Handbook of Thermal 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)This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been 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.For permission to photocopy or use material electronically from this work, please access www.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.Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for 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: VBalamugundanviPrefaceThe whole range of steel thermal processing technology, from casting and plastic forming towelding and heat treatment, not only produces workpieces of the required shape but also optimizesthe end-product microstructure. Thermal processing thus plays a central role in quality control,service life, and the ultimate reliability of engineering components, and now represents a funda-mental element of any companys competitive capability.Substantial advances in research, toward increasingly accurate prediction of the microstructureand properties of workpieces produced by thermal processing, were based on solutions of partialdifferential 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 themselvesrelied on empiricism and traditional practice. The level of inaccuracy inherent in computationalpredictions hindered both materials performance improvements and process cost reduction.Since the 1970s, the pace of development of computer technology has made possible effectivesolution of PDEs in complicated calculations for boundary and initial conditions, as well as non-linear and multiple variables. Mathematical models and computer simulation technology havedeveloped rapidly; currently well-established mathematical models integrate fundamental theoriesof materials science and engineering including heat transfer, thermoelastoplastic mechanics, uidmechanics, and chemistry to describe physical phenomena occurring during thermal processing.Further, evolution of transient temperature, stressstrain, concentration, microstructure, and owcan 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 datehave yielded enormous advantages for some engineering applications. Continued research in thisdirection attracts increasing attention now that the cutting-edge potential of future developments isevident. Increasingly profound investigations are now in train globally. The number of importantresearch papers in the eld has risen sharply over the last three decades. Even so, the existingmodels are regarded as highly simplied by comparison with real commercial thermal processes.This has meant that the application of computer simulation has thus far been relatively limitedprecisely because of these simplifying assumptions, and their consequent limited computationalaccuracy. 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 main objectives of this book are, therefore, to provide 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. A practical 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 to everyone for their contribution, assistance, encouragement, and constructive criticismthroughout the preparation of this book.Here, we also extend our sincere gratitude to Dr. George E. Totten (Totten Associates and aformer president of the International Federation for Heat Treatment and Surface Engineering[IFHTSE]) and Robert Wood (secretary general, IFHTSE), whose initial encouragement madethis book possible, and to the staff of CRC Press and Taylor & Francis for their patience andassistance 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. He is also the director of the Welding Tech-nology and Nondestructive Testing Research and ApplicationCenter at the same university. Professor Gr has publishednumerous papers on a wide range of topics in materials scienceand engineering and serves on the editorial boards of nationaland international journals. His current research includes simula-tion of tempering and severe plastic deformation processes,nondestructive evaluation of residual stresses, and microstruc-tures obtained by various manufacturing processes.Jiansheng Pan is a professor in the School of Materials Scienceand Engineering at Shanghai Jiao Tong University, Shanghai,China. He was an elected member of the Chinese Academy ofEngineering in 2001. Professor Pans expertise is in chemicaland thermal processing of steels (including nitriding, carburiz-ing, and quenching) and their computer modeling and simula-tion. He has established mathematical models of these processesintegrating heat and mass transfer, continuum mechanics, uidmechanics, numerical analysis, and software engineering. Thesemodels have been used for computational simulation to designand 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,Professor Pan was the president of the Chinese Heat Treatment Society (20032007) and is thechairman of the Mathematical Modeling and Computer Simulation Activity Group of the Inter-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: VBalamugundanxii1 Mathematical 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 ....................................................................................................................................... 59Steels are usually under the action of multiple physical variable elds, such as temperature eld,uid eld, electric eld, magnetic eld, plasm eld, and so on during thermal processing. Thus, heatconduction, diffusion, phase transformation, evolution of microstructure, and mechanical deform-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 PDES FOR HEAT CONDUCTION AND DIFFUSIONThe rst step of computer simulation of thermal processing is to establish an accurate mathematicalmodel, i.e., the PDEs and boundary conditions that can quanticationally describe the relatedphenomena.The PDE describing the temperature eld inside a solid is usually expressed as follows:@@x l@T@x_ _ @@y l@T@y_ _ @@z l@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. The rstitem on the left-hand side of the equation is the net heat ux input to the innitesimally smallelement along axis x, i.e., the difference between the heat ux entering dQxin and the heat uxeffusing dQxout. The second and third items are the net heat ux along axes y and z, respectively(Figure 1.1). The intensity of the internal heat source Q may be caused by different factors, such asphase transformation, plastic work, electricity current, etc. The right-hand side of the equationstands for the change in heat accumulating in the innitesimal element per time unit due to thetemperature 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 eachtime unit, so it functions in accordance with the energy conservative law. The heat 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.The rst 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 surfaceThe third boundary condition S3: The heat transfer coefcient between the workpiece andenvironment 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 workpieceh is the overall heat transfer coefcient, representing the heat quantity exchanged betweenthe 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 should also be considered in other complicated ones, such as gas quenching andheating 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-ture eld inside the workpiece at different times, the so-called unsteady temperature eld, can beobtained by solving Equation 1.1. When the temperature eld inside the workpiece does not changewith time any more, it arrives at the steady temperature eld, and the left-hand side of Equation 1.1becomes zero.The unsteady concentration eld inside the workpiece subjected to carburizing or nitriding isusually governed by the following PDE.@@x D@C@x_ _ @@y D@C@y_ _ @@z D@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 mass ux 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)Although the diffusion and heat conduction PDEs describe different physical phenomena, theirmathematical expression and solving method are exactly the same.1.1.2 SOLVING METHODS FOR 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 certain eld 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 practical manufacture environment, which are featuredwith complicated boundary conditions and a high degree of nonlinearity.The numerical solution, also named approximate solution, is applicable for different kindsof boundary conditions and can cope with nonlinear problems. It is the most basic simulationmethod in engineering. Up to now, the nite-element method (FEM) and nite-difference method(FDM) are the most widely used methods in simulation of the process, and their commoncharacteristic is discretization of continuous functions, thus transforming the PDEs into largesystems of simultaneous algebraic equations and solving the large algebraic equation group nally(Figure 1.2).yf1f0f1x1 x0 x1y = 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 INTRODUCTION OF 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@xxx0 f00 f1f0Dx Dx2 f000 % f1f0Dx (1:13)Equation 1.13 is the rst-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@xxx0 f00 f0f1Dx Dx2 f000 % f0f1Dx (1:14)This is the rst-order backward difference with its truncation error of V(Dx).Subtracting Equation 1.12 from Equation 1.11 yields@f@x f00 f1f12 2(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)2 2(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 can be observed that the truncation error, originating from the replacement of the partialderivatives by nite-difference quotients, makes the FDM solution an approximate one; however,the accuracy can be improved by reducing the step size.1.2.2 FDM FOR ONE-DIMENSIONAL HEAT CONDUCTION AND DIFFUSIONIn this section, two simple cases are taken to elucidate the FDM to solve the PDEs in engineering.The rst case is the unsteady, one-dimensional heat conduction PDE without an internal 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 with nite-difference quotients, the differenceequation can be obtained as follows:Tni12Tni Tni1(Dx)2 1aTn1i TniDt (1:18)wherei is the running index in the x directionn is the running index in the t directionWhen one of the independent variables is a marching variable, such as t in Equation 1.17, it isconventional to denote the running index for this marching variable by n and to display this index asa superscript in the nite-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 Tn1i is 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 onlyone unknown Tn1i appears in Equation 1.19. Hence, Equation 1.19 allows for the immediatesolution of Tn1i from the known temperatures at time level n.Equation 1.19 is one of the so-called explicit nite-difference approaches, which provide astraightforward mechanism to 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 2Tn1i Tn1i1(Dx)2 Tn1i TniDt (1:20)With some rearrangement, Equation 1.21 can be written asF0Tn1i1 (2F01)Tn1i F0Tn1i1 Tni (1:21)Observing Equation 1.21, the unknown Tn1i is not only expressed in terms of the known temper-atures at time level n, namely Tni , but also in terms of other unknown temperatures at time leveln 1, namely, Tn1i1 and Tn1i1 . In other words, Equation 1.21 represents one equation with threeunknowns, namely Tn1i1 , Tn1i , and Tn1i1 . Hence, Equation 1.21 applied at a given grid point i doesnot stand alone; it cannot by itself result in a solution for Tn1i . Rather, Equation 1.21 must bewritten at all interior grid points, resulting in a system of algebraic equations from which theunknowns Tn1i for 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 arrayed at a given time level. Because of this need to solve 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 1.21and 1.22, which are the only two of many difference representations of the original PDE. Forexample, writing the spatial difference on the right-hand side in terms of average temperaturesbetween time level n and n 1, Equation 1.17 can be represented bya Tni12Tni Tni12(Dx)2 Tn1i1 2Tn1i Tn1i12(Dx)2_ _Tn1i TniDtF0Tn1i1 2(1 F0)Tn1i F0Tn1i1 F0Tni12(1 F0)Tni F0Tni1 (1:22)This special type of differencing employed in Equation 1.22 is called the CrankNicolson form,which is also unconditionally stable and has a small truncation error of V(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@nx0(1:23)The spatial discretization is shown in Figure 1.4, generating m1 nodes from surface (node 0) tocenter (node m). Here, because of the symmetry half of the slab can only be 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 Ta12 Tn10 Tn0_ __ _ l2Dx Tn10 Tn11 Tn0 Tn1_ _Dxrcp2Dt Tn10 Tn0_ _ (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)Tn10 F0Tn11 (1 F0Bi)Tn0 F0Tn1 2BiTa (1:25)where Bi hDt=Dx.For the central node (i m), the adiabatical boundary @T=@x 0 is input, and the differenceequation is(1 F0)Tn1m1F0Tn1m1 (1 F0)TnmF0Tnm1 (1:26)The nite-difference form for 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 temperature eld can usually be obtained by a mature algorithm.The one-dimensional diffusion PDE (Equation 1.27) and its difference equations have the sameform as that of the unsteady, one-dimensional heat conduction without an internal heat resource.Hence, the corresponding equations are briey repeated and then entered into solving of thealgebraic equation group.D@2C@X2 @C@t (1:27)The boundary conditions at the surface node and the center node are the third type of boundarycondition and the adiabatical condition, respectively, and are listed as follows:D@C@Xx0 b(CgCs)D@C@Xxm 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)Cn1i F0Cn1i1 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)Cn10 F0Cn11 (1 F0Bi)Cn0F0Cn12BiCg (1:30)For the central node (i m), the difference equation is(1 F0)Cn1m F0Cn1m1 (1 F0)CnmF0Cnm1 (1:31)The algebraic equation group constituted by Equations 1.29 through 1.31 can be expanded andwritten in the following matrix form:d0 a0b1 d1 a1.........bi di ai.........bm1 dm1 am1bm dm____C0C1...Ci...Cm1Cm____d00 a00b01 d01 a01.........b0i d0i a0i.........b0m1 d0m1 a0m1b0m d0m____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 obtaind0 a0b1 d1 a1.........bi di ai.........bm1 dm1 am1bm dm____C0C1...Ci...Cm1Cm____n1F0F1...Fi...Fm1Fm____(1:33)The coefcient matrix on the left-hand side of Equation 1.33 is a tridiagonal matrix, dened ashaving 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 the rst term of eachequation, replacing the coefcient of the main-diagonal term by Equation 1.34, and replacing theright-hand side with Equation 1.35.di* di bidi1* ai1 (i 1, 2, 3, . . . , m 1, m) (1:34)Fi* Fi bidi1* Fi1* (i 1, 2, 3, . . . , m 1, m) (1:35)Then Equation 1.33 transforms into an upper bidiagonal form as follows:d0 a0d1* a1.........di* ai.........dm1* am1dm*____C0C1...Ci...Cm1Cm____n1F0F1*...Fi*...Fm1*Fm*____(1:36)The elements with the superscript asterisk are those subjected to Gaussian elimination. The lastequation in Equation 1.36 has only one unknown. Solving Cn1m from Equation 1.37, all otherunknowns are found in sequence from Equation 1.38, starting from Cn1m1 to Cn11 .Cn1m Fm*dm* (1:37)Cn1i Fi*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 Cn10 can be obtained byCn10 F0a0Cn11_ _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 applied but for objects with relatively simple shapes due to its limitation in coping withcomplicated-shape boundaries. Hence, the FEM method is mainly used in the simulation oftemperature elds and concentration elds with three-dimensional complicated-shape parts.1.3 FINITE-ELEMENT METHODThe FEM, sometimes referred to as nite-element analysis (FEA), is a computational technique usedto obtain approximate solutions of boundary value problems in engineering. Simply stated, aboundary value problem is a mathematical problem in which one or more dependent variablesmust satisfy a differential equation everywhere within a known domain of independent variablesand satisfy specic conditions on the boundary of the domain. Usually, FEM divides the denitiondomain into reasonably dened subdomains (element) by hypothesis and supposes the unknownstate 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 functionscan be harmonized at element nodes or certain joint points, the unknown function can approxi-mately be expressed in the whole denition domain.Because of the extraordinary exibility of element division, the FEM elements can t well toobjects with complex shape and the boundaries with complex curved surfaces. For example,complex three-dimensional regions can be effectively lled by tetrahedral elements, similar totriangular elements lling 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, uid ow, 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 the nal 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,boundary conditions, loading, and materials information. It is generally described as dening themodel 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, the nite-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 heat ow.1.3.1.3 Stage 3: PostprocessingAnalysis and evaluation of the solution results is referred to as postprocessing, so the postprocessingstage deals with the presentation of results. Typically, the deformed conguration, mode shapes,temperature, and stress distribution are computed and displayed at this stage. Postprocessor softwarecontains sophisticated routines used for sorting, printing, and plotting selected results from a nite-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 solution data can be manipulated in many ways in postprocessing, the most importantobjective is to apply sound engineering judgment in determining whether the solution results arephysically 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 FEM FOR TWO-DIMENSIONAL UNSTEADY HEAT CONDUCTIONThe method of weighted residuals, especially the embodiment of the Galerkin FEM, is a powerfulmathematical tool that provides a technique for formulating a nite-element solution approach topractically any problem for which the governing differential equation and boundary conditions canbe written.Here, two-dimensional unsteady heat conduction is taken as an example to derive the FEMformulations by Galerkins weighted residual method.The governing PDE for unsteady, two-dimensional heat conduction with an internal heatresource isl @2T@x2 @2T@y2_ _Q rcp@T@t 0 (1:40)The initial condition, supposing the temperature eld 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 the temperature eld, is substituted. On the contrary, the approximate solution T makes theresidual 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__DWi l @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 exist and the techniques vary primarily inhow the weight functions are determined or selected. The most common techniques are pointcollocation, subdomain collocation, least squares, and Galerkins method [4].The second-order differential item in Equation 1.43 can be transformed into a rst-order item byintegration in parts. Therefore, the second-order differential item can be expressed as__DWi l @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 on boundary S1 is 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 Wi rcp@T@t Q_ _dx dy _S2Wil@T@n ds _S3Wil@T@n ds (1:46)The element analysis is to establish the formulations by which the continuous function in thesubdomain DD can be expressed by the node values, i.e., the function of the node values. Supposingthere are n elements in the solution domain, and there are m nodes in each element with thetemperature Ti (i 1, 2, . . . , m), the unknown temperature Te(x, y, z) dened in the element can beexpressed 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 xjxiAlthough the shape functions for different types of elements have different forms, they havecommon 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).Equation 1.46 is applicable in the whole solution domain D, so for its subdomains, i.e., eachelement DD, we have__DDl @Wi@x@T@x @Wi@y@T@y_ _dx dy Wi rcp@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 Ni rcp@T@t Q_ _dx dy_DS2Nil@T@n ds _DS3Nil@T@n ds (1:51)Substituting Equation 1.47 into Equation 1.51, and introducing the boundary conditions fromEquations 1.2 through 1.4, we have__DDl@Ni@x@Ni@x Ti@Nj@x Tj@Nm@x Tm_ _l@Ni@y@Ni@y Ti@Nj@y Tj@Nm@y Tm_ _ _ Nircp Ni@Ti@t Nj@Tj@t Nm@Tm@t_ _NiQ_dx dy_DS2qNids _DS3hNi(NiTiNjTjTa)ds (1:52)Taking WjNj(x, y) and WmNm(x, y), and deriving in exactly the same way, can be obtainedas follows:__DDl@Nj@x@Ni@x Ti@Nj@x Tj@Nm@x Tml@Nj@y_ _ @Ni@y Ti@Nj@y Tj@Nm@y Tm_ _ _ Njrcp Ni@Ti@t Nj@Tj@t Nm@Tm@t_ _NiQ_dx dy_DS2qNjds _DS3hNj(NiTiNjTjTa)ds (1:53)__DDl@Nm@x@Ni@x Ti@Nj@x Tj@Nm@x Tm_ _l@Nm@y@Ni@y Ti@Nj@y Tj@Nm@y Tm_ _ _ Nmrcp Ni@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 _DS3hNiNi NjNi 0NiNj NjNj 00 0 0____ds (1:56)Substituting Equation 1.48 into Equation 1.56, and integrating, we obtain[K]e l4Ab2i c2i bibjcicj bibmcicmbjbicjci b2j c2j bjbmcjcmbmbicmci bmbjcmcj b2mc2m____hlij62 1 01 2 00 0 0____ (1:57)whereA is the area of element DDlij 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__DDrCpNiNi NjNi NmNiNiNj NjNj NmNjNiNm NjNm NmNm____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, heat ux in the second kind ofboundary condition, and the heat transfer coefcient in the third kind of boundary condition,respectively.The item {pQ}ecan be obtained by{pQ}e_DDQNi dx dy _DDQNj dx dy _DDQNm dx 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}e A122Qi Qj QmQi 2qj QmQi Qj 2Qm____ (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_DS2qNi ds _DS2qNj ds _DS2qNm ds (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_DC3hTaNi ds _DC3hTaNj ds _DC3hTaNm ds 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, the continuous function Te(x, y, t) on the subdomains, i.e., the element 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)whereGeneral stiffness matrix: [K]

[K]e(1:69a)Heat capacity matrix: [C]

[C]e(1:69b)Heat flux matrix: {P}

{P}e(1:69c)The solving of Equation 1.69 brings the solution of the temperature eld for the unsteady two-dimensional heat conduction. The evolution of temperature can be observed by recording the nodetemperature at each time level, and the heating (or cooling) curves at special points can also beextracted from the result les.1.3.3 FEM FOR THREE-DIMENSIONAL UNSTEADY HEAT CONDUCTIONThe principles and procedure in the FEM for three-dimensional unsteady heat conduction areidentical 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 PDE of three-dimensional unsteady heat conduction with an internal heatresource is repeated here.l @2T@x2 @2T@y2 @2T@z2_ _Q rcp@T@t (1:70)Supposing that there are total n elements (m nodes in each element) and p nodes after thediscretization of the whole solution domain, Equation 1.70 can be transformed into an algebraicequation 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 boundariesIn the unsteady temperature eld, the variable temperature T(x,y,z,t) is a function of spatiallocation and time. Equation 1.71 is the spatial discretized formulation with the time differentialitem @T=@t, which can be discretized by the difference method in a uniform format as follows:u @T@t_ _t(1 u) @T@t_ _tDt 1Dt(TtTtDt) (1:72)If u 1, the backward difference format of time discretization is@T@t_ _t 1Dt(TtTtDt) (1:73)Substituting Equation 1.73 into Equation 1.71, we get[K] 1Dt[C]_ _{Tt} 1Dt[C]{TtDt} {P} (1:74)If u 1=2, the CrankNicolson difference format (also named central difference format) of timediscretization 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 algebraic equation 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 afashion similar to the FEM, in which the smaller the elements are, physically, the better is thesolution, the FDM converges more rapidly to the true solution as the time step is decreased.When the temperature eld at time level t Dt is known, and input into Equation 1.74 orEquation 1.76, the temperature {Tt} on all nodes at time level t can thus be solved, and themarching solution can progress in time until a steady state is reached. The nodal temperatures,recorded step by step at different time levels, constitute the evolution history of the three-dimen-sional temperature eld.1.4 CALCULATION OF TRANSFORMATION VOLUME FRACTION1.4.1 INTERACTIONS BETWEEN PHASE TRANSFORMATION AND 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 temperature eld inside. The interactions between phase transformationand temperature are not one-way, but bilateral, increasing the complexity in the accurate numericalsimulation of the thermal processing.The latent heat per time unit, expressed as the internal heat source in the heat 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 step DtThe calculation of the volume fraction is the key to predict the evolution of temperature andmicrostructure during the thermal processing, as well as the nal microstructure constituents andrough mechanical property. Since phase transformations are usually classied into two categoriesaccording to their mechanisms, diffusion transformation (for example, pearlite transformation),and nondiffusion transformation, i.e., martensitic transformation, the mathematical models of thetransformation kinetics are very different.In this section, the numerical method to calculate the transformed volume fraction of a newphase is mainly introduced.1.4.2 DIFFUSION PHASE TRANSFORMATIONA timetemperaturetransformation (TTT) diagram describes the relationship between the trans-formation starting, ending, and the transformed volume fraction during the isothermal process atdifferent temperatures. The isothermal kinetics equation, namely JohnsonMehl equation [8],provides a solid base for numerical simulation of thermal process although it cannot be directlyapplied to calculate the volume fraction due to the nonisothermal process of the practical heatingor cooling process. Until now, the method proposed by Fernandes et al. [9] has been widelyaccepted, in which the practical nonisothermal process is considered as many isothermal stageswith a tiny time duration as shown in Figure 1.5, and the effect of these stages can be summedtogether 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 b and n at different temperatures are usually calculated from the experimentallyobtained 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-atures can be calculated by Equations 1.78 through 1.80. Most experimental data of phase trans-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 or

ni1DtitTTTi 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 processtTTTi is the incubation period under a certain isothermal temperatureDti is the time stepIt can be understood that the transformation in the nonisothermal process occurs when thesummation of the relative fractions Dti tTTTi_ reaches unity.Scheils additivity rule has also been applied in volume fraction calculation. For example, 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 to transform the fraction jm isothermally at the current isotherm Tt.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* andctitious transformed volume fraction fi* are commonly developed [9]. These two variables canbe calculated byti* ln(fi1)bi_ _1=ni(1:83)fi* 1 exp bi ti*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 1.78 through 1.85 constitute the basic frame of volume fraction calculation for 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]and Reti and Felde [13] held the same viewpoint that the additivity rule sometimes seriouslyoverestimates the incubation time.Taking the eutectoid steel as the example, the practical incubation period tCCTs under certaincooling 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 23from CCT and TTT diagrams, respectively. Hence, the relative durations of each tiny isothermalstage Dt tTTTs_ can be summed stage by stage from zero to tCCTs byx

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 averagecooling rate during arbitrary time step Dti on the practical cooling curve, that is, the slope of thetangent 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 curvedddd ViAclStarting 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 curvetCCTs in 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)The modication coefcient wi reects the difference in incubation period between the CCT andTTT 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 duration Dti=tTTTs(i) for the time step Dti can be modied as (Dti=tTTTs(i) )(1=fi).Hence, for the transformation starting time, namely, the incubation period for the isothermalprocess, the criterion is modied as follows:

DtitTTTs(i) 1wi 1 (1:89)The theoretical derivation of this modication is still in the process of consummation, and theapplication 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 neededto get a fraction of 0.25 of the new phase). In the algorithm 2 based on the modied Avramiequation, t0.5 and t0.75 are the time to transform the new phase of the volume fraction of 0.5 and0.75, respectively, both of which include the incubation period ts and can be got from the TTTdiagram. With the help of Equations 1.90 through 1.92, the coefcients n, b, and t0.25 are alsocalculated. All these data are listed in Table 1.2. It is obvious that the modied Avrami equationseems more reasonable and ts 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 in the hypoeutectoid steel can be explained by Figure 1.7. When the steel iscooled rapidly until the temperature is below line SE0, the stable ferrite a and cementite Fe3Cprecipitate simultaneously and form the quasieutectoid. However, if the temperature is abovepoint a, for example, at point a0, the nucleus of ferrite appears rst at the austenite grain boundary,and the ferrite grows up as the time goes on. At the same time, carbon diffuses into the neighboringaustenite grain due to its low dissolvability in ferrite, increasing the carbon concentration inaustenite. When the concentration reaches point b located on the boundary of two-phase areaaFe3C, 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.The nucleation sites and the mode of the proeutectoid ferrite are theoretically different fromthose of eutectoid pearlite. Although their growth is controlled by diffusion, the growth mode anddiffusion route of carbon are different. These factors support the viewpoint that the fraction ofproeutectoid ferrite and pearlite should be calculated separately, that is, two sets of independentTABLE 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 thereis enough space for growth at the initial stage. However, with the growth of the new phase, 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 hardto determine because the carbon content of austenite varies with the precipitation of proeutectoidferrite. 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.Assuming that the steel is isothermally kept for enough long time to precipitate the proeutectoidferrite only and the pearlite transformation does not occur, the nal volume fraction of ferrite fFendunder certain temperature Ti can be calculated by lever rule asfFend a0b0cb0 (1:95)Therefore, the denition domain of the Avrami equation describing the proeutectoid ferrite is[0, fFend]. However, the practical situation is that the pearlite transformation usually occurs beforethe volume fraction of proeutectoid ferrite reaches fFend. Thus, there exists a maximum volumefraction of proeutectoid ferrite fFmax at a certain temperature, which can be calculated asfFmax ba0cb (1:96)When the volume fraction of proeutectoid ferrite reaches fFmax, the carbon concentration in theretained 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: the rst 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:If fi < fFmax, then fFi fifPi 0_ (1:97)If fi ! fFmax, then fFi 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

tCCTetCCTsDtitTTTei tTTTsi(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-ition process of austenite in hypoeutectoid steel can be reasonably described by one Avramiequation.1.4.3 MARTENSITIC TRANSFORMATIONIn most cases, the transformed volume fraction of martensite is independent of the cooling rate, but afunction of temperature. The KoistinenMarburger [18] equation is always adopted in thermalprocess simulation, and it isfM 1 exp a(MsT) (1:100)FerritePearliteFerriteVolume fractiontsF tsPTimePearlite10fmaxFfendFFIGURE 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)whereV is the average volume of martensite lamellaw is the ratio of the martensite number newly formed to the driving force in the austenite ofunit volumeDGg!Mv is 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 thecarbon diffusion. Therefore, the KoistinenMarburger equation should be modied for the lowcarbon steel, and it becomesf 1 exp b(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, the volume of martensite in the carbon steel and alloy steel is calculated usingEquations 1.100 and 1.102, and the Ms point can be obtained by experiment and the followingempirical 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, and so on, occurheretofore, changing the composition of the parent phase or consuming the embryos. It can beGur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 29 3.11.2008 3:17pm Compositor Name: BManiMathematical Fundamentals 29observed from the CCT curves that the more the austenite decomposes, the more the Ms pointdecreases. The phenomenon has not been generally reected in the additivity rule in calculation ofvolume fractions using TTT curves. Hence, the calculation model, i.e., the KoistinenMarburgerequation, for martensite is in great need of further amendment based on transformation mechanismsto improve its universality and accuracy.1.4.4 EFFECT OF STRESS STATE ON PHASE TRANSFORMATION KINETICSThe stress state has considerable effects on the phase transformation kinetics irrespective of thediffusion 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,the research in this eld is still far from satisfactory. Some mathematical models proposed bydifferent 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 and Wang [26] developed the 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 constantDenis et al. [27] introduced the effect of stress into the coefcients (n and b) in the Avramiequation (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 TransformationFor martensitic transformation, the stress state in the steel has much greater effects on the trans-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 exp f(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 exp a(MsAsmBsiT) AsmBs1=2i_ _ _ _ (1:113)Compared to Equation 1.112, this equation additionally considers the effect of stress on the Mspoint, which can be clearly expressed as an extra item DMs [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 EquationIn a solid under deformation, the stress depends on the current strain state, and is a single valuefunction 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 Ceijkl is the elastic tensor, the superscript e means elasticGur/Handbook of Thermal Process Modeling of Steels 190X_C001 Final Proof page 31 3.11.2008 3:17pm Compositor Name: BManiMathematical Fundamentals 31Such a material model is called linear elastic constitutive equation. The incremental form of theabove equation isdsij Ceijkl dekl (1:117)Usually metals can be taken as isotropic, linear elastic materials or ideal elastic materials. For suchmaterials, the elastic tensor is a fourth-order isotropic tensor, including only two independentparameters, as follows:Ceijkl 2G dikdjl v1 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)In nite deformation, to meet the requirement of objectivity, the Jaumann rate of Cauchy stresstensor and the rate of deformation tensor, which possess objectivity, are adopted, and the elasticconstitutive equation is rewritten as^ sij Ceijkldekl (1:119)For the convenience of nite-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:Ce 2G1 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 symbol s 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 2nE sm1.5.1.1.2 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 isCeC11 C12 C12 0 0 0C12 C11 C12 0 0 0C12 C12 C11 0 0 00 0 0 C44 0 00 0 0 0 C44 00 0 0 0 0 C44____(1:121)1.5.1.2 Hyperelastic Constitutive EquationSome materials such as rubbers, have highly nonlinear stresstrain relationships; they remainelastic 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 densit