Handbook ofThermal Process Modeling of SteelsGur/Handbook of Thermal Process Modeling of Steels 190X_C000 Final Proof page i 6.11.2008 6:02pm Compositor Name: VBalamugundanGur/Handbook of Thermal Process Modeling of Steels 190X_C000 Final Proof page ii 6.11.2008 6:02pm Compositor Name: VBalamugundanHandbook ofThermalProcess Modelingof SteelsEdited byCemil Hakan GrJiansheng PanGur/Handbook of Thermal Process Modeling of Steels 190X_C000 Final Proof page iii 6.11.2008 6:02pm Compositor Name: VBalamugundanCRC PressTaylor & Francis Group6000 Broken Sound Parkway NW, Suite 300Boca Raton, FL 33487-2742 2009 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa businessNo claim to original U.S. Government worksPrinted in the United States of America on acid-free paper10 9 8 7 6 5 4 3 2 1International Standard Book Number-13: 978-0-8493-5019-1 (Hardcover)Thisbookcontainsinformationobtainedfromauthenticandhighlyregardedsources.Reasonableeffortshavebeen made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid-ity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint.Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or uti-lized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopy-ing, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers.Forpermissiontophotocopyorusematerialelectronicallyfromthiswork,pleaseaccesswww.copyright.com(http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For orga-nizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.TrademarkNotice:Productorcorporatenamesmaybetrademarksorregisteredtrademarks,andareusedonlyfor identification and explanation without intent to infringe.Visit the Taylor & Francis Web site athttp://www.taylorandfrancis.comand the CRC Press Web site athttp://www.crcpress.comGur/Handbook of Thermal Process Modeling of Steels 190X_C000 Final Proof page iv 6.11.2008 6:02pm Compositor Name: VBalamugundanContentsPreface............................................................................................................................................. viiEditors .............................................................................................................................................. ixContributors ..................................................................................................................................... xiChapter 1 Mathematical Fundamentals of Thermal Process Modeling of Steels...................... 1Jiansheng Pan and Jianfeng GuChapter 2 Thermodynamics of Thermal Processing................................................................ 63Sivaraman GuruswamyChapter 3 Physical Metallurgy of Thermal Processing ........................................................... 89Wei ShiChapter 4 Mechanical Metallurgy of Thermal Processing .................................................... 121Boo SmoljanChapter 5 Modeling Approaches and Fundamental Considerations ..................................... 185Bernardo Hernandez-MoralesChapter 6 Modeling of Hot and Warm Working of Steels ................................................... 225Peter Hodgson, John J. Jonas, and Chris H.J. DaviesChapter 7 Modeling of Casting.............................................................................................. 265Mario RossoChapter 8 Modeling of Industrial Heat Treatment Operations.............................................. 313Satyam Suraj SahayChapter 9 Simulation of Quenching ...................................................................................... 341Caner S ims ir and C. Hakan GrChapter 10 Modeling of Induction Hardening Processes ........................................................ 427Valentin NemkovGur/Handbook of Thermal Process Modeling of Steels 190X_C000 Final Proof page v 6.11.2008 6:02pm Compositor Name: VBalamugundanvChapter 11 Modeling of Laser Surface Hardening.................................................................. 499Janez GrumChapter 12 Modeling of Case Hardening................................................................................ 627Gustavo Snchez Sarmiento and Mara Victoria BongiovanniChapter 13 Industrial Applications of Computer Simulation of HeatTreatment and Chemical Heat Treatment ............................................................. 673Jiansheng Pan, Jianfeng Gu, and Weimin ZhangChapter 14 Prospects of Thermal Process Modeling of Steels................................................ 703Jiansheng Pan and Jianfeng GuIndex............................................................................................................................................. 727Gur/Handbook of Thermal Process Modeling of Steels 190X_C000 Final Proof page vi 6.11.2008 6:02pm Compositor Name: VBalamugundanviPrefaceThewholerange of steel thermal processingtechnology, fromcastingandplastic formingtowelding and heat treatment, not only produces workpieces of the required shape but also optimizestheend-product microstructure. Thermal processingthusplaysacentral roleinqualitycontrol,servicelife, andtheultimatereliabilityofengineeringcomponents,andnowrepresentsafunda-mental element of any companys competitive capability.Substantial advances in research, toward increasingly accurate prediction of the microstructureandpropertiesofworkpiecesproducedbythermalprocessing,werebasedonsolutionsofpartialdifferential equations (PDEs) for temperature, concentration, electromagnetic properties, and stressand strain phenomena. Until the widespread use of high-performance computers, analytical solutionof PDEs was the only approach to describe these parameters, and this placed severe limitations interms of prediction for engineering applications so that thermal process developments themselvesreliedonempiricismandtraditional practice. Thelevel ofinaccuracyinherent incomputationalpredictions hindered both materials performance improvements and process cost reduction.Since the 1970s, the pace of development of computer technology has made possible effectivesolutionof PDEs incomplicated calculations forboundary and initialconditions, as well asnon-linear andmultiple variables. Mathematical models andcomputer simulationtechnologyhavedeveloped rapidly; currently well-establishedmathematical models integratefundamentaltheoriesofmaterialsscienceandengineeringincludingheattransfer,thermoelastoplasticmechanics, uidmechanics, andchemistrytodescribephysical phenomenaoccurringduringthermal processing.Further, evolutionoftransient temperature, stressstrain, concentration, microstructure, andowcan now be vividly displayed through the latest visual technology, which can show the effects ofindividual process parameters. Computation=simulation thus provides an additional decision-making tool for both the process optimization and the design of plant and equipment; it acceleratesthermal processing technology development on a scientically sound computational basis.The basic mathematical models for thermal processing simulation gradually introduced to datehaveyieldedenormousadvantagesforsomeengineeringapplications.Continuedresearchinthisdirection attracts increasing attention now that the cutting-edge potential of future developments isevident.Increasingly profound investigationsarenowintrainglobally.The number ofimportantresearchpapersinthe eldhasrisensharplyover thelast threedecades. Evenso, theexistingmodelsareregarded ashighlysimpliedby comparison withrealcommercialthermalprocesses.Thishasmeant that theapplicationof computer simulationhasthusfar beenrelativelylimitedpreciselybecauseof thesesimplifyingassumptions, andtheir consequent limitedcomputationalaccuracy. Extensive and continuing research is still needed.This book is now offered as both a contribution to work on the limitations described above andas an encouragement to increase the understanding and use of thermal process models andsimulation techniques.The mainobjectives of this bookare, therefore, toprovide a useful resource for thermalprocessing of steels by drawing together.An approach to a fundamental understanding of thermal process modeling.A guide to process optimization.An aid to understand real-time process control.Some insights into the physical origin of some aspects of materials behavior.What is involved in predicting material response under real industrial conditions not easilyreproduced in the laboratoryGur/Handbook of Thermal Process Modeling of Steels 190X_C000 Final Proof page vii 6.11.2008 6:02pm Compositor Name: VBalamugundanviiLinked objectives are to provide.A summary of the current state of the art by introducing mathematical modeling method-ology actually used in thermal processing.Apractical reference (industrial examples and necessary precautionary measures areincluded)It is hoped that this book will.Increase the potential use of computer simulation by engineers and technicians engaged inthermal processing currently and in the future.Highlight problems requiring further research and be helpful in promoting thermal processresearch and applicationsThis project was realized due to the hard work of many people. We express our warm appreciationto the authors of the respective chapters for their diligence and contribution. The editors are trulyindebted toeveryonefortheir contribution,assistance,encouragement,andconstructivecriticismthroughout the preparation of this book.Here,wealsoextendoursinceregratitudetoDr.GeorgeE.Totten(TottenAssociatesandaformer president of the International Federationfor Heat Treatment andSurface Engineering[IFHTSE]) andRobert Wood(secretarygeneral, IFHTSE), whoseinitial encouragement madethisbookpossible, andtothestaff of CRCPressandTaylor &Francisfor their patienceandassistance throughout the production process.C. Hakan GrMiddle East Technical UniversityJiansheng PanShanghai, Jiao Tong UniversityGur/Handbook of Thermal Process Modeling of Steels 190X_C000 Final Proof page viii 6.11.2008 6:02pm Compositor Name: VBalamugundanviiiEditorsC. Hakan Gr is a professor in the Department of Metallurgicaland Materials Engineering at Middle East Technical University,Ankara, Turkey. HeisalsothedirectoroftheWeldingTech-nologyandNondestructiveTestingResearchandApplicationCenter at the same university. Professor Gr has publishednumerous papers on a wide range of topics in materials scienceandengineeringandservesontheeditorialboardsofnationaland international journals. His current research includes simula-tion of tempering and severe plastic deformation processes,nondestructiveevaluationofresidual stresses, andmicrostruc-tures obtained by various manufacturing processes.Jiansheng Pan is a professor in the School of Materials ScienceandEngineeringat Shanghai JiaoTongUniversity, Shanghai,China.HewasanelectedmemberoftheChineseAcademyofEngineeringin2001. Professor Pansexpertiseisinchemicalandthermal processingofsteels(includingnitriding, carburiz-ing, andquenching)andtheircomputermodelingandsimula-tion. He has established mathematical models of these processesintegratingheatandmasstransfer, continuummechanics, uidmechanics, numerical analysis, and software engineering. Thesemodelshavebeenusedforcomputationalsimulationtodesignand optimize thermal processes for parts with complicated shape.Pan and his coworkers have published extensively in these areasand have been awarded over 40 Chinese patents. In addition to anumber of awards for scientic and technological achievements,ProfessorPanwasthepresident oftheChineseHeat Treatment Society(20032007)andisthechairmanoftheMathematical ModelingandComputerSimulationActivityGroupoftheInter-national Federation for Heat Treatment and Surface Engineering.Gur/Handbook of Thermal Process Modeling of Steels 190X_C000 Final Proof page ix 6.11.2008 6:02pm Compositor Name: VBalamugundanixGur/Handbook of Thermal Process Modeling of Steels 190X_C000 Final Proof page x 6.11.2008 6:02pm Compositor Name: VBalamugundanContributorsMara Victoria BongiovanniFacultad de IngenieraUniversidad AustralBuenos Aires, ArgentinaandFacultad de Ciencias Exactas y NaturalesUniversidad de Buenos AiresBuenos Aires, ArgentinaChris H.J. DaviesDepartment of Materials EngineeringMonash UniversityMelbourne, Victoria, AustraliaJanez GrumFaculty of Mechanical EngineeringUniversity of LjubljanaLjubljana, SloveniaJianfeng GuSchool of Materials Science and EngineeringShanghai Jiao Tong UniversityShanghai, ChinaC. Hakan GrDepartment of Metallurgical and MaterialsEngineeringMiddle East Technical UniversityAnkara, TurkeySivaraman GuruswamyDepartment of Metallurgical EngineeringUniversity of UtahSalt Lake City, UtahBernardo Hernandez-MoralesDepartamento de Ingeniera MetalrgicaUniversidad Nacional Autnoma de MxicoMexicoPeter HodgsonCentre for Material and Fibre InnovationInstitute for Technology Research andInnovationDeakin UniversityGeelong, Victoria, AustraliaJohn J. JonasDepartment of Materials EngineeringMcGill UniversityMontreal, Quebec, CanadaValentin NemkovFluxtrol, Inc.Auburn Hills, MichiganandCentre for Induction TechnologyAuburn Hills, MichiganJiansheng PanSchool of Materials Science and EngineeringShanghai Jiao Tong UniversityShanghai, ChinaMario RossoR&D Materials and TechnologiesPolitecnico di TorinoDipartimento di Scienza dei Materiali eIngegneria ChimicaTorino, ItalyandPolitecnico di TorinoSede di AlessandriaAlessandria, ItalySatyam Suraj SahayTata Research Development and Design CentreTata Consultancy Services LimitedPune, Maharashtra, IndiaGur/Handbook of Thermal Process Modeling of Steels 190X_C000 Final Proof page xi 6.11.2008 6:02pm Compositor Name: VBalamugundanxiGustavo Snchez SarmientoFacultad de IngenieraUniversidad de Buenos AiresBuenos Aires, ArgentinaandFacultad de IngenieraUniversidad AustralBuenos Aires, ArgentinaWei ShiDepartment of Mechanical EngineeringTsinghua UniversityBeijing, ChinaCaner SimsirStiftung Institt fr Werkstofftechnik (IWT)Bremen, GermanyBoo SmoljanDepartment of Materials Science andEngineeringUniversity of RijekaRijeka, CroatiaWeimin ZhangSchool of Materials Scienceand EngineeringShanghai Jiao Tong UniversityShanghai, ChinaGur/Handbook of Thermal Process Modeling of Steels 190X_C000 Final Proof page xii 6.11.2008 6:02pm Compositor Name: VBalamugundanxii1Mathematical Fundamentalsof Thermal ProcessModeling of SteelsJiansheng Pan and Jianfeng GuCONTENTS1.1 Thermal Process PDEs and Their Solutions.......................................................................... 21.1.1 PDEs for Heat Conduction and Diffusion .................................................................. 21.1.2 Solving Methods for PDEs ......................................................................................... 51.2 Finite-Difference Method....................................................................................................... 61.2.1 Introduction of FDM Principle ................................................................................... 61.2.2 FDM for One-Dimensional Heat Conduction and Diffusion ..................................... 61.2.3 Brief Summary.......................................................................................................... 121.3 Finite-Element Method ........................................................................................................ 121.3.1 Brief Introduction...................................................................................................... 121.3.1.1 Stage 1: Preprocessing ................................................................................ 131.3.1.2 Stage 2: Solution ......................................................................................... 131.3.1.3 Stage 3: Postprocessing............................................................................... 131.3.2 Galerkin FEM for Two-Dimensional Unsteady Heat Conduction........................... 141.3.3 FEM for Three-Dimensional Unsteady Heat Conduction ........................................ 191.4 Calculation of Transformation Volume Fraction................................................................. 211.4.1 Interactions between Phase Transformation and Temperature ................................. 211.4.2 Diffusion Phase Transformation ............................................................................... 211.4.2.1 Modication of Additivity Rule for Incubation Period .............................. 231.4.2.2 Modication of Avrami Equation ............................................................... 251.4.2.3 Calculation of Proeutectoid Ferrite and Pearlite Fraction........................... 261.4.3 Martensitic Transformation....................................................................................... 281.4.4 Effect of Stress State on Phase Transformation Kinetics ......................................... 301.4.4.1 Diffusion Transformation............................................................................ 301.4.4.2 Martensitic Transformation ......................................................................... 301.5 Constitutive Equation of Solids ........................................................................................... 311.5.1 Elastic Constitutive Equation.................................................................................... 311.5.1.1 Linear Elastic Constitutive Equation........................................................... 311.5.1.2 Hyperelastic Constitutive Equation............................................................. 331.5.2 Elastoplastic Constitutive Equation .......................................................................... 361.5.2.1 Introduction ................................................................................................. 361.5.2.2 Yield Criterion............................................................................................. 361.5.2.3 Flow Rule .................................................................................................... 371.5.2.4 Hardening Law............................................................................................ 381.5.2.5 Commonly Used Plastic Constitutive Equations ........................................ 39Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 1 3.11.2008 3:17pm Compositor Name: BMani11.5.2.6 Elastoplastic Constitutive Equation............................................................. 441.5.2.7 Thermal Elastoplastic Constitutive Equation .............................................. 451.5.3 Viscoplastic Constitutive Equation........................................................................... 471.5.3.1 One-Dimensional Viscoplastic Model ........................................................ 471.5.3.2 Viscoplastic Constitutive Equation for General Stress State ...................... 491.5.3.3 Commonly Used Viscoplastic Models........................................................ 491.5.3.4 Creep ........................................................................................................... 501.6 Basics of Computational Fluid Dynamics in Thermal Processing...................................... 531.6.1 Introduction............................................................................................................... 531.6.2 Governing Differential Equations for Fluid.............................................................. 531.6.2.1 Generalized Newtons Law......................................................................... 531.6.2.2 Continuity Equation (Mass Conservation Equation) .................................. 541.6.2.3 Momentum Conservation Equation............................................................. 551.6.2.4 Energy Conservation Equation.................................................................... 551.6.3 General Form of Governing Equations..................................................................... 561.6.4 Simplied and Special Equations in Thermal Processing........................................ 561.6.4.1 Continuity Equation for Incompressible Source-Free Flow ....................... 571.6.4.2 Euler Equations for Ideal Flow................................................................... 571.6.4.3 Volume Function Equation ......................................................................... 581.6.5 Numerical Solution of Governing PDEs .................................................................. 58References ....................................................................................................................................... 59Steelsareusuallyundertheactionofmultiplephysicalvariableelds,suchastemperatureeld,uideld, electriceld, magneticeld, plasmeld, and so on during thermal processing. Thus, heatconduction, diffusion, phase transformation, evolution of microstructure, and mechanicaldeform-ation are simultaneously taken place inside. This chapter includes the mathematical fundamentals ofthe most widely used numerical analysis methods for the solution of partial differential equations(PDEs), and the basic knowledge of continuum mechanics, uid mechanics, phase transformationkinetics, etc. All these are indispensable for the establishment of the coupled mathematical modelsand realization of numerical simulation of thermal processing.1.1 THERMAL PROCESS PDEs AND THEIR SOLUTIONS1.1.1 PDESFOR HEAT CONDUCTIONAND DIFFUSIONTherst step of computer simulation of thermal processing is to establish an accurate mathematicalmodel, i.e., thePDEs andboundaryconditions that canquanticationallydescribetherelatedphenomena.The PDE describing the temperatureeld inside a solid is usually expressed as follows:@@xl@T@x @@yl@T@y @@zl@T@z Q rcp@T@t(1:1)whereT is the temperaturet is the timex, y, z are the coordinatesl is the thermal conduction coefcientr is the densitycp is the heat capacityQ is the intensity of the internal heat resourceGur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 2 3.11.2008 3:17pm Compositor Name: BMani2 Handbook of Thermal Process Modeling of SteelsEquation 1.1 has a very clear physical concept, and can be illustrated as in Figure 1.1. Therstitemontheleft-handsideoftheequationisthenet heat uxinput totheinnitesimallysmallelement alongaxisx, i.e., thedifferencebetweentheheat uxenteringdQxinandtheheat uxeffusingdQxout. The second and third items are the net heat ux along axesy andz, respectively(Figure 1.1). The intensity of the internal heat source Q may be caused by different factors, such asphasetransformation, plasticwork, electricitycurrent, etc. Theright-handsideof theequationstandsforthechangeinheat accumulatingintheinnitesimal element pertimeunit duetothetemperature change. Equation 1.1 shows that the sum of the heat input and heat generated by theinternal heat source is equal to the change in heat accumulating for an innitesimal element in eachtimeunit, soit functionsinaccordancewiththeenergyconservativelaw. Theheat conductioncoefcient l, density r, heat capacity cp, and the intensity of the internal heat source are usually thefunctions of temperature, making Equation 1.1 a nonlinear PDE.There are three kinds of boundary conditions for heat exchange in all kinds of thermalprocessing technologies.Therst boundary condition S1: The temperature of the boundary (usually certain surfaces) isknown; it is a constant or function of time.Ts C(t) (1:2)The second boundary condition S2: The heat ux of the boundary is known.l@T@n q (1:3)where@T=@n is the temperature gradient on the boundary along the external normal directionq is the heat ux through the boundary surfaceThethirdboundaryconditionS3: Theheat transfer coefcient betweentheworkpieceandenvironment is known.zdQz outdQz indQy outdQz indQy indQx outxyFIGURE 1.1 Heat ux along coordinates subjected to an innitesimal element.Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 3 3.11.2008 3:17pm Compositor Name: BManiMathematical Fundamentals 3l@T@n h(TaTs) (1:4)whereTa is the environment temperatureTs is the surface temperature of the workpiecehistheoverallheattransfercoefcient, representingtheheatquantityexchangedbetweenthe workpiece surface and the environment per unit area and unit time when their tempera-ture difference is 18CIt is worth mentioning that only convective heat transfer occurs in some cases; however, radiationheat transfer shouldalsobeconsideredinother complicatedones, suchasgasquenchingandheating under protective atmosphere. Hence, the overall heat transfer coefcient h should be the sumof the convective heat transfer coefcient hc and the radiation heat transfer coefcient hr. Therefore,we haveh hchr(1:5)The radiation heat transfer coefcient hr can be obtained as follows:hr s(T2a T2s )(TaTs) (1:6)where is the radiation emissivity of the workpieces is the StefanBoltzmann constantThe boundary condition can be set according to the specic thermal process, and the tempera-tureeld inside the workpiece at different times, the so-called unsteadytemperatureeld,can beobtained by solving Equation 1.1. When the temperatureeld inside the workpiece does not changewith time any more, it arrives at the steady temperatureeld, and the left-hand side of Equation 1.1becomes zero.Theunsteady concentrationeldinsidetheworkpiecesubjectedtocarburizingornitriding isusually governed by the following PDE.@@xD@C@x @@yD@C@y @@zD@C@z @C@t(1:7)whereC is the concentration of the element being penetrated (carbon or nitrogen)D is the diffusion coefcientThe boundary conditions can also be classied into the following three kinds.Boundary s1: The surface concentration is known.Cs C (1:8)Boundary s2: The massux through the surface is known.D@C@n q (1:9)Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 4 3.11.2008 3:17pm Compositor Name: BMani4 Handbook of Thermal Process Modeling of SteelsBoundary s3: The mass transfer coefcient between the workpiece surface and environment(ambient media) is known.D@C@n b(CgCs) (1:10)whereD is the diffusion coefcientb is the mass transfer coefcientCg is the atmosphere potential of carbon (or nitrogen)Cs is the surface concentration of carbon (or nitrogen)Althoughthediffusionandheat conductionPDEsdescribedifferent physical phenomena, theirmathematical expression and solving method are exactly the same.1.1.2 SOLVING METHODSFOR PDESUsually, there are two methods to solve the PDEs, analytical method and numerical method. Theanalytical method, taking specic boundary conditions and initial conditions, can obtain the analyticalsolution by deduction (for example, variables separation method), which is a type of mathematicalrepresentation clearly describing certaineld variables under space coordinates and time.The analytical solution has the advantage of concision and accuracy, so it is also called exactsolution. Although it plays an important role in fundamental research, it is only applicable to veryfew cases with relatively simple boundary and initial conditions. Therefore, the analytical solutioncannot cope with massive problems under practicalmanufacture environment, which are featuredwith complicated boundary conditions and a high degree of nonlinearity.Thenumerical solution, alsonamedapproximatesolution, is applicablefor different kindsof boundaryconditionsandcancopewithnonlinear problems. It isthemost basicsimulationmethod in engineering. Up to now, thenite-element method (FEM) andnite-difference method(FDM) are the most widely used methods in simulation of the process, and their commoncharacteristic is discretizationof continuous functions, thus transformingthe PDEs intolargesystems of simultaneous algebraic equations and solving the large algebraic equation groupnally(Figure 1.2).yf1f0f1x1x0x1y = f (x)xFIGURE 1.2 Discretization of the continuous function.Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 5 3.11.2008 3:17pm Compositor Name: BManiMathematical Fundamentals 51.2 FINITE-DIFFERENCE METHOD1.2.1 INTRODUCTIONOF FDM PRINCIPLEFirst, for a continuous function of x, namely f(x), f1, f0, and f1 are retained as the values of f at x1,x0, and x1, respectively (Figure 1.2). When the function has all its derivatives dened at x0 and f1, f1can be expressed by a Taylor series as follows:f1 f0Dx f00(Dx)22!f000 (Dx)33!f0000(Dx)iV4!fiV0 (1:11)f1 f0Dx f00(Dx)22!f000 (Dx)33!f0000(Dx)iV4!fiV0 (1:12)Truncating the items after (Dx)2, Equation 1.11 can be written as@f@x

xx0 f00 f1f0DxDx2f000 %f1f0Dx(1:13)Equation 1.13 is therst-order forward difference with its truncation error of V(Dx). Here V(Dx) is aformal mathematical notation, which represents terms of order Dx.In the same way, another difference scheme from Equation 1.12 can be obtained as follows:@f@x

xx0 f00 f0f1DxDx2f000 %f0f1Dx(1:14)This is therst-order backward difference with its truncation error of V(Dx).Subtracting Equation 1.12 from Equation 1.11 yields@f@x f00 f1f122(Dx)23!f0000%f1f12(1:15)Equation 1.15 is the second-order central difference with its truncation error of V(Dx2).Summing Equations 1.11 and 1.12, and solving for @2f=@x2, we have@2f@x2 f000 f12f0f1(Dx)22(Dx)24!fiV0%f12 f0f1(Dx)2(1:16)Equation 1.16 is the second-order central second difference with its truncation error of V(Dx2).It canbeobservedthat thetruncationerror, originatingfromthereplacement of thepartialderivatives bynite-difference quotients, makes the FDM solution an approximate one; however,the accuracy can be improved by reducing the step size.1.2.2 FDMFOR ONE-DIMENSIONAL HEAT CONDUCTIONAND DIFFUSIONIn this section, two simple cases are taken to elucidate the FDM to solve the PDEs in engineering.The rst caseis theunsteady, one-dimensional heat conductionPDEwithout aninternal heatresource item, and the second one is the one-dimensional diffusion PDE.The governing PDE for the unsteady, one-dimensional heat conduction without an internal heatresource item has the following concise form:a@2T@x2 @T@t(1:17)Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 6 3.11.2008 3:17pm Compositor Name: BMani6 Handbook of Thermal Process Modeling of Steelswhere al= rcp, x and t are independent variables. Equation 1.17 has two independent variables,x and t.Replacing the partial derivatives in Equation 1.17 withnite-difference quotients, the differenceequation can be obtained as follows:Tni12Tni Tni1(Dx)21aTn1iTniDt(1:18)wherei is the running index in the x directionn is the running index in the t directionWhenoneoftheindependent variablesisamarchingvariable, suchastinEquation1.17, it isconventional to denote the running index for this marching variable by n and to display this index asa superscript in thenite-difference quotient (Figure 1.3), Tni1, Tni , and Tni1 are the temperatures onnode i 1, i, and i 1 at time level n, respectively, and Tn1iis the temperature on node i 1 at timelevel n 1.With some rearrangement, this equation can be written asTn1i F0Tni1F0Tni1(1 2F0)Tni(1:19)whereF0 aDt(Dx)2 l Dtrcp(Dx)2Equation 1.19 is written with temperatures at time level n on the right-hand side and temperaturesat time level n 1 on the left-hand side. Within the time-marching philosophy, all temperatures atlevel n are known and those at level n 1 are to be calculated. Of particular signicance is that onlyoneunknownTn1iappears inEquation1.19. Hence, Equation1.19allows for theimmediatesolution of Tn1ifrom the known temperatures at time level n.Equation1.19is oneof theso-calledexplicit nite-differenceapproaches, whichprovideastraightforwardmechanismto accomplish this time marching (Figure 1.3). However,this approachTxi 1 i + 1 ixn + 1nxFIGURE 1.3 Illustration of discretization and time marching.Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 7 3.11.2008 3:17pm Compositor Name: BManiMathematical Fundamentals 7has the limitation that only when the stability criterion F0 1=2 is met, the converged solution can beobtained. The truncation error can be estimated as V(Dx2)(Dt).Writing the spatial difference on the right-hand side of Equation 1.17 in terms of temperatures attime level n 1, we can obtaina Tn1i1 2Tn1iTn1i1(Dx)2Tn1iTniDt(1:20)With some rearrangement, Equation 1.21 can be written asF0Tn1i1 (2F01)Tn1iF0Tn1i1 Tni(1:21)Observing Equation 1.21, the unknown Tn1iis not only expressed in terms of the known temper-aturesattimeleveln,namelyTni ,butalsointermsofotherunknowntemperaturesattimeleveln 1,namely, Tn1i1andTn1i1 .Inotherwords,Equation1.21representsoneequationwiththreeunknowns, namely Tn1i1 , Tn1i, and Tn1i1 . Hence, Equation 1.21 applied at a given grid point i doesnot standalone; it cannot byitselfresult inasolutionforTn1i. Rather, Equation1.21must bewrittenat all interior gridpoints, resultinginasystemof algebraicequations fromwhichtheunknowns Tn1ifor all i can be solved simultaneously.Equation 1.21 is one of the so-called implicit nite-difference approaches, in which the unknownmust be obtained by means of simultaneous solution of the difference equations applied at all gridpoints arrayedat a given timelevel.Becauseofthis need tosolve large systems of simultaneousalgebraic equations, implicit methods are usually involved with the manipulations of large matrices.One advantage of these methods lies in the fact that they are unconditionally stable, i.e., they canalways get the converged solution. Their truncation error can also be estimated as V(Dx2) (Dt).There are different difference equations that can represent Equation 1.17 except Equations 1.21and1.22, whicharetheonlytwoof manydifferencerepresentationsof theoriginal PDE. Forexample, writingthespatial differenceontheright-handsideintermsof averagetemperaturesbetween time level n and n 1, Equation 1.17 can be represented byaTni12Tni Tni12(Dx)2Tn1i1 2Tn1iTn1i12(Dx)2 Tn1iTniDtF0Tn1i1 2(1 F0)Tn1iF0Tn1i1 F0Tni12(1 F0)Tni F0Tni1(1:22)ThisspecialtypeofdifferencingemployedinEquation1.22iscalledtheCrankNicolsonform,which is also unconditionally stable and has a small truncation error ofV(Dx)2(Dt)2.As a typical case of one-dimensional heat conduction without an internal heat resource, an inniteplate with a thickness of d is subjected to the boundary condition that can be expressed as follows:h(TaTs) l@T@n

x0(1:23)The spatial discretization is shown in Figure 1.4, generating m1 nodes from surface (node 0) tocenter(nodem).Here,becauseofthesymmetryhalfoftheslabcan onlybe considered,and thesymmetry axis can be set as the adiabatical boundary.For the surface node (i 0), the boundary condition is introduced and the CrankNicolson formcan be obtained as follows:h Ta12Tn10Tn0 l2DxTn10Tn11Tn0 Tn1 Dxrcp2DtTn10Tn0 (1:24)Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 8 3.11.2008 3:17pm Compositor Name: BMani8 Handbook of Thermal Process Modeling of SteelsWith some rearrangement, Equation 1.24 can be written as(1 F0Bi)Tn10F0Tn11 (1 F0Bi)Tn0 F0Tn1 2BiTa(1:25)where Bi hDt=Dx.Forthecentralnode(i m), theadiabaticalboundary@T=@x 0isinput, andthedifferenceequation is(1 F0)Tn1m1F0Tn1m1 (1 F0)TnmF0Tnm1(1:26)The nite-difference formfor transient heat conduction in the innite plate is composed byEquations 1.22, 1.25, and 1.26, providing m1 algebraic equations for m1 unknown Ti for allnodes. The unique solution for the temperatureeld can usually be obtained by a mature algorithm.The one-dimensional diffusion PDE (Equation 1.27) and its difference equations have the sameformasthatoftheunsteady,one-dimensionalheatconductionwithoutaninternalheatresource.Hence, the correspondingequations are brieyrepeatedandthenenteredintosolvingof thealgebraic equation group.D@2C@X2 @C@t(1:27)Theboundaryconditionsatthesurfacenodeandthecenternodearethethirdtypeofboundarycondition and the adiabatical condition, respectively, and are listed as follows:D@C@X

x0 b(CgCs)D@C@X

xm 0

(1:28)d2T0 1 2 i m x/FIGURE 1.4 Spatial difference scheme of nodes for an innite plate.Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 9 3.11.2008 3:17pm Compositor Name: BManiMathematical Fundamentals 9For the inner nodes (i 1 m1), the CrankNicolson difference form can be written asF0Cn1i1 2(1 F0)Cn1iF0Cn1i1 F0Cni12(1 F0)Cni F0Cni1(1:29)where F0 (D Dt)(Dx)2.For the surface node (i 0), the CrankNicolson form is(1 F0Bi)Cn10F0Cn11 (1 F0Bi)Cn0F0Cn12BiCg(1:30)For the central node (i m), the difference equation is(1 F0)Cn1mF0Cn1m1 (1 F0)CnmF0Cnm1(1:31)ThealgebraicequationgroupconstitutedbyEquations1.29through1.31canbeexpandedandwritten in the following matrix form:d0a0b1d1a1.........bidiai.........bm1dm1am1bmdm

C0C1...Ci...Cm1Cm

d00a00b01d01a01.........b0id0ia0i.........b0m1d0m1a0m1b0md0m

C0C1...Ci...Cm1Cm

n2lC00...0...00

(1:32)whered0 1 F0Bid1 d2 dm1 2(1 F0)dm 1 F0b1 b2 bm F0a0 a1 am1 F0d00 1 F0Bid01 d02 d0m1 2(1 F0)d0m 1 F0b01 b02 b0m F0a00 a01 a0m1 FGur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 10 3.11.2008 3:17pm Compositor Name: BMani10 Handbook of Thermal Process Modeling of SteelsWhen the concentration of the nodes at time level n is known, the right-hand side of Equation 1.32can be simplied as a column matrix. Thus, we obtaind0a0b1d1a1.........bidiai.........bm1dm1am1bmdm

C0C1...Ci...Cm1Cm

n1F0F1...Fi...Fm1Fm

(1:33)Thecoefcient matrixontheleft-handsideofEquation1.33isatridiagonal matrix, denedashaving nonzero elements only along the three diagonals, which are marked with three dashed lines.The solution of the system of equations denoted by Equation 1.33 involves the manipulation of thetridiagonal arrangement; such solutions are usually obtained using Thomas algorithm, which hasbecome almost standard for the treatment of tridiagonal systems of equations. A description of thisalgorithm is given as follows.First, Equation 1.33 is changed into an upper bidiagonal form by dropping therst term of eachequation, replacing the coefcientof the main-diagonalterm by Equation 1.34, and replacing theright-hand side with Equation 1.35.di* dibidi1*ai1(i 1, 2, 3,. . . , m 1, m) (1:34)Fi* Fibidi1*Fi1*(i 1, 2, 3,. . . , m 1, m) (1:35)Then Equation 1.33 transforms into an upper bidiagonal form as follows:d0a0d1*a1.........di*ai.........dm1*am1dm*

C0C1...Ci...Cm1Cm

n1F0F1*...Fi*...Fm1*Fm*

(1:36)Theelementswiththesuperscript asteriskarethosesubjectedtoGaussianelimination. ThelastequationinEquation1.36hasonlyoneunknown. SolvingCn1mfromEquation1.37, all otherunknowns are found in sequence from Equation 1.38, starting from Cn1m1to Cn11.Cn1mFm*dm*(1:37)Cn1iFi*aiCn1i1 di*(i m 1, m 2,. . . , 2, 1) (1:38)Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 11 3.11.2008 3:17pm Compositor Name: BManiMathematical Fundamentals 11The concentration of the surface node Cn10can be obtained byCn10F0a0Cn11 d0(1:39)So far, the concentration of all nodes at time level n 1 has been calculated. Sometimes, the activityis used instead of the concentration in diffusion problems such as the nitriding process.1.2.3 BRIEF SUMMARYThe main advantage of FDM lies in its rigorous mathematical derivation, and it is very simple whenapplied in one-dimensional problems. For the two- and three-dimensional problems, FDM can alsobe appliedbut for objects withrelativelysimple shapes due toits limitationincopingwithcomplicated-shape boundaries. Hence, the FEMmethod is mainly used in the simulation oftemperatureelds and concentrationelds with three-dimensional complicated-shape parts.1.3 FINITE-ELEMENT METHODThe FEM, sometimes referred to asnite-element analysis (FEA), is a computational technique usedtoobtainapproximate solutions of boundaryvalue problems inengineering. Simplystated, aboundaryvalueproblemisamathematical probleminwhichoneor moredependent variablesmustsatisfyadifferentialequationeverywherewithinaknowndomainofindependentvariablesand satisfy specic conditions on the boundary of the domain. Usually, FEM divides the denitiondomainintoreasonablydenedsubdomains(element)byhypothesisandsupposestheunknownstate variable function in each element approximately dened, so that the approximate solution ofboundary value and initial value problems is thus obtained. Since the respectively dened functionscanbeharmonizedat element nodesorcertainjoint points, theunknownfunctioncanapproxi-mately be expressed in the whole denition domain.Because of the extraordinaryexibility of element division, the FEM elements cant well toobjects withcomplexshape andthe boundaries withcomplexcurvedsurfaces. For example,complexthree-dimensional regions canbeeffectively lledbytetrahedral elements, similar totriangular elementslling a two-dimensional region. Therefore, the FEM is the most widely usedmethod in heat treatment numerical simulation so far.FEM has been dissertated in detail in related monographs [13]. Hence, a brief introduction ispresented in this section.1.3.1 BRIEF INTRODUCTIONNo matter what the physical nature of the problem, the standard FEM is always performed with asequential series of steps. Certain steps in formulating an FEA of a physical problem are common toall such analyses, whether structural, heat transfer,uidow, or some other problems. The steps aredescribed as follows:1. Denition of problem and its denition domain2. Discretization of the denition domain3. Determination of all kinds of state variables4. Formulations of the problem5. Establishing of coordinate system6. Construction of the approximate function for elements7. Derivation of element matrix and equationGur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 12 3.11.2008 3:17pm Compositor Name: BMani12 Handbook of Thermal Process Modeling of Steels8. Coordinate transformation9. Assembly of element equation10. Introduction of the boundary condition11. Solution to thenal algebraic equation12. Explanation of the resultsWhen FEM for a specic engineering problem is performed by computer (these steps are embodiedin commercial nite-element software packages), it usually involves three stages of activity:preprocessing, solution, and postprocessing.1.3.1.1 Stage 1: PreprocessingThe preprocessing stage involves the preparation of data, such as nodal coordinates, connectivity,boundaryconditions,loading,andmaterialsinformation.Itisgenerallydescribedasdeningthemodel and includes the following:.Dene the geometric domain of the problem.Dene the element type(s) to be used.Dene the material properties of the elements.Dene the geometric properties of the elements (length, area, and the like).Dene the element connectivities (mesh of the model).Dene the physical constraints (boundary conditions).Dene the loadings1.3.1.2 Stage 2: SolutionThe solution stage involves stiffness generation, stiffness modication, and solution of equations,resulting in the evaluation of nodal variables. Other derived quantities, such as gradients for stresses,may be evaluated at this stage. In other words, thenite-element software assembles the governingalgebraic equations in matrix form and computes the unknown values of the primary eld variable(s).The computed values are then used by back substitution to compute additional derived variables, suchas reaction forces, element stresses, and heatow.1.3.1.3 Stage 3: PostprocessingAnalysis and evaluation of the solution results is referred to as postprocessing, so the postprocessingstagedealswiththepresentationofresults. Typically, thedeformedconguration, modeshapes,temperature, and stress distribution are computed and displayed at this stage. Postprocessor softwarecontains sophisticated routines used for sorting, printing, and plotting selected results from anite-element solution. Examples of operations that can be accomplished include.Sort element stresses in order of magnitude.Check equilibrium.Calculate factors of safety.Plot deformed structural shape.Animate dynamic model behavior.Produce color-coded temperature plotsWhile solutiondata canbemanipulatedinmanyways inpostprocessing, themost importantobjectiveistoapplysoundengineeringjudgmentindeterminingwhetherthesolutionresultsarephysically reasonable.Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 13 3.11.2008 3:17pm Compositor Name: BManiMathematical Fundamentals 131.3.2 GALERKIN FEMFOR TWO-DIMENSIONAL UNSTEADY HEAT CONDUCTIONThe method of weighted residuals, especially the embodiment of the Galerkin FEM, is a powerfulmathematicaltoolthatprovidesatechniqueforformulatinganite-elementsolutionapproachtopractically any problem for which the governing differential equation and boundary conditions canbe written.Here, two-dimensional unsteadyheat conductionistakenasanexampletoderivetheFEMformulations by Galerkins weighted residual method.The governingPDEfor unsteady, two-dimensional heat conductionwithaninternal heatresource isl@2T@x2 @2T@y2 Q rcp@T@t 0 (1:40)The initial condition, supposing the temperatureeld is known, can be written ast 0: T T0(1:41)The three types of boundary conditions have been listed by Equations 1.2 through 1.4.The right-hand side of Equation 1.40 equals zero when the column vector T, the exact solutionof thetemperature eld, issubstituted. Onthecontrary, theapproximatesolutionTmakestheresidual error:R l@2T@x2 @2T@y2 Q rcp@T@t(1:42)The basic idea of the weighted residual method is to construct a suitable weight function so that theintegration of products by residual error and weight function equals zero; the approximate solutionon the whole domain can thus be obtained. Therefore, we have

DWil@2T@x2 @2T@y2 rcp@T@t Q dx dy 0 (1:43)where Wi is the weight function.Several variations of the weighted residual method existand the techniques vary primarily inhowtheweight functions aredeterminedor selected. Themost commontechniques arepointcollocation, subdomain collocation, least squares, and Galerkins method [4].The second-order differential item in Equation 1.43 can be transformed into arst-order item byintegration in parts. Therefore, the second-order differential item can be expressed as

DWil@2T@x2 @2T@y2 dx dy

Dl@Wi@x@T@x @Wi@y@T@y dx dy

SWil@T@n ds (1:44)and

SWil@T@n ds

S1Wil@T@n ds

S2Wil@T@n ds

S3Wil@T@n ds (1:45)Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 14 3.11.2008 3:17pm Compositor Name: BMani14 Handbook of Thermal Process Modeling of SteelsSince the temperature onboundaryS1is known, we have S1Wil(@T=@n)ds 0. SubstitutingEquations 1.44 and 1.45 into Equation 1.43, we can get

Dl@Wi@x@T@x @Wi@y@T@y dx dy Wircp@T@t Q dx dy

S2Wil@T@n ds

S3Wil@T@n ds (1:46)Theelement analysis is toestablishtheformulations bywhichthecontinuous functioninthesubdomain DD can be expressed by the node values, i.e., the function of the node values. Supposingtherearenelements inthesolutiondomain, andtherearemnodes ineachelement withthetemperatureTi(i 1, 2, . . . , m), theunknowntemperatureTe(x, y, z)denedintheelement canbeexpressed as the interpolation function of each node. Therefore, we haveTe(x, y, z) N1T1N2T2 NmTm [Ni]{Ti} (1:47)where Ni is the shape function, which is a function of the components of each element nodes (xi, yi, zi)and those of the location (x, y, z).For example, the shape functions of a triangle element with three nodes can be simply derivedand expressed asNi(x, y) 12A(aibix ciy)Nj(x, y) 12A(ajbjx cjy)Nm(x, y) 12A(ambmx cmy)(1:48)whereA 12(bicjbjci)ai xjymxmyj;aj xmyixiym; am xiyjxjyibi yjym; bj ymyi; bm yiyjci xmxj; cj xixm; cm xjxiAlthoughthe shape functions for different types of elements have different forms, theyhavecommon characteristics and can be obtained by the geometrical method. Especially the coefcientsin the shape functions are the functions of the coordinates of each node, that is,Ni(x, y, z) F(xi, yi, zi, x, y, z), i 1, 2,. . . , m (1:49)Obviously, the shape functions are determined only by the coordinates of each node and the type ofelement. Hence, the temperature on certain points in an element can be expressed as the function ofnode temperature, namely, the column vector {Ti} (see Equation 1.47).Equation1.46 is applicable inthe wholesolution domain D, so forits subdomains, i.e.,eachelementDD, we have

DDl@Wi@x@T@x @Wi@y@T@y dx dy Wircp@T@t Q dx dy

DS2Wil@T@n ds

DS3Wil@T@n ds (1:50)Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 15 3.11.2008 3:17pm Compositor Name: BManiMathematical Fundamentals 15In Galerkins weighted residual method, the weight functions are chosen to be identical to the trialfunction. Here, the shape functions are taken as the weight functions, for example, WiNi(x, y), andsubstitute into Equation 1.50. Hence, we have

DDl@Ni@x@T@x @Ni@y@T@y dx dy Nircp@T@t Q dx dy

DS2Nil@T@n ds

DS3Nil@T@n ds (1:51)SubstitutingEquation1.47intoEquation1.51, andintroducingtheboundaryconditions fromEquations 1.2 through 1.4, we have

DDl@Ni@x@Ni@xTi@Nj@xTj@Nm@xTm l@Ni@y@Ni@yTi@Nj@yTj@Nm@yTmNircp Ni@Ti@t Nj@Tj@t Nm@Tm@t NiQ

dx dy

DS2qNids

DS3hNi(NiTiNjTjTa)ds (1:52)TakingWjNj(x, y)andWmNm(x, y), andderivinginexactlythesameway, canbeobtainedas follows:

DDl@Nj@x@Ni@xTi@Nj@xTj@Nm@xTml@Nj@y @Ni@yTi@Nj@yTj@Nm@yTm NjrcpNi@Ti@t Nj@Tj@t Nm@Tm@t NiQ

dx dy

DS2qNjds

DS3hNj(NiTiNjTjTa)ds (1:53)

DDl@Nm@x@Ni@xTi@Nj@xTj@Nm@xTm l@Nm@y@Ni@yTi@Nj@yTj@Nm@yTm NmrcpNi@Ti@t Nj@Tj@t Nm@Tm@t NiQdx dy

DS2qNmds

DS3hNm(NiTiNjTjTa)ds (1:54)For the interior elements, the right-hand side of Equations 1.52 through 1.54 equals zero respect-ively; while for the boundary element, only the right-hand side of Equation 1.54 equals zero with theassumption that only node i and j are on the boundary.An equation group constructed by Equations 1.52 through 1.54 contains only the threeunknowns, Ti, Tj, and Tm. After rearrangement, it can be written in matrix form as follows:[K]e{Te} [C]e@@t{T}e {r}e(1:55)Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 16 3.11.2008 3:17pm Compositor Name: BMani16 Handbook of Thermal Process Modeling of Steelswhere[K]eis the element stiffness matrix{T}eis the column vector of temperature on element nodes (unknown)[C]eis the element heat capacity matrix{r}eis the element constant vector itemThe element stiffness matrix assembled from Equations 1.52 through 1.54 is[K]e

DDl@Ni@x @Ni@x @Ni@y @Ni@y@Nj@x @Ni@x @Nj@y @Ni@y@Nm@x@Ni@x @Nm@y@Ni@y@Ni@x @Nj@x @Ni@y @Nj@y@Nj@x @Nj@x @Nj@y @Nj@y@Nm@x@Nj@x @Nm@y@Nj@y@Ni@x @Nm@x@Ni@y @Nm@y@Nj@x @Nm@x@Nj@y @Nm@y@Nm@x@Nm@x@Nm@y@Nm@y

dx dy

DS2qNiNj0

ds

DS3hNiNiNjNi0NiNjNjNj00 0 0

ds (1:56)Substituting Equation 1.48 into Equation 1.56, and integrating, we obtain[K]el4Ab2i c2ibibjcicjbibmcicmbjbicjcib2j c2jbjbmcjcmbmbicmcibmbjcmcjb2mc2m

hlij62 1 01 2 00 0 0

(1:57)whereA is the area of elementDDlij is the length of exterior boundary; lij0 for the interior elementThe coefcients bi, bj, bm, ci, cj, cm are determined by the coordinates of nodes. Obviously, everyelement in [K]eis determined.For the element heat capacity matrix, we have[C]e

DDrCpNiNiNjNiNmNiNiNjNjNjNmNjNiNmNjNmNmNm

dx dy (1:58)Substituting Equation 1.48 into Equation 1.58, and integrating, we obtain

DDNiNj dx dy

DDNiNm dx dy

DDNjNm dx dy A12

DDNiNi dx dy

DDNjNj dx dy

DDNmNm dx dy A6(1:59)Thus, Equation 1.58 can be rewritten as[C]ercpA122 1 11 2 11 1 2

(1:60)Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 17 3.11.2008 3:17pm Compositor Name: BManiMathematical Fundamentals 17The element constant vector item {r}ein Equation 1.55 can be expanded as{p}e {pQ}e{pq}e{ph}e(1:61)where {pQ}e, {pq}e, {ph}eoriginate from the internal heat resource, heatux in the second kind ofboundarycondition, andtheheat transfer coefcient inthethirdkindof boundarycondition,respectively.The item {pQ}ecan be obtained by{pQ}e

DDQNidx dy

DDQNjdxdy

DDQNmdx dy (1:62)When the internal heat resource intensity Q is a constant, we have{PQ}eAQ3111

(1:63)When the internal heat resource intensity Q is a linear function, we have{PQ}eA122QiQjQmQi2qjQmQiQj2Qm

(1:64)where Qi, Qj, and Qm are the internal heat resource intensity on node i, j, and m, respectively.The item {pq}edue to the heat ux on the boundary can be obtained by{pq}e

DS2qNids

DS2qNjds

DS2qNmds (1:65)When the heat ux through the boundary q is a constant, we have{pq}elij2 q110

(1:66)When the heat ux through the boundary q is a linear function, we have{pq}elij6 q2qiqjqi2qj0 0

(1:67)The item {ph}edue to the heat transfer coefcient on the boundary can be obtained by{ph}e

DC3hTaNids

DC3hTaNjds

DC3hTaNmds lij2 hTa110

(1:68)Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 18 3.11.2008 3:17pm Compositor Name: BMani18 Handbook of Thermal Process Modeling of SteelsTherefore, thecontinuous functionTe(x, y, t)onthesubdomains, i.e., theelement DD, has beenexpressed by the node temperature, and the algebraic equation group with the unknown variablesof node temperature has been established.For the whole solution domain, the algebraic equation groups on each element are assembledtogether so that a large-scale equation group can be obtained as follows:[K]{T} [C]@@t{T} {P} 0 (1:69)whereGeneralstiffnessmatrix: [K] [K]e(1:69a)Heatcapacitymatrix: [C] [C]e(1:69b)Heatfluxmatrix: {P} {P}e(1:69c)ThesolvingofEquation1.69bringsthesolutionofthetemperatureeldfortheunsteadytwo-dimensional heat conduction. The evolution of temperature can be observed by recording the nodetemperatureat eachtimelevel, andtheheating(orcooling)curvesat special pointscanalsobeextracted from the result les.1.3.3 FEMFOR THREE-DIMENSIONAL UNSTEADY HEAT CONDUCTIONTheprinciples andprocedureintheFEMfor three-dimensional unsteadyheat conductionareidentical to those in the two-dimensional one, as presented in the previous section. The derivationof its FEM formulations is more complex and can be found in related monographs [57]. Here, onlythe derived results are briey outlined.The governing PDEof three-dimensional unsteady heat conductionwithaninternal heatresource is repeated here.l@2T@x2 @2T@y2 @2T@z2 Q rcp@T@t(1:70)Supposing that there are total n elements (mnodes in each element) and p nodes after thediscretizationofthewholesolutiondomain, Equation1.70canbetransformedintoanalgebraicequation group as follows:[K]{T} [C]@T@t {P} 0 (1:71)[K] [K]e(1:71a)[C] [C]e(1:71b){P} {P}e(1:71c)[K]e

Ve[B]T[D] [B]dv

Seh[N]T[N]ds (1:71d)Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 19 3.11.2008 3:17pm Compositor Name: BManiMathematical Fundamentals 19[C]e

Vercp[N]T[N]dv (1:71e)[p]e

VeQ[N]Tdv

SehTa[N]Tds (1:71f)[D] l 0 00 l 00 0 l

(1:71g)[B] @N1@x@N2@x. . .@NP@x@N1@y@N2@y. . .@NP@y@N1@z@N2@z. . .@NP@z

(1:71h)[N] [N1N2 NP] (1:71i)whereVeis the element volumeSeis the exterior boundariesIntheunsteadytemperature eld, thevariabletemperatureT(x,y,z,t)is afunctionof spatiallocationandtime. Equation1.71isthespatial discretizedformulationwiththetimedifferentialitem @T=@t, which can be discretized by the difference method in a uniform format as follows:u@T@t t(1 u)@T@t tDt1Dt(TtTtDt) (1:72)Ifu 1, the backward difference format of time discretization is@T@t t1Dt(TtTtDt) (1:73)Substituting Equation 1.73 into Equation 1.71, we get[K] 1Dt[C] {Tt} 1Dt[C]{TtDt} {P} (1:74)Ifu 1=2, theCrankNicolsondifferenceformat (alsonamedcentral differenceformat)oftimediscretization can be obtained as12@T@t t@T@t tDt 1Dt(TtTtDt) (1:75)Substituting Equation 1.75 into Equation 1.71, the corresponding FEM algebraicequation can beobtained as follows:[K] 2Dt[C] {Tt} 2Dt[C] [K] {TtDt} 1Dt{P}t{P}tDt (1:76)Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 20 3.11.2008 3:17pm Compositor Name: BMani20 Handbook of Thermal Process Modeling of SteelsThe CrankNicolson difference format is, in general, more accurate than the backward differencemethod, in that it does not give preference to either temperature at time level t Dt or time level tbut, rather, gives equal credence to both.In FDM of time, the key parameter governing solution accuracy is the selected time step Dt. In afashionsimilartotheFEM, inwhichthesmaller theelementsare, physically, thebetteristhesolution, the FDM converges more rapidly to the true solution as the time step is decreased.Whenthetemperature eldat timelevel t Dtisknown, andinput intoEquation1.74orEquation1.76, thetemperature{Tt}onall nodes at timelevel tcanthus besolved, andthemarchingsolutioncanprogressintimeuntil asteadystateisreached. Thenodal temperatures,recorded stepbystepatdifferenttime levels,constitutetheevolutionhistoryofthethree-dimen-sional temperatureeld.1.4 CALCULATION OF TRANSFORMATION VOLUME FRACTION1.4.1 INTERACTIONSBETWEEN PHASE TRANSFORMATIONAND TEMPERATUREThe latent heat releases as an internal heat source inside the solid when phase transformation occurs.Thus, the kinetics of phase transformation strongly depends on the temperature history of steel parts,but also strongly affects the temperatureeld inside. The interactions between phase transformationand temperature are not one-way, but bilateral, increasing the complexity in the accurate numericalsimulation of the thermal processing.Thelatent heat per timeunit, expressedastheinternal heat sourceintheheat conductionequation, i.e., Equation 1.1, is usually calculated by the following Equation 1.77:Q DHDVDt(1:77)whereDH is the enthalpy difference when a new phase of unit volume forms from the parent phaseDV is the change of transformation fraction in the time stepDtThecalculationofthevolumefractionisthekeytopredicttheevolutionoftemperatureandmicrostructureduringthethermalprocessing,aswellasthenalmicrostructureconstituentsandrough mechanical property. Since phase transformations are usually classied into two categoriesaccordingtotheir mechanisms, diffusiontransformation(for example, pearlitetransformation),andnondiffusiontransformation, i.e., martensitictransformation, themathematicalmodelsofthetransformation kinetics are very different.Inthissection, thenumerical methodtocalculatethetransformedvolumefractionofanewphase is mainly introduced.1.4.2 DIFFUSION PHASE TRANSFORMATIONAtimetemperaturetransformation(TTT)diagramdescribestherelationshipbetweenthetrans-formationstarting,ending, andthetransformedvolumefractionduringtheisothermalprocessatdifferent temperatures. The isothermal kinetics equation, namely JohnsonMehl equation [8],providesasolidbasefornumerical simulationofthermal processalthoughit cannot bedirectlyappliedto calculate the volume fraction due to the nonisothermal process of the practical heatingor coolingprocess. Until now, themethodproposedbyFernandes et al. [9] has beenwidelyaccepted, inwhichthepractical nonisothermal processisconsideredasmanyisothermal stageswithatinytimedurationasshowninFigure1.5, andtheeffectofthesestagescanbesummedtogether according to Scheils additivity rule.Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 21 3.11.2008 3:17pm Compositor Name: BManiMathematical Fundamentals 21Due to the strict limits of JohnsonMehl equation, Avrami [10] proposed an empirical equation,which has a simpler form and has been widely used, as follows:f 1 exp(btn) (1:78)wheref is the volume fraction of a new phaset is the isothermal time durationb is a constant dependent on the temperature, composition of parent phase, and the grain sizen is also a constant dependent on the type of phase transformation and ranges from 1 to 4Coefcient bandnat different temperaturesareusuallycalculatedfromtheexperimentallyobtained TTT diagrams by the following equations:n(T) ln[ ln(1 f1) ln(1 f2)]ln t1ln t2(1:79)b(T) ln(1 f1)tn(T)1(1:80)Therefore, the relationship between the volume fraction and time under certain isothermal temper-aturescanbecalculatedbyEquations1.78through1.80.Mostexperimentaldataofphasetrans-formation agree well with the Avrami equation.Only with the help of Scheils additivity rule can the transformation starting time, namely, theincubation period, for a nonisothermal cooling or heating process be determined. The general formof Scheils additivity rule can be written as

ts0dttTTTi 1 orni1DtitTTTi 1 (1:81)TemperatureTime timet2, Fictitioust1t1t2t1, FictitiousVolume fraction, ff2f1T2f1f0t2T1T1T2FIGURE 1.5 Schematic model for fraction calculation of diffusion transformation during continuous coolingprocess.Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 22 3.11.2008 3:17pm Compositor Name: BMani22 Handbook of Thermal Process Modeling of Steelswherets is the incubation period to be determined for the nonthermal processtTTTiis the incubation period under a certain isothermal temperatureDti is the time stepIt canbeunderstoodthat thetransformationinthenonisothermal processoccurswhenthesummation of the relative fractions DtitTTTi

reaches unity.Scheilsadditivityrulehasalsobeenappliedinvolumefractioncalculation.Forexample,tond the time (tm) needed to attain a certain fraction transformed (jm) in a nonisothermal transform-ation with timetemperature path Tt, Scheils additivity rule is expressed as

tm0dttjm(T) 1 (1:82)where tjm(T) is the time totransformthe fraction jmisothermallyat the current isothermTt.According to Equation 1.82, the total time required for completion of the transformation is obtainedby adding the absolute durations of time Dt spent at each temperature T, until the sum of the relativedurations Dt=tjm(T) becomes unity. The time tjm(T), to reach transformed fraction jm, can be obtainedfrom the kinetics equation, i.e., the Avrami equation (Equation 1.78).To explicitly demonstrate its calculation method, the concept of ctitious time t* andctitioustransformedvolumefractionfi*arecommonlydeveloped[9]. Thesetwovariablescanbe calculated byti*ln(fi1)bi 1=ni(1:83)fi* 1 exp biti*Dt ni (1:84)Hence, the practical transformed fraction isf fi*fgi1fi1 fmax(1:85)wherefgi1 and fi 1 are the austenite fraction and transformed fraction at the end of the previous timestepfmax is the maximum possible transformed fraction for this type of transformationEquations 1.78through1.85constitutethebasicframeof volumefractioncalculationfor thediffusion transformation, whereas it should be carefully applied in specic case studies andnecessary modications be made.1.4.2.1 Modication of Additivity Rule for Incubation PeriodBased on theoretical analysis and experimental results, Hsu [11] points out that the additivity ruleis not always accurate enough to be applicable in incubation period prediction. Hawbolt et al. [12]andReti andFelde[13] heldthesameviewpoint that theadditivityrulesometimes seriouslyoverestimates the incubation time.Takingtheeutectoidsteel astheexample, thepractical incubationperiodtCCTsundercertaincooling rate, as well as the incubation periods for different isothermal stages tTTTs, can be obtainedGur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 23 3.11.2008 3:17pm Compositor Name: BManiMathematical Fundamentals 23fromCCTandTTTdiagrams,respectively.Hence,therelativedurationsofeachtinyisothermalstageDt tTTTs

can be summed stage by stage from zero to tCCTsbyx tCCTs0DttTTTs(1:86)Beyond expectation, the summation x, listed in Table 1.1, is much less than unity.Another approach has been proposed to modify the additivity rule for prediction of incubationperiod using TTT and CCT diagrams simultaneously [14].The starting curves of TTT and CCT diagram are plotted together as Figure 1.6. The averagecoolingrateduring arbitrarytimestepDti onthepracticalcoolingcurve,thatis,theslopeofthetangent line dd drawn at the middle point of time step Dti, can be calculated byVi Ti1TiDti(1:87)TABLE 1.1Summation of Relative Durations in NonisothermalProcess of Eutectoid SteelAverage CoolingRate, 8C=sSummation ofRelative Durations, x Source7.5 0.2 [12]2.0 0.2338.5 0.245.3 0.43 [14]21.2 0.3147.6 0.28Time0Practical cooling curveddddViAclStarting curve of CCTStarting curve of TTTTemperatureTi 1tsTititCCTs(i)tTTTs(i)tCCTsTTTFIGURE 1.6 Schematic sketch of the modication of additivity rule for incubation period.Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 24 3.11.2008 3:17pm Compositor Name: BMani24 Handbook of Thermal Process Modeling of SteelsCooling from the critical point Ac1 with the constant rate of Vi, the cross point with the starting curvetCCTsin the CCT diagram gives the incubation period tCCTs(i), during which the modication coefcientwi of relative durations in this time step for the TTT diagram can be calculated bywi

tCCTs(i)0dttTTTs(1:88)Themodicationcoefcient wireectsthedifferenceinincubationperiodbetweentheCCTandTTT diagram. When wi equals unity, there is no difference. Usually, it is bigger than unity since it isvery common that the starting curve in CCT lags behind that in the TTT diagram. Therefore, therelative time durationDti=tTTTs(i)for the time stepDti can be modied as(Dti=tTTTs(i))(1=fi).Hence, forthetransformationstartingtime, namely, theincubationperiodfortheisothermalprocess, the criterion is modied as follows:DtitTTTs(i)1wi 1 (1:89)Thetheoretical derivationof thismodicationisstill intheprocessof consummation, andtheapplication and the accuracy also need to be further validated.1.4.2.2 Modication of Avrami EquationIt is very clear that the time in the Avrami Equation 1.78 counts since the transformation occurs, thatis, the incubation period is not included. Therefore, Hawbolt et al. [12] suggested that the modiedform of Avrami equation should bef 1 exp b(t ts)n (1:90)The calculation of coefcients n and b is also changed asn(T) ln ln(1 f1) ln(1 f2) ln(t1ts) ln(t2ts)(1:91)b(T) ln(1 f1)(t1ts)n(T)(1:92)where ts is the incubation period of the isothermal process at a certain temperature.To demonstrate the difference between these two methods, again, the eutectoid steel (T8 steel) istaken as an example. In the algorithm 1 based on the Avrami equation, the transformation startingtime ts (approximate as t0.01) and ending time te (approximate as t0.99) read from the TTT diagramare input to Equations 1.78 through 1.80 to calculate the coefcients n, b, and t0.25 (the time neededtoget afractionof 0.25of thenewphase). Inthealgorithm2basedonthemodiedAvramiequation,t0.5andt0.75 arethetimetotransformthenewphaseofthevolumefractionof0.5and0.75, respectively, bothofwhichincludetheincubationperiodtsandcanbegot fromtheTTTdiagram. WiththehelpofEquations1.90through1.92, thecoefcientsn, b, andt0.25arealsocalculated. All thesedata arelisted inTable 1.2. It isobviousthat the modied Avramiequationseems more reasonable andts better with the TTT diagram than the unmodied one.Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 25 3.11.2008 3:17pm Compositor Name: BManiMathematical Fundamentals 251.4.2.3 Calculation of Proeutectoid Ferrite and Pearlite FractionThe transformations inthe hypoeutectoidsteelcan be explainedby Figure 1.7. Whenthesteel iscooledrapidlyuntil thetemperatureisbelowlineSE0, thestableferriteaandcementiteFe3Cprecipitate simultaneouslyandformthe quasieutectoid. However, if the temperature is abovepoint a, for example, at point a0, the nucleus of ferrite appearsrst at the austenite grain boundary,and the ferrite grows up as the time goes on. At the same time, carbon diffuses into the neighboringaustenite graindue toits lowdissolvabilityinferrite, increasingthe carbonconcentrationinaustenite. Whentheconcentrationreaches point blocatedontheboundaryof two-phaseareaaFe3C, the retained austenite starts to decompose into pearlite.The mechanism of the transformation process has been investigated intensively and has almostreached common understanding, while the fraction calculation of proeutectoid ferrite and pearlite isan unsettled dispute.Thenucleationsitesandthemodeoftheproeutectoidferritearetheoreticallydifferent fromthose of eutectoid pearlite. Although their growth is controlled by diffusion, the growth mode anddiffusionrouteof carbonaredifferent. Thesefactorssupport theviewpoint that thefractionofproeutectoidferriteandpearliteshouldbecalculatedseparately, that is, twosetsofindependentTABLE 1.2Coefcients n, b, and t0.25 Calculated by Different MethodsTemperature(8C)Data from TTT Diagram ofEutectoid Steel (T8 Steel)Algorithm 1: AvramiEquationAlgorithm 2: ModiedAvrami Equationts(s) t0.25(s) t0.5(s) t0.75(s) te(s) n(T) b(T) 105t0.25(s) n(T) b(T) 105t0.25(s)700 12.5 39 63 90 120 2.71 1.07 43.1 1.62 121 41.9650 2.7 6.5 8.8 11.5 18.5 3.18 42.5 7.7 1.89 2271 6.5450 1.4 5.5 8.0 11 20 2.13 463 6 2.05 1460 5.7400 3 30 4.3 65 100 1.75 147 20.5 1.58 200 26GP + + Fe3C + Fe3CCSPEGEa baC (%)T (C)TibFIGURE 1.7 Schematic sketch of the quasiequilibrium diagram of Fe-C.Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 26 3.11.2008 3:17pm Compositor Name: BMani26 Handbook of Thermal Process Modeling of SteelsAvrami equations should be used. Therefore, Wang et al. [15] proposed that the volume fraction ofproeutectoid ferrite and pearlite in steel 1025 be calculated by two Avrami equations asfF 1 exp bF(t tSF)nF (1:93)fP 1 exp bP(t tSP)nP (1:94)On the other hand, the proeutectoid ferrite precipitates along the austenite grain boundary, and thereisenoughspaceforgrowthattheinitialstage.However, withthegrowthofthenewphase, thevolume fraction increases, and the transformation rate decreases because of the encounter of the newphase. Usually, the pearlite transformation occurs when the proeutectoid ferrite precipitation is stillin process. Hence, the proposal [12,15] that the proeutectoid ferrite precipitation should bedescribed by one dependent Avrami equation seems to lack convincing evidence.In addition, the starting temperature of pearlite transformation during continuous cooling is hardtodetermine becausethecarboncontentofaustenite varies with theprecipitation ofproeutectoidferrite. The pearlite transformed is usually the quasi-eutectoid in which the carbon content decreaseswith the decrease in temperature. These are not in agreement with the hypotheses of the additivity rule.Based on the above analysis, Pan et al. [16] developed a set of new approaches that uses onecombined Avrami equation to calculate the total volume fraction of proeutectoid ferrite and pearlite.Assumingthatthesteelisisothermallykeptforenoughlongtimetoprecipitatetheproeutectoidferrite only and the pearlite transformation does not occur, thenal volume fraction of ferrite fFendunder certain temperature Ti can be calculated by lever rule asfFend a0b0cb0(1:95)Therefore, thedenitiondomainof theAvrami equationdescribingtheproeutectoidferriteis[0,fFend].However,thepracticalsituationisthatthepearlitetransformationusuallyoccursbeforethevolumefractionof proeutectoidferritereaches fFend. Thus, thereexistsamaximumvolumefraction of proeutectoid ferrite fFmaxat a certain temperature, which can be calculated asfFmax ba0cb(1:96)Whenthevolumefractionof proeutectoidferritereaches fFmax, thecarbonconcentrationintheretained austenite reaches the composition range of quasieutectoid. The decomposition of austeniteenters the stage of pearlite transformation. The accurate kinetics of austenite decomposition, i.e., thesolid line in Figure 1.8, can be split into two parts: therst part is the transformation of proeutectoidferrite from [0, fFmax], and the second part is that of pearlite from [fFmax,1]. The whole curve is stillS type and can be united into one Avrami equation.The total volume fraction of proeutectoid ferrite fFi and pearlite fPi can be calculated referring toprevious sections, and their separation is performed:Iffi< fFmax, thenfFi fifPi 0

(1:97)Iffi ! fFmax, thenfFi fFmaxfPi fifFmax

(1:98)In order to test this approach, the TTT and CCT diagram of 45# steel was used to calculate the volumefraction of proeutectoid ferrite fFi and pearlite fPi, which were then input into the additivity rule asGur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 27 3.11.2008 3:17pm Compositor Name: BManiMathematical Fundamentals 27xi tCCTetCCTsDtitTTTeitTTTsi(1:99)The results listed in Table 1.3 indicate that the summation of relative durations almost reaches unityunder different cooling rates, demonstrating the feasibility of the approach discussed here.Related experimental and simulation results [17] also demonstrated that the whole decompos-itionprocess of austenite inhypoeutectoidsteel canbe reasonablydescribedbyone Avramiequation.1.4.3 MARTENSITIC TRANSFORMATIONIn most cases, the transformed volume fraction of martensite is independent of the cooling rate, but afunctionof temperature. TheKoistinenMarburger [18] equationis always adoptedinthermalprocess simulation, and it isfM 1 exp a(MsT) (1:100)FerritePearliteFerriteVolume fractiontsFtsPTimePearlite10fmaxFfendFFIGURE 1.8 Schematic sketch of isothermal transformation in hypoeutectoid steel.TABLE 1.3Results of Check Computation of the Additivity Rule in 45# SteelAverage Cooling Rate (8C=s) tCCTs(s) tCCTs(8C) tTTTe(s) xi0.6 113.3 717 288.3 0.988.1 9.5 689 635 0.9725 2.7 670 625 0.94Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 28 3.11.2008 3:17pm Compositor Name: BMani28 Handbook of Thermal Process Modeling of SteelswherefM is the transformed volume fraction of martensiteT is the temperatureMs is the start temperature of martensitic transformationa is the constant reecting the transformation rate and varying with the steel compositionFor carbon steel with the carbon content lower than 1.1%, we have a0.011.Assuming that the average volume of martensite lamella is constant, Magee [19] theoreticallyderived the constant a asa Vw@DGg!Mv@T(1:101)whereVis the average volume of martensite lamellaw is the ratio of the martensite number newly formed to the driving force in the austenite ofunit volumeDGg!Mvis the free energy difference between austenite and martensiteHsu [20] inferred that the KoistinenMarburger equation is only applicable in high and mediumcarbon content steels in which the carbon does not diffuse when martensite transform occurs. Whilein the low carbon steel and the medium carbon steel with strip austenite the alloy element affects thecarbondiffusion. Therefore, theKoistinenMarburger equationshouldbemodiedfor thelowcarbon steel, and it becomesf 1 expb(C1C0) a(MsTq) (1:102)whereC0 is the carbon concentration in austeniteC1 is the carbon concentration in martensitea and b are constants dependent on the materials; the former can be obtained according to Equation1.101, while the latter can be calculated as follows:b Vw@DGg!Mv@C(1:103)where C is the carbon concentration.Generally, thevolumeof martensiteinthecarbonsteel andalloysteel iscalculatedusingEquations1.100and1.102, andtheMspoint canbeobtainedbyexperiment andthefollowingempirical equations:Ms(

C) 520 (C%) 320 (1:104)Ms(

C) 512 453C 16:9Ni 15Cr 9:5Mo 217(C)271:5(C)(Mn) 67:6(C)(Cr) (1:105)The Ms point is considered as a constant in KoistinenMarburger equation, while it varies if othertransformations, for example, pearlite transformation, bainite transformation, andsoon, occurheretofore, changingthecompositionof theparent phaseor consumingtheembryos. It canbeGur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 29 3.11.2008 3:17pm Compositor Name: BManiMathematical Fundamentals 29observedfromtheCCTcurvesthat themoretheaustenitedecomposes, themoretheMspointdecreases. The phenomenon has not been generally reected in the additivity rule in calculation ofvolumefractionsusingTTTcurves.Hence, thecalculationmodel,i.e., theKoistinenMarburgerequation, for martensite is in great need of further amendment based on transformation mechanismsto improve its universality and accuracy.1.4.4 EFFECTOF STRESS STATEON PHASE TRANSFORMATION KINETICSThestressstatehasconsiderableeffectsonthephasetransformationkineticsirrespectiveofthediffusion transformation or that of nondiffusion. Numerous investigations [2125] have been carriedout to clarify the mechanism and to establish effective models for accurate simulation. Up to now,theresearchinthis eldisstill far fromsatisfactory. Somemathematical modelsproposedbydifferent groups are only applicable for some specic steels, and some need to be further validated.In this section, representative models are briey introduced.1.4.4.1 Diffusion TransformationFor the pearlite transformation, Inoue andWang[26] developedthe model of transformationkinetics under stress. In his work, the JohnsonMehl equation in the stress-free state is expressed asf 1 exp

t0F(T)t t 3dt

(1:106)The JohnsonMehl equation under stress state was modied asf 1 exp

t0exp (csm)F(T)t t 3dt

(1:107)whereF(T) is the temperature functionsm is the mean stressc is a constantDenisetal. [27]introducedtheeffectofstressintothecoefcients(nandb)intheAvramiequation (Equation 1.78) as follows:ns n (1:108)bs b(1 Cse)n(1:109)wherese is the equivalent stressC is a constant1.4.4.2 Martensitic TransformationFormartensitictransformation,thestressstateinthesteelhasmuchgreatereffectsonthetrans-formation process. The coefcient a in the KoistinenMarburger equation (Equation 1.100), as wellas the Ms point, is strongly dependent on the stress state.Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 30 3.11.2008 3:17pm Compositor Name: BMani30 Handbook of Thermal Process Modeling of SteelsObserving Equation 1.101, the coefcient a can be obtained from the driving force in which thestress plays an important role. Thus, Hsu proposed an empirical equation as follows, indicating therelationship between a and equivalent stress si.a a0a1si(1:110)where the coefcients are about a01.2430 102and a16.9752.Inoue and Wang [26] expressed the KoistinenMarburger equation without stress asf 1 expf(T) (1:111)while the kinetics equation under stress was modied asf 1 exp AsmBJ122 f(T) (1:112)wheresm is the mean stress, i.e., hydrostatic pressureJ2 is the second stress invariantcoefcients A and B are the material constantsSimilarly, Denis et al. [27] developed the model of martensitic transformation under stress, andher equation for volume fraction of martensite isf 1 expa(MsAsmBsiT) AsmBs1=2i (1:113)ComparedtoEquation1.112, thisequationadditionallyconsiderstheeffect ofstressontheMspoint, which can be clearly expressed as an extra itemDMs [28]. Therefore, we haveMss MsDMs(1:114)DMs AsmBsi(1:115)1.5 CONSTITUTIVE EQUATION OF SOLIDS1.5.1 ELASTIC CONSTITUTIVE EQUATION1.5.1.1 Linear Elastic Constitutive EquationInasolidunderdeformation,thestressdependsonthecurrentstrainstate,andisasinglevaluefunction of the strain. The elastic deformation will be recovered, when the load is released, i.e., theelastic deformation is reversible. Usually elastic constitutive relation is taken as linear.1.5.1.1.1 Isotropic ElasticityIn the isothermal, small deformation process of solids, the stress components are a linear function ofthe components of the small strain tensor, as follows:sij Ceijklekl(1:116)where Ceijklis the elastic tensor, the superscripte means elasticGur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 31 3.11.2008 3:17pm Compositor Name: BManiMathematical Fundamentals 31Suchamaterial model iscalledlinearelasticconstitutiveequation. Theincrementalformoftheabove equation isdsij Ceijkl dekl(1:117)Usually metals can be taken as isotropic, linear elastic materials or ideal elastic materials. For suchmaterials, theelastictensor is afourth-order isotropic tensor, includingonlytwoindependentparameters, as follows:Ceijkl 2Gdikdjlv1 2vdijdkl (1:118)whereG is the shear modulusv is the Poissons ratioG is related with Youngs modulus E by the following equation:G E2(1 v)Innitedeformation, tomeet therequirement ofobjectivity, theJaumannrateofCauchystresstensorandtherateofdeformationtensor,whichpossessobjectivity,areadopted,andtheelasticconstitutive equation is rewritten as^ sij Ceijkldekl(1:119)For the convenience ofnite-element formulation, the stress and strain are usually rewritten fromsecond-order tensors into vectors, e.g.,s [s11s22s33s12s23s31]T [112233212223231]TAccordingly, the fourth-order tensor Ceis rewritten into a matrix:Ce2G1 2n1 n n n 0 0 0n 1 n n 0 0 0n n 1 n 0 0 00 0 0 (1 2n)=2 0 00 0 0 0 (1 2n)=2 00 0 0 0 0 (1 2n)=2

(1:120)Then the elastic constitutive equation can be written in the matrix form:s CeIn this chapter, the symbols is used to represent the stress both in tensorial form and in vectorialform. Similar usage is also applied for and Ce. The exact meaning of the symbol can be deducedfrom the context.Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 32 3.11.2008 3:17pm Compositor Name: BMani32 Handbook of Thermal Process Modeling of SteelsElastic deformation can be decomposed into a volumetric component,em e11e22e33 =3e0ij eijemdije0ij 12Gs0ijem 1 2nEsm1.5.1.1.2 Orthotropic ElasticitySome materials, such as a single crystal, show different elastic behavior along different directions. Ifthe principal axes are mutually orthogonal, it is called orthotropic elasticity. If along each principalaxis, the elastic properties are identical, then the elastic matrix isCeC11C12C120 0 0C12C11C120 0 0C12C12C110 0 00 0 0 C440 00 0 0 0 C4400 0 0 0 0 C44

(1:121)1.5.1.2 Hyperelastic Constitutive EquationSome materials suchas rubbers, have highlynonlinear stresstrainrelationships; theyremainelastic even when they go through a huge deformation. Such materials are described by hyperelasticconstitutive equation.If a material possesses the strain energy density function, w, which is an analytic function of thestrain tensor, and the increment of which is the work done by the stress, then it is called hyperelasticmaterial. For hyperelastic materialw 1r0Sij _Eij(1:122)wherer0 is the mass density in initial congurationSij is the second PiolaKirchhoff stress tensor_Eij is the Green strain tensorOn the other hand, the strain energy function is the analytic function of strain tensor; then_ w @w@Eij_Eij(1:123)1r0Sij@w@Eij _Eij 0 (1:124)Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 33 3.11.2008 3:17pm Compositor Name: BManiMathematical Fundamentals 33If the components of_Eij are mutually independent, from the arbitrariness of_Eij, we haveSij r0@w@Eij(1:125)Equation 1.125 is the hyperelastic constitutive equation.If the volume change in deformation process is negligible, then the components of_Eij are notindependent, but should meet the incompressible condition:ii @Xm@xi@Xn@xi_Emn 0 (1:126)Comparing Equations 1.124 and 1.126, it is known that1r0Sij@w@Eij@Xi@xm@Xj@xmThat is,Sij 1r0@w@Eijh@Xm@xi@Xn@xi(1:127)where h is the hydrostatic pressure, determined by the boundary conditions. Equation 1.127 is theconstitutive equation of incompressible hyperelastic materials.For isotropic hyperelastic materials, the deformation energy density function could be expressedas a function of the invariables of Green deformation tensor:w w(I1, I2, I3)whereI1 Cii 2Eii3I2 CiiCjjCijCji =2 2EiiEjj4Eii2EijEij3I3 detC det@xm@Xi@xm@Xj

(1:128)SoSij r0@w@I1@I1@Eij@w@I2@I2@Eij@w@I3@I3@Eij (1:129)where@I1@Eij 2dij@I2@Eij 2(dijdrsdirdjs)(2Ersdrs)@I3@Eij 2 @Xi@xm@Xj@xmI3

(1:130)Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 34 3.11.2008 3:17pm Compositor Name: BMani34 Handbook of Thermal Process Modeling of SteelsFor an incompressible hyperelastic material, I3 need not be considered. Then,Sij r0@w@I1@I1@Eij@w@I2@I2@Eij h @Xi@xm@Xj@xm(1:131)By dening different deformation energy density functions, different hyperelastic material modelscould by dened. Some commonly used models are given below:1. Neo Hooke model [3,29]w C1(I13) (1:132)2. Moony model [3032]W C1(I13) C2(I23) (1:133)where C1 and C2 are the material constants.The deformationenergydensityfunctions of theabove models are denedbytheinvariables of Green deformation tensor. Because Cii l2i ,li is the principal stretch; thedeformation energy density functions of the models are functions of even power of li, sotheybelongtotheRivilinmaterials. For theRivilinmaterials, thedeformationenergydensity functions can be chosen as functions of higher-order power of I1 and I2. With anincrease in the order, or the number of the items, the agreement of the constitutive equationwith the behavior of real materials will be improved, but the parameters to be determinedincrease, and the computation becomes more complicated. It is found that I3 is not includedin the deformation energy density functions of the above models, so they are incompress-ible hyperelastic materials.3. Ogden model [33,34]Volume change in deformation is taken into account in this model. Its deformation energydensity function isw Nn1mnanJan3lan1lan2lan3 3 4:5K J131 2(1:134)wheremn and an are the material constantsK is the voluminal modulusJ is the ratio of volume change expressed as J l1l2l3Equation 1.134 should be applied in case the ratio of volume change is in the order of 0.01. Ifvolume change is large, the following generalized compressible Ogden model could be used:w Nn1mnanlan1lan2lan33 Nn1mnbn(1 Jbn) (1:135)wheremn, an, and bn are the material constants. Usually N2 or N3 is appropriate.Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 35 3.11.2008 3:17pm Compositor Name: BManiMathematical Fundamentals 351.5.2 ELASTOPLASTIC CONSTITUTIVE EQUATION1.5.2.1 IntroductionTheplasticbehaviorofthemetal at normal temperaturedependsondeformationhistoryofthematerial, hence the plasticow theory, that establishes the relationship between stress rate and strainrate, should be normally used.In establishing the constitutive equation, rst a small deformation problem is considered, andthe relationship betweenCauchy stress and small strain is established. Then the constitutiveequation is extended tonite-deformation problem; here Cauchy stress and small strain should besubstituted by corresponding appropriate stress rate and strain rate. The appropriate stress rate couldbe chosen as Jaumann rate of Cauchy stress, ^ s, and the objective strain rate could be chosen as therate of deformation tensor, d.The elastoplastic deformation is decomposed into a recoverable elastic deformation and an un-recoverableplasticdeformation. Insmalldeformation, thestrainincrementdisdecomposedasfollows:d deadpCorrespondingly, innite deformation, the strain rate, d, is decomposed asd deadpwheresuperscriptse andp mean the elastic and plastic components, respectivelya is the loading factorIn pure elastic deformation, including unloading, a0; in elastoplastic deformation, a1.1.5.2.2 Yield CriterionWhen a metal sample is deformed under uniaxial loading, if the deformation is small, the material isin elastic state, and the relation between stress and strain is linear. When the stress reaches the yieldpoint, ss, thematerial yieldsanddeformsplastically, andtherelationbetweenstressandstrainbecomes nonlinear.Inadeformingbody, thestressdistributionisgenerallynot uniform. For aparticleinthematerial, when the stress components meet a certain condition, the particle will yield. This conditioniscalledyieldcriterion.Theyieldcriterioncanbewritteninthefollowinggeneralmathematicalexpression:F f (sij) ss 0 (1:136)wheref(sij) is the function of the stress componentsss is the initial yield stressTheyieldcriterionmaybegeometricallyrepresentedbyasurfaceinthestressspace, calledyield surface. If the stress, sij, locates inside the surface, i.e., F (sij) 0.Gur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 36 3.11.2008 3:17pm Compositor Name: BMani36 Handbook of Thermal Process Modeling of Steels1.5.2.3 Flow Rule1.5.2.3.1 Associated Flow RuleForordinaryengineeringmaterials, theowstressisanincreasingfunctionoftheplasticstrainin plastic deformation; such material is called a stable material, which meets the conditionDsijDij>0 during plastic deformation. Consequently, successive yield surface, or loading surface,F(sij, Y) is convex. Let thestress spacecoincidewiththestrainspace; thentheplasticstrainincrement must point in the outward normal direction of the yield surface F(sij, Y), i.e.,_ ij _l@F@sij(1:137)where@F=@sij, the geometric meaning of the partial derivative, is the component of the unit outwardnormal vector of the yield surface F, in the stress space_l is the scalar factorAdopting the concept of plastic potential theory, F is taken as the plastic potential function. Whentheyieldcriterionistakenastheplasticpotentialfunction,associatedowrulecanbeobtained.Usingadifferent yieldfunction, different plasticstressstrainrelationshipscanbederivedfromassociatedow rule. If the plastic potential function is not chosen as the yield criterion, the resultingplastic stressstrain relationship is called nonassociatedow rule. For metals, associatedow rule isnormally used.1.5.2.3.2 LoadingUnloading ConditionWorkhardeningusual