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Thermo-mechanical modelling of hot flat-rolling of components with curved profiles Xiao, Ming 1993-12-31

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We accept this thesis as conformingto the required standardTHERMO-MECHANICAL MODELLING OFHOT FLAT-ROLLING OF COMPONENTS WITH CURVED PROFILESByMing XiaoB. ASc. (Electronic-Mechanical Engineering)Guilin Electronic Industry College, 1984A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESMECHANICAL ENGINEERINGTHE UNIVERSITY OF BRITISH COLUMBIAJune 1993© Ming Xiao, 1993In presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.Mechanical EngineeringThe University of British Columbia2075 Wesbrook PlaceVancouver, CanadaV6T 1Z1Date:/ 9 9 3ABSTRACTA complete mathematical model for implementing the VGR process(a flat rolling processfor producing components with variable thickness) under hot working conditions has beenestablished. The work is based on an idea that a combination of separate submodels,such as, deformation, flow stress, roll force, temperature, etc., should be used, so thatnew development in each of these areas can be easily incorporated.A deformation submodel, based on the upper bound theorem, is given for the analysisof three-dimensional deformation of the workpiece. A simple velocity field is proposed. Topreserve the theoretical consistency, an equivalent coefficient of friction is adopted for theroll force calculation. Furthermore, the basic assumption, i.e., the rigid perfectly-plasticmaterial assumption, is modified by introducing a concept of an isotropic rate-dependentmaterial. Satisfactory results were obtained in spread, torque and force prediction.To obtain the mean temperature of the workpiece, a temperature submodel is formu-lated based on one-dimensional transient flow in the roll-bite and two-dimensional flowoutside the roll-bite. The model is capable of predicting the through-thickness temper-ature distribution in the roll-bite, estimating the mean temperature of the deformingbody, and roughly calculating the mean temperature distribution along the workpiece.To characterize the high temperature behaviour of steels, the well-known unified creeprelationship is chosen as the flow stress submodel. Reasonably accurate prediction of theflow stress is achieved by using some experimental data reported in the literature.1 1Table of ContentsABSTRACT^ iiList of Figures^ viiNotation^ xiACKNOWLEDGEMENT^ xiii1 INTRODUCTION 11.1 Deformation Analysis of Rolling Process — An Overview ^ 11.2 Manufacturing of Single Leaf Springs ^ 21.3 Aim and Approach ^ 42 BASIC CONSIDERATIONS 62.1 Available Analytical Methods ^ 62.2 Assumptions ^ 102.3 Coordinate Systems ^ 123 CONSTRUCTION OF VELOCITY FIELDS 143.1 The Dual-Functional Method ^ 143.2 A Brief Derivation of the Solution to the Velocity Fields ^ 153.3 The Derivation of the Velocity Fields ^ 163.4 Discussion of the General Solution 203.5 Further Considerations ^ 22Hi4^3.5.1^Theoretical Aspect ^^3.5.2^Technical Aspect FORMULATION THROUGH UPPER BOUND APPROACH2223264.1 The Strain Rates ^ 274.2 The Energy Rates 284.2.1^The energy dissipation rate due to deformation(Ed) ^ 284.2.2^The energy dissipation rate due to friction between roll surfacesand workpiece(Ef) ^ 294.2.3^The energy dissipation rate due to velocity discontinuities (E,) 304.3 The Minimization Procedure ^ 314.3.1^The derivatives of the geometry ^ 324.3.2^The neutral point ^ 324.3.3^Solutions ^ 335 FLOW STRESS MODEL 345.1 Dynamic Flow Stress ^ 345.2 Incorporation of Rate Dependency ^ 365.3 Mean Values of Strain and Strain Rate 385.3.1^Mean Strain ^ 385.3.2^Mean Strain Rate 405.4 Temperature Prediction ^ 415.4.1^Basic Considerations 415.4.2^Heat Conduction Model ^ 435.4.3^Assumptions ^ 435.4.4^Initial and Boundary Conditions ^ 465.4.5^Solution to the Heat Conduction Equation ^ 47iv5.4.6 Heat Transfer Coefficient at Workpiece-Roll Interface ^ 495.4.7 Convection and Radiation ^  525.4.8 Mean Temperature of the Deforming Body in the Roll-Bite ^535.5 Flow Stress Models ^  545.5.1 Model for Steels  ^545.5.2 Using Lead as a Modelling Material ^  556 MODEL VERIFICATION^ 576.1 Flow Stress Model Verification  576.1.1 Empirical Constants in Flow Stress Model ^  576.1.2 Determination of Flow Stress at any Desired Conditions ^ 606.2 Temperature Model Verification ^  616.2.1 Verification Using Data from Conventional Rolling ^ 616.2.2 Simulation of First Pass in VGR Process ^  626.3 Deformation Model Verification ^  636.3.1 The Side Spread Estimation  636.3.2 Torque Estimation ^  646.3.3 Roll Separating Force Estimation ^  646.4 Concluding Remarks ^  666.4.1 Summary and Conclusions ^  666.4.2 Suggestions for Further Work  68Bibliography^ 77Appendices^ 82A ANALYTICAL SOLUTION TO THE UPPER BOUND EQUATIONS 82A.1 The neutral point ^  82A.1.1 The derivative of Ed against w1 ^  82A.1.2 The derivative of Ef against w1  ^84A.1.3 The derivative of t, against w1  ^85A.1.4 The derivatives of td, E1 and ti, against xn ^ 86B TRUE STRESS & STRAIN CORRELATION FOR C-Mn STEELS 88viList of Figures1.1 The schematic illustration of the profiles of single leaf springs. a. Taperedprofile. b. Parabolic profile. c. Arbitrary profile ^32.1 The schematic illustration of ring-rolling configuration  ^72.2 (a) The illustration of fishtail-shaped sides in ring-rolling. (b) The illus-tration of the slip-line field for flat-end tool forging.  ^82.3 Schematic illustration of the part of a bar in rolling  ^112.4 The coordinates system ^133.1 Schemetic illustration of the top view of a rectangular bar between the rolls 183.2 Side views(a, c) and plan views(b, d) of 1S alloy specimens deformed at40% reduction; height: 26 mm, width: 26 mm ^  193.3 The schematic illustration of the simplified side profile of the section be-tween the rolls  ^203.4 The schematic illustration of a parabolic profile in rolling configuration  ^244.1 The schematic illustration of the velocity vector space in conventionalrolling process  ^305.1 Schematic flow curve for metals at elevated temperature. From point 0 to1, the plastic strain rates increase rapidly. Note that the scale for the strainis arbitrarily chosen from typical C-Mn steels[27] to illustrate a possiblerange in applications  ^355.2 Schematic setting of plane compression test  ^38vii5.3 Schematic relation between mean yield stress and mean equivalent strainin plane compression test ^  395.4 Thermal conductivity as a function of temperature for 3 carbon content0.08 C, 0.4 C, 0.8 C in austenite phase. (From Devadas, 1989) ^ 455.5 Specific heat as a function of temperature for 3 carbon content 0.08 C, 0.4C, 0.8 C in austenite phase. (From Devadas, 1989) ^ 465.6 Thermal conductivity as a function of temperature for C-Mn steels .^475.7 Specific heat as a function of temperature for C-Mn steels ^ 485.8 Example for heat transfer coefficient at workpiece-roll interface (lubricant:water) ^  505.9 Flow stress model: the correlation between yield-stress and reduction ratioof pure lead. Data source: [56]  ^566.1 Relationship between ln(sinh(ao-)) and ln e for a C-Mn steel (0.03 C 0.62Mn) at a strain of 0.35. Experimental data: o — (i = 2s-1), * — (i =-20s-1) and + — (e = 140s-1). Data source: [53]  ^586.2 Relationship between ln(sinh(ao-)) and 1/T for a C-Mn steel (0.03 C 0.62Mn) at a strain of 0.35. Experimental data: o — (i = 2s-1), * — (i =20s-1) and + — (i = 140s-1). Data source: [53]  ^596.3 Flow stress model verification: C-Mn steel (0.03 C 0.62 Mn). Data source:[53]. +, *, x — experimental data. Solid or dotted lines — model predictions. 696.4 Temperature distribution in roll-bite. Data source:[37]. Top: 3D-meshplot. Middle: grid used. Lower left: front view of the 3D-mesh. Lowerright: surface temperature comparison. Reduction ratio: 35%. Entrytemperature: 1012 °C.  ^706.5 Sample calculation: mean temperature distribution of the workpiece forthree different rolling conditions. Workpiece dimension(before rolling):21.6 x 21.6 x 800mm. Reduction ratio: 8% - 50.8%. Furnace temperature:1200 °C. Top(case A), time for passl: 6.015 sec. Middle(case B), time forpassl: 3.008 sec. Bottom(case C), time for passl: 3.008 sec.   716.6 Sample calculation: effects of changes in geometry and roll velocity onmean temperature of the deforming body in the roll-bite during the firstpass. Furnace temperature: 1200 °C. Workpiece dimension(before rolling):21.6 x 21.6 x 800mm. Reduction ratio: 8% - 50.8%  726.7 Spread prediction - comparison between simulation results and experi-ments[551. Roll diameter: 101.6 mm(4 in). Specimen dimension: 9.525 x9.525 mm(0.375 x 0.375 in)   736.8 Torque prediction: comparison between simulation results and experi-ments[57]. Roll diameter: 127 mm(5 in). Specimen dimension: 12.7 x 19.05mm(0.5 x 0.75 in)   746.9 Schematic illustration of the roll force vectors layout in flat rolling process 756.10 Roll force prediction: comparison between simulation results and experi-ments[57]. Roll diameter: 127 mm(5 in). Specimen dimension: 12.7 x19.05mm(0.5 x 0.75 in)   76B.1 True- stress and strainare experimental dataB.2 True- stress and strainare experimental dataB.3 True- stress and strainare experimental data ^relationships for 0.01 C 0.19 Mn steels.899091relationships for 0.03 C 0.62 Mn steels.relationships for 0.19 C 0.64 Mn steels. ixB.4 True- stress and strain relationships for 0.38 C 0.64 Mn steels. * 0 X2^2^2 +are experimental data ^  92xNotation0) 1^subscripts for entrance and exit, respectivelyi) f^subscripts indicating the initial and final conditionssurface area of the workpiece, m2Stefan-Boltzmann constant, 5.67x 10- 8 wm-2K-4control parameter in the roll gap functionspecific heat of the workpiece, Jkg-1K-1Ed, E1, E3 energy dissipation rates due to deformation, friction andshear, respectivelyroll separating forceFT^frictional force at the interface between rolls and workpieceFR^overall radial force on the rollG(C)^the roll gap functionconvection heat transfer coefficienth(x)^the thickness functionH(r)^heat transfer coefficientJ.^upper-bound on powerthe shear yield stressk,^thermal conductivity of the workpiece, kWm-2K-11^the length of the projected arc in contactfriction shear factorheat generationthe prescribed flow factorxiQ.^activation energy, kJ/molthe reduction ratio: r = (to — ti )/t0Rg^the universal gas constant, 8.31 J/mol Kroll radiusthe thickness functionthe rolling torqueconstant temperature, with different subscripts (used in Chapter 5)T,.^roll surface temperature, °CT,^surface temperature of the workpiece, °CT(x, 7-)^temperature field in workpiece, °C^ volume of the workpieceVR^roll peripheral velocityVx, Vy, V,^velocity components in the x, y, z directionsw(x)^the width functionXn^neutral pointAV^velocity discontinuityemmisivity of the workpiece, Jkg-1K'strain rate tensorsmean strainmean strain rate/leg^equivalent coefficient of frictionP.^density of the workpiece, (for steel, 7600 kgm-3)the flow stress of the materialTo^the friction shear stresstime variablexiiACKNOWLEDGEMENTI would like to express my sincere thanks to Professor F. Sassani for taking the author asa graduate student and providing the inspiration for this work. His valuable discussionsand helpful suggestions throughout the project are particularly apperciated. I would liketo thank Professor I. Yellowly for his suggestion of using the upper bound approach inthis work at first place, and his stimulating discussions at the times when I was in needof help.Financial asistance from NSERC is gratefully acknowledged. I would also like tothank computer system managers, G. Rohling and A. Steeves, for their help in using thecomputers in the Department.I wish to express my appreciations for my fellow students (also my friends) in theDepartment for their constant interactions and encouragement, in particular, RaminArdekani and Fariba Aghdasi.I take this opportunity to express my gratitude for the support and encouragementshown by my parents, to which I am in debt for a lifetime.TO MY PARENTSxivChapter 1INTRODUCTION1.1 Deformation Analysis of Rolling Process — An OverviewThe history of the rolling process can be traced to more than four hundred years back,whereas, the theoretical analysis, mainly within the plane-strain assumption, of the pro-cess began much later, probably in the 1920's[1].Apparently because the plane-strain analysis was much simpler and was able to meetthe requirements in steel production, the most widely known and used theoretical modelsto date were derived by assuming plane-strain deformation, such as Orowan's model [1]and Von Karman's equation [1).Usually, all the earlier passes in the hot rolling of slabs were scheduled by meansof empirical methods, and only the few final passes were of greater concern, where theparameters of the products were obtained. In such situations, the plane-strain assumptionworked very well, since in most cases, the width-to-thickness ratio of the material inrolling is much larger than 10 and thus the spreads in width can be ignored. It wasobserved in the operation of the strip mills that the spreads in width seldom exceed 1 to2 percent [2].With increasingly stringent demands for accuracy, innumerable efforts have been putinto the development of theoretical models for flat rolling during the last seven decades.Almost all the successful comprehensive(namely, various factors were included, and usu-ally also combined hot and cold rolling) models for the strips and plates rolling were1Chapter 1. INTRODUCTION^ 2evolved from plane-strain assumption, not only because it was most reasonable, but alsobecause the resulting models were readily adaptable for engineering use and rather con-venient for the programming of control computers. It would not be an exaggeration thatif one would state that the plane-strain analysis in the rolling process has culminatedwith the help of the rapid advances in the computer technology in the recent years. Orin other words, if one were seeking a solution to a plane-strain problem in the traditionalrolling process, where the roll-gap is constant, most likely he would find out, amongstthe vast literature on the related subjects, that the work has already been done for sometime.Although the research and development of the rolling process have reached such aencouraging stage, however, all the theoretical works, including quite a few successfulattempts made in the last three decades, where three dimensional deformation analyseswere worked out, say, [3-6], were originated from constant roll-gap geometries, for auniform thickness of the products has been one of the primary goals in flat rolling.Having considered this situation, one may realize that a detailed review on the literaturewould not be helpful, and therefore is omitted here.1.2 Manufacturing of Single Leaf SpringsAt the beginning of 70's, the automobile industry in Britain began to make tapered leafsprings using the flat rolling process, where the roll-gap was designed as a variable so thatit could be changed continuously while rolling and produced a tapered leaf, which hadthickness variations along its longitude, figure 1.1a. Ever since, the world-wide demandfor the tapered leaf springs has been increasing. The distinct advantage of such a processis its flexibility, in producing an arbitrary set of parameters of the tapered shape, thatChapter 1. INTRODUCTION^ 3can not be achieved by any other tooling structure.It has been well-known for a long time that the most efficient type of leaf springs isthose with a parabolic profile [7], figure 1.1b, but the preparation for such shaped materi-als is much too difficult. The reason that the industry wound up in the manufacturing ofthe tapered leaf springs is that the tapered leaf is a compromise between the difficultiesinvolved in the manufacturing and its efficiency.CFigure 1.1: The schematic illustration of the profiles of single leaf springs. a. Taperedprofile. b. Parabolic profile. c. Arbitrary profile.Although, in the last two decades, a number of mills were set up around the worldspecifically for making of the tapered leaf springs, no theoretical work on the subject hasappeared in the literature.The only research work on this subject that one may find out in the literature, [8],[9] and [10], was based on the empirical formula for side spreads prediction, which wasoriginated from factor-analysis in 1950's, proposed by Sparling[11] in 1961 and improvedby El-Kalay and Sparling[12] in 1968. It is well-known that empirical formulae are oftenChapter 1. INTRODUCTION^ 4simple and handy to use. Chitkara and Johnson[13] found out that the formula gavereasonable results within the range of experimental conditions under which they weredevised, "however, their accuracy under different conditions is open to question." Thepresent author considers that it could be a good idea to take advantage of the formula,if the purpose of the research is only to give fair estimates in the deformation, that is,the spreads and the elongation. As a matter of fact, good results in their work werereported[8][9][10]. However, the formula may not be adequate for machine set-up andprocess design, where a consistent model is needed in order to calculate deformation alongwith the force and torque variations during production. Moreover, temperature effectsare dominant in hot rolling, to which much attention should be given. It would be mostdesirable to have a model capable of incorporating the most recent developments, sayin tribology and material science, so that a thorough knowledge of the deformation andforce of the process may be obtained, thereby confidence may be gained for the designpurpose.After a careful study of the important previous works on the flat rolling process, theauthor believes that any reasonable theoretical analysis of the variable gap rolling (VGR)process' has to start with the ever-changing geometry of the workpiece, and one has togive up the idea of making a direct use of any available theoretical models in flat rollinganalysis.1.3 Aim and ApproachAs mentioned in Section 1.1, only in cases where the spread in width is negligible, theproblems can be considered in plane-strain.'From now on, for the sake of convenience, the rolling process with a variable roll-gap geometry willbe referred to as the VGR process.Chapter 1. INTRODUCTION^ 5It is conceivable that two schemes may be used in the design of the production rollingmills for leaf springs. One is that the rolling is to start with a rectangular bar, of whichthe width-to-thickness ratio may be 1 to 3, as those required by some already existingproduction mills for tapered leaf springs. In this scheme, the leaf spring is produced oneat a time, and the main problems are that the width of leaf must be constant and alsothe thickness variation is usually from 30% to 70%[14]. Apparently, in such cases thedeformation analysis has to be considered in three dimensions, except when consideringedge-rolling for the purpose of width correction(see Section 2.1).The other scheme is to start with a wide slab, the width-to-thickness ratio may belarger than 10. In this case, the slab is to be rolled to the desired dimensions and morethan 10 leaves can be obtained at a time by cutting the rolled blank. Accordingly,the spread in width is at least no longer a serious problem, and the process may beconsidered in plane-strain. However, the machine set-up in this case may be much morecostly. Besides, at present, cutting process is usually detrimental to the final propertiesof the products in terms of their strength and fatigue resistance, and therefore is notpreferable in leaf spring production.The present work is aimed to propose a generic theoretical model, three dimensionalstrains will be considered. The model should be capable of giving a good descriptionof the variation in deformation, force and torque, under conditions of hot working andvarying geometry. Thus, the model may also be used in producing such componentsshown in figure 1.1c.Chapter 2BASIC CONSIDERATIONS2.1 Available Analytical MethodsIn the past half century, a number of approximate analytical methods have been devel-oped and applied to the rolling process, amongst which the most well known are thefollowing.1. The slab analysisThis method is very restricted by its nature and is usually used for a rough pre-diction of the required load, with a reasonable stress distribution; but it is not asuitable method for a rigorous deformation analysis.2. The slip line fieldSince its derivation is based on the plane-strain assumption for perfectly plastic ma-terials, generally speaking, it is not reasonable to apply this method to three dimen-sional problems. In addition, the construction of slip-line fields is still quite limitedin predicting results that offer good correlations with experimental evidence[16].One of the applications of the method that should be mentioned here is the oneused by Hawkyard et al [9] in ring-rolling. See figure 2.1. Despite its formidable-seeming complexity, that is, ring-rolling has the similarities with asymmetric rollingprocesses, it could still be tackled since the work-roll feed rate is generally smalland is a known constant. It is worth noting that the reduction is usually very6Chapter 2. BASIC CONSIDERATIONS^ 7Roll velocityFeeding velocityFigure 2.1: The schematic illustration of ring-rolling configurationsmall for each turn and the ratio of 1/h is usually large, figure 2.1. These lead tosuch an effect that the plastic deformation between the rolls is far from penetratingthe ring thickness, and instead of the usual bulges in rolling, fishtail-shaped sidesresult, figure 2.2a.Based on the above observations, they defined the plastic deformation betweenthe rolls by the kind of slip-line field applicable to flat-end rigid tools, which wasdeveloped by Hill [17], when forging a block whose depth is larger than that of thetool width, figure 2.2b, was considered.Apparently, such a method is not applicable to the deformation in the presentprocess, but it may be a good choice for the edge-rolling, which is usually used toobtain a uniform width.Chapter 2. BASIC CONSIDERATIONS^ 8(a)(b)Figure 2.2: (a) The illustration of fishtail-shaped sides in ring-rolling. (b) The illustrationof the slip-line field for flat-end tool forging.3. The visioplasticityThis approach is experimental and could be useful if theoretical or numerical rela-tions cannot be worked.4. Hill's general method[3]This method is derived from the virtual work-rate principle, and will be cited inthe related part of the next chapter.5. The finite - element method(FEM)The applications of this method to the analysis of three dimensional rolling processwere started in early 80's [18]. One of the main advantages of the method is thatit can provide detailed information, point to point, of the deforming body betweenthe rolls, which, however, requires a step-by-step computational procedure andChapter 2. BASIC CONSIDERATIONS ^ 9therefore, requires a much higher data storage and much longer computing time.Fortunately, the conventional rolling process is a steady-state process, and only afew samples of the deformation analysis are needed to obtain a good prediction ofthe whole process.The formidable difficulty in the analysis of the VGR process is that the patternof the deformation between the rolls changes continuously throughout the process,and inevitably, a calculation for every point along the product has to be done.For instance, to roll a one-meter long component, the step to generate a set ofreasonable interpolation data for the digital controller of the machine is 0.0005meter, then, at least 2000 steps is needed (for one pass), which means that if finite-element method were to be used, the demands for data storage and computingtime would increase a thousand times! Incidentally, Grober [19] reported in 1986that a simulation of transient hot rolling of steel was performed, where a CPUtime of 7 hours on a microVAX II computer was consumed for the simulating ofa total length of 51.25 mm, and only a two-dimensional finite-element model wasused. This, most likely, were the very reason that no significant theoretical workon the subject, ie., the VGR process, by using FEM (nor by any other method)was found in the English publications(a rather great amount of efforts has been putinto the search). Conscious of the foreseeable great demands for the computation,the method is not to be used in the present problem.6. The upper bound approach (UBA)1The upper bound approach is valuable to mechanical or production engineers, sinceit offers rather good results in the roll mill operations with lower costs, and from theviewpoint of machine design, the data produced by the method, though somewhat1This name was first used by W. Johnson and was considered being well-expressed[55].Chapter 2. BASIC CONSIDERATIONS^ 10approximate in nature, are certainly on the safe side. It can also be found inthe literature that many researchers used this approach in conjunction with othermethods, such as least squares, finite-element, viscoplasticity and the weightedresiduals, etc., with which the difficulties to obtain proper velocity fields may bereduced.Perhaps the earliest application of the UBA to the three dimensional analysis ofthe conventional rolling process was made by Oh and Kobayashi in 1975[5], wherequite accurate solutions to the average spread and the torque requirements weregiven. By extending the work of Oh and Kobayashi[5], Lahoti, et al, developeda production software package for the rolling of plates and airfoils in 1978 [20],and Sevenler, et al, managed to analyze shape rolling with flat rolls [21]. Withthe method, bulge predictions were also attempted by Kennedy[6] in 1986, thoughfairly noticeable difference between the theoretical predictions and the experimentalresults, cited in the work, exists. The major difference between the model by Ohand Kobayashi[5] and the one by Kennedy[6] is that two totally different solutionsto the velocity fields were used.Considering all the facts involved, it is reasonable to use the upper bound approach inthe analysis of the present problem.2.2 AssumptionsThe present analysis is based on the mathematical consideration proposed by C. S. Yih[22] in 1957. To be practical, the following assumptions have to be made.1. The material is isotropic and incompressible.26i^II-/////,2t12w ,Chapter 2. BASIC CONSIDERATIONS ^ 11Z^2tFigure 2.3: Schematic illustration of the part of a bar in rolling2. The material assumes perfect rigid-plastic properties, that is, the flow stress of thematerial a is constant2.3. The deformations meet Mises yield criterion and assume unrestricted plastic flow,therefore, the Levy-Mises equations can be used. The inertial forces are negligible,since in this type of rolling lower speeds are used.4. The friction shear stress To is constant in the roll-workpiece interface and is givenby a constant friction shear factor m defined by the relation, To = m • a/./a.5. The bar to be rolled can be considered as a set of flow solenoids with differentrectangular cross sections, figure 2.3 shows one such element. For the sake ofsimplicity, any line in z-direction is assumed to remain straight. Since one can neverhave the exact solution to the flow patterns in question, good approximation hasto be dependent upon two properties — theoretical tenability and mathematicalfeasibility. In order to avoid overwhelming mathematical difficulty, it would be2This assumption is only based on the strict restraint in the upper-bound theorem. In fact, in realapplications, this restriction has to be relaxed and the concept of instantaneous yield conditions has tobe adopted. Detailed discussion may be found in Sections 5.2 and 5.3.Chapter 2. BASIC CONSIDERATIONS^ 12better to give up the bulge prediction at this stage. 36. The profile of the product can be expressed in a continuous function or may bedefined in several continuous intervals with different functions, and the slopes ofthe function, have to be very small with no sharp change on the boundaries of theintervals, if any.2.3 Coordinate SystemsIt can be seen that the cartesian coordinate system,(x, y, z), is convenient in developingthe velocity fields, figure 2.4. The x, y and z directions are defined along the length,width and thickness of the product, respectively.For convenience in design and control of the machine, the coordinates (X ,Y, Z), figure2.4, are used. And C is a constant dependent upon the dimensions of the products andthe configuration of the rolling mill. Henceforth, the roll gap function can be simplyexpressed as G(x + C) in the analysis coordinate system (x, y, z).3Actually the author considers that it should be a good choice to make use of the available methodsfor bulge prediction with this model, amongst which there are several good methods. Besides, the authorbelieves that accurate bulge prediction that is really helpful in production is more an art than a science,by its nature.x ( X )Figure 2.4: The coordinates systemChapter 2. BASIC CONSIDERATIONS^ 13Chapter 3CONSTRUCTION OF VELOCITY FIELDS3.1 The Dual-Functional MethodAs mentioned Chapter 2, the analysis will start with Yih's mathematical considera-tions[16], of which the related results are to be re-introduced here, however, detaileddiscussions should be referred to the original paper.Yih derived that for a continuous flow, the velocity vector can be expressed as thefollowing:V = (grad x (grad g)^ (3.1)where f and g are two artificially chosen functionals that may represent stream surfacesin which streamlines are embedded, and along any streamline the two functionals assumethe following form,f(x,y,z) = a^g(x, y, z) = b^ (3.2)where a and b are some constants.In addition, since the tool surface is the actual flow surface, f can be so chosen as tomake a solid boundary an [surface, and g the free sides of the being rolled material andcan be solved by the equation,V • (grad g) = 0^ (3.3)which is one of the properties of continuous flow.'lYih did not indicate this last approach explicitly, which, however, can be easily derived from theoriginal idea.(See Section 3.2)14Chapter 3. CONSTRUCTION OF VELOCITY FIELDS^ 153.2 A Brief Derivation of the Solution to the Velocity FieldsIncidentally, a much shorter and simpler mathematical derivation of the aforementionedidea was discovered during this work, which is the following.Assuming an incompressible material(nondiffusible), the following holds,^ay. av^avz^ + Y^+ ^ = 0Ox^ay az (3.4)which can be written in the following form,div V = 0^ (3.5)Note that we have the following mathematical relationship,div (A x B) = B - (curl A) — A - (curl B)^(3.6)curl (grad 0) = 0^ (3.7)where A, B and 0 are any differentiable functionals. Suppose that there are two contin-uous functionals, f (x, y, z) and g(x,y, z), the gradients of which exist. LetA = grad f^B = grad g^ (3.8)Substituting Eq.3.8 into 3.6, and using Eq.3.7, the following results,div [(grad f) x (grad g)] =(grad g). [curl (grad f)] — (grad f) x [curl (grad g)] = 0^(3.9)Comparing Eq.3.9 with 3.5, we haveV = (grad f) x (grad g)^ (3.10)Eq.3.10 suggests that V could be constructed by two functionals, and the only re-straint to these two functionals is that their gradients exist. In other words, f and gChapter 3. CONSTRUCTION OF VELOCITY FIELDS ^ 16need to be constants. Henceforth, f and g can be artificially chosen. Naturally, f can bechosen as the tool surface, meanwhile, adopting the prescribed flow factor (see the deriva-tions below); and g can be so chosen as to represent the free surfaces of the material.The latter property leads to the following,V   n = 0 (3.11)where n is the normal vector on free surface of the material between the rolls.Since n and grad g have the same direction, Eq.3.11 can be replaced byV (grad g) = 0 (3.12)which is the same as Eq.3.3 and can be used to determine g.3.3^The Derivation of the Velocity FieldsUsing the notation of gradient,Va fBy^azag^BgBy^azEq.3.11+may be writtenaLLaz^axag^agaz^asj+as(2LOx^Byag^agOs^By(3.13)where i, j, k are unit vectors along x, y and z directions, respectively. Thus, the velocityfields can be expressed as the following.(3.14)With the idea described in previous Section in mind, it can be easily understood thatthe assumption 5 in Section 2.2 is theoretically tenable. The remaining problem will beChapter 3. CONSTRUCTION OF VELOCITY FIELDS ^ 17to choose the proper functionals, f and g, which should assume the simplest forms butsatisfy Eq.3.3.As indicated earlier, the functional f can be chosen as the tool surface. From figure2.4, under assumption 5 the roll surface can be taken as part of a cylinder, and expressedasz = t(x) = G(x +C)-F R — VR2 — ( 1 — x)2^ (3.15)Obviously, t(x) is a non-zero function in this case and Eq.3.15 therefore can be writtenast(x)Since we are talking about flow, it will be seen that to relate Eq.3.15 to the rollperipheral velocity at this stage is a reasonable choice (also see next section), thus,Eq.3.16 becomesQz t(x) = Q^ (3.17)where Q is an arbitrary constant related to the flow and its physical meaning will turnout later.Now defineQ zf(x,y,z) =^ (3.18)t(x)which then meets the restraint that f should be a constant.Before using Eq.3.3 to determine functional g, some explanation relating to the as-sumption 5 should be given. As can be seen in figure 3.1, g(x,y, z) is assumed as the flowsurfaces constituted by the free sides of the part of a bar being rolled and are parallel toz-direction. Note that the curvature of the sides is usually very large (the dotted line infigure 3.1) in reality and very close to a straight line (the solid line in figure 3.1), whichmay be confirmed by many photographs in the literature reflecting the deformation of theworkpiece between the rolls that showed the projected profiles in the x-y plane remain2=11 (3.16)Chapter 3. CONSTRUCTION OF VELOCITY FIELDS ^ 18g(x,y,z)Figure 3.1: Schemetic illustration of the top view of a rectangular bar between the rollsalmost a straight line, and which, in fact, is not unexpected as a result of the requirementfor keeping the deformation as uniformly as possible in the roll-pass design, and figure3.2b and 3.2d, from [17], may serve as a good example amongst many.However, it seemed that researchers preferred to attempt an exact estimation of thecurvature in the calculations with a complicated model. It is imaginable that the con-tribution of such estimation to the overall value of the energy dissipation would be verysmall, which leads to the idea that assuming a straight line as the side profile in thepresent problem is a good approximation. The main purpose of such consideration, ofcourse, is to reduce as much as possible the difficulty in the mathematical handling andthe computational demands.Now, the equation for a straight taper profile, figure 3.3, can be written as the fol-lowing,, ‘ wi — woy . wkx ) = 1 x + woSuppose that g(x, y, z) will assume the form,(3.19)4Y^g(x,y, z) =^= qw(x) (3.20)171/ = tW2v. = _-QqtwQqyw'(3.22)Chapter 3. CONSTRUCTION OF VELOCITY FIELDS ^ 19Skta *ad pkka viatqa at 1$ mitoy ay.amainens do-far:nod at 40% reductlea; St ^mm OV/h,.::^ .^.Figure 3.2: Side views(a, c) and plan views(b, d) of 1S alloy specimens deformed at 40%reduction; height: 26 mm, width: 26 mmwhere q is some constant, therefore Eq.3.20 meets the constancy restraint of streamsurface functional.Substituting Eq.3.14, Eq.3.18 and Eq.3.20 into Eq.3.3, the following results,Q Yw'e^Yw' 0( ^^tw w2^w2^ (3.21)which indicates that q can be any constant.Substituting Eq.3.18 and Eq.3.20 into Eq.3.14, the following are obtained,Qqzti=wt2Through observation, it can be decided that q = — 1 is the best choice, simply be-cause Vx can never be negative in the current coordinate. Hence, the velocity fields areVs =--Vz =--QtwQYWIVY = tW2Q zt'wt2(3.23)Chapter 3. CONSTRUCTION OF VELOCITY FIELDS^ 20Figure 3.3: The schematic illustration of the simplified side profile of the section betweenthe rollsdetermined as3.4 Discussion of the General SolutionAt this point, it would be convenient to make a few comments about history. Hill[3] firstproposed a similar velocity field solution for the flat rolling and bar drawing in 1963, inwhich researchers have shown great interests ever since. He must have found a particularway to reach his solution, but, unfortunately, he didn't indicate it2. Lahoti, et al, con-sidered Hill's work as a "unfinished" one, and continued the work in 1973[4]. However,'For the sake of convenience, Hill's solution is quoted in the following."The simplest class of approximating velocity fields with these properties (i.e., the similar assumptionsas those of 1, 2 and 5 in Section 2.2, and the same coordinate system, (x,y,z), as in Section 2.3.) hasChapter 3. CONSTRUCTION OF VELOCITY FIELDS ^ 21as Hill himself didn't explain how to relate the solution to the rolling velocity(VR), nei-ther did Lahoti, et al. This, however, becomes significant when a solution to the rollingprocess, as general as it is in the present work, is sought.It can be noted from Eq.3.23 that for flat rolling, the following holds,Q vo = vrl..0 =^ (3.25)and this may be rearranged into,Q , VotoW0^ (3.26)which indicates an interesting fact that the constant in the solution is the discharge.Accordingly, in the case of(a) a bar pushed in between two idle rolls,Q , Votowo^ (3.27)where Vo is a known constant.(b) a bar pulled through two idle rolls(Steckel rolling, see reference[1]),Q = vit(ow(i)^ (3.28)where V1 is a known constant.(c) the conventional rolling process,Q = 0(xn, VR)^ (3.29)where 0 is some functional. Therefore, in the case of (a) and (b) , there is no need toworry about the estimation of the neutral points. This conclusion is different from thecomponents(here, h =2.),1^YID'^zh' (3.24)tly = h—, Vy = hw2) vz = Wh2within the region bounded by the planes of entry and exit, the faces of the dies, and the lateral surfacesy = ±w(x),   Such fields are clearly solenoidal and preserve the configuration;   The immaterialconstant of proportionality(the emphasis is added here) has been chosen so that the total rate of flowacross each quarter-section is unity.  "towoChapter 3. CONSTRUCTION OF VELOCITY FIELDS ^ 22one by Oh and Kobayashi[5]. There are some velocity field solutions for the conventionalrolling, say, [6], in which the relation, Q = Votowo, was used. This can be consideredobscure, because in the conventional rolling, 170 is an unknown constant before startingthe process, besides, I/0 may be temperature dependent.Thus, it is reasonable and instructive to adopt the prescribed factor Q in the solution,especially when considering the variation of the rolling geometry in the VGR process,where Q may no longer be a constant. In other words, Q may be an instantaneousconstant (See next Section).3.5 Further Considerations3.5.1 Theoretical AspectIn the foregoing sections, the solution to the velocity fields has been treated in a generalmanner. It should have been noted that the derivation of Eq.3.23 is started from theoriginal geometry without any restriction on the motion of the rolls. This is one ofthe nice things about the solution, as it would be very awkward and unnecessary toconstruct the component for the up-or-down velocity of rolls at this stage, which couldbe non-linear. The implication is that when Eq.3.15 is used, which includes a roll-gapfunction, the variations in the velocity components will naturally be taken into account.It should also be noted here that in the VGR process, owing to the ever-varyinggeometry, the velocity vectors in the deformation space become time-dependent, thus, itis a non-steady state process, which is the essential difference from the traditional rollingprocess, where the geometry is fixed and therefore it is a steady state process. In thepresent case, however, during the whole process, the geometrical resemblance is basicallymaintained, which may be the theoretical validity that one set of formulation can be usedChapter 3. CONSTRUCTION OF VELOCITY FIELDS^ 23in the modelling of the present process. Accordingly, it is both reasonable and necessaryto divide the process into a series of small pseudo-steady state deformation steps, thatis, by focusing attention on each sampling instant, the geometry may be frozen, then aset of instantaneous solutions can be obtained. And by updating the geometry, a setof approximating numerical solutions may be finally produced, which may serve as adescription of the dynamic characters of the process under study. This idea should alsobe considered as the basis for the discretization procedure in the design of the machinecontrol system.3.5.2 Technical AspectIn the VGR process, when the conventional rolling scheme is used, the discharge Q is avariable if the roll peripheral velocity is to be designed as a constant, which, apparently,is preferable; when Steckel rolling scheme is used, if Q were to be designed as a constant,the pulling mechanism would be very complicated. In any case, since Q is preferred tobe a variable of the geometry, it is clear that the geometrical parameters have to bedetermined in advance in order to update Q, which will be a crucial issue in the controlscheme.The roll-gap function GFrom machine design standpoint, it is important to consider the feasibility of a simplecontrol scheme at this stage.In figure 3.4, the two coordinate systems, (x, y, z) and (X ,Y, Z), have the followingChapter 3. CONSTRUCTION OF VELOCITY FIELDS ^ 24Figure 3.4: The schematic illustration of a parabolic profile in rolling configurationsimple relationship,X = C(3.30)Z^zHere, in order that a typical set of data may be produced, a parabolic type of producthas been chosen, although the roll-gap is generically defined.In coordinates (X,Y,Z), the profile assumes the following form,(X — a)2 = —2p(Z — h)(3.31)which can be reduced toa2(X — a)2 = b h(Z — h)where,(3.32)a2 = —2p(b — h)^ (3.33)Chapter 3. CONSTRUCTION OF VELOCITY FIELDSi.e.,Letthen Eq.3.32 becomesZ = e(X — a)2 + hUsing Eq.3.30, Eq.3.36 has the following form in coordinates (x, y, z),z = e(C — a)2 + hFor notational consistency, Eq.3.37 should be written asG(x + C) = e(C — a)2 + ha2P =2(b — h )b — he =a225(3.34)(3.35)(3.36)(3.37)(3.38)It can be seen that Eq.3.38 ensures a simple but reliable control scheme. However,detailed discussion is beyond the scope of present work and is left for later; for now,Eq.3.38 is adequate for the present mathematical analysis.The length of plastic zoneNote that in present problem, the length of the projected arc in contact is also a variable— a function of the roll-gap and to, i.e., /(x). From figure 2.3,1(x) = VR2 — KR— to) — GI2^ (3.39)With the above geometrical relationships determined, a new velocity field solution isestablished, which will be the basis for the next chapter.Chapter 4FORMULATION THROUGH UPPER BOUND APPROACHThe upper-bound theorem states that among all kinematically admissible strain ratefields, the actual one minimizes the following expression, using the same notation in [24],p.419,J* = a,*343dV + ElD k^idyl; _^TitcisT^(4.1) Srwhere, ê j is the strain-rate field derivable from v:;^is a kinematically admissible veloc-ity field; lAV*1 is the amount of velocity discontinuity along the surfaces of discontinuitySi); and o-:3 satisfies the yield criterion and is associated with i:i. In present case, onlythe conventional rolling scheme is considered, therefore, the back and front tensions arezero, i.e., T, = 0. SD is the surface where velocity discontinuities occur, and consistsof three parts in present problem: the friction surfaces, S f , between the rolls and thematerial; the entrance and exit planes, Sdo Sdi, where shearing occurs; accordingly,SD = Sf Sdo Sdi)J* =u,dV +^TolAV sjelS f +^klAVdIdSd3V JsfOr using the conventional notations, Eq.4.2 may be written as,= Ed +Ef +E8where,Ed = J èdV26Eq.4.1 becomes(4.2)(4.3)(4.4)Chapter 4. FORMULATION THROUGH UPPER BOUND APPROACH^27=mkIVsIdSfSI=kIAV didSd19.10+SdlDefinitions of various terms in Eqn.4.4 through 4.6 is given in the following sections.4.1 The Strain RatesThe six strain tensors are defined as the following,v.^a x1 /ay,/ ay.,E. = ^ax isv — 2 ax + ay^avy^.^1 (avy ay.,Ey= ^evz = --‘ az + ay 'ay^ay.^..^1 , ay. ay.,ez =^Ezx — k ^ + —)^az 2 az^ax(4.7)Substituting Eq.3.23 into Eq.4.7 and re-arranging it, the following results,i t ^w'ex = — Q —(--- + —)tw t= Q wtw2(4.8)(4.5)(4.6)Q ywi^211/21w2 t^) wQ z^2t'212 [tEIZXiyz =2 [(u1)2 1_ (t')2 Hhw)^tw=(t' 2w)M = ^t ^w)IT t"^t' 12t'^w'\T^-t )Substitution of Eq.4.10 into Eq.4.9 results the following,Ed = 40.fift,weQ Nip 4. 1NI y 2 + N2z2dydzdxo o o^3 twAfter integration against y and z, Eq.4.12 becomesEd = 2 .1—Qa Z(x)dx3^owhere,(4.11)(4.12)(4.13)Chapter 4. FORMULATION THROUGH UPPER BOUND APPROACH ^284.2 The Energy Rates4.2.1 The energy dissipation rate due to deformation(Ed)Using the Levy-Mises equations, Eq.4.4 may be written as,•^2Ed = —0 \11    J 2 EljEljdV (4.9)Since,1 .2—E.^.E2 E2 _4_ 22 y^zj^; zs -r- c yz1  Q2 2^2 22 t2w2 " M^N2z2) where,(4.10)Z(x) =^ 12 + m2w2 Ni^+ m2w2 + N212V12 M2y2 N2z2 ^ inNt^v12 m2w2212MNtw [1,1(02 m2w2, Mw, Nt) —^, 0 , NO]-F Nm 2 tw (02 + m2w2, Mw, Nt) — (/, 0, NO] (4.14)Chapter 4. FORMULATION THROUGH UPPER BOUND APPROACH ^29where,focPi(a, b, c)^ln(b + Va2 + x2)dx= cln(b + Va2+ c2) -c-f-bln (Va2 + C2 +a ^C) (4.15) + 2 va 2 - b2 tan' iVa2 - b2 Wa2 + c2 - alL a + b^c10.P2(a, b, c)^x2 ln(b + Va2 + x2)dx1  ^1= d {c3 ln(b + Va2 + c2) - c3 -I- c(a2 - b2)1   1^(Va2-F-bcVa2 + c2 - -b(3a2 - 2b2)1n '^+ c2 _4_ c)2^2 a (4.16)- 2(a2 - b2) tan-1^a(Va2 - 112 Wa2 + c2 - all-I - b^c4.2.2 The energy dissipation rate due to friction between roll surfaces andworkpiece(Éf)From figure 4.1, the velocity discontinuity between the surfaces of roll and material canbe expressed as the following,IIEf = 4mkQJ fwo Jo \,\2[(YW  ) + (VR 1 wt/1 + t'2) 21 (1 + t12)dxdyAVs =-- Vy + VR - (V. + V.) (4.17)and the magnitude,1,6Ars I = VVy2 + ( VR — 1/1/2 + 1/2 )2QN1 (ywi 2) ^(VR V1 + t' 2 \ 2^tI.V2 ) ^I\ Q^wt )Note that on the surface, z = t(s). Then Eq.4.5 becomes,(4.18)(4.19)- (V1 + 111 )--i - - -V, + Vy + V,^AV xV, + Vz,/^I11Chapter 4. FORMULATION THROUGH UPPER BOUND APPROACH^30Figure 4.1: The schematic illustration of the velocity vector space in conventional rollingprocessand after integration against y and z, Eq.4.19 becomesEf -= 2mkQ 11 F(x)0 + ts2dxowhere,w (wi ) 2 ( VR Wt^VRV1 + 0 ) 2 V]. + ti2 ) 2 .F(x) = —21\ tto) + Q + ws^Q(wt^) wt^)(4.20)in [{ Wi^()2 (V^R ^N/1 + ti2)2 . VR^■,/1. + t'2Wi +^iW + Q^wt^  Q^wt^(4.21)4.2.3 The energy dissipation rate due to velocity discontinuities (E',)From figure 4.1, at the entrance plane,AVdo = V — V, = V1, + V,^ (4.22)Chapter 4. FORMULATION THROUGH UPPER BOUND APPROACH ^31and the magnitude,1AVdol = VVy2^ (4.23)And at the exit plane, since the width of the deforming body is almost the same asthe width of the rigid body, we may assume Vy = 0; thus, from figure 4.1, the followingholds,AVdi = V — Vz = V, = 0^ (4.24)Substitution of Eq.4.23 and Eq.4.24 into Eq.4.6 gives, note that, Vo = Q/woto,= 4k j V(Vy2 1/2).=ods,= 4kVoft.() ft.Jo^\(ytv, ) 2 (Zii)2dzdywx=o^t(4.25)The integration of Eq.4.25 results in the following,kQS(x,wi)^ (4.26)where,t12^+ 012 + W12S(X, WI)^[0'2 +^ln4^F I^I^2w'2^1 Itlw7P2(t t^) z=o (4.27)4.3 The Minimization ProcedureSince the unknown parameters in the velocity fields solution are w1 and xn, the problemis reduced to solving the following non-linear equations,awl^'^ax„^ (4.28)Chapter 4. FORMULATION THROUGH UPPER BOUND APPROACH^32By substituting Eq.4.3 into Eq.4.28, the equations to be minimized are obtained,atd atf at,^, + ^ + ,^ =o (4.29)owl mill owlatd DE1 at,^+ ^ + ^ = 0 (4.30)axn axn axn4.3.1 The derivatives of the geometryFrom Eq.3.19, the following is obtained,wi = (w1 — w0)-11,along withaw x^,awl^1andOw' _ 1awl — /through which the following results readily,(aaww:).=0 = 1and Eq.4.31 also gives the following,1,^1^\wo --= kwi — wo )—17(4.31)(4.32)(4.33)(4.34)(4.35)4.3.2 The neutral pointApparently, in present case, the neutral point is ever-changing. However, the followingstill holds instantaneously for each frozen moment, figure 4.1,YR = Yx(Xn) + Yz(Xn)^ (4.36)V1 + ti(x,)w(x)t(x) cl ( x n ) = vR (4.39)Chapter 4. FORMULATION THROUGH UPPER BOUND APPROACH ^33Using Eq.3.23 and note that, z = t(x„), we have the following,4? /VR =--- - V 1 + iiWi X=Xn(4.37)or it may be written as,Or,VR V1 + e(x)Q^w(x)t(x) (4.38)4.3.3 SolutionsThe analytical solution to Eqn.4.28 is too complicated and may be found in AppendixA.The numerical solution is obtained by coding in MATLAB. A CPU time of 15 secondsfor each cycle of calculation is achievable on Sparc-I SUN workstation. The results aregiven in Chapter 6.Chapter 5FLOW STRESS MODELIn the foregoing chapters, a fundamental geometric description of the process has beengiven. To achieve a reasonable control of the roll-gap in the VGR process, a detailedand accurate roll force change prediction is required. It should also be noted that, inorder that the newest developments in material science may be adopted in the calcula-tion, the assumption that the materials are rigid-perfectly plastic is in need of furthermodification. In other words, it is necessary to establish a correlation describing thechanges in flow stress as the workpiece moves through the rolls and between the passes,so the changes in rolling forces may be predicted. An adequate mathematical modelof flow stress incorporating the thermal characteristics of materials would be of greatimportance in implementing the VGR process and in controlling the final properties ofthe rolled products.5.1 Dynamic Flow StressUnlike cold working, the hot deformation of steels at elevated temperatures (above therecrystalization temperature) may be divided into three transient stages[2511261[27] —1. Microstrain deformation. This is the interval during which the plastic strain rateincreases from zero to the approximate forming rate, and the stress in the workpiecerises rapidly, Fig. 5.1. The loading slopes are rate controlling in this interval.34True stressWork hardening]Dynamic recoveryHigh strain rateDynamicrecrystallizationLow strain rateChapter 5. FLOW STRESS MODEL ^ 35_0 0.2 0.4 0.6 True strainFigure 5.1: Schematic flow curve for metals at elevated temperature. From point 0 to 1,the plastic strain rates increase rapidly. Note that the scale for the strain is arbitrarilychosen from typical C-Mn steels[27] to illustrate a possible range in applications2. Plastic yielding. The region of the yield stress marks the beginning of work hard-ening. The slope is temperature and rate sensitive, but in general, is graduallydecreasing.3. Dynamic restoration. This stage is most complicated due to the concurrent opera-tions of the strain hardening and softening. Generally speaking, if the strain rate islow and the hardening effect is only offset by the dynamic recovery process alone,a steady state will quickly be reached, so a simple flow curve results.The alternative restoration process is dynamic recrystallization, which modifies theflow curve in the following ways. At high strain rates, the flow stress rises to amaximum at peak strain(point 2 in figure 5.1), and then it diminishes to a valueintermediate between the yield stress and the peak stress(See figures in AppendixChapter 5. FLOW STRESS MODEL^ 36B). At low strain rates, the dynamic recrystallization process causes a softeningfollowed by a oscillatory type of flow curve before a steady state is reached, figure5.1.In [27], I. Tamura, et al, stated the following.For most commercial steels, dynamic recovery behaviour is observedin iron and carbon steel deformed in the ferritic region, or in ferritic alloysteels such as silicon iron. Dynamic recrystallization takes place in thehot-deformation of -y including HSLA steels and austenitic alloy steels.This may serve as a general guideline, although care should be taken for each differentmaterial.The reduction ratio in the VGR process can be anything between 8% and 50%, whichcorresponds to the true strain range from 0.051 to 0.693. From figure 5.1, it is clear thatthe flow stress will vary significantly during each pass. In other words, in reality, thematerials are rate dependent within the strain range of hot rolling. Accordingly, insteadof using a flow stress value corresponding to the steady state, which itself may result inconflicts, it is necessary to have a flow stress model to count in such variations.5.2 Incorporation of Rate DependencyIt has long been concluded that the flow stress model that gives a complete descriptionof the hot deformation of steels in the roll-bite assumes the form[28]:= f(ccrmposition of the alloying elements,T,e,e)^(5.1)where T is temperature.As is stated in Section 3.5, the VGR process is a nonsteady state forming process, andin the last Section we concluded that it is not reasonable to use a constant yield stressChapter 5. FLOW STRESS MODEL ^ 37throughout the whole process. A yield function similar to the above form would be veryimportant in obtaining a solution close to reality. Therefore, modifications are neededsince, theoretically, the upper bound approach is only valid in rigid perfectly plasticmaterials. It is clear that, unless the strain is beyond the steady state point, which isnot the case in rolling process, the state of rigid-perfectly plastic deformation will neverbe reached. In order that rate dependent' effects can be incorporated, the concept ofinstantaneous yield conditions has to be adopted and further assumptions must be made.First of all, it is interesting to note that the available flow stress data are usuallyobtained from three sources: (1) uniaxial compression tests; (2) thin-walled tortion tests;(3) plain strain compression tests. A close examine of those tests tells one that it isimpossible to get rid of the rate effects. Therefore, the results from these tests areusually similar to those in figure 5.1. Figures B.1—B.4 in Appendix B give some resultsfrom experiments that clearly show irregularities in flow stress due to the restorationprocess. This indicates that even though a perfectly plastic material is assumed, flowstress value at the steady state can not be used. Instead, one has to use the data atcertain strain points. The implication is that the solution calculated from the availableexperimental yield stress data can only be an approximation. Nevertheless, this is theusual practice in modelling of the conventional rolling process.Then, the usual assumption that the material is perfectly plastic may be modified,that is, the appropriate assumption in the real situation may be that the material has aisotropic rate dependency'. Thus, the material is still a Mises material, and the yield lociibecome a set of uniform expansions of the previous one, i.e., a set of circular cylinders.Furthermore, a similar assumption adopted by Hi11[17], when an extrusion process wasconsidered, should also be made. That is, the mean strain of the deforming body remains1The definition for rate dependency may be given differently by different authors. In this work, theterm, "rate", should include the temperature decreasing rate, the rate of restoration and the strain rate.2The condition of isothermal is implied.Normal platen pressureSpecimen stripPlatenChapter 5. FLOW STRESS MODEL^ 38Figure 5.2: Schematic setting of plane compression testunchanged regardless of the rate dependent characteristics of the workpiece material.This implies that the mean strain of such a rate dependent material is the same as thatof a perfectly plastic material, since isotropy and volume constancy are assumed. Basedupon this assumption, the concept of mean equivalent strain, or effective strain, E, andof mean strain rate can be introduced.5.3 Mean Values of Strain and Strain RateIn Eqn.5.1, it should be clear that for a practical purpose, a, e, é, and T can only begiven in mean values.5.3.1 Mean StrainTo explain the concept of mean equivalent strain, it is helpful to look at the approachapplied in the plane compression test. In figure 5.2, the lateral strain may be takenas zero, i.e., ez = 0, and volume constancy is assumed. The thickness strain ez isChapter 5. FLOW STRESS MODEL^ 39Figure 5.3: Schematic relation between mean yield stress and mean equivalent strain inplane compression testnumerically equal to longitudinal strain ey. These results may then be used in the Levy-Mises relationship,1 ^E = DI2[(e s — Eyr + (Ey - Ezy + (Ez - E.)212vaezwhich indicates that the mean equivalent strain is proportional to the strain in thedirection of reduction. So the mean flow stress am in such tests is usually given as= F1(e) = F(e)^ (5.3)where F is determined by the test. The results would be similar to those in figure 5.3.The above approach reveals an important idea. For an isotropic rate dependentmaterial, the mean yield stress may be obtained by establishing first an appropriateprocedure to calculate mean equivalent strain. This idea was successfully used by manyresearchers in the past, such as, Kudo[29][30], when upper bound approach was applied.Johnson[31] and Dodeja and Johnson[32] used it in an analysis of forging and extrusionprocess in an upper bound solution, where the E was calculated through the estimationof the total power consumed while assuming the workpiece is perfectly plastic.= (5.2)Chapter 5. FLOW STRESS MODEL^ 40For the present case, since no bulge is assumed, the following method described in[33][34] can be used in estimating the mean strain,E^j1 VA.^A.1 0^2 2^r2 dx^ (5.4)where the components, ex, e ^ez are taken as the principle strains.The assumptions in Section 2.2 imply proportional straining, i.e.,EX + Ey Ez = 0^ (5.5)which leads to—1I ‘IE2 E E^e2 dx1 0^y^Y z zwhere the instantaneous strains are usually defined as,ez^ln t(x) = ln w(x)to^woNote that sometimes the stress-strain tests were carried out in uniaxial compression.Then, the following conversion should be done before using the results from Eqn.5.4,eu^2 (5.8)It is noticed in the literature that the stress testing data for rolling process were tradi-tionally arranged in a correlation between stress and reduction ratio, which is convenientin application. In such cases, a mean reduction ratio may be calculated first as follows,r„, 1 — ee^Or rmu 1 — elu^ (5.9)(5.6)(5.7)5.3.2 Mean Strain RateIn the present work, a similar hypothesis in calculating the mean strain described byChakrabarty[35] will be used, which is the following.Chapter 5. FLOW STRESS MODEL ^ 41Considering the second invariant of the plastic strain increment tensor, an equivalentplastic strain increment is defined as2=3–i (5.10)Therefore,=^dV—1 IIn V V (5.11)By substituting Eqn.5.11 into Eqn.5.3, a rate dependent hypothesis, which is exactlythe same as the strain-hardening hypothesis described in [35], may be introduced as thefollowing,crm=F(ëm) (5.12)where F should also be considered as a function of the other two variables(the mean strainand the mean temperature). Accordingly, the strain rate increments may be calculatedby firstly assuming that the material is a perfectly plastic, so Eqn.4.3 can be used toobtain a reasonable mean strain.5.4 Temperature PredictionAlthough it is known that temperature has a dominant effect on the magnitude of theflow stress, not many models incorporating temperature changes have been formulatedfor the purpose of predicting the flow stress[36]. A large amount of research work on thethermal effects has been done in the past two decades. An excellent review may be foundin reference [37].5.4.1 Basic ConsiderationsSince the VCR process is designed for the rolling of rather small workpieces, the tem-perature of such small workpiece will drop very quickly. Accordingly, drastic changes inflow stress during rolling will be expected.Chapter 5. FLOW STRESS MODEL^ 42The important variables that have great influence on temperature changes are:• Processing time per pass.• Interpass time.• Temperature of the rolls (determined by cooling methods).• Conditions of the roll-workpiece interface.It is clear that the VGR process planning will largely depend on the above variables.A valid correlation is needed.The present work is based on the idea that a combination of separate submodels, suchas, deformation, flow stress and force, etc., should be used, so that new developments canbe easily adopted. Fundamental analysis is preferred, since all the empirical formulaewere meant for large mill operations and are difficult to make necessary changes.All the available fundamental analyses, e.g., [37][38][39], are similar in that a heattransmission from the furnace to the finish stands was considered. In the present case, arelatively small machine is considered, and the temperature will drop due to contactingthe rolls and the run-out table, convection to the spray and the air, and radiation.It is conceivable that in such a heat transfer chain, the temperature distribution of therolls is affected by too many factors, such as the cooling method, the size and structure,the surface condition and the number of rolls, etc. It is almost impossible to give aaccurate prediction due to the variation of the interface area, even if one has a detailedspecifications of the process. Therefore, the present author believes that, when the VCRprocess is considered, the temperature of the rolls has to be obtained from some sensorreading.Chapter 5. FLOW STRESS MODEL^ 435.4.2 Heat Conduction ModelThe heat conduction in a. isotropic solid, the rectangular bar being rolled, may be mod-elled by the following equation:a2T .92T 52T).8^ay2^2aTk (— +^+^+ q = Pweiwax2 az (5.13)The coordinates are the same as before. The present work will only consider onedimensional transient heat transfer, and necessary simplifications to Eqn.5.13 have to bemade.5.4.3 Assumptions1. In order to simplify the calculation in the temperature model, it is assumed that theworkpiece is always a rectangular bar when convection and radiation are considered,that is to say that the influence on heat transfer due to the changes in the geometryis ignored. Mean values of the dimensions will be used in calculation.2. Heat conduction along the length of the product is negligible in comparison withthe heat transfer by the movement of the product.3. The temperature gradients in the roll-gap along the width direction is negligiblesince no contact is made between the sides and the rolls. In the roll-bite, onlyone-dimensional transfer will be considered. It is assumed that the small amount'of heat generated by the deformation, and of heat generated by friction betweenthe rolls and the workpiece will somewhat offset the heat loss from the two opensides in the roll-bite.31t can be easily found out that the heat generated by deformation in the roll-bite is very small. Someresearchers, say, [41], calculated the amount of temperature raised due to deformation in plate and striprolling and the result is only about 20 °C. The contribution of such a variation in temperature to thechange in flow stress is negligible.Chapter 5. FLOW STRESS MODEL ^ 44Then, it may be assumed that 4 = 0. Thus, Eqn.5.13 is reduced to:02T^OTk. ^ pwCwaz2^Or (5.14)4. The temperature gradient at the centerplane through the thickness is assumed aszero. Previous work[40] showed that no temperature difference exists between thetop and bottom surfaces of the workpiece. This indicates that no heat flows acrossthe centerplane. That is,aT(0,T)Oz — (5.15)5. The effect of the scale layer on the surfaces of the workpiece can be ignored. Thisis a usual practice in the research works, due to the lacking of experimental data.6. It is assumed that the stages of cooling under consideration are: radiation andconvection before entry, cooling by contacts to the rolls, radiation and convectionafter exit. The effect of contacting to the entry and exit tables is ignored, becauseno suitable data can be applied. Besides, when the workpiece is out of the roll-bite, the body temperature gradient is ignored, only surface temperature will beconsidered.7. Thermal conductivities, k., and specific heat, G, of steels with various carbon con-tents as function of temperature was established by Devadas[371 by fitting availabledata from BISRA Tables[42]. The results are shown in figures 5.4 and 5.5. Thesetwo figures indicate that, with the temperature increasing to about 1200 °C, theeffect of varying the carbon contents on k, and CI, decreases. In cases of 0.4 C and0.8 C, the differences are negligible.Carbon Content  0.08%C+ 0.4%C* 0.8%C0.1Chapter 5. FLOW STRESS MODEL^ 453130292:262524E5PE2223750 850 950 1050 1150TEMPERATURE (°C)Figure 5.4: Thermal conductivity as a function of temperature for 3 carbon content 0.08C, 0.4 C, 0.8 C in austenite phase. (From Devadas, 1989)Figure 5.6 and 5.7 show the results from fitting some data in [43] for two differentC-Mn steels. When choosing the appropriate functions for thermal conductivityand specific heat, reference may be made to these results:= —1.11024(10-4)T2 + 0.3670T + 399.5600(for 0.19C 1.39Mn steel)= 0.2511T + 411.4217^(for 0.37C 1.44Mn steel)(5.16)(5.17)= —0.0001T2 + 0.4103T + 390.9195 (for 0.23C 1.51Mn steel) (5.18)k, --- —0.0246T + 49.202^(for all C > 0.15 C — Mn steels)^(5.19)Chapter 5. FLOW STRESS MODEL^ 46690680 -670 -660650 -640630 -620610600 -590 -580 ^600553.867+0.164T-5X 10E-5T211652.886-0.1927T+17.3X10E-5T211431.25+0.247T+2.2X10E-5T2800^1000Carbon Content  0.08% C+ 0.4%C* 0.8%C1200TEMPERATURE COFigure 5.5: Specific heat as a function of temperature for 3 carbon content 0.08 C, 0.4C, 0.8 C in austenite phase. (From Devadas, 1989)5.4.4 Initial and Boundary ConditionsThe initial temperature of the workpiece is taken as that when it is moved out of thefurnace. i.e.,T(z, 0) = To^ (5.20)In addition to Eqn.5.15, Eqn.5.14 is subjected to the following conditions, which canbe derived from heat balance.Owing to symmetric cooling, at the interfaces between roll and workpiece, only con-sider convection, we have the following,aTaZ^= 11(T)[T(i, T) Tr]^(5.21)z=t(x)where 1/(r) is the heat transfer coefficient, to which further definition is given in Section5.4.6.Experimental results:o --- 0.23 C 1.51Mn steel+ --0.37 C 1.44 Mn steelChapter 5. FLOW STRESS MODEL ^ 47Temperature -- Celsius degreeFigure 5.6: Thermal conductivity as a function of temperature for C-Mn steels5.4.5 Solution to the Heat Conduction EquationIn terms of computing speed, it is preferred to have an analytical solution to Eqn.5.15.The following is a brief account of the method used for finding the solution. A detaileddescription may be found in [51].1. Assuming the solution has two parts — one is steady and the other is transient,T(z,r) = U(z)+ v(z,r)^ (5.22)then, using the boundary conditions, the steady part, U(z), may be determined asU() Tr^ (5.23)2. Setting v(z,r) = T(z,r)— U(z), using Eqn.5.14 and the boundary conditions, thefollowing results,av(0, r )^0azChapter 5. FLOW STRESS MODEL^ 48Temperature -- Celsius degreeFigure 5.7: Specific heat as a function of temperature for C-Mn steelsav(t, r)^ -I- hkv(t, r) = 0azv(z,0) = To — U(z)where,hk = H(r) K = k,ks^p.,,,Cu,3. Eqn.5.14 with homogeneous conditions of Eqn.5.24 has a solution as(5.24)(5. 25)v(z,r) = f(z)e —AKI-^ (5 .26)where,K =  kapwC.,and A is an eigenvalue and f(z) is an eigenfunction of Sturn-Liouville problem:(5.27)02faz2 + Af = 0, with 494°)^ _ 0az — - (5.28)v(z, 7-) = E BnCOS(ZIZI)e-n=1^AnKr00(5.31)Chapter 5. FLOW STRESS MODEL^ 49The general solution to Eqn.5.29 is f(z) = Asin(z\a) Bcos(zVX). Using theboundary condition, one may obtain Nasin(t VX) = hkcos(tV)'), then we havef(z) Bcos(zA^ (5.29)VAT, = hkcot(tX) (n = 1, 2, 3 ...)^(5.30)From Eqn.5.26, Eqn.5.29, Eqn.5.30,4. When T = 0, from Eqn.5.25,To — U(z) E BnCOS(ZVAn)^ (5.32)n=1Using orthogonality,Jot [To — U(z)]cos(z,VAn)dz = Bn cos2(zOn)dz^(5.33)Bn may be decided as,^B ^2(T0 — Tr)sin(t0,-,)^n +So, finally the desired solution isv(z,T) = E 2(To — Tr)isin(tt, ) cos(z\IZ.,)e-AnKr^n t^+ s n(2VAT)(5.34)(5.35)5.4.6 Heat Transfer Coefficient at Workpiece-Roll InterfaceRecent experiments[46] showed that the heat transfer coefficient at the workpiece-rollinterface is a variable. In [46], when summarizing the experiments, it is stated that,The heat transfer coefficient is seen to vary from 17 to 57 kW/m2K duringthe first 0.015 second in the roll gap and then to remain constant thereafter.Chapter 5. FLOW STRESS MODEL ^ 5065 ^60555045'4- 4°353025200.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02Time in secondsFigure 5.8: Example for heat transfer coefficient at workpiece-roll interface (lubricant:water)The progressive increase in the heat-transfer coefficient in the first 0.015 sec-ond is thought to be due to the decrease in contact resistance with the increasein specific rolling pressure, which reaches a maximum at the neutral point.This may be the reason that the coefficient is expressed as a function of time in a morerecent work[471 as the following,H(T) = 19.561 + 42.47(102)-r — 1.09(105)7-2 (kW/m2K) (5.36)where T is in seconds. Figure 5.8 shows the plot. It is clear that the transient periodis very short. Therefore, in present study the coefficient may be taken as constantcorresponding to the mean steady state value.It should be noted that lubricants have significant effects on the magnitude of thecoefficient. Using the results from compression tests in studying the contacts betweenhigh temperature (780 °C) specimens and low temperature specimens, Murata and his150Chapter 5. FLOW STRESS MODEL^ 51Table 5.1: Heat-transfer coefficients at roll-gap interface for different lubricantsLubrication conditionHeat-transfer coefficient,No scalekWm-2K-1ScaleNo lubricant 29.1 - 34.9 7 - 10.6Water 23.3 - 81.4 10.6Hot-rolling oil 200 - 460 5.8Hot-rolling oil+20% CaCO3 69.8 - 175 12.8 - 23.3Hot-rolling oil+40% CaCO3 12.79 - 17.4 ...KPO3 5.8 ...coworkers[49] calculated the heat transfer coefficients at the interface under differentlubrication conditions, and suggested some useful values for hot rolling processes. Theresults are reproduced in Table 5.1.Devadas et al[36} used 37 kWm-2K-1in their work. They stated that according toStevens et al's estimation[50], the heat-transfer coefficient at the roll-gap interface is37.6 kWm-2K-1during the first 30ms and 18 kWm-2K-1 thereafter. They examined thecoefficient in the range 30-50 kWm-2K-1, and concluded the following.• A 25% variation in the coefficient causes a 2% change in surface temperature.• 40% CaCO3 in a hot-rolling oil reduces the coefficient to 13 kWm-2K-1from anaverage value of 50 kWm-2K-1for water lubrication. The difference of the surfacetemperature at the exit for the two cases is 175 °C.These results should be taken into account in the VCR process design and the produc-tion. The present study uses water as coolant and an average value of 50 kWm-2K-1asthe coefficient, so the roll surface temperature may be taken as around 100 °C.After integration and some tedious manipulation, the following results,11-iT = {th {e2hcs,T' (1^hc 713 EB^1Chapter 5. FLOW STRESS MODEL^ 525.4.7 Convection and RadiationWhen the workpiece is moved out of furnace, temperature starts dropping because ofconvection and radiation. The method for estimating heat loss by convection and ra-diation used in this work is similar to that suggested in [44][45], which consists of twosteps.1. Establishing a model for one-dimensional flowAccording to assumption 6 in Section 5.4.3, when taking its two pairs of the twoparallel surfaces as infinitely large, an instantaneous heat balance of the bar in open airholds for each pair:Rate of heat loss from surfaces = —rate of heat accumulating in the barThus, we have the following equation,2ha.A.(T. — Ta) 26.13A3(T: — 11) = — pCialiddTs ^(5.37)where Ta is the ambient temperature which, in comparison with the workpiece tempera-ture, may be taken as zero. AssumingSb = pC —V. = pCLAwhere L will be substituted by either t or w. Eqn.5.37 becomes,2(haT. EBT) = S- drby rearranging,Tf^dT.—=7, dr IT, 2h,T, E.1371,4(5.38)(5.39)(5.40)(5.41)Chapter 5. FLOW STRESS MODEL ^ 53where, AT is the time delay for temperature dropping to Ti; h, is convection heat transfercoefficient and is chosen to use forced convective heat transfer correlations for flow overa plate.Since the rolling velocity is not high and the cooling can be treated as laminar flowover isothermal plates, the average heat transfer coefficient is given by[48]:h, = 0.664R.0.5 po.33kcr Lc(5.42)where Re and Pr are the Reynolds and Prandtl numbers that are defined as usual; L, isthe characteristic length, determined as the length of the plate in contact with the airflow; Ice is the thermal conductivity of the air.2. Establishing a model for two-dimensional flowUsing the idea introduced in [45], due to J. S. Langston, the following may be written,(Tf)= (Tf)^Tf\^total^width^) thickness(5.43)Substituting to and wo or ti and w1 into Eqn.5.38, respectively, corresponding valuesof Sb may be obtained. From values of Sb, one will be able to compute (Tf)wzdth and(Tf)thidknese before the entry and after the exit. Thereafter, Eqn.5.43 may be solvedunder different conditions.5.4.8 Mean Temperature of the Deforming Body in the Roll-BiteWith all the above relationships established, it is very convenient to compute the desiredmean temperature for each pass.Eqn.5.35 gives the temperature distribution along the z-direction at certain instant.By chosing a finite number of points equally spaced along z-axis, i.e., ;, and time-axis —same as the x-axis —, i.e., T3 , the mean temperature may be computed in the followingChapter 5. FLOW STRESS MODEL^ 54manner,1 1 niT„, =^E E T(zi,r;)^ (5.44)ni ni J=1 i=1which is similar to finite element analysis.In this study, it is considered that accurate enough solution should be obtained byusing i = 10 and j = 20.5.5 Flow Stress Models5.5.1 Model for SteelsThe current analysis will consider low to medium operation speed with a strain raterange of 1-50s-', while the temperature may vary in the commercial hot rolling range,i.e., 800-1250 °C. The flow stress model used should apply to this conditions.It can be easily found out in the literature that numerous researchers utilized theunified creep relationship that works well in hot rolling operations, which is the following,or rearrange it,= A sinh(aa) exp^QaR T (5.45)Z = i exp Q.R ) = A sinh(ao-) ^ (5.46)gTwhere A, n and a are constants independent of temperature, Z is the well-known Zener-Holloman temperature compensated strain rate parameter.A reasonable value of the activation energy, Qa, for a wide range of C-Mn steels(0.05-0.68 C, 0.44-1.64 Mn), has been reported[52] to be approximately 312 kJ/mol, althoughit was observed that some variations did exist, such as 270 and 286 kJ/mol cited in [37].Devadas[37] showed that both A and Q. are variables of strain, though the variationsare small.Chapter 5. FLOW STRESS MODEL^ 55Eqn.5.46 may be linearized as,Qa lni = lnA nln[sinh(aa)]RgT (5.47)The empirical constants may be determined by holding either the strain rate or thetemperature as a constant and using the experimental data(further explanation on theprocedures to determine the constants is given in Section 6.1.1). For C-Mn steels, thedata published in [53] are used in present study.In order to incorporate the data in the computer algorithm, a fairly large amountof data is read from [53], and then fitted by least square approximation into some 80different polynomials, each of which is within 5th to 7th order and is in a matrix form.Figures B.1—B.4 in Appendix B show the fitting results. Such a wide range of datais needed to suit the variation in the C and Mn composition, and the variation of thetemperature, strain and strain rates. Such a data-base is also convenient for interpolationor extrapolation in determining Eqn. Using Lead as a Modelling MaterialIt is undisputed that an accurate flow stress model is absolutely necessary for a desirablesolution to the hot rolling problem. Unfortunately however, since it is generally notfeasible to obtain such information from production mill trials, a complete set of data onthe deformations of various metals under the processing conditions does not exist. "It wasobserved that lead strain hardens in a similar manner to steel at elevated temperatures,and that the static softening behavior can be modelled using equations similar to thesefor C-Mn and stainless steels." [54], Hence the laboratory techniques of simulation andmodelling have been developed by using commercially pure lead [54][55].It should be noted here that lead can be only used to simulate the dimensional changesduring hot rolling, but not the temperature effects on the rolling loads. However, usingChapter 5. FLOW STRESS MODEL^ 56i302826z2422.cg-0Ti20''-,18161401^0.15^0.2^0.25^0.3^0.35^0.4^0.45^0.5^0.55Reduction ratio --- (Ho - h) / HoFigure 5.9: Flow stress model: the correlation between yield-stress and reduction ratioof pure lead. Data source: [56]lead as a modelling material may be a good approach to reduce the experimental errorsto minimum caused by the thermal effects and the difficulties in handling when operatingat high temperature.The data used in [55] came from [56], which was the only source that can be foundat this time providing a wide range of compression testing data. By using least squaredata fitting technique, the following stress model is obtained,^a = 742.5r4 — 1028.9r3 -I- 485.2r2 — 55.8r + 17.2^(5.48)and figure 5.9 shows the fitting results.In Eqn.5.48, instead of using strain as variable, the reduction ratio, r, is used, whichwas the testing results before conversion.Chapter 6MODEL VERIFICATIONIn the above chapters, a complete model for implementing the VGR process has beenfully described, which consists of the deformation model, the flow stress model and thetemperature model. It is important that these models be verified in experiments beforethe machine design process may be carried out. Due to the lacking of the experimentalequipments, a complete set of testing data, regarding a certain workpiece composition,temperature changes, strain-stress correlations, roll force and torque, etc., is not available.Nevertheless, the model may be verified separately, by using some appropriate datareported in the literatures.6.1 Flow Stress Model Verification6.1.1 Empirical Constants in Flow Stress ModelThe linearized flow stress model, Eqn.5.47, is used to determine the empirical constants,A, n and a. Due to the availability of experimental data, C-Mn steel is used for a samplecalculation. A value of 312 kJ/mol for the activation energy, Qa, is chosen, as is explainedin Section 5.5.1.By holding T as a constant, n and A may be determined graphically by the relation-ship between ln(sinh(acr)) and In i. The results are shown in figure 6.1. Or similarly,by holding i as a constant, figure 6.2, n and A may be determined by the relationshipbetween ln(sinh(cw)) and 1/T. It is observed, after many testing runs in the simula-57Chapter 6. MODEL VERIFICATION^ 585550454035302520151050.L. 0-_____5^1^1 5^5^3 5^4^45,C cc = 0.15in t Ccc = 0.01--0.5 -  ^_________0 -^ -4-1.5^1.5^4^4.5^5In eFigure 6.1: Relationship between ln(sinh(acr)) and in i for a C-Mn steel (0.03 C 0.62Mn) at a strain of 0.35. Experimental data: o — (i = 2s-1), * — (i = 20s-') and + —(i = 140s-1). Data source: [53].tion with different values of a, that the magnitude of a has no effect on the final stresscalculation. In other words, the value for a may be arbitrarily chosen.Note that, since the experimental data shows significant irregularity in the flow stressof this type of steel, the slopes in different regions of the relationship, ln(sinh(acr)) — inor ln(sinh(aa)) — 1/T, are not the same. So n and A are no longer constants. Thisresults in a complication in the modelling.It is also confirmed that, when holding a as a constant, both n and A are non-linearChapter 6. MODEL VERIFICATION^ 59555045403530ez 252015105 10000/T --- 1/K2.521.5t5:44^1.-E 0.50—0.5—1 6.5 .s^g.s^4.5^101 0000rr --- 1/K10.5Figure 6.2: Relationship between ln(sinh(aa)) and 1/T for a C-Mn steel (0.03 C 0.62Mn) at a strain of 0.35. Experimental data: o — = 2s-1), * (i = 20s-1) and + —(i = 140s-1). Data source: [53].functions of strain and temperature, which can not be written in a simple analyticalexpression. In order to calculate the instantaneous values of n and A, the two non-linearfunctions were established by a computer program based on a simple surface fittingalgorithm[59]. A brief description of the algorithm follows.1. Assuming the surface is a two dimensional polynomial,[n lnA] f(e,T) (6.1)= ao + ale a2T a3e2 a4ET a5T2Chapter 6. MODEL VERIFICATION^ 60„,2rp^,,3 E3T2 4-(2,6c.^--r- u7c.. -r as_^, ..• (6.2)where a, (i = 0,1,2, ...) are constants to be determined, and by holding temperatureas a constant, an one dimensional polynomial may be obtained,[n lnA] = bo bie b2E2 b3E3^ (6.3)where b, (i = 0,1,2, ...) are constants to be determined.2. Use Eqn.6.3 to generate enough points at certain strains, including the desiredstrains points.3. Repeat the first two steps at different typical temperatures.4. By holding strain at the desired value to interpolate n and lnA along temperaturedirection, n and lnA as functions of temperature at the desired strain is thenestablished.6.1.2 Determination of Flow Stress at any Desired ConditionsWith the establishment of functions for n and A, Eqn.5.46 may be rearranged as thefollowing,—nh1 i-1a^{i exp (Q„k1T-1)a = s ]A (6.4)Now, by using Eqn.6.1 to calculate the values of n and A at the required strain andtemperature, the parameters in Eqn.6.4 may all be determined. So a may be determinedfor that point, (e,i,T). Note that a value for a has to be chosen when determiningEqn.6.1.Figure 6.3 shows a comparison between the model prediction and experimental datafor a certain steel. For the working range of the VGR process, i.e., a strain rateChapter 6. MODEL VERIFICATION^ 61range of 1-50s-', a temperature range of 800-1250 °Cand a strain range of 0.051—0.693(reduction ratio 8% - 50%), the errors in the model prediction are within 5%. It isnoticed that at 700 °C, the errors are larger. This may be a confirmation to a remark byHosford and Caddell[60]: such correlations as Eqn.5.46 "may break down if applied overtoo large a range of temperatures, strains, or strain rates".6.2 Temperature Model Verification6.2.1 Verification Using Data from Conventional RollingThe testing data in [37] for workpiece surface temperature and the same set of data of therolling conditions have been used in order to solve for the temperature model, Eqn.5.22,5.41 and 5.43. It is found out that the converging speed of Eqn.5.35 is largely dependentupon the size of time steps and the magnitude of the rolling speed. The smaller the sizeof the time step and the faster the rolls rotate, the slower Eqn.5.35 converges, and viceversa. The calculation of the first node converges very slowly. As the calculation moveson to the second and the rest nodes, the equation converges faster and faster.Figure 6.4 shows the simulation results. For the grid shown and for a convergingaccuracy of 0.1 °C, the number of terms taken in calculating Eqn.5.35 for the first nodeis 97, and for the last node is 9. This results in a relatively long time in computation.The mesh plot shows the roll chilling effects clearly. The temperature gradients onlyexist in the 4-5 layers to the surface. Such a shallow gradient is due to the high rollingspeed(note that only about 33 milliseconds of in-bite time).In the surface temperature prediction, large errors, compared to the experimentaldata, exist in the first few nodes, which most likely are caused by taking a contact of50(kWm-2K-1) for the heat transfer coefficient at the interface between the workpieceand rolls, H(r). As is indicated in figure 5.8, H(T) is very small at the beginning, whichChapter 6. MODEL VERIFICATION ^ 62would result in a very small temperature gradient at the first few nodes, as confirmed bythe experimental data. For the rest of the nodes, good agreement is obtained.It should be indicated that such a fine grid as was used in simulation is not necessaryfor real applications. Besides, for the VGR process, the rolling speed will be much slower.These would contribute to a significant reduction in computing time.6.2.2 Simulation of First Pass in VGR ProcessIn order to further verify the capability of the model, simulations for the mean temper-ature distribution along a whole workpiece during and after the first pass of the VCRprocess have been carried out.The simulation follows the sequence in a real production. That is, first, allowing5 seconds for transferring the workpiece from furnace(1200 °C) to the mill, the actualentrance temperature may be calculated by the radiation and convection model. Thenrolling starts and the temperature at each node in the roll-bite is calculated by theconduction model, while continuing the estimation of temperature drop in the workpieceoutside the entrance.At the exit, the mean temperature is estimated by averaging the temperature at thenodes in vertical direction. And then, right out of the exit, starts the radiation andconvection calculation again to count in the temperature drop. Finally, allows 5 secondshandling time for the workpiece to be transferred to the entrance for second pass.Figures 6.5 and 6.6 are the results from three different runs, showing the effects ofvarying rolling conditions on a same workpiece. The same heat transfer coefficient(50kWm-2K-1) at the interface between workpiece and rolls used in the last Section waschosen. Some enlightening observations for this particular workpiece and working tem-perature range may be summarized in the following.Chapter 6. MODEL VERIFICATION^ 631. The rate of temperature drop due to radiation and convection is about 2 - 3 °C/sec.2. The mean temperature drop is mainly due to conduction in the roll-bite. This againconfirms that the roll chilling effects is dominant. The higher reduction ratio, thelarger the amount of mean temperature drops. Since higher reduction ratio implieslonger time contacting to the rolls.3. Increasing the rolling speed reduces the contacting time between the workpiece andthe rolls, which results in smaller amount of temperature drop.4. There are two different ways of increasing the rolling speed. One is to increaseroll diameter(case B in figure 6.5). The other is to increase roll velocity(case C infigure 6.5). It is clear from figure 6.5 that the second method(case C) is preferred,because in that way the contact length 1 is not increased.5. Figure 6.6 shows the mean temperature in the roll-bite along the whole pass. Again,it shows that case C(the top curve) is preferred, as smaller changes in mean tem-perature is resulted, which, in turn, would result in smaller variations in flow stress.The calculations were performed by Eqn.5.22 and 5.44.The above observations are important and may serve as guidelines in the VGR processimplementation.6.3 Deformation Model Verification6.3.1 The Side Spread EstimationIn Chapter 4, a deformation model based on upper bound theorem has been workedout. As has been explained in Section 5.2, when looking merely at the deformation, thematerial may be assumed rigid perfectly-plastic. This means that the flow stress becomesChapter 6. MODEL VERIFICATION^ 64a proportional constant and has no influence on the amount of side spread. Figure 6.7shows the results from the simulation run and a comparison is made with the data from[55]. The agreement is excellent.6.3.2 Torque EstimationThe total net power consumption is given by the minima of Eq.4.3, and torque may becalculated from the following equation,M = —.I* •VR min(6.5)Figure 6.8 gives the results from the simulation run and a comparison is made withthe data from [57]. The agreement is very good, as is expected that the results fromEq.4.3 should be the upper bound.6.3.3 Roll Separating Force EstimationTraditionally, the upper-bound approach can not be directly used in determining therolling force. The reasons are apparent — the frictional force at the interfaces changesdirection due to the changes in relative speeds between the rolls and the workpiece, andthere is very little knowledge on this particular subject available.It is unfortunate that, one may find out in the literature, many researchers had touse the slab method to calculate the roll force, when everything else had been workedout by the upper bound approach. Since slab method can only be used under plainstrain condition, some researchers, such as [20][6], after devoted painstaking efforts inthree dimensional solution through upper bound approach, chose to further simplify theproblem into a two dimensional one so the slab method can be used for the solutionto the roll force. This resulted in inconsistency in the theoretical analysis and may beconsidered a shortcoming of the UBA approach.Chapter 6. MODEL VERIFICATION^ 65In order to overcome such a difficult situation, a concept called equivalent coefficientof friction(peq), similar to the one proposed by D. Y. Yang and J. S. Ryoo[58] when ringrolling process was investigated, will be adopted.This coefficient is defined as the ratio of the frictional force(FT), which is the differencebetween the foreward friction and the backward friction, to the overall radial force(FR)upon the rolls, figure 6.9. That is,FTpegR(6.6)where FT is also the tangential force on the rolls. Accordingly,(6.7)Therefore, if /leg were determined by experiments, the roll separating force may be de-termined by using Eqn.6.5, 6.6 and 6.7 as,F = V.F1, +^ (6.8)Since the actual friction is affected by various factors, such as surface conditions, rollsize, temperature, workpiece dimension, etc., it is virtualy impossible to obtain the ex-act distribution of friction. By considering the overall friction effect on the roll forceand torque, the extremely complicated situation is simplified. The value of peg may bedetermined from a set of measurements for the roll force and torque, as suggested in [58].Note that geometrically(figure 6.9)peg^tanO ^X d^ (6.9)VR2 —then, there must be 0 < xd < 1. As the normal pressure is not uniformly distributedalong the contact length(/), the value of xd has to be determined experimentally. Thismay suggest another approach in computation.Chapter 6. MODEL VERIFICATION^ 66In the present study, the experimental results for roll force and torque reported in[57] are used as a demonstration, and the values of peg are computed for the same setof data as was used in computing the torque(Section 6.3.2). Figure 6.10 shows thesimulation output. For the same reason as described in Section 5.5.2, lead was chosen asthe experimental material, and the flow stress model, Eqn.5.48, is used. For reductionratios ranging from 8% — 50%, the best results are obtained when the values of pegwere chosen as a variable within 0.108 — 0.135, which gives the values of the ratio, xd//,within 0.8506 — 0.4285.6.4 Concluding Remarks6.4.1 Summary and ConclusionsA complete mathematical model for implementing the VCR process under hot work-ing conditions has been established through a fundamental analysis. The model takesinto account the three major aspects of the process that the machine designers and theproduction engineers have to deal with.1. The deformation model is capable of producing the upper bounds of the rollingloads, namely, the roll separating force and the torque, that will be needed inmachine design task. Using the model, the instantaneous rolling loads, which areimportant information for the roll-gap control, can also be obtained with the helpof accurate flow stress data.The force calculation is based on a new concept called the equivalent coefficientof friction, peg. The adoption of this concept leads to two significant advantages.Firstly, the tests for the actual friction coefficient at the interface between the rollsand the workpiece, as in the traditional rolling process, may be avoided. It is knownChapter 6. MODEL VERIFICATION^ 67that there is no reliable method for such testing. Besides, the available methodsusually involve many facilities that are costly, time consuming and inconvenient inpractice. The test(See Section 6.3.3) to the /leg, however, will be very simple andreliable.Secondly, this concept leads to a theoretical consistency in the application of theupper bound approach. Further simplifications are no longer needed, as were doneby some researchers(See Section 6.3.3).Another new feature related to the deformation model is that a simpler velocityfield solution has been proposed. This largely reduced the work in derivation andcomputation. By such a velocity field, the side spread predictions were shown tobe successful.2. A temperature model has been formulated based on one-dimensional flow assump-tions in the roll-bite and two-dimensional flow outside the roll-bite. The model iscapable of predicting the through-thickness temperature distribution in the roll-bite, estimating the mean temperature of the deforming body that is one of theimportant inputs to the flow stress model, and roughly calculating the the meantemperature distribution along the workpiece.Based on the mean temperature of the workpiece, roll pass scheduling may beperformed, along with the side spread estimation. Some preliminary estimation ofthe final properties of the rolled products may also be possible.3. The modelling of flow stress for steels is very involved. An accurate flow stressmodel, which is of great importance in implementing the VCR process, has beenchosen. It has been shown that the unified creep relationship, Eqn.5.45, works verywell, as is widely reported in the literature.Chapter 6. MODEL VERIFICATION ^ 68It is observed that, for the certain type of C-Mn steel used in this work, the param-eters, n and A are non-linear functions of strain and temperature. Such functionsmay be determined by numerical methods.Also, the value of 312 kJ/mol for the activation energy, Qa, has been selected, aswas reported in the literature. It worked well in this study.6.4.2 Suggestions for Further Work1. Further study of the equivalent coefficient of friction by experiments is needed.Without experiments, it is difficult to decided the actual range of this variable.2. A temperature model for the rolls is needed in order to estimate accurately thetemperature changes of the rolls. Although many reasons that in this study thetemperature of rolls was assumed as a constant is given in Section 5.4.1, attemptscan still be made.3. The final microstructure of the rolled products will be another issue in the VGRprocess planning. A mathematical submodel of this purpose may be incorporatedin the present model.Strain rate: 20/s400350c" 3001:*250g 20015010050ea300250col200150E-• 100500Chapter 6. MODEL VERIFICATION 69200 18016014012010080604020 0Strain rate: 2/s14.1^.2^0.3^6.40 0 0 0 0 0 0 ()opt* 6.s 6.6^0.7True strain( ot = 0.025)Figure 6.3: Flow stress model verification: C-Mn steel (0.03 C 0.62 Mn). Data source:[53].^*, x — experimental data. Solid or dotted lines — model predictions.Chapter 6. MODEL VERIFICATIONFigure 6.4: Temperature distribution in roll-bite. Data source:[37]. Top: 3D-mesh plot.Middle: grid used. Lower left: front view of the 3D-mesh. Lower right: surface temper-ature comparison. Reduction ratio: 35%. Entry temperature: 1012 °C.70110010801090104-0al 1 0200.1^0.2^0.3^04^0.5^0.0Position along the workplace. rn960oaria0.7 0Start1000980Roll diameter 0.0635 truster (5 ni)Flail velocity: 20 r-r-np (Q.133 m/s)emperatureSt the. skit of the first pass-- temperature. St the en-hence to the saccinci pass. 1080cr.:g 100010-401 02010000Encl0.1^0.0^0.3^0.4^0.5^0.0Position along the vvorkpiece0.7 0Start11201100it-1)ctimmeter: O.127 rneter (1-0velocity: 20 rma (0.206 m/$)FlailFl alltamperispAre. St trvia exit of the first passtamper-Attire aat^entriarsCa 1.0 the. sec.:v.-1cl•pass1110C..1 1 100: 1 090.4MES' 1 oso10701060105010400S rid0.1^0.2^0.3^0.4^0.5^0.61=.c.sition elc.ing the. workplace0.7 0Start11301100exit of ths first passtemperature. St thsFlail c9airneter: p.osas rrie.t.ar (s it-I)Flail -velocity: 4.0 rrrip (o.kss m/s)emparahrre St the entreince to 'Wis siscc.nd peaseChapter 6. MODEL VERIFICATION^ 71Figure 6.5: Sample calculation: mean temperature distribution of the workpiece for threedifferent rolling conditions. Workpiece dimension(before rolling): 21.6 x 21.6 x 800mm.Reduction ratio: 8% — 50.8%. Furnace temperature: 1200 °C. Top(case A), time forpassl: 6.015 sec. Middle(case B), time for passl: 3.008 sec. Bottom(case C), time forpassl: 3.008 sec.Roll -diameter: .0.Roll velocity: 20Roll- diameter-:-Roll velocity: 20.RolL diameter:. a.Roll velocity: 400635 meter (5 in)ma-1p (0.133 m/s).127.rneter (10 in)rmp (0.266 m/s).0635 -meter -(5• iri)rmp (0.266 m/s)1150C-),th,1140a>a>15 1120.1=a>a 00cT, 1090a>1080107010600End0.1^0.2^0.3^0.4^0.5^0.6Position along the workpiece m1130111 00.7^08StartChapter 6. MODEL VERIFICATION ^ 72Figure 6.6: Sample calculation: effects of changes in geometry and roll velocity on meantemperature of the deforming body in the roll-bite during the first pass. Furnace temper-ature: 1200 °C. Workpiece dimension(before rolling): 21.6 x 21.6 x 800mm. Reductionratio: 8% — 50.8%.Chapter 6. MODEL VERIFICATION 731.451.41.351.3g^0.1^0.15^0.2^0.25^0.3^0.35Reduction ratioFigure 6.7: Spread prediction — comparison between simulation results and exper-iments[55]. Roll diameter: 101.6 mm(4 in). Specimen dimension: 9.525 x 9.525mm(0.375 x 0.375 in).0.4^0.45^05Chapter 6. MODEL VERIFICATION^ 740.1^0.15^0.2^0.25^0.3^0.35^0.4^0.45^0.5Reduction ratioFigure 6.8: Torque prediction: comparison between simulation results and experi-ments [57]. Roll diameter: 127 mm(5 in). Specimen dimension: 12.7 x 19.05 mm(0.5 x 0.75in).Chapter 6. MODEL VERIFICATION^ 75Figure 6.9: Schematic illustration of the roll force vectors layout in flat rolling processChapter 6. MODEL VERIFICATION^ 76Figure 6.10: Roll force prediction: comparison between simulation results and experi-ments[57]. Roll diameter: 127 mm(5 in). Specimen dimension: 12.7 x 19.05 mm(0.5 x 0.75in).Bibliography[1] Roberts, W. L., Cold Rolling Of Steel, Marcel Dekker, INC.1978.[2] Theocaris, P.S., "A Study of the Contact Zone and Friction Coefficient in Hot-Rolling", Metal Forming and Impact Mechanics, Pergamon Press, Ed. by S.R. Reid,1985[3] Hill, R., "A General Method of Analysis for Metalworking Processes", Mech. Phys.Solids, Vol. 11, 1963, pp. 305.[4] Lahoti, G. D., Kobayashi, S., "On Hill's General Method of Analysis of MetalworkingProcesses", International Journal of Mechanical Sciences, Vol. 16, 1974, pp. 521-540.[5] Oh, S. I., Kobayashi, S., "An Approximate Method For A Three-Dimensional Anal-ysis of Rolling", International Journal of Mechanical Sciences, Vol. 17, 1975, pp.293-305.[6] Kennedy, K. F., "A Method for Metal Deformation and Stress Analysis in Rolling",Doctoral Dissertation, The Ohio State University, 1986.[7] Design And Application Of Leaf Springs — SAE J788a Handbook Supplement, 1978.[8] Sepehri, N., "Computer-Aided Rolling of Parts With Variable Rectangular Cross-Section", MASc. Thesis, The University of British Columbia, 1986.[9] Sassani, F., Sepehri, N., "Prediction of Spread in Hot Flat Rolling under VariableGeometry Conditions", J. Materials Shaping Texhnology, Vol. 5, No. 2, pp.117-123,1987.[10] Sassani, F., Sepehri, N., "Computer-Aided Process Planning for Rolling of PartsHaving Smoothly Varying Rectangular Cross-Section", Int. J. Mach. Tools Manu-fact., Vol. 29, No. 2, pp.257-266, 1989.[11] Sparling, L. G. M., "Formula for Spread in Hot Flat Rolling", Proc. Inst. Mech.Engrs., Vol. 175. pp. 604-640, 1961.[12] El-Kalay, A. K. E. A., Sparling, L. G. M., "Factors Affection Friction and theirEffects upon Load, Torque, and Spread in Hot Flat Rolling", J. Iron Steel Inst.,Feb. 1968, No. 206, pp. 1110-1117.77Bibliography^ 78[13] Chitkara, N. R. and Johnson, W., "Some Experimental Results Concerning Spreadin the Rolling of Lead", Journal of Basic Engineering, Vol. 88, pp. 489-499, 1966.[14] Takahashi, J., Sato, T., Sakai,Y., and Ayada, M., "Production of Taper Leaf SpringBy S.P.M.",UDC 621.771.07, 1980.[15] Hawkyard, J. B., Johnson,W., Kirkland, J., Appleton, E., 1973 "Analyses for rollforce and toque in ring-rolling, with some supporting experiments", Int. J. Mech.Sci. 15, pp.873-893.[16] Kobayashi, S., Oh, S., Altan, T., Metal Forming and the Finite-Element Method,Oxford University Press, 1989.[17] Hill, R., The Mathematical Theory of Plasticity, Clarendon Press, Oxford, 1950.[18] Li, G.-J., Kobayashi, S., "Spread Analysis in Rolling by the Rigid-Plastic Finite El-ement Method", Numerical Methods in Industrial Forming Processes, p.777, Piner-idge Press, Swansea UK, 1982.[19] Grober, H., "Finite Element Simulatoin of Hot Flat Rolling of Steel", NUMIFORM86 — Numerical Methods in Industrial Forming Processes, ed. by Mattiasson, K.,et al, p.225, 1986.[20] Lahoti, C.D., Akgerman, N., Altan, T., "Computer-aided analysis and design ofthe shape rolling process for producing turbine engine air-foils", Final report, 1978December, Contract Nas 3-20380.[21] Sevenler, S., Raghupathi, P. S., Altan, T., "Spread and bulge in Bar and Rod RollingUsing Flat Rolls", Iron and Steel Engineer, March 1986, pp. 57-62.[22] Yih, C. S., "Stream Functions in Three-Dimensional Flow", La Houille Blanche,Vol. 12, 1957, pp. 445-450.[23] Sheppard, T., Wright, D. S.,"Parameters affecting lateral deformation in slabbingmills", Metals Technology, February 1981, pp. 46-57.[24] W. Johnson, P.B. Mellor, Engineering Plasticity, Ellis Horwood Limited,1983[25]M. J. Luton, J. J. Jonas, Proc. Int. Conf. Strength Metals Alloys, 2nd, 1970, pp.1100-1105.[26]H. J. McQueen, J. J. Jonas, "Recovery and Recrystallization during High Temper-ature Deformation", Plastic Deformation of Materials Vol. 6, Ed. R. J. Arsenault,1975, pp. 393-493.Bibliography^ 79[27] I. Tamura, C. Quchi, T. Tanaka, H. Sekine, Thermomechanical Processing of HighStrength Low Alloy Steel, 1988.[28] W. J. McG. Tegart, A. Gittins, "The Hot Deformation of Austenite", The HotDeformation of Austenite, Ed. J. B. Ballance, 1977, pp.1-46.[29] H. Kudo, "Upper Bound Approach To Metal Forming Processes — To Date AndIn The Future", Metal Forming and Impact Mechanics, Ed. S.R. Reid, PergamonPress, 1985, pp. 19-45.[30] H. Kudo, "Some Analytical And Experimental Students of Axi-symmetric cold forg-ing and Extrusion—I", International Journal of Mechanical Sciences, Vol. 2, 1960,pp. 102-127.[31] W. Johnson, J. Inst. Met., Vol. 85, 1956, pp. 403.[32] L. C. Dodeja, W. Johnson, J. Mech. Phys. Solids, Vol. 5, 1957, pp. 281.[33] Avitzur, B., Handbook of Metal-Forming Processes, John Wiley & Sons, 1983.[34] Thomsen, E. G., Yang, C. T., Kobayashi, S., Mechanics of Plastic Deformation inMetal Processing, the Macmillan Ltd., 1965.[35] J. Chakrabarty, Theory of Plastisity, McGraw-Hill, 1987.[36] C. Devadas, I. V. Samaradekera, "Heat transfer during hot rolling of steel strip",Ironmaking and Steelmaking, Vol.13, No. 6, 1986, pp.311-321.[37] C. Devadas, "The Prediction of the Evolution of Microstructure During Hot Rollingof Steel Strips", Ph.D. thesis, the University of British Columbia, June 1989.[38] D. Partington, L. Talbot, "Computer-aided draftiong control in plate rolling", HotWorking and Forming Processes, Ed. C. M. Sellers, et al, 1979, pp716-180.[39] G. F. Bryant, M. 0. Heselton, "Roll-Gap Temperature Models for Hot Mills", MetalsTexhmology, Vol. 9, Dec. 1982, pp.469-477.[40] F. Hollander, "A model to calculate the complete temperature dsitribution in steelduring hot rolling", The Iron and Steel Institute, London, 1969, pp.46-78.[41] Roberts, W. L., Hot Rolling Of Steel, Marcel Dekker, INC.1983.[42] B.I.S.R.A., Ed., Physical Constants of some Commercial Steels at Elevated Temper-atures, Butterworths, London, 1953, pp.1-38.Bibliography^ 80[43] J. Woolman, R. A. Mottram, The Mechanical and Physical Properties of the BritishStandard EN Steels, the Macmillan Co., 1964.[44] R. I. L. Guthrie, Engineering in Proess Metallurgy, Oxford Science Publications,1989.[45] J. P. Holman, Heat Transfer, 6th. edn., McGraw-Hill, 1986.[46] C. Devadas, I. V. Samaradekera, and E. B. Hawbolt, "The Thermal and Metal-lurgical State of Steel Strip during Hot Rolling: Part I. Characterization of HeatTransfer", Metallurgical Transactions, Vol. 22A, Feb. 1991, pp.307-319.[47] Ashok Kumar, I. V. Samaradekera, and E. B. Hawbolt, "Roll-bite deformation furingthe hot rolling of steel strip", Journal of Materials Processing Technology, Vol.30,1992, pp.91-114.[48] F. Kreith, W. Z. Black, Basic Heat Transfer, Harper and Row, New Youk, 1980.[49] K. Murata, H. Morise, M. Mitsutsuka, H. Haito, T. Komatsu, and S. Shida, Trans.Iron Steel Inst. Jpn., Vol. 24, (9), 1984, pp.B309.[50] P. G. Stevens, K. P. Ivens, and P. Harper, Journal of Iron and Steel Institute, 1971,Vol. 209, pp.1-11.[51] M. A. Pinsky, Introduction to Partial Differential Equations with Application,McGraw-Hill Book, 1984.[52] C. M. Sellars, "The Physical Metallurgy of Hot Working", Hot Working and FormingProcesses, Ed. C. M. Sellers, et al, 1979, pp.3-15.[53] M. J. Stewart, "Hot Deformation of C-Mn Steels from 1100 to 2200 °F (600 to 1200°C) with Constant True Strain Rates from 0.5 to 140 s'", The Hot Deformation ofAustenite, Ed. J. B. Ballance, 1977, pp.47-67.[54] S. F. Wong, P. D. Hodgson, P. F. Thomson, "Lead As A Model Material For TheHot Rolling Of Steel", Mathematical Modelling of Hot Rolling of Steel, Ed. S. Yue,1990, pp.281-289.[55] Chitkara, N. R. and Johnson, W., "Some Experimental Results Concerning Spreadin the Rolling of Lead", Journal of Basic Engineering, Vol. 88, pp. 489-499, 1966.[56] Loizou, N. and Sims, R. B., "The Yield Stress of Pure Lead in Compression", Journalof the Mechanics and Physics of Solids, Vol. 1, pp. 234 to 243, 1953.[57] Sims, R. B., "The Calculation of Roll Force and Torque in Hot Rolling Mills", Proc.Inst. Mech. Engng, Vol. 168, pp. 191. 1954.Bibliography^ 81[58] Yang, D. Y., Ryoo, J. S., "An Investigation into the Relationship between Torqueand Load in Ring Rolling", Transactions of ASME, Journal of Engineering for In-dustry, Vol. 109, pp. 190-196, Aug. 1987.[59] Gerald, C. F., Wheatley, P. 0., Applied Numerical Analysis, 4th edn. 1989.[60] Hosford, W. F., Caddell, R. M., METAL FORMING — Mechanics and Metallurgy.Prentice-Hall, Inc., 1983.S=--XnAppendix AANALYTICAL SOLUTION TO THE UPPER BOUND EQUATIONSA.1 The neutral pointDifferentiating Eqn.4.39 gives,aQ(s,i) = VRt(xo) Ow(x)awi^t' (x„) awiOr,5Q (x ) ^ aw(x,i)awl^w(x) awland5Q(x) n (1  Si + 1 aW^t' at'ax„^t\t axn w ax„ 1 + axn)A.1.1 The derivative of Ed against w1(A.4)where, for the sake of simplicity, define Eq.4.14 as Z(x)^Z3(x) Z4(x) Z5(x), andafter some manipulations, one may have the following,Z3(x ) =Z4( x) =(/2^ln  + Z2a^3^f332 {z2 Ay(2.12 + B2)](A.5)(A.6)Z5(x) = 12 I( 2/ —1) tan_1 D T 2) tgia^3^a 3'y (A.7)From Eq.4.13 and Eq.A.2, the following results,aEd 1^aw(xn) E 2 (az3 az, az5 clsawl dw(x)^awl Q o,.,owl owl owl82Appendix A. ANALYTICAL SOLUTION TO THE UPPER BOUND EQUATIONS83where, let-y = Nt^a = Mw^ (A.8)0 = V/2 +72 (A.9)B = VI2 + a2^ (A.10)(Z2 – B)ID =  (A.11)(B + a)-yZ2 --= VB2 + 72^ (A.12)A = in '..r + Z2 B (A.13)tg= tan- ^ (A.14)7and differetiation of Eq.A.8 — A.14 against w1 results in the following,ON— ton^ = tn^ (A.15)uwi^uwi0p 1 (aI2^ON),=^+ 27t (A.16)owl^20 awi^awlOa am A, Ow^ „ (A.17)awl^w awl + 1" awlaB2^ar^aa,^. ,, + 2a ^ (A.18)owl^owl^owl.0z21^ (aB2^ON),.,^= + 2-yt (A.19)awl^2Z2 atpi^0w1aA, ^1 (3y az2) 1 aB awl^7 + Z2 alVi + aWi ) Bawl^(A.20)atg 1 ^f 03 1)  a-y^7 [-yt ON + 2 (i.^1)0121}Owl^72 + (0 –1)2 1^awl^1. (A.21)0 awl^/3 I) awlap aB) a'  I.  ,^+(z2 B) ,wawl^[(B +142)7]2 {(B + a)7 [I (0,Z2owl owl ol–(Z2 – B)I [ry ( aawB 4_ Oa \ + (B + a)  0-y  11 (A.22)1^awl)^awl.) jNow, with the help of the Eq.A.8 — A.21, the partial derivative of Eq.A.5, A.6 andQ^wt(p4 = NIA + (PiW(P2 -WtVR V 1 + t '2(P3 =Appendix A. ANALYTICAL SOLUTION TO THE UPPER BOUND EQUATIONS84A.7 may be expressed as the following,az3 1 { Z3  Oa + (312 +,-), al ) ln a + Z2=awl^a^awl \awl 3 awo^0+ (12 + -12.) r  1 ^( Oa  + 3z2) 1 aol 1 (A.23)3 ) [a + Z2 W1 OW1 ) 13 awl jj0Z4^2 1 3z2 +1 [(212 + B2) (aA^ A a-y ),^+ A (2°,,I2 + 3,B2)] }(A.24)=awl^3 awl 'I^awl / owl owl owlaz5 25 ar + 12 ( 1 { 1 (^aa^1= c, . a arawi 1 atvi) [2 tan-1 D + (3-7 — 2) tgi0wi^/2 awl+I Ill _ 2) Otg^tg a-y 1 1 ^1 ^(21 1) aD) (A.25)R3-y^) Owl 323J f 1 + D2 a 3) awiThus, Eq.A.4 is completely determined.A.1.2 The derivative of Ef against wi.From Eq.4.20, one may have the following,anEf —M k (naQ  11 F(x)Vi +^t'2dx + Q II V1 + t12 !F dX)01111^OW1 0^ 0^OW1where, let,(A.26)(A.27)(A.28)(A.29)then, Eq.4.21 becomes,F(x) =w (v4+ C-a-1 + ln CP2 + (P4cp2^l(P3iPartial differetiating of Eq.A.27 — A.30 against w1 results the following,a(p2 = 1 ( 312^,Owawl^tw2 ell awl w atvi )(A.30)(A.31)Appendix A. ANALYTICAL SOLUTION TO THE UPPER BOUND EQUATIONS85VR a Q^awQ2 awl +  tw2 awl1^5992^aso3 (104 402^ +403+493atoi)(404+^+ln Co2 + (iO4 Ow^1 ^(9:03)21(102^l^ (P2(P31^awl + (104(2CO3^1 3403 +(+ ^1 ^5404k 402^403) °W1^402 + (P4) OW1 1note that for Eq.A.30, when co3 <0, 14031 = —w3, then,595,3atvi3994 awlSFawl(A.32)(A.33)5402 awl(A.34)514031 _ _aw3 (A.35)awl — awltherefore, we have,1 314031 _ 1 3403 (A.36)1403I awl^(,03 awlA.1.3 The derivative of È, againstFrom Eq.4.26 and Eq.4.27, one may have the following,^aQ^as kQ (A.37)awl^Q awi^awlwhere, let,S = 1/1D'2 t'2^ (A.38)then,Os^w' Ow'„^ (A.39)awl^s owland the following may be obtained with the help of the above two equations,as^ , 1= 1 ^w 2t 7-=e ( [2 ln(ti + s) — w^s)sawl^3 Appendix A. ANALYTICAL SOLUTION TO THE UPPER BOUND EQUATIONS86w + s+(ti )2 ( ^1 ^+ 1) + 2 in s —^ inktvi)^ivi(tvi+s)^s)^3^w'^It'l2 ( 1 \ 1^&Lilo'3 .s(s — ts) w1) fs=o awl (A.40)A.1.4 The derivatives of Ed, Ef and E, against xn1. From Eq.4.13, one may have the following,atd^[ 5Qaxn^3 axn o= —a-- Z(x)dx„by substituting Eq.A.3 into the above equation, one may obtain,aEd^(1  at^1 aw^t' at' )axn^t axn + w axn 1 + t'2 axn Ed2. From Eq.4.20, one may have the following,atf^c2 F(x)i+ti2dx +Q vq+ti2aF  dx)axn mn' aOxn 0V^0^axnand note that from Eq.A.28 and Eq.A.29,VR aQ Q2 ax„VR (p3 aQQ2 (,04axnX=Xnap3awlacp4 awl(A.41)(A.42)(A.43)(A.44)(A.45)Now, with the help of Eq.A.44 and Eq.A.45, partial differentiation of Eq.4.21 resultsthe following,SF VR CO3 ( _4_^1 2W3 1 1 aQaxn 111(22 L(P4 (P2 + CO4 + (P2 (p3.1 axnthus, by substituting Eq.A.46 and Eq.A.3 into Eq.A.43, one may obtain,(A.46)atf^(1  Si^1 aw^t' at')^{"i — VR \ + t2  Ox,.^aXn + W aXn^ii2 aXn )^0X=Xn[(p3 (1 + ^1 ^)^4)3^1 I cis}CO4^CO2 + (004^(102^(t03(A.47)Appendix A. ANALYTICAL SOLUTION TO THE UPPER BOUND EQUATIONS 873. Partial differetiating Eq.4.26 and then using Eq.A.3, one may have the following,at,^ac2= kSax„^ax„^(1 at^1 aw^t'^at'=t ax,, +w ax„ 1 + t'2 ax„ .... t„^(A.48)Appendix BTRUE STRESS & STRAIN CORRELATION FOR C-Mn STEELS8830 20700 C 18251620 14Strain rare: 20/s151012_.1000 C 8^10^ ^ 800C. —7-77 ....   ^'900.C.^81200 C 65 420.1 0.2 0.3^0.4 03 0.6 oo07 0.1 0.2 0.3^0.4 0.60.5 07True strain True strain3022 20 -18 -16 r14 -121-10-......... : ... . r .900C _1000 C................... 800 C151200 C50 2 Oo0.20.1 0.3^0.4 0.603 07 07True strain True strainStrain rate:Strain rate: 2/s----^...... . .  ^....^.... .  .... .  0.3 0.4 0.50.1 0.225 100.1 0.2 0.3^0.4'True strain----------  900C ^ ^ 800 C ^_^- - -a ---^-Haw -0.1 05 0.60.2 070.3^0.4403530201510700 C. ------------ -----   -- --Strain rate: 20/s1200 C True strain55 700 CStrain rate: 140/s900C  __^-^----------------      ^ 1860 C.............15v."3550 05 070.61100^_......^.........   ---------- ---------1200 coo 07  40. 1 0.2700 C   ^ /57..ifif .5.Strain rate: 0.5/s0.3^0.4True strain20 0.5 800 C0.6 0 7181614121064900 C 1000 C..,-^,o'w^l'.....^x,.. a. .....___,.....--.4. ...........- -^...... 1000 C800 C.....   .....   ..... , ....^_,. .......;% .....4.-..4-..-*-.- 4-------------.------.-1-1.°-.0 Cl?' f:!14.i ...''.4.--.--&----- ____, ...... ..... . .... .^ ^  1?..00 C True strain3025201510530if 800 C• Strain rate: 20/s0.6 070.1^0.2^0.3^0.4True strain454035252015105045'  0 6050q=-30coco 420010oo700 C0.1^0.2^0.3^0.4^05^0.6^07True strain C900C700 CStrain rate: 1401s ................^........^ 1000 Ca 15 e"7?(.:10 -Is# ........oo........ -&1000 C1100 C.......... . • .. .......^1200^_5 -Strain rate: Vs0.1 0.2 0.3^0.4 0.5 0.70.630700 C....*, .......... .100._.......900 C2520True strain_________^.•^ 900 C•CO345^ cao2535Strain rate: 20/s30. .^1000C^1 20 . 900C^iI1100 C -a 10(10.0a . 410151200C10105^0.6^07 0.1^0.2^0.3^0.4 0.5^0.6^07True strain Trite strain60oo 0.1^0.2^0.3^0.435Strain rate: 140/s SOO C50 30ao...... - -- ^.1000 30 ..^• • .......^. ......^....... . .^900 C^_.^ 1000C V.20 .1100C1200 C10........... . ^)200 C 00 0.1^0.2^0.3^0.4True strain0.5 0.6^07 0.3^0.4 0.6 07True strain.......................... .......25


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