Modeling and Simulation for Material Selection and Mechanical Design

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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

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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

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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

26.


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


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


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


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


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.


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


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.


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


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


6

Designing Fastening Systems Christoph Friedrich


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


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.


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


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


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.


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


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


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


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.


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


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


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


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


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].


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


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


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


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


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


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


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


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.


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


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.


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


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


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.


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.


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.


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.


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.


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


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


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


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


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


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.


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


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


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


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.


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.


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.


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.


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,


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.


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


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.


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


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


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


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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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.



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.


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 ( 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


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


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


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.


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.


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


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


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.


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-


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-


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


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


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


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.


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


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.


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.


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


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


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


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.


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


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


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


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


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


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


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


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


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


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.


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.


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.


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.


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.


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 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

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