Modeling and Simulation for Material Selection and Mechanical Design Part 1 ppsx

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 NEW YORK 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 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved ENGINEERING A Series of Textbooks and Reference Books Founding Editor L L Faulkner Columbus Division, Battelle Memorial Institute and Department of Mechanical Engineering The Ohio State University Columbus, Ohio 1 2 3 4 5 Spring Designer's Handbook, Harold Carlson Computer-Aided Graphics and Design, Daniel L Ryan Lubrication Fundamentals, J George Wills Solar Engineering for Domestic Buildings,William A Himmelman Applied Engineering Mechanics: Statics and Dynamics, G Boothroyd and C Poli 6 Centrifugal Pump Clinic, lgor J Karassik 7 Computer-AidedKinetics for Machine Design, Daniel L Ryan 8 Plastics Products Design Handbook, Patf A: Materials and Components; Patf 6 : Processes and Design for Processes, edited by Edward Miller 9 Turbomachinery:Basic Theory and Applications, Earl Logan, Jr 10 Vibrations of Shells and 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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 3 Design Dimensioning with Computer Graphics Applications, Jerome C 1 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 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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 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 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 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157 Gear Noise and Vibration: Second Edition, Revised and Expanded, J Derek Smith Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 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 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved In Memoriam During the preparation of this book, one of our most valued authors and mentors passed away on April 29, 2003 Professor George C Weatherly (1941–2003) graduated from Cambridge University in 1966 He began his career as a research scientist in the Department of Metallurgy at Harwell In 1968 he moved to Canada where he worked for the University of Toronto for 22 years as a professor in the Department of Metallurgy and Material Science In 1990 he became a professor of Materials Science and Engineering at McMaster University He was Director of Brockhouse Institute for Material Research from 1996–2001 and a Chair of the Department of Materials Science and Engineering Dr Weatherly has published over 200 papers in different areas of Materials Science He was Fellow for the Canadian Institute for Mining and Metallurgy and Fellow of ASM International George was a devoted scientist in the field of electron microscopy and an educator with a distinguished career at McMaster University and the University of Toronto He will be cherished by his friends, colleagues, and students for the richness of his life, his quiet humor, his humanity and care for others, and above all for his unfailing honesty His contributions were many and are written clearly in the lives of those with whom he taught and worked This book is dedicated to his memory Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Preface In every industry survey, development and use modeling, and simulation technology are cited among the top five critical needs for manufacturing industries to remain viable and competitive in the future This is particularly true for materials and component design To address this need, various research programs are currently underway in government, academic, and industry laboratories around the world This book addresses a number of selected, important areas of computer model development Effective material and component design procedures are vitally important with increasing pressures to improve quality at lower production costs for all traditional industrial markets Advanced design procedures typically involve computer modeling and simulation if the necessary algorithms are sufficiently advanced or by using advanced empirical procedures The objective is to be able to make design decisions based on numerical simulations as an alternative to time-consuming and expensive laboratory or production experimental process development In fact, advanced engineering processes are becoming increasingly dependent on advanced computer modeling and design procedures This book addresses various aspects of the utilization of modeling and simulation technology Some of the topics discussed include hot-rolling of steel, quenching and tempering during heat treatment, modeling of residual stresses and distortion during forging, casting, heat treatment, mechanical property prediction, modeling of tribological processes as it relates to the design of surface engineered materials, and fastener design These chapters summarize and demonstrate key numerical relationships used in computer Copyright 2004 by Marcel Dekker, Inc All Rights Reserved model development and their application at various stages in the material production process In particular, the material covered in this text includes:       Modeling and simulation of microstructural evolution and mechanical properties of steels during the hot-rolling process, calculation of metallurgical phenomena occurring in steel during hot-rolling, and prediction of mechanical properties from microstructure Heat treatment processes such as quenching and tempering is an active area for process model development Models used to simulate the kinetics of multicomponent grain boundary segregations that occur in quenched and tempered engineering steels are discussed These models permit the evaluation of the effect of alloying elements and various tempering parameters on hydrogen embrittlement, stress-corrosion cracking, and other phenomenon Of all the various problems associated with component design and production, none are more important that residual stress and distortion Chapter 3 discusses the metallo-thermo-mechanical theory, numerical modeling and simulation technology, coupling of temperature, inelastic behavior and phase transformation and solidification involved with elastic-plastic, viscoplastic and creep deformation as they relate to quenching, forging, and casting processes Modeling and simulation of mechanical properties, in particular, material behavior during plastic deformation, low-cycle fatigue, creep, and impact strength This discussion includes the importance of the determination and implementation of adequate material data, consideration of inelastic material behavior, and the formulation of physically founded material models Chapter 5 discusses the role played by physico-chemical interactions in modifying and controlling friction and wear of critically loaded tribo-couple surfaces during high-speed cutting operations A comprehensive overview of one of the most important processes in manufacturing is presented in Chapter 6 Threaded fastener selection and design is addressed with many equations and figures included to aid in the design process Chapters 1 through 4 describe advanced computer modeling and simulation processes to predict microstructures, material process behavior, and mechanical properties Chapters 5 and 6 describe more empirical process design procedures for tribological and fastener design Copyright 2004 by Marcel Dekker, Inc All Rights Reserved This book will be an invaluable resource for the designer, mechanical and materials engineer, and metallurgist Thorough overviews of these technologies seldom encountered in other handbooks for materials design are provided The book is an excellent textbook for advanced undergraduate or graduate engineering courses on the role of modeling and simulation in materials and component design We are indebted to the vital assistance of various international experts Special thanks to our spouses for their infinite patience with the various time-consuming tasks involved in putting this text together We extend special thanks to the staff at Marcel Dekker, Inc including Richard Johnson, Rita Lazazzaro, and Russell Dekker for their invaluable assistance Without their assistance, this text would not have been possible George E Totten Lin Xie Kiyoshi Funatani Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Contents Preface Contributors 1 A Mathematical Model for Predicting Microstructural Evolution and Mechanical Properties of Hot-Rolled Steels Masayoshi Suehiro 2 Design Simulation of Kinetics of Multicomponent Grain Boundary Segregations in the Engineering Steels Under Quenching and Tempering Anatoli Kovalev and Dmitry L Wainstein 3 Designing for Control of Residual Stress and Distortion Dong-Ying Ju 4 Modeling and Simulation of Mechanical Behavior Essam El-Magd 5 Tribology and the Design of Surface-Engineered Materials for Cutting Tool Applications German Fox-Rabinovich, George C Weatherly, and Anatoli Kovalev Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 6 Designing Fastening Systems Christoph Friedrich Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 1 A Mathematical Model for Predicting Microstructural Evolution and Mechanical Properties of Hot-Rolled Steels Masayoshi Suehiro Nippon Steel Corporation, Futtsu-City, Chiba, Japan I INTRODUCTION A model for calculating the mechanical properties of hot-rolled steel sheets from their processing condition makes it possible not only to design chemical compositions and processing conditions of steels through off-line simulation but also to guarantee the mechanical properties of hot-rolled steels through on-line simulation From this point of view, some attempts have been made to develop a mathematical model for calculating the evolution of austenitic microstructure of steels during hot-rolling process and their transformations during cooling subsequent to hot-rolling [1–3] The mathematical models basically consist of four models for calculating metallurgical phenomena occurring in hot-strip mill and a model for predicting mechanical properties from the microstructure of steel calculated by the metallurgical models In this chapter, the basic idea and several applications of the mathematical model will be presented Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 1 Schematic illustration of a hot-strip mill II THE OVERALL MODEL Since mechanical properties of hot-rolled steels are determined by their microstructure, a model for calculating the mechanical properties of hot-rolled steels is composed of two kinds of models: one for calculating microstructure of steels from their processing conditions, and the other for calculating their mechanical properties from their microstructure There are several kinds of hot-rolled steel products: sheet and coil, plate, beam, wire, rod, bar, etc Although the processing conditions are dependent upon each process, each product is produced through the processes such as heating, hot-working, and cooling Figure 1 shows the schematic illustration of a hot-strip mill Hot-rolled steel sheets are produced through slab reheating, rough hot-rolling, finish hot-rolling, cooling, and coiling Table 1 shows the typical thickness and temperature changes in this process and the metallurgical phenomena occurring through this process In the slab-reheating process, transformation from ferrite and pearlite to austenite and grain growth take place The Table 1 The Changes in Thickness and Temperature of Steels and Metallurgical Phenomena in a Hot-Strip Mill Thickness (mm) Temperature (8C) Slab reheating 250 1200 Rough rolling Finish rolling Cooling Coiling !40 1200–1000 !3 1000–850 3 3 — 600–700 Process Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 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 À expðÀktn 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 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 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 À expðÀXe 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 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 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 À tÞ2 À y2 Š1=2 r¼0 for y < Gðt À tÞ for y ! Gðt À 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 À tÞ2 À y2 Š for y < Gðt À tÞ for y > Gðt À 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 tÀy=G Z ½G2 ðt À tÞ2 À y2 Š dt dYe ¼ pIs 0 ð5Þ 0 By exchanging y=Gt for x, this equation leads to ! 3 2 3 1Àx 2 À x ð1 À 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 À expðÀYe 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 1Àx 2 V0 ¼ 2 Y dy ¼ 2Gt À x ð1 À xÞ 1 À exp ÀpIs G t dx 3 0 0 ð8Þ Multiplying V0 by the area of nucleation site, the extended volume fraction is obtained as À1=3 Xe ¼ SV0 ¼ bs fs ðas Þ Copyright 2004 by Marcel Dekker, Inc All Rights Reserved ð9Þ where Is Ns ¼ 8S3 G 8S4 G &  '! 3 3 1Àx 2 1 À exp Àpas dx À x ð1 À xÞ 3 as ¼ ðIs G2 Þ1=3 t; 1 Z fs ðas Þ ¼ as bs ¼ ð10Þ 0 and Ns the nucleation rate for unit volume The actual volume fraction X is expressed as À1=3 X ¼ 1 À expðÀbs 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 À expðÀp=3Ns G3 t4 Þ as 51 ð12Þ X ¼ 1 À ðÀ2SGtÞ 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 À expðÀp=3N G 3 t4 Þ X ¼ 1 À expðÀ2SGtÞ X ¼ 1 À expðÀpLG2 t2 Þ X ¼ 1 À expðÀð4p=3ÞCG 3 t3 Þ Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 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 ma ¼ mg i i 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 À g Á ð15Þ xa mi À ma DGm ¼ i i 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 À Cg D 2r Cg À Ca Copyright 2004 by Marcel Dekker, Inc All Rights Reserved ð16Þ 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 À 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 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 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 Dg Cyg À Cga ¼ c dg Cga À Ca dt ð19Þ where t is the time, Dg the diffusion constant of C in austenite, and Cg, Cga c 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 d2 À d2 ¼ k 2 t g g0 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 À 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 À r3 ¼ k3 t 0 ð23Þ where k3 is the factor related to temperature, interfacial energy and the diffusion coefficient of an alloying element controlling the Ostwald ripening of Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 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 À ec 2 ð24Þ Xdyn ¼ 1 À exp À0:693 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 À t0 2 Xdyn ¼ 1 À exp À0:693 ð25Þ t0:5 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved calculated from the average dislocation density which is obtained by calculating the changes in the dislocation density in the region dynamically recovered, rn, and in the region recrystallized dynamically, rs, according to time independently This method makes it possible to calculate the changes in grain size and dislocation density Table 3 shows the summary of equations used in the model developed by Senuma et al The numbers of phenomena in Table 3 correspond to those in Fig 5 Figure 6 shows an example of calculation of the changes in grain size and dislocation density [14] Figure 7 shows the calculation result of the effect of the initial austenite grain size on the final microstructure in the finishing stands of a hot-strip mill, which shows that the initial austenite grain size does not affect very much the final grain size This model can be applied to the prediction of the resistance to hot deformation as well and it can contribute to the improvement of the accuracy in thickness In this method, the average values concerning the grain size and the accumulated dislocation density are used taking the fraction recrystallized into consideration This averaging can be applied to the hot-strip mill because the total thickness reduction is large enough to recrystallize their microstructure In the case of plate rolling, the use of the average values is unsuitable because the reduction at each pass is small and the total thickness reduction is not enough to recrystallize the microstructure of steels The model applicable to this case has been developed by dividing the microstructure into several groups [26] This type of modeling was carried out for Nb-bearing steels [19,21,25], Ti- and V-bearing steels [21], Ti- and Nb-bearing steels [22], Ti-, Nb-, and Vbearing steels [27] as well as C–Mn steels In these steels, the recovery and recrystallization are retarded by alloying elements This retardation might be caused by the pinning effect due to fine precipitates or by the solute-drag effect This effect can be considered by modifying the values of fitting parameters from experimental data D Transformation Model 1 Basic Idea of the Modeling In the cooling process subsequent to hot-rolling, steels transform from austenite phase to ferrite, pearlite, bainite, and=or martensite phases Transformation model predicts the microstructural change during cooling and the final microstructure of steels after cooling The modeling of transformation kinetics can be performed by obtaining the parameters k and n in Avrami equation [29–31], formulating new equations corresponding to transformation kinetics obtained experimentally [32], and adopting the nucleation and growth theory [33–36] Copyright 2004 by Marcel Dekker, Inc All Rights Reserved  ... Totten, Lin Xie, and Kiyoshi Funatani 16 5 Structural Analysis of Polymeric Composite Materials, Mark E Tuttle 16 6 Modeling and Simulation for Material Selection and Mechanical Design, George E... Meadows 11 3 Handbook of Materials Selection for Engineering Applications, edited by G T Murray 11 4 Handbook of Thermoplastic Piping System Design, Thomas Sixsmith and Reinhard Hanselka 11 5 Practical... York 12 7 01, U.S.A tel: 800-228 -11 60; fax: 845-796 -17 72 Eastern Hemisphere Distribution Marcel Dekker AG, Hutgasse 4, Postfach 812 , CH-40 01 Basel, Switzerland tel: 41- 61- 260-6300; fax: 41- 61- 260-6333