated from the average dislocation density which is obtained by calculating the changes in the dislocation density in the region dynamically recovered, rn, and in the region recrystallized dynamically, rs, according to time independently. This method makes it possible to calculate the changes in grain size and dislocation density. Table 3 shows the summary of equations used in the model developed by Senuma et al. The numbers of phenomena in Table 3 correspond to those in Fig. 5. Figure 6 shows an example of calculation of the changes in grain size and dislocation density [14]. Figure 7 shows the calculation result of the effect of the initial austenite grain size on the final microstructure in the finishing stands of a hot-strip mill, which shows that the initial austenite grain size does not affect very much the final grain size. This model can be applied to the prediction of the resistance to hot deformation as well and it can contribute to the improvement of the accuracy in thickness. In this method, the average values concerning the grain size and the accumulated dislocation density are used taking the fraction recrystallized into consideration. This averaging can be applied to the hot-strip mill because the total thickness reduction is large enough to recrystallize their microstructure. In the case of plate rolling, the use of the average values is unsuitable because the reduction at each pass is small and the total thickness reduction is not enough to recrystallize the microstructure of steels. The model applicable to this case has been developed by dividing the microstructure into several groups [26]. This type of modeling was carried out for Nb-bearing steels [19,21,25], Ti- and V-bearing steels [21], Ti- and Nb-bearing steels [22], Ti-, Nb-, and Vbearing steels [27] as well as C–Mn steels. In these steels, the recovery and recrystallization are retarded by alloying elements. This retardation might be caused by the pinning effect due to fine precipitates or by the solute-drag effect. This effect can be considered by modifying the values of fitting parameters from experimental data. D.

Transformation Model

1. Basic Idea of the Modeling In the cooling process subsequent to hot-rolling, steels transform from austenite phase to ferrite, pearlite, bainite, and=or martensite phases. Transformation model predicts the microstructural change during cooling and the final microstructure of steels after cooling. The modeling of transformation kinetics can be performed by obtaining the parameters k and n in Avrami equation [29–31], formulating new equations corresponding to transformation kinetics obtained experimentally [32], and adopting the nucleation and growth theory [33–36].

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Table 3 Equations Used for Calculating Microstructural Change During Hot Deformation Phenomena 1.

2. 3.

4.

Critical strain for dynamic recrystallization Grain size of dynamically recrystallized grain Fraction dynamically recrystallized Dislocation density in dynamically recrystallized grain Dislocation density Grain growth of dynamically recrystallized grain Grain size of statically recrystallized grain Fraction statically recrystallized

5. 6.

Change in dislocation density due to recovery Grain growth

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Calculation model ec ¼ 4:76  104 expð8000=TÞ ðaÞ ddyn ¼ 22600½_e expðQ=RTÞ0:27 ¼ Z0:27 ; Q ¼ 63800 cal=mol ðbÞ Xdyn ¼ 1  exp½0:693ððe  ec Þ=e0:5 Þ2  ðcÞ e0:5 ¼ 1:144  103 d00:28 e_ 0:05 expð6420=TÞ ðdÞ rso ¼ 87300½_e expðQ=RTÞ0:248 ¼ 87300Z0:248 ðeÞ rs ¼ rso exp½90 expð8000=TÞt0:7  ðfÞ re ¼ ðc=bÞð1  ebe Þ þ r0 ebe ðgÞ dy ¼ ddyn þ ðdpd  ddyn Þy ðhÞ dpd ¼ 5380 expð6840=TÞ ðiÞ y ¼ 1  exp½295_e0:1 expð8000=TÞt ðjÞ dst ¼ 5=ðSveÞ0:6 ðkÞ Sv ¼ ð24=pd0 Þð0:491ee þ 0:155ee þ 0:1433e3e Þ ðlÞ Xst ¼ 1  exp½0:693ððt  t0 Þ=t0:5 Þ2  ðmÞ t0:5 ¼ 0:286  107 Sv0:5 e_ 0:2 e2 expð30000=TÞ ðnÞ rr ¼ re exp½90 expð8000=TÞt0:7  ðoÞ d 2 ¼ dst2 ¼ 1:44  1012 expðQ=RTÞt

ðpÞ

Figure 6 passes.

Changes in grain size and dislocation density during hot rolling of six

When using the Avrami equation, the fitting parameters, k and n, can be obtained from the Avrami plot based on the isothermal transformation kinetics as shown in Fig. 8. The effect of chemical composition of steels and the austenitic grain size before transformation on transformation kinetics can be taken into consideration by obtaining the dependence on the values of k and n from experiments. The first attempt of this type of modeling was carried out by Kirkaldy [29]. In his study, the prediction of mechanical properties was also tried. In order to increase the generality of the transformation model, it should be necessary to take the nucleation

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Figure 7 Effect of initial grain size on the change in grain size during hot-rolling in a hot-strip mill.

and growth theory into consideration. Utilizing the Johnson–Mehl type equation [5] or Cahn’s equation [6]. Cahn’s equation can be recognized as being the most general one because the approximation of the equations leads to the Avrami and Johnson–Mehl type equations. The modeling based on the nucleation and growth theory [33] will be explained in the following paragraphs. Assuming that the nucleation site is the surface of grain boundaries and the rates of nucleation and growth are independent of time, the transformation rate can be expressed for two cases [6]. One is the case where both the nucleation and the growth of new phase occur and the other is the case where only the growth of new phase occurs after nucleation sites are covered by new phase. The first case is described by   X ¼ 1  exp p=3ISG3 t4 ð26Þ

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Figure 8 Avrami’s plot. T1, T2, and T3 show different temperatures.

and the second case is described by X ¼ 1  ð2SGtÞ

ð27Þ

Transformation rates can be obtained by differentiating the equations as  3=4

p1=4 dX 1 1=4 3=4 ¼4 ðISÞ G ln ð1  XÞ dt 3 1X

ð28Þ

dX ¼ 2SGð1  XÞ dt

ð29Þ

For obtaining Eq. (28), the term of time is replaced by the fraction transformed on the assumption of the additivity of transformation with regard to the changing temperature. Equation (29) essentially holds the additivity of transformation. By using this type of equations, it is possible to obtain fitting parameters from continuous cooling transformation

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kinetics. It should be noticed that it is difficult to obtain accurate TTT diagrams of low-carbon steels which are the primary products in a hot-strip mill. This is due to their very rapid transformation kinetics and it causes difficulty in determining the fitting parameters from TTT data. From this point of view, this type of equation is very useful. In a case where it is possible to obtain the accurate TTT data, we can use the equations in Table 2 for determining the parameters. Transformations from austenite to ferrite, pearlite, and bainite can be calculated by using these equations. The start of each transformation is assumed as follows. The start of ferrite transformation is when the temperature of steel drops to Ae3. Carbon content in austenite increases during ferrite transformation and pearlite transformation starts when the carbon content in austenite achieves the amount shown on the Acm line. Bainite transformation starts when the temperature of steel drops to Bs temperature. Ae3 and Acm can be calculated from thermodynamics. Although Bs can be calculated on the same assumptions relating to T0 temperature [37], the equation obtained from experimental data is used in this model because the calculated Bs temperature still does not fit with experimental data. 2. Ferrite Transformation The nucleation site of transformation from austenite to ferrite is mainly the surface of austenite grains and its nucleation would be completed at the beginning of the transformation. Accordingly, the transformation from austenite to ferrite in the early stage is calculated by Eq. (28) and that in the latter stage is calculated by Eq. (29) in this model. The change in transformation kinetics, i.e., from the nucleation and growth to the site saturation, is assumed when the transformation rates calculated by both equations coincide with each other. Although the dissipation of incubation time for nucleation is generally used for the condition of the start of phase transformation, it is quite unclear theoretically. In this model, g=(a þ g) temperature, A3, is used for the starting condition of the calculation of phase transformation. There are two methods for calculating g=(a þ g) temperature; one is the condition called para-equilibrium [38] and the other is ortho-equilibrium. In the ortho-equilibrium, all elements are partitioned between ferrite and austenite; on the other hand, in the paraequilibrium condition, only carbon is partitioned. The idea of the para-equilibrium comes from that the diffusion of carbon which occupies interstitial sites in steel which is much more rapid than other substitutional alloying elements such as Mn, Si, and so on. There is an idea of NP–LE (No Partition–Local Equilibrium) [39], where interstitial atoms are partitioned between ferrite and austenite, the substitutional elements are locally

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

in the state of equilibrium between ferrite and austenite and the partition between ferrite and austenite does not occur after transformation. The ortho-equilibrium should be used when the transformation occurs at a relatively high temperature and the cooling rate is very slow. The paraequilibrium and the LE–NP should be used when the transformation occurs at a relatively low temperature as in rapid cooling. For the phase transformation in the continuous hot-rolling mill where the cooling rate is rapid, the para-equilibrium or the NP–LE should be used. The A3 temperature calculated based on the para-equilibrium and will be used in this model. For this calculation, C, Si, and Mn concentrations which are commonly included in steels are taken into consideration. According to the classical nucleation theory, the nucleation rate, I, can be described by equation [40]   DG  I ¼ Nb Z exp  ð30Þ kT where N is the number of nucleation sites, b is the rate of solute atoms arriving at the surface of new phase, Z (the Zeldovich factor) characterizes the annihilation of nucleation, T the temperature, k is the Boltzmann constant and DG is the driving force for forming the nucleus with a critical size. Z and b can be described by the following equations, respectively Z ¼ aT1=2

ð31Þ

b ¼ bD

ð32Þ

where D is the diffusion coefficient of solute atoms, a the variable related to interfacial energy and b is the variable related to the nearest atomic distance. DG is described by the equation DG ¼

cs3 DG2V

ð33Þ

where s is the interfacial energy, DGV is the free-energy difference between ferrite and austenite and c is the configurational coefficient. DGV is calculated by the thermodynamic parameters. Although there are many reports concerning interfacial energy, its value is still unclear. The values of Z and b are also unclear. In this model, we introduced two parameters for the nucleation rate as   k2 I ¼ k1 T1=2 D exp  ð34Þ RTDG2V

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and they are evaluated from experimental data to fit the calculation result to that obtained by experiment. As mentioned above, DGV can be calculated with thermodynamic parameters from chemical compositions of steels. In the equation, R is the gas constant and D is the diffusion coefficient of C in austenite. The diffusion coefficient of C can be calculated by the equation reported by Kaufman et. al. [41] which is expressed as       QD D cm2 =sec ¼ 0:5 exp 30Cg exp  ð35Þ RT QD ðcal=molÞ ¼ 38300  1:9  105 C2g þ 5:5  10

ð36Þ

where Cg is the carbon content (mole fraction) in austenite. For growth rate, the Zener–Hillert equation [8,9], Eq. (16) is used. This equation is formulated for the lengthening growth of needle-shaped ferrite based on the idea that the growth of ferrite is controlled by the diffusion of carbon in austenite. It has been reported that this equation well describes the growth of ferrite and bainite phases into austenite [41]. The tip radius of growing ferrite affects the carbon content in austenite and ferrite at phase interface and the carbon content is calculated by the method reported by Kaufman et al. [41] in this model. The carbon content in austenite increases during the transformation to ferrite due to the small solubility of carbon in ferrite and the increase in carbon content in austenite affects the growth rate of ferrite. The carbon content in austenite, Cg, can be calculated by the equation Cg ¼

C0  XF Ca 1  XF

ð37Þ

where C0 is the initial carbon content, XF is the fraction transformed to ferrite, and Ca is the carbon content in ferrite. By putting this value for Cg into Eq. (16), the change of the growth rate of ferrite by the progress of ferrite transformation can be considered. Parabolic growth equation, G ¼ at1=2, can be used instead for Eq. (16). This relation between growth rate and time can be obtained from experiment and the theory for isothermal transformation kinetics. This is due to the change in carbon content ahead of interface into austenite during transformation. This situation is considered by using Eqs. (16) and (37) [33]. Growth equations explained above can consider only the partition of carbon between ferrite and austenite during transformation. The partition of substitutional elements like Mn and Si cannot be considered. Recently, Enomoto and Atkinson [42–44] and A˚gren [45] have analyzed the partition of substitutional elements and its effect on transformation in detail. The calculation method by A˚gren [46] is based on the local equilibrium theory and

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has shown that chemical composition in ferrite and austenite changes from ‘‘para’’ to ‘‘ortho’’ during transformation. 3. Pearlite Transformation In the case of low-carbon steel, transformation to pearlite occurs subsequent to that of ferrite and can be assumed to start when the carbon content in austenite calculated by Eq. (37) reaches the extrapolated Acm line, as shown in Fig. 9. This starting condition of pearlite transformation was assumed to correspond to the site saturation case because the determination of kinetics by experiments for low-carbon steel is almost impossible. The equation formulated by Hillert [47] is used for the growth rate which is expressed as GP ¼

 kP  D Cga  Cgb Sl

ð38Þ

where Sl is the lamella spacing, Cgb the carbon content in austenite at the interface between cementite and austenite, and kP is the constant. The lamella spacing has a linear relation with the inverse of the undercooling temperature below Ae1, DT, and Eq. (38) can be expressed as   ð39Þ GP ¼ kP DTD Cga  Cgb

Figure 9 Starting condition of each transformation in the equilibrium diagram.

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The effect of a chemical composition on the growth rate of pearlite is considered by using Cga, Cgb and DT calculated by the thermodynamic parameters. As the transformation from austenite to pearlite is a eutectoid reaction, the change of the chemical composition in untransformed austenite during the progress of transformation does not occur. Equation (38) shows that pearlite transformation is controlled by the volume diffusion of carbon. On the other hand, it has been recently reported that the mechanism of pearlite transformation is somewhere between the volume diffusion of carbon in austenite and the interfacial diffusion of substitutional elements [48]. 4. Bainite Transformation Transformation to bainite is assumed to start when steels are cooled to the bainite-start temperature, Bs. In this report, Bs formulated from the observation of microstructures of steels (0.05–0.15 mass% C–0.5–1.5 mass% Mn–0–1.0 mass% Si steels) which are transformed isothermally is used, which is expressed as Bs ¼ 717:5  425½mass%C  42:5½mass%Mnð CÞ

ð40Þ

Bainite transformation in low-carbon steels occurs subsequent to ferrite or pearlite transformation. No flexion point between ferrite and bainite transformations is observed in the transformation curve. This result indicates that bainite transformation subsequent to ferrite transformation can be treated by the site saturation. The nucleation site in this case would be the interface between austenite and ferrite. The progress of bainite is calculated by Eq. (29), and the transformation rate is calculated by the Zener–Hillert equation (Eq. (16)). 5.

Summary of Equations and Parameters Used for the Transformation Model Table 4 shows the summary of equations and fitting parameters. Although the fitting parameters were obtained from the experimental data of one steel, this set of equations can simulate transformations of steels with different chemical compositions due to the application of thermodynamics to the calculation of the growth rate and the free-energy difference between ferrite and austenite. Figures 10 and 11 show the calculation results of the effects of chemical composition and austenite grain sizes on transformation kinetics [33]. The calculation of the fraction of each phase after transformation can be determined. By introducing the thermodynamic parameter of other alloying elements, the applicability would be easily extended [49]. Recently, the calculation of transformation for a 10 element system was reported [50]. This

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Table 4 Equations and Parameters Used for the Transformation Model

Transformation Ferrite

Pearlite

Basic equation of transformation rate Nucleation and Growth

1=4 dx ¼ 4:046 kt 6=dg4 IG 3 dt  3=4 1 ln ð1  xÞ 1x Site saturation dx 6 ¼ k2 Gð1  xÞ dt dg

Bainite

Factor corresponding to nucleation rate and growth rate  I ¼ T 1=2 D exp  G ¼

1 Cga  Cg D 2r Cg  Ca

k3 RT DGv2

G ¼ DTDðCga  Cgb Þ

G ¼

1 Cga  Cg D 2r Cg  Ca

Coefficient 

k1 ¼ 17; 476

  21100 k2 ¼ 8:933  1012 exp T k3 ¼ ðcal3 =mol3 Þ ¼ 1:305  107 K2 ¼ 6:72  106

k2 ¼ 6:816  104 exp

  3431:5 T

Note: dg: austenite grain size, D: diffusion coefficient of carbon in austenite, Cg: carbon content in austenite, Ca: carbon content in ferrite, Cga: carbon content in austenite at g=a boundary, Cgb: carbon content in austenite at g/cerm boundary, DT: undercooling below Ae1, G : Zener–Hillert equation (the value was calculated with the method by Kaufman et al.), and r: radius of curvature of advancing phase.

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Figure 10

The effect of chemical compositions on transformation kinetics.

extension of the model is possible only when alloying elements affect transformation kinetics through the change in phase stability. From the viewpoint of the effect of the chemical composition on transformation kinetics, other effects such as the solute drag, the pinning etc. should be taken into consideration. Kinsman and Aaronson [51] reported that transformation in C–Mo steels is retarded due to the solute drag-like effect. There is a report that Nb retards transformation due to solute-drag effect [52]. The solute-drag effect in Fe–C–X systems was investigated as well [53–56]. The solute-drag effect and others are not considered in the model. Instead of the application of these theories, the fitting parameters for the rates of nucleation and growth are introduced [57]. The introduction of these theories into the mathematical models is the remaining problem. 6. The Calculation of Ferrite Grain Sizes After Transformation The prediction of ferrite grain size is necessary for the calculation of mechanical properties. The calculation can be carried out using the austenite grain size before transformation and cooling rate [22]. This type of formulae is useful for the calculation because the prediction can be carried out

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Figure 11 The effect of austenite grain size on transformation kinetics of 0.15%C– 0.5%Si–1%Mn steels.

without knowing the accurate transformation kinetics. It, however, could not be applied for the case where cooling rate is not constant. The next equation can be applied for the case where the cooling rate changes during cooling. On the assumption that the shape of ferrite grains is spherical, the average ferrite grain size after cooling, da, has a relation with a fraction transformed to ferrite and the number of ferrite grains and the relation is expressed as   4p da 3 ð41Þ XF ¼ N 3 2 where N is the number of ferrite grains in the unit volume and XF is the volume fraction of ferrite. The equation can be transformed into   6XF 1=3 da ¼ ð42Þ pN Since XF can be obtained from the transformation model explained above, the calculation of ferrite grain size can be carried out if we know

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the number of ferrite grains. The formulation of the equation for calculating the number of grains is based on the idea that the number of ferrite grains is determined in the early stage of ferrite transformation. Figure 12 shows the relationship between the number of ferrite grains and the temperature at 5% transformation, T0.05, which was calculated by the above-mentioned transformation model. This experiment was carried out for (0.1–0.15) mass% C–0.5 mass% Si–(0.5–1.5) mass % Mn steels. The temperature T0.05 is used as a representative temperature indicating the early stage of ferrite transformation. This figure shows that the number of ferrite grains is dependent upon the temperature at the early stage of ferrite transformation and the austenite grain size before transformation. Based on this result, the number of ferrite grains N(mm3) was formulated as   21430 11 1:75 exp N ¼ 3:47  10 dg ð43Þ T0:05 where dg is the austenite grain size before transformation. Using Eqs. (42) and (43), we can obtain the equation expressing the ferrite grain size da as
  1=3 21430 exp  ð44Þ da ¼ 5:51  1010 d1:75 XF g T0:05

Figure 12 The relationship between the number of ferrite grains and the temperature, T0.05.

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where da and dg are in units of mm. Figure 13 compares the ferrite grain sizes calculated versus those observed. A good agreement between those calculated and observed is found in this figure. Umemoto et al. [58] have derived theoretical equations for ferrite grain size in an isothermally transformed steel as da ¼ 0:564ðI=GÞ1=6 d2=3 g

ð45Þ

for the austenite grain edge nucleation of ferrite, and as da ¼ 0:695ðI=GÞ2=9 d1=3 g

ð46Þ

for the grain surface nucleation. In the present study, ferrite grain size is expressed by Eq. (45), in which the relationship between da and dg is da ¼ kd1:75=3 g

Figure 13

Comparison between calculated and observed ferrite grain sizes.

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ð47Þ

where k is the coefficient depending on the transformation temperature and ferrite fraction. This result indicates that the nucleation of ferrite transformation in low-carbon steels takes place mainly at the surface of austenite grains. 7. The Effect of Residual Strain on Transformation Kinetics The austenitic microstructure with which we start the calculation using the transformation model is a fully recrystallized austenite. In the hot-rolling mill, the austenite before transformation could contain many imperfections such as dislocations, deformation bands, and so on, and they may affect the nucleation rate and the growth rate. This effect would cause deterioration in predicting the accuracy of the microstructure after cooling. At this time, it is difficult to account for the effect of imperfections on transformation perfectly on theory. In this section, we thus will consider it using the dislocation density calculated by the hot-deformation model will be considered [20].

Figure 14 Comparison between calculated and observed ferrite grain sizes with and without consideration of the effect of the dislocation density.

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Figure 14 shows the difference between the calculated and the observed ferrite grain sizes as a function of residual dislocation densities. The dislocation density was estimated from the hot-deformation model [20]. In this figure, open circles show the case when the above-mentioned model is used, i.e., the effect of the residual strain is not considered, while solid circles show the case when the effect of residual strain is considered by the method which will be explained below. In the case where the effect of residual strain is not considered, the difference between the ferrite grain sizes measured and those calculated is small for low dislocation density and large for high dislocation density. Using the observed ferrite grain sizes, T0.05 and Eq. (44), we can estimate the austenite grain size upon which the correct ferrite grain size for a whole range of dislocation density can be given, and this austenite grain size can be called the effective austenite grain size. Figure 15 shows the difference between the austenite grain size, dg, calculated from the hot-deformation model, and the effective austenite grain

Figure 15

Effect of the dislocation density on the effective austenite grain size.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

size estimated as a function of the dislocation density. From this figure, the following relationship can be obtained: dgeff ¼ dg =ð1 þ 1011 r1:154 Þ

ð48Þ

where r is the dislocation density before transformation in cm2. The ferrite grain sizes calculated from the transformation model using dgeff instead of dg agree well with those observed as shown in Fig. 14. Figure 16 compares the ferrite fractions calculated with those observed when either dg or dgeff is used. The use of dgeff provides better agreement than the use of dg. These results prove that the effect of stored strain on transformation kinetics can be predicted quantitatively.

Figure 16 Comparison between the ferrite fractions calculated and observed while taking=not taking into account the effect of dislocation density.

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Umemoto et al. [59] studied the effects of the residual strain on the rate of growth and nucleation separately in pearlite transformation and showed that only the change in nucleation rate is caused by the stored strain. In the transformation model explained above, the effect of austenite grain size on transformation kinetics is considered by using dg. The change in the area of nucleation sites and the change in the rate of nucleation and growth would be taken into consideration by using Eq. (48), although it is not clear which is the dominant factor. For quantitative evaluation of each of these effects, conducting an experiment similar to the work done by Umemoto et al. is necessary, although it is difficult to conduct such an experiment with regard to low carbon steels because of their rapid transformation. E.

Precipitation Model

Precipitation of carbides and=or nitrides in austenite phase could affect recovery, recrystallization, and grain growth after hot deformation, and transformation during cooling. Precipitation in ferrite phase could affect mechanical properties. Accordingly, it is necessary to take precipitation into consideration in the model. It is not necessary to consider the hard impingement for the modeling of precipitation. For this reason, nucleation and growth can be calculated independently and the amount, number, and size distribution of precipitates can be directly calculated. Although formulations based on Avrami’s equation have been reported [60–62], modeling based on the nucleation and growth theory has been attempted from the above point of view [63–67]. Okamoto and Suehiro [67] reported a model which can be applied from the beginning to the end (the Ostwald ripening) of precipitation. The feature of this model is in the calculation method of growth rate of precipitates. This calculation method will be explained briefly as follows. The velocity of the interface between matrix and precipitates can be expressed from the flux balance of each chemical element as v¼0

JNb JC JN ¼ ¼ CNb b CNb 0 CC b CC 0 CN b CN

ð49Þ

where Jj is the flux of each element, 0Cj and bCj the content of element j in precipitates and matrix at the interface between matrix and precipitates, respectively. For the calculation of the content of element j in matrix at the interface, the local equilibrium condition is normally applied. In this model, the content of element j is calculated considering the radius of

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precipitate using the next equation. This relationship is called the Gibbs–Thompson effect mNbCN þ 2sVNbCN =R ¼ mM j j

ð50Þ

where the second term of the left side of this equation is the increase in the Gibbs free energy, Esurf, due to the interfacial energy, s. The Ostwald ripening of precipitates appearing at the latter stage of precipitation in which the small size of precipitates would dissolve and the large size of precipitates would grow can be calculated by considering this term. Figure 17 shows the isothermal section of equilibrium diagram of three element system. From this figure, the effect of the radius of precipitates can be understood. Figure 18 compares the calculation result of the average diameter and the mole fraction of precipitates with the experiments. Figure 19 shows the calculation result of the average diameter and the number of precipitates which shows that the average diameter increases with a half power of time at the beginning of precipitation (region II) and with one-third power of time at the latter stage (region IV). Region IV would correspond to the Ostwald ripening.

F.

Relationship Between Strength and Microstructure of Steel

The mechanical properties to be predicted are dependent upon the type of products. Sheets and coils require YS (yield strength), TS (tensile strength), and El (elongation). Plates require toughness other than YS, TS and El. Wire and rods require TS and the reduction of section area. The prediction of these mechanical properties was carried out by formulating regression equations for strengths with respect to chemical compositions and grain sizes of ferrite [68]. Other formulations for strength were based on the volume fractions of ferrite and pearlite phases, ferrite grain sizes, and lamella spacing of pearlite for steels consisting of ferrite and pearlite in phases [69]. For toughness, some regression formulae based on the ferrite grain size and=or chemical compositions were reported [70–72]. Various reports on this type of formulations are available [32–74]. Irvine and Pickering [73] showed that the tensile strength of ferritepearlite steel or bainite steel was determined from the transformation temperature of steels [73]. In their experiment, only the content of alloying elements was a variable of the transformation temperature, but the result indicated that the change of transformation temperature due to processing variables such as cooling rate had a similar influence on the strength of steels

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Figure 17 Isothermal section of equilibrium diagram of Fe–Nb–C system; (a) the beginning and (b) the latter stage of precipitation.

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Figure 18 Comparison of the calculated results with the observed ones. Steel A: 0.006%C–0.14%Nb–0.0022%N, steel B: 0.018%C–0.052%Nb–0.0041%N.

consisting of ferrite, pearlite, and=or bainite. A recent study confirmed this result with regard to the accelerated cooled steels of a substantially ferritic transformation structure [75]. Therefore, the determination of a more general relationship between strength and transformation temperature applicable to the individual microconstituent in a mixed microstructure is required. Since the strength

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Figure 19 model.

Change in the size and the number of precipitates calculated by the

of steel is generally proportional to its hardness, by assuming the law of mixture for hardness, the tensile strength, TS, can be expressed as TS ¼ a½XF ðHF þ bda1=2 Þ þ XP HP þ XB HB 

ð51Þ

where X is the fraction of each transformed phase (F: ferrite, P: pearlite, and B: bainite), H the hardness of each microconstituent, da the ferrite grain size in mm, and a and b are the constants. If the relationship between strength and transformation temperature for each constituent of steel is established, the tensile strength can be calculated from Eq. (51). Figure 20 shows the relation between the hardness of each microconstituent and its average transformation temperature, TM, calculated from Z Z ð52Þ TM ¼ T dX= dX

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where the transformation temperature, T, is obtained for an infinitely small transformation product, dX, from the transformation model. A linear relationship is found between the hardness and average transformation temperature for both ferrite and bainite. Such a relationship is not found for pearlite, although it is presumed. This is probably because of the narrow temperature range of transformation in the steels used. Silica has a strong effect on solid-solution hardening while C and Mn have a very small effect. Further, the hardness does not depend on the cooling rate. From these results, the hardness of each microconstituent is expressed as HF ¼ 361  0:357TF þ 50½mass%Si HP ¼ 175

ð53Þ

Figure 20 The relationship between the measured microhardness of each microconstituent and its calculated mean transformation temperature.

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and HB ¼ 508  0:588TB þ 50½mass%Si

ð54Þ

where H is the hardness and T is the average transformation temperature (8C). Other Methods of attempting to predict YS, TS, n-value, etc. have been reported. Tomota et al. [76] attempted the prediction of YS, TS, and n-value, etc. based on the prediction of flow stress curves. Shikanai et al. [77] and Iung et al. [78] utilized the analysis by the finite element method in order to consider the effect of morphology of microstructure. For steels containing chemical elements forming precipitates such as Nb, Ti, V, etc., the precipitation hardening should be taken into consideration. It would be possible by using the precipitation model. It might be possible to predict the precipitation hardening from the alloying elements in solid solution at austenite region before cooling [79,80].

VI.

PREDICTION OF STRENGTH OF HOT-ROLLED STEEL SHEETS

Using the model mentioned above, we can predict the strength of hot-rolled steel sheets from its composition and processing conditions such as hot-rolling condition, cooling condition, and so on. Figure 21 shows the flow chart of the calculation. Figure 22 compares the calculated and observed transformed fraction of each phase in 0.2 mass% C-0.2 mass% Si-0.5 mass% Mn steels hot rolled in a two-stand laboratory mill from 40 to 2.4 mm by six passes after being soaked at 11008C for 30 min. The microstructure is ferrite–pearlite in the sample (a) cooled at around 108C=sec and ferrite–bainite in (b) cooled at around 608C=sec. The values calculated by the present model are in good agreement with those measured. Tensile strengths were calculated using Eqs. (51), (53), and (54) with the constant a of 3.04 and the constant b of 2.55. An agreement between the calculated and the observed tensile strengths is good for various steels (C: 0.1– 0.2 mass%, Si: 0.006-0.5 mass%, Mn: 0.5-1.5 mass%) as shown in Fig. 23. The present integrated model has been applied to the prediction of the microstructures and strengths of steel hot rolled in a production mill. 0.15 mass% C-0.1 mass% Si-0.6 mass% Mn steel was hot rolled. In the hot-rolling, the finish rolling temperature and the coiling temperature varied lengthwise as shown in Fig. 24. Figure 25 shows the calculated and observed ferrite grain size and ferrite fraction of the steel sheet. Figure 26 compares the strengths calculated with those measured. These figures show that the values

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Figure 21 Flowchart of the calculation of microstructural change and mechanical properties of hot-rolled steels.

calculated by this model have a good agreement with those measured. These results indicate that this type of simulation could be a very efficient tool for designing chemical compositions and processing conditions in order to obtain required mechanical properties. Many empirical equations have been developed to predict the strength of hot rolled steel products. These equations combine the parameters of chemical compositions such as C, Mn, and Si, fraction of each microconstituent and cooling rate. In the present model, Eqs. (51), (53) and (54) for calculation of the strength of hot-rolled steel products have no explicit parametric terms of C, Mn and cooling rate. Therefore, it seems as if the content of C and Mn and the cooling rate do not influence the strength of these products. As mentioned above, the microstructure and the hardness of each microconstituent predicted based on the hot-deformation and transformation models are strongly affected by the C- and Mn-content and the cooling condition. The results shown in Fig. 20 indicate that the solid-solution hardening by C and Mn reported in the literature includes the variation of the microstructure with the change in the transformation temperature. Since

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Figure 22 Comparison of the calculated microstructure with that observed for steels cooled at about (a) 108C=sec and (b) 608C=sec.

the cooling rate does not appear explicitly, the present model is suitable for the prediction of strength of steels processed by thermomechanical treatment.

VII.

A.

APPLICATION OF THE MODEL TO THE PREDICTION OF TEMPERATURE OF HIGH CARBON STEELS DURING COOLING AFTER HOT DEFORMATION Modification of the Model to the Application to High Carbon Steels

Steels containing more than 0.3mass%C carbon (high -carbon steels) show a remarkable evolution of latent heat of transformation during cooling. This

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Figure 23 Comparison between the calculated and observed tensile strengths of various steels.

evolution makes on-line control of temperature difficult and thus affects mechanical properties of final products. Thus, accurate calculations of temperature of steel during cooling prior to production in a mill are desirable to enable the control and the investigation of suitable processing conditions. In order to calculate the temperature accurately, the prediction of transformation is particularly important.

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Figure 24 Finish rolling and coiling temperatures for experiments in production mill.

Figure 25 Variation of ferrite fraction and ferrite grain size from top to tail of the coil of 0.16mass%C–0.015 mass%Si–0.73mass%Mn steel hot rolled in a production mill.

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Figure 26 Variation of tensile strength from top to tail of the coil of 0.16mass% C–0.015mass%Si–0.73mass%Mn steel hot rolled in a production mill.

As mentioned above, a mathematical model for predicting transformation of low-carbon steels during cooling while taking ferrite transformation into consideration has been developed because the principal transformation product in low-carbon steels is ferrite. In high-carbon steels, however, the main transformation product is pearlite, so that the development of a model for calculating pearlite transformation accurately is necessary. In this section, the model for high-carbon steels and its application [81] will be explained.

B.

Transformation Model

1. Start of Pearlite Transformation The calculation procedure for ferrite and bainite transformations is the same as that explained in Sec. 5. The treatment for pearlite transformation is modified. The calculation of ferrite transformation which starts when the temperature drops to the equilibrium temperature Ae3 is calculated using thermodynamic parameters. Transformation from austenite to ferrite is controlled by the volume diffusion of carbon into austenite so that carbon in austenite increases with the progress of ferrite transformation. This point is explained in the Sec. 5.

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Pearlite transformation is generally assumed to start when the carbon content in austenite meets the extrapolated Acm line in the phase equilibrium diagram. The observed ferrite fraction of carbon steels transformed isothermally from austenite, however, indicates that the pearlite transformation starts earlier than the above expectation at the temperature range below 960 K shown in Fig. 27. This deviation might be due to the distribution of carbon between ferrite and austenite at the g=a phase boundary because the carbon content in austenite at=near the phase boundary can exceed the value on the extrapolated Acm line below Ae1 temperature. In this model, the start of pearlite transformation is dealt with based on this result. 2. Kinetics of Pearlite Transformation For low-carbon steels, there has been no experimental data which clarifies the mode of pearlite transformation kinetics. In 0.5mass%C steels, however, the experimental results of pearlite transformation kinetics show that pearlite transformation conforms to nucleation and growth case [81]. Accordingly, Eq. (28) is used and the fitting parameter [81] was obtained from the experimental results of 0.5mass%C steels as shown in Table 5.

Figure 27 Deviation of maximum ferrite fraction measured in isothermal transformation experiments from the value calculated thermodynamically.

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Table 5 Equations and Parameters for Pearlite Transformation of High-Carbon Steels Basic equation of transformation rate Nucleation and growth

p1=4 dX ¼4 ðk1 ISÞ1=4 dt 3  3=4 1 G3=4 ln ð1  XÞ 1X

C.

Factor corresponding to nucleation rate and growth rates Coefficient   k3 I ¼ T 1=2 D exp  k1 ¼ 2.01  1013 RTDGV2 k3¼2.27  109 (J3=mol3) S¼6=dg   GP ¼ DTD Cga  Cgb

Method for Calculating Temperature of Steels During Cooling

The transformation model developed was coupled with the model for calculating the temperature of steels during cooling by a two-dimensional finite element method in order to take into consideration the evolution of latent heat [15]. The heat conduction equation used here is as follows: mCaP

@T @2T @2T ¼l 2 þl 2 þQ @t @x @y

ð55Þ

where m is the density of steel, l the heat conductivity, and CPa is the specific heat of ferrite on the assumption of nonexistence of magnetic transformation as shown in Fig. 28. Q is the rate of latent heat evolution accompanying transformation and can be formulated by Eq. (56), in which the latent heat is divided into that of lattice transformation, ql, and that of magnetic transformation, qm:
 @X @ þ ðqm XÞ ð56Þ Q ¼ m ql @t @t where X is the fraction transformed and calculated by the above-mentioned transformation model. The value of ql used is 16.7 J=g and qm is expressed as Z910 qm ¼



 CP  CaP dT

T

where CP is the specific heat of steel. Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

ð57Þ

Figure 28

D.

Specific heat and latent heat of magnetic transformation of steel.

Calculation Results

1.

Accuracy of the Model for Predicting Temperature of Steels During Cooling Figure 29 shows examples of simulation in which the temperature and the progress of transformation of 0.5mass%C steels on the run-out table of a hot-strip mill were calculated simultaneously for three different cooling conditions. In this calculation, the initial state and the hot-deformation models were also used and the average austenite grain size before transformation was calculated as being about 15 mm. A good agreement between calculated and measured temperatures was obtained. This implies that transformation kinetics on the run-out table are accurately predicted by the model. 2. Improvement of Productivity To improve the productivity in a hot-strip mill, it is necessary to increase the traveling speed of the hot-strip and intensify the cooling rate. Figure 30(a) shows the calculated results of temperature and transformation behavior of 0.5mass%C steel of 2 mm thickness cooled at a heat transfer coefficient,

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Figure 29 Simulation of temperature and progress of transformation of 0.5 mass%C steel under different cooling conditions on run-out table of a hot-strip mill.

a, of 1670 kJ=m2 h K for water cooling. In this calculation, the finish rolling temperature is 1123 K, the coiling temperature is 873 K and transformation is completed before coiling. The traveling time through the run-out table can be shortened from 11.5 to about 5 sec by increasing the heat transfer coefficient up to 5020 kJ=m2 h K, as shown in Fig. 30(b). The figure, however, shows the undesirable situation where the temperature of steel drops to about 800 K which is below the bainite-start temperature (about 823 K for 0.5mass%C steel) and bainite which deteriorates the quality of steel might appear. This situation can be avoided by changing the cooling condition. Figure 30(c) shows the suitable cooling condition in which the water cooling is stopped just before the transformation start and restarted at about 20% transformation.

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Figure 30 Temperature change of 0.5mass%C steel on the run-out table under three different cooling conditions.

The results show that the calculation by the model can be used for determining a suitable cooling pattern on the run-out table for attaining high productivity without conducting an experiment in a production mill.

3.

Effect of the Finish Rolling Temperature on Mechanical Properties In a hot-strip mill, the finish rolling temperature varies widthwise. This variation affects the transformation behavior during cooling due to the changes in austenite grain size and dislocation density in austenite, and it also changes the temperature range of water cooling on the run-out table as well. Figure 31 shows the calculated cooling curves of 0.5mass%C steel coil of 4 mm thickness and 1 m width at two different positions; the center along the width and the position of 12.5 mm apart from the edge of strip. The temperature difference between these two positions is about 408C. By changing the water-cooling condition, the temperature difference can be reduced as shown in Fig. 32. This change contributes to the reduction of the fluctuation of mechanical properties as shown in Fig. 33, in which the index of vertical

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Figure 31 Temperature changes of steel at two different positions in the direction of width on the run-out table.

Figure 32 Temperature change of steel at two different positions in the direction of width on the run-out table. Cooling condition was modified from that in Fig. 31.

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Figure 33 Changes in the index of hardness, (Ae1TP)1=2, in which TP is the mean pearlite transformation temperature, along the width under two different conditions.

axis, (Ae1TP) 1=2 where TP is the mean transformation temperature of pearlite, representing hardness of pearlite because the pearlite hardness depends on the lamella spacing of pearlite which depends upon the under-cooling from Ae1 temperature. Conditions A and B in Fig. 33 correspond to those in Figs. 31 and 32. This result indicates that the fluctuations of properties due to the variation of temperature along the width can be compensated by controlling the water-cooling intensity. 4.

Effect of Fluctuations of Coiling Temperature on Mechanical Properties Coiling temperature is fluctuated widthwise by the fluctuations of water cooling intensity even though the finish rolling temperature is constant, and this coiling temperature fluctuation affects the mechanical properties due to the change in the transformation temperature. Figure 34 shows an example of the calculated results for 0.5mass%C steel sheets with thickness

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Figure 34 Effect of thickness of steel on the relationship between the fluctuations of coiling temperature, DCT, and those of the index of pearlite hardness, (Ae1TP)1=2.

of 4 and 6 mm under the condition that the finish rolling temperature is 1122 K and the coiling temperature fluctuates between 843 K and 933 K. This result indicates that fluctuations of the index of hardness, D(Ae1TP)1=2, depend on those of coiling temperature, DCT, and their relationship is influenced by the thickness of steel sheets. The fluctuations of transformation temperature are mostly dependent on the cooling rate just before and=or at the beginning of transformation. Since the water-cooling intensity for the steel of 4 mm thickness is the same as that for steel of 6 mm thick in this calculation, the water-cooling time for steel of 6 mm thickness is longer than that for 4 mm steel because of the mass effect. Hence, the change in cooling rate, which causes a certain value of DCT, becomes smaller as steel strip thickens. This is the reason why the thickness of steel sheets causes fluctuations in hardness. Figure 35 shows the effect of traveling speed on the fluctuations of transformation temperature. The calculation shown was carried out for three different traveling speeds (300, 400, and 500 mpm) with the finish rolling temperature of 1123 K and the coiling temperature of between

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Figure 35 Effect of traveling speed of steel through the run-out table on the relationship between the fluctuations of coiling temperature, DCT, and those of the index of pearlite hardness, (Ae1TP)1=2.

853 K and 913 K. The fluctuations of the index of hardness do not vary in the order of traveling speed; the fluctuations for 400 mpm are the largest among three traveling speeds. This variation is also related to the cooling rate before and during transformation. The faster the traveling speed, the longer the water-cooling time, in the case where the intensity for water cooling is constant regardless of the traveling speed. This is due to the change in the water-cooling temperature range depending on the finish rolling temperature, the coiling temperature, the traveling speed, and the thickness. In this calculation, the water-cooling time for 300 mpm is too short to affect the transformation behavior greatly. This is the reason why the fluctuations in transformation temperature for 400 mpm are greatest in Fig. 35. These results depend on the chemical composition of steel and the capacity of the hot-strip mill, such as the water-cooling intensity and the length of the run-out table. Accordingly, the pre-calculation by this type

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of mathematical model is very useful for designing cooling facilities and determining cooling conditions for obtaining a more uniform quality product. VIII.

CONCLUSION

In this chapter, mathematical models for predicting microstructural evolution during hot deformation and subsequent cooling, and mechanical properties from the resultant microstructure of steels were explained. These models include some empirical parameters although they are based on theory. This is because mechanisms of some phenomena are still unclear; for instance, solute-drag effect on recrystallization and transformation. The model calculating mechanical properties from microstructure is much more phenomenological. To extend the applicability, the empirical parameters should be replaced by those obtained from theories. Although the models include some empirical parameters, they are very useful for investigating production conditions such as chemical compositions, processing conditions and so on. The accuracy of predicted mechanical properties is satisfactory. It is noted that these models should be widely used for off-line simulation of designing steel compositions and processing condition and on-line simulation for guaranteeing mechanical properties of steels. REFERENCES 1. Proceedings of International Conference on Physical Metallurgy of Thermomechanical Processing of Steels and Other Metals; ISIJ: Tokyo, 1988. 2. Yue, S., Ed. Proceedings of International Symposium. on Mathematical Modelling of Hot Rolling of Steel; CIM: Quebec, 1990. 3. ISIJ Int., 1992, 32. 4a. Avrami, M.J. Chem. Phys. 1939, 7, 1103. 4b. Avrami, M.J. Chem. Phys. 1940, 8, 212. 4c. Avrami, M.J. Chem. Phys. 1941, 9, 177. 5. Johnson, W.A.; Mehl, R.F. Trans. AIME 1939, 135, 416. 6. Cahn, J.W. Acta Metall. 1956, 4, 449. 7. Umemoto, M. Proceedings of International Symposium on Mathematical Modelling of Hot Rolling of Steel; Yue, S., Ed.; CIM: Quebec, 1990; 404. 8. Zener, C. Trans. AIME 1946, 167, 550. 9. Hillert, M. Jernkontrets Ann. 1957, 141, 757. 10. Kaufman, L.; Vernstein, H. Computer Calculation of Phase Diagrams; Academic Press: New York, 1970. 11. Hillert, M. In Hardenability Concepts with Applications to Steel; Doane, D.V.; Kirkaldy, J.S., Eds.; The Metallurgical Society of AIME: 1978 ; 5.

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12. Hillert, M.; Staffansson, L.I. Acta Chem. Scand. 1970, 24, 3618. 13. Sundman, B.; Jansson, B.; Andersson, J.-O. CALPHAD 1985, 9, 153. 14. Suehiro, M.; Sato, K.; Tsukano, Y.; Yada, H.; Senuma, T.; Matsumura, Y Trans. Iron Steel Inst. Jpn. 1987, 27, 439. 15. Suehiro, M.; Sato, K.; Yada, H.; Senuma, T.; Shigefuji, H.; Yamashita, Y. Proceedings of International Conference on Physical Metallurgy of Thermomechanical Processing of Steels and Other Metals; ISIJ: 1988; 791. 16. Yoshie, A.; Fujioka, M.; Morikawa, H.; Onoe, Y. Proceedings of International Conference on Physical Metallurgy of Thermomechanical Processing of Steels and Other Metals; ISIJ: 1988; 799. 17. Nishizawa, T. Tetsu-to-Hagne 1984, 70, 1984. 18. Sellars, C.M.; Whiteman, J.A. Met. Sci. 1979, 14, 187. 19. Sellars, C.M. International Conference on Working and Forming Processes; Sellars, C.M., Davies, G.J., Eds.; Met. Soc.: London, 1980, 3. 20. Senuma, T.; Yada, H.; Matsumura, Y.; Futamura, T. Tetsu-to-Hagane 1984, 70, 2112. 21. Yoshie, A.; Morikawa, H.; Onoe, Y.; Itoh, K. Trans. Iron Steel Inst. Jpn. 1987, 27, 425. 22. Roberts,W.; Sandberg, A.; Siwecki, T.; Werlefors, T.; Werlefors, T. Proceedings of International Conference on Technology and Applications of HSLA Steels; ASM: 1984, 67. 23. Wiliams, J.G.; Killmore, C.R.; Harris, G.R. Proceedings of International Conference on Physical Metallurgy of Thermomechanical Processing of Steels and Other Metals; ISIJ: Tokyo, 1988, 224. 24. Komatsubara, N.; Okaguchi, H.; Kunishige, K.; Hashimoto, T.; Tamura, I. CAMP-ISIJ 1989, 2, 715. 25. Saito, Y.; Enami, T.; Tanaka, T. Trans. Iron Steel Inst. Jpn. 1985, 25, 1146. 26. Anan, G.; Nakajima, S.; Miyahara, M.; Nanba, S.; Umemoto, M.; Hiramatsu, A.; Moriya, A.; Watanabe, T. ISIJ Int. 1992, 32, 261. 27. Siwecki, T. Proceedings of International Conference on Physical Metallurgy of Thermomechanical Processing of Steels and Other Metals; ISIJ: Tokyo, 1988, 232. 28. Sakai, T.; Jonas, J.J. Acta Metall. 1983, 32, 100. 29. Kirkaldy, J.S. Metall. Trans. 1973, 4, 2327. 30. Umemoto, M.; Komatsubara, N.; Tamura, I. Tetsu-to-Hagane 1980, 66, 400. 31. Hawbolt, E.B.; Chau, B.; Brimacombe, J.K. Metall. Trans. A 1983, 14, 1803. 32. Choquet, P.; Fabregue, P.; Giusti, J.; Chamont, B. Proceedings of International Symposium on Mathematical Modelling of Hot Rolling of Steel; Yue, S., Eds.; CIM: Quebec, 1990; 34. 33. Suehiro, M.; Senuma, T.; Yada, H.; Matsumura, Y.; Ariyoshi, T. Tetsu-toHagane 1987, 73, 1026. 34. Saito, Y. Tetsu-to-Hagane 1988, 74, 609. 35. Reed, R.C.; Bhadeshia, H.K.D.H. Mater. Sci. Tech. 1992, 8, 421. 36. Jones, S.J.; Bhadeshia, H.K.D.H. Metal. Mater. Trans. A 1997, 28, 2005. 37. Bhadeshia, H.K.D.H Acta Metall. 1981, 29, 1117.

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38. Hultgren, A. Trans. ASM 1947, 39, 915. 39. Hillert, M. Internal Report; Swedish Institute for Metal Research: Stockholm, Sweden, 1953. 40. Aaronson, H.I.; Lee, J.K. In Lectures on the Theory of Phase Transformations; Aaronson, H.I., Ed.; TMS-AIME: 1975; 28. 41. Kaufman, L.; Radcliffe, S.V.; Cohen, M. In Decomposition of Austenite by Diffusional Processes; Zackay, V.F.; Aaronson, H.I., Eds.; Interscience Publishers: New York 1962. 42. Enomoto, M. ISIJ Int. 1992, 32, 297. 43. Enomoto, M.; Atkinson, C. Acta Metall. Mater. 1993, 41, 3237. 44. Enomoto, M. Tetsu-to-Hagane 1994, 80, 653. 45. A˚gren, J. ISIJ Int. 1992, 32, 291. 46. Anderson, J.-O.; Ho¨glund, L.; Jo¨nsson, B.; A˚gren, J. In Fundamentals and Applications of Ternary Diffusion; Purdy, G.R., Ed.; Pergamon Press: New York, 1990, 153. 47. Hillert, M. In Decomposition of Austenite by Diffusional Processes; Zackay, V.F.; Aaronsson, H.I., Eds.; Interscience Publishers: New York, 1962; 313. 48. Jo¨nsson, B. TRITA-MAC-0478, Internal Report, Division of Physical Metallurgy, The Royal Institute of Technology, S-10044 Stockholm, Sweden, 1992. 49. Suehiro, M.; Yada, H.; Senuma, T.; Sato, K. Proceedings of International Symposium on Mathematical Modelling of Hot Rolling of Steel; Yue, S., Ed.; CIM: Montreal, 1990; 128. 50. Nanba, S.; Katsumata, M.; Inoue T. Nakajima, S.; Anan, G.; Hiramatsu, A.; Moriya, A.; Watanabe, T.; Umemoto, M. CAMP-ISIJ 1990, 3, 871. 51. Kinsman, K.R.; Aaronson, H.I. Transformation and Hardenability in Steels; Climax Molybdenum Co.:Ann Arbor, MI, 1967; 39. 52. Suehiro, M.; Liu, Z.-K.; Agren, J. Acta Mater. 1996, 44, 4241. 53. Purdy, G.R.; Brechet, Y.J.M. Acta Metal. Mater. 1995, 43, 3763. 54. Enomoto, M. Acta Mater. 1999, 47, 3533. 55. Enomoto, M.; Kagayama, M.; Maruyama, N.; Tarui, T. Proceedings of the International Conference On Solid–Solid Phase Transformation ’99 (JIMIC-3); Koiwa, M., Otsuka, K., Miyazaki, T., Eds.; JIM: 1999; 1453. 56. Suehiro, M. Proceedings of the International Conference on Solid–Solid Phase Transformation ’99 (JIMIC-3); Koiwa, M., Otsuka, K., Miyazaki, T., Eds.; JIM: 1999; 1465. 57. Fujioka, M.; Yoshie, A.; Morikawa, H.; Suehiro, M. CAMP-ISIJ 1989, 2, 692. 58. Umemoto, M.; Ohtsuka, H.; Tamura, I. Acta Met. 1986, 34, 1377. 59. Umemoto, M.; Ohtsuka, H.; Tamura, I. Tetsu-to-Hagane 1984, 70, 238. 60. Saito, Y.; Shiga, C.; Enami, T. Proceedings of International Conference on Physical Metallurgy of Thermomechanical Processing of Steels and Other Metals; ISIJ: Tokyo, 1988; 753. 61. Park, S.H.; Jonas, J.J. Proceedings of International Symposium on Mathematical Modelling of Hot Rolling of Steel; Yue, S., Ed.; CIM: Montreal, 1990, 446. 62. Okaguchi, S.; Hashimoto, T. ISIJ Int. 1992, 32, 283.

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63. Dutta, B.; Sellars, C.M. Mater. Sci. Technol. 1987, 3, 197. 64. Liu, W.J.; Jonas, J.J. Metal. Trans. A 1989, 20, 689. 65. Akamatsu, S.; Matsumura, Y.; Senuma, T.; Yada, H.; Ishikawa, S. Tetsu-to Hagane 1989, 75, 993. 66. Akamatsu, S.; Senuma, T.; Hasebe, M. Tetsu-to Hagane 1992, 78, 102. 67. Okamoto, R.; Suehiro, M. Tetsu-to-Hagane 1998, 84, 650. 68. Gladman, T.; Holmes, B.; Pickering, F.B. JISI 1970, 208, 172. 69. Gladman, T.; McIvor, I.D.; Pickering, F.B. JISI 1972, 210, 916. 70. Petch, N.J. Phil. Mag. 1958, 3, 1089. 71. Duckworth, W.E.; Baird, J.D. JISI 1969, 207, 854. 72. Pickering, F.B. Towards Improved Toughness and Ductility; Climax Molybdenum Co.:Greenwich, CT, 1971; 9. 73. Irvine, K.J.; Pickering, F.B. JISI 1957, 187, 292. 74. Yoshie, A.; Fujioka, M.; Watanabe, Y.; Nishioka, K.; Morikawa, H. ISIJ Int. 1992, 32, 395. 75. Morikawa, H.; Hasegawa, T. In Accelerated Cooling of Steel; Southwick, P.D., Ed.; TMS-AIME: Warrendale, 1986; 83. 76. Tomota, Y.; Umemoto, M.; Komatsubara, N.; Hiramatsu, A.; Nakajima, N.; Moriya, A.; Watanabe, T.; Nanba, S.; Anan, G.; Kunishige, K.; Higo, Y.; Miyahara, M. ISIJ Int. 1992, 32, 343. 77. Shikanai, N.; Kagawa, H.; Kurihara, M.; ISIJ Int. 1992, 32, 335. 78. Iung, T.; Roch, F.; Schmitt, J.H. International Conference on Thermomechanical Processing of Steels and Other Materials; Chandra, T., Sakai, T., Eds.; TMS: 1997, 2085. 79. Sato, K.; Suehiro, M. Tetsu-to-Hagane 1991, 77, 675. 80. Sato, K.; Suehiro, M.; Tetsu-to-Hagane. 1991, 77, 1328. 81. Suehiro, M.; Senuma, T.; Yada, H.; Sato, K. ISIJ Int. 1992, 32, 433.

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2 Design Simulation of Kinetics of Multicomponent Grain Boundary Segregations in the Engineering Steels Under Quenching and Tempering Anatoli Kovalev and Dmitry L. Wainstein Physical Metallurgy Institute, Moscow, Russia

I.

INTRODUCTION

The basic factors controlling grain boundary segregations (GBS) in engineering steels are discussed. In contrast to single-phase alloys, in engineering steels, the multicomponent segregation is developed simultaneously with undercooled austenite transformations and martensite decomposition. Based on these reasons, the influence of steel phase composition and kinetics on concurrent segregations is discussed. It is established that grain boundary enrichment by harmful impurities (S and P) is possible after carbon and nitrogen segregation dissolution. Two models of GBS are described. The dynamic model of segregation during quenching is based on the solution of independent diffusion and adsorption–desorption equations for various impurities in steel. The model of multicomponent segregation under tempering considers the influence of alloying and tempering parameters on concentration and thermodynamic activity of carbon in the a-solid solution.

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II.

GRAIN BOUNDARY SEGREGATION AND PROPERTIES OF ENGINEERING STEELS

Chemical composition and structure of the grain boundary influences various properties of engineering steels. The following are some of these properties: inclination to temper and heat brittleness, resistance to hydrogen embrittlement, corrosion, delayed fracture, and creep. The intergranular fracture is the main reason for decrease of many steel exploitation properties. Application of modern physical experimental or calculation methods has successfully helped in solving the old metallurgical problem of intergranular fracture. The affinity of various kinds of intergranular brittleness is associated with two main unfavorable factors that decrease intergranular bonds. The impurity segregation to grain boundaries (GBS) and localization of internal microstresses are necessary and sufficient conditions that could initiate embrittlement [1]. Despite the common features, certain kinds of steel brittleness are distinguishable from each other and are stipulated by complex interaction of these factors. The concentration of internal stresses on grain boundaries could be an effect of martensite transformation, hydrogen accumulation, or carbide precipitation; and grain boundary segregations could appear during the equilibrium or non-equilibrium processes of element redistribution in steel. The concentration of microstresses on grain boundaries cause the initiation of cracking and acts as the primary reason for brittleness. The enrichment of grain boundaries by harmful impurities is a major and common condition for development of various intercrystalline brittleness phenomena and it specifies crack propagation entirely along grain boundaries at low stresses. The concept of intercrystalline internal adsorption [2] that was confirmed by theoretical [3], and experimental work [4], the thermodynamic analysis of chemical element interaction during equilibrium grain boundary segregation [5], and investigations of quenched and tempered steel [6,7]. This made it possible to interpret the tempering embrittlement phenomenon. Elemental impurities enrich grain boundaries in thin layers up to several atoms and change the type and value of interatomic bonds that lead to intercrystalline fracture. Embrittlement power is commonly attributed to the elements of the 3rd to 5th periods of groups IV to VI in the periodic system [1]. Sulfur, phosphorus, arsenic, selenium, tellurium, antimony, bismuth, and oxygen are the most harmful impurities that segregate in grain boundaries. The concentration in grain boundaries could reach several atomic percentages exceeding the volume one Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

by several hundreds. Intercrystalline brittleness, as caused by GBS, due to harmful impurities is observed, as a rule, in BCC metals and alloys. The austenite alloys are significantly more resistant to this kind of fracture. The motonic increase of plasticity that is expected after martensite decomposition due to tempering of engineering steels is disturbed by two anomalies resulting in a relative decrease of impact strength. These anomalies are accompanied by intergranular fracture. Steels may be susceptible to embrittlement when they are heated for prolonged period in the temperature range 350–5508C, or when slowly cooled through it. Depending on the heattreatment cycle, the phenomenon is called either temper embrittlement (3508 embrittlement) or reversible temper embrittlement (5508 embrittlement). Common indications of embrittlement are a loss of toughness, segregation of harmful impurities to grain boundaries and the fracture path usually along prior austenite grain boundaries, and the impact transition temperature (FATT—the fracture appearance transition temperature) which is displaced towards higher values. The irreversible temper embrittlement of low-alloyed steels (<5% of alloying elements) is developed during tempering of quenched steel in the temperature range 300–4008C. The reversible temper embrittlement of medium-alloyed steels is developed during tempering in the temperature range 500–6008C. This kind of brittleness is observed after tempering and slow cooling of annealed, normalized, or quenched steel. It is now established that GBS of small impurities play a decisive role in development of these phenomena.

A.

Brittle Fracture of Steel After Quenching and Medium Tempering

The quenched medium-alloyed engineering steel is subjected to tempering in the temperature range 250–4008C to achieve high strength and plasticity. But after tempering at 300–4008C, one can see an abnormal drop in impact strength. At higher tempering temperatures, the impact strength increases again (see Fig. 1) [8]. The intergranular fracture of steel tempered at 300–4008C is due to the action of two unfavorable factors: enrichment of grain boundaries by P during austenitization and formation of lamellar Fe3C particles along the primary austenite grains and martensite packs (Fig. 2a, b). The P segregations decrease the cohesion within boundaries significantly, and carbides block the dislocation movement. This is the reason for peak stresses under plastic

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Figure 1 Impact strength at room temperature of samples with V-shape cut for several melts of industrial steel 4340 after quenching (8508C, 1 hr) in oil and 1 hr tempering at various temperatures. (From Ref. 8.)

Figure 2 (a) Lamellar parts of cementite Fe3C on primary austenite grains and (b) martensite packs boundaries. Steel 0.35C–1.5Mn–0.1P. Tempering at 3508C, 1 hr (TEM, replicas).

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Figure 3 (a) Brittle fracture through primary austenite grains and (b) martensite pack boundaries in steel 0.35C–1.5Mn–0.1P. Tempering at 3508C, 1 hr (SEM).

deformation and, consequently, formation of grain boundary cracks. The brittle crack develops, in this case, within boundaries of primary austenite grains and martensite packs (Fig. 3a, b). The significant enrichment of grain boundaries by phosphorus is confirmed by Auger spectroscopy. The decisive role of austenitization when compared with tempering in achieving equilibrium GBS is shown by low diffusion mobility of harmful impurities’ atoms (P, As, S, and Sb) for medium tempered steel. Calculations show that only nitrogen has significant diffusion mobility in this temperature range which is sufficient for diffusion at a distance of 10 mm for 1 hr at 3508C [1]. The temperature of steel for quenching is sufficiently high for intensive diffusion of P in austenite with the formation of equilibrium GBS [9], and quenching fixes this enrichment. The level of P segregation depends on the austenitization temperature and increases when the temperature decreases below 10508C. This is related to the decrease of P solubility in austenite. It is confirmed by a significant decrease of the quenched steel delay fracture resistance with respect to temperature in the austenite region (Fig. 4). Phosphorous content in steel influences its embrittlement at 3508C tempering. Its concentration in GBS is several hundred times higher than in the volume, and this harmful element segregates in austenite even at low concentrations, about 0.01% mass, leading to a significant decrease in the impact strength (Fig. 5). Change of the relative energy of grain boundaries results in segregation enrichment by impurities during austenitization. The interferometry is one such direct technique for grain boundary energy determination. For this

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Figure 4 Delayed fracture diagram of quenched steel 0.35C; 1.5Mn; 0.1P after austenitization at 1423K, 3 hr and 30 min interim cooling to temperatures: 1, 1123K; 2, 1223K; 3, 1273K and cooling in water. (From Ref. 10.)

purpose, the opening angle of GB slot (y) is determined on metallographic grinds after vacuum etching at various temperatures. The grinds are austenitized in vacuum at high temperature, then subjected to interim cooling to the desired temperature and then quenched at a high cooling rate. The relative energy of GB is calculated by the equation gb y ¼ 2 cos 2 gs

ð1Þ

where symbols b and s correspond to boundary and bulk, respectively. Figure 6 shows the change of relative surface energy with respect to the interim cooling temperature during austenitization and P content in the 0.35% C, 1.5% Mn. For the steel with low phosphorus content, the GB energy regularly decreases slowly as the temperature increases within the austenite region. Significant segregation enrichment of grain boundaries by phosphorus in the steel with 0.1% P does not show this dependence and decreases the surface energy of GB with decreasing of the temperature. The decrease of P solubility in austenite with decreasing temperature assists its adsorption on the GB and decreases significantly its surface tension energy. At the same time, one can observe another process: P enriches

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Figure 5 Temper embrittlement of steel Fe–0.3C–3.5N–1.7Cr: 1, high purity steel; 2, 0.01 mass% P; 3, 0.03 mass% P; 4, 0.06 mass% P. (From Ref. 8.)

non-metallic inclusions. The dependence of P content on Mn sulfides in steel 0.35% C, 1.5% Mn, 0.1% P from austenitization temperature is shown in Fig. 7. When annealing temperature decreases, P redistributes between the bulk and grain boundaries enriching them and the non-metallic inclusions. Steel grain size reduction can decrease significantly its tendency to temper embrittlement. Increase of the specific surface of the GB at dispersion of the steel structure decreases the concentration of the impurity in the grain boundary that leads to growth of steel brittle fracture resistance (Fig. 8) [3]. Phosphorous segregations decrease the surface energy of intergranular cohesion. Using the approach proposed in Ref. [11], one can estimate the role of phosphorus in change of surface energy of intergranular cohesion for development of this kind of embrittlement.

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Figure 6 Change of relative surface energy of grain boundaries in dependence of temperature of austenitization. Fe–0.35C–1.5Mn steel: 1, 0.03% P; 2, 0.1% P.

Figure 7 Phosphorus concentration in manganese sulfides in 0.35C–1.5Mn–0.1P steel vs. austenitization temperature.

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Figure 8 FATT (T50) of steel vs. grain size d1=2 and specific grain boundary surface P S. Phosphorus content: 1, 0.03%; 2, 0.1%. (DT50 ¼ T50(0.1% P)T50(0.03% P)).

Influence of grain size on ductile–brittle transition temperature FATT is determined from the well known Petch–Hall equation sf ¼ so þ Ky d1=2

ð2Þ

where Ky ffi aðGgÞ1=2 ; a is a constant (1–3); g is the surface energy of intergranular cohesion for generation of cracks on grain boundaries; G is the shear modulus. It is described by the equation dT50 1 ¼ b dðd1=2 Þ

ð3Þ

where b is determined as a tangent of inclination angle of straight lines in Fig. 8. Taking into account that

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dT50 dT ðK0y Þ; ¼ dðd1=2 Þ dsf

ð4Þ

it is possible to use the change of the inclination angle of lines 1 and 2 in Fig. 8 to determine the influence of P on GBS by the coefficient b and the effective intergranular cohesion surface energy g that is proportional to b2. The values of b for steel samples containing 0.03% and 0.1% P at the temper embrittlement state are equal to 0.54 and 0.22 sec, respectively. Taking into account dependencies (2) and (3), one can conclude that effective intergranular cohesion surface energy in steel with higher P concentration decreases in g0:03 =g0:1 ¼ b20:03 =b20=1 ¼ 6:04 times. B.

Ductile Intergranular Fracture of Overheated Steel

The ductile fracture of steel as well as brittle fracture could be characterized by the lowest energy capacity. Such a fracture occurs when the inclusions are located along grain boundaries occupying a very large volume near the boundary. The intergranular microvoid fracture is observed in this case due to overheating of the steel. The samples are exposed to high temperature heating to dissolve inclusions. As a rule, the large and lamellar oxysulfides that did not embrittle steel are dissolved. After their dissolution, segregation of O, S, P, and precipitation, disperse particles during steel cooling occurs. In low-alloyed steels, precipitates could be sulfides of chromium and manganese (MnS, CrS), and aluminum nitride, AlN. These particles build a dense network on grain boundaries. The fracture occurs at higher or room temperatures by intergranular microvoid coalescence at low stress intensity. The micrometer scale cavities nucleate on the intergranular fine dispersion of sulfide or nitride particles (Fig. 9). The segregation of harmful impurities is observed on grain boundaries in this case. C.

Reversible Temper Embrittlement

The reversible temper embrittlement (RTE) is observed in engineering steel alloyed by carbide-forming elements after quenching and high tempering (500–6008C). This phenomenon is developed in steels of industrial purity. It consists of a large decrease in the steel impact strength after slow cooling, but after rapid cooling at 6508C, the steel has a standard impact strength. The RTE phenomenon was identified for the first time in 1883 [12], when blacksmiths observed that some steels had to be water quenched after tempering, to avoid embrittlement. This decrease of impact strength is not accompanied by a change of physical or other mechanical properties of

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Figure 9 Ductile fracture of steel through grain boundaries (SEM).

steel. The duration of the exposure time within a definite range of tempering temperatures (brittleness zone) plays a decisive role in the development of RTE. The brittle fracture goes through the primary austenite grains (Fig. 10). The duration of the exposure time of the normalized or annealed steel within the dangerous temperature range also leads to this kind of embrittlement. Steels sensitive to RTE are subjected to rapid cooling from 6508C during different heat treatments. The GBS of phosphorus is the main reason for this kind of brittleness. The RTE of alloyed steels is mainly sensitive to two factors: the chemical composition of the grain boundaries, and the mechanical and microstructural parameters of the alloy. The direct correspondence of embrittlement kinetic features and GBS of P at steel tempering has been established. Figure 11 shows the iso-FATT curves for temper embrittlement of Ni–Cr steel [13].

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Figure 10 SEM image of intergranular fracture of Fe–0.35C–1.5Mn–0.1P steel after quenching with temperature of 9508C, 1 hr and tempering at 5508C, 2 hr.

At a constant temperature of embrittlement tempering, the FATT increases with time. The embrittlement is reversible. It can be rejected by short heating above the nose of the C-curve in the ferrite range. Renewed aging and slow cooling of a de-embrittled steel in the critical temperature region gives reembrittlement. These processes are accompanied by a redistribution of impurities on the grain boundaries.

III.

FACTORS DETERMINING MULTICOMPONENT INTERFACE ADSORPTION IN ENGINEERING STEELS, AND THE METHODS OF ITS CALCULATION

The intercrystalline internal adsorption, or grain boundary segregation phenomenon, means the increased concentration of small impurities on

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Figure 11 Diagram of time–temperature FATT, 8C: (1) 60; (2) 55; (3) 50; (4) 45; (5) 40; (6) 35; (7) 30; (8) 25; (9) 20; (10) 15; (11) 10; (12) 5; (13) 0; (14) þ5; (15) þ10; (16) þ15; (17) þ20; (18) þ25; (19) þ30.

the grain boundary (compared to bulk) is caused by decreasing boundary energy. The energy of impurity segregation corresponds to energy gain in the ‘‘bulk-boundary’’ system that accompanies the transition of one impurity atom from bulk to boundary. Thermodynamic description of this phenomenon is based on the Gibbs theory of the equilibrium segregation on free surface. The decrease of redundant energy of GB is the thermodynamic stimulus to change its chemical composition compared to bulk. The impurities that decrease the energy of interfaces are surface-active. These elements could form equilibrium segregations under favorable conditions. The impurities that increase surface tension escape from surface. Gibbs’ adsorption isotherm for grain boundary segregation in a solid binary system, where the matrix obeys Raoult’s law and Henry’s law of diluted solutions, may be expressed Gb ¼

Xc dgb kT dXc

ð5Þ

where Gb is the surplus concentration on the GB, mol=m2; Xc is bulk mole part of impurity; dgb=dXc is the tendency to adsorb. Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

McLean [14] has developed a statistical model of intercrystalline internal adsorption. The main points of this theory are the following: A definite number of adsorption centers (one monolayer) are present on the GB. They have equal adsorption potential. 2. The impurity atoms are adsorbed independently on each of the centers. 3. The adsorption decreases at temperature growth. 1.

This model adequately describes the GBS process for two-component system:   Xb Xc DG ¼ exp  ð6Þ RT X0b  Xb 1  Xc where Xb and Xc are the impurity concentrations on boundary and in bulk, respectively; X0b is the ultimate equilibrium concentration of the impurity on boundary; DG is the segregation energy. Hondros and Seah [15] have established the interrelation of GB enrichment and solubility limit X0c   Xb Xc DG ¼ exp ð7Þ RT X0b  Xb X0c The enrichment factor, determined as the GB content=bulk concentration ratio, is of the order of magnitude 104 and 101 for impurity and alloying elements, respectively (see Fig. 12). A.

Binding Energy of Impurities with Grain Boundaries

One can determine the adsorption energy as an alteration of the system free energy during transition of the dissolved atom from the grain bulk to boundary E ¼ ðEb  Eb0 Þ  ðEc  Ec0 Þ

ð8Þ

where Eb and Ec are the free energy of the system with impurity atoms on the grain boundary or in the bulk, respectively; Eb0 and Ec0 are the free energies of boundary and bulk in the pure solvent. The value of energy E is tied with the elastic (dimensional) and chemical interaction of impurity with boundary. The influence of dimensional discordance of atoms is accounted by the Eshelby equation [16] 16 E ¼ pGr3 ðs  1Þ2 ð9Þ 3 where G is the shear modulus of the solvent; r is the solvent atom radius; s is the ratio of impurity and solvent atoms radii.

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Figure 12 Calculated and experimental enrichment coefficients for surface segregations. (From Ref. 15.)

The decrease of elastic distortion energy during the transition of the impurity atom from ideal bulk lattice to distorted boundary lattice is the driving force adsorption. According to Ref. [17], one can describe the chemical part of adsorption Ech ¼ DZðe12  e11 Þ

ð10Þ

where DZ is the difference of the atom coordination numbers in bulk and on boundary; eij is the energy of interaction between the nearest neighbors (1 for the solvent, 2 for impurity).

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The Fowler–Guggenheim theory [18] accounts for the interaction of atoms adsorbed on GB:   E1  ZoXb =X0b Xb Xc ¼ exp ð11Þ KT X0b  Xb X0c where Z is the coordination number for impurity atoms; o is the interaction energy of nearest atoms. The portion of segregation energy ZoXb=X 0b corresponds to interaction of the same and different atoms on boundary, and depends on the degree of neighbor centers filled. At o > 0, the atoms repulse mutually, and they attract at o < 0. The attraction increases the adsorption energy substantially. Interaction of atoms on interface influences segregation enrichment and promotes formation of 2D phases with ordered atomic structure. B.

Impurities’ Concurrence During Adsorption

None of the existing adsorption theories adequately describe the micromechanism of impurities’ concurrence on the adsorption centers. This is related to the peculiarities of adsorption from gaseous phase to the free surface to describe the grain boundary segregation mechanism [19–21]. According to this point of view, all segregating elements (for example N, C, S, and P) occupy equal positions on GB, described by the ‘‘site competition’’ term. The peculiarity of GBS formation consists of diffusion of alloying elements and impurities from bulk to interface. The migration mechanisms for substitial and interstitial impurities are different. The reason for this is that the adsorption centers on interface are different for these two kinds of impurities. Therefore, the interstitial impurities (C, N) are located in interstices, but S, P, Sb, Bi, etc. occupy substitution position on the GB. Adsorption of any surface-active impurity on GB decreases its free energy and lowers the thermodynamic stimulus for adsorption of other impurity in a similar way. This causes a site competition between atoms on the GB [22]. The concurrence of impurities A and B segregating at the same temperature and occupying the same positions in crystalline lattice (lattice points or interstices) with different binding energies to GB was investigated in Ref. [23]. The equilibrium concentration of competing impurities A and B could be calculated using equations: XA b ¼

A XA C expðEseg =kTÞ B A B 1 þ XA C expðEseg =kTÞ þ XC expðEseg =kTÞ

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ð12Þ

XBb ¼

XBC expðEBseg =kTÞ 1þ

XA C

B B expðEA seg =kTÞ þ XC expðEseg =kTÞ

ð13Þ

where Eseg is the segregation energy; XbI is the GB concentration of element I A B > Eseg , the (atomic fraction). As one can see from these equations, at Eseg concentration of element A decreases with increasing temperature. In this case, the adsorption level of the element B reaches its maximum at critical temperature: Tcr ¼

EA seg A B A k lnðEA seg =ðEseg  Eseg ÞXb Þ

ð14Þ

The grain boundaries are enriched in this case by element B at low temperatures, and by element A at high temperatures.

C.

Thermodynamic Calculations of the Segregation Energy

The segregation energy calculations are based on various models of solid solution electronic structure or quasi-liquid model of grain boundary. Thermodynamic properties of the solid solution determine firstly the surface activity of small impurities in the formation of equilibrium GBS. The components of the alloy influence the energy of interaction with grain boundaries significantly. The parameters of such interaction are determined by thermodynamic calculations, phase equilibrium diagram analysis, computer modeling of GBS. The heat of solution of different atoms in solid solution is the thermodynamic measure of their interaction. The model establishing interrelationship segregation energy and heat of solutions is proposed in Ref. [24] 2=3

Eseg ¼ F Hsol  PðgA  gB ÞVA

ð15Þ

where F and P are empirical coefficients; Hsol is the heat of solution of A in B [25–30]; gA, gB is the surface enthalpy of elements A and B [31,32]; VA is the molar volume of A. Using the liquid grain boundary model approximation, the segregation energy of impurities (Eseg) could be determined from an analysis of the solidus and liquidus curves on phase equilibrium diagrams: Eseg ¼ KT LnðK0 Þ ¼ KT LnðCL  CS Þ

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ð16Þ

where K0 is the coefficient of equilibrium distribution of the element between solid and liquid phases [33]; T is the melting temperature of the pure solvent; CL and CS are concentrations of impurity in the liquid and solid solutions, respectively. An experimental method for determination of binding energy of impurity atoms to grain boundary is used. The analysis of large number of phase equilibrium diagrams has led to the establishment of the basic property of two-component solid solutions consisting of periodic variation of the segregation formation energy of an element as a function of its location in the periodic table (atomic number). As seen in Fig. 13, the impurities could have positive or negative surface activity, or be neutral. The elements So, Mo, Ni, and Co are neutral in two-component alloys with Fe. At the same time, it is well known that molybdenum is the surface-active element in steels and it reduces the tendency of steel to the reversible temper embrittlement. This change of surface activity is observed only in multicomponent alloys, and it is due to the mutual influence of elements on its thermodynamic activity. The binding energy of an impurity to the GB depends significantly on boundary structure. The wide spectrum of Eseg exists for the given substance analogous to the spectrum of the GB energy. This circumstance explains the wide dispersion of the segregation energy for various impurities that are listed in literature sources. Based on this reasoning, it is useful, for segregation modeling, to apply the unified approach for determination of the generalized characteristic of definite impurity segregation in definite solvent. The thermodynamic calculations of segregation energy are the most suitable way for its estimation. Auger electron spectroscopy (AES) for investigation of segregation kinetics on the free surface of polycrystalline foils is a reliable experimental technique for the averaged Eseg determination. The part of elastic and chemical interaction in GBS process could be estimated experimentally based on concentration dependencies of segregation energy. These dependencies were determined for alloys whose compositions are listed in Table 1. The segregation energy was determined based on AES of equilibrium free surface segregations of phosphorus. The polycrystalline foil samples were tempered at 823K for 4 hr in a work chamber of electron spectrometer ESCALAB MK2 after quenching from austenitization temperature of 1323K. The segregation energy of P was determined using Eq. (6) based on surface and bulk impurity concentration. Figure 14 shows the dependence of the P segregation energy and its bulk content in alloy. For the diluted solid solutions, Eseg is independent of concentration or temperature. It is caused only by elastic distortions that are formed by impurity atoms in

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Figure 13 Change of calculated Eseg of impurities in Fe-base alloys in accordance to its number in periodic system. Calculations were based on Hsol in the following publications: (a) Refs. 25 and 26; (b) Refs. 27 and 28; (c) Ref. 33.

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Table 1 Composition (at.%) of the Fe–P and Fe–P–Mo Alloys Chemical composition, at.% C 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

S

P

Mo

0.002 0.003 0.002 0.003 0.002 0.002 0.005 0.014

0.017 0.10 0.15 0.093 0.033 0.14 0.09 0.074

0 0 0 3.1 3.1 3.1 0.3 0.02

bulk and on the interface. As seen in Fig. 14, the elastic interaction energy of the P atoms with grain boundaries in iron is equal to 0.53 eV=at and decreases significantly at molybdenum alloying to 0.24 eV=at in the alloy Fe–3.1at.% Mo. Decrease of segregation energy of the impurity at its

Figure 14 Change of Eseg of phosphorus with its volume concentration in Fe (1) and Fe–3.1 at.% Mo–P alloys (2). Auger electron spectroscopy of free surface segregations at 823K.

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volume concentration growth is caused by chemical pair interaction of the atoms in alloy. Using the example of the Fe–P system, we could determine chemical interaction of elements by applying the approach proposed in Ref. [34]. Analyzing the solidus and liquidus equilibrium (volume and GB) on the equilibrium phase diagram at three temperatures permits the construction of a system of three equations that describe this equilibrium   100  Xs ð17Þ ¼ X2s W0  X2l W00 þ kqa Ta kT qa  ln 100  Xl where k is the Boltzmann constant; Ta is the melting temperature of Fe; qa is melting entropy per atom divided by Boltzmann constant; W0 and W00 are the mixing energies in solid and liquid states; Xs and Xl are the impurity concentration in solid and liquid phases at the temperature T. Solving these equations for the phase diagram of Fe–P binary system [35], the sign and value of mixing energy in liquid phase equal 0.425 eV=at were determined. The positive value (in accordance with physical sense) means that binding force of P–P and Fe–Fe atoms is higher than for Fe–P atoms: 1 W ¼ WFeP  ðWFeFe þ WPP Þ 2

ð18Þ

emphasizing the tendency for solid solution tendency for stratification or intercrystalline internal adsorption.

D.

Effect of Solute Interaction in Multicomponent System on the Grain Boundary Segregation

Guttman has expanded the concept for synergistic co-segregation of alloying elements and harmful impurities at the grain boundaries. His theory is very important for analysis of steels and alloys that contain many impurities and alloying elements. In accordance with the theory, the interaction between alloying elements and the impurity atoms could be estimated from enthalpy of formation of the intermetallic compounds (NiSb, Mn2Sb, Cr3P, etc.). The alloying elements could influence on the solubility of impurities in the solid solution. Only the dissolved fraction of the impurity takes part in the segregation [36]. When preferential chemical interaction exists between M (metal) and I (impurity) atoms with respect to solvent, the energy of

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segregation becomes functions of the intergranular concentrations of I and M: DGI ¼ DG0I þ

bbMI b baMI a Y  X M Cb Ca M

ð19Þ

bbMI b baMI a Y  a XI ab I a

ð20Þ

DGM ¼ DG0M þ

where Cb and ab are the fractions of sites available in the interface for I and M atoms, respectively ðab þ C b ¼ 1Þ; Yb is the partial coverage in the interface; Xa is the concentration in the solid solution a; bMI is the interaction coefficient of M and I atoms in a-solid solution (a) or on the grain boundary (b). For a preferentially attractive M–I interaction, the bMIare positive and the segregation of each element enhances that of the other. If the interaction is repulsive, the bMI are negative and the segregations of both elements will be reduced. For a high attractive M–I interaction in the a-solid solution, the impurity can be partially precipitated in the matrix into a carbide, or intermetallic compound. The interface is then in equilibrium with an a-solution where the amount of dissolved I, XIa, may become considerably smaller than its nominal content. In the ternary solid solutions, the segregation of impurity (I) could be lowered or neglected at several critical concentrations of the alloying element (M) whose value (CM a) depends on surface activity of each compoI,M ) and interaction features of the dissolved atoms (bMI): nent (ESeg CM a ¼

EISeg bMI ðexpðEM Seg =RTÞ  1Þ

ð21Þ

The critical concentration of alloying element is accessible for segregation of I;M > 0 and repulsion of different atoms impurity and alloying element ESeg M < 0 and with bMI > 0; or without segregation of alloying element ESeg attraction of different atoms bMI < 0. In this case, the dependence of EI,M Seg on the dissolved element concentration is not taken into account. Indeed, for systems with limited solubility, the alteration of value and sign of segregation energy is possible at a definite content of alloying element. The phase equilibrium diagram analysis allows the determination of mutual influence of components on their surface activity. The equilibrium distribution of solute elements between solid and liquid phases in iron-base ternary system (distribution interaction coefficient K0) is known to be an important factor in relation to microsegregation during the solidification of steels. As it was shown above, these analogies are useful for the prediction of GBS and for impurity segregation energy

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Figure 15 Change of the equilibrium distribution coefficient of some elements with carbon concentration in Fe–C-based ternary systems. (From Ref. 37.)

determination in the given solvent. The K0 of some elements, especially in multicomponent systems, is considered to be different from those in binary systems because of the possible existence of solute interactions, but the mechanisms are so complicated that detailed information has not yet been obtained. Therefore, it would be very useful if the effect of an addition of an alloying element on the distribution could be determined by the use of a simple parameter. Equilibrium distribution coefficient K01 of various elements in Fe–C base ternary system is calculated from equilibrium distribution coefficient in iron-base binary systems [40–43]. In Fig. 15, the calculated results are compared with the measured values by various investigators. The changes of the K01 of P and S with various alloying elements are shown in Fig. 16(a, b) in Fe–P and Fe–S base ternary system, respectively. These data could be applied for calculation of phosphorus segregation energy change under the alloying element influence in Fe–Me–0.1at.% P alloys (Fig. 17) or for calculation of the segregation energy change of alloying elements with concentration of carbon in Fe–0.1Me–C alloys (Fig. 18). For the growth of carbon volume content, the segregation energy of C and P decreases which means lowering of the segregation stimulus for these elements.

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Figure 16 (a) Change of the equilibrium distribution coefficient of phosphorus with the concentration of alloying elements; and (b) change of the equilibrium distribution coefficient of sulfur with the concentration of alloying elements. Solid line: a-phase. Chain line: g-phase. (From Ref. 38.)

E.

Kinetics of Segregation

The existing models of multicomponent adsorption do not analyze in detail the kinetics of the process. But in reality, GBS forms only during a limited time of the heat treatment process. The difference of segregation level from the equilibrium one depends on temperature and time. At low temperatures and limited time of heat treatment, segregation is controlled by diffusion. As the temperature increases, segregations with lower equilibrium concentra-

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Figure 16

(Continued)

tion are developed, but rich segregations dissolve. Distinguishing diffusion mobility and mutual influence of elements on their diffusion coefficients determines much of their segregation ability. Amplification or suppression of adsorption could be due to a kinetic factor. This peculiarity determines the fundamental factor of distinguishing adsorption from gas phase to free surface when comparing it to intercrystalline internal adsorption: GBS is controlled by diffusion during heat treatment of steels and alloys. Many GBS features in multicomponent systems cannot be predicted adequately using the equilibrium segregation thermodynamic accounting basis. Particularly, the thermodynamic concept of the cooperative (synergis-

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Figure 17 Change of Eseg of phosphorus with the concentration of alloying elements in Fe–Me–0.1% P alloys.

tic) adsorption of elements is disturbed when they do not segregate at the same temperature. The concurrence of impurities at GBS could be tied not only due to their attractive or repulsive interaction, but also with higher diffusion mobility of some impurities. In many cases, the determination interatomic interaction on grain boundary that is proposed in Guttmann’s theory has a significantly lower effect for segregation prediction than accounting of mutual influence of elements on their thermodynamic activity in the grain bulk. Mutual influence of the alloy components on their surface activity is caused by their interaction in solid solution in the bulk. The interaction on grain boundaries could be analyzed only for those elements that segregate in near temperature ranges. Many postulates of the thermodynamic theory of equilibrium grain boundary segregation could not be applied simply for heat treatment of multicomponent alloys. This is especially important for steels, which have complex phase transformations during treatment that accompany change of the solid solution composition. Auger electron spectroscopy permits the investigation of multicomponent adsorption kinetics. The composition of grain boundaries on the intercrystalline fracture surface made under high vacuum is analyzed for this purpose. In these cases, the experimental modeling of GBS is widely used. The chemical composition of free surface of thin poly-crystalline foils that are heated in situ is investigated using Auger electron spectrometers. The P grain boundary adsorption isotherms for samples of three Fe–Cr–Mn

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Figure 18 Change of Eseg of alloying elements (Me) with the concentration of carbon in Fe–0.1% Me–C alloys.

steels after quenching from 1273K and tempering at 923K for 25 min, 1 and 2 hr with air cooling are presented in Fig. 19. Dissolution of Ti and V carbonitrides after steel quenching promotes enrichment of the solid solution by these elements. They have high values of Gibbs energy for phosphide formation, decrease the thermodynamic activity of phosphorus in solid solution and reduce its GBS. Most models of kinetics are classically analyzed in terms of the law derived by McLean [14] for binary alloys " #
pffiffiffiffiffiffiffi  Xb ðtÞ  Xb ð0Þ 4Di t 2 Di t ¼ 1  exp erfc Xb  Xb ð0Þ ðXb =Xai Þ ðXb =Xai Þ2 d2

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ð22Þ

Figure 19 Kinetics of P GBS in steel 0.3C–1.6Mn–0.8Cr–008P (1) with adds of 0.047Ti (2) or (0.07Ti and 0.026V) (3), quenched from 1273K and tempered at 923K.

where Xb(t) is the interfacial coverage of element, at time t; Xb(0) — is its initial value and Xb its equilibrium value as defined by Eq. (7); Xia — is its volume concentration; Di is the bulk diffusivity of i and d is the interface thickness. Assuming Xb=Xia ¼ const, using Laplace transformation for (22), one can obtain the approximate expression rffiffiffiffiffiffiffiffiffiffi Xb ðtÞ  Xb ð0Þ 2Xai FDti ¼ ð23Þ Xb d p Xb  Xb ð0Þ where F ¼ 4 for grain boundaries and F ¼ 1 for free surface. The kinetics of segregation dissolution could be described by these equations (22) and (23). But, in this case, the variables Xb(0) and Xb exchange places. The influence of Mo, Cr, and Ni additions on kinetics of P segregation has been studied in six Fe–Me–P alloys, whose base compositions are listed in Table 1. These materials were austenitized for 1 hr at 1323K and quenched in water. The tempering of foils at 773K was carried out in a work chamber of an electron spectrometer ESCALAB MK2 (VG). The kinetics of P segregation studied for Fe–Me–P alloys (Figs. 20–22) show that equilibrium is reached within several hours. Based on the starting position of adsorption isotherms, the phosphorus diffusion coefficients in these alloys were calculated using Eq. (22). The data are presented in Table 2. Molybdenum reduces significantly P surface activity and decelerates its diffusion. Nickel is not a surface-active element in carbonless alloys, Fe– P–Ni. It increases sharply P thermodynamic activity and equilibrium GB concentration, and accelerates its diffusion. Chromium segregates poorly

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Figure 20 Kinetics of P segregation on free surface in Fe–P–Mo alloys with different relative concentration of Mo=P at 773K.

in these alloys. It also, as Ni, increases diffusion mobility of P and its grain boundary adsorption. The adsorption isotherms of various elements have non-monotonous shape in multicomponent alloys. The isodose thermokinetic diagrams present the averaged information on segregation of all components. Such

Figure 21 Kinetics of P and Cr segregation on free surface in Fe–0.04P–2.3 Cr alloy at 773K.

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Figure 22 Kinetics of P segregation on free surface in Fe–P–Ni alloys with different relative concentration of Ni=P at 773K.

diagrams for steel with 0.2–0.3% C alloyed by Cr, Mo, Ni, Mn, V, and Nb are presented in Figs. 23–27. Chemical composition of steel is listed in Table 3. The T–t diagrams are obtained based on adsorption isotherms on free surface of foils that were heat treated in Auger spectrometer ESCALAB MK2 at vacuum about 1010 Torr. The isodose curves characterizing time of definite segregation level access depending on temperature are shown in these diagrams. At elevation of an isothermal exposition temperature, the mobility of impurities increases, and time for reaching of definite segregation level decreases. The lower branch of isodose curve means decrease of segregation formation Table 2 Composition (at.%) of the Fe–Me–P Alloys and Kinetics Characteristics of P Free Surface Segregation, Deduced from the Segregation Kinetics Chemical Composition, at.% P

Mo

Ni

Cr

Surface activity Xb=Xia

Bulk diffusivity of P DP  1018 (m2=sec)

0.07 0.03 0.04 0.16 0.21 0.15

0 0 0 1.0 2.1 3.1

0.9 3.1 0 0 0 0

0 0 2.3 0 0 0

285 2530 1450 380 250 80

5.34 1048 160 27 2.28 0.3

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Figure 23 The isodose C-curves of multicomponent interface segregation in 0.3C– Cr–Mo steel (see Table 3). Auger electron spectroscopy of free surface segregations.

time at increasing temperature. With temperature increase, the solubility of impurity in solid solution increases, and its GB concentration reduces. It follows that the probability to form the segregation with high impurity content reduces, and time for such segregation increases extensively. The upper branch of isodose curves corresponds to dissolution of rich segregations and access to new equilibrium with lower impurity concentration. The

Figure 24 The isodose C-curves of multicomponent interface segregation in 0.2C– Cr–Mn–Ni–Si steel (see Table 3) under its tempering. Auger electron spectroscopy of free surface segregations.

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Figure 25 The isodose C-curves of multicomponent interface segregation in 0.3C– Cr–Mn–Nb steel (see Table 3) under its tempering. Auger electron spectroscopy of free surface segregations.

adsorption patterns for engineering steels have common as well as individual features. As a rule, carbon segregates at temperatures lower than 523K, nitrogen—in 523–623K range, phosphorus—in 523–823K range, sulfur segregates at temperatures higher than 723K. The substitual and interstitial element concurrence promotes blocking of adsorption centers by mobile impurities and impedes P segregation at

Figure 26 The isodose C-curves of multicomponent interface segregation in 0.3C– Cr–Mn–V steel (see Table 3) under its tempering. Auger electron spectroscopy of free surface segregations.

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Figure 27 The isodose C-curves of multicomponent interface segregation in 0.3C– Cr–Mn–Si–Ti steel (see Table 3) under its tempering. Auger electron spectroscopy of free surface segregations.

temperatures lower than 523–673K. The preferential enrichment of GB by P and S becomes possible after dissolution of C and N segregations. The alloying elements change significantly the segregation stability regions for various elements. Fig. 28(a,b) shows the P adsorption isotherms in the investigated steels at 723K. Molybdenum sharply slows down the P segregation formation. The differences in diffusion mobility of elements and temperature intervals of segregation stability are the reasons for nonequilibrium enrichment of grain boundaries. The rich segregations are formed at the initial stage of isothermal exposition, and they are dissolved after longer exposition. Comparing the behavior of 0.3C–Cr–Mn–Nb (1) and 0.22C–Cr–Mn–Si–Ni (3) steels at 673K tempering, one can see that small (lower than 20 min) expositions 0.3C–Cr–Mn–Nb, and longer ones (about 1 hr 20 min) are dangerous for 0.22C–Cr–Mn–Si–Ni steel. Analyzing Table 3

Chemical Composition of Steels Concentration of elements, wt.%

No. 1 2 3 4 5

Steel

C

Si

Mn

Cr

V

Al

Ti

Nb

Ni

Mo

S

P

0.3C–Cr–Mn–V 0.3C–Cr–Mn–Nb 0.3C–Cr–Mo 0.3C–Cr–Mn–Si–Ti 0.2C–Cr–Mn–Ni–Si

0.32 0.29 0.33 0.28 0.22

0.25 0.33 0.23 0.61 0.43

0.88 1.04 0.56 1.15 0.92

0.92 1.07 0.96 0.75 0.89

0.088 0 0.003 0 0

0.014 0.007 0.014 0.029 0.030

0.024 0.036 0.025 0.016 0.015

0 0.025 0 0 0

0 0 0 0.32 0.91

0 0 0.25 0 0

0.016 0.014 0.005 0.013 0.015

0.027 0.027 0.004 0.022 0.025

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Figure 28 Influence of alloying on the kinetics isotherms of P free surface segregation at 723K. The following steels were investigated (see Table 3): 1, 3C–Cr– Mn–Nb; 2, 3C–Cr–Mn–Si–Ti; 3, 2C–Cr–Mn–Ni–Si; 4, 3C–Cr–Mo; 5, 3C–Cr– Mn–V.

the thermokinetic diagrams for ternary Fe–Me–P alloys based on Eqs. (23) and (6), the mutual influence of elements on their binding energy to GB was determined [36] Mo B EPseg ¼ 20:6 þ 183CPa  4:8CAl  3:4CNi a  7:2Ca a  7141Ca S Mo N þ 4:9CCr a  444Ca  183Eseg  87Eseg

ð24Þ

P Sn ESseg ¼ 6:9  151CSa  1:5CAl a þ 14:5Ca  39Eseg

ð25Þ

Al Mo Ti Mo EN þ 4:2CCr seg ¼ 16  2:6Ca þ 3Ca a  2625Ca þ 175Eseg

ð26Þ

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Al Mo N P EC þ 676CBa þ 1:2CCr seg ¼ 7:9  1:4Ca þ 5Ca a  130Eseg þ 116Eseg

ð27Þ N P EMo seg ¼ 0:7 þ 32Eseg  28Eseg

ð28Þ

C P ETi seg ¼ 17 þ 3Ca  Eseg

ð29Þ

Al EAl seg ¼ 1:4Ca

ð30Þ

Ni B Cu ESn seg ¼ 21; Eseg ¼ 14; Eseg ¼ 54; Eseg ¼ 20 kJ/mol

where EIsegis segregation energy of the I element, Caj is bulk concentration of j impurity.

F.

Stability of the Segregation

The equilibrium GBS dissolves as temperature increases. Analysis of the kinetic development of the equilibrium segregation level of P shown in Fig. 29 gives the T–t plot of segregation directly. Obviously that segregation level close to the maximum exists only within a specific temperature range. This range is characterized by a maximum temperature stability Tmax, over which the intensive dissolution of the segregates is observed. This temperature can be calculated by computer analysis of Eq. (7) at dCbmax=dT ¼ 0. The temperature Tmax depends on Eseg and temperature dependencies of solubility limits, which can be determined from analysis of phase equilibrium diagrams [43]. Using these dependencies as a generalizing criterion, it is possible to simplify the analysis of data on element segregation kinetics in iron alloys. The interrelationship of maximum temperature of stability (Tmax) of rich equilibrium segregations and segregation energies of different elements is presented in Fig. 30. The common features of kinetics show the following groups: enriching grain boundaries at low- and medium-tempering temperatures—B, C, N, and Cu; 2. co-segregating with P at high tempering—P, Sn, Ti, and Mo; 3. segregating at high temperatures—S and Al. 1.

Phosphorus in Fe alloys has abnormally weak dependence of Tmax on Eseg in reversible temper embrittlement temperature range. In other

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Figure 29 The calculated segregation level of P as a function of temperature according to Eq. (7). Tmax is the maximal temperature of stability of rich segregation level. (From Ref. 42.)

words, this means that the temperature of P segregation stability in the RTE development interval weakly depends on segregation energy or alloy composition. This circumstance is associated with the specific shape of the temperature dependence of P solubility in Fe. The established regularity allows to explain the difficulties with rational alloying of engineering steels for RTE suppression. G.

Nature of Reversibility of Temper Embrittlement

The reversibility of temper embrittlement is usually associated with precipitation or dissolution of carbide phase at various modes of quenched steel heat treatment below Ac1 [44–46]. The complex character of multicomponent GB adsorption—namely interrelation of two opposite processes: concurrence between impurities, and their cooperative segregation—is not taken into account using this approach.

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Figure 30 The interconnection of Tmax-segregation stability temperature and Eseg-energy of impurities segregation in Fe-base alloys.

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Figure 31 Thermo-kinetics diagrams of multicomponent segregation on free surface in steel 0.35C–1.58Mn–0.1P–0.6Al.

Figure 31 presents the thermokinetic diagram of element segregation in 0.35C–1.5Mn–0.1P–0.6Al steel. The chemical composition of free surface segregations was determined by AES for a set of isothermal conditions in the spectrometer ESCALAB MK2 (VG). The temperature–time interval of preferential segregation of chemical elements is the result of different diffusion mobility and binding energy of elements with GB. The temperature interval of P preferential segregation is caused by concurrence of this impurity with mobile interstitial elements C and N. This process determines temperature and exposition necessary for RTE development. Direct investigation of grain boundary composition by AES confirms the conclusion about the prevailing role of concurrent segregation in RTE. The composition of several grain boundaries on brittle intercrystalline fracture of 0.35C–Mn–Al steel after heat treatment: quenching from 1223K, tempering at 923K for 1 hr with rapid (a) and slow (b) cooling is presented in Fig. 32 [47]. These data are in good correspondence with those in Fig. 31. Accelerated cooling of steel, does not provide enough time for the development of segregations with high P content, and GB are enriched by C. During slow cooling, phosphorus has enough time to enrich the grain boundaries. In this case, the carbon concentration on GB is sufficiently lower than at rapid cooling of steel. Carbon segregations are unstable at temperatures higher than 500–673K, and they are dissolved. At slow cooling, P segregates to grain boundaries, decreasing the GB redundant energy. This circumstance lessens the thermodynamic stimulus for carbon segregation as the temperature decreases. Carbon and phosphorus in steels are responsible for RTE development. They have high surface activity and diffusion mobility that

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Figure 32 Chemical composition of GB in steel 0.35C–1.58Mn–0.1P–0.6Al (AES); (a) tempering at 923K, water cooling; (b) tempering at 923K, cooling with furnace. (From Ref. 47.)

predetermines their segregation on GB at heat treatment. Difference of diffusion mobility as well as difference of maximum temperature of segregation stability is the reason for preferential segregation of an impurity. This is the reason for the characteristic temperature region of RTE development. Reversibility (i.e. disappearance) of temper embrittlement is associated with full dissolution of rich GBS of phosphorus at high temperatures and

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enrichment of GB by carbon at rapid cooling [48]. Undoubtedly, carbide transformation, internal stresses, substructure transformations are very important for RTE. One should take into account such circumstances where kinetics of C and P segregation are dependent significantly on steel alloying.

IV.

DYNAMIC SIMULATION OF GRAIN BOUNDARY SEGREGATION

A.

Interface Adsorption During Tempering of Steel

1. Decomposition of Martensite The common laws of multicomponent GBS and analysis of experimental diagrams on elements segregation kinetics in iron alloys are used to develop the computer models of these processes. The exact solution of McLean’s diffusion Eq. (21) accounting for temperature dependant of diffusion and element solubility is a complex problem. In low-alloyed steels, the concentration of surface-active impurities (S, P, and N) is rather small, and based on this reason, it is possible to analyze the diffusion of each element separately. The model takes into account mutual influence of bulk and surface concentration of elements with respect to segregation energies. Carbon in solid solution has maximum influence on phosphorus GBS kinetics. Concentration of C in martensite changes significantly during quenched steel tempering and mainly depends on alloying element content. Based on this reason, one should take into account the solid solution composition altering segregation processes modeling during tempering. Investigations of martensite tetragonality at alloyed steel tempering [6,7] are the basis for calculations of mutual influence of alloying elements on martensite decomposition kinetics and carbon content in solid solution. The carbon content change in solid solution during tempering of engineering steels is well described by equation   
DXC Q n a ¼ 1  exp KD t exp  o XC RT a ð0Þ

ð31Þ

where C C DXC a ¼ Xa ð0Þ  Xa ðtÞ

XaC(0)

XaC(t)

ð32Þ

and are the carbon content in quenched steel and after a time t; Do is the carbon diffusion coefficient; Q is the activation energy associated with the interstitial diffusion of carbon atoms; K is the constant associated

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Table 4

Coefficient A in Eq. (28) for Low-Alloying Engineering Steels Alloying element

Coefficient A

Ni

Si

Mn

Cr

Mo

433.56

1,432.54

726.35

2,898.91

971.51

with the nucleation; n is the constant independent of both temperature and XaC(0); R is the gas constant and T is the temperature. Influence of C and alloying elements on parameters Q, K, and n in Eq. (31) is determined for various steels. The activation energy Q in lowalloyed steel depends on the concentration of carbon and alloying elements in solid solution: Me Qðcal=molÞ ¼ 8571:5XC a þ A Xa þ 18; 000

ð33Þ

where XaC and XaMe are concentrations of C and alloying elements, mass%; A is a constant depending on alloying element. The values of coefficients in Eq. (31) are presented in Tables 4 and 5. The diffusion activation energy of

Table 5 Influence of Carbon and Alloying Elements on Parameters Q, K, and n in Eq. (31) Steel, wt.% 0.4C–0.24Ni 0.39C–3.0Ni 0.37C–5.6Ni 0.4C–0.32Mn 0.4C–1.32Mn 0.4C–2.43Mn 0.4C–0.2Cr 0.4C–2.1Cr 0.4C–3.6Cr 0.4C–6.7Cr 0.4C–0.37Si 0.38C–1.75Si 0.4C–2.75Si 0.4C 1.4C 1.2C–2.0Mo

Q, cal=mol

Ln K

n

21,532 22,643 23,599 21,196 20,298 19,406 20,848 15,348 10,992 2,005 21,929 23,764 25,368 21,429 30,000 26,343

15.364 17.481 18,575 15.737 14.241 13.713 15.366 10.076 5.481 1.698 16.351 15.234 10.050 16.72 40.881 29.768

0.26 0.22 0.24 0.24 0.22 0.24 0.21 0.24 0.42 2.32 0.19 0.22 0.15 0.24 0.07 0.08

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Figure 33 Change of carbon concentration in solid solution with temperature and time of tempering. Steel 0.43C–2.43 Mn (mass%). Isodose curves for: 1, 1 at.% C; 2, 0.5 at.% C; 3, 0.1 at.% C; 4, 0.05 at.% C; 5, 0.03 at.% C.

carbon decreases on the growth of carbide-forming element (Mn, Cr, and Mo) concentration. The contrary effect is observed for Ni and Si. Obviously, it is associated with the different influence of these elements on thermodynamic activity of carbon in ferrite. These dependencies are basic for calculations of segregation kinetics of C since carbon is the element that influences on P segregation highly. The kinetics of carbon content in solid solution change during tempering of quenched steel 0.43C–2.43Mn (mass%) are shown in Fig. 33. These data are obtained by computer modeling using Eqs. (31–33) and those from Tables 4 and 5. This model provides the possibility of calculating the influence of alloying on cementite formation temperature interval, growth rate of its particles, and many other parameters of martensite decomposition at tempering [49]. Fig. 34 presents the calculation results of effective growth rate of Fe3C nucleus at tempering of engineering alloyed steels. The calculations were carried out using expression [49]: Vmax R ¼ ð27D=256pÞN

ð34Þ

where R is the cementite particle radius; N is the right part of Eq. (30). Manganese decreases martensite stability significantly promoting its decomposition at low temperatures. Silicon, at a concentration greater than 1%, activates martensite decomposition at 700–800K and inhibits it at lower

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Figure 34 Change of effective growth rate of Fe3C nucleus with alloying of 0.4% C steel: 1, unalloyed steel with 0.4C(mass%); 2, alloyed with 0.35Si; 3, alloyed with 2.1Cr; 4, alloyed with 1.75Si; 5, alloyed with 2.43Mn.

content. Chromium does not change the temperature of intensive cementite growth. 2.

Calculation of Thermokinetic Diagrams of Impurities’ Segregation During Tempering of Steel Modeling of multicomponent adsorption kinetics is carried out using a sequence of computer calculations. At the initial stage, thermodynamic characteristics of surface activity in Fe-base binary and ternary alloys are determined. Analysis of phase equilibrium diagrams permits the determination of i the impurity segregation energy Eseg and temperature dependence of ultimate solubility X8c. These two parameters are very important for determination of equilibrium GB concentration of impurity Xbi using Eq. (17). Examples of such calculations for binary and ternary alloys have been presented. Mutual influence of alloy components on their surface activity could be refined experimentally. The equations for binding impurity segregation energy with solid solution composition could be obtained by regression analysis of multicomponent adsorption diagrams. These experiments allow the determination of the effective diffusion coefficient of elements. The diagram of the equilibrium impurity concentration calculation on grain boundary in engineering steel is presented in Fig. 35. Carbon concentration in martensite changes drastically during tempering, as it depends on chemical composition of steel, temperature, and duration

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Figure 35 Calculation scheme of equilibrium impurity GBS. Xai (0) is the initial concentration of ith element in the steel; XaC(T,t) is the running carbon concentration in martensite during its tempering; Xbi (T) is the maximal equilibrium GBS of ith i element; Eseg Fe–i is the segregation energy of ith element in two-component Fe–I i alloy; Eseg Fe–i–j is the segregation energy of ith element in multicomponent alloy; Di(T) is the diffusion coefficient of ith element in austenite, martensite, and ferrite.

of treatment. This factor influences on thermodynamic activity of all steel components and on their energy of GB segregation. The second important stage of GBS modeling includes calculation of C volume concentration in martensite XaC(T), depending on steel chemical composition Xai (0) and parameters of tempering. New segregation energy values of each element at changing of treatment temperature or time and new equilibrium GBS level have been calculated in this way (see Fig. 35). The final stage of modeling includes a set of independent calculations of various element diffusion to GB zone, and their desorption. The limited capacity of boundary and its effective width (about 0.5 nm) are shown. It is assumed that interstitial and substitial impurities occupy different positions on GB. Time t of reaching the definite concentration of impurity in segregation Xb(t) at given temper temperature T is calculated by (22), and it is controlled by diffusion Di(T). Adsorption in multicomponent system is accompanied by concurrence: arrival of some surface-active impurity decreases GB energy and, in this way, the thermodynamic stimulus for segregation of other impurities. Dissolution of segregations is observed at increasing temperature. Impurity desorption to grain bulk is analogous to adsorption, however it is tied not with concentration Xi(0) but with Xb(t), and it is also controlled by diffusion Di(t). Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

The model is restricted to initially homogeneous bulk concentrations Xib ð0Þ ¼ Xia

ð35Þ

The kinetics of segregation to surfaces or grain boundaries from the bulk are determined by volume diffusion of impurities with bulk concentrations Xia(t) which can be treated as a one-dimensional problem. Since both bulk concentrations are very small, Arrhenius type diffusion coefficients: Di ¼

Di0

  Qi exp  RT

ð36Þ

can be used which are independent of Xia(t). In the case of site competition, the GB impurities concentration is qi ¼

1

Xi P

  Ei exp  KT J Xj

ð37Þ

The equations describing the time evolution of segregation for homogeneous initial condition [60] are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z t Z t X0i 1 qi ðt0 ÞDi ðt0 Þ ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Di ðt0 Þ dt0  pffiffiffi Xi ðtÞ ¼ Xi ð0Þ þ 2 pffiffiffi ð38Þ R t0 pd 0 pd 0 00 Þ dt00 D ðt i t In the case of constant temperature (i.e. Di ¼ const), Eq. (38) can be simplified: pffiffiffiffi pffiffi 2 D 0 Xi ðtÞ ¼ Xi ð0Þ þ pffiffiffi ½Xi  qi ðtÞ t pd

ð39Þ

Diffusion coefficient for impurities in Fe and Fe-base alloys in ferrite interval is present in Table 6. The calculated diagrams of multicomponent adsorption in steels 0.3C– Cr–Mo, 0.3C–Cr–Mn–V, 0.3C–Cr–Mn–Si–Ti (see Table 3) are presented in Figs. 36–38. Comparing these diagrams with the experimental ones (Figs. 24, 26, and 27), a good correlation of segregation kinetic features for various elements is observed, that confirms the basic principles of the proposed model of GBS in steels. According to this model, the main role of carbide precipitation in GBS consists of changing solid solution

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Table 6 Solute C C P P P P P P P P P P P P P S Sn Cr Co C C C C C C C C C

Coefficients of Diffusion for Impurities in a-Fe and Steels System

Temperature, K

0.3C–10Ni Martensite a-Fe a-Fe a-Fe a-Fe a-Fe a-Fe Fe–2.1Mn Fe–Ni–P a-Fe 0.1P–0.15Cr 0.1P–0.13Si 0.1P–0.17Mn 0.1P–0.14Mo 0.1P–0.14Ni Fe–3Si a-Fe Fe–Cr Fe–6.8Co Fe–0.79Si Fe–0.79Si Fe–0.79Si Fe–0.6Ni Fe–0.6Ni Fe–0.6Ni Fe–0.56Mo Fe–0.56Mo Fe–0.56Mo

723–873 623 723 748 773 798

a a a a a 973 973–1303 1048 903–1073 803 873 973 803 873 973 873 923 973

D, m2Sec1

Q, D0 (m2Sec1) kcal=mol Reference 5.26  105

5.3 2.8 7.7 2.0 4.8

    

1014 1019 1019 1018 1018

0.108 exp(288=RT) 0.336 exp(296=RT) 18 exp(329=RT) 0.235 exp(292=RT) 3.23 106 exp(434=RT) 43.9 exp(336=RT)

3.6  1012 1.4  1011 1.9  1010 4  104 16  104 6.9  104 7.2  104 21  104 55  104

15.2

9.55  106 1.43  104 0.51  104

50.6 54.2 55

1.7  102 5.4 2.33  104 4.69  105

61.2 55.5 57.1 44.7

[50] [51] [52] [52] [52] [52] [53] [53] [54] [55] [55] [55] [55] [55] [55] [56] [57] [58] [59] [60] [60] [60] [60] [60] [60] [60] [60] [60]

composition, as it is exactly this factor that controls mutual influence of elements on their surface activity. Only such elements that segregate in near temperature ranges mutually influence GBS. The computer calculations of segregation kinetic diagrams predict these effects with small changes of steel chemical composition. Figures 39 and 40 present the modeling data on influence of sulfur content in 0.3C–Cr–Mn–Si–Ti steel on phosphorus segregation kinetics. Sulfur and Phosphorus are strong surface-active elements, and they can compete at grain boundaries. Desulfurization of steel significantly slows down GBS of S. Indeed, P adsorption increases with a decrease of S content. According to calculations (see Fig. 40), the time of 6% P GBS formation exceeds 4000 sec at a sulfur content more than 0.02 at.%. This time it is significantly longer than the usual duration of quenched steel

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 36 The isodose C-curves of multicomponent interface segregation in 0.2C– Cr–Mn–Ni–Si steel (see Table 3) under its tempering. Computer simulation.

tempering. Deeper cleaning of steel by S activates GB adsorption of P, drops down time of segregation formation, and increases the maximum temperature of segregation stability (see Figs. 39 and 40). Such calculations are very useful for the design of optimal alloying and purification degree on harmful impurities, since they permit the determination of the influence of alloying on ultimate concentration of harmful impurities.

Figure 37 The isodose C-curves of multicomponent interface segregation in 0.3C– Cr–Mn–V steel (see Table 3) under its tempering. Computer simulation.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 38 The isodose C-curves of multicomponent interface segregation in 0.3C– Cr–Mn–Si–Ti steel (see Table 3) under its tempering. Computer simulation.

B.

Interface Adsorption During Quenching of Engineering Steels

Mathematical models of GBS [61] and phase transformations permit the analysis of heat treatment with respect to the accompanying phenomena in a greater detail than that of a simple summary of the experimental knowledge.

Figure 39 Dependence of Tmax of P GBS as a function of sulfur containing in 0.3C–Cr–Mn–Si–Ti steel under its tempering. Computer simulation.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 40 Dependence of time of 6 at.% GBS of phosphorus and sulfur as a function of sulfur concentration in 0.3C–Cr–Mn–Si–Ti steel during its tempering at 700K. Computer simulation.

The results of mathematical modeling provide backgrounds for reasonable planning of full-scale experiments when seeking for the optimum technological procedures and steel composition and they enable the extrapolation of the consequences of variations in the technological conditions even outside the boundary of the empirical experience we have available. Interaction of GB segregation enrichment and phase transformations during heat treatment of steels in the austenitic region is hard to imagine. Nb and V carbonitride precipitation in microalloyed austenite, precipitation of free ferrite, change chemical composition of austenite, and influence on GBS kinetics to a large extent. The experiments show that nonequilibrium grain boundary phenomena occur for a rather short time up to 100 sec. The minimum time of 5% volume fraction of Nb and V carbonitride precipitation is about 1000 sec [62,63]. Precipitation of free ferrite needs from several seconds to several minutes depending on steel chemical composition. Therefore, the non-equilibrium GBS in steels with a wide region of undercooled austenite stability independently from phase transformations. This computer model has some limitations but redistribution of harmful impurities between grain bulk and boundaries permits the analysis of steel quenching. The modeling of non-equilibrium GB phenomena allows during investigation of such short-time changes of chemical composition that could not be measured experimentally and that has an extreme importance for modern heat-treatment processes with high heating and cooling velocities in controlled media.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 41 The presentation of C-curve on simulating TTT diagrams. (Scheme.) Parameters U and S correspond to Table 7.

1. Phase Transformations of Undercooling Austenite At present, many computer models of evolution of structure and phase composition of steels during quenching have been developed. Most of them are based on physical models of phase transformations [64–66]. But physical models cannot describe adequately all kinetic features of undercooled austenite transformations. The computer models based on regression analysis of experimental data can best predict steel phase composition changes during steel cooling. It was introduced directly by Davenport and Bain [67] and the time– temperature-transformation (TTT) diagram was the predominant tool to describe the isothermal decomposition kinetics of supercooled austenite. In most TTT diagrams, general S- or C-curves are used to represent the kinetics of a number of isothermal transformation products: ferrite, pearlite, upper bainite, lower bainite, and martensite. Conversely, many experimental results demonstrate that each type of transformation product has a separate C-curve. To build a mathematical model, all TTT diagrams published in Refs. [68–71] were analyzed. The rationalization of the kinetics of isothermal decomposition of austenite permitted the establishment of a metastable product (phase) diagram of a number of steels of different compositions with 6% of total content of all alloying elements.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Most isothermal transformations take place by nucleation at the austenite grain boundaries, so the original austenite grain size will affect the isothermal decomposition kinetics of austenite. From the total number of factors characterizing austenite matrix, the present day experimental knowledge allows only an approximate examination of the statistically recrystallized proportion and estimation of the size of deformed austenite grains. The grain growth kinetics satisfy the law [73]   Q dðtÞ ¼ d0 þ kt exp  RT

ð40Þ

where d(t) is averaged grain size at moment t; d0 is initial grain size; Q is activation energy; k is a constant. The algorithm of calculating the size of austenite grains is described in Ref. [73]. The procedure for calculation of the structural proportions of anisothermal decomposition of austenite at engineering steel cooling is given in Tables 7 and 8 and shown in Fig. 41. The cooling curve is approximated partially by a constant function and at the individual time intervals Dt and the rate of decomposition is calculated as isothermal transformation corresponding to the mean temperature of that interval. The required kinetic data are available from the TTT diagrams [68–71] that can be digitized (see Table 8) by procedures shown in Fig. 41, using equation
1=2
 S  S0 1 U  U0 1=2 U  U0 ¼e exp  SN  S0 UN  U0 2 UN  U0

ð41Þ

where S ¼ Int-time interval, s; U ¼ 1000=(T þ 273). Since it is necessary to distinguish between the parts of the C-curves representing the formation of ferrite, pearlite, and bainite, only those diagrams having readily distinguishable component curves were used in the analysis. The calculation method includes the effect of the size of austenite grains on the kinetics of phase transformations. The main precondition is knowledge of this effect on the course of C-curves showing the start and end of transformations in the graph of isothermal decomposition of austenite for the relevant steel.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Table 7

The Algorithm of Calculation of the Structural Proportions

1. Temperature of start of transformations yAc3, yAc1, yBa, yMs 2. t 0; y(0) y0; V1(0) 1; Vi(0) 0, i ¼ 2, 3, 4, 5; i ¼ 1-austenite, 2-ferrite, 3-pearlite, 4-bainite, 5-martensite 2.1 Mean temperature at the interval of ht, t þ Dti y ¼ (y(t) þ y(t þ Dt))=2; if y < ¼ yMs pass to 3; if y < ¼ min (yjs) then n j; 2.2 for i ¼ 2, . . . , n carried out as follows: – calculation of the transformable proportion of austenite Vmi(t) for i ¼ 2: Vm2(t)¼0; for y ¼ > yA3, 0 yA3 y Vm2 ðtÞ ¼ Vm2 yA3 yA1 ; for yA1 < y < yA3 ;

0 Vm2 ðtÞ ¼ Vm2 ; for y <¼ yA1 ; Sð % CÞXð %CÞ ; for i > 2 and V ¼ 1 0 Vm2 ¼ mi Sð%CÞPð%CÞ – calculation of the start and end of transformation tsi, tfi and exponent ki for y ki ¼ 6.127=ln (tsi=tfi); for ferrite k2 ¼ 1; – the fictive volume fraction of the transformed proportion Xi ¼ Vi(t)=[Vi(t) þ Vi(t) Vmi(t)]; – the fictive time of isothermal transformation required for reaching the proportion Xi

ki  t lnð1X Þ t0i ¼  si b i 1=ki ; S

– the fictive volume proportion of the structural component at time t þ Dt


k  ti þDt i ; Xi ðti þ DtÞ ¼ 1  exp b2 tsi – the volume proportion of the structural component at time t þ Dt Vi ðt þ DtÞ ¼ Xi ðt0i þ DtÞ½Vi ðtÞ þ Vi ðtÞVmi ðtÞ; 2.3 the new value of residual content of austenite P V1 ðt þ DtÞ ¼ 1  ni¼2 Vi ðt þ DtÞ; if V1 ðt þ DtÞ <¼ 0 the transformation is finished; 2.4 t t þ Dt; pass over 2.1 3. The martensite transformation for yMf < ¼ y < ¼ yMs V5(y) ¼ (1V2V3V4)[1exp(0.011(yMsy))] P 4. The residual content of austenite at y < ¼ yMf: Vi ¼ 1  5i¼2 Vi ; I ¼ 1, 2, 3, 4, 5—austenite, ferrite, pearlite, bainite, martensite; S(%C), P(%C), X(%C)containing carbon in points S and P of Fe–C phase diagram and in the steel, consequently.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Table 8 The Constants Blk for Calculations of the YL Parameters of C-Curves k 1

2

3

4

5

6

7

8

Ferrite-start

T8C Time

Pearlite-start

T8C Time

Pearlite-finish T8C Time Bainite-start

T8C Time

Bainite-finish T8C Time MS

YL TOFS TNFS IntOFS IntNFS TOPS TNPS IntOPS IntNPS TOPF TNPF IntOPF IntNPF TOBS TNBS IntOBS IntNBS TOBF TNBF IntOBF IntNBF TMS

Const

10

11

12

13

14

15

16

Const C

Mn

Si

Ni

Cr

Mo

GS

48 1.92 2.58 0 6 0.03 0.03 107 107 5.44 1.40 76 76 7.73 1.98 0

30 57 2.80 0.20 0 32 10.12 1.31 0 4 14.08 1.31 12 30 1.46 1.29 12 30 0.75 1.17 0

66 25 3.55 1.10 15 20 3.55 0.20 15 5 0.76 1.56 –5 35 1.64 1.64 5 28 5.44 5.84 0

22 15 1.50 0.04 5 5 3.30 0.20 5 5 1.99 0.65 –5 5 0.40 0.20 7 7 0.40 0.02 0

70 15 3.00 2.55 5 27 3.00 2.55 5 36 3.00 0.98 –5 36 3.49 2.04 5 40 0.90 2.47 0

5 25 3.50 3.50 22 23 4.20 2.50 22 25 4.35 4.35 –40 2 3.00 6.60 51 96 6.88 4.00 0

0 3.90 0.0096 0.021 0 3.20 0.096 0.041 0 1.31 0.0096 0.021 0 0.37 0.0626 0.0248 0 0.37 0.0813 0.0340 0

Dlk

Blk Transformation

9

C

727 572 10.13 1.04 727 572 10.13 1.04 727 577 10.55 0.15 570 485 3.52 0.96 570 488 6.90 0.15 539 423

Mn

Si

Ni

Cr

Mo

GS

13 33 2.5 0.6 13 33 2.5 0.6 13 2 6.86 2.0 12 12 2.14 1.30 12 48 5.23 3.00 30.4

9 44 2.84 0.19 9 44 2.84 0.19 9 35 3 0.19 16 2 1.07 0.30 16 25 2.8 2.8 0

17 9 0.46 0.07 17 9 0.46 0.07 17 7 1.81 0.07 12 7 0.05 0.02 7 7 0.7 0.7 17.7

22 25 4.8 4.8 22 25 4.8 4.8 22 53 4.8 2.65 20 40 2.3 2.2 22 47 6.2 3 12.1

3.5 50 5.9 5.9 3.5 50 5.9 5.9 3.5 58 12.3 10.6 22 32 3.1 2.2 22 61 9.5 5.9 7.5

0 1.01 0.095 0.038 0 1.01 0.095 0.038 0 0.52 0.095 0.038 0 0.50 0.070 0.024 0 0.52 0.070 0.024 0

229 48 1.92 2.58

Ck

YL ¼

P8

k¼1 ½Blk

C1

C2

C3

C4

C5

C6

C7

C8

1

%C

%Mn

%Si

%Ni

%Cr

%Mo

2

þ Dlk þ 8ð0:8  C2 ÞCk ; GS—grain size (ASTM).

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

The program involves the calculation of temperatures of transformations of bainite and twinned, athermal and lamellar martensite [74] BS ¼ 720  585:63ðCÞ þ 126:6ðCÞ2  66:34ðNiÞ þ 6:06ðNiÞ2  31:66ðCrÞ þ 2:17ðCrÞ3  91:68ðMnÞ þ 7:82ðMnÞ2  42:37ðMoÞ þ 9:16ðCoÞ  0:125ðCoÞ2  36:02ðCuÞ

ð42Þ

2 MTM S ¼ 420208:33ðCÞ72:65ðNÞ43:36ðNÞ 16:08ðNiÞ

þ0:78ðNiÞ2 0:025ðNiÞ3 2:47ðCrÞ33:428ðMnÞþ1:296ðMnÞ2 þ30:0ðMoÞþ12:86ðCoÞ0:2665ðCoÞ2 7:18ðCuÞ

ð43Þ

¼ 540  356:25ðCÞ  260:64ðN  24:65ðNiÞ þ 1:36ðNiÞ2 MLM S  17:82ðCrÞ þ 1:42ðCrÞ2  47:59ðMnÞ þ 2:25ðMnÞ2 MA S

þ 17:5ðMoÞ þ 21:87ðCoÞ  16:52ðCuÞ ¼ 820  603:76ðCÞ þ 247:13ðCÞ2  55:72ðNiÞ þ 3:97ðNiÞ2

ð44Þ

 31:1ðCrÞ þ 2:348ðCrÞ2  66:24ðMnÞ  24:29ðMoÞ  0:196ðCoÞ þ 0:165ðCoÞ2  31:88ðCuÞ

ð45Þ

The size of ferritic grain is expressed as follows [75] 0:5 da ¼ 11:7 þ 0:14dg þ 37:7VC

ð46Þ

where dg is the size of austenitic grain, (mm); VC is the cooling speed, (8C min1). The interlamellar distance of pearlite can be estimated as follows [75]: " # X S¼ 18:0DVP ðyi Þ=ð996  yi Þ =VP ð47Þ i

where DVP(yi) is the volume proportion of pearlite transformed at yi temperature. The thickness of the ferritic and carbide lamellae of pearlite is approximately lf ¼ 0.885S; lc ¼ 0.115S The size of martensitic and bainitic particles is identical with the original size of the austenite grains. The course of the anisothermal decomposition of austenite in several steels has been calculated by the

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

just-described method and by applying the digitized TTT diagrams of Table 7. 2.

Determination of the Kinetics of Carbonitride Precipitation in Austenite Microalloying of steels with Ti, V, Nb, and Zr affects decomposition of supercooled austenite, its recrystallization and grain boundary segregations of harmful impurities. These changes of material properties are associated with carbonitride precipitation and changes of austenite chemical composition. Based on these reasons, the modeling of the kinetics of carbonitride precipitation is important. The nucleation time t of carbonitrides per unit volume N at any temperature T, can be expressed as [76] ! !     Q B Q B exp exp t ¼ C exp t ¼ C exp RT RT T3 ðLnKS Þ2 T3 ðLnKS Þ2 ð48Þ where C ¼ 6  1013 for homogeneous nucleation; activation energy of Nb diffusion Q ¼ 270 kJ=mol; B ¼ 16pg3 V2m N0 =3R3

ð49Þ

Vm ¼ 1.28  105 m3=mol; g ¼ 0.5 J m2; N0 are numbers of nucleus by radius R per molar volume Vm; KS is supersaturation [77] LgðKS Þ ¼ 

A þB T

ð50Þ

where thermodynamics parameters A and B for various carbides and nitrides are calculated in Ref. [78] and presented in Table 9. Thus, the calculation of carbonitride nucleation time necessary to reproduce the C-curves

Table 9

Thermodynamics Parameters in Eq. 50 (From Ref. 77) Chemical compound

Parameter

AlN

VC

VN

TiC

TiN

NbC

NbN

ZrC

ZrN

A B

7,130 1.463

9,500 6.72

7,985 3.09

8,872 4.04

15,573 3.82

7,714 3.27

10,440 3.87

8,464 4.96

13,968 3.08

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corresponds to start or finish of this transformation in austenite under heat treatment of steel. 3.

Calculation of Thermokinetic Diagrams of Impurity Segregation During Quenching of Steel Computation of grain boundary multicomponent adsorption kinetics could be simplified for steels with high undercooled austenite stability. The GBS develops in this case in austenite in short time and has no dependence on phase and structure transformations at steel quenching. Enrichment of grain boundaries by various impurities as well as their desorption is treated as a result of multicomponent diffusion of impurities from near-boundary volume to the boundary. Impurity binding energy with GB includes mutual influence of elements in grain bulk and on the boundary in accordance with Guttmann’s theory [Eqs. (18) and (19)]. Auger electron spectroscopy is the technique for experimental investigation of GBS kinetics. These experiments are basic for analysis of correlation of impurity segregation energy with the content of other elements in the bulk and on boundaries (see Section 2.5, Eqs. (23)–(29). Adsorption and desorption of impurities on GB (qi) at steel quenching is modeled well using the equation 2q0i ffi qi ¼ qi ð0Þ þ pffiffiffiffiffi pd

ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z t

Di

1

Zt

ðt0 Þ dt0  pffiffiffiffiffiffi

0

pd

0

Cia ðt0 ÞDi ðt0 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dt0 Rt 00 00 t0 Di ðt Þ dt

ð51Þ

where d is the grain boundary thickness; Di(t0 ) is the diffusion coefficient of impurity which depends on the temperature and phase composition (austenite, martensite, and ferrite); in the case of adsorption Cia ¼

CiGB expðGi =kTÞ P 1  j CjGB

ð52Þ

i is the element i concentration on grain boundary; Cai is the conwhere CGB centration of ith element in the adjacent bulk layer; Gi is segregation energy. Desorption is determined by GB concentrations of impurities, and in this case, the parameter Cai in Eq. (51) is equivalent to GB concentration XbI in Eqs. (12) and (13). The change of temperature at cooling or isothermal exposition is described by equation

TðtÞ ¼ ½Tð0Þ  Tð/Þ expðrtÞ þ Tð/Þ

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ð53Þ

Table 10

Cooling Rates for Some Metallurgical Technologies

Name of the treatment Quenching Controlled cooling Air cooling of hot-rolled metal Cooling with furnace Controlled cooling of large-size forging

Cooling rates r (K sec1) 100–10 10 10–0.1 0.01 0.001

Table 10 presents cooling rates r for heat-treatment processes. The block diagram of multicomponent intercrystalline adsorption model is shown in Fig. 42. Adsorption of P, C, and S is determined by parameters K1, K2, K3, and their desorption by parameters K2, K4, K6. The parameters Ci are equivalent to GB concentration of element i. This model allows the computation of the condition when there is change of GB composition in steels and alloys at preselected arbitrary mode of cooling including isothermal exposition. Given below are the examples of investigation of phosphorus and sulfur grain boundary adsorption in Cr–Ni–Mo steel (see Table 11). The components of steel mutually influence their diffusion mobility and GBS activation energy. Based on this reason, one should take into account the stochastic fluctuations of diffusion flows of various impurities on GBS kinetics. For this purpose, the random fluctuation of diffusion coefficients up to 30% of its mean value was used in the model. Figures 43 and 44 present the GBS kinetics calculation results at cooling of various purity steels cooling that were carried out using the stochastic model. As one can see, the self-regulation of adsorption is observed which is developing despite significant short-time oscillations of impurity concentration on grain boundaries. The significant non-equilibrium enrichment of GB by impurities is observed at initial stage of the heat treatment. This effect is determined by cooling velocity as well as impurities content. Increasing cooling velocity from 0.001 to 1000K s1 decreases the non-equilibrium GBS of P and S. Formation of non-equilibrium rich GBS of harmful impurities at small cooling times could be established only by using computer modeling methods. The experimental verification of such phenomena needs special techniques which allow to open grain boundaries: hydrogenation of quenched samples or delayed fracture tests. Since these techniques are conducted in air and could not be applied in the vacuum chamber of electron spectrometer; for most of engineering steels, the regularities of non-equilibrium GBS formation at quenching could only be estimated by a computer experiment.

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Figure 42 carbon).

Table 11

Calculation scheme of three-component GBS (phosphorus, sulfur, and

Chemical Composition of Cr–Ni–Mo Steels Chemical composition, mass%

Smelting number 82 83

C

S

P

Ni

Cr

Mo

Time of austenite stability at 600 hr

0.38 0.38

0.027 0.01

0.054 0.006

3.95 4.02

3.0 3.0

0.51 0.50

2.0 2.0

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Figure 43 The change of GBS during quenching of Cr–Ni–Mo steel containing 0.027S and 0.054P (mass%). Computer simulation of fast (a) and slow (b) cooling.

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Figure 44 The change of GBS during quenching of Cr–Ni–Mo steel containing 0.01S and 0.006P (mass%). Computer simulation of fast (a) and slow (b) cooling.

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Figure 45 Chemical composition of GB in Cr–Ni–Mo steel containing 0.027S and 0.054P (mass%) after austenitization at 1373K (30 min), interim cooling up to 873K and quenching in water (a) and in furnace (b). Auger electron spectroscopy of intergranular fracture.

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Figure 46 Chemical composition of GB in Cr–Ni–Mo steel containing 0.01S and 0.006P (mass%) after austenitization at 1373K (30 min), interim cooling up to 873K and quenching in water (a) and in furnace (b). Auger electron spectroscopy of intergranular fracture.

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The validation of calculation reliability was done for steel composition 82 and 83 (see Table 10) by Auger spectroscopy. The samples after austenitization at 1373K (30 min) were in the interim cooled to 873K with further cooling in water or with furnace cooling. The undercooled austenite in this steel has high stability and does not transform in ferrite region for 2 hr. After cooling the samples had martensite–baintite structure. To investigate the chemical composition of grain boundaries by Auger spectroscopy, special samples were crushed in the electron spectrometer ESCALAB MK2 at vacuum at about 108 Pa at temperature 83K. The fields with intercrystalline fracture type were investigated on the fracture surface. The variation of phosphorus and sulfur content in GBS in Cr–Ni–Mo steel of several melts after heat treatment is shown in Figs. 45 and 46. At accelerated cooling the GB are significantly enriched by carbon. The P concentration in GB increases only at slow cooling of samples, and P segregation is strongly suppressed in pure steel. A good correspondence of calculated and experimental results is observed for all cases to be analyzed. The results of numerical modeling give information about the equilibrium and non-equilibrium character of a GB adsorption processes, which are frequently unavailable from experiments. Moreover, these simulation methods explain the phenomenon of reverse temper embrittlement as the result of non-equilibrium concurrent GBS of carbon and phosphorus. These results explain many questions in the multicomponent GB adsorption kinetics in engineering steels that were dynamically developed in the last 10 years. Further investigations in this direction are required especially for competitive internal adsorption in engineering steels treated by using newest schemes of heat treatment. REFERENCES 1. Briant, C.L.; Banerji, S.K. Intergranular failure in steel. Int. Met. Rev. 1978, 4, 164–196. 2. Arharov, V.I.; Ivanovskaya, S.I.; Kolesnikova, K.M.; Farafonova, T.A. The nature of phosphorous influence on temper embrittlement. Fiz. Met. Metallioved. 1956, 2, 57–65. 3. Hondros, E.D.; Seah, M.P. Segregation to interfaces. Int. Met. Rev. 1977, 22, 12,261–12,303. 4. Seah, M.P. Grain boundary segregation. J. Phys. F. 1980, 10 (6), 1043–1064. 5. Guttmann, M. Equilibrium segregation in ternary solution: a model for temper embrittlement. Surf. Sci. 1975, 53, 213–227. 6. Kaminskii, E.Z.; Stelletskaya, T.I. Kinetic of martensite dissolution in carbon steel. Problems of Fisical Metallurgy; Metallurgy: Moscow, 1949, 192–210. 7. Bokshtein, S.Z. Structure and Mechanical Properties of Alloyed Steel; Metallurgy: Moscow 1954.

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8. Briant, C.L.; Banerji, C.K. Intergranular failure of ferrum alloys in inagressive environment. In Treatise on Materials Science and Technology, Vol. 25; Embrittlement of Engineering Alloys; Briant, C.L., Banerji, S.K., Eds.; Academic Press: 1983; 29–59. 9. Erhart, H.; Paju, M. Phosphorus segregation in austenite. Scripta Met. 1983, 17, 171–174. 10. Kovalev, A.I. Influence of grain boundaries phosphorus segregation and steel microstructure on fracture after quenching and middle tempering. Fiz. Met. Metalloved. 1980, 49, 818–826. 11. Glikman, E.; Cherpakov, Ju.; Bruver, R. Dependence of fracture toughness and surface energy of grain boundaries on grain size of Si-steel under reversible temper embrittlement. Fiz. Met. Metalloved. 1978, 42, 864–870. 12. Howe, H.M. Proc. Inst. Mech. Eng. 1919, Jan–May, 405. 13. Carr, F.L.; Goldman, M.; Jaffe, L.D.; Buffin, D.C. Trans. AIME. 1953, 197, 998. 14. McLean, D. Grain Boundaries in Metals; Oxford University Press: London 1957. 15. Hondros, E.D.; Seah, M.P. The theory of grain boundary segregation in terms of surface adsorption analogues. Met. Trans. A 1977, 8, 1363. 16. Abraham, F.F.; Brundle, C.R. Surface segregation in binary solid solutions: a theoretical and experimental perspective. J. Vac. Sci. Technol. 1981, 18 (2), 506–519. 17. Abraham, F.F. Bond and strain energy effects in surface segregation. Scripta Met. 1979, 13 (5), 307–311. 18. Fowler, R.H.; Guggenheim, E.A. Statistical Thermodynamics; University Press: Cambridge 1939. 19. Grabke, H.J. Adsorption, segregation and reactions of non-metal atoms on iron surfaces. Mat. Sci. Eng. 1980, 42, 91–99. 20. Grabke, H.J. Surface and grain boundary segregation on and in iron. Steel Res. 1986, 57, 4178–4185. 21. Grabke, H.J. Surface and Grain Boundary segregation on and in Iron and Steels. ISIJ Int. 1989, 29, 7,529–7,538. 22. Kovalev, A.I.; Mishina, V.P.; Stsherbedinsky, G.V.; Wainstein, D.L. EELFS method for investigation of equilibrium segregation on surfaces in steel and alloys. Vacuum. 1990, 41 (7–9), 1794–1795. 23. Bruver, R.E. Investigation of impurities influence on intergranular low temperature failure of metallic solid solution. Ph.D. Dissertation, Tomsk, 1970. 24. Chelikowsky, J.R. Predictions for surface segregation in intermetallic alloys. Surf. Sci. 1984, 139, L197–L203. 25. Miedema, A.R.; Boer, F.R. et al. Enthalpy of formation of transition metal alloys. Calphad 1977, 1, 4341–4359. 26. Miedema, A.R. On the heat of formation of plutonium alloys. In Plutonium and Other Actinides; Blank, H.; Linder, R., Eds.; North-Holland Publ.: Amsterdam, 1976; 3–20. 27. Bennett, L.H.; Watson, R.A. A database for enthalpies of formation of binary transition metal alloys. Calphad 1980, 5 (1), 19–23.

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28. Watson, R.E. Optimized prediction for heats of formation of transition metal alloys. Calphad 1981, 5 (1), 25–60. 29. Birnie, D.; Machlin, E.S.; Kaufman, C.; Taylor, K. Comparison of pair potential and thermochemical models of the heat of formation of BCC and FCC alloys. Calphad 1982, 6 (2), 93–126. 30. Machlin, E.S. Correlation terms to pair potential model values of the energy of formation for transition element–polyvalent element phases. Calphad 1980, 5 (1), 1–17. 31. Zadumkin, S.N. Modern theory of surface energy of pure metals. In. Surface Phenomena in Liquid metals and Solid Phases; Kabardino-Balkarskoe Publ. Press: Nalchik 1965; 12–28. 32. Zadumkin, S.N. Surface energy on the interface of metals. In Surface Phenomena in Liquid metals and Solid Phases; Kabardino-Balkarskoe Publ. Press: Nalchik, 1965; 79–88. 33. Morita, Z.; Tanaka, T. Effect of solute-interaction on the equilibrium distribution of solute between solid and liquid phases in iron base. Trans. ISIJ 1984, 24, 206–211. 34. Kamenetskaya, D.S. Influence of molecular interaction on phase diagrams. Problems of Physical Metallurgy and Metal Science; Metallurgia: Moscow 1949; 113–131. 35. Toshihiro, T. Equilibrium distribution coefficient of P in Fe-base alloys. J. Iron Steel Inst. Jpn. 1984, 70 (4), 220. 36. Guttmann, M.; Dumolin, Ph.; Wayman, M. The thermodynamics of interactive co-segregation of phosphorous and alloying elements in iron and temper-brittle steels. Met. Trans. A 1982, 13, 1693–1711. 37. Morito, Z.-I.; Tanaka, T. Effect of solute-interaction on the equilibrium distribution of solute between solid and liquid phases in iron base ternary system. Trans. ISIJ. 1984, 24, 206–211. 38. Morito, Z.-I.; Tanaka, T. Equilibrium distribution coefficient of phosphorus in iron alloys. Trans. ISIJ 1986, 26, 114–120. 39. Okamoto, T.; Morito, Z.; Kagawa, A.; Tanaka, T. Partition of carbon between solid and liquid in Fe–C binary system. Trans. ISIJ 1983, 23, 266–271. 40. Yamada, K.; Kato, E. Effect of dilute concentrations of Si, Al, Ti, V, Cr, Co, Ni, Nb and Mo on the activity coefficient of P in liquid iron. Trans. ISIJ 1983, 23, 51–55. 41. Michina, V.P. Atomic interaction at grain boundary segregation in Fe alloys. Author’s Abstract of Doctor’s Thesis, Moscow, 1988. 42. Seah, M.P. Grain boundary segregation and the Tt dependence of temper brittleness. Acta Met. 1977, 25, 345–357. 43. Bannih, O.A.; Budberg, P.B.; Alisova, C.P. Equilibrium diagrams of two- and poly-component systems on base of Fe. Handbook; Metallurgia: Moscow 1986. 44. Arharov, V.I.; Konstantinova, T.S. The nature of reversible temper embrittleness of 0.35C–Cr–Mn–Si and 12C–Cr–Ni steels. Fiz. Met. Metalloved. 1974, 38 (1), 169–175.

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45. Otani, H.; Feng, H.C.; McMahon, C.J. New information on the mechanism of temper embrittlement of alloy steels. Met. Trans. 1974, 5 (2), 516–518. 46. Lei, T.C.; Tang, C.H.; Su, M. Temper brittleness of 0.3C–Cr–Mn7–Si–2Ni steel with various initial microstructures. Metal Sci. 1983, 17, 75–79. 47. Kovalev, A.I.; Michina, V.P. Role of grain boundary segregation in RTE of steels. Physical basis of construct of physical and mechanical properties of steels and alloys. In Proceedings of CNIICHERMET; Bardin, I.P., Ed., Metallurgia: Moscow, 1990, 43–46. 48. Kovalev, A.I.; Mishina, V.P.; Stsherbedinsky, G.V. Features of intergranular adsorption of carbon and phosphorus in Fe-alloys. Fiz. Met. Metalloved 1986, 62 (1), 126–132. 49. Zuyao, X.; Siwei, C. Mechanism of embrittlement in tempered martensite. Mat. Sci. Technol. 1985, 1, 1025–1028. 50. Zemskii, S.V.; Litvinenko, D.A. Diffusion of C in two-phase system. Fiz. Met. Metalloved 1971, 32 (3), 591–596. 51. Golikov, V.M.; Matosyan, M.A.; Estrin, E.I. Influence of pressure on carbon diffusion. Protect. Coat. Met. 1971, 4, 74–78. 52. Guttmann, M.; Dumolin, Ph.; Wayman, M. The thermodynamics of interactive co-segregation of phosphorus and alloying elements in iron and temper-brittle steels. Met. Trans. A 1982, 13, 1693–1711. 53. Grabke, H.J.; Hennesen, K.; Moller, R.; Wei, W. Effect Mn on the grain boundary segregation, bulk and grain boundary diffusitivity of P in ferrite. Scr. Met. 1987, 21 (10), 1329–1334. 54. Yongbin, I.M.; Daniluk, St. A surface segregation study in P and S-doped type 304 stainless steel. Met. Trans. A. 1987, 18 (1), 19–26. 55. Mastsuyama, T.; Hosokawa, H.; Suto, H. Tracer diffusion of P in iron alloys. Trans. JIM 1983, 24 (8), 589–594. 56. Gruzin, P.L.; Mural, V.V. S -diffusion in 3%Si–Fe alloy. Fiz. Met. Metalloved 1971, 32 (1), 208–212. 57. Treheus, D.; Marchive, D.; Delagrange, J. Determination of the coefficient of diffusion of Sn of infinite dilution in a-Fe. Compt. Rend. Acad. Sci. C 1972, 274 (13), 1260–1262. 58. Huntz, A.M.; Guiraldeng, P.; Aucouturier, M.; Lacombe, P. Relation between the diffusion of radioactive Fe and Cr in Fe–Cr alloys with 0–15% Cr and their a=g transformation. Mem. Sci. Rev. Met. 1969, 66, 85–104. 59. Hirano, K.; Cohen, M. Diffusion of Co in Fe–Co alloys. Trans. JIM 1972, 13 (2), 96–102. 60. Gruzin, P.L.; Babikova, Yu.F.; Borisov, E.B.; Zimskii, S.V., et al. Investigation of diffusion of C in steel and alloys by C14 isotope. Problems of Metals Science and Physical Metallurgy; Metallurgy: Moscow 1958; 327–365. 61. Militzer, M.; Wieting, J. Theory of segregation kinetics in ternary systems. Acta Met. 1986, 34 (7), 1229–1236. 62. Dutto, B.; Sellars, C.M. Effect of composition and process variables on Nb(CN) precipitation in niobium microalloyed austenite. Mat. Sci. Technol. 1987, 3 (3), 197–206.

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63. Rios, P.R. Expression for solubility product of niobium carbonitride in austenite. Mat. Sci. Technol. 1988, 4 (4), 324–327. 64. Adrion, H. Thermodynamic model for precipitation of carbonitrides in high strength low alloy steels containing up to three microalloying elements with or without additions of aluminum. Mat. Sci. Technol. 1992, 8 (5), 406–420. 65. Reed, R.C.; Bhadeshia, H.K.D.H. Kinetics of reconstructive austenite to ferrite transformation in low alloy steels. Mat. Sci. Technol. 1992, 8, 421–436. 66. Rees, G.I.; Bhadeshia, H.K.D.H. Bainite transformation kinetics Part 1. Mat. Sci. Technol. 1992, 8, 985–993. 67. Davenport, E.S.; Bain, E.C. Trans. AIME. 1930, 90, 117–144. 68. Isothermal Transformation Diagrams, 1943, 1st Ed., 1963, 2nd Ed.; United States Steel Corporation: Pittsburgh, PA. 69. Atlas of Isothermal Transformation Diagrams of BS En Steels, 1949, 1st Ed., 1956, 2nd Ed.; The Iron and Steel Institute, British Iron and Steel Research Association: London. 70. Popov, A.A.; Popova, L.E. Isothermal and Thermokinetics Diagrams of Decomposition of Supercooled Austenite; Metallurgia: Moscow 1965. 71. Atlas of Isothermal Transformation and Cooling Transformation Diagrams; ASM:Metals Park, OH: 1977. 72. BISRA Atlas of Isothermal Transformation Diagrams of BS En Steels, Special Report no. 56, 2nd Ed.; The Iron and Steel Institute: London, 1956. 73. Priestner, R.; Hodson, P.D. Ferrite grain coarsening during transformation of thermomechanically processed C–Mn–Nb austenite. Mat. Sci. Technol. 1992, 8 (10), 849–854. 74. Sellars, C.M. The physical metallurgy of hot working. In Hot Working and Forming Processes, Proceedings of International Conference, University of Sheffield, July 17–20, 1979; The Metalls Soc: London, 1980, 3–15. 75. Zhao, J. Continuous cooling transformations in steel. Mat. Sci. Technol. 1992, 8 (11), 997–1003. 76. Licka, S.; Wozniak, J. Mathematical model for analyzing the technological conditions of hot rolling of steel. Hutnicke Aktuality 1981, 22 (9), 1–49. 77. Dutta, B.; Sellars, C.M. Effect of composition and process variables on Nb(C,N) precipitation in niobium microalloyed austenite. Mat. Sci. Technol. 1987, 3, 197–206. 78. Adrion, H. Thermodynamic model for precipitation of carbonitrides in high strength low alloy steels containing up to three microalloying elements with or without additions of aluminum. Mat. Sci. Technol. 1992, 8, 406–420.

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3 Designing for Control of Residual Stress and Distortion Dong-Ying Ju Saitama Institute of Technology, Okabe, Saitama, Japan

I.

INTRODUCTION

Residual stresses in materials are often produced from metallurgical processes, such as casting, forging, welding and quenching processes, and so on. Usually, the production of residual stresses during metallurgical process depends on changes of thermal sources and volume due to microstructure. Generally, residual stresses of two types can be considered, i.e., macro-residual stresses and micro-residual stresses [1,2]. The macro-type depends on the plastic deformation of solid materials due to rapid non-uniform cooling. And, the strain and deformation due to phase transformation and change of microstructure are the sources of the micro-type residual stresses. We also know that the distortion due to thermal and elastic–plastic deformation and strain as well as change of phase transformation and texture in manufactured materials are important no matter the type of residual stress. Therefore, one of the many important problems is how to control and utilize residual stresses and distortion due to variations of macro- and microstructure in materials for increasing and ensuring strength and quality of products after metallurgical process. To improve the mechanical properties of materials, it is important to know how to raise compressive residual stress and reduce tension residual stress in materials as it is known that the compressive residual stress can increase fatigue strength of materials. Therefore, measurement method and analysis of residual stresses are always developed

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as a conspicuous technology. Many measurement methods of residual stress such as x-ray diffraction, neutron diffraction, rapid drilling method, and so on have been developed in industrial technology [3–8]. However, each measurement technique always has some limitations and problems when used for measuring, for example, the x-ray method only measures residual stresses on surface of materials; on the other hand, in the method of neutron diffraction, a major problem is how to measure the standard lattice spacing d0 of strain-free materials, which consider distortion and strain due to microstructure or texture. In order to address thermal–mechanical behavior in the entire material process, a so-called metallo-thermo-mechanical theory is proposed by Inoue et al. [9–18]. Based on this theory, computer simulation of metallurgical processes can be used as a new technology. This is a critical component of the worldwide effort to develop virtual metallurgical capabilities in order to acquire a better understanding of processing operations and optimize processes with a view to improving quality and reducing production costs. By using this technique, residual stress and distortion in metallurgical process also can be predicted; hence some useful theory and simulation methods can be proposed and introduced in the simulations of various metallurgical processes [19–23]. From these research results, we can obtain useful knowledge to seek more perfect designing for metallurgical process. In this chapter, the metallo-thermo-mechanical theory, numerical modeling and simulation technology considered with coupling of temperature and phase transformation or solidification as well as inelastic behavior involved with elastic–plastic, viscoplastic and creep deformation will be introduced. The theory and simulation technology also are used in metallurgical process, such as heat treatment, continuous casting, and thermal spray coating as well as forging for the control of residual stress and distortion in these processing. This chapter not only presents various simulation methods and results of the residual stress, distortion and thermal– mechanical behavior as well as microstructure according to the peculiarity of each metallurgical process, but also the thermo-mechanical modeling that was verified by comparison with the experimental data, such as the measured temperature, residual stresses and distortion in various metallurgical processes. In the final part of the chapter, some conclusions and remarks which are used in the designing of metallurgical process for control of residual stress and distortion are presented. From these conclusions, we summarize many, yet expected problems and subjects in future metallurgical process and material industry for reference of research.

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II.

METALLO-THERMO-MECHANICAL MODELING FOR CONTROL OF RESIDUAL STRESS AND DISTORTION

There are many physical and chemical phenomena that occur in metallurgical processes. These phenomena do not always act alone on materials; the metallic structures incorporating phase transformation, temperature and stress=strain are also strongly coupled in the process. The metallo-thermomechanics theory are considered with the coupling effect on temperature, stress=strain and phase transformation fields as schematically illustrated in Fig. 1. The coupling effects are indicated as follows: 1. Thermal Stress. The thermal expansion caused by such a temperature gradient is restricted by the shape of a solid body, thus generating thermal stress. 2. Heat Generation due to Deformation. When stress=strain that appears in the case of large inelastic deformation is applied to materials, the energy is partially discharged as heat. 3. Temperature-dependent Phase Transformation. Temperature is the major factor, which determines phase transformation start time. However, in the case of diffusion-type transformations of ferrite, pearlite, and bauxite, the temperature history also affects the phase transformation. 4. Latent Heat due to Phase Transformation. Latent heat generated in the course of phase transformation affects the temperature field.

Figure 1 Metallo-thermo-mechanical coupling during processes involving phase transformation.

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5. Stress (or Strain)-Induced Transformation. The phase transformation behavior is also affected by stress=strain existing in the solid. For example, pearlite transformation time is shortened under tensile stress, and vice versa. Martensite transformation is generated even though a material is processed at a temperature higher than the martensite transformation temperature under the applied stress or strain. 6. Transformation Stress and Transformation Plasticity. Volume dilatation in the work is caused by the phase transformations. When this volumetric dilatation is inhomogeneous depending on the complicated shape of the body, stress and strain are induced, it is defined as transformation stress and strain. The level of such induced stress is comparable to the thermal stress. The effect of transformation plasticity is sometimes important. The details of introducing the governing equations in the framework of thermo-mechanical behavior for describing temperature and stress=strain fields incorporating metallic structures in the heat treatment process are already reported elsewhere, and are applied in welding, casting and so on [24–31]. Based on the theory, a simulation program called ‘‘HEARTS’’ [32,33] for heat treatment process was developed to predict the temperature field, phase transformation, residual stress, and distortion during heat treatment process. The fundamental equations which can be applied in the metallurgical process are summarized in the following.

A.

Mixture Rule

When a material point undergoing a metallurgical treatment process is assumed to be composed of multi-phase structure, an assumption is made that a material parameter w is described by the mixture law [34] w¼

N X I¼1

wI xI

and

N P

xI ¼ 1

ð1Þ

I¼1

where xI denotes the volume fraction for the Ith phase. B.

Heat Conduction Equation

The local energy balance or the first law of thermodynamics is usually given in terms of the internal energy e ¼ gþTZ þ (sijeij)=r, as Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

re_ ¼ sij e_ ij 

@hi @xi

ð2Þ

with stress power sij e_ ij . Here, r and hi are the density and heat flux, respectively. Z is the entropy of the thermodynamic state. Introducing the expression for specific heat c ¼ T(Z=T), Eq. (2). is reduced to a heat conduction equation   X @ @T _ k rI lI x_ I ¼ 0 ð3Þ  sij e_ pij þ rcT  @xi @xi where k and lI denote the coefficients of heat conduction and the latent heat produced by the progressive Ith constituent. The boundary conditions of heat transfer on the inner surface are assumed to be k

@T ni ¼ hðTÞðT  Tw Þ @xi

ð4Þ

where h(T) is a function dependent on temperature. Tw denotes the heat transfer coefficient and the temperature of coolant on heat transfer boundary with unit normal ni, respectively. C.

Diffusion Equation of Carbon Content

Carbon content during carburizing process is arrived at by the diffusion equation   @ @C _ C¼ D ð5Þ @xi @xi where C is the content in the position xi-direction, D is the diffusion constant determined by the boundary condition being specified by the reaction across the surface layer D

@C ni ¼ hc ðC  Cs Þ @xi

ð6Þ

where hc and Cs are the surface reaction rate coefficient and the known content of the external environment, respectively. D.

Constitutive Equation

In order to simulate distortion due to variations of temperature, phase transformation, and inelastic deformation in metallurgical processes, many

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constitutive models which are capable of representing the relation of stress and strain including macro- and microstructures are proposed and established [35–52]. A few constitutive equations to be used in simulation of metallurgical processes are introduced in this chapter. Total strain rate e_ ij is assumed to be divided into elastic, plastic, thermal strain rates and those by structural dilatation due to phase transformation and creep such that tp c e_ ij ¼ e_ eij þ e_ pij þ e_ Tij þ e_ m ij þ e_ ij þ e_ ij

ð7Þ

Elastic and thermal strains are normally expressed as eeij ¼

1þn n sij  skk dij E E

ð8Þ

and eTij ¼ aðT  T0 Þdij

ð9Þ

with Young’s modulus E, Poisson’s ratio n and thermal expansion coefficient a, respectively. T0 is the initial temperature of material. Strain rates due to structural dilatation and transformation plasticity depending on the Ith constituent are em ij ¼

N X

ð10Þ

bI xI dij

I¼1

and e_ tp ij ¼

N X

BI hðxI Þx_ I sij

and

hðxI Þ ¼ 2ð1  xI Þ

ð11Þ

I¼1

where bI stands for the dilatation due to structural change. BI denotes the material parameters depending on phase transformation. The plastic strain rate is reduced to the form when employing temperature-dependent material parameters ! N @F @F @F _ X @F _ @F p ^ Tþ xI s_ kl þ ¼G ð12Þ e_ ij ¼ l @sij @skl @T @xI @sij I¼1 with a temperature dependent yield function F ¼ Fðsij ; ep ; k; T; xI Þ

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ð13Þ

with hardening parameter k, where   1 @F @F @F smn ¼ p þ ^ @k @s @e G mn mn

ð14Þ

The creep strain rate is assumed to follow a simple Norton creep law as 3
ðnmÞ=m
ecðm1Þ=m sij s e_ cij ¼ A1=m 2 c

ð15Þ


and
e are deviatoric stress, equivalent stress, and equivalent Here, sij, s strain, respectively. Either isotropic or kinetic hardening type of yield function F is available to be used in the section. Symbol Ac, n and m are material parameters based on Norton creep law. Either isotropic or kinetic hardening type of yield function F is to be used in this section. In casting and welding processes, we often should consider inelastic deformation incorporating the solidifying behavior of materials. In the solidifying process of materials, there has been substantial development of the so-called ‘‘unified’’ theories of viscoplasticity in which all aspects of inelastic behavior are intended to be represented by the same variable. Consider three types of constitutive models which have some possibility of describing the experimental behavior obtained above. Generally, the total strain rate e_ ij is expressed as the sum of the elastic strain e_ eij and the viscovp e plastic strain rate e_ vp ij , i.e., e_ ij ¼ e_ ij þ e_ ij , where the elastic strain rate is given by Hook’s law. The models for inelastic model are Perzyna’s model [42,43] based on the excess stress theory, and superposition model [52] which are introduced below. The excess stress theory assumes that the viscoplastic strain is produced as e_ vp ij ¼ LhcðFÞi

@F @sij

ð16Þ

where L denotes a viscosity constant of the material. c(F) is a function of the static yield function ðsij ; evp ij ; k; TÞ in stress space involving inelastic , the hardening parameter k and temperature T. As a proposal strain evp ij of the viscoplastic constitutive theory, Perzyna has proposed four kinds of functions for c(F), among which a special case c(F) ¼ F is employed to give the viscoplastic strain rate e_ vp ij , * +n 1=2 3l 3 sij vp e_ ij ¼ skl skl sy ð17Þ 2 2 ð3J2 Þ1=2 where J2 is the second invariant of deviatoric stress sij, and sy denotes the yield stress. In order to deal with a viscous fluid, we choose l ¼ 1=3 m with

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

the viscosity m. When the value of parameter n tends to unity, eqn. (17) which is reduced to a uniaxial stress–strain relation under monotonic tension is then expressed as s ð18Þ e_ vp ¼ lðjsj  sy Þ jsj from which the parameter l, or viscosity m is to be identified. n When the Norton’s equation (e_ c¼As
) is adopted to describe creep behavior of material associated eqn. (18), the model can deal with more complex inelastic behavior as a superposition model. E.

Kinetics of Phase Transformation in Metallurgical Processes

In the case of quenching, two kinds of phase transformation are anticipated: one is governed by the diffusionless or martensite mechanism. From a thermodynamic consideration, the formula for this type of reaction from austenite is assumed to be governed by modified Magee’s [53] rule as xM ¼ 1  exp½fðT  TM Þ  jðsij Þ

ð19Þ

with 1=2

jðsij Þ ¼ A1 sm þ A2 J2

ð20Þ

where TM is the martensite-start temperature under vanishing stress. The parameters A1 and A2 can be identified if we have the data of the martensitic transformation depending on the applied stress. The other type of phase transformation is controlled by diffusion mechanism, and the volume fraction of developing phase such as pearlite may be expressed by modifying the Johnson–Mehl [54] relation as xp ¼ 1  expðVe Þ

ð21Þ

where Ve is defined by Zt Ve ¼

f
ðT; sij Þðt  tÞ3 dt

ð22Þ

0

Here, we separate the function f
(T, sij) into two independent functions of temperature and stress as f
ðT; sij Þ ¼ f1 ðTÞf2 ðsij Þ

ð23Þ

Since the time–temperature-transformation TTT diagram under the applied mean stress sm in logarithmic scale deviates from the one without

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

stress which is represented by the function f(T), the kinetic equation of diffusion type is often applied to the variations of pearlite or ferrite structure in quenching processes. An identification of the function f(T) can be made possible by the use of some experimental data of the structure change. In casting processes, it is important to determine a moving boundary due to solidification as two phase structure with solidus and fluidus [55,56]. However, when we consider solidification of an alloy material, the mixture law also can be applied in the casting simulation. To identify the volume fraction of the solid phase xs during solidification appearing in eqns. (1) and (3), a phase diagram for the alloying system is employed, and the volume fraction of the solid phase xs is assumed to be determined by the Scheil equation [57,58]: xs ¼ 1  f1ð1k0 Þ

and

xl ¼ 1  xs

ð24Þ

Here, f is a dimensional function that is dependent on temperature: f¼

ðT  Ts Þ=ms ðT  Ts Þ=ms þ ðTl  TÞ=ml

ð25Þ

where ml and ms are the gradients of the liquidus and solidus temperature Tl and Ts with respect to the alloying element on the phase diagram, respectively, and k0 is a distribution coefficient representing the segregation effect. When the segregation effect is assumed to be neglected as in the present article, Eq. (24) can be reduced to the simple form xs ¼

ðTl  TÞ=ml ðT  Ts Þ=ms þ ðTl  TÞ=ml

ð26Þ

which is called the lever rule. F.

Algorithm of Finite Element Analysis

The formulated finite element equation system, considering the coupling between increment of nodal displacement fDug and temperature fTg as well as volume fraction of structure xI, can be expressed as ½PfT_ g þ ½HfTg ¼ fQðxI Þ; sij Þg

ð27Þ

½Kðui ÞfDui g ¼ fDFðT; xI Þg

ð28Þ

and

Here, [P], [H] and [K] represent the matrices of heat capacity, heat conduction and stiffness, respectively, and the vectors fQg and fDFg are heat flux, and increments of thermal load. These equations are strongly

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

nonlinear, which are derived by the use of the expression of stress increment vector as ! N N X X 3d
ec
fdsg ¼ ½D fdeg  fsg  f
ag dT  fbg dxI 2
s I¼1 I¼1 ! N X 1 @
s2 @
s2 dT þ dx fsg þ ð29Þ @xI I S0 @T I¼1 where
ec and fsg are equivalent creep strain and deviatoric stress vector. S0 and [D] denote a parameter depending on material hardening and an elastic–plastic matrix based on Mises’ type yield function. Here, the func
can be tions depending on temperature and phase transformation a
and b written as  N  X @ @aI xI dTf1g ð½De 1 fsgÞ þ f
ag ¼ aI xI f1g þ ð30Þ @T @T I¼1 and
g ¼ fb

@ ð½De 1 fsgÞ þ bI f1g @xI

ð31Þ

aI and bI are the coefficients of thermal expansion and dilatation due to Ith phase transformation, respectively. [De] denotes the elastic matrix of materials. In order to treat the unsteady coupled nonlinear FEM equations dependent on time, a time integration scheme ‘‘step-by-step time integration’’ method and a modified ‘‘Newton–Raphson’’ method are introduced in numerical calculation, while an incremental method is used for deformation and stress analysis. Because the heat transfer coefficient is dependent on the variation of the temperature on the boundary of heat transfer, we also use the time step of non-uniform cooling to calculate temperature, phases transformation and deformation fields [59,60]. In the simulation of continuous casting, the finite element method based on the Eulerian coordinate is applied to analyze the velocity, displacement rate vector fug, strain rate e_ ij and stress rate s_ ij . Displacement ui, strain eij, and stress sij can be evaluated by integrating the rate e_ ij , and s_ ij along flow line of velocities which have been verified by Ju and Inoue [61]. A finite volume computational method [62] to simulate large deformation in forging is adopted in this chapter. The advantages of the finite element and the finite volume approaches are combined: it employs a fixed finite volume mesh for tracking material deformation and an automatically refined facet surface (material surface) to accurately trace the free surface of

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

the deforming material. This is particularly suitable for large three-dimensional material deformation such as forging since remeshing techniques are not required. By means of this finite volume method, an approach based on the metallo-thermo-mechanics to simulate metallic structure, temperature, and stress=strain in the forging process associated with strain-induced phase transformation has been developed. The material is considered as elastic–plastic and takes into account the phase transformation effects on the yield stress. The temperature increase due to plastic deformation, heat conduction in the workpiece and dies, heat transfer between workpiece= die and ambient and thermal stress has been analyzed simultaneously. Strain-induced phase transformation, latent heat, transformation stress, and strain are included. This approach has been implemented in the commercial computer program MSC.SuperForge [62].

III.

DESIGNING OF HEAT TREATMENT PROCESS FOR CONTROL OF RESIDUAL STRESS AND DISTORTION

The purpose of the quenching and carburizing quenching of steel parts is to get desired metallurgical structures, hardness and strength of steel parts increased by the cooling rate and change of microstructure in quenching. So, we need to decide the compromise between maximizing hardness and minimizing distortion must often be made. Therefore, it is very important and difficult to select the proper quenching conditions, i.e., quenchant temperature, the method, and the cooling intensity of the quenchant media, etc. [63–65]. In order to obtain the optimum conditions, computer simulation is very useful for the determination of the quenching conditions, because the application of metallo-thermo-mechanical theory is capable of describing the interaction among temperature field, stress=deformation field and microstructure changes in quenching so that the simulation technology has been developed. By using this method based on continuum thermodynamics and the finite element method, several numerical simulations were carried out for the quenching process [66–71]. The simulation of quenching process is so complicated that most of the current research focused on the validity of numerical modeling for designing steel parts. On the other hand, to obtain improved mechanical properties and fatigue strength of machine components, such as gears, and shafts carburizing–quenching process is often used for surface hardening in the industry. However, carburizing– quenching is usually accompanied with carbon concentration and leads to distortion, residual stresses, and hardness of steel parts. The effect of carbon content is especially significant on phase transformation behavior.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

The first part of this section will present the thermal–mechanical behavior included with residual stress and distortion by using quenching conditions on cylinder and ring of carbon steel, which are simulated by an FEM program ‘‘HEARTS’’ [32]. Several techniques [63,71] were used for the estimation of the heat transfer coefficients during water and polymer quenching. The effect of the heat transfer coefficients on the quench distortion and the simulation result will be discussed. The simulation accuracy will be verified by the comparison between the experimental data of the distortion and the simulated results. The second part of the section will present the simulation results of the residual stress field and distortion of a steel gear in carburizing–quenching process. The computer simulation is based on the metallo-thermomechanical theory involving the diffusion of carbon and considering the effects of coupled temperature, phase transformation, and stress=strain fields on carbon component. In carburizing–quenching process, the fields of metallic structures and stress=strain as well as temperature affect each other [67]. A series of theoretical models, taking into consideration the effect of carbon diffusion and distribution is introduced. The accuracy of simulation is also verified by comparison with the experimental data. The predicted results validated the improvement of the hardness and strength of the gear component in carburizing–quenching process. Based on the comparison with the deformation experimental work, the thermomechanical predictions are found to be in good agreement with the experimental results. A.

Residual and Distortion in Carburizing Quenching of Cylinder

1. Experimental Method of Carburizing Quenching Cylinder specimens of 20 mm diameter and 60 mm length were used to verify the accuracy of the simulation. The carbon content of carburizing environment was 1.3% and the aim is to reach 0.9% carbon content on the surface of the cylinder. To detect the temperature changes during the process, four metallic sheathed thermocouples were attached to the specimen which are shown in Fig. 2. The entire carburizing–quenching process is shown in Fig. 3. The specimen was heated and carburized in a furnace and then quenched in the oil bath. In the cooling process, the temperatures were detected by the thermocouples attached to the specimen, and the signals were fed into a personal computer after the A=D conversion. The changes of the thickness and the diameter of the specimen were measured with a digital indicator and a micrometer caliper.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 2 The specimen and sheathed thermocouples.

Figure 3 The carburizing–quenching process.

2. Models of Simulation The simulation model in carburizing–quenching process is a steel cylinder of 20 mm diameter, 60 mm length and 0.45% carbon. It is assumed that the model is located in a uniform coolant. Then, the finite element model belongs to an axisymmetrical problem. As an initial step of the heat treatment process, calculations for the heating and carburizing process were used to simulate thermal stress field, thermal distortion as well as carbon distribution in the model. The quenching process of the model was started from the initial temperature 8508C and the model was cooled to 308C with oil. The heat transfer coefficients h during quenching were calculated as a function of temperature by the methods mentioned below and were used for the simulation as the surface boundary condition in Eq. (4). 3. Identification of Heat Transfer Coefficients It is important to determine the heat transfer coefficients in the quenching process of metal parts for numerical simulation. However, it is rather difficult to evaluate heat transfer coefficients during quenching of actual steel

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

parts which depend on not only the quenchant but also the shape, size, surface condition, and thermal properties of parts, etc. It is, therefore, very difficult to evaluate the heat transfer coefficients in quenching of steel parts. Some approximate methods are estimating the coefficients from the cooling curve data of standard probes which are used for evaluation of the cooling power of liquid quenchants. We have already reported the availability of the lumped-heat-capacity method for the estimation of the heat transfer coefficient from the cooling curve data of the JIS silver probe (pure silver solid cylinder of 10 mm diameter by 30 mm length, Japanese industrial standard K 2242 [72]) which has a high thermal conductivity. A computer program ‘‘LUMPPROB’’ based on the lumped-heat-capacity method was developed [73]. On the other hand, it was confirmed that the inverse method is more suitable for estimating the heat transfer coefficients from quenching data of the ISO probe (Inconel 600 alloy solid cylinder of 12.5 mm diameter by 60 mm length, International standard ISO 9950 [74]), because of its low thermal conductivity. We developed a computer program ‘‘InvProbe-2D’’ [75], which uses both a lumped-heat-capacity method and a two-dimensional inverse method with the least residual method. In this section, we used these programs for the estimation of the heat transfer coefficients during quenching. Furthermore, more precise heat transfer coefficients were estimated by a trial-and-error method, in which the calculation of cooling curves and modification of the surface boundary condition were repeated until the simulated cooling curves gave good agreement with the measured cooling curves of the steel specimen. 4. Carbon Diffusion and Distribution Fig. 4(a) shows the changes of carbon content with time in different positions during the carburizing process. The carbon content in the surface of the steel cylinder increases from 0.45% to 0.9% in 250 min. Fig. 4(b) and (c) compares the difference of carbon content of the steel cylinder before and after the carburizing process. The carbon content decreases being reserved in heating furnace for 35 min after carburization, which shows the effect of diffusion on carbon content distribution. 5. Heat Transfer Coefficients and Cooling Curves The heat transfer coefficients used for the simulation are shown in Fig. 5. The coefficients are estimated by using the inverse method program ‘‘InvProbe-2D’’ and the cooling curve data of the ISO Inconel 600 alloy probe. The cooling curves that were calculated with these heat transfer

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 4 The distribution of carbon after carburizing and heating.

Figure 5 The heat transfer coefficient.

coefficients are shown in Fig. 6. The cooling curve of the center point in the steel specimen shows good agreement with the measured one. And those near the boundary have a little difference from the measured one due to the influence of boundary of the steel cylinder specimen. 6. Prediction of Martensite Distribution and Equivalent Stress Martensite distribution and equivalent stress with consideration of the transformation plasticity in carburizing–quenching process are predicted and shown in Fig. 7. From the result of Fig. 7(a), we know that martensite is mainly generated near the surface due to the increase of the carbon

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 6 Calculated and measured cooling curves in different position.

Figure 7 Distribution of (a) martensite and (b) equivalent stress.

content. On the other hand, depending on measured hardness as shown in Fig. 8, distribution of martensite after carburized–quenching also is verified. 7. Distortion During Quenching Figure 9 shows the distortion of the calculated and the measured diameter of the steel cylinder after carburizing–quenching. Except for the influence of surface boundary condition, the calculated distortion of the center part of the cylinder is in good agreement with the measured value as shown in Fig. 9. However, because identification of the heat transfer coefficient on the corner of the cylinder is difficult, prediction of the distortion on the corner remains to be solved.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 8 Measured hardness in center and surface.

Figure 9 The calculated and experimental diameter expansion of the steel cylinder after carburizing–quenching.

8. Effect of Transformation Plasticity To check the effect of the transformation plasticity on the simulation result, simulation results which were calculated with or without the transformation plasticity are compared. In Figs. 10 and 11, it is shown that residual stresses considering the transformation plasticity are much less than without considering transformation plasticity. These results show the importance of considering transformation plasticity in the simulation of carburizing– quenching process.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 10

The calculated and experimental residual stress on the surface.

Figure 11 Comparison of residual stresses with and without consideration of transformation plasticity.

B.

Residual Stress and Distortion in Carburizing–Quenching of Gear

Based on the series of governing equations above, a finite element program called ‘‘HEARTS’’ was developed to predict the temperature field, carbon diffusion, phase transformation and distortion during carburizing–quenching process. The simulation model in carburizing–quenching process is a JISSCM420 steel gear with edge circle diameter of 36 mm, teeth number 16 and module 2 mm as seen in Fig. 12. Figure 13 shows the variation of TTT-curves of the material when carbon content is changed to 0.8% by using

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 12

Illustration of gear specimen.

Figure 13

Variation of TTT curves depending on carbon component.

the carburizing–quenching process as seen in Fig. 14. The identified heat transfer coefficient is shown in Fig. 15. It is assumed that the model (Fig. 16) is located in a uniform heating and cooling process as in Fig. 14. Then, the finite element model belongs to an axisymmetrical problem. As an initial step of the heat treatment process, calculations for the heating and carburizing process were used to simulate thermal stress field, thermal distortion as well as carbon distribution in the model. The quenching process of the model was started from the initial temperature 8508C and the model

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 14

Process conditions of carburizing–quenching.

Figure 15

Heat transfer coefficient depending on temperature.

was cooled to 308C with oil. The heat transfer coefficients h during the quenching were calculated as a function of temperature by the methods mentioned below and used for simulation as the surface boundary condition. 1. Carbon Diffusion and Distribution Fig. 17 shows the changes of carbon content in different positions with time during the carburizing process. The carbon content in the surface of the steel gear increases from 0.45% to 0.9% in 250 min. Fig. 18(a) and (b)

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 16

FEM model of gear.

Figure 17

Variation of carbon contents in different position.

compares the difference of carbon content of the steel gear before and after the carburizing process. The carbon content decreases since it is present in heating furnace for 35 min after carburization, which shows the effect of diffusion on the carbon content distribution. 2. Prediction of Martensite Distribution Martensite distributions in carburizing–quenching process are predicted and are shown in Fig. 19. Fig. 19 shows that martensite is mainly

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 18

Distribution of carbon after carburizing and heating.

Figure 19

Simulation of martensite distribution.

formed near content.

the

surface

due

to

an

increase

of

the

carbon

3. Residual Stresses In order to predict the residual stresses in the gear, the simulation results were calculated taking into account the strain due to phase transformation. From Fig. 20, we come to know that the residual stresses are greater at the corner of the gear surface. These results show the importance of considering the transformation plasticity in the simulation of carburizing–quenching process.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 20

Equivalent stress.

4. Distortion after Quenching Figure 21 shows the distortion of the calculated and the measured diameter of the steel gear after carburizing–quenching. And except for the influence of surface boundary condition, the calculated distortions of the center part of the gear are in good agreement with the measured value as shown in Fig. 22. However, because identification of the heat transfer coefficient on the corner of the gear is difficult, prediction of the distortion on the corner remains to be solved.

Figure 21

Deformation of gear.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 22

IV.

Deformation on surface of gear teeth.

DESIGNING OF CONTINUOUS CASTING FOR CONTROL OF RESIDUAL STRESS AND DISTORTION

In order to optimize material metallurgical processes, such as casting and welding during solidification, it is important to determine the residual stress after these processes for evaluating the mechanical properties and strength of materials and to optimize the operative conditions in manufacturing [76–78]. However, the formation of residual stress not only depends on the thermo-mechanical behavior and processing effect due to the variation of in micro-=macro-structures during the manufacturing of materials, but also on the interaction of heat conduction, change of phase transformation due to solidification, and stress=strain. Especially, to describe variation of the mechanical behavior from viscous fluid to solid and the mixture domain in the material due to solidification, a unified constitutive model incorporating with solids and fluids of metal is proposed [64]. In order to solve the above-mentioned problems, this chapter presents some developments of the thermo-mechanical theory and numerical analysis method incorporating solidification of material to simulate the residual

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

stress formation during casting. A unified inelastic constitutive relationship capable of describing both elastic–viscoplastic solids and viscous fluids to apply simulation of the casting process was proposed and verified by experimental and numerical results. On the other hand, a proposal based on the finite element method to couple temperature, stress fields as well as deformation during solidification was presented. Depending on the simulations of the continuous or semi-continuous casting, the mechanism of the residual stress formation during these casting processes can be represented. The thermo-mechanical modeling was also verified by a comparison with the experimental data, such as the measured residual stress and variation temperature in casting. Vertical semi-continuous direct chill casting process is one of most efficient methods to produce ingots of aluminum alloys and other metals. It is beneficial for optimizing the operating conditions to simulate thermo-mechanical field in the solidifying ingot. So many reports have been published concerning such analyses of the temperature distribution incorporating solidification by finite element method, but a few papers treat the induced stress=strain field. Simulations of thermal stress in continuous casting slab were made by using elastic–plastic constitutive models [79,80], and viscoplastic stresses [81–84] were simulated based on the solidification analysis by Williams et al. [31]. However, in their studies, the influence of casting speed was neglected, so that the numerical simulation along with the variation of casting conditions could not be realized. In order to solve this problem, Ju and Inoue [62] proposed a numerical simulation method by the Eulerian coordinate, and application to the continuous casting process of steel slab was performed. A.

Residual Stress Formation During Semi-continuous Casting

The aim of this section is to apply the coupled method of temperature and stresses incorporating solidification developed for semi-continuous direct chill casting of aluminum alloys. When the bottom block plate is located at the upper position and the length of the growing ingot is small, the temperature, liquid–solid interface, and stresses in the ingot vary with time, both in the sense of space and of material. However, when the ingot becomes long enough, the physical field in the upper part is regarded to be time-independent or steady in the spatial coordinate fixed to the system. In the first part of this section, a steady heat conduction equation with heat generation due to solidification is formulated in a spatial coordinate system when considering the material flow. A numerical calculation for the temperature in the solidifying ingot as well as the simulation of the location of liquid–solid interface is carried out by a finite element technique.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Most metallic materials at low temperature may be treated as an elastic–plastic solid. However, if they are heated beyond the melting point, the materials can be regarded as a viscous fluid, and they behave in a timedependent inelastic manner at high temperature close to the melting temperature. Therefore, a unified constitutive model needs to be established to describe the elasto-plastic and viscoplastic behavior of the solidified part of the ingot as well as the viscous property of the liquid state. Taking into account the effects of such phenomena, a modification of Perzyna’s constitutive model similar to the one in other sections is presented in the second part of this section, and some experimental results of the viscosity appearing in the model are presented for a Al–Zn type alloy. By using the model, elastic–viscoplastic stresses are calculated for the ingot to establish the residual stress distribution, and are verified by the measured data from a hole-drilling strain-gauge technique. Finally, results of a numerical simulation are presented on the influence of operating conditions on temperature and stresses, such as ingot size, casting speed, and initial temperature, to provide fundamental data for optimizing the operating condition.

1. Finite Element Model and Casting Conditions The theory and the procedure developed above are now applied to the simulation of the vertical semi-continuous direct chill casting process shown schematically in Fig. 23. The material treated is a Al–Zn type alloy with 5.6% zinc and 2.5% magnesium. A quadrilateral finite element mesh pattern of 600 elements with 1941 nodes illustrated in Fig. 24 is employed for both analyses of temperature and stress fields. The boundary condition for heat conduction is assumed in such a way that the temperature of the meniscus of molten metal is prescribed to be w0, and that heat is insulated along the central line and the bottom of ingot as well as the surface contacted with the refractory. The cylinder facing the mold is regarded as the boundary Sq on which heat flux is given. The other part of the surface Sh is given by a heat transfer boundary due to the cooling of water. Figure 25 depicts the measured heat flux q absorbed by the mold, and heat transfer coefficient h depending on flow rate of water TW is shown in Fig. 26. Other data used for temperature calculation incorporated with solidification are shown in Table 1. Characteristic results of calculated temperature and residual stresses for an ingot of 1 m in length with the diameter of 240 mm are compared with experimental data to verify the method. Simulations in other cases of different operating conditions such as casting velocity, size of the ingot, and cooling rate are also made.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 23

Schematic view of semi-continuous vertical direct chill casting.

2. Results of Simulation Figure 27 indicates a bird’s eye view of the isothermal representation of calculated temperature distribution. The lines denoted by Tl and Ts in the figures are the liquidus and solidus temperature, respectively. The data is replotted to give the temperature change at the center and on the surface, while the circles represent the measured temperature by thermocouples as shown in Fig. 28. The fact that the calculated temperature on the surface coincides well with the measured data may indicate the validity of the simulation method. Change of the volume fraction xs at four characteristic points along the distance from meniscus is represented in Fig. 29, which may give information on the progressing mode of solidification and thickness of solidified shell.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 24 Finite element mesh pattern for semi-continuous vertical direct chill casting.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 25

Variation of heat flux from top of the mold.

Simulated results of temperature and mode of solidification presented in the above analysis of stresses was carried out on the entire area of the ingot, including the molten metal, using the finite element method based on the elastic–viscoplastic constitution model. The displacement mode is depicted in Fig. 30. The contour of radial, tangential, and axial stress distributions sr, sy, sz are shown in Fig. 31. The contours of stress distribution are represented in Fig. 32. Examples of the calculated stress distribution by elastic–viscoplastic constitutive model and the one by time-independent elastic–plastic model are shown in Fig. 33. When compared with each other, the viscoplastic stress analysis gives smaller results, at least on the surface, than elastic–plastic stresses. The stresses are found to be generated at the location where the solidification starts (see Figs. 31 and 32), and the radial distribution becomes steady owing to the flatter temperature distribution. In order to examine this effect, the stress distributions at several locations are given in Fig. 34, in which the distribution at the end of the ingot (Fig. 34d) can be regarded as residual stresses. Circles in the figure indicate the experimentally measured residual stresses by a hole-drilling strain gauge method [7]. The fact that the experimental data coincide well with the analytical results suggests the validity of the simulation procedure based on viscoplasticity. As is seen in Fig. 34 where the shear stresses are insignificant in the area

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 26 Distribution of heat transfer coefficient depending on discharge of cooling water.

of molten state, the normal stresses sr, and sz (see Figs. 31 and 33) are regarded to be hydrostatic stresses, which means that the constitutive equation employed here reveals to modify deformation of the liquid. 3. Simulations for Other Operating Conditions This procedure may be applied to other cases of operating conditions. Hereafter, the focus is on the effect of temperature and stress distributions on the casting speed, size of the ingot and cooling condition. Figures 35 and 36 represent examples of temperature profiles for different casting speed and the radius of ingots with various cooling rate. The effect of discharge of cooling water on thickness of the mussy zone at the center is summarized in Fig. 37. The effects of casting speed and ingot radius on the stress s are represented in Figs. 38 and 39, respectively, and the relation between stresses and casting speed is summarized in Fig. 40. The simulated stresses varying with the discharge of cooling water are also plotted in Fig. 41. The calculated results shown above seem to simulate the characteristic of temperature and stresses depending on the operating

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Table 1

Material Properties of Aluminum 7075

Physical item Heat conductivity Density Specific heat Latent heat due to solidification Casting speed Liquidus temperature Solidus temperature Gradient of liquidus line Gradient of solidus line Young’s modulus Poisson’s ratio Viscosity

Initial yield stress Hardening coefficient Thermal expansion coefficient Dilatation due to solidification

Value kl ¼ 0.0425; ks ¼ 0.0125 þ 0.0583T r ¼ 2.8  103 cl ¼ 0.015 þ 0.00173T cs ¼ 0.083 þ 0.00267T ls ¼ 93.16

Unit cal=(mm 8C) g=mm3 cal=(g 8C)

V ¼ 80.0 Tl ¼ 638

cal=g mm=min 8C

Ts ¼ 600 ml ¼ 3.69

8C 8C=%

ms ¼ 9.09

8C=%

El ¼ 500.0 Es ¼ 75000.0 – 76.3T n ¼ 0.33 m ¼ 3700.0
0.007178T2 21.1698T 10160.7 sv0 ¼ 2.0 v0 150.0 – 0.428T H0 ¼ 350.0 – 0.13333T H0 ¼ 330.0–1.683(T150) H0 ¼ 0.132 al ¼ 33  106 as ¼ 21.8  106 þ 0.2T  107 b ¼ 7.5%

Temperature range

MPa MPa s

T  3468C T 3468C

MPa

T  3468C T < 3468C T 1508C 150 < Ta < 3468C T  3468C

MPa

l=8C

conditions. If the data of such a simulation at different operating conditions are accumulated as shown in Figs. 35–39, the possibility of optimizing the design of the system would be realized.

B.

Residual Stress Formation During Strip Continuous Casting by Twin-Roll Method

The continuous strip casting technology by twin-roll method is a promising technology that not only saves energy, but also reduces production costs in the manufacturing of material. However, there are many difficulties in

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Figure 27

View of the calculated temperature profile.

Figure 28

Temperature variation at the center and surface of the ingot.

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Figure 29

Volume fraction of solid along the distance from meniscus.

controlling the quality of the strip because of the existence of the deformation of the strip itself, due to thermal expansion or thermal stress. There are two key points: firstly, if the solidification is completed before the liquid reaches the minimum clearance point between the rolls, then the strip will occur at a fixed gap. Hence, one of the key points is controlling of

Figure 30

View of deformation.

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Figure 31

View of stress distributions.

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Figure 32

Iso-stress contours.

solidification. Another key point is that the viscoplastic deformation incorporating material flow must be considered in this thermo-mechanical process. 1. Continuous Casting System by Twin-Roll Method The twin-roll continuous casting system is schematically illustrated in Fig. 42(a). In this process, molten metal is between the two rolls rotating in opposite directions with same angular velocity. The level of the molten metal is always kept constant by overflowing the excess molten metal from the nozzle. As soon as the molten material is poured into the rolls, solidification takes place on the roll surface, which is cooled by circulating water inside the roll. Therefore, the problem then is to find this steady solidification profile and the distribution of temperature and fluid velocities in both the liquid phase and the solid phase. On the other hand, due to the symmetry to the central line, half of the model shown in Fig. 42(b) is treated for the analysis.

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Figure 33 Calculated stresses by (a) elastic–viscoplastic model and (b) elastic– plastic models.

2. Analytical Models and Parameters The procedure developed above is now applied to the simulation of the thin slab casting process under the same operating conditions. The results are summarized as follows. Figure 43 represents the finite element descritization of the whole region of the strip and roll. The surface of roll as well as the contacted boundary with the roll and strip is assumed to belong to

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Figure 34

Stress distribution in several sections of ingot.

heat transfer boundary and the surface of the strip to the heat radiation boundary. In order to verify the numerical analysis method proposed in Section 2, continuous casting of SUS304 steel is taken into consideration in this section. In continuous casting process of SUS304, the thickness of the slab is 1 mm, and two casting speeds are used Vc ¼ 400 and 600 mm=sec. 3. Calculated Results Simulated results of steady temperature field both in the strip and roll is shown in Fig. 44(a) and (b) for the casting speeds of 400 and 600 mm=sec.

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Figure 35

Isothermal lines by effect of casting speed.

sec. Temperature distribution along the central line and surface is plotted in Fig. 45. The fraction of solid phase in solidified region of the strip with liquidus and solidus temperatures Tl ¼ 14608C and Ts ¼ 13998C is depicted in Fig. 46. In the early stage of rapid cooling by the roll, the solidified shell is seen to grow gradually toward the central part. The change in casting speed is known to affect the distribution of temperature and shell thickness. As the casting speed becomes faster, the position of the liquidus line moves downstream. Figs. 45 and 46, shows that the temperature distribution on surface of slab at the roll outlet presents violent fluctuation

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Figure 36

Isothermal lines by effect of ingot radius.

due to latent heat generation by the solidification domain. When the casting speed Vc is raised, the temperature fluctuation towards a small range is present on the surface of the slab. The distribution of the horizontal stress sx from meniscus to the kissing point at the roll outlet is represented in Fig. 47. From these figures, the high-level stress field by rapid cooling is found to occur near the surface of the shell, and the equivalent stress distribution on the surface of the slab at roll outlet also presents violent fluctuation due to the temperature fluctuation. The equivalent stress analysis results based on several constitutive equations are plotted in Fig. 48. In the section of slab at the roll outlet, calculated equivalent stress by three types of constitutive equations is

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Figure 37 Relation between thickness of mussy zone at the center depending on discharge of cooling water.

approximated to the same distribution. However, when the casting speed is lower, the effect of creep strain is evident. V.

DESIGNING OF THERMAL SPRAY COATING FOR CONTROL OF RESIDUAL STRESS

Plasma-spray coating is an important method by which a new functional layer can be produced on material surface [85,86]. In the spray coating process, solidified particles form thin lamellae whose microstructure depends on the particle cooling rate during solidification. Indeed, lamination of several kinds of materials deposited by laser thermal spray coating leads to an increase of the strength of the structural components, such as plates, and cylinders, especially in the case when exposed to severe temperature gradient. However, the residual stresses at an interface between multilayer after coating play an important role that affects thermo-mechanical properties of the materials, since it is associated with the growing liquid boundary due to the supply of the molten material and also from the moving liquid–solid interface depending on the progress of phase transformation. It is also necessary to consider the effect of coupling between

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Figure 38

Stress distribution by effect of casting speed.

temperature and stress=strain fields and the phase transformation, or solidification in this case. Some open problems still remain in the development of a mathematical model capable of treating the growing boundary of molten state and a moving interface between liquid and solid phases as well as the evaluation of residual stresses. There are three complicated aspects to be considered: The first one is that the spray coating process is nonlinear problem with respect to time and location, and the second is that the solidifying process plays an important role in the simulation of temperature and stresses field. The third is how to deal with the difference in mechanical and physical properties of the material in each layer. To solve the first and second problems, a scheme for numerical analysis by the finite element method has already been proposed by the authors [86]. This chapter shows the simulation of temperature and stress=strain induced in the layers of the plate with progressive domains based on the metallo-thermo-mechanics, and a finite element scheme is proposed to evaluate the change of both fields. Heat input due to successively pouring

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Figure 39 Stress distribution depending on effect of ingot diameter and cooling water rate.

molten metal is introduced in the heat conduction equation coupled with mechanical work and latent heat generation by solidification. Simultaneously, a method of stress analysis using an elastic–viscoplastic constitutive relationship capable of describing the mechanical behavior of both solid and liquid is proposed. Examples of the numerical calculation of temperature with solidification mode and residual stresses in the multi-layer plate in the course of the spray coating are presented, and the validity of the calculated results is discussed in comparison with the x-ray experimental results. A.

Experimental Procedure of Spray Coating

Generally, from the viewpoint of continuum thermomechanics, the spray coating process of material has many complicated phenomena which are affected by the interaction between temperature, stresses, and solidification incorporating rapid cooling, and the state in the solidifying material termed

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Figure 40

Relation between stresses at the center of ingot and casting speed.

as ‘‘mushy zone’’ which is a mixture of solid and liquid phases. Figure 49 illustrates an example of spray coating system, which is developed to fabricate laminated plate. In most cases of the spray coating process, after the spray layer is solidified completely, the other kind of material is poured onto the substrate so that the inner boundary grows and the interface of molten state moves toward spray layer direction. In the spray coating process, solidifying material in the growing domain undergoes a variation of mechanical quantities, such as mass, momentum and energy as well as the change of material properties due to phase transformation from liquid to solid state. The complicated interaction between temperature and inelastic deformation, in this case, is to be taken into consideration. B.

Numerical Model and X-Ray Residual Stress Measurement

1. Modeling of Numerical Simulation To verify the validity of the theory and the procedure stated above, simulation of the fields of temperature and solidification mode and the stress

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Figure 41 water.

Relation between stresses at the center of ingot and discharge of cooling

distribution are performed over the course of the spray coating process of laminated plates with two layers. The, assumption is made that both ends of laminated plate are constrained during the spray coating, and that the laminated plates are large enough, so that a growing model of two-dimensional finite element is proposed to interpret the experimental phenomena of the spray coating process and to compute the interfacial thermo-mechanical behavior between the impinging particles and the surface of substrate just beneath them. A model of the spraying layer represented by a flat disk of 30 mm diameter and 0.5 mm thickness of initially uniform temperature which is put into contact with the substrate elements is shown in Fig. 50. When the numerical analysis is started, the growth elements incorporated with the impinging particle from the spraying direction were put into the substrate. In this analysis, the growth model is represented by an axisymmetrical problem. Thermal flux entering and leaving each element as well as the latent heat liberated within the elements themselves during the solidification process is evaluated and the resulting element temperature is computed after each successive time increment. To simulate the spray coating processes, it is assumed that the different materials are successively poured into the substrate, i.e., the different

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Figure 42

View of (a) twin-roll casting system and the (b) model for simulation.

material is supplied after the material in the outer layer is completely solidified. On the outer surface along spraying direction, the heat radiation boundary conditions was set on the initial step of the coating, and heat transfer condition was set on the cylinder surface of the axisymmetrical model, respectively. C.

Properties and Coating Condition of Specimen Materials

Laminated layers are deposited on a stainless steel (SUS304) substrate of 5 mm thickness. In this work, the layer thickness and layer materials produced were: 0.50 mm for stainless steel. The specified thickness was obtained when spraying was performed 10 times. Table 2 shows the components of these wires which are generally used for carbon dioxide arc welding. Table 3 shows the conditions of the spray coating process. The thermo-physical properties of the wire materials used for temperature and stress calculation incorporated with solidification are shown in Tables 4 and 5, respectively [69]. It is assumed that the properties of substrate are the same as those of the wire. The heat transfer coefficient

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Figure 43

Mesh of finite element model.

for air on the surface of the model is chosen to be h ¼ 2.78  103 (cal=(mm2 sec deg)). The thermal radiation coefficient G ¼ 7.028  107 (cal=(mm sec K)) is used for the model. Depending on the inelastic constitutive model of Section 2, the inelastic strain rate can be given by a viscoplastic relationship. Here, the viscosity which described the viscoplastic model of wire material (stainless steel) is shown in Fig. 50.

D.

X-Ray Residual Stress Measurement

To measure the residual stresses in the spray coating process, CrKa characteristic x-rays were used. Diffraction planes and angles were (2 1 1) and 2y0 ¼ 1568 for the stainless steel SUS304. Surface roughness of the coated layer was about 6.5 mm. These values might be too large for x-ray stress measurement to provide reliable results. However, the parallel beam method could give stress values with sufficient accuracy on such rough surface. Before x-ray stress measurement, electropolishing conducted to remove an

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Figure 44

Distribution of temperature in strip and roll.

oxide-film from the layers. Stresses were measured parallel and perpendicular to the spray traveling direction. The x-ray measuring conditions are shown in Table 6. The full width at the half-maximum method was used to determine peak positions. We measured the residual stresses in a phase with 2 1 1 diffraction and g phase with 2 2 0 diffraction. The stresses were obtained by the sin2 c method. The c-diffractometer method was used for the measurement of residual stresses. E.

Verification and Discussion of Simulation Results

An example was used for simulating thermo-mechanical behavior and residual stress during the spray coating. Figure 51 shows the variation of temperature distribution on the central element of spraying surface and outside surface of the layer. From these results, we arrive at the temperature difference between the central element and the outside surface of the spraying layer due to the solidifying process. The distribution of temperature on the total domain dependent on time is shown in Figs. 52 and 53. The volume fraction of solid and the variation of the solidified thickness are depicted in Figs. 54 and 55, respectively. These figures, show that the temperature on the central element tends to decrease slowly by the latent heat generation

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Figure 45

Temperature variations at center and surface of strip.

Figure 46

Distribution of solid fraction.

due to solidification and also by the heat supply by the successively poured material followed by the rapid temperature decrease at the end of solidification. Difference in heat conductivity due to the temperature difference reveals the influence on the cooling rate and mode of solidification. As for the results of stress analysis, residual stress is represented in the following figures, in which increasing stress reduces fluctuation or jump depending on the solidification and growing domain. Distribution of residual stresses sr on the radial direction is shown by the lines in Fig. 56. Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 47

Distribution of stress sx.

The data are compared with the experimental results represented by the same condition of the process. Relatively reasonable agreement between both values is seen even in the region with fluctuation of stresses on the interface boundary between the spraying layer and the substrate. The distributions of residual stress sy and sz are shown in Figs. 57 and 58. Here, the jump behavior of stresses on the interface is presented by these simulated results. Thus, it is important to reveal the damage of the spraying layer based on the theory and numerical method.

VI.

DESIGNING OF FORGING PROCESS FOR CONTROL OF INTERFACIAL STRESS

A.

Basic Description

A typical industrial metal component may be manufactured by forging and heat treatment. In the design of forging processes, information such

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Figure 48

Dependence of constitutive relationship on stress distribution.

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Figure 49

Schematic diagram of laser spraying.

as material flow in the dies, level of die fill, defects, strain, stress, temperature distribution in the work pieces and the dies, and forging force is necessary. In the subsequent heat treatment operations, information on combination of microstructure, residual stresses, and dimensional accuracy in the final product is also very important. Such information may be obtained by numerical simulation [87–90]. The currently available commercial (and also academic) codes of forging are usually based on the finite element method. For a forging process where the metal is plastically deformed under high pressure into machine parts with high-strength performance, the finite element method sometimes exhibits weaknesses which must be carefully monitored


Finite element meshes usually become over-distorted; hence, autoremeshing is then necessary to complete the simulation. But the auto-remeshing technology for three-dimensional problems is not so robust and is also very time consuming.

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Figure 50




Coefficient of viscosity based on viscoplastic model.

Even for two-dimensional elastic–plastic problems, the remeshing may lead to an erroneous result [70]. Each remeshing step will involve a loss in volume, which is not acceptable for a considerable forging simulation.

On the other hand, the finite element method is suitable for stimulating the heat treatment process after forging because it is principally a relatively small deformation process [88]. But when deformation-induced phase transformation during the process of metal forming (large deformation like forging) is taken into account, the finite element method will meet the same problems of remeshing and loss of volume. In this section, a method to simulate forging and subsequent heat treatment processes is described. The advantages of the finite element and the finite volume approach are combined: it employs a finite volume mesh for tracking material deformation and an automatically refined facet surface (material surface) to accurately trace the free surface of the deforming material. This is particularly suitable for large three-dimensional material deformation such as forging because remeshing techniques are not required. Table 2

Substrate Wire

Composition of the Substrate and Wire (wt%) C

Mn

Si

Cr

Ni

0.06 0.04

– 1.90

– 0.46

18 20.1

0.4 8

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Table 3

Conditions of Spray Coating

Laser output Wire feed Inner gas pressure Out gas pressure Spraying distance Nozzle Assist gas

3.0 (KW) 5.1 (m=min) 5.0 (kg=cm2) 5.0 (kg=cm2) 100 (mm) Double nozzle Argon

This finite volume method provides an approach based on the ‘‘metallo-thermo-mechanics’’ to simulate metallic structure, temperature and stress=strain coupled in the heat treatment process such as quenching. The material surface utilizes the advantages of the finite element method to ensure a level of accuracy for the small deformation process in the frame of the finite volume method. This approach also provides latitude for implementation of phase transformation analysis coupled with forging process in the future.

B.

Validation Example

In this section, we consider forging followed by subsequent quenching of an S45C carbon steel rod used in a ship engine. The outline of the basic numerical model is shown in Fig. 59. In the first process, the rod is forged from a round billet of 524 mm diameter and 500 mm length between two closed dies. A threedimensional model is necessary for the simulation. In the second process, the rod goes through the water spray quenching. The problem is an axial symmetric model. (1) In the forging process, the initial temperature of the billet is 12508C. Since the material is above the recrystallization temperature, the influence of strain upon flow stress is insignificant, while the influence of

Table 4

Thermal Properties of Wire

Heat conductivity (cal=(mm s deg) Density (g=mm3) Specific heat (cal=(g deg)) Latent heat (cal=g) Temperature of solidus and liquidus (deg)

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k ¼ 1.83  103þ1.02  106T rs ¼ 7.9  106; rl ¼ 7.4  106 c ¼ 65.2 þ 9.77  106T ls ¼ 65.0 Ts ¼ 1432; Tl ¼ 1449

Table 5

Mechanical Properties of Wire

Young’s modulus (MPa) Yield stress (MPa)

Poisson’s ratio Thermal expansion (L=deg) Dilatation due to solidification (%)

Es ¼ 2.5  1051.73  102T; El ¼ 2264 sy0 ¼ 280.35–1.4T þ 5.75  103T21.1  103T3 þ 9.8  109T4  4.14  1012T5þ6.7  1016T6 n ¼ 0.35 as ¼ 12.76  108; al ¼ 12.67  1012 b ¼ 0.25

strain rate becomes increasingly important. The flow stress for hot forging is defined as follows: sy ¼ C_eM

ð32Þ

where C and M depend on equivalent strain and temperature listed in the literature [90]. The heat transfer between workpiece and die plays an important role in hot forging. Several methods are available to determine the contact heat transfer based on data reported in the literature. Here, 5000 W=(m2 K) are used according to Ref. [88]. Other material properties of heat conduction in the workpiece and dies and the heat transfer between the workpiece=die and ambient are also taken from Ref. [88] as seen in Table 7. The punch velocity is described in Fig. 60, shows the different stages of forging, holding, and ejecting. Figure 61(a) and (b) shows a quarter model of the rod in its initial position and at the end of the ejection stage.

Table 6 X-ray Measuring Conditions Characteristic X-ray Tube voltage Tube current X-ray optics Divergence slit Receiving slit Sampling time Filter Diffraction plane Irradiated area

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CrKa 35 kV 9 mA Parallel beam 0.58 0.58 Unifixed V a Phase 211  Phase 220 2  8 mm2

Figure 51

Temperature variation on center and surface of layer.

The simulated shape of the rod is in good agreement with the experiment. The burr can be clearly seen here and is also confirmed by experiment. The residual stress and equivalent plastic strain obtained at the end of the ejecting stage is depicted in Fig. 62. The sample points are taken from the center to the surface in the middle of the rod along the punch moving direction. The residual stress will be taken into account in the subsequent quenching process as the initial condition. The temperature in the center of the rod increases during forging because of heat generation due to mechanical work. But the temperature

Figure 52

Temperature distribution at t ¼ 0.35 sec.

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Figure 53

Temperature distribution at t ¼ 0.75 sec.

near the surface drops because of die chilling. In the ejection stage, the temperature near the surface may rise again slightly since the heat transfer from the workpiece to ambient is much lower than that of the die, so that the surface can be heated again at the center. Figure 63 represents the temperature variation with time at the center and the surface in the middle of the rod during forging. (2) In the quenching process, the temperature range is from 9008C to room temperature (208C). The material at 9008C is in the austenite phase.

Figure 54

Volume fraction of solid phase at t ¼ 0.35 sec.

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Figure 55

Volume fraction of solid phase at t ¼ 0.75 sec.

Fig. 64, shows that when the rod is continuously cooled down, the austenite can transform into pearlite through diffusion transformation. If the cooling rate is high enough, austenite may transform into martensite when the temperature is decreased below the martensite transformation temperature. The properties of the material are highly dependent upon the content of each phase and are found in Ref. [34]. To obtain a relatively effective cooling medium for water spray quenching, 9100 W=(m2 K) are used as the heat transfer coefficient, and

Figure 56

Distribution of residual stress sr.

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Figure 57

Distribution of residual stress sy.

208C is used as a constant ambient temperature. Figures 63 and 64 show the temperature changes with respect to time taken at the center to the surface in the middle of the rod during quenching and illustrates the volume fraction of metallic structures formed. Figures 65 and 66

Figure 58

Distribution of residual stress sz.

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Figure 59

Table 7

Outline of basic numerical model.

Material Properties of Heat Conduction Heat conductivity

Die Workpiece

31 W=m K 35 W=m K

Specific heat 470 J=kg K 800 J=kg K

Heat transfer with ambient 20 W=m2 K 52 W=m2 K

illustrate equivalent stress and axial residual stress distribution from the center to the surface in the middle of the rod after quenching, respectively. In Fig. 66, two cases are shown for axial residual stress distribution: one case is quenching without taking into account the residual stress from the forging process; and the other is quenching, taking account of

Figure 60

Punch velocity of forging, holding, and ejecting.

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Figure 61

The rod in its (a) initial position and at the (b) end of ejecting stage.

Figure 62

Residual stress at the end of ejecting stage.

Figure 63

Cooling curve.

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Figure 64

Volume fraction of metallic structures.

the residual stress from the forging process. Experimental results measured by the Sacks method are also shown in the same figure. In both cases, the axial residual stress is in tension near the center and in compression near the surface. Because of the effect of the residual stress from the forging process, less tension near the center and more compression near the surface were obtained as seen in Fig. 66. The axial residual stress coming from the forging process does not make a big difference with the quenching calculation,

Figure 65

Effective stress after quenching.

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Figure 66

Axial residual stress after quenching.

because the axial residual stress coming from the forging process is quite small.

VII.

CONCLUSION AND REMARKS

A general discussion on the framework of thermo-mechanical theory incorporating phase transformation and solidification is described in this chapter, and a series of practical analytical schemes based on the finite element method are introduced. These schemes are applied to simulate thermomechanical fields in quenching, casting, coating, and forging processes. A coupling method to simulate phase transformation and solidification and temperature as well as stress distribution in these metallurgical processes is formulated, and the implementation by finite element calculation is presented in this chapter as examples of the application of the theory and procedure developed in the handbook. Elastic–plastic constitutive equations and a modification of Perzyna’s viscoplastic constitutive relationship were used, which reflects actual phase transformation and solidification during these metallurgical processes. The results of simulation are also verified by comparison with experimental data, and application of the technique to other operating conditions to obtain the fundamental data of optimum design of the system. From the discussion of theory, simulation method and results presented in the chapter, the conclusions and remarks obtained are summarized as follows:

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1. Comparing with calculated results and experimental data for temperature, distortion and residual stress in these metallurgical processes, the metallo-thermo-mechanical theory and simulation method proposed in Section II are verified. 2. Effects of cooling curves, distortion, and residual stresses on the occurrence of the phase transformation in quenching are proved by simulations of quenching and carburizing-quenching processes. 3. It is important to identify the heat transfer coefficients of quenchants with respect to the quenching process are obtained from simulation results. 4. The unified inelastic constitutive equation may describe the stress and deformation in the whole region of the solidifying process including liquid and solid state. 5. In the simulation of continuous casting, the development of stresses from solidifying domain is presented. On the other hand, the effects of distortion on solidification also shown to be an important factor. 6. In the simulation of coating process, the jump behavior of stresses on the interface between substrate and spraying layer is shown. Thus, it is important to reveal the damage of the spraying layer based on the metallo-thermo-mechanical theory and numerical method. 7. The advantage of finite volume technique over the finite element method in the simulation of forging was shown. From this point of view, the finite volume method is expected to be a powerful tool in the simulation of metal forming processes. REFERENCES 1. Noyan, C.; Cohen, J.B. Residual Stress, Springer Verlag, New York, 1987. 2. Mura, T. Residual stresses due to thermal treatments. Res. Rep. Faculty Eng. Meiji Univ. 1995, 10, 14–27. 3. Hanabusa, T. Japanese Standard for X-ray Stress Measurement, Proceedings of 6th International Conference on Residual Stresses, Oxford, England, July 10–12, 2000; 181–188. 4. Standard Method of X-ray Stress Measurement, JSMS, 1997 5. Bacon, G.E. Neutron Diffraction; 3rd Ed. Oxford University Press, Oxford, England, 1975. 6. Pyzalla, A. Determination of the residual stress state in components using neutron diffraction. J. Neutron Res. 2000, 8, 187–213. 7. Redner, S.; Perry, C.C. Factors affecting the accuracy of residual stress measurements using the blind hole drilling method. Proceedings of 7th International Conference on Experimental Stress Analysis, New York, August; 1982; 604–616.

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38. Inoue, T.; Nagaki, S. A constitutive modeling of thermo-viscoelastic-plastic materials. J. Therm. Stresses 1978, 1, 53–61. 39. Kujawski, D.; Mroz, Z. A viscoplastic material model and its application to cyclic loading. Acta Mech. 1980, 36, 213–230. 40. Matsui, H.; Ju, D.Y.; Inoue, T. Inelastic behavior and unified constitutive equations of SUS304 at high temperature. J. Soc. Mater. Sci. (JSMS) 1992, 41, 1153–1159. 41. Ju, D.Y.; Inoue, T.; Matsui, H. Visco-plastic Behaviour of SUS304 Stainless Steel at Ultra-high Temperature, Advances in Engineering Plasticity and its Application; Lee, D., Ed.; 1993; 521–528 42. Perzyna, P. The constitutive equations for rate sensitive plastic materials. Arch. Mech. Stos. 1968, 20, 499–512. 43. Perzyna, P. Thermodynamic Theory of Viscoplasticity, in Advance of Applied Mechanics; Academic Press: New York, 1971; vol. 11, 313–354. 44. Sorimachi, K.; Brimacombe, J.K. Improvements in mathematical modeling of stresses in continuous casting of steel. Iron Steelmaking 1977, 4, 240–245. 45. Chan, K.S.; Bodner, S.R.; Walker, K.P.; Lindholm, U.S. A survey of unified constitutive theories. Proceedings of 2nd Symposium on Nonlinear Constitutive Relations for High Temperature Applications, Cleveland; 1984; 108–112. 46. Bodner, S.R.; Partom, Y. Constitutive equations for elastic-viscoplastic strain hardening materials. J. Appl. Mech., Trans. ASME. 1975, 42, 385–389. 47. Bodner, S.R.; Merzer, A. Viscoplastic constitutive equations for copper with strain rate history and temperature effects. J. Appl. Mech., Trans. ASME. 1978, 100, 388–394. 48. Chaboche, J.L.; Rousseler, G. On the plastic and viscoplastic constitutive equation (part 1). Trans. ASME, PVT. 1983, 105, 153–158. 49. Chabache, J.L.; Rousselie, G. On the plastic and viscoplastic constitutive equations (part 2). Trans. ASME, PVT. 1983, 105, 158–164. 50. Ackermann, P.; kurtz, W.; Heinemann, W. In situ tensile testing of solidifying aluminium and Al–Mg shells. Mater. Sci. Eng. 1985, 75, 79–86. 51. Suzuki, T.; Tacke, K.H.; Schwerdtfeger, K. Influence of solidification structure on creep at high temperatures. Metall. Trans. 1988, 19, 2857–2859. 52. Inoue, T. et al., Benchmark Project on the application of inelastic constitutive relations in plasticity-creep interaction condition to structural analysis and the prediction of fatigue-creep life, part-I. Proceedings of Subcommittee Inelastic Analysis of High Temperature Materials, JSMS. 1991; Vol.1, 1–5. 53. Magee, C.L. Nucleation of Martensite; ASM: New York, 1968. 54. Johnson, W.A.; Mehl, R.F. Reaction kinetics in processes of nucleation and growth. Trans. AIME 1939, 135, 416–458. 55. Boley, A; Weiner, J.H. Theory of Thermal Stresses; Wiley: New York, 1960. 56. Rubenstein, L.I. The Stefan Problem; American Mathematical Society: Tennessee, 1971. 57. Flemings, M.C. Behavior of metal alloys in the semisolid state. Metal. Trans. A 1991, 22A, 957–980. 58. Battle, T.P.; Pehlks, R.D. Mathematical of microsegregation in binary metallic alloys. Metal. Trans. 1990, 21B, 357–375.

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59. Zienkiewicz, O.C. The Finite Element Method; McGraw-Hill: New York, 1977. 60. Peirce, D.; Shih, D.F.; Needlemen, A. A tangent modulus for rate dependent solid. Int. J. Computer Struct. 1984, 18, 875–887. 61. Williams, J.R.; Lewis, R.W.; Morgan, K. An elastic–viscoplastic thermal stress model with applications to the continuous casting of metals. Int. J. Num. Meth. Eng. 1979, 14, 1–9. 62. Ju, D.Y.; Inoue, T. Metallo-mechanical simulation of centrifugal casting process of multi-layer roll. J. Mater. Sci. Res. Int., 1996, 2 (1), 18–25. 63. Ju, D.Y.; Ichitani, K.; Nakamura, E.; Mukai, R. Simulation and experimental verification of residual stresses and distortion in quenching process with stirring, heat treatment and surface engineering. AIM 1998, 2, 283–291. 64. Narazaki, M.; Ju, D.Y. Simulation of distortion during quenching of steel effect of heat transfer in quenching. Proceedings of 18th ASM Heat Treating Society Conference Including the Liu Dai Memorial Symposium; ASM International: Ohio, 1998; 629–638. 65. Narazaki, M.; Ju, D.Y. Influence of transformation plasticity on quenching distortion of carbon steel. Proceedings of the 3rd International Conference on Quenching and Control of Distortion, Prague, March; 24–26 ASM International: Ohio, 1999; 405–415. 66. Ju, D.Y. Analysis of residual stresses during quenching process of large steel shaft. J. Saitama Institute Technol. 1994, (3), 17–21. 67. Ju, D.Y. Computer prediction of thermo-mechanical behavior and residual stresses during induction hardening of notched cylinder. J. Mater. Sci. Forum, Trans Tech Publications 2000, 347–349, 352–357. 68. Ju, D.Y.; Narazaki, M. Simulation and experimental verification of residual stresses and distortion during quenching of steel. Proceedings of 20th ASM Heat Treating Society Conference Including the Prof. J.B. Cohen Memorial Symposium; ASM International: Ohio, 2000; 441–447. 69. Ju, D.Y.; Narasaki, M.; Kamisugi, H. Computer predictions and experimental verification of residual stresses and distortion in carburizing–quenching of steel. J. Shanghai Jiaotong University 2000, E-5(1), 165–172. 70. Ju, D.Y. Computer prediction of residual stresses and distortion in carburizing–quenching of gear. Proceedings of 6th International Conference on Residual Stresses; IOM Communications: London, England, 2000; Vol. 1, 550–556. 71. Ju, D.Y.; Liu, C.C. Numerical modeling and simulation of carburized and nitrided quenching. Proceedings of International Conference on Advances in Materials and Processing Technologies, Ed. J. M. Torralba, Universidad Calos III De Madrid, Spain, Madrid, Sept 18–21; 2001, Vol. 3, 1025–1032. 72. Japanese Industrial Standard, Heat Treating Oils, JIS K 2242–1980. Japanese Standards Association: Tokyo, Japan, Analysis of Quenching Processes Using Lumped-Heat-Capacity Method, 1980. 73. Narazaki, M.; Kogawara, M.; Shirayori, A.; Fuchizawa, S. Analysis of Quenching Process Using Lumped-Heat-Capacity Method. Proceedings of the 6th International Seminar of IFHT, Kyongju, Korea; ASM International: Ohio, 1997; 428–435.

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74. Industrial Quenching Oil—Determination of Cooling Characteristics—Nickel– Alloy Probe Test Method, International Standard, ISO 9950 1995 (E). 75. Narazaki, M.; Hiratsuka, H.; Shirayori, A.; Fuchizawa, S. Examination of Methods for Obtaining Heat Transfer Coefficients by Quenching Small Probes. Proceedings of the Asian Conference on Heat Treatment of Materials, Beijing, 1998; 269–274. 76. Ju, D.Y.; Inoue, T. A. Thermomechanical model incorporating moving liquid= solid interface and its application to solidification process. Proceedings of the 6th International Conference on Mechanical Behaviour of Materials, JSMS, July 28, Kyoto, 1991; Vol. 5, 119–120. 77. Ju, D.Y.; Oshika, Y.; Inoue, T. Simulation of solidification and temperature in the centrifugal casting process (in Japanese). J. Soc. Mater. Sci. (JSMS) 1991, 40, 12–18. 78. Ju, D.Y.; Takemura, S.; Inoue, T. Analysis of coupled mode of solidification and stresses the centrifugal casting process. J. Soc. Mater. Sci. (JSMS) 1992, 41 (464), 751–757. 79. Sham, T.L.; Chow, H.W. A finite element method for an incremental viscoplasticity theory based on overstress. Int. J. Comp. Mech. 1989, 34, 143–156. 80. Grill, A.; Brimacombe, J.K.; Weinberg, F. Mathematical analysis of stresses in continuous casting of steel. Iron Steelmaking 1976, 1, 38–47. 81. Miyazawa, K.; Szekely, J. A mathematical model of the splat coolong process using the twin-roll technology. Metall. Trans. 1981, 12A, 1047–1057. 82. Ju, D.Y.; Inoue, T.; Yoshihara, N. Simulation of vertical semi-continuous direct chill casting process of cylindrical ingot of aluminum alloy (in Japanese).Trans. Jpn. Soc. Mech. Eng. (JSME) 1989, 55 (513), 1236– 1243. 83. Ju, D.Y.; Inoue, T. Simulation of solidification and Evaluation of residual stresses during centrifugal casting. Proceedings of the 3rd International Conference on Residual Stresses, Elsevier Science: Holland, 1991; Vol. 1, 220–225. 84. Ju, D.Y.; Inoue, T. Simulation of solidification and viscoplastic deformation in the twin roll continuous casting process (in Japanese), Trans. Jpn. Soc. Mech. Eng. (JSME) 1991, 57, 1147–1154. 85. Gassot, H.; Junquera, T.; Ji, V.; Jeandin, M.; Guipont, V.; Coddet, C.; Verdy, C.; Grandsire, L. A Comparative Study of Mechanical Properties and Residual Stress Distributions of Copper Coatings Obtained by Different Thermal Spray Processes, Surface Modification Technologies; IOM Communications, London, England, 2000; 16–23. 86. Ju, Y.; Nishida, M.; Hanabusa, T. Simulation of the thermo-mechanical behavior and residual stresses in the spray coating process. J. Mater. Process. Technol., 1999, 92–93, 243–250. 87. Kirara, S. Evaluation for the influence of press speed on the working load and die temperature. The Proceedings of the 44th Japanese Joint Conference for the Technology of Plasticity (in Japanese); J. JSTP 1993; 6–9.

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88. Kato, T.; Akai, M.; Tozawa, Y. Thermal analysis of cold upsetting. J. JSTP (in Japanese) 1987, 28 (319), 791–798. 89. Kennedy, K.F.; Lahoti, G.D. Review of flow shess date. Battle columbus laboratories, 1981. 90. Nakanishi, K.; Nonoyama, F.; Sawamura, M.; Danno, A. Evaluation of interface heat transfer coefficient for thermal analysis in forging. J. JSTP (in Japanese) 1996, 37 (421), 207–212. 91. Isogawa, S.; Mori, I.; Tozawa, Y. Determination of basic data for numerical simulation—analysis of multi-stage warm forging sequence for austenitic stainless steel I. J. JSTP (in Japanese) 1997, 38 (436), 84–89. 92. Ding, P.; Ju, D.Y.; Inoue, T.; de Vries, E. Numerical Simulation of forging and subsequent heat treatment of a rod by a finite volume method. Third International Conference on Physical and Numerical Simulation of Materials and Hot Working (ICPNS’99), 1999; 270–280.

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4 Modeling and Simulation of Mechanical Behavior Essam El-Magd Aachen University, Aachen, Germany

With the rapid increase of the capacity and speed of computers, work stations, and even personal computers, numerical methods can now be applied to solve easily many complex engineering problems, for example, in the fields of metal forming, strength of materials, and reliability studies of parts, components or systems. Some of the conventional methods of stress analysis, such as photo-elasticity, have nearly disappeared. Also, the interest in analytical methods like the elementary theory of plasticity and the slip line theory is decreasing to some extent with the increasing accuracy of the numerical methods. At first, great effort had to be done in developing the numerical methods themselves to insure stability, convergence, and accuracy of the computation process. In the mean time, many powerful codes that carry over the major part of this responsibility are commercially available. Especially for engineers, the main field of activities changed in another direction, namely to the development of adequate constitutive equations that describe well the behavior of the material in the macroscopic, microscopic, and even in the atomistic scale. From the practical point of view, it is not urgently required to increase numerical accuracy by some 0.1%, while the material data for plastic deformation processes deviate by more than 2% from reality or when the data for creep or fatigue life scatter by a factor between 12 and 2. There are still several important improvements to be done in the numerical codes, for example, in order to achieve an accurate consideration of large deformations. However, the determination and implementation of adequate material data seem to be one of the most important tasks. Due

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to lack of material data, the computations were carried out in the past mostly assuming an elastic material behavior using the common values of the modulus of elasticity and the Poisson ratio, even if it was well known that the material behavior is inelastic under the conditions considered. Complex procedures were developed in order to estimate the inelastic behavior using the elastic computational results. This is changing monotonically towards the consideration of the inelastic material behavior with current codes. The formulation of the material law is decisive for the experimental effort and costs needed to determine its parameters. Empirical relations may be helpful when only few variables of the process are considered and when these variables vary within limited ranges. Otherwise, physically founded material models may be more suitable, as they a priori define the tendency of the relations to be determined. However, the experimental determination of each parameter of these models according to its exact physical meaning may require such a great effort that this procedure remains restricted to academic research activities. For the practical application, a compromise is gaining increasing interest, according to which the functions are taken from physical and microstructure-mechanical models, but their parameters are determined by curve fitting of the experimental data. In the following, some examples are represented for modeling and simulation of the material behavior during plastic deformation, low cycle fatigue, creep, and impact strength.

I.

PLASTIC BEHAVIOR

Flow curves represent the relationship between the true stress and the true stain during plastic deformation at constant strain rate and temperature. They are usually determined in tensile, compression, or torsion tests. When a cylindrical rod of an initial length L0 and initial cross-sectional area A0 is loaded by a force F, its dimensions change to L and A. If the force is further increased by an increment dF, the length increases by dL and the area decreases by dA. The corresponding increments of the engineering stress and strain are defined by dS ¼ dF=A0 and de ¼ dL=L0, while the increments of the true stress and strain increments are defined by ds ¼ dF=A and de ¼ dL=L. At an arbitrary time point during the test, the stresses and strains are given by s ¼ F=A0 ;

e ¼ ðL  L0 Þ=L0

s ¼ F=A;

e ¼ lnðL=L0 Þ

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ð1Þ

If the deformation is uniformly distributed along the bar and volume constancy can be assumed, the true and the conventional stresses and strains are related by s ¼ Sð1 þ eÞ and e ¼ lnð1 þ eÞ. In addition to the physical relevance, the use of true stresses and strains allows for: P (a) a simple addition of the strains of different deformation steps e ¼ ei , (b) a simple formulation of the plastic volume constancy by exx þ eyy þ ezz ¼ 0, and (c) equal absolute values in tension and compression if specimen length is increased or decreased by the same factor. Under service conditions, engineering materials are usually subjected to a multiaxial stress state. On the other hand, material data are determined in laboratory tests under almost uniaxial loading. For comparison, an equivalent uniaxial stress is to be define for the multiaxial case. In the case of isotropic incompressible materials, the equivalent stress is given according to von Mises by

seq

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

 ðsxx  syy Þ2 þ ðsyy  szz Þ2 þ ðszz  sxx Þ2 ¼ þ 3 t2xy þ t2yz þ t2zx 2 ð2Þ

No plastic deformation takes place, as long as this equivalent stress is lower than the flow stress sY of the material. When the loads are so increased that the equivalent stress reaches the flow stress, the material starts to yield. For the strain state, which is represented by the plastic strain tensor epij , an equivalent strain increment is defined by depeq

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 1h i 2h p 2 ðdexx Þ þ ðdepyy Þ2 þ ðdepzz Þ2 þ ðdgpxy Þ2 þ ðdgpyz Þ2 þ ðdgpzx Þ2 ¼ 3 3 ð3Þ

where gij ¼ 2eij with i 6¼ j. The plastic strains epij , which are the irreversible response of the material to the applied stresses, depend not only on the current values of the stresses, but also on the total loading history. For isotropic incompressible materials, the plastic strain increments are proportional to the deviatoric stresses, i.e., the normal stresses reduced by their mean value ðsxx þ syy þ szz Þ=3 and the shear stresses unchanged, or in short form Sij ¼ sij  ð1=3Þdij skk . If the material follows the Mises yield criterion, the plastic strain increments are given by depij ¼

3 depeq Sij 2 seq

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ð4Þ

A.

Flow Curves

In order to characterize the strain hardening behavior of metallic materials during plastic deformation, one has to determine experimentally the relation sY ¼ sY ðeeq ; e_ eq ; TÞ that defines the dependence of the flow stress sY on the plastic parts of the equivalent strain eeq , the equivalent strain rate e_ eq and on the temperature T. Flow curves are defined as the relation sY ¼ sY ðeeq Þ determined for e_ eq ¼ const: at a constant temperature. They are often determined in compression test, taking into consideration the influence of friction. They are also to be determined in tension test up to the ultimate force assuming uniform deformation. 1. Empirical Relations The flow curves are almost described by power laws. The oldest of these relations, introduced in 1909 by Ludwik [1], is given by sY ¼ K0 þ Ken

ð5Þ

This relation allows a good description of the flow curves of materials having a finite elastic limit. For a plastic strain ðe ¼ 0Þ, the flow stress equals K0 . It leads, however, to an infinite value for the slope of the curve @ sY =@ e at the yield point. A simplified form of this equation sY ¼ Ken

ð6Þ

was suggested by Hollomon [2]. Because of its simplicity, it is till now the most common relation applied for the description of the flow curve. However, no yield point is considered by this relation as sY ¼ 0 for e ¼ 0. Especially for materials with a high yield point or materials previously deformed, the flow stress cannot be described well by this relation in the region of small strains. A more adequate description is achieved by the Swift relation [3] sY ¼ KðB þ eÞn

ð7Þ

For e ¼ 0, a yield point is considered with a value of sY ¼ KBn . An alternative description sY ¼ a þ b½1  expðceÞ

ð8Þ

was introduced by Voce [4] and is well applicable for the range of small strains. Figure 1 shows the optimum fit achieved by the four equations (5–8) for the flow curves of an austenitic steel at different temperatures in the range of relatively small strain up to 0.2. The figure shows that the Swift relation and the Voce-relation describe well the flow curves in the relative

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Figure 1 Description of the flow curves of the austenitic steel X6CrNi18-11 at e_ ¼ 8  104 sec1 by empirical relations.

small strain range considered. Also, a good description is achieved by the Ludwik relation when the parameter K0 is considered as an arbitrary constant and is chosen to be much smaller than the actual yield point. 2. Microstructure Mechanical Relations In one-parametrical models, the flow stress depends only on the total dislocation density r which is considered as the single internal variable of the material according to pffiffiffi s ¼ aGb r ð9Þ where G is the shear modulus, and b is the Burger vector. If the rates of creation and annihilation of dislocation are known, an evolution equation can be determined for the flow stress. Mecking and Kocks [5] introduced the following evolution equation for the total dislocation density: dr pffiffiffi ¼ k1 r  k2 r ð10Þ de and determined the slope of the flow curve by
 ds k2 k1 ¼ s ð11Þ de 2 k2 aGb

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If the parameters k1 and k2 are considered to be constants, the flow curves follow by: s ¼ s0 þ ðs1  s0 Þ½1  expðe=e Þ

ð12Þ

This equation is identical with the empirical Voce relation. In the range of relatively small strains, it fits the experimental data very well. However, it fails to describe the flow curves in the range of high strains because the experimental results for the flow stress do not asymptotically approach a definite value [6]. The following modification can be suggested, to yield an evolution equation that describes well the strain hardening in the range of high pffiffiffi strains. The parameter k1 ¼ 1=ðl rÞ, where l is the dislocation free path. This parameter can be considered as a function of strain and may be expressed as k1 ¼ kð1 þ ceÞ. The evolution equation of the flow stress becomes
 ds k kc K2 s ð13Þ ¼ þ e 2 de 2aGb 2aGb

Figure 2 Flow curve of the austenitic steel X8CrNiMoNb16-16, described by Eqs. (13) and (14).

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The solution of this differential equation is s ¼ C1 þ C2 e þ C3 ½1  expðC4 eÞ

ð14Þ

where C1 is the yield stress, C2 ¼ kc=ðk2 aGbÞ, C3 ¼ kð1 þ 2c=k2 Þ=ðk2 aGbÞ; and C4 ¼ k2 =2. This equation is identical with the empirical relation introduced in Ref. [7]. It is found to give the optimum fit for the experimental results of several materials (Fig. 2). However the determination of its parameter needs some more effort. It should be mentioned that also the empirical Swift relation given by Eq. (5) fits well the experimental data in this strain range. B.

Influence of Strain Rate and Temperature

Figure 3a shows an example for the influence of increasing temperature on the flow stress for given values of strain and strain rate [8]. Considering the slope ds=dT, three different temperature ranges can be defined: (A) range of low temperatures, between absolute zero and about 0.2 of the absolute melting point, where the influence of the temperature on the flow stress is great. The material behavior is governed by thermally activated glide, (B) range of intermediate temperatures between 0.2 and 0.5 of the absolute melting temperature. Only a slight influence of strain rate and temperature on the flow stress is usually observed in this range, and (C) range of temperatures higher than 0.5Tm in which the flow stress depends highly on the temperatures because of the dominance of diffusion-controlled deformation processes. The influence of the strain rate variation [9] is represented in Fig. 3b. Three different strain rate ranges can also be recognized according to the

Figure 3 (a) Temperature influence on the yield stress of NiCr22Co12Mo9 at e_ ¼ 3  104 sec1 [8]. (b) Influence of stain rate on shear yield stress of mild steel [9].

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variation of @s=@ ln e_ : (I) range of low strain rates with only a slight influence of the strain rate due to athermal glide processes, (II) range of intermediate and high strain rates with relatively high strain rate sensitivity due to thermal activated glide mechanisms, and (III) range of very high strain rates where internal damping processes dominate and a very high strain rate sensitivity is observed. The boundary between the ranges (I) and (II) depends on the temperature. Overviews concerning the mechanical behavior under high strain rates are represented, e.g. in Refs. [10,11]. To estimate the mechanical behavior over wide ranges of strain rate and temperature, constitutive equations must be established taking the time dependent material behavior into consideration. A visco-plastic behavior is often assumed by using, for example, the Perzyna equation [13] e_ ij ¼

S_ ij 1  2v @f s_ kk dij þ 2ghFðFÞi þ 2m 2E @sij

ð15Þ

where m is the shear modulus, f is square root of the second invariant of the stress deviator Sij and F ¼ (f=k) pffiffiffi 1 is the relative difference between f and the shear flow stress k ¼ sF = 3. The function FðF Þ is often estimated using simple rheological models assuming FðF Þ ¼ F and leading to linear relation of the type s ¼ sF ðeÞ þ Z_e which is acceptable for metals only at strain rates >103 sec1. 1. Empirical Relations Different empirical relations could be implemented in Eq. (15). With FðF Þ ¼ expðF =aÞ  1 or FðF Þ ¼ F 1=m , the corresponding relations between stress and stress rate in the uniaxial case are identical with the empirical relations introduced 1909 by Ludwik [14] s ¼ sF ðe; TÞ½1 þ a lnð1 þ e_ =aÞ

ð16Þ

s ¼ sF ðe; TÞ½1 þ ð_e=a Þm 

ð17Þ

The influence of temperature on the flow stress is also described by different relations of the type s ¼ sðe; e_ Þf ðT=Tm Þ where Tm is the absolute melting point of the material, such as s ¼ s0 ðe; e_ Þ exp½bT=Tm 

ð18Þ

or according to Ref. [15] s ¼ s0 ðe; e_ Þ½1  ðT=Tm Þv 

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ð19Þ

On applying such empirical relations, the flow stress is usually represented by s ¼ ft ðeÞf2 ð_eÞf3 ðTÞ as a product of three separate functions of strain, strain rate and temperature. This is a rough approximation especially in the case of moderate strain rates of e_ < 103 sec1 . However, the basic problem is that nearly all the parameters of these empirical equations can only be regarded as constants only within relatively small ranges of e, e_ , and T. The determination of the functional behavior of the parameters requires a great number of experiments. Therefore, constitutive equations based on structure-mechanical models are gaining increasing interest as they can improve the description of the mechanical behavior in wider ranges of strain rates and temperature and may, if carefully used, allow for the extrapolation of the determined relations.

2. Structure-Mechanical Models The macroscopic plastic strain rate of a metal that results from the accumulation of sub-microscopic slip events caused by the dislocation motion is given by e_ ¼ brm v=MT

ð20Þ

In this equation, the Burger vector b and the Taylor factor MT are constants for a given material whereas the mobile dislocation density rm is mainly a function of strain. The relation between the dislocation velocity v and the stress was experimentally determined for several materials [16]. It can be represented in the range of low stresses by a power law approaches v ¼ v0 ðs=s0 ÞN . At very high stresses, the dislocation velocity qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

asymptotically the shear wave velocity cT and s ¼ avn =

1  ðv=cT Þ2 .

3. Athermal Deformation Processes In the range of intermediate temperatures and low strain rates (combined ranges B and I), and at relatively low temperatures, i.e., less than 0.3 of the absolute melting point Tm, the influence of strain rate and temperature depends on the e_ -range of the deformation process. Below a specific value of the strain rate, that depends on temperature, only a slight influence of strain rate and temperature on the flow stress is observed. In this region I, athermal deformation processes are dominant, in which the dislocation motion is influenced by internal long range stress fields induced by such barriers as grain boundaries, precipitations, and second phases. The flow stress varies with temperature in the

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same way as the modulus of elasticity. The influence of strain rate can be described by s¼C

EðTÞ m e_ EðT0 Þ

ð21Þ

where E is the modulus of elasticity and m is of the order of magnitude of 0.01. 4. Thermally Activated Deformation In the ranges of low temperatures (A) and intermediate to high strain rates (II), the dislocation motion is increasingly influenced by the short range stress fields induced by barriers like forest dislocations and solute atom groups in fcc-materials or by the periodic lattice potential (Peierls-stress) in bcc materials. If the applied stress is high enough, these barriers can immediately be overcome. At lower stresses, a waiting time Dtw is required until the thermal fluctuations can help to overcome the barrier. A part of the dislocation line becomes free to run, in the average, a distance s until it reaches the next barrier within an additional time interval Dtm . The mean dislocation velocity is given by v ¼ s =ðDtw þ Dtm Þ. The waiting time Dtw equals the reciprocal value of the frequency n of the overcoming attempts. If the strain rate is lower than ca. 103 sec1, it can be assumed that Dtw 4 Dtm . The relation between strain rate and stress is then given by e_ ¼ e_ 0 ðeÞ exp½DG=kT  where e_ 0 ¼ brm n 0 s =MT . The activated free enthalpy DG depends on the difference s ¼ s  sa between the applied Rstress and the athermal stress according to kT lnð_e0 =_eÞ ¼ DG ¼ DG0  V  ds where V  ¼ bl  s =MT is the reduced activation volume. For given stress and strain, the value of T lnð_e =_eÞ is constant for all temperatures and also for all strain rate values between e_ 0 exp½DG0 = ðkTÞ and e_ 0 . This means that the increase of stress at constant strain with decreasing temperature or with increasing strain rates is the same, as long as the values of DG ¼ kT lnð_e =_eÞ are equal in both cases. Depending upon the formulation of the function V  ðs Þ; different relations for e_ ¼ e_ 0 ðsÞ were proposed in Refs. [17–21]. The most common are the relation introduced by Vo¨hringer [19,20] and by Kocks et al. [21]

 q  DG0 s  sa p 1 e_ ¼ e_ 0 ðeÞ exp  kT s0  s a

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ð22Þ

and that by Zerilli and Armstrong [22,23]
    e_ 0 DG0 k ln s  sa ¼  exp  b0 þ T DG0 V0 e_

ð23Þ

5. Transition to Linear Viscous Behavior At strain rates higher than some 103 sec1, the stress is high enough to the extent that Dtw vanishes. Only the motion time Dtm is to be considered. The dislocation run with high velocity throughout the lattice and damping effects dominate. The dislocation velocity v ¼ s =Dtm can then be given by v ¼ bðt  th Þ=B according to Ref. [24]. The flow stress follows the relation s ¼ sh ðe; TÞ þ Z_e

ð24Þ

with Z ¼ MT B=ðb2 Nm Þ. This relation is validated experimentally in Ref. [9] as well as by Sakino and Shiori [25], as shown in Fig. 4a. A continuous transition takes place, when the strain rate is increased from the thermal activation range (II) to the damping range (III). This can be described in two different ways: regarding the dislocation velocity to be equal to v ¼ s =ðDtw þ Dtm Þ, the strain rate can be represented by
 
 q  1 DG0 s  sa p x _e ¼ e_ 0 exp 1 ð25Þ þ kT s0  sa s  sh where x is a function of strain. Alternatively, the continuous transition can be described by an additive approximation. The stress is regarded to be the sum of the athermal, the thermal activated and the drag stress components. According to this approximation, s  sa þ sth þ Z_e where sth is the thermal activated component of stress determined from Eq. (22) or (23).

6. Diffusion-Controlled Deformation In the range of high temperatures (C), the deformation is governed by strain hardening and diffusion-controlled recovery processes


s n Q1 exp  e_ ¼ e_ 0 ð26Þ RT G At very high temperatures and low stresses
 sO 1 pdDB Dv 1 þ e_ d;e ¼ 14 kT d2 dDv

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ð27Þ

Figure 4 Dependence of flow stress on the strain rate. (a) In the range of damping controlled deformation, described by Eq. (24) [25]. (b) In the transition range between thermally activated and damping controlled deformation ranges, described by Eq. (25) [26].

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C.

Material Laws for Wide Ranges of Temperatures and Strain Rates

Material laws that describe the flow behavior over very wide ranges of temperatures and strain rates are needed for the simulation of several deformation processes, such as high-speed metal cutting. In this case, different physical mechanisms have to be coupled by a transition function. Fig. 5 shows the dependence on the stress with the strain rate at different temperatures for a constant strain. Three main mechanisms can be distinguished: (a) diffusion-controlled creep processes with e_ cr / sNðTÞ in the region (1) of low strain rates and high temperatures, (b) dislocation glide plasticity with mðTÞ s / e_ pl in the region (2) of intermediate temperatures and strain rates, and (c) viscous damping mechanism with s ¼ sG þ Zð_e  e_ G Þ in the region of very high strain rates e_ > 1000 sec1 in the region (3). 1. Visco-plastic Material Law For a continuous description over the different ranges, the strain rates have to be combined [27] to obtain   ð28Þ e_ ¼ ð1  MÞ e_ kr þ e_ pl þ M_edamping with the transition function M ¼ 1  exp½ð_e=_eG Þm . The complete strain rate range can be described by  NðTÞ  1=mðTÞ ! s s e_ ¼ ð1  MÞ e_  þ s0 ðT; eÞ sH ðT; eÞ   s  sG ðT; eÞ þ e_ G ðT; eÞ þM ð29Þ Z with e_  ¼ 1 sec1. The parameters and functions s0(T, e), sH(T, e), m(T), N(T), and Z have to be determined by curve fitting in the individual regions (1)–(3), whereas the parameters sG and e_ G are determined requiring that the derivative @s=@ e_ follows a continuous function in the transition region: sG ¼ sH ð_eG =_e Þm and e_ G ¼ ðmsH =ZÞ1=ð1mÞ . The values of the parameter used are given in Ref. [28]. An exception of the rule of the reduction of flow stress with increasing temperature is the influence of dynamic strain hardening observed in ferritic steel at temperatures between 2008C and 4008C, where the flow stress increases towards a local maximum. It is caused by the interaction between moving dislocations and diffusing interstitial atoms. The additional stress can be described by Ds ¼ að_eÞ exp½fðT  bð_eÞÞ=cð_eÞg2 , With this additional term, the dependence of flow stress of steel Ck45 (AISI 1045) on temperature and strain rate is determined [28] and represented in Fig. 6.

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Figure 5 True stress vs. true strain rate at different temperatures [27]. Markers: experimental results, curves: calculations according to Eq. (29).

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Figure 6 Influence of temperature and strain rate on the flow stress of unalloyed steel with 0.45% C. (From Ref. 27.)

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

2. Adiabatic Softening Flow curves determined in the range of high strain rates are almost adiabatic, since the deformation time is too short to allow heat transfer. The major part of the deformation energy is transformed to heat while the rest is consumed by the material to cover the increase to internal energy due to dislocation multiplication and metallurgical changes. On strain increase by de, the temperature increases according to dT ¼ k

0:9 s de rc

ð30aÞ

where the factor 0.9 is the fraction of the deformation work transformed to heat, s is the current value of the flow stress which is already influenced by the previous temperature rise and k is the fraction of energy remaining in the deformation zone. At low strain rate, there is enough time for heat transfers out of the deformation zone and the temperature increase is negligible. In this case, k ¼ 0. On the other hand, the deformation process is almost adiabatic at high strain rate and k ¼ 1. A continuous transition from the isothermal deformation under quasi-static loading to the adiabatic behavior under dynamic loading can be achieved considering k as a function of strain rate in the form.

Figure 7 Quasi-static and adiabatic flow curves of unalloyed fine grained steel.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

  1 4 e_ kð_eÞ ¼ þ arctan 1 3 3p e_ ad

ð30bÞ

The transition strain rate e_ ad depends on the thermal properties of the material. If the temperature of the surroundings is the room temperature, e_ ad is around 10þ1 sec1. As the flow stress usually decreases with increasing temperature, the flow curve shows a maximum (Fig. 7). A thermally induced mechanical instability can take place leading to a concentration of deformation, a localization of heat and even to the formation of shear bands. An overview of different criteria for the thermally induced mechanical instability is presented in Ref. [29]. The adiabatic flow curve can be determined numerically for an arbitrary function sðe; e_ ; TÞ for the shear stress which has been determined in isothermal deformation tests. In order to obtain a closed-form analytical solution demonstrating the adiabatic flow behavior, the simple stress–temperature relation s ¼ siso ðe; e_ ÞCðDTÞ can be used [30,31]. In this case, the change of temperature can simply be determined by separation of variables and integration. For example,
 T  T0 _ s ¼ siso ðe; eÞ 1  m ; Tm




s ¼ siso

0:9km exp  rcTm



Z siso de


1  Z T  T0 0:9kb s ¼ siso ðe; e_ Þ exp b siso de ; s ¼ siso 1 þ Tm rcTm

ð31Þ




ð32Þ


and
c are the mean values Tm is the absolute melting point of the material, r of density and specific heat in the temperature range considered. Around room temperature, the product rc lies between 2 and 4 MPa=K for most of the materials. For a rough approximation, it can be assumed that (rcTm =0.9)  3Tm in MPa using Tm in K. Many experimental investigations e.g. Ref. [32] were carried out in order to determine the temperature dependence of the flow stress. Up to a homologous temperature of 0.6, the stress–temperature relation can be described better by Eq. (35) than by Eq. (34), showing values of b between 1 and 4. Therefore, only Eq. (35) will be considered in the following discussion. If the isothermal stress can be simply described by siso  Ken Fð_eÞ

ð33Þ

the flow stress, determined in an adiabatic test with constant e_ , is then given by

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sad ¼ Ken Fð_eÞ 1 þ

ka Ke1þn Fð_eÞ ð1 þ nÞTm

1 ð34Þ

where a ¼ 0:9b=ð
r
cÞ. The parameter a can be considered as approximately constant represented by its mean value over the deformation process which is of the order of magnitude of 1 K=MPa. The flow curve shows a maximum smax at the critical strain ec , where

 nð1 þ nÞTm 1=ð1þnÞ KFð_eÞ nð1 þ nÞTm n=ð1þnÞ ; smax ¼ ð35Þ ec ¼ 1 þ n k aKFð_eÞ k aKFð_eÞ and the parameters K and a can be estimated by K¼

1 þ n smax ; Fð_eÞ enc

ka ¼

nTm smax ec

ð36Þ

the remaining unknown parameter n can be determined by fitting the curve the adiabatic flow curve [12]. Similar to the process of neck formation in a tensile specimen, the existence of a stress maximum leads to mechanical instability. Especially after reaching the stress maximum, a great part of the specimen is unloaded elastically causing further deformation localization. In dynamic torsion tests, the deformation localization leads to a heat concentration and hence a higher local temperature rise and a high shear strain concentration. Coffey and Armstrong [33] introduced a global temperature localization factor which is the ratio of the plastic zone volume to the total specimen volume. The influence of inhomogeneity on the strain distribution has been demonstrated by using a simple model [34] which represents the torsion specimen by two slices, a reference one and another slice with slight deviations in strength or dimensions. Furthermore, the deformation localization could be traced during the torsion test by observing the deformation of grid lines on the specimen surface by means of high-speed photography [35,36]. The influence of adiabatic softening can be illustrated in the case of compression test at high strain rates. Due to friction between the cylindrical specimen and the loading tools, a compression specimen becomes a barrel form during the test. In an etched cross-section of a quasi-statically tested specimen, two conical zones of restricted deformations can often be recognized after quasi-static upsetting. The deformed geometry is symmetrical about the midplane (Fig. 8). An FE-simulation is carried out for a compression test with e_ ¼ 0.001 sec1 considering stain hardening according to Eq. (8) and friction at the upper and lower surfaces by a coefficient m ¼ 0.1. The computational results indicate that the maximum values of equivalent stress as well

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Figure 8 Etched longitudinal section of a cylindrical compression specimen of steel 9SMnPb36 loaded quasi-statically (_e ¼ 0:001 sec1 ).

as that of the equivalent strain lie in the center of the cylindrical compression specimen (Fig. 9). In the case of dynamic compression with a strain rate of e_ ¼ 5000 sec1 , only one compression cone exists (Fig. 10). This is due to the influence of the mass inertia forces, which cannot be neglected at such high strain rates. The loads proceed in the form of mechanical waves propagating through the material. The stress at the upper surface which is impacted by a hammer is much lower than the stress at lower part of the specimen due to wave superposition after reflection from the lower surface which contacts the fixed tool. In the FE simulation, the mass inertia forces have to be considered. The deformation process is adiabatic. The local temperature increase is computed according to Eqs. (30a) and (30b) and its influence on the local flow stress is considered according to Eq. (32). The computation verifies the non-symmetry of the stress and strain distributions (Fig. 11). In the regions of high strain values, the critical strain is exceeded and the stress decreases with increasing strain. Therefore, low stress values are determined in these regions. 3. Extrapolation to Very High Strain Rates A large deformation concentration is expected in piercing and cutting process. The shear process can be dealt with as localized plastic deformation process. In the deformation zone, very high strains and high strain rates arise. The experimental tests are carried out using the so-called hat specimens, which is loaded by a compression force at its the upper surface

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Figure 9 Quasi-static compression test on cylindrical specimens with e_ ¼ 0:001 sec1 . (a) Etched section of a SMnPb-steel. (b) Distribution of Mises’ stress. (c) Distribution of the equivalent plastic.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 10 Etched longitudinal section of a cylindrical compression specimen of Armco iron loaded dynamically (_e ¼ 5000 sec1).

(Fig. 12a). The combination of experiment and finite-element simulation allows examining the possibility of extrapolation of materials laws to the range of very high strains and strain rates [28]. In addition, valuable information can be obtained for the optimization of the width B in shear

Figure 11 Distribution of Mises’ stress and equivalent plastic strain in a compression specimen after dynamic test with e_ ¼ 5000 sec1 , friction coefficient m ¼ 0.1.

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Figure 12

Hat-shaped specimen [37] and network of the shear zone [28].

processes of different materials. The specimens are tested dynamically using a split Hopkinson bar arrangement with a mean test velocity of 34 m=sec. The force as well as the displacements are recorded as functions of time. The process is simulated using a commercial explicit code taking mass inertia forces into consideration. Coulomb friction is considered with a coefficient of m ¼ 0.1. As loading conditions, the experimentally determined

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Figure 13 Distribution of the equivalent stress and the equivalent stress in the shear zone of a hat-shaped specimen [28]. (a) Equivalent strain. (b) Equivalent stress in MPa according to von Mises. Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

displacement–time function of the upper surface is applied to upper the nodes. In order to reduce the total number of elements, the lower die is idealized using the so-called infinite elements. The material law is determined in compression tests at different temperatures with strain rates up to 7500 sec1. As discussed above, one can assume a linear viscous behavior according to s ¼ sh ðeÞ þ Z_e, when e_ > 2000 sec1 and the damping mechanism dominates. The simulation should examine the accuracy of the reproduction of the force–displacement curves determined experimentally for this geometry. The distributions of the von Mises equivalent stress and equivalent strain, represented in Fig. 13, show a great non-uniformity. High strain concentrations exist at the two diagonally opposite corners of the deformation zone. In these regions, the strain rate is so high that the influence of adiabatic softening is more than compensated and high stress values are determined there. Examples for the force–displacement curves determined experimentally and computed by the FEM are shown in Fig. 14 for different values of the shear zone width B. The deviation of the computed curves from the experimental ones is relatively small. Therefore, it can be assumed that the material law determined in the range of high strain rates e_ > 2000 sec1

Figure 14 Force–displacement curves of the hat specimen. Markers: experimental results, curves: FE simulation. (From Ref. 28.)

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

can be extrapolated to much higher strain rates assuming the dominance of the viscous damping mechanism according to Eq. (24). This result is consistent with the experimental results of Sakino and Shiori (Fig. 4a). Such material laws allow the simulation of different metal forming as well as metal cutting processes. They can be validated by high-speed metal cutting tests [38]. Structural damage during high rate tensile deformation can be accounted for by introducing a damage function [39]. II.

CYCLIC DEFORMATION BEHAVIOR

A.

Phenomenological Approach

If a specimen is extended with a constant strain rate e_ 0 , the stress increases first according to Hooke’s law of elasticity till the elastic limit is reached. Then, a plastic deformation begins accompanied with a non-linear hardening. After reaching an arbitrary total strain e1tot ¼ e1el þ e1pl , the strain rate changes to (_e0 ). At first, the material is unloaded elastically and is then compressed until a total strain of (e1tot ), as represented in Fig. 15. It can be clearly observed that the plastic compression begins at an = elastic limit Re , whose absolute value much smaller than the initial value

Figure 15 Stress–strain diagram of an experiment with a change of the loading direction.

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Re . In general, it can be stated that a previous tensile deformation reduces the compression elastic limit. Also, a compression deformation reduces the subsequent elastic limit under tension. This phenomenon,known as Bauschinger effect, is characteristic for the behavior of the material under cyclic loading in the low cycle fatigue range. With further cyclic loading, the stress range increases usually due to strain hardening (Fig. 16). If the material is highly pre-deformed or hardened, a cyclic softening takes place and the stress range decreases with increasing number of cycles. The rate of change of the stress range Ds decreases with the number of cycles and approaches a stationary value and the hysteresis loop remains unchanged. The strain hardening phenomena under cyclic loading can be classified in two terms: (a) Isotropic deformation resistance sF , that includes the yield point

as well as the isotropic change of the flow stress. It increases (or decreases) monotonically with the number of Rcycles, depending upon the specific plastic P deformation work s de, or on the accumulated plastic strain jdej. Its variation with strain can be considered as a result of the increase of the density of immobile

Figure 16 Stress–strain diagram of an austenitic steel under cyclic loading with a constant range of the total strain of De ¼ 0:0066 at 6508C.

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dislocation with an additional influence of the changing microscopic residual stress state. (b) Kinematic hardening or internal back stress si that depends on the direction of the deformation and the loading history and accounts for the Bauschinger effect. It may result from the reversible interactions of mobile dislocations with obstacles, such as in the cases of pile ups or bowing between particles. Figure 17 shows the influence of these stress parts in the biaxial case on the form of the Mises-ellipse. The isotropic hardening leads to an equal increase of the ellipse in all directions, while the kinematic hardening shifts the ellipse in the loading direction. Usually, both of these hardening types are to be expected during plastic deformation. During uniaxial cyclic deformation, the influence of the strain rate can be considered in two ways: ðs  si Þ2 ¼ s2F Fð_epl Þ ð s  si Þ 2 ¼



2 sF þ Cð_epl Þ

ð37aÞ ð37bÞ

If the direction of the strain rate, and hence its sign, is suddenly changed, the sign of the isotropic material resistance sF changes at once. In contrast, the value of internal back stress si changes gradually with increasing deformation approaching asymptotically a stationary value sis with the same sign as the strain rate. In the first cycle, the stress equals sF0 þ si0 , at the beginning of the plastic deformation where si0 is approximately equal to 0 for annealed materials. With increasing strain, both of sF and si increase approaching the stationary

Figure 17 Mises’ ellipse after: (a) isotropic hardening, (b) kinematic hardening, and (c) mixed mode hardening.

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values sF1 and sis . On reaching the maximum strain of Detot =2, the maximum stress is given by smax1 ¼ sF1 þ si1 . If the loading direction is changed from tension to compression, the isotropic material resistance changes from (þsF1 ) to (sF1 ) at once. The material is first unloaded and the stress drops by the amount of sF1 . With further reduction of length, plastic compression start when the stress is reduced by 2sF1 . During this short time, the internal back stress si remains unchanged at the value si1 . After the beginning of plastic compression, it starts to decrease gradually approaching a new stationary value ðsis Þ that corresponds to the new strain rate of (_e ). Each time when the strain rate changes from (þ_e) to (_e) in an arbitrary cycle, the stress drops during the elastic deformation by DsF ¼ 2sF and then gradually by Dsi during the plastic deformation of the half cycle. On the next reverse (_e) to (þ_e), a stress decreases first by DsF and then gradually by Dsi . The stress ranges is Ds ¼ DsF þ Dsi . The maximum and the minimum stress smax ¼ sF þ Dsi =2 and smin ¼ sF  Dsi =2. The range DsF is defined as the difference between the maximum stress and the elastic limit in the subsequent compression phase. Especially in cyclic deformation, it is rather difficult to exactly determine the elastic limit, i.e., the transition point between the elastic and plastic deformation ranges because this transition is almost gradual. However, this point can be easily estimated if the hysteresis loop (Fig. 18a) is differentiated and ds=detot is represented as a function of s (Fig. 18b). Starting at minimum stress (point 1), the slope of the curve remains approximately constant and equals the modulus of elasticity till point 2 is reached. Then the slope decreases to a

Figure 18 (a) Hysteresis loop, and (b) the derivative with respect to the total strain as a function of the stress.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

relatively low value at point 3 of the maximum stress. The stress–strain curve rotates to point 4 of maximum strain. The slope decreases to 1, changes to þ1 and decreases again to the value of the modulus of elasticity. This value should remain unchanged till point 5, where plastic compression begins and the slope decreases again till reaching point 6. Between point 6 (minimum stress) and point 1 (minimum strain), the value of slope changes to þ1 then to þ1 and decreases to the value of the modus of elasticity. The relation between ds=detot and s can be linearized in the plastic ranges between the points 2 and 3 as well as between points 5 and 6. The intersection of these linear relations with the elastic relation ds=detot ¼ E defines the value of range DsF of the isotropic material resistance. The range of the internal back stress is the defined by Dsi ¼ Ds þ DsF : Repeating the procedure represented in Fig. 18 for the different cycles, one obtains Fig. 19a. For each half cycle, the value of DsF and the linear relation between ds=de and s can be determined. The isotropic component sF as well varies monotonically and continuously with increasing number of cycles. However, it can be assumed that its value is constant within arbitrary half cycles and changes only at the beginning of the next one, if the total number of cycles is great enough. In this case dsi ds ¼ de de within each half cycle. The linear relation
 dsi si  si0 ¼E 1 detot sis  si0

ð38Þ

ð39Þ

can be written for the internal back stress as well. The stationary value sis varies with the number of cycles. The isotropic material resistance sF as well as the stationary value sis of the internal back stress si are represented in Fig. 19b as functions of the accumulated strain. These relations may be described by sF ¼ sF0 þ ðsF1  sF0 Þ½1  expðCF eacc Þ ð40Þ sis ¼ sis0 þ ðsis1  sis0 Þ½1  expðCi eacc Þ

ð41Þ

yielding the evolution equations dsF ¼ CF ðsFs  sF Þ deacc

ð42Þ

dsis ¼ Ci ðsis1  sis Þ deacc

ð43Þ

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Figure 19 (a) Description of the derivative ds=detot by a linear function of stress in the transition range between the ranges of elastic and plastic deformation. (b) sF and sis as function of the accumulated plastic strain.

Within each half cycle, si approaches the stationary value sis as shown in Fig. 20a. The transition range from the elastic to the plastic deformation ranges is fairly good described by Eq. (39). On the other hand, relatively large deviations are observed at low values of dsi =detot , which corresponds to high stress and plastic strain values. A more accurate description is

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Figure 20 Relation between stress derivative and internal back stress si . (a) Derivative with respect to the total stress. (b) Derivative with respect to plastic strain.

achieved by considering two internal variables for the internal back stress instead of only one as considered here [40]. With depl ¼ detot  dsi =E, the derivative of the internal back stress with respect to the plastic strain can be obtained from Eq. (39) as
 dsi sis  si0 ¼E 1 depl si  si0

ð44Þ

A comparison of this relation with the experimental results is represented in Fig. 20b.

B.

Constitutive Equation of Cyclic Behavior

In contrast to these experimental facts, serious simplifications are usually made to reduce the number of the material parameters involved in computation. The hyperbolic function of Eq. (8) is simply linearized yielding dsi ¼ C  gsi depl

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ð45Þ

where C and g are material constants. The stationary value sis is considered constant assuming that the material follows the Masing-rule and Eq. (5) is reduced to sis ¼ C=g

ð46Þ

Lemaitre and Chaboche [41,42] introduced a non-linear isotropic= kinematic hardening model, which provides predictions that are near to the experimental evidence. This model is applicable for isotropic incompressible materials. The yield surface is defined by the function F ¼ fðsij  Xij Þ  s0 ¼ 0

ð47Þ

where s0 is the yield stress that is equivalent to the isotropic material resistance sF and Xij is the tensor of the internal back stress denoted sis in the uniaxial case. The function f ðsij  Xij Þ equals the equivalent Mises stress when the back stress X is taken into consideration: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 0 ðs  X0ij Þðs0ij  X0ij Þ f¼ 2 ij

ð48Þ

where s0ij is the deviatoric stress tensor and is the Xij0 deviatoric part of the back stress tensor. The associated plastic flow is given by e_ pl ij ¼

@F
pl 3 ðs0ij  X0ij Þ
pl e_ e_ ¼ @sij 2 f

ð49Þ

where e_ pl represents the rate of plastic flow and
e_ pl is the equivalent plastic strain rate rffiffiffiffiffiffiffiffiffiffiffiffiffi
e_ pl ¼ 2 e_ pl e_ pl ð50Þ 3 ij ij The size of the elastic range, s0 , is a function of the equivalent plastic strain
epl and the temperature. For a constant temperature, it is written similar to Eq. (40) as 

ð51Þ s0 ¼ s0 þ Q1 1  expðb
epl Þ where sj0 is the yield surface size at zero plastic strain, and Q1 and b are additional material parameters that must be determined from cyclic experiment.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

The evolution of the kinematic component of the model, when temperature and field variable are neglected, is defined as
0  sij  X0ij 2 pl

pl _ X_ ij ¼ C_epl  X ¼ C  gX e ij ij e_ 3 ij s0

ð52Þ

where C and g are material parameters. C.

Application to Life Assessment

The assessment of the fatigue life under cyclic elasto-plastic deformation requires an accurate determination of the strain ranges of the individual loading cycles in the region of maximum local deformation. For this reason, FE simulation is often needed especially when the analytical solutions are not available or when they include unacceptable simplifications. For example, the fatigue life of notched machine parts is often predicted using approximation formulas [43–45] that have been driven using the Neuber rule [46]. However, the accuracy of these methods remains lower than that of the inelastic FE analysis, when adequate materials lows are implemented. Figure 21a illustrates the distribution of the axial stress sxx in a notched 3-point bending specimen. The mesh is built of three-dimensional continuum solid elements. Around the notch, a refined mesh is chosen so that the critical zone at notch root would cover several elements. The material considered is the AlZnMgCu alloy AA7075. The material parameters determined in uniaxial cyclic experiments are: sj0 ¼ 310 MPa, Q ¼ 75 MPa, b ¼ 36.6, C ¼ 14,844 MPa and g ¼ 86.3. As a loading condition, a line load with a total compressive force F is applied at the midlength of the upper surface. The force follows a sinusoidal time function with Fmin =Fmax ¼ 0:1. The specimen is supported at two parallel lines on the lower surface with a span width of 80 mm. The computational results of the local strain components at notch root are represented in Fig. 21b as functions of time. For an arbitrary loading cycle, the time functions eij ðtÞ are used to determine an equivalent strain range D
e for the cycle according to different approaches [47–52]. With an additional damage accumulation rule, a representative periodically repeated strain range can be computed which should lead to the same fatigue life. Experimentally, fatigue life is determined as the number of cycles, at which a technically detectable crack is initiated. Often, a direct current potential drop system [53,54] is used to determine the potential drop across the notch as time function. This is then converted to a relation between crack length and number of cycles to initiate cracks.

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Figure 21 (a) Distribution of the bending stress sxx in a 100  20  10 mm3 specimen with a notch radius or 6 mm. (b) FE computation results for the variation of local strain components at notch root.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 22 Representative strain range as function of the number of cycle to failure. (From Ref. 54.)

The equivalent strains, determined according to Ref. [48] for different notch radii vs. the number cycles to fracture are plotted in Fig. 22. The experimental results obtained for alternating tension–compression loading on smooth bars are also represented in Fig. 22 and are described by the usual function Detot ¼ Depl þ Deel with Depl according to the Manson– Coffin equation and Deel following a similar one: d Detot ¼ aNb i þ cNi

ð53Þ

The results of the notched bending specimens, obtained by the combination of FE analysis, strength hypothesis, and experimental life determination, lie in a scatter band around the uniaxial data. D.

LCF of Metal Matrix Composite Materials

The fatigue life of metal matrix fiber composites is found to be strongly reduced in the range of low cycle fatigue due to the formation of kink bands, at which fatigue cracks initiate. A cross-section of one of these materials is represented in Fig. 23a. This composite consists of a pure copper matrix and continuous fibers of

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Figure 23

Continued

the austenitic steel X5CrNi18-12. It is used mainly in the electrical industry as a contact material. The mean fiber diameter dm ¼ 200 mm and the volume fraction equals 40%. Under alternating elasto-plastic strains, the slim fibers buckle plastically within the softer matrix during the compression phase of the load cycle and then expand to the straight form in the tension phase. However, if the material is unloaded and the elastic part of the deformation diminishes, the fibers do not remain completely straight. This promotes lateral deflection in the next compression cycle and the fiber buckling increases from one cycle to the other. The fiber takes an S-shape in the first few cycles. With increasing number of cycles, bending localiza-

"

Figure 23 Copper reinforced by austenitic steel fibers. (a) Etched cross-section. (b) Initiation of fatigue cracks. (c) Crack growth.

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tion takes place at two points of each fiber, and a kink band is observed, which is inclined to the load direction [59–59]. Within the kink band, the matrix suffers high cyclic shear deformation, which may cause crack initiation. If the fibers are brittle, fiber fracture occurs at the kink band boundaries. Figure 23b shows the initiation fatigue cracks in a specimen subjected to alternating low cycle fatigue loading. The specimens were tested strain controlled with an alternating total strain value by a strain rate e_ ¼ 0:0017 sec1 [60]. A two-dimensional idealization is chosen in order to get a clear idea about the deformation process even when the results are only of a qualitative character. The material is supposed to consist of plain layers of cupper and austenitic steel. To simulate buckling, an imperfection must be introduced to allow for mechanical instability. This is done by bringing in an inclination with a small angle b (for example 28) which may represent a deviation between the load and the fiber directions resulting from a non-accuracy of the specimen geometry or a non-alignment of the testing machine axis. The computed distribution of the equivalent stress is shown in Fig. 24a for a small elastic tensile strain of 0.001. The volume fraction of the harder material component equals 40%. Due to the difference in the modulus of elasticity of the material components, higher stresses arise in the elements of the stiffer materials and they appear brighter in the plot. The gradient of the stress in the lateral (horizontal) direction is related to bending caused by stretching of the inclined network. Fig. 24b shows the stress distribution after reaching the maximum strain of 0.024 in the first cycle. The displacements are exaggerated in the plot. Both the material components are plastically deformed. The harder material appears in the mean brighter than the softer one. Fourteen cycles later, the mesh and the stress distribution look completely different, as shown in Fig. 24c and d for the time instants of reaching the maximum compressive strain of the 14th cycle and the maximum tensile strain of the next cycle. The originally softer materials show higher isotropic hardening due to the greater amount of accumulated strain. An inclined kink band can be easily recognized with high stresses in the matrix. The successive fiber buckling with increasing number of cycles can be more obviously observed under pulsating compressive stresses. A characteristic phenomenon is the initiation of an inclined shear band accompanied by a reduction in the specimen deformation resistance, deformation localization in the matrix, and hence a reduction in the fatigue life. Figure 25 shows a comparison between the 2D-FEM results for the deformed mesh with the fiber configuration in longitudinal section of

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Figure 24 The von Mises equivalent stress distribution in the deformed mesh. The darker the element, the lower the stress. Displacements are exaggerated. (a) Under partial elastic loading. (b) At the maximum elasto-plastic tensile strain of the first cycle. (c) At maximum compression strain of the 14th cycle. (d) At the maximum tensile strain of the 14th cycle.

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Figure 25 FEM-plots and etched longitudinal sections of cylindrical specimens subjected to pulsating compressive loads up to: (a) 135 cycles, (b) 157 cycles, and (c) 284 cycles.

specimens that have been subjected to the same number of cycles in pulsating compression LCF-tests. In both the experiment and simulation, the specimens are subjected to a compressive stress pulsating periodically between 450 MPa and –45 MPa. The influence of friction at the top and the bottom of the specimen is not taken into consideration in the simulation. The fibers are continuous with a volume fraction of 20%. The fibers do not appear as continuous dark lines in the optical photographs because they do not necessarily lie completely in the plane of the prepared section. The similarity of both configurations allows a better understanding of the failure mechanism as far as the history of deflections, local deformations, residual stresses, multiaxiality, and similar values can be followed up through out the test simulation.

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III.

CREEP BEHAVIOR

In contrast to plasticity, a long-time high-temperature creep exposure causes a continuous change in the constitution of the materials. Beside hardening by the increasing dislocation density, several microstructural events take place such as initiation of subgrains, precipitation, ripening, and coagulation of particles, oxidation, high-temperature corrosion or even phase transformation. The creep strain is accompanied by a slowly increasing damage process that covers a great fraction of the creep life. Constitutive equations based on a combination of overstress concept [62,62] and threshold stress concept [5,63] allow an adequate description of the materials behavior if successive damage is taken into consideration. The current value of the strain rate depends on the current values effective stress, which is the difference between the applied stress s and the internal back stress si , the particle deformation resistance sp , the material creep resistance sF and the degree of damage D. For long-time creep under low stresses, the creep rate can be represented by
 s  si n sgnðs  si Þ e_ ¼ C sF ð1  DÞ

ð54Þ

where in case of high-temperature creep   Q C ¼ e_ 0 exp  RT

ð55Þ

This relation is applicable for true stress and true strain rate. If the creep tests are carried out with a constant force, the applied engineering stress value is to be multiplied by the factor ð1 þ eÞ in order to account for the increase of true stress due to reduction of area. The engineering strain rate is to be divided by the same factor. In many cases of the modeling of high-temperature material behavior, it is highly recommended to use the same set of equation to describe timedependent creep and the time-independent plastic behavior. As discussed above, the low cycle fatigue can be described by s  si ¼ 1 sF ð1  DÞ

ð56Þ

With decreasing strain rate, a transition takes place towards a timedependent creep behavior described by Eq. (54), which can be rewritten in the form.

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Figure 26 Transition between the ranges of time-dependent (creep) and timeindependent (plasticity) ranges. (From Ref. 40.)

s  si ¼ sF ð1  DÞ

 1=n e_ C

A continuous transition function may be given by  m =n !1=m s  si e_ ¼ þ1 sF ð1  DÞ e_ 0

ð57Þ

ð58Þ

as represented in Fig. 26. In contrast to low cycle fatigue behavior, with its relative short life, the parameter sF is considered in the case of long-time creep exposure as a constant reference stress and can be set equal to 1 MPa. In this case, only the overstress concept is considered. The kinematic hardening may include several components [64]: sd þ sS þ sp . The first term sd accounts for the variation of the dislocation density. An additional material resistance sS that accounts for subgrain formation can be taken into consideration, considering the material as a composite consisting of hard subgrain boundaries and soft subgrain interior [65]. The particle stress sp accounts for the interaction between mobile dislocaCopyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

tions and precipitates. It depends on the mean distance between the particles which may change in the course of creep exposure (Figs. 27). For engineering construction and life assessment, it is required to reduce the number of parameters to the amount that is essential for the description of the mechanical behavior and that can be determined with a tolerable experimental effort. Therefore, most engineering materials may be described well using si ¼ sd þ sS with a unique function of strain. The particle is then treated separately and eq. (54) is rewritten as
 s  si  s p n e_ ¼ C sgnðs  si Þ ð59Þ sF ð1  DÞ

A.

Damage Function

Different formulations can be applied for the damage function D. Kachanov and Rabotnov [67,68] introduced the relation dD a ¼ ; dt ð1  DÞp

  t 1=ðrþ1Þ D¼1 1 tf

ð60aÞ

where tf is the fracture time (Fig. 28). Other applicable functions are dD ¼ bDm ; dt dD ¼ c þ gD; dt

D ¼ ðt=tf ÞM D¼

expðgtÞ  expðgt0 Þ expðgtf Þ  expðgt0 Þ

ð60bÞ ð60cÞ

As the damage increases very rapidly with time in the late tertiary stage, a more accurate description can be achieved by formulating damage as a function of strain instead of the time function used above. In this case, a modified Kachanov and Rabotnov relation: D ¼ 1  ½1  ðe=ef Þ1=ðmþ1Þ or further function such as   D ¼ ðe ef Þ1 ð1nÞ  D ¼ ½expðbe=ef Þ  1 ½expðbÞ  1 can be applied.

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ð61aÞ

ð61bÞ ð61cÞ

Figure 27 Main factors affecting internal back stress. (a) Dislocations. (b) Subgrain boundary. (c) Particles. (From Ref. 66.)

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 27

B.

Continued

Internal Back Stress

If the applied stress is reduced from s0 to sR at a time point t0 , the strain decreases directly by ðs0  sR Þ=E to the strain value e0 , and starts to increase or to decrease depending on the value of the reduced stress sR . After a transition time, the steady state is reached again and the strain increases with normal strain rate expected for the stress sR . The relation between the subsequent creep strain e  e0 and the time t  t0 after the partial unloading is shown in Fig. 29. There is a certain value of the reduced stress sR at which the creep rate is reduced to 0 immediately after load reduction. This value of sR is equal to the current value of the internal back stress si according to Eq. (54). For materials with negligible particle resistance, it is equal to the internal back stress. If the particle stress sP and the additional substructure resistance sS can be neglected, the internal back stress si is only influenced by the evolution of the dislocation density and can be set equal to sd . In this case, the evolution of the internal back stress in the course of creep exposure under constant or varying stresses and temperatures can be formulated [70]. Even if the assumptions mentioned do not exactly hold, such a formulation can help in determining the creep behavior as a phenomenological description.

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Figure 28 Creep damage. (a) Crack and voids initiated in the tertiary range. (b) Description of damage as a function of creep strain.

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Figure 29 Ref. 69.)

Creep strain after partial unloading as a function of time. (From

Similar to Eq. (9), the internal stress component related to dislocation pffiffiffi density is defined as sd ¼ aGb r. With the evolution Eq. (10) of the dislocation density [5], the variation of the internal back stress is described by the relation dsi C1 ¼ ðsis  si Þ de e1

ð62Þ

which is validated experimentally, e.g. in Ref. [69]. In this equation,sis is the quasi-stationary value of the internal back stress and e1 is the corresponding creep strain. Under constant stress and temperature, the internal back stress increases in the primary stage according to
 si e ¼ 1  exp C1 ð63Þ sis e1 with e1 as the strain at the end of the primary creep stage. This relationship is shown in Fig. 30 for different materials stresses and temperatures. The quasi-stationary value sis depends on the applied stress (Fig. 29). With increasing creep stress, sis increases approaching a saturation value siss . The experimental data for the secondary creep rate are usually well described by the Norton–Bailey relation e_ s ¼ AsN [71]. On the other hand, e_ s should follow the relation e_ s ¼ C ðs  sis Þn . It can be shown that Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 30 Development of the internal back stress in the primary creep stage. (From Ref. 70.)

s=siss < N=ðN  nÞ;

s=siss  N=ðN  nÞ;

" #   sis s n N  n s N=n1 ¼ 1 siss siss N N siss

ð64aÞ

sis ¼1 siss

ð64bÞ

as shown in Fig. 31. In these equations, N is the usual Norton–Bailey stress exponent N ¼ @ ln e_ =@ ln s. The parameter n depends on temperature. It can be set equal to 4 in the temperature range of the practical service conditions of high-temperature engineering materials. The saturation value siss depends on the temperature and can be described by the exponential function: siss ðTÞ ¼ k0 expðb=TÞ

ð65Þ

where k0 and b are material constants. As a rough approximation, siss can be considered as proportional to the creep strength Rm=1000 for a fracture time of 1000 hr and set equal to about 1:4Rm=1000 . Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 31 Relation between the quasi-stationary internal back stress and the applied stress.

After a sudden increase of the applied stress, the internal back stress starts to increase gradually approaching the quasi-stationary value (Fig 32a). If the applied stress is then reduced to the original value, a gradual reduction of the internal back stress towards the corresponding quasi-stationary value is observed (Fig. 32b). As the strain rate depends on the difference between the applied load and the internal back stress, a load enlargement leads to a very high strain rate that reduces gradually to normal value. In the same way, a load drop causes a severe strain rate reduction, even to negative strain rate values when s declined to a value lower than si . Figure 33a shows the creep rate curve of the austenitic 18=11 Cr–Nisteel under cyclic creep loading. The stress is changed periodically between 150 and 125 MPa. The period is equal to 96 hr. The influence of fatigue can be neglected and the material behavior can be described as a pure cyclic creep. Under cyclic stress, the influence of the creep strain transients is found to reduce the creep life. Compared with life values calculated by the linear damage accumulation rule

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Figure 32 Variation of the internal back stress after a sudden change of the applied stress. (a) Load increase. (b) Load reduction. (From Ref. 70.)

Figure 33 Description of strain transients under pulsating stress by the overstress model. (From Ref. 70.)

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Xt ¼L tf

ð66Þ

the value of L decreases to about 0.6 in case of cyclic stress at constant temperature and about 0.8 in the case of pulsating temperature under constant stress [72].

C.

Influence of Particles

1. Behavior of Dispersion-Strengthened Materials Dispersion-strengthened materials are usually produced by powder metallurgical techniques, especially mechanical alloying. They include very fine oxide or carbide particles embedded within the grains. The particles obstacle the dislocation motion and increase the resistance to deformation. The strength depends mainly on the size and the volume fraction of the dispersoids as well as on the consolidation process and the matrix material [73,74]. Contrary to precipitation hardening, the dispersoids are thermally stable and do not ripen or coagulate during long-time high-temperature exposure [75,76]. Therefore, such materials are predestined for applications under high-temperature creep conditions. Their behavior is studied under tensile and compressive loads, e.g. in Refs. [77–79]. Fig. 34a shows the creep rate curves of the Aluminum alloy AlSi20 which was produced from its powder without any additions by cold pressing and hot extrusion. This material includes an oxygen content of less than 0.5 mass%. The corresponding curves in Fig. 34b are determined for a dispersion-strengthened version AlSi20C1O2 produced by mechanical alloying. Carbon powder is added to the matrix powder and the mixture undergoes intensive milling before cold pressing and hot extrusion. The material includes a volume fraction of 4% of Al2O3 and 4% of Al4C3 as dispersoids with a mean particle size of 150 nm. The creep rate can be described by e_ ¼ f ðs  sp Þ where sp is the additional resistance to deformation caused by the particles (Fig. 35a). The following simple relation can be used for the estimation of the minimum creep rate hs  s iN p expðQ=RTÞ ð67Þ e_ s ¼ C E where E is the modulus of elasticity at the creep temperature, N is the Norton–Bailey stress exponent of the matrix material, Q is the activation energy for self-diffusion of the base element of the matrix. Fig. 35b shows that the creep strength is increased by a constant value which depends only on the volume fraction of the particles and their morphology.

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Figure 34 Creep rate curves of Aluminum Alloy AlSi20 [78]: (a) without, and (b) with dispersoids.

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Figure 35 Influence of dispersoids on creep stress [78], for arbitrary values of: (a) minimum creep rate, and (b) creep life.

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The particle resistance may be estimated by sp  sO where sO is the Orowan stress given by   MG 2b Ld ln sO ¼ 0:84 ð68Þ 4pð1  nÞ L  d 2b In this equation, M is the Tailor factor, G is the shear modulus depending on temperature, L is the mean distance between particles, and b is the Burger vector. A precise description of the experimental data in the range of low creep stresses and high fracture times allows a model introduced by Reppich et al. [80]. According to this model, the additional particle resistance sp depends not only on the particle morphology but also on the applied stress. In the range of high creep stresses, sp approaches an upper limit sp , which can be set equal to the Orowan stress sO . With decreasing creep stress, the dislocation can overcome the particle resistance by partial climb, and the particle resistance is assumed to be directly proportional to the applied stress. Fig. 36 shows the relation between the relative particle resistance sp =sp as a function of the normalized creep stress s=sp . A continuous

Figure 36 Increase of additional particle resistance with increasing creep stress. (From Ref. 79.)

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

transition from the range of low stresses to the range of higher once may be described by a function of the type " " #m !#1=m sp s ¼ 1  exp   ð69Þ sp sp The upper limit of the particle resistance can be set equal to the Orowan stress: sp  sO . 2. Precipitation Hardening The high-temperature creep behavior of precipitation hardenable industrial alloys is influenced by the kinetics of the precipitation and the ripening processes. Creep specimens are found to exhibit a longer creep life after solution treatment compared to those ones additionally aged before testing [81]. This phenomenon is attributed to the precipitation of fine particles during the early stages of creep [82], which strengthen the material and reduce the creep rate (Fig. 37). With increasing the creep time, the particle coarsening leads to an increase of the interparticle spacing and to an acceleration of creep strain rate.

Figure 37

Precipitates and dislocations in Alloy 800HT. (From Ref. 66.)

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The increase of the size of an existing precipitate can be considered according to the Oswald-ripening mechanism. However, the precipitation process under creep loads is rather complex and is expected to be dependent not only on the temperature but also on the change in the degree of supersaturation of the matrix as well as on the defect structure and dislocation density. Therefore, the rate of precipitation is assumed to depend on creep conditions. The particles strengthen the material by exerting an internal, or threshold, stress sp on the moving dislocations. According to the theory of Brown and Ham, reported by Martin [83], it is assumed that the high-temperature yield stress of particle-hardened material is controlled by local climb of dislocations over the particles sp ¼

S L

ð70Þ

where S is a material constant and L is the planar interparticle spacing. From geometrical considerations, it can be shown that the planar interparticle p spacing ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL varies with Vp and d according to the relationship L ¼ p=ð6Vp Þd. The particle stress can be rewritten as rffiffiffi pffiffiffiffiffiffi Vp 6 sp ¼ ð71Þ S d p An analytical model [84] that considers the influence of a continuous precipitation process accompanied by particle coarsening on the creep behavior of metallic materials will be discussed briefly in the following lines. The volume of particles per unit volume V p that precipitate out of the supersaturated matrix can be written as p ð72Þ Vp ¼ Np d3 6 where Np is the number of nuclei or the number of carbide particles per unit volume and d is the average particle diameter. The number of growing nuclei per unit volume, Np, that form during precipitation out of a supersaturated solid solution is known to depend on the degree of supersaturation, the temperature, and the defect structure, especially the dislocation density. At a given temperature, the value of Np can be written as Np ¼ Np0 þ fðtÞgðtÞ

ð73Þ

where Np0 is the initial number of nuclei and can be neglected in case of solution annealed materials, f(t) is a function of the temperature and the defect structure, g(t) is a function of the remaining supersaturation, which

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decreases during the precipitation process. During the primary creep stage, the dislocation density depends on the creep strain and f(t) can be written as fðtÞ ¼ f½a1 f1 ðTÞ þ a2 ðsn2 eQ2 =RT ÞM2 tgM

ð74Þ

where a1 , a2 , n2 , M2 , and M are constants, s, t, and T are stress, time, and temperature in Kelvin, respectively, Q2 is the creep activation energy and is close to the activation energy of self-diffusion. The first part of f(t) including f1(T) represents the nucleation under static conditions when no stress is acting on the material while the second part represents the effect of the applied creep stress. As precipitation progresses, the supersaturation of the matrix decreases. The function g(t) can be represented as a hyperbolic function of time gðtÞ ¼ ½1 þ ðBtÞn 

m

ð75Þ

where n and m are parameters that control the rate of precipitation, B is a function of stress and temperature. The increase in particle diameter d with time due to Oswald ripening can be represented by the function [85]

1=m d ¼ dm0 þ Ct  ðCtÞ1=m

ð76Þ

where d0 is the original particle diameter and may be neglected, m is a constant whose value depends on the particle growth mechanisms and C is a function of temperature given by C ¼ C0 expðQ1 =RT Þ, where Q1 is the activation energy of the coarsening process, e.g. for carbon in iron. Based on the experimental results, the first part of the function f(t) in Eq. (74) is expected to be small compared with the second part so that it may be neglected for creep loading. With this approximation, the volume fraction Vp can be written as Vp ¼

ðp=6Þ½a2 ðsn2 eQ2 =RT ÞM2 M C3=m tMþ3=m ½1 þ ðBtÞn  m

ð77Þ

With increasing time, Vp approaches asymptotically a final value Vp1 that depends only on the initial supersaturation, but not on a function of time or stress, M þ ð3=mÞ must be equal to mn and the volume fraction Vp can be written as
m ðBtÞn ð78Þ Vp ¼ Vp1 1 þ ðBtÞn

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As M must be positive, mn > 3=m. Taking m ¼ 3, mn > 1. Based on the above assumptions, the parameter B can be expressed as a function of stress and temperature as follows:
 Qa na B ¼ B0 s exp  ð79Þ RT where Qa ¼ ð3Q1 =m þ M2 MQ2 Þ=ðmnÞ is an apparent activation energy without a specific physical interpretation and na ¼ n2 M2 M=ðmnÞ. Substituting Eqs. (76) and [78] in eq. (71), sp can be written as rffiffiffiffiffiffiffiffiffiffiffi
m=2 S 6 ðBtÞn Vp1 sp ¼ 1=m t1=m ð80Þ p 1 þ ðBtÞn C The condition for maximum strengthening due to precipitation, e_ ¼ e_ min , is achieved when sp reaches a maximum value,sp max , after time t , thus   nmm  2 1=n 1 ð81Þ t ¼ 2 B

Figure 38 Time to minimum creep rate of Alloy 800HT for different creep stresses and temperatures. (From Ref. 84.)

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Figure 39 (a) Experimental data (markers) and (b) model results (curves) for the creep rate of Alloy 800HT as a function of time or creep strain. (From Ref. 66.)

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Figure 40 Creep rate curves of AA2024 in two heat treatment conditions. (a) Highly coarsened precipitates. (b) Solution annealed. (From Ref. 86.)

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sp max

S ¼ 1=m C

rffiffiffiffiffiffiffiffiffiffiffi
m=2 6 2 Vp1 1 þ t1=m p mmn  2

ð82Þ

Considering the stress dependence of the parameter B, the following relationships are determined: t ¼ A0 sna eQa =RT

ð83Þ

This relation is found to fit well experimental results for precipitation hardening austenitic steel (Fig. 38) and Aluminum AA2024. The variation of sp with creep time is given by the relationship sp sp max




m=2 1=m t ¼ n  t ðnmm=2Þ  1 þ ðt=t Þ nmm 2

ð84Þ

Figure 39 shows a comparison between experimental results and model results for Alloy 800HT. Similar results, obtained for the Aluminum Alloy AA2024, are presented in Fig. 40. In order to determine the matrix behavior, the material is over-aged for a long span of time until the precipitates coagulate. The particle diameter and the distance between them are so increased that the particle stress can be neglected. This matrix behavior is described by Eq. (54). For the solution annealed condition, the influence of the particle stress is considered using Eqs. (59) and [84]. For practical applications, some of the parameters can be set equal to certain values. Assuming that ripening takes place on the basis of volume diffusion, the parameter m is to be set equal to 3. Furthermore, the variation of the volume fraction of the precipitates with time, according to Eq. (78), can be well described with m ¼ 2 as far as the parameter n can be determined by best fit of the experimental results (Fig. 41). With these values, the particle stress can be given by

t 1=3 sp 3n ¼ ð85Þ n  t sp max 3n  1 þ ðt=t Þ

D.

Simulation of Creep Microcrack Growth

Under high-temperature creep exposure, failure takes place due to the initiation and growth of inter-crystalline cavities, in the form of voids or intercrystalline microcracks. In the case of high temperatures, low stresses and small grain size of the material, the inter-crystalline void initiation, growth and coalescence is the dominant damage mechanism. Several well-founded

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 41

Variation of the particle stress with time according to Eq. 85.

models were introduced to estimate the rate of void growth. At lower stresses, the voids grow by diffusion of atoms out of the void surface into the grain boundary. At higher stresses, the void growth takes place by a creep deformation of the surrounding material [87–94]. On the other hand, higher creep stresses can be allowed by lowering temperature or by increasing grain size. In this case, wedge type inter-crystalline microcracks are observed. A successive damage is caused by the growth of the existing cracks as well as by the continuous initiation of new ones. Most of these microcracks are extended over only few grain boundary facets (Fig. 42). In contrast to the long technical cracks, whose growth rate can be calculated by the methods of the non-linear fracture mechanics, there are only a few investigations about the initiation and growth of wedge type inter-crystalline creep microcracks. The micrograph of Fig. 42 shows a representative example for the form of the wedge type inter-crystalline creep microcracks. Few typical shapes and a preferred orientation can be identified. The microcracks start usually at grain triple points and extend along such grain boundaries, that are nearly perpendicular to the direction of the applied stress. All the different crack shapes can be explained by the fact that the initiation and growth of the cracks result from the grain boundary sliding.

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Figure 42 (a) Microcracks in a longitudinal section of a creep specimen of steel X6CrNi18-11 after creep fracture (s ¼80 MPa, T ¼ 7008C). (b) The role of grain boundary sliding in crack initiation.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 42

Continued

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The estimation of the rate of growth of such microcracks can be achieved mainly by two methods: (a) the determination of the statistical distribution of the microcrack length and its variation with the creep time, and (b) finite-element simulation of the creep behavior regarding the material as a composite consisting of grains separated by thin grain boundary layer with different properties. 1. Statistical Model The fundamental concept of the statistical model is that a small fraction of short cracks and a high fraction of long cracks are expected when the growth rate is high, and vice versa [95–98]. In order to achieve reliable results using statistics, a great number of cracks have to be classified. Over a period of several years, about 50,000 cracks were classified in the steel X6 CrNi18-11 and more than 60,000 cracks in the steel X8CrNiMoNb16-16 for different temperatures and stresses. Based on the results of metallographic investigations, the following assumptions are introduced: (a) A crack grows quickly along the grain boundary from one triple point to the next, where it rests for a longer time before it grows again to the next triple point, (b) The crack length is always an integral multiple n of grain boundary facets and (c) every crack is initiated in the length class n ¼ 1 and grows step by step to next higher length classes. Let Z be the total number of cracks per unit area and Yn the number of cracks having a length n. In a time unit, Vn,n þ 1 cracks grow out of the length class n into the next higher length class (n þ 1). In the same time, Vn  1,n cracks grow from the lower length class (n  1) into the considered class n. The mean rate of growth is given by dn=dt ¼ Vn;nþ1 =Yn and the rate of change of dYn =dt ¼ Vn;nþ1  Vn1;n . As all cracks initiate with the length n ¼ 1, the rate V0;1 represents the rate of crack initiation and must be equal to rate dZ=dt of the increase of the total crack number. Therefore, following relation can be deduced: n dZ X dYi ð86Þ  Vn;nþ1 ¼ dt dt i¼1 Denoting the fraction of cracks with the length of n by Xn, so that dYn ¼ Xn dZ þ ZdXn , the mean rate of growth ðdn=dtÞn of the cracks of the length class n can be written as ! P  
  n dn 1  ni¼1 Xi 1 dZ 1 X dXi ¼  ¼ FðnÞGðtÞ  Hðn; tÞ dt n Z dt Xn i¼1 dt Xn ð87Þ

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 43

Statistical distribution of the inter-crystalline creep microcracks.

The first term denoted F(n) is determined by the statistical distribution of the microcrack length. This distribution can be described by Xn ¼ ð1  qÞqn1

ð88Þ

as represented in Fig. 43 for two different austenitic steels. Deviations are mainly observed in the range of long microcracks and low population. These deviations can be avoided by adding a second term including a Leibnitz series, but it will be neglected here. The function F ðnÞ is then given by F¼

q 1q

ð89Þ

The parameter q depends on stress, temperature, and the constitution of the material, but not on crack length. Therefore, approximately no influence of the crack length on the growth rate arises from this term. The function G, which is the relative rate of crack initiation, depends on the material and the creep conditions. In order to determine this func-

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

tion, several creep tests are to be carried out until different stages of damage in tertiary creep stage are reached. Using the results of metallographic investigations and digital image analyzing systems, the function Z(t) is found to be described adequately by the Kachanov–Rabotnov-relation given in eq. (60a), as well as by the empirical relation. ðZ=Zf Þ ¼ exp½gðtf  tÞ=tf 

ð90Þ

where the index f indicates the value at fracture (Fig. 44). According to this relation g G¼ ð91Þ tf with g depending on material constitution, stress, and temperature. The quantity G remains constant during the creep test, as long as Eq. (90) is valid. The function H is determined by the rate of change of the statistical distribution (Fig. 45) and can be written as HðtÞ ¼ nq_ =ð1  qÞ, where q(t) is described by a power law according to q ¼ qf ðt=tf Þm . Hence, H can be written as nm q ð92Þ H¼ t 1þq

Figure 44 Number of cracks Z per unit area, related to its value at fracture as a function of the life fraction.

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Figure 45 Decrease of the fraction of short microcracks (n ¼ 1) with increasing creep exposure time.

The rate of growth reads

  dn g qf ðt=tf Þm m ¼ 1þ n dt tf 1  qf ðt=tf Þm gt=tf

Just before fracture (t  tf ), the creep crack growth is   dn g qf m ¼ 1þ n dt tf 1  qf g

ð93Þ

ð94Þ

The model indicates that the ratio between the growth rates of long and short cracks is smaller than that expected by applying the fracture mechanics concept. 2. FEM Simulation The grain boundary can be considered as a thin amorphous layer surrounding the grains. Its thickness equals few atomic distances. At low temperatures, the grain boundary is harder than the crystalline grains. With increasing temperature, the grain boundary becomes more softer than the

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 46

Idealization of the grain=grain boundary combination.

Figure 47

(a) Regular and (b) randomly modified idealization.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

grains. At higher temperatures, the grain boundary behaves as a viscous layer with much higher strain rate sensitivity than the grains. In the FEM analysis, two different material elements are used for the idealization of grains and grain boundaries with different material parameters (98), as shown in Fig. 46. The creep behavior of the grain interior and the grain boundary layers is described by the Norton–Bailey creep law e_ ¼ Cðs=s ÞN

ð95Þ

with s just equal to the stress unit. The parameters C and N are set approximately equal to the values determined for the entire in the secondary creep stage, neglecting the influence of the grain boundaries in this stage due to their small volume fraction. The grain boundary zone can be considered as a linear viscous Newton solid. Its stress exponent is set equal to unity as first approximation. A suitable thickness and the parameter C of the grain boundary layers are determined iteratively. Their values are varied till the fracture time computed for different creep stresses coincides with the experimentally determined values. In order to avoid all grain boundaries having the same orientation fracture simultaneously, the size of each individual element in the network is stochastically changed by adding random values to the grain node coordinates (Fig. 47). The network determined in this manner has to be considered as a quarter of the idealized body and to be symmetrically mirrored, so that no additional anisotropy is induced. The whole network can also be rotated by an angle between 0 and 608 to exclude preferred orientations for crack initiation. Two different crack initiation criteria are tried out: a strain criterion and a stress one. According to the strain criterion, a crack initiates as soon as the equivalent creep strain reaches a critical value. In this case, the grain boundary element is not totally eliminated but its thickness is reduced by a factor of 1=1000. Such a weakened element behaves during further deformation like a crack. The second criterion which is based on the maximum principle stress or the maximum shear stress instead of the equivalent strain is found to be non-applicable because the experimentally determined stress-life function could not be achieved with this criterion. With increasing extension of the whole mesh under constant load forces, the crack opening criterion is fulfilled first at a single grain boundary facet. The next crack opens at a different grain far from the first crack, but at a place where the orientation of the grain boundary facet is favorable. After the initiation of several individual cracks having a length of one grain

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 48 Comparison between experimental data and computational results for the increase of the number of cracks with increasing creep strain.

boundary facet, the total creep extension of the mesh is high enough to induce crack growth along the neighboring facets which are steeply inclined to the load direction. In this way, cracks of length class n ¼ 2 initiate at different locations. With further growth, the individual cracks start to coalescence resulting in a great additional extension of the mesh. Fracture is considered to take place as soon as the total creep extension of the mesh reaches a predefined value, and the computation is stopped. Figure 48 presents the ratio of the number of cracks Z to that of cracks at fracture Zf as a function of relative strain e=ef determined by the finiteelement simulation and by the creep experiment. The comparison shows that most of the data from the finite-element simulation lie in the same scatter band as those of the experimental investigation. Figure 49 shows that the fraction X1 of short cracks having a length of one grain boundary facet slightly increases with increasing nominal stresses as determined in experiments and by the finite-element simulation. With these results, the main reason for crack initiation and growth seems to be the relatively high local strains, and not the local stress, in the neighborhood of the grain boundaries. Metallographic investigation confirms the existence of such deformations in the neighborhood of the

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 49 Influence of creep stress on the fraction of short microcracks at fracture.

grain boundaries. The same procedure can be applied to estimate creep damage and fracture time of the technical geometries. An example is represented for the simulation of the creep behavior of a notched specimen [99]. Figure 50 shows the deformed mesh as well as a metallographic section.

IV.

BEHAVIOR UNDER IMPACT LOADING

During plastic deformation at high strain rates, such as in the case of forging or automobile crash conditions, the mechanical behavior of metallic materials is influenced by mass inertia forces, an increased strain rate sensitivity, and the adiabatic character of the deformation process. An accurate simulation of the material behavior needs an adequate consideration of the mechanical wave propagation, steeping and reflection phenomena [100]. Under impact loading, stress waves propagate through out the material and the strain distribution is time dependent and highly non-uniform. Figure 51 shows a long rod with a free front surface at x ¼ 0. The far end of the rod is fixed. The cross-sectional area is constant over the whole length and is denoted as A. If a force F is applied quasi-statically on the free

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 50

Creep microcrack initiation and growth in a notched specimen.

front surface, one can assume that the stress F=A induced is uniformly distributed over the whole rod. On the other hand, if the rod is impacted, e.g. by a hammer at the front surface, the mass inertia forces cannot be neglected. The rod front is pushed forward by a velocity v. An arbitrary cross-section at a distance x from the free end dose not start immediately to move with the same velocity, before all masses between the front surface and the cross-section considered have been accelerated to the velocity v. This

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Figure 51

Material element in an impacted bar.

needs a certain time interval Dt. The longer the distance x, the longer the time interval. This explains why displacements, strains, and stresses propagate throughout the material in the form of mechanical waves with the characteristic wave properties, such as reflection at surfaces. At an arbitrary time point, the cross-section at the distance x is displaced by u. At a neighboring cross-section x þ dx, the displacement is u þ ð@u=@xÞdx. The strain in the material element dx is given by e ¼ @u=@x. The forces acting on the element are As and A½s þ ð@s=@xÞ dx. The mass inertia force is rA dxð@ 2 u=@x2 Þ. Therefore, Að@s=@xÞ dx ¼ rA dxð@ 2 u=@x2 Þ

ð96Þ

In the case of elastic behavior, @s @s @2u ¼E ¼E 2 @x @x @x

ð97Þ

and the following differential equation is obtained for the local displacement: @2u E @2u ¼ @t2 r @x2 Any function fðx  ctÞ or fðx þ ctÞ fulfills this condition, when sffiffiffiffi E c¼ r

ð98Þ

ð99Þ

A certain value of the displacement u ¼ fðx0  ct0 Þ that is observed at the distance x0 at the time point t0 arises at the distance x0 þ Dx after Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

the time interval Dt, yielding fðx  ctÞ ¼ f½x þ Dx  cðt þ DtÞ and hence, Dx ¼ cDt. Therefore, c is the propagation velocity of the longitudinal wave. If the load is applied in the lateral direction or if the load is a torsion moment, a transversal wave is induced that propagates with a velocity of pffiffiffiffiffiffiffiffiffi cT ¼ G=r, where G is the shear modulus. When plastic deformation takes place, the modulus of elasticity E and the shear modulus G are to be replaced by the tangent modules sffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi @s=@e @t=@g c¼ ; cT ¼ ð100Þ r r While these equations are essential for analytical modeling, they have not to be necessarily externally considered in the numerical simulation when adequate computation codes are used. These codes must account for the mass of the material, for example, by considering point masses lumped at the nodes of the finite elements. Beside the FEM, the finite difference method and the method of characteristics are often applied. A.

Non-uniformity of Strain Distribution

If a tensile specimen is chosen too long or the impact energy input is relatively low, the local strain at the impacted specimen end is found experimentally to be much lower than that measured at the far end of the specimen. Such phenomena can be explained by an FE-simulation using a code for transient dynamic problems. The loading time function and the idealization of the impact tensile test arrangement are shown in Fig. 52. The material is considered as strain hardening and strain rate sensitive. Immediately after loading the specimen, an elastic and a plastic wave propagate along the axial direction of the specimen. The elastic wave is much faster than the plastic one. An elastic deformation propagates along the specimen to the far specimen head, where the elastic wave reflects. It runs back towards the near specimen head, where it reflects again. This

Figure 52

Input load time function and idealization of impact tension test.

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Figure 53 Variation of the distribution of the plastic strain in a tensile specimen at different time point after dynamic loading.

process is repeated many times during the propagation of the plastic wave, representing an elastic vibration superimposed plastic deformation process. The plastic wave propagates first throughout the specimen and is then reflected from the far specimen head. Due to superposition of the advancing and the reflected wave, high stresses and strains are induced at the far end of the specimen. If the impact energy is completely consumed by the plastic deformation, a permanent non-uniform strain distribution remains in the specimen (Fig. 53). With increasing impact energy, the plastic wave can run several times along the specimen, reflecting at both ends, before the impact energy W is completely consumed by the plastic deformation of the material. In this case, the strain distribution is approximately uniform over the whole gauge length (Fig. 54). B.

Fiber Composites Under Dynamic Compression

Under quasi-static compressive loading of a composite material, the slim fibers buckle within the softer matrix leading to a global plastic bending of the work piece. In order to simulate this behavior, an imperfection is

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Figure 54 Computational and experimental results for the local strain distribution along an impact–tension specimen of steel X6CrNi18-11, tested by different values W of impact energy.

introduced in form of a small inclination of 28 of the specimen axis, which showed that the specimens must buckle in the form observed experimentally (Fig. 55). Under dynamic loading, the specimens are found to get the usual barrel form of compression specimens. However, etched sections show that the fibers have undergone a buckling process. When the mass inertia forces are taken into consideration by regarding the material mass to be lumped at the nodes, the material behavior can first be understood [101,102]. A specimen buckling needs a lateral motion of the upper and lower contact surfaces in opposite directions. This motion is now obstacled by the inertia forces in the radial direction. The fibers buckle without causing a global bending of the specimen (Fig. 56). C.

Dynamic Notch Sensitivity

With increasing deformation rate, the strain rate sensitivity increases. This acts stabilizing on a tensile deformation process. As soon as neck formation starts, the local strain rate in the neck zone increases rapidly. The local flow

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Figure 55 Fiber buckling under quasi-static loading of copper reinforced by 45% volume fraction of austenitic steel fibers with 0.2 mm diameter. (From Ref. 101.) Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 56 Fiber buckling in a composite material under dynamic loading. (From Ref. 101.)

stress increases as well, so that higher tensile forces are needed for the continuation of extension. Other specimen regions undergo additional deformation, so that the uniform elongation increases with increasing strain rate sensitivity and extension rate. On the other hand, the adiabatic character

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 57

Idealization of perforated plates.

of the deformation process reduces the flow stress and promotes instability. Mass inertia in the lateral direction arises in connection with radial acceleration due to the reduction of area. This causes the initiation of either lateral tensile or lateral compressive stresses depending on the time function of specimen elongation. In addition to these ductility considerations, an increased notch sensitivity is observed under dynamic loading. One of the reasons is that the local fracture strain decreases with increasing strain rate. This will be discussed later on in this chapter. The other reason lies in the interaction between

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 58 Stress distribution around voids at different time points after impact loading: (a) t¼10 ms, smax¼598 MPa, (b) t¼18 ms, smax¼647 MPa, and (c) t¼24 ms, smax¼661 MPa.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

mechanical waves and notches. Figure 57 shows the idealized part of a perforated plate used in a study of the wave notch interaction [102,103]. The holes are chosen as circular or elliptical with different axes ratio and orientation. Also, the distance between the holes is variable. The variation of the stress distribution with increasing time, numerically computed, shows the propagation of the mechanical wave through the material (Fig. 58). Stress peaks are observed at the notch roots, before the maximum loading stress reaches this points. High stress values remain at the peaks, even when the maximum lading stress has passed through. Compared with the notch effect under quasi-static loading, the dynamic notch effect is characterized by higher stress and strain concentrations, greater strain gradients, lower stress relief by neighboring voids and lower influence of the orientation in the case of elliptical voids. V.

DUCTILE FRACTURE

The ductile fracture results usually from nucleation, growth, and coalescence of microvoids, that initiate mostly around inclusions. In accordance to its appearance of the fracture surface, ductile fracture can be classified into two cases [105]. In the case of softer materials, void nucleation at inclusions followed by marked void growth with internal necking and shear fracture of the intervoid matrix. The fracture surface shows a structured configuration of dimples often orientated perpendicular to the loading direction (Fig. 59). In case of high strength materials, shear fracture takes place without distinctive void growth. The matrix fails due to instabilities like shear bands forming between voids resulting in fracture with nearly no necking, promoted by low strain hardening material, high stress multiaxiality, and regions of high porosity [106]. A.

Failure Criteria

Beside macromechanical empirical failure criteria [107,108], several mesoscopic mechanical models are introduced. The failure criterion is defined by the local failure strain
ef ðsm =
sÞ as a function of the ratio of the local
. mean stress sm to the equivalent stress s For the nucleation of microvoids, different models have been deduced considering an energy criterion [109–111], critical stress [112–114], or critical strain [115–119]. Rice and Tracy [120] deduced a closed-form solution for the rateof-change of the mean radius of a void, in an ideal plastic material, as a function of the current value of the radius and of the ratio between the mean stress and the effective stress

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Figure 59 SEM micrograph of fracture surface of highly over aged Aluminum AA7075 after tensile impact loading at room temperature. (From Ref. 104.) Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.


¼ const:; s

1 dR ¼ 0:28 expð3sm =2
sÞ R de

ð101Þ

Hancock and Mackenzie [121] showed that the failure strain is assumed to be inversely proportional to the relative cavity growth rate (d ln R=de). The strain at fracture can be deduced from the Rice and Tracy criterion and be expressed as sÞ ef ¼ en þ a exp½3sm =ð2


ð102Þ

where en is the effective strain before void nucleation. The Rice and Tracey model has been used, e.g., in Ref. [122] and was verified by Thomason [123–125] in numerical simulations. Experimental results of Marini et al. [126] showed that the factor 0.28 of Eq. (101) should be replaced by higher values according to the volume fraction of inclusions. In Ref. [121], the local plastic strain which leads to coalescence of cavities was found to be highly influenced by the volume fraction of inclusions fN. Using special treatments for ferritic steels, different residual sulfur-concentrations were realized by Holland et al. [127] which were found to affect the fracture strain (Fig. 60a). These results were described by the modified relation s ef ¼ en þ a exp½bsm =


ð103Þ

where instead of the factor 3=2, a parameter b is introduced with values ranging between 5 and 23. The degree of purity had a drastic influence on en , which was affirmed by the investigation of further materials and treatments (Fig. 60b). Based on the models of McClintock [128] and of Rice and Tracey, Gurson [112] deduced a yield function for materials with randomly distributed voids of a volume fraction f. In this model, the flow rule according to Mises is extended by two additional terms including the porosity f. In more detailed investigations carried out by Tvergaard and Needleman [129–131], the Gurson model is modified yielding a plastic potential in the form   h i 3 q2 skk   1 þ ðq1 f  Þ2 ¼ 0 f ¼ 2 Sij Sij þ 2q1 f cosh 2sY 2sY

ð104Þ

In this equation, Sij is the stress deviator given by Sij ¼ sij  dij skk =3 where dij is the second order unit tensor. sY is the yield stress of the matrix and skk is the sum of the normal stress components. f  is a function of the volume fraction f of the voids according to f fc ;

f ¼f

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ð105aÞ

Figure 60 Influence of sulfur content in steel on: (a) the local effective strain at fracture as a function of ratio of mean stress to flow stress, and (b) the fracture strain for high multiaxiality as a function of the true strain in the neck zone of unnotched specimens, (Z ¼ reduction of area at fracture). (From Ref. 107.) Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

f > fc ;

 1 f  fc f ¼ fc þ  fc q1 fF  fc 



ð105bÞ

where fc is the volume fraction at the beginning of void coalescence and fF is the volume fraction at fracture. The rate-of-change f_ of the void volume fraction, is considered as the sum of three different contributions: (1) the growth rate of existing voids, which is proportional to (1f ) and to the local strain rate, (2) the nucleation rate of new voids depending on the effective strain rate
e_ in the matrix, and (3) the nucleation rate of new voids which is proportional to the rate of change of the mean stress sm ¼ dij skk =3. When the third contribution is neglected, the following relation is used for the evolution of fv :
pl f_ ¼ f_growth þ f_nucleation ¼ ð1  f Þdij e_ pl ij þ Ae_

ð106aÞ

A non-zero value of A is only used if
epl exceeds its current maximum in the time increment considered. In this case
  fN 1
epl  eN ð106bÞ A ¼ pffiffiffiffiffiffi exp  sN 2 2psN where fN is volume fraction of particles that may nucleate voids, eN is the mean value strain for nucleation, and sN is the corresponding standard deviation. B.

Influence of Strain Hardening and Strain Rate Sensitivity

In an early study on the growth of cavities by plastic deformation of the surrounding material, McClintock [128] deduced a closed-form analytical solution for the rate-of-growth of cylindrical cavities of elliptical cross-section with the semi-axes a and b in a strain-hardening material which is "pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # pffiffiffi 3ð1  nÞ sa þ sb 1 dR 3 n ¼ sinh s ¼ Ce ; ð107Þ
2 s R de 2ð1  nÞ Where R¼(aþb)=2 is the mean cross-sectional radius and sa and sb are the normal stresses in the direction of the ellipse axes. Because of its simplicity, the Hancock–Mackenzie relation is also applied to the range of high strain rates after introducing correction factors considering the influences of strain rate. Carroll and Holt [132] introduced a visco-plastic modification of the Hancock–Mackenzie model. Johnson and Cook [133] considered the strain rate sensitivity as well as the influence of the temperature on the local fracture strain ef ¼ ½D1 þ D2 expðD3 sm =
sÞ½1 þ D4 lnð_e=_e0 ½1 þ D5 T=Tm 

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

ð108Þ

with e_ 0 ¼ 1 sec1 and Tm the absolute melting point. As sm =
s; e_ and T change during deformation, it is assumed that fracture takes place when a damage parameter D ¼ S½De=ef ðsm ; e_ ; TÞ reaches the value of 1. A failure criterion for void growth considering non-linear visco-plastic behavior of a strain-hardening and rate-sensitive material can be obtained using an analytical solution [134]. The void growth is to be determined by
¼ K
en ð
e_ =_e Þm , with equivalent stress means of flow stress described by s _
, equivalent strain rate
e, reference strain rate e_  ¼ 1 sec1 , equivalent plass tic strain
e and the material constants K, n and m. A spherical void of radius R is considered to exist at the center of a metallic sphere (Fig. 61). At the outer radius L of this hollow sphere, a radial stress component srL is acting which is set equal to the mean stress sm ¼ ðs1 þ s2 þ s3 Þ=3, which leads to a visco-plastic deformation of the material and hence to an increase in void volume. For an arbitrary void radius r, the tangential strain rate is given by e_ t ¼ r_ =r. Under consideration of the plastic volume constancy,

Figure 61

Spherical void growing in a hollow sphere matrix.
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