ABSTRACT LAGRANGIAN KINEMATICS IN THE PLANE-STRAIN EXTRUSION OF A REAL METAL by William Herbert Ginman Co-Chairmen: Robert M. Caddell, Thomas A. Despres This dissertation presents a kinematic investigation of the direct cold plane-strain extrusion process. The deformations are described in terms of the differences in the path lines for the flow of 6061-0 aluminum through tapered dies as compared to an ideal flow. Actual deformations are also compared to the deformations predicted by the slip line theory. The deformations are determined experimentally by examining the changes in a grid pattern that is marked on the halves of a split billet before extrusion. Two different die angles and reductions in area are investigated. A measure of the rotationality of the aluminum flow field is determined, and an analytical model is created for the actual velocity field. The deformations predicted by the analytical model of the actual velocity field are compared to the actual deformations.

LAGRANGIAN KINEMATICS IN THE PLANE-STRAIN EXTRUSION OF A REAL METAL by William Herbert Ginman A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mechanical Engineering) in The University of Michigan 1974 Doctoral Committee: Professor Robert M. Caddell, Co-Chairman Professor Thomas A. Despres, Co-Chairman Professor Anthony G. Atkins Professor Joseph Datsko Professor William F. Hosford Professor Tsung Y. Na

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ACKNOWLEDGEMENTS I wish to express my appreciation to the people and organizations which contributed to the success of this research: To Professors Robert M. Caddell and Thomas A. Despres, Co-Chairmen of my doctoral dissertation committee, for their guidance and valuable advice. To the members of my doctoral dissertation committee, Professors Anthony G. Atkins, Joseph Datsko, William F. Hosford, and Tsung Y. Na for their interest, participation and cooperation. To members of the Mechanical Engineering Instrument Shop, James Allen, William Galer, Donald Haidys and Arnold Solstad. To the National Science Foundation and the Horace H. Rackham School of Graduate Studies Graduate Student Research Fund for financial support of this project. To the Mechanical Engineering Department for providing computer time. To my wife and family for their faith in me. ii

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS...................................................iii LIST OF TABLES..................vi.................. LIST OF FIGURES............................,........,..,.......vii LIST OF SYMBOLS....x....................x Chapter I. INTRODUCTION....................1........., Object of Investigation................................... 4 Historical Review............................................ 4 Scope of Investigation...................................... 10 II. EXPERIMENTAL PROCEDURE, TEST EQUIPMENT, AND EXPERIMENTAL RESULTS.................................................13 Introduction.............................................13 Test Equipment............15................15 Extrusion Billet..............................20 Lubrication................................................. 22 Grid System.............................................. 24 Experimental Results..................................26 Experimental Capabilities................................31 III. ANALYTICAL PROCEDURE FOR MODELING REAL METAL DEFORMATION FIELDS IN DIRECT TWO DIMENSIONAL COLD EXTRUSION OF ALUMINUM.................................................... 34 Introduction and Assumptions................................ 34 Reference Frame and Normalization........................... 35 Reference Continuum.........42 Path Lines, Time Lines, and Strain Ellipses................45 Computer Implemented Model of Perfect Fluid Deformation Field in the Selected Reference Frame..................52 Comparison of the Perfect Fluid Deformation Field with the Real Metal Deformation Field.......................60 Modeled Real Metal Path Lines........................... 63 Modeled Real Metal Velocity Fields.......................... 75 Computer Implemented Model of Real Metal Deformation Fields...................................81 Slip Line Generated Deformation Fields......................85 iii

Page Chapter IV. COMPARISON AND DISCUSSION OF MODELED DEFORMATION FIELDS AND EXPERIMENTAL RESULTS....................................88 Comparison of the Perfect Fluid Deformation Fields with the Real Metal Deformation Fields......................88 Modeled Real Metal Velocity Fields..........................95 Comparison of Perfect Fluid Deformation Fields with Modeled Real Metal Deformation Fields.................. 96 Comparison of Modeled Real Metal Deformation Fields with Real Metal Deformation Fields.....................98 Comparison of Slip Line Deformation Fields with Real Metal Deformation Fields..............................102 Perturbations of the Developed Model for the Real Metal Deformation Fields............................. 105 Potential Flow and Plane-Strain Extrusion.................110 Conclusions...............................................110 Future Applications..............................111 APPENDIX A. SELECTED PROPERTIES OF 6061-0 ALUMINUM.................112 Hardness...................................................112 Yield Strength Versus Percent Cold Work.................... 113 Tensile Behavior.......................................113 Modeled Tensile Behavior...................................113 Anisotropic Behavior.....................................114 APPENDIX B. RAM LOADS AND VELOCITIES DURING THE DIRECT TWO DIMENSIONAL COLD EXTRUSION PROCESS.........................115 APPENDIX C. THE SCHWARZ CHRISTOFFEL TRANSFORMATION FOR THE GENERAL QUADRILATERAL WITH TWO VERTICES EXTENDED, OF THE EXTENDED VERTICES, ONE CONTAINS A SOURCE AND THE OTHER CONTAINS AN EQUAL SINK...........................118 APPENDIX D. COMPUTER AIDED ANALYSES AND PLOTTING PROGRAMS..........127 General Purpose Subprograms...........................127 Orthogonal Curvilinear Coordinate Computation and Plot.....136 Perfect Fluid Deformation Field Computation and Plot.......138 Experimental Modeling Coefficients Computation.............146 Experimental Function Definition Using a Cubic Spline Fit...................................................148 Modeled Real Metal and Perfect Fluid Velocity Hodograph Plots.......................................152 Modeled Real Metal Deformation Field Computation and Plot................................................154 BIBLIOGRAPHY......................................................162 iv

LIST OF TABLES Page TABLE I GEOMETRIES INVESTIGATED.................................11 TABLE II PRESS SPECIFICATIONS....................................15 TABLE III MARKING EQUIPMENT SPECIFICATIONS........................20 TABLE IV LUBRICANT SPECIFICATIONS................................ 23 TABLE V FORESHORTENING RATIOS................................... 31 TABLE VI EXPERIMENTAL POINTS FOR DETERMINING MODEL COEFFICIENTS...........................................72 TABLE VII MODEL COEFFICIENTS.....................................73 TABLE VIII EXPERIMENTAL POINTS FOR DETERMINING MODEL COEFFICIENTS..............................92 TABLE IX MODEL COEFFICIENTS.................................. 93 TABLE X SELECTED MECHANICAL PROPERTIES 6061-0 ALUMINUM.........112 TABLE XI YIELD STRENGTH VERSUS PERCENT COLD WORK..............113 TABLE XII FIN THICKNESSES..............................117

LIST OF FIGURES Figure Page 1. Two dimensional extrusion through tapered dies.................10 2. Exposed internal view of the extrusion machine.................16 3. Die pieces for altering extrusion geometry.....................17 4. Essential elements of ram travel control.......................18 5. Clamps on assembled extrusion machine..........................19 6. Extrusion billet. Material 6061-0 aluminum....................21 7. Lubrication for extrusion billet...............................23 8. Real metal deformation field. Material 6061-0 aluminum........25 9. Real metal deformation field...................................28 10. Real metal deformation field....................29 11. Real metal deformation field...................................30 12. Orthigonal curvilinear coordinate systems showing two different spacings.3..........................................36 13. Physical plane..................................37 14. The Schwarz Christoffel upper half plane where point D is extended and contains sink.................................38 15. The rationalizing plane. Point C is extended................39 16. The complex potential plane where p2 equal to a constant is a streamline and P1 equal to a constant is a potential line. Points A and D are extended..........................40 17. The velocity plane. Point C is extended.................44 18. Images of an array of perfect fluid path lines in the perfect fluid velocity plane................44 19. Schematic distortion of ellipse showing important parameters...47 20. Local coordinate systems C ane 1; Intermediate transformation coordinate system..........................................49 vi

Figure Page 21. Incremental array of perfect fluid path lines used to define the time function in the physical plane.................54 22. Example array of z(I,J) and z(I,K) positions...................57 23. Perfect fluid deformation field.........................3......58 24. Perfect fluid deformation field superimposed on the real metal deformation field........................................61 25. Schematic representation of perfect fluid potential lines, real metal path lines, and perfect fluid path lines in the z, p, and r planes..64 26. Schematic representation of the difference between the perfect fluid path lines and the real metal path lines AP..............68 27. Schematic synthesis of function A...........................69 28. Schematic synthesis of function AP............................69 29. Model coefficients versus real metal path line................74 30. Images of an array of modeled real metal path lines in the modeled velocity plane.....................................79 31. Images of an array of modeled real metal path lines in the perfect fluid velocity plane...............................79 32. Modeled real metal deformation field...........................84 33. Slipline field and velocity hodograph..........................86 34. Deformation field of a perfectly plastic solid................86 35. Perfect fluid deformation field superimposed on the real metal deformation field................................ 89 36. Perfect fluid deformation field superimposed on the real metal deformation field..................................90 37. Orthogonal curvilinear coodinate system........................91 38. Model coefficients versus real metal path line, r2............. 94 39. Images of an array of modeled real metal path lines in the modeled real metal velocity plane v.....................95 40. Images of an array of modeled real metal path lines in the perfect fluid velocity plane q.............................96 vii

Figure Page 41. Perfect fluid deformation field superimposed on the modeled real metal deformation field.......................97 42. Perfect fluid deformation field superimposed on the modeled real metal deformation field.98 43. Modeled real metal deformation field superimposed on the real metal deformation field............................100 44. Modeled real metal deformation field superimposed on the real metal deformation field.............................101 45. Slip line deformation field superimposed on the real metal deformation field.....................................102 46. Slip line deformation field superimposed on the real metal deformation field................................. 103 47. Slip line deformation field superimposed on the real metal deformation field..................................... 104 48. Modification 1, model coefficients versus real metal path line r2..106 49. Modeled deformation field based on Modification 1 superimposed on the modeled deformation field developed in Chapter III................................107 50. Modification 2, model coefficients versus real metal path line, r2....................108 51. Modeled deformation field based on Modification 2 superimposed on the modeled deformation field developed in Chapter III...................9............. 52. True stress versus true plastic strain.....................114 53. Ram loads and velocities during the direct cold two dimensional extrusion of 6061-0 aluminum....................115 54. Physical plane z...........................................119 55. The Schwarz Christoffel upper half plane....................21 56. The complex potential plane...................... 122 57. The velocity plane, q....................................... 122 58. The rationalizing plane, s..................124 59. Two additional equivalent physical configurations for the Schwarz Christoffel transformation......................126 viii

LIST OF SYMBOLS Symbol Description A Model coefficient in Equation 71 A Defined by Equation 71 A09 Al Differential areas at times tO and tl, respectively a Model coefficient in Equation 71 aT First transformation constant B Model coefficient in Equation 72 B Defined by Equation 72 b Model coefficient in Equation 72 bT Second transformation constant c Constant defined by Equation 46 c1 Skew symmetry coefficient in Equation 71 C2 Skew symmetry coefficient in Equation 72 D Displacement vector dl, d2 Semi-major diameter and semi-minor diameter, respectively Vector describing strain ellipse e Unit vector normal to a perfect fluid path nq line e IUnit vector tangent to a perfect fluid path q line H Outlet to inlet dimension ratio H. Inlet dimension to extrusion dies H Outlet dimension to extrusion dies 0o I Index identifying a particular path line i The imaginary constant from complex variables (i/CT) ix

it An integer derived from time function J Index identifying a particular position on a path line K Constant in plastic stress, strain equation K1, K2 Constants in Equation 51 K3 Constant in Equation 52 k1, k2 Unit vectors in the z plane k3 Unit vector normal to the z plane L Effective foreshortening ratio Los L1 Differential lengths at times tO and tl, respectively M Integer 11 1 2 Unit vectors in the C local coordinate system ml, m2 Unit vectors in the n intermediate transformation coordinate system N Integer n Constant in plastic stress strain equation n1' n2 Unit vectors in the W local coordinate system p Potential plane Pi' P2 Dimensions in the potential plane or Lhe potential function and the stream function, respectively PM (I) Selected starting position on a perfect fluid path line P2(1) Perfect fluid path line that coincides with a real metal path line in the limit up and down stream from the tapered die section AP Represents the difference in position in the complex potential plane between the real metal path line and a perfect fluid path line x

AP The model for AP P max The maximum deviation AP min The minimum deviation q Velocity plane q1,' q2 Dimensions in velocity plane for velocity components of a perfect fluid particle q Perfect fluid velocity vector R Reduction in area R Position vector Ro Position when time is zero r Modeled real metal path plane rl, r2 Dimensions in the modeled real metal plane or the potential function and the modeled real metal path line function, respectively r' (I) Selected starting position on a modeled real metal path line S Distance s Rationalizing plane S1, s2 Dimensions in rationalizing plane t Time at Time interval V Magnitude of velocity Vo, V1 Magnitudes of velocities at times tO and t1, respectively Vi Inlet velocity V Velocity vector VA Normalized velocity in the limit up stream from the tapered die section v Modeled real metal velocity vector xi

V1, v2 Components of the modeled real metal velocity vector w Schwarz Christoffel upper half plane w1, w2 Dimensions in Schwarz Christoffel upper half plane x Dimension x1, x2 Dimensions in plane of measurement for billet x ref. Reference dimension z Physical plane Z1, z2 Dimensions in the physical plane Greek Letters Symbol Description Model coefficient in Equation 71 Model coefficient in Equation 72 r Constant in Equation 47 Y Semi-die angle Differential length True plastic strain FIEe2 Maximum and minimum principal natural strains Local coordinate system E1'42 Dimensions in local coordinate system Intermediate transformation coordinate system Tll s r)2 Dimensions in intermediate transformation coordinate system ~~~~0 ~Constant in Equation 47 e Angle between xl axis and path line xii

ow Angle in the w plane using polar representation Local coordinate system Dimensions in local coordinate system Constant in Equation 47 P Density CY Tensile stress a Stress in the x direction w ayy Stress in the y direction + Angle in the C local coordinate system Shear angle Angle between maximum principle strain axis and path line Vorticity vector xiii

Chapter I INTRODUCTION This dissertation presents a kinematic investigation that begins to answer a question that arose during a graduate seminar on metal forming. This discussion may have transpired as follows: Professor: Today's topic will be slip line theory as applied to non-homogeneous plane strain deformations of rigid-perfectly plastic isotropic solids. Questioning Student: From our studies of the mechanical properties of metals, we learned that for many annealed materials such as brass, aluminum and low carbon steel, the yield strength could change by a factor of two within 10 percent cold work. How well does the slip line theory apply to forming these materials, considering that a perfectly plastic material has a constant yield strength? Professor: Given the differences between the more common metals and the perfectly plastic solid, slip line theory has proved to be a surprisingly useful concept that has led to a better understanding of many metal forming processes. But before answering your question, let us review some properties of slip lines. Slip lines can be considered curvilinear coordinates that coincide with the directions of the planes acted (1) upon by the maximum shear stresses. When substituting expressions for the stresses into the stress equilibrium equations, -x + a —yx= 0 = __y + 3_xy ax ay 3y ax

2 an expression for the yield strength is included. If the shear yield strength is a constant, the partial differentiations in the stress equilibrium equations are performed easily, resulting in the commonly accepted slip line relations. These relations are derived from kinetical considerations together with the material model for the perfectly plastic solid. The kinetical expression of stress equilibria and selection of a curvilinear coordinate system coincident with the maximum shear stress planes are equally valid and useful when analyzing deformations of work hardenable materials. However the material models employed for work hardening metals usually require the deformation histories. Deformation histories imply the ability to identify material particles and determine the strain state as a function of time along the particle's path during the forming process. Questioning Student: Wouldn't this introduce enormous complexities when trying to perform the indicated partial differentiations from the stress equilibrium equations? Professor: Precisely. A general problem statement for this slip line approach for work hardenable materials has been rMlade, (2) but solutions are difficult. Student: It appears that a somewhat general kinematical model that is stated in terms of a particle's path could be useful when trying to analyze the forming of work hardening materials from this generalized slip line approach. Professor: Perhaps a research project studying the kinematical relations of a specific metal forming process in terms of the

3 particles' path line would be fruitful. As a result of this discussion, the extrusion process was selected for investigation and presentation in this dissertation. The parameters completely characterizing extrusion can be classified as belonging to two groups: (1) The parameters characterizing the extrusion process. (2) The parameters characterizing the extruded product. The forces, energies, powers required, etc. for the process are directly related to extruded material. The hardness distribution, inhomogeneity in the yield strength, residual stresses, etc. in extruded product are directly related to the extruded material. The analysis of the extrusion process, therefore, depends on the accuracy of the material model. The analysis of the problem can be thought to have three parts: 1. the kinematical part 2. the kinetical part 3. the material model part which relates the kinematics to the kinetics for the extrusion problem. For material models that depend on the previous deformation, a particle's deformation history must be part of the analysis. The particle's path line must be determined and the deformation of a particle as a function of time relative to the path line is required to meet this historical (3) requirement. Prandtl points out that this type of problem statement has been a traditional source of difficulty. The kinematical relationships of a flow can be described from two points of view: (1) The Eulerian point of view, which is to describe the flow in the neighborhood of a fixed point in space.

4 (2) The Lagrangian point of view, which is to identify a neighborhood of particles at some time, to, and describe the neighborhood and its path through space as a function of time. OBJECT OF INVESTIGATION The object of this investigation is to provide a kinematic modeling method for the flow of a work hardenable material during the extrusion process; the method must be Lagrangian and give path lines, velocity fields, and deformations relative to path lines as a function of time for various die geometries. Flow fields can be classified as either irrotational or rotational, the flow of a real metal in an extrusion process being rotational. A Lagrangian measure of this rotationality is defined by the modeling method and the rotationality is characterized by this measure. In addition, the kinematic modeling method must be adaptable to further work relating the kinetics and kinematics through a material model that depends on deformation history. HISTORICAL REVIEW The extrusion process is studied using the analytic principles of plasticity. With the exception of shop practice, the history of the analytic investigations in plasticity started with M. H. Tresca in 1864, when he (4,5) attempted to answer the question, "What is the stress state under which a metal begins to plastically deform?" The maximum shear stress theory for the yielding of ductile metals was the result of his investigation. Saint-Venant, in applying this theory to the problem of determining stresses in a partly plastic cylinder, recognized there is no one-to-one relationship between stress and plastic strain. (7) Levy, adopting a Saint-Venant concept of an ideal plastic material, proposed three dimensional relations between stress and plastic strain

5 (8) rate, while Guest investigated the yielding of materials under combined stresses obtaining results broadly agreeing with Tresca's maximum shear stress theory. These are the major accomplishments in the 19th Century in attempting to develop a plasticity theory for ductile metals. By the beginning of the 20th Century, the directions for future investigations had been charted: (1) Refinement of the yield criteria. (2) Continued material studies. (3) Differentiating between small scale and large scale deformations. (4) Solutions to specific problems or approximate solutions to specific problems. von Mises(9) proposed a yield criterion that is analytically more tractable than Tresca's, with experimental data usually bounded by these two yield criteria. Hencky(10) interpreted von Mises' yield criterion to be a maximum distortion energy theory, while Nadai(11) interpreted this yield criterion in terms of octahedral shear stresses. Al(12) though Lode's experimental results were in agreement with von Mises (13) to a first approximation, Taylor and Quinney examined some of the deviations in Lode's experimental work and determined that real metals have regular deviations from the von Mises theory. Yoshimura and Takendaka(14) in their paper propose a theory that is an extension of the von Mises' theory of plasticity, for isotropic work-hardening materials. This recent theory accounts for some of the regular deviations as observed by Taylor and Quinney, however, it doesn't include the effects of temperature or strain rate which can be important at high temperatures and strain rates. This is not mentioned as serious criticism, but to indicate the complexities encountered describing material properties under all environmental conditions.

6 With the advent of dislocation theory by G. I. Taylor et al. in th. 20's and early 30's, investigations in material behavior have proceeded from two points of view. The yielding phenomenon and the plastic behavior of metals can be viewed as either macroscopic or microscopic. The dislocation theory uses dislocation mechanics to explain the microscopic behavior of individual crystals. On a sufficiently large scale with proper averaging to account for the polycrystalline nature of metals, the microscopic and macroscopic points of view in the limit should be the same. Unfortunately, at present the respective points of view have not developed to this level. Common metals and alloys are susceptible to work hardening as judged by the tabulated results of yield strength versus amount cold work in the Metals Handbook.(15) The current state of the art in material science is such that the material model must be judiciously selected when specific plasticity problems are investigated. The history of plasticity problems starts in 1920 and 1921 when Prandtl(16) showed that the two dimensional plastic problem for a perfectly plastic solid is hyperbolic. The general theory underlying Prandtl's w (17) special solutions was supplied by Hencky in 1923. During this time, Nadai investigated the plastic zones in a twisted prismatic bar of arbitrary contour both experimentally and theoretically. This work by Nadai is best summarized in his book.(11) In 1925, von Karman(18) analyzed the state of stress in rolling using approximate techniques. Siebel and then Sachs soon followed with a similar analysis for wire drawing.(1920) The analysis of plasticity renewed interest in the concept of strain. P. Ludwik(21) used Hencky's concepts of natural or logarithmic(22) strain to compare taensile test and compressive test curves on the basis of natural strains. He found that the stress strain relationships are

7 nearly coincident. This was the first modification of the concept of strain to come into general use. Prior to this time the accepted theory of strain was Cauchy's infinitesimal strain theory. Cauchy's equations together with Cauchy's other analytical work in the 1820's form the elastic theory now admitted for isotropic solid bodies with small deformations. (Reference 23) Love(23) references the history of the development of strain theory for general displacement while summarizing the results in his (24) book. Green and Zerna also summarize the theory of strain for general displacements emphasizing the general curvilinear tensoral properties of the theory. There is an important analytical distinction between plasticity problems that can be characterized as involving small deformations, such as the yielding of bars, and problems characterized by large plastic deformations, such as extrusions. The plasticity problems characterized by small deformations have plastic and elastic strains of approximately the same order of magnitude. Reuss(25) produced a genral plasticity theory, to allow for both components of strain. Plasticity problems characterized by large deformations usually consider the elastic strain negligible; this invokes the assumption that the material is incompressible. Prandtl used this last assumption together with the isotropic, rigid, plastic, and nonhardening material to describe large deformation plasticity problems as hyperbolic. The characteristics of the solution of these hyperbolic plasticity equations are called slip lines. Geiringer(26) developed the equations governing the variation of velocity along these slip lines. Since the slip line theory of Prandtl, Hencky, and Geiringer, several other ways have been suggested for the analysis of plastic processes with large deformations. These methods are sometimes classified as follows:(27)

8 (19) a. The Siebel energy method. b. The Sachs slab method. (20) c. The Johnson-Kudo-Kobayaski upper-bound method. (282930) (31) d. The Thomsen visioplasticity method. When compared to experiment, slip line and the first three methods predict power requirements and forming forces that were within engineering accuracy, provided appropriate values for the effective stresses have been cho(27) sen. These methods have been modified to approximate the variation in the effective yield stress (the Sachs method), or predict a stress distribution (the slip line method). Those parameters characterizing the process can be modeled quite accurately. On the other hand, the accurate prediction of the distribution of deformations cannot be made for many plastically deformed products. The Thomsen visioplasticity method is a thorough experimental examination of a plastic process which is capable of correlating the process variables to the yield strength distribution after the fact. However, it is not predictive. This method consists of measuring the deformation and calculating the strain field by using a split billet; with the assumption of steady state operation, the strain rate field is determined and with the aid of a material model, the stress field can be determined from the strain and/or the strain rate field. This can then be compared with experimentally measured loads. In the plastic analysis of large deformations, the material model provides the means of relating the mechanics of deformation to loading. (32) Bridgman, in his survey of material models, discusses the constraints a model must exhibit if it is to display realistic behavior. At present, a general model does not exist. Therefore, the material model selected

9 must be in agreement with observed behavior of the test material in the range of process variables that are encountered in the specific plastic process studied. The elevated temperatures and high strain rates of many commercial processes are those very conditions where a rigid-plastic material model can be effectively used. On the other hand, at lower temperatures and slower strain rates the work hardening behavior of materials can predominate. The work hardening behavior of metals is complex, yet reasonable ap(33) approximations have been developed. Hodierne, for example, using the well known power law, found that a = K(e), is a good approximation for the plastic portion of the stress strain be(34) havior of 15 different metals. Barclay in 1965 used the same relationship in the experimental examination of hardening mechanisms in AISI type (35) 301 stainless steel. Datsko uses this power law in the elementary analysis of forming operations. Caddell and Atkins, (36,37) in a study of redundant work factors for rod drawing, found that the redundant work factor can be related to the strain hardening characteristics of the metal (38) being drawn. Caddell, Needham, and Johnson cold rolled rings of aluminum and compared experimentally determined yield strengths in three mutually perpendicular directions to the response of the annealed metal if it (39) were given equivalent uniaxial reductions. Avitzur includes effective strain hardening in an upper bound solution of plastic flow through conical converging dies. The cold extrusion of metals requires the analysis of the deformation histories, if an accurate material model is to be included; such models

10 are required to relate deformations to product properties. Because of engineering interest in product properties, the Lagrangian point of view for kinematic analysis of the cold extrusion of aluminum is used. This analysis of the extrusion process is explicitly related to deformation histories. SCOPE OF INVESTIGATION The extrusion process used in this investigation is the direct cold plane strain extrusion through tapered dies; The work hardenable material used in this investigation is 6061-0 aluminum. (See Appendix A.) Die angles and reductions in area used in this study exclude those that would form dead metal regions within the die cavity. In this dissertation the phrases "plane-strain extrusion" and "two-dimensional extrusion" will be used interchangeably. C1-1o~~ =X IDIRECTION OF RAM TRAVEL D TI INLET DIMENSION = Hi DIE OUTLET DIMENSION = Ho SEMI-DIE ANGLE = Y Figure 1 Two dimensional extrusion through tapered dies.

11 The reduction in area, R = (Hi - H )/Hi, (1) and the semi-die angle, y, are investigated for values shown in Table I. The outlet to inlet dimension ratio, H = H /H. (2) o 1 is related to the reduction in area, H = 1 - R. (3) TABLE I GEOMETRIES INVESTIGATED Semi-die Angle Reduction in Area Dimension Ratio Y (R) (H) 22.50 0.276 0.724 22.50 0.451 0.549 45.00 0.449 0.551 Within the limitations of these constraints, the kinematic modeling of the direct two dimensional extrusion process takes the following steps: 1. A split billet is steadily extruded for the geometries listed in Table I.

12 2. A reference frame or coordinate system is created for the geometries listed in Table I. 3. A reference continuum is chosen and the path lines and deformation field are determined for the steady flow through the geometries listed in Table I. The reference continuum chosen is the perfect fluid. The deformation field of a perfect fluid is the result of an irrotational flow field. 4. The real metal deformation field is compared to the perfect fluid deformation field. The term real metal is used to indicate a direct relation to the experimentally determined kinematic information. 5. The selected measure of real metal rotationality is the differences between real metal path lines and perfect fluid path lines. 6. The measure of rotationality for the real metal flow is modeled. 7. Using stream function theory and the modeled measure of rotationality, the modeled velocity field is determined. 8. The modeled real metal deformation fields are determined from the modeled path lines and velocity field. 9. The modeled real metal deformation field is compared to the real metal deformation field. 10. The slip line theory deformation field is compared to the real metal deformation field. These steps are taken to meet the objective of providing a kinematic modeling method for the flow of a work hardenable material during the extrusion process.

Chapter II EXPERIMENTAL PROCEDURE TEST EQUIPMENT, AND EXPERIMENTAL RESULTS. INTRODUCTION It seems that small changes in system geometry can result in large changes in the entire flow pattern during plastic deformation processes. These small changes in system geometry can make the difference between: (1) Die break through during stamping or no die break through. (2) Columnar plastic instability during cold heading or no columnar instability. (3) Dead metal region formation during extrusion or no dead metal region formation. It is nearly impossible to predict flow patterns for all conditions from theoretical considerations alone; other approaches are needed. One of the most effective methods of approach is flow visualization. Such direct observation of the internal flow in metals is in conflict with metallic opaqueness, but the split billet technique partially overcomes this difficulty for the steady two dimensional extrusion of metals. In this technique, the billet is split normal to the third dimension and a lattice is marked on these internal surfaces. The split billet i:; then extruded as a single billet. Two dimensional extrusion assumes no deformation gradient in the third dimension. During the extrusion process the billet is contained in a die cavity, so there is inevitable contact between the billet and the outer die wall. As the extrusion process proceeds, the frictional effects between the outer die wall and the billet can introduce deformation gradients in the third dimension. Proper experimental techniques and lubrication can minimize these frictional effects, but if any plane in the 13

14 third dimension is to be considered representative of the flow, a negligible deformation gradient in the third dimension must be demonstrated experimentally. To do this, different splitting planes relative to the third dimension in the billet to be extruded are selected. The deformations of the lattices marked on these different planes are compared and the magnitude of the deformation gradient in the third dimension is assayed. Flow visualization is an important tool for establishing flow models as a basis for mathematical models. But in order to interpret pictures of the flow field, it is necessary to understand four concepts that relate the pictures to the kinematic description of the flow field, These four concepts, from continuum mechanics, are: (1) Path lines; the path a particle takes through space. (2) Streak lines; the locus, at a given instant, of all particles which have passed through or will pass through a fixed point in space (coincident with (1) for steady state (s.s.)). (3) Stream lines; the curves in space always tangent to the velocity vectors of the flow field (coincident with (1) for s.s.). (4) Time lines; the level curves of a time function defined for the flow by identifying particles passing a particular line in space. These lines and curves are surfaces in general three dimensional flows. In general, the time function is a path dependent function and the time lines are most useful when the particles are identified at a particular line normal to the flow directions. For steady flows, the path lines, streak lines and stream lines are coincident, that is, the paths do not change with time. All particles passing a particular point continue on

15 the same path and, consequently, if the particles stay on the same path the velocity must always be tangent to that path. TEST EQUIPMENT The equipment used to implement the experimental investigation can be related to three areas: (1) that used to perform the extrusion process, (2) that used to prepare the extrusion billets, (3) that used to mark the lattices on the internal surfaces of the split plane in the billets. The specific extrusion process used requires a press and an extrusion machine. The specifications for the press are tabulated in Table II. TABLE II PRESS SPECIFICATIONS Company: Forney's Incorporated, New Castle, Pennsylvania Model Number: QC-500 Serial Number: 62175 Operation: Hydraulic Daylight Distance: The maximum distance between the lower platen and upper frame cross member is 22-1/2 inches. Load Range: 0 to 500,000 pounds Load Indicator: Bourbon Tube type, 1,000 pound increments from 0 to 500,000 pounds. Piston Velocity Range: 0 to 0.5 inches per minute. Maximum Piston Stroke: 3 inches

16 The extrusion machine, Figure 2, was developed specifically for the split billet technique of flow visualization as applied to the direct two dimensional cold extrusion of aluminum. The final design of the machine can be characterized as having met the following constraints: (1) It fits within the 22-1/2 inches maximum daylight clearance in the press. (2) Allowance is made for ram travel, space for the billet before extrusion, and space for the extruded product without requiring modification of press cross members. (3) It is easily modified with respect to reduction in area and semi-die angle. (4) Ram travel is controlled accurately. (5) The machine is easily disassembled for removal of the partially extruded billet without damage. (6) It withstands the loads during the extrusion of 6061-0 aluminum. MOUNT FOR RAM -- RAM TRAVEL CONTROLLER CLEARANCE FOR RAM TRAVEL LEAD IN FOR RAM GUIDANCE BILLET SHOWING SPLIT PLANE SLOT FOR CLAMP OVER TAPERED SECTION!' DIE OF THE DIE DIE PEDESTAL CNSPACE FOR THE EXTRUDED PRODUCT CONTE WL FOOTPRINTS OF MISSING......... CONTAINER WALLS 6-INCH RULE Figure 2 Exposed internal view of the extrusion machine

17 The relationships between the extrusion machine elements for changing extrusion geometry are shown in Figure 3, where it can be seen that two new die pieces are required to change extrusion geometry. While the scope of this investigation is limited to tapered die geometries, the possible die geometries to be studied are limited only by fabrication techniques. CONTAINER WALLS 2.4 INCHES -DIE TO BE INSERTED DIE IN PLACE DIE PEDESTAL E AISI 1018 COLD ROLLED STEEL AISI 6150 STEEL (HARDENED, TEMPERED, GROUND) Figure 3 Die pieces for altering extrusion geometry The control of ram travel (as shown in Figure 4) is required because the piston travel is limited to three inches. As indicated in Figure 2, the billets are fabricated so as to originally fill the tapered portion of the die cavity before extrusion. In the first stage of extrusion, the ram travel is limited to clearing the tapered section of the die of original material and establishing a steady flow. The length of ram travel required to establish steady flow is determined by a previous visual

18 observation of the path lines on the split plane of an extruded billet downstream from the tapered section. After the first stage the press is stopped. After placing a spacer block between the ram and press crosshead, the second stage of the extrusion is now accomplished. This produces a steadily extruded billet within the piston travel limitation of three inches. ADJUSTABLE ROD RAM SET SCREW FOR ADJUSTABLE ROD XL i CONTAINER WALLS PRESET RAM LU TRAVEL DISTANCE t4 MICROSWITCH FOR CONTROLLING RAM TRAVEL CLAMP OVER SPACE FOR BILLET Figure 4 Essential elements of ram travel control Clamping on the extrusion machine is shown in Figure 5. The clamp over the lead-in region for ram guidance is held together with tapered pins. This provides added rigidity to the container walls in the lead-in region, but low clamping pressure so as to minimize binding the ram. Tapered pins are threaded to accommodate a pin removal nut. The clamps over the space for the billet and over the tapered die section are held together with two, two-inch diameter threaded studs. Through the use of

19 the threaded tapered pins and threaded studs, the extrusion machine can be disassembled without damaging the partially extruded billet. Large clamping forces on the container walls are produced by the clamps over the space for the billet and over the tapered die section. These forces are the result of evenly tightening the nuts on the threaded studs to withstand over two thousand foot pounds of tightening torque. The clamping forces must be larger than the extrusion forces,which tend to separate the container walls, if finning of the billet into the spaces between the container walls is to be prevented. During the development of the extrusion machine, the magnitude of the required clamping forces was initially underestimated. However, with the present setup, the extrusion machine can withstand the loadings during the extrusion of 6061-0 aluminum. (See Appendix B.) RAM NUT AND TREADED TAPERED PIN CLAMP OVER LINE FROM MICROSWITCH TO CLAMP OVER TAPERED RELAY CONTROLLING POWER T / SECTION OF THE DIE HYDRAULIC PRESS PULAMP OVER mbled e uio a

20 EXTRUSION BILLET The billets are annealed, machined on a vertical milling machine, then the surfaces of the billet are finished by hand with a single cut mill smooth file. Finally, the process used to mark the lattice on the billets is electrochemical marking and is accomplished with commercially available equipment. The specifications of the marking equipment are tabulated in Table III. TABLE III MARKING EQUIPMENT SPECIFICATIONS Company: The Lectroetch Company, East Cleveland, Ohio Power Unit: Model V45A with heavy duty cord set (nominally 0-25 volts AC/DC, 0-45 amperes) Power Unit Serial Number: 274 Rocker Pad Assembly: Model 3-1/2" x 7" RP3 Stencil: Model 3L25055, 5" x 9" Heavy Duty. Stencil Cleaner: Type 3L Electolyte: Type #210A Cleaner: Type #3 The space for the billet in the extrusion machine is nominally 1 inch by 2.4 inches wide by 6 inches long, and the billet shown in Figure 6 is half as thick as the billet cavity. Two halves, as shown in Figure 6, are required to make up the split billet for the flow visualization study.

21 The bar stock is originally in a T-6511 temper condition. The 6061 Aluminum is heated at 7750F for 3 hours and then cooled at a maximum of 500F/Hr until 5000F to give it an "0" temper designation as specified in (15) the Metals Handbook. The "0" temper designates a full anneal. 3 INCHES 1 INCH MATERIAL REMOVED BY MILLING FROM BAR STOCK t- 2.4 INCHES," CROSS SECTION OF 0,5 INCH FINISHED BILLET. 1 — 6 INCHES 1 Hi H INLET DIMENSION = Hi = 2.4 INCHES OUTLET DIMENSION = Ho SINGLE-DIE ANGLE = y Figure 6 Extrusion Billet. Material: 6061-0 aluminum Material is removed from the billet as shown in Figure 6 on a vertical mill with successively smaller cuts, the final cut always being less than 0.003 inches. This machining procedure is adopted to minimize the material affected by deformation during machining.

22 The objects of the finishing procedure are: (1) to further remove material deformed by machining and the milling marks (2) to prepare a surface that can be electrochemically marked by the marking equipment (3) to prepare a surface that can be photographed well to record the visual flow pattern. The billets are hand finished with a file; this is followed by light buffing with crocus cloth. The draw filing procedure meets all three objectives, however two hours of filing time are required for each billet prepared for extrusion. LUBRICATION The effect of lubrication is twofold; both extruded product and the loads during the extrusion process are influenced. Three different types of lubricants, as specified in Table IV, are used and Figure 7 shows the areas of application for the lubricants. Inadequate lubrication can result in a poor surface finish on the extruded product and is indicative of frictional effects that can cause deformation gradients in the third dimension. The sacrificial lead foil together with coatings of lubricant types A and B effectively reduces the frictional effects in the third dimension. The diminished frictional effects reduce the ram load and the extrusion forces on extrusion machine container walls, the latter leading to

23 easier restraint by the clamping forces. More effective clamping forces tend to reduce finning and increase overall quality of the extruded product. TABLE IV LUBRICANT SPECIFICATIONS Lubricant A: A grease gear lubricant (Mobil Oil, Mobilplex) Lubricant B: Mixture by volume: i) 5 parts, Gear Lube (Mobilplex) ii) 2 parts, Vinyl Stearate Powder iii) 1 part, Flake Graphite Lead Foil: Commercially pure lead billet rolled to 0.008 inch foil LUB. B LUB. A LUB. B 0,008 INCH LEAD FOIL SPLIT PLANE EXTRUSION BILLET 0,008 INCH LEAD FOIL Figure 7 Lubrication for extrusion billet. The billet surfaces adjacent to the die walls were coated with lubricant A.

24 GRID SYSTEM The regular lattice marked on the billet prior to extrusion will now be called the grid; (40) it is especially chosen to relate to the important kinematic variables of the steady two dimensional extrusion process. Figure 8 shows the photographic record of the change in the grid during the extrusion process and this record is called the real metal deformation field. The grid chosen is a lattice of numbered concentric circles, 0.1 inch and 0.2 inch diameters respectively, enclosed within 0.2 inch squares. The sides of these squares, being coincident with the velocity direction upstream from the tapered die section, are then stream lines. For steady flow, the path lines, streak line and stream lines are coincident. Since the flow of aluminum through the extrusion machine is steady, the paths do not change with time, and all particles passing a particular point continue on the same path. A steady flow is assured when the sides of the deformed squares again are coincident with the direction of the steady velocity downstream from the tapered die section as shown in Figure 8. For this steady flow the sides of the squares are the path lines, streak lines, and the streamlines. Upstream from the tapered die section the lines normal to the streamlines are moving with the steady velocity of the ram. The distance between the normal lines is a constant 0.2 inches in this uniform velocity region. The inlet velocity is Vo. If the material is incompressible, the time interval for these normal lines to pass a point fixed in space is also constant, At = 0.2 inches/Vi. (4) 1~

25 Figure 8 Real metal deformation field. Material: 6061-0 aluminum For the large deformations being studied here, the aluminum is considered incompressible. If the line between rows of circles numbered 163, 164,..., 174 and circles numbered 181, 182,..., 192 in Figure 8 is considered coincident with a fixed line in space at which the particles are identified every At (Equation 4), the sides of the squares normal to the flow directions at the line, are identified at equal time intervals. This defines a time function, for the steady flow field, in

26 the x plane, (See Figure 9.), t = t(xl,x2) (5) The level curves of this function are the sides of the squares and are defined by t(x1,x2) = ~NAt (6) where N = 0, 1,..., K and where At is defined by Equation (4). Down stream where the flow is faster, the distance between time lines is greater so that the time interval between time lines passing a fixed point remains constant. Therefore, the sides of the squares originally normal to the flow direction upstream from the tapered die section are time lines for the entire flow field. Stream lines proceeding through the periphery of an infinitesimal area at some time t, will form a tube; this is called a stream tube. Figure 8 shows 6 stream tubes on each side of the center line and the changing of the circles to ellipses describes the deformations. Within a stream tube the next ellipse downstream shows the change in deformation of the ellipse during an increment of time, At. The procession of circles changing to ellipses down each stream tube describes the deformation history within the stream tube and this history is for discrete increments of time. The twelve stream tubes complete the incremental representation of the entire deformation field from the Lagrangian point of view. EXPERIMENTAL RESULTS The experimental results are summarized in Figures 9, 10, and 11. In these figures are the photographic records of the steady flow pattern

27 for 6061-0 aluminum during the direct cold plane strain extrusion through tapered dies. These flow patterns which are called "real metal deformation fields" are for the geometries listed both in Table I and on the figures. The photographic records are for both the split plane and the plane adjacent to the die wall as indicated on the figures. During the extrusion process, the aluminum displaces some of the lead in the foil (See Figure 7) and the clamps deflect due to the extrusion pressure. The combined effects of displacing lead and deflecting clamps results in an increase in thickness of the extruded billets. This is a change in dimension normal to the x1, x2 plane shown in Figures 9, 10, and 11. Due to the very nearly conserved volume during these plastic deformations the squares in the grid are foreshortened in the direction of flow. In Table V, the effective foreshortening ratio, ZL _Shorter length due to increased thickness (7) Length without increased thickness for geometries shown in Figures 9, 10, and 11 are listed. This results in the interpretation that a somewhat thicker foreshortened billet underwent extrusion.

P-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~4 X2 xl Split plane Plane adjacent to diewl Semi-die angle: y =22.3) Reduetion in area: R = 0.276 Outlet to inlet dimension ratio: H O.7 0.24 aterial: 6061-0 aluminum. Figure 9 Real metal deformation field

;~G('e~-ctE'eee x liltil x2 Split plane Plane adjacent to di wl Semi-die angle: y 22.5u Reduction in area: R = 0.451 Outlet to inlet dimension ratio: H= 0.549 Material: 6061-0 aluminum Figure 10 Real metal deformation field

:..~,' L c if-' C) Lw __ o ~~~~~~~~~~~' ~~ (3 6"' o, -~':~r'~k a~,u9 qj ~t.~ X'.:'L v]~, ~ ~ ~:49 (Q5,t Q,'~..t:,~~~2~/ i', ISBN~~~~~~~ Split plane Plane adjacent to die wall'014 ~ ~ ~ Semi-dilae Pange adacn to5~ Semidie angle: y = 45 Reduction in area R = 0.449 Outlet to inlet dimension ratio:' H = 0.551 Material: 6061-0 aluminum Figure 11 Real metal deformation field

31 TABLE V FORESHORTENING RATIOS Figure Geometry Foreshortening Ratio Y = 22.50 9 H = 0.724 L = 0.975 R = 0.276 Y = 22.50 10 H = 0.594 L = 0.942 R = 0.451 Y = 450 11 H = 0.551 L = 0.960 R = 0.449 EXPERIMENTAL CAPABILITIES The aim of the experimental portion of this investigation is to provide a capability for the direct visualization of kinematic variables which are the objects modeled. The steady extrusion results in the path, 1. Path lines 2. Streak lines 3. Stream lines 4. Time lines 5. Deformations. These are the objects modeled. The steady extrusion results in the path,

32 streak and stream lines being coincident. The split billet technique, the selected grid and the specially developed extrusion machine are the experimental techniques used to implement the flow visualization. If any plane in the third dimension is to be representative of the flow, different planes must be compared to determine the magnitude of any deformation gradients introduced by friction. Assuming a deformation gradient exists, the greatest difference in deformation exists between a split plane at the center of the billet where the frictional effects are the least and the plane adjacent to the die wall where the frictional effects are the greatest. A split plane at the center of the billet and the plane adjacent to the die wall are shown for the geometries listed in Table I in Figures 9, 10, and 11. In each case, the planes adjacent to the wall show evidence of local scraping where lubrication was inadequate. Both planes in each figure have the same scale. The existence of deformation gradients is evidenced by changes in location and shape of the path lines, time lines, and deformation ellipses between the central split plane and the plane adjacent to the die wall. By comparing the differences in location and shape of the path lines, time lines, and deformation ellipses between the split plane and the plane adjacent to the die wall in Figures 9, 10, and 11, the deformation gradients are assayed to be negligible. The central split plane is taken as most representative of this extrusion process. In conclusion it is felt that the experimental aim of providing a capability for the direct visualization of path lines, time lines and deformation ellipses is met. The real metal deformation fields that result from the application of the experimental capability described in this chapter are the objects of a Lagrangian kinematic

33 modeling method that mathematically describes the deformation of a work hardenable material as a function of time, relative to the path line.

Chapter III ANALYTICAL PROCEDURE FOR MODELING REAL METAL DEFORMATION FIELDS IN DIRECT TWO DIMENSIONAL COLD EXTRUSION OF ALUMINUM INTRODUCTION AND ASSUMPTIONS Real metal deformation fields shown in Figures 9, 10, and 11 are to be modeled. The content of this chapter will show, by example for the deformation field in Figure 9, how to use this kinematic information as the basis for a mathematical model. The presentation of the example follows these steps: 1. selection of a reference frame 2. selection of a reference continuum 3. determination of the path lines, time lines, and deformation ellipses for the reference continuum in terms of the reference frame for the geometry of Figure 9 4. determination of the kinematic variables in step 3 on the computer and implementation of computer aided plots of these variables in the format of the information shown in Figure 9 5. comparison of the plotted deformation field of the reference continuum with the real metal deformation field 6. modeling of the real metal path lines 7. modeling of the real metal velocity field 8. modeling of the real metal deformation field using the models from steps 6 and 7 9. implementation of the computer aided plots of the modeled real metal deformation field. Four assumptions are made: 1. The metal can be modeled by a continuum. 2. The continuum representing the metal can be modeled as 34

35 incompressible for the large deformations encountered during this two dimensional extrusion process. 3. The metal is bounded and follows the die geometry. 4. The flow is symmetric with respect to the center line. The first assumption neglects the microscopic crystalline nature of metals, while the second neglects changes in density. The third and fourth assumptions limit the investigation to symmetric extrusion processes where dead metal regions are not formed. REFERENCE FRAME AND NORMALIZATION The selected reference frame is the orthogonal curvilinear coordinate system shown in Figure 12. The selected reference frame and the reference continuum are directly related. The reference frame is the result of the Schwarz Christoffel transformation for the general quadrilateral with two vertices extended. One of the extended vertices contains a source and the other contains an equal sink and from the source to the sink flows a perfect fluid. The perfect fluid is the selected reference continuum. The complex analysis for this Schwarz Christoffel transformation is presented in Appendix C. The generation of the Schwarz Christoffel transformation for the geometry of Figure 9 starts with the extrusion geometry being transformed to the upper Schwarz Christoffel half plane by conformal transformations through the rationalizing plane. The solution for a perfect fluid flowing in the transformed geometry is described by the conformal transformation from the complex potential plane to the Schwarz Christoffel half plane. Since conformal transformations of solutions remain solutions to flow problems in the transformed geometry, the conformal transformation

36 of the flow solution in the Schwarz Christoffel half plane to the physical plane is a solution to the flow problem in the physical plane. In the complex potential plane, streamlines and potential lines are defined. The images of a family of these lines in the physical plane results in the orthigonal curvilinear lattice shown in Figure 12. -INLET TO OUTLET-r INLET TO OUTLET -DIMENSION RATIO- DIMENSION RATIO -H = 0.549 -H = 0,724 __ 11111111111 REDUCTION IN AREA REDUCTION IN AREA R = 0.451 R = 0,276 P2 = 2.62 = 2,09 P2 = 1.57_ P2 - Figure 12 Orthigonal curvilinear coordinate systems showing two different spacings The physical plane is shown in Figure 13. Due to the symmetry of the extrusion process selected, only half of the flow field is needed for ized physical dimensions of the extrusion billet (Figure 6),

37 z = Z + iz2 (8) where z1 Xl/Xref.; 2 x2/ ref (9) and Xref. = 1.2 inches/ (10) With this definition the geometry becomes dimensionless. The inlet dimension becomes 2m and the distance to the center line becomes aT, as shown in Figures 12 and 13. In the limit upstream and downstream from the tapered die section are the extended points A and D, while the tapered die section is defined by points B and C in Figure 13. ~~i / / ANALYTIC REGION FOR OTHER CONIFORMAL +to / \\ TRANSFORMATIONS, | -y+1r v WHERE 0 H < 1 B Z1 Y = 22,5 H = 0.724 Figure 13 Physical plane The derivative defining the Schwarz Christoffel transformation is in terms of w, the complex variable defining the Schwarz Christoffel upper half plane as shown in Figure 14. The derivative form of the transformation used for the extrusion geometry shown in Figure 9 is 1/8 dz Hlw-l1 dw - wW-H/ (11)

38 The transformation defined by Equation (11) applies to all possible outlet to inlet dimension ratios H, when the semi-die angle is 22.50 or 7T/8 radians. The images of the points defining the die geometry in the physical plane are shown in Figure 14. iW2 W PLANE ANALYTIC REGIONS FOR OTHER CONFORMAL TRANSFORmAT IONS, SOURCE Z,S,P,q A B W Figure 14 The Schwarz Christoffel upper half plane where point D is extended and contains sink Equation (11) cannot be integrated directly, but by defining a new variable 1/8 = w —H1 (12) so that now w H 8 ) (13) 1 - s the Schwarz Christoffel transformation in terms of the rationalizing plane becomes, dz 8Hs6 8 8 8Hs (11(1-s) - 1/(H s )) (14) ds Hs

39 which can be integrated. The images of the points defining the die geometry in the rationalizing plane are shown in Figure 15. Equations (11), (12), (13), and (14) are conformal or define conformal transformations. I S2 H 1 S1 ANALYTIC REGION FOR OTHER CONFORMAL S PLANE | TRANSFORMATIONS, Figure 15 The rationalizing plane. Point C is extended. The conformal transformation describing a perfect fluid flowing from a source to an extended sink as shown in Schwarz Christoffel upper half plane (Figure 14) is p = V Log w, (15) or conversely w = exp (p/VA), (16) where the term VA is the normalized uniform velocity in the limit upstream from the tapered die section, VA = Actual Velocity/Reference Velocity (17) where Reference Velocity = 0.5 inches/minute (18) The term Log w (a complex function) is defined, Log w = log wl + w2 + iw

40 where 0 = tan (w2/wl), 0 is limited 0 < < for this problem, and w 21 w w log is a logarithm to the base e (2.7183...). The complex potential plane is shown in Figure 16. iP2 P PLANE iVA'r P2 = CONSTANT ANALYTIC REGION FOR OTHER CONFORMAL TRANSFORMATIONS, z,s,w,q P1 = CONSTANT VALOG H8 B C P Figure 16 The complex potential plane where P2 equal to a constant is a streamline and P1 equal to a constant is a potential line. Points A and D are extended. The complex potential plane, P = P1+ iP2 can be interpreted as representing two important kinematic concepts for the flow of a perfect fluid. The image of P1 equaling a constant in the z plane, P1 (Zl'Z2) = constant, (19) is a potential line for the perfect fluid. The image of P2 equaling a constant in the z plane,

41 p2(Z1lZ2) = constant, (20) is a streamline for the perfect fluid. The potential function is defined as that scalar function whose gradient defines the velocity field. Since a conformal transformation conserves angular relations between lines, the conformal transformation of a family of streamlines that are orthogonal to the potential lines in the p plane remain orthogonal for the extrusion geometry in the physical plane, z. Upon integration of Equation (14), the generation of the Schwarz Christoffel transformation is finished. The integrated form is 1/2 (-i) 1/2H+s 1/2 i 1/2H+s iH+s H+s z = i Log 1i)/2 + (-i) Log 1/2 + iLogiH-s - LogH-s (-i) H-s i H-s F1/2 11 1/2 (-i) +s 1/2 i1'2+s i+s l+s (lH) 1/2r/ i1/s l2s -H) where for the two possible values of i1/2 and (-i)/2 tan 22.50 the specific values required are i1/2 = V/2 + i/2 and (-i)1/2 =- /2 + iV-/2. 1/2 1/2 Both values of i and (-i) are in the upper half plane. The Equations (12), (16), and (21) are combined to provide an explicit closed form of the analytical expression for the conformal transformation from the complex potential plane to the physical plane. Each point

42 in the analytic region of the complex potential plane has a unique point in the analytic region of the physical plane. Therefore, the conformal transformation from the complex potential plane to the physical plane is a coordinate transformation and the family of streamlines and potential lines from the potential plane forms the basis of an orthogonal curvilinear coordinate system in the physical plane. Three properties of this orthogonal curvilinear coordinate system are explicitly used in this investigation. The first property used is that boundaries and boundary conditions can be more economically described in this somewhat more natural curvilinear coordinate system. For example, the die shape is decribed by the expression, P = 0. The second property used concerns differentiation. Christoffel symbols of the first and second kind are used when differentiating in general curvilinear coordinate systems. The concepts and distinctions embodied in Christoffel symbols for general curvilinear coordinate systems are not required for differentiations in the two dimensional orthogonal coordinate system shown in Figure 12. The third property used is the direct relationship between the coordinate system and the kinematics of the reference continuum. REFERENCE CONTINUUM The reference continuum selected is the perfect fluid; it is considered inviscid, incompressible and has only density, i.e., inertia. The perfect fluid is an analytical concept and is conceived to flow steadily in a symmetrically bounded channel as shown in Figure 12. This channel is of similar geometry to the die configuration for the steady direct two

43 dimensional cold extrusion of 6061-0 aluminum shown in Figure 9. The extrusion of the aluminum is referenced to this flowing perfect fluid; in essence, the flowing perfect fluid is the standard to which the actual flow is compared. Any differences between the actual flow and the standard are the result of differences between: 1. the environmental or external influences 2. the process variables 3. the flowing materials. Since the flow of the perfect fluid is steady, the path lines, stream lines and streak lines are coincident. The stream lines for the steady flow of a perfect fluid in the geometry of Figure 12 are described by letting Pi = constant, in Equations (12), (16), and (21). These equations also describe the path lines. The time lines are determined from the path lines when the velocity with respect to the path line is known. For the perfect fluid, the velocity is defined by the conformal transformation q = conjugate (22) For the geometry of Figure 12 the expression for the velocity is, q = (VA/H) conjugate (s). (23) The velocity plane is shown in Figure 17 together with the images of the points defining the die geometry. Equations (12), (16), and (23) directly relate the position on a path line to the velocity. The images of an array of perfect fluid path lines in the perfect fluid velocity plane is shown in Figure 18. The velocity at a particular position along a path

44 line is determined by the coordinate values of that path line point in the velocity plane. An example for a particular path line point is shown in Figure 18. ANALYTIC REGION FOR q PLANE OTHER CONFORMAL TRANSFORMAT IONS, ZSWP \ D\A 8 I l- - VA/H - - V Figure 17 The velocity plane. Point C is extended. qq INLET TO OUTLET I q? DIMENSION RATIO: H = 0.724 REDUCTION IN AREA: R = 0,276 P2 = 2.618 2\ 2. 5= \P2 ql q1 _j- __9 VA/H Figure 18 Images of an array of perfect fluid path lines in the perfect fluid velocity plane

45 PATH LINES, TIME LINES, AND STRAIN ELLIPSES The path lines and time lines are related. For the discussion of these relationships, the path and the velocities on the path lines are assumed to be known. Experimentally the time lines can be identified when the grid system is selected as discussed in Chapter II. The mathematical expression for the time lines (two dimensional case) follows. Let there be a velocity field. The curve x = X(xl,X2) = constant (24) identifies all particles flowing through or on that curve. The particle's identity is assured if the position vector of that particle R = R(t) (25) is a solution to Equation (24) when time t, is zero. That is, if R = D(t) + R0 (26) where D(t) is the displacement vector, then D(t) = 0 when t = 0 (27) and R0 is a solution to Equation (24). The differential equation, RdR _dS dt = R-dR = Vs (28) LRt2 V where R = dR/dt, dS = R.dR/ and V = IR,

46 together with the boundary conditions S = 0 (29) tO R t=0 =R (30) defines the time function for this process. The specific R0 associated with a particle, identifies that particle's path. Time is a path function as defined for this process and upon integration, time becomes, t = t dS/V. (31) Let the expression t = t(RoS) (32) emphasize the path dependent nature of this time function. The expression t(R0,S) = constant (33) identifies the position of any particle on its path at some time equaling a constant. The locus of these positions for all particles R0, is a time line in continuum mechanics. Equation (33) is the mathematical expression for a time line. For a steady incompressible flow, the level curves of this time function are identical with the concept defined by Equation (6) in Chapter II. Equation (33) expresses the relationship between the path lines, the velocities on the path lines, and the time lines. The deformation ellipses are related to the path lines, time lines, and the velocities on the path lines which are assumed to be known. As shown in Figure 19 the identifying surface, x(x1,x2) = constant is chosen normal to the flow direction. The differential length 6, identifying the neighborhood of points about R0 is small compared to the radius of curvature characterizing the change in the path line and the radius of curvature characterizing the change in the time line at R0.

47 NORMAL TO THE PATH LINE TIME LINE Xf _g t(XRXS) = t MINOR DIAMETER AXIS PATH LINE \2 MAJOR DIAMETER AXIS \ _, i1 X1 Figure 19 Schematic distortion of ellipse showing important parameters These characteristic radii of curvature of the time line and path line represent the magnitude of the second order differential geometrical relations for the neighborhood of points. The differential length 6, is chosen to be small compared to these characteristic radii for an displacement, D, of interest. That is to say that the changes in the relative displacements within this neighborhood are sensibly homogenous, are of first differential order, and that these changes are described locally by linear transformations. Within this neighborhood, by virtue of these linear transformations, any straight line is transformed into a straight line and any ellipse is transformed into an ellipse. These relative displacements describe the strained neighborhood and the ellipses characterizing the deformations are called strain ellipses.

48 The descriptions of the strained neighborhoods proceed from the X(Xl,x2) = constant, time equal zero surface. By definition, this is the zero strain state surface. For two dimensional flow, the zero strain state ellipse is the circle. If the zero strain surface is to represent all previous strain history, the path lines and the velocities along the path lines must be consistent with the motion of a rigid body previous to the intersection of the path lines and the zero strain surface. The condition of incompressibility assures that volume is conserved, that is LAO A 11 (34) in terms of a differential length L, along the path line and a differential area A, normal to the path line. The subscripts 0 and 1 refer to time t=O, and some later time t = tl, respectively. In terms of continuity along the path line, PV0AO = V1A1. (35) Since density is constant, from Equations (34) and (35), L /L = V/V. (36) The strain ellipse at time t=0, is a circle and is transformed into an ellipse at some later time t=t1. Relative to the path line, this ellipse can be thought to have been formed by two transformations, an elongation and a shear. If the shear transformation is performed last, the shear angle and the angle between the time line and the normal to the path line are the same. The local coordinate systems, Figure 20, have the same scale as

49 the x1, x2 coordinates in Figures 9, 10, 11 and 19. The 1,' q11, and Ail axis together with their respective unit vectors 11, ml, and n1 are tangent to the path line. The 2' q2' and p2 axis together with their respective unit vectors are normal to the path line. X(Xi,X2) = CONSTANT TIME LINE, t(Ro,SS)= O m2 b.'2 TIME LINE MAJOR DIAMETER AXIS MINOR DIAMETER AXIS Figure 20 Local coordinate systems C and P; Intermediate transformation coordinate system q The ellipse at time, t = 0, is a circle, e 1 1 2 2 (37) where = 6cos f 52 = 6sin t, or e 6 (cos t11 + sin ~12). (38)

50 The first transformation is a simple elongation with respect to the pathline, 1 = aT51' (39) q2 = (1/aT) C2 The transformation constant is defined from Equation (36) aT = L1/L0 * (40) The second transformation is a simple shear with respect to the pathline 1 l1 + bTn2, (41) IJ = 1 1 2 2 The second transformation constant is defined bT = tan ~, (42) where P is the shear angle shown in Figures 19 and 20. The ellipse in terms of the first transformation is, e = 6(aTcostml + (1/aT)sintm2). (43) The final expression for the ellipse after both transformations is e = [(aTcost + (bT/aT)sint)nl + (1/aT)sintn2] (44) Since the Jacobians for transformations represented by Equations (39) and (41) are equal to one, the area of neighborhood 6 is conserved. The semi-major diameter dl, and the semi-minor diameter d2, of the final ellipse as described by Equation (44) are

51 dl,2 = 6 /(c c - 4 )1/,5) where dl and d2 are defined by the minus sign and the plus sign respectively and where 2 2 2 c = aT + [(b + 1)/aT]. (46) The angle P from the path line to the major diameter axis is 4 2 = (r/8)r(1 -0)(1 -E) + 1/2 arctan[2bT/(aT + bT 1)] (47) where r = bT/ IbTI and r = 1 when bT = 0 where 0 = (aT-1)/IaT- 1l and 0 = 1 when aT = 1 4 2 4 where = (aT + bT + bT 1 and 3 = 1 when aT + b = 1. The strain ellipse as described by Equations (44), (45), (46), and (47) together with the angle of rotation 0, of the path line relative to the x1 axis, is completely determined. This strain ellipse describing the deformed neighborhood of points is completely determined for any shear angle -900 < < 90 and any extension O < a < ~ T

52 where 0 > aT > 1 describes contractions with respect to the path line. The maximum and minimum principal natural strains c1 and ~ 2 respectively, are defined ~ = loge (dl/6), (48) and C2 = loge (d2/6) (49) The C1 and c2 principal strain axes relative to the path line are defined by ~ and i+W/2 respectively. The cl and E2 principal strain axes relative to the fixed X1 axis are 1 + Gand P +e+rI/2 respectively, as indicated in Figure 19. The natural elongation in the strained state for any line at angle + in the unstrained state is:=loge (I T/6) (50) where e is defined as in Equation (44). The path lines, the velocities along the path lines, and the time lines defined or discussed in this section are for general two dimensional continuous flows. The strain ellipse is defined relative to the path line for general two dimensional incompressible flows and these kinematic concepts describe the flow from a Lagrangian point of biew. COMPUTER IMPLEMENTED MODEL OF PERFECT FLUID DEFORMATION IN TIlE SELECTED SELECTED REFERENCE FRAME Since the path lines and the velocities along the path lines are known for the steadily flowing perfect fluid in the geometry of Figures 12 and 13, the kinematic variables from a Lagrangian point of view for this flow are defined. The purpose of this section is to present the method of evaluation and the method of graphically representing these kinematic variables. The digital computing facilities at the University

53 of Michigan(41) are used to implement the evaluation and graphical procedures. This presentation is limited to the algorithms used and the results of these algorithms; however, the program listings and example non-graphical output are presented in Appendix D. This manner of presentation is used for all computer aided procedures presented in the text of this dissertation. The incremental array of perfect fluid path lines used to define the time function is shown in Figure 21 where the time function for the flow field is incrementally defined by numerically integrating Equation (31) along each perfect fluid path line in the array shown. The index I identifies the particular path line, while the index J identifies the particular position along each path line. Each point on a path line has an image in 1. the physical plane, z(I,J) 2. the complex potential plane, p(I,J) 3. the Schwarz Christoffel half plane, w(I,J) 4. the rationalizing plane, s(I,J) 5. the velocity plane, q(I,J). The path lines (stream lines) are defined in the complex potential plane, P2 (1) = 0 P2(2) = K1 P2(I) N(K2) + K1 (51) where N = I-2, and I = 3,..., 25 resulting when I = 25, in P2(25) = VAT.

54 Z(P(25,1)) Z(P(25,J)) Z(P(25,41)) Z(P(I,J-1))i Z(P(I,J+1)): -Z 4 I ~Z(P (I1) - 4 1 ))J) Z(P(1, 1)) Semi-die angle: y = 22.5 Outlet to inlet dimension ratio: H = 0.549 Reduction in area: R = 0.451 Figure 21 Incremental array of perfect fluid path lines used to define the time function in the physical plane The resulting path lines from this definition correspond to dividing the experimental streamtubes in Figure 9 into fourths in the limit up and down stream from the tapered die section. Every fourth path line defined by Equation (51) corresponds with an experimental path line in the limit up and down stream. The positions along the path lines are also defined in the complex potential plane, P1(J) = p (I) + M(K3) where M = J - 1 and J = 1, 2,..., 41. (52) When J = 1, p (1) = pl(I) The index, I, in p1(I) identifies the starting point along each path

55 line as being uniquely determined for that path line. The pl(I)'s are determined by the intersection of the time equaling zero line in the physical plane with the perfect fluid path lines. The time equaling zero line for the perfect fluid is chosen to correspond to the experimentally selected time line between circles numbered 199, 200,..., 209, 210 and circles numbered 217, 218,..., 227, 228 in Figure 9. This selected time line facilitates the comparison of the perfect fluid deformation field with the real metal deformation field. The points plotted in Figure 21 are defined z(I,J) = z(p(I,J)) (53) indicating the functional character of this definition. Path lines are represented by the straight line segments between the images of path line points in the physical plane as shown in Figure 21. The velocity of the perfect fluid particle at each point in Figure 21 is defined q(I,J) = q(p(I,J)). (54) Equation (54) defines the images of the perfect fluid path lines in the velocity plane, (See Figure 18.) As in Equations (51), (52), and (53), the index I identifies a path line and the index J indicates a position along the path line. Functional subprograms are defined for the functions as represented by Equations (53) and (54), (See ZF and QF, Subprograms, Appendix D.) An increment of time is defined using the trapezoidal rule between points on a path line z(I,J) and z(I,J+l), (See DT, Subprograms, Appendix D.),

56 At= 1/2[(1/Jq(IJ) ) + (l/Iq(IJ+l))][jz(I,J+l) - z(I,J)|] (55) Starting with t(I,l) = O and summing the time increments along the incremental perfect fluid path line, each point on the path line z(I,J) is assigned a time t(I,J). Upon integration along the entire array of incremental perfect fluid path lines, the time function for the steady flow field is determined for the array shown in Figure 21. A new variable is employed to determine the level curves of this time function. The new variable defined for the flow field is it(I,J) = integer (t(I,J)/At), (56) where At = V Tr/12 if twice as many level curves are desired as those shown in Figure 9. This conversion from a real number to an integer drops all digits after the decimal. While traveling downstream on path line I, if it(I,J) + 1 = it(I,J+l), the time function has acquired the value t = NAt where N = it(I,J+l) in the interval between z(I,J) and z(I,J+l). The position z(I,K) = t(IJ+l) - t(I,J) [z(I,J+l) - z(I,J)] (57) + z(I,J) where K = it(I,J+l), is defined at the location where the time function t = NAt, (See RINT, Subprograms, Appendix D.) That is to say that the position z(I,K) is

57 determined by the linear interpolation with respect to time along the straight line segment between z(I,J) and z(I,J+l), and the index K identifies the position where level curve K intersects path line I. All values of t(I,J) are examined in this manner, which determines the array of positions z(I,K). The index K is thus associated with particular time lines. The time lines are represented by straight line segments between the images of time line points in the physical plane as shown in Figure 22. K= 4 /1, ^K_- -eD 7.z-D(I K+ I:8-eJAj 7 1, K-1 ) ZKen _'~,I= 7 1=7- /7Z(I-1K) re= 6 1=5 I/- g by/< IK=4 K=2 K=3 z(I,K) are time line segments, when K is constant z(I,J or K) are path line segments, when I is constant Figure 22 Example array of z(I,J) and z(I,K) positions An example array of z(I,K) and z(I,J) positions is shown in Figure 22, where time lines are defined for z(I,K) when K is a constant integer, and path lines are defined for z(I,J) or z(I,K) when I is a constant integer. In the limit up or down stream from the tapered section the I = 5 and the I = 9 perfect fluid path lines are straight and correspond to the first and second experimentally selected path lines in from the wall as shown in Figure 9. The perfect fluid time lines K=2 and K=4 correspond to the same time line increment as shown in Figure 9. There are twice ias many perfect fluid time lines as experimentally selected time lines. The additional time lines are used to determine

58 strain ellipses. In Figure 22 the solid lines correspond to those selected experimental lines shown in Figure 9. To implement the computer aided plots of the time lines and the path lines in the format of those kinematic variables shown in Figure 9, every fourth perfect fluid path line and every other perfect fluid time line is plotted, (See Perfect Fluid Deformation Field Computation and Plot, Appendix D.) The plotting is done on a CALCOMP 780/763 digital plotter from plot descriptions generated on the IBM 360 model 67 computer at the University of Michigan. (42) The perfect fluid deformation field resulting from the computer aided plot is shown in Figure 23; this field includes strain ellipses.'l~iuuoo~ / iDOWNSTREAMo Semi- die angle: y = 22.50 Outlet to inlet Reduction in area: R = 0.276 dimension ratio: HI = 0.724 Figure 23 Perfect fluid deformation field

59 Each strain ellipse is plotted using similar information as that reprsented in Figure 22, together with the velocity vector at the center of the deformed squares. In Figure 22 the center of the deformed square is z(I,K) = z(7,3) and the velocity vector in the complex form is q(I,K) = q1(7,3) + iq2(7,3) The inclination of the path line to the z1 axis at point z(I,K), (Angle 0, see Figure 19.) is determined from the velocity vector, since this vector is tangent to the path line at point z(I,K). The elongation ratio (the transformation constant "aT", Equation (40)) is determined from the velocity q(I,K), aT = {q(I,K) /VA. (58) The tangent to the segmented time line at z(I,K) is represented by the directed time line segment in complex form, z(I+l,K) - z(I-l,K) = z(8,3) - z(6,3) (59) The angle between this directed time line segment and the normal to tihe path line at z(I,K) is shear angle f. From the shear angle ~, the transformation constant is determined, bT = tan. (42) The untransformed radius 6, is selected to represent strain ellipses intermediate in size between those strain ellipses that would result from the concentric circles in the selected grid system discussed in Chapter II. The collection of strain ellipses plotted at the centers of the array of deformed squares is shown in Figure 23. The major and minor

60 diameters together with the major diameter axes are determined from ile representative array of points associated with each deformed square in a manner analogous to the example for the array of points shown in Figure 22. This collection of strain ellipses is an incremental representation of the strain field for the flowing perfect fluid. The kinematic variables for the steadily flowing perfect fluid, i.e., (1) the path lines, (2) the time lines, (3) the strain ellipses are presented in Figure 23 in the same format as the experimentally determined kinematic variables of the real metal deformation field shown in Figure 9. Both this analytical and the experimental deformation fields are defined and interpreted from the Lagrangian point of view. COMPARISON OF THE DEFORMATION FIELD OF A PERFECT FLUID WITH THE DEFORMATION FIELD OF A REAL METAL The deformation field of Figure 23 is superimposed on the deformation field of Figure 9 in Figure 24. The composite in Figure 24 allows the perfect fluid deformation field to be compared to the real metal deformation field of 6061-0 Aluminum. The selected time equaling zero line for the perfect fluid corresponds to the line between circles numbered 199, 200,..., 209, 210 and circles numbered 217, 218,.. 227, 228 of the real metal deformation field. This selected time line does not define a zero strain state for the perfect fluid, since the motion of the perfect fluid is not entirely consistent with that of a rigid body previous to the selection time line. However, the selected time line can be said to identify both perfect fluid particles and real metal particles at equal time intervals and at the same location in the flow field. Subsequent deviations in location between the real metal time lines and the perfect fluid time lines as shown in Figure 24

61 represent the kinematical differences in the flow fields. The time increments in the perfect fluid deformation field have been corrected to account for the effect of foreshortening in the real metal deformation field, (See Table V.), At corrected = LAt. (60) DOWNSTREAm Semi-die angle: Y = 22.50 Reduction in area: R = 0.276 Outlet to inlet dimension ratio: H = 0.724 Material: 6061-0 aluminum Figure 24 Perfect fluid deformation field superimposed on the real metal deformation field. (Note that the real metal deformation field has numbered ellipses.)

62 The perfect fluid flows steadily with a velocity field that is desecribed as irrotational, whereas the real metal flows steadily in this extrusion process with a velocity field that is described as rotational. Whether a velocity field is rotational or irrotational depends on the vorticity of the velocity field. The vorticity is defined by the vector operator, u = 1/2 curl(V) (61) where V is a general velocity field. For an irrotational velocity field the vorticity is everywhere equal to zero. Since vorticity is defined at a point fixed in space, vorticity is defined from an Eulerian point of view. From a Lagrangian point of view the effects of the rotationality of the real metal velocity field manifest themselves in the deviations between the families of time lines and path lines of the real metal deformation field and those same families of lines in the irrotational perfect fluid deformation field. The measure of rotationality for the real metal flow field is selected to be the deviations of the real metal path line from the perfect fluid path line, when both path lines would be coincident in the limit upstream and downstream from the tapered die section. The effects of the rotationality are now defined as a function of time along the real metal path line and therefore, the selected measure of rotationality fulfills the requirements of a Lagrangian measure. The environment of the extruding aluminum is that of a cold extrusion, i.e., the 0.5 inch per minute ram speed results in a negligible increase in billet temperature above room temperature. The environment of the perfect fluid is conceptual and is assumed to be compatible with

63 that of the extruding aluminum, i.e., the perfect fluid flows steadily in a similar geometry. The flowing aluminum is subject to frictional effects at the die and container walls, but the flowing perfect fluid is inviscid. Aluminum has all the properties of a metal, e.g., density, hardness, yield strength, shear strength, tensile strength, elasticity, ductility, etc. With respect to the large deformations encountered in this extrusion process, the aluminum is assumed to be incompressible and though the flowing perfect fluid is inviscid, it is also incompressible with density. Therefore, within the given extrusion environment, relations between the frictional effects and those metallic properties other than density and incompressibility which can account for the deviations between the families of path lines, remain to be determined. MODELED REAL METAL PATH LINES The Lagrangian model of the real metal flow field requires that the real metal path lines are modeled. To accomplish this, these lines are described in terms of the normalized complex variable r, where r = r + ir2 (62) Real metal path lines are defined when, r2 = constant (63) and the position along a real metal path line is determined by a value of rl. Figure 25 shows the schematic relationship between the perfect fluid path lines, the real metal path lines, and the perfect fluid potential lines.

64 Z PLANE UNIQUE COORDINATE TRANSFORMATION REGION _ = k ~ DIE\ DIE WALL DIE WALL Z1 Z Z(P) Ip2 i VA7 P PLANE UNIQUE COORDINATE TRANSFORMATION REGION DIE WALL DIE WALL P1 P = P(r) 2 iVA7r r PLANE UNIQUE COORDINATE TRANSFORMATION REGION DIE WALL DIE WALL r, PERFECT FLUID PATH LINES REAL METAL PATH LINES PERFECT FLUID POTENTIAL LINES EXTEND FROM DIE WALL TO CENTER LINE Figure 25 Schematic representation of perfect fluid potential lines, real metal path lines, and perfect fluid path lines in the z, p, and r planes

65 The explicit relation, z = z(p) is defined by Equations (12), (16), and (21), while the explicit relation between the r plane and the complex potential plane p, is p = p(r) (64) where pl r= and P2 = P2 (rlr2) or P2 = r2 + AP (65) The flow of a real metal is rotational during the extrusion process and no conformal representation can exist for the path lines (stream lines) of a steady rotational flow. Thus, Equations (64) and (65) are not conformal representations of the transformation from the r plane to the p plane. The real metal path line and the perfect fluid path line are coincident at the center line, at the die wall, and in the limit upstream and downstream from the tapered die section (See Figure 24.) That is, P2(rl'r2) has the following constraints: lim P2 (rr2) = r rl + 2 1 2 2' (66) lim P2(rl'r2) = VA' r2 +*VATr and lim p (r,r2) = 0. r2+ 2 1

66 If p2(rl,r2) is expressed by Equation (65), AP must be equal to zero at the center line, at the die wall, and in the limit up and downstream from the tapered die section. That is, AP has the following constraints: lim AP = 0, r L1_ + i0 (67) lim AP =, r2 +VA 7 and lim AP = 0 r2 0 The functional notation p = p(r) implies a corrdinate transformation which, if it is to be unique, i,e., one-to-one, the following constraint must be met. As r2 increases monotonically from 0 to VAT p2(rl,r2) must increase monotonically from 0 to VAT for any value rl. That is, from Equation (65), aP2(rlr2)/3r2 > O (68) or AP/ ar2 > -1 Equations (66), (67), and (68) express constraints on the functional relation, p = p(r). Let AP = [P2(rl,r2) - P2(1)]r = constant (69) where P2(1) = lim 2 (r1,r2) = r2 This definition of AP satisfies constraint Equation (67). The perfect

67 fluid path line P2(1), is coincident with the real metal path line r2, in the limit up and down stream from the tapered die section. AP is measured along the perfect fluid potential line, since P = rl The perfect fluid potential line pi, is orthogonal to the perfect fluid path line p2(1). AP represents the difference in position in the complex potential plane between the real metal path line r2, and the perfect fluid path line p2(1). Therefore, APis the selected Lagrangian measure of rotationality for the real metal velocity field of this extrusion process. Ap is defined in the complex physical plane and the coordinate system shown in the right-hand side of Figure 12 facilitates the determination of positions in the physical plane in terms of complex potential plane coordinates. Since every fourth perfect fluid path line in Figure 12 is coincident with the real metal path lines of Figure 9 in the limit up and down stream, when the incremental array of real metal path lines in Figure 9 is superimposed on the right-hand coordinate system of Figure 12, the values of AP can be experimentally determined. Deviations of the real metal path lines from the perfect fluid path lines as shown in Figure 24 have a regular pattern on traveling downstream. The real metal path line first deviates toward the die wall and then deviates toward the center line. In Figure 26 is shown a schematic plot of these regular deviations in terms of the experimentally determined AP's. The model selected for the experimental AP's has the form, AP = A - B, (70) m

68 where A = A/(a (r1e ) + e - rl-a)) (71) and B = B/(b(rl-~) + ec2(rl -)) (72) Modeling AP is equivalent to modeling the real metal path lines. AP 0.2 0 0 0 0 EXPERIMENTAL DATA MODELED A p 0.1 r2= CONSTANT PM -~4.0 -2.0 0.0 2.0 r PMIN -0.1 Figure 26 Schematic representation of the difference between the perfect fluid path lines and the real metal path lines AP. The synthesis of function A is diagrammed in Figure 27. The four coefficients A, a, a, and c1 are closely related to different geomet~lc properties of the nonlinear function A. The coefficient A, is closely related to the maximum value of A and the coefficient a, is closely related to the decay for increasing values rl from the maximum in the function A. The coefficient a, is closely related to the location rl, of the maximum in the function A, while the coefficient cl is related to the skewed symmetry of the function A. The function B is similar to the function A except for the opposite sign in skewed symmetry term. When the coefficient 8, is less than the coefficient a, the synthesis of the model for AP is shown in Figure 28.

69 1arl-a;2 -cl(rl-a1 a br, r, a.. (a' rl-a)2+ e-C1 (rl-a)) a r, a r, C. d. Figure 27 Schematic synthesis of function A 3 fa a r, Figure 28 Schematic synthesis of function A P m Since AP A Pm the modeled real metal path lines have the form, P = rl, (64) and = 2 + mP (73) and ~~~~~~P2 2 m

'u Alternate models exist for the experimental AP's, but the relative accuracy of these models is not within the scope of this investigation. The accuracy of the selected model is assayed by comparing the real metal deformation field with the resulting modeled real metal deformation field. The coefficients for AP are determined from the location of the m four expermientally determined points AP AP 1/3AP, and max min max 1/3AP. as shown in Figure 26, together with the constraints that min dA P m 0 o (74) and mdA = 0. (75).jr7 d r APmin 1 min When Equation (73) is evaluated at the four experimental points, and when Equation (74) and (75) are evaluated at APmax and APmin respectively, the resulting six nonlinear equations can be solved for six of the eight coefficients in AP, if two of the coefficients are known. The skew m symmetric coefficients cl, and c2 are selected to meet two criteria. The first criterion is that the nonlinear equation solving algorithm used in the computer program converges, (See Experimental Modeling Coefficients Computations, Appendix D.) whereas the second is that the resulting AP, is representative of the flow field. The decision with respect to the satisfaction of the second criterion is made by comparing the resulting modeled real metal deformation field with the real metal deformation field; a schematic representation of the modeled AP's, AP, is shown in Figure 26. The selected skewed symmetry coefficients cl and c2 are geometrically most closely related to the fit of the model APm, between A Pmin and A Pmax

71 The nonlinear equation solving computer program is typical in that an initial estimate of the resulting model coefficients must be made. If the initial estimate is not sufficiently close to the final result, the computational algorithm may not converge. The geometric relationship between the experimental values ofA Pand the model coefficients is useful in making the initial estimates of the model coefficients. After some experience is gained with the successful solution of the set of nonlinear equations, plots of the resulting coefficients versus their most closely related geometrical properties are additionally useful in making initial estimates. The experimental points for determining the model coefficients for the five interior path lines of the real metal deformation field in Figure 9 are tabulated in Table VI. The resulting coefficients A, a, c, B, b, and 1 from the nonlinear equation solving computer programs are tabulated in Table VII, when the selected skewed symmetry terms have values c! = (4/T)r2+1 and c = (4/T)r +3. 1 2 2 2 The modeled coefficients are a function of the real metal path line, r2. In Figure 29 the model coefficients are plotted as a function of the path line, r2. These five interior values for the model coefficients are calculated from experimentally determined information. The values of the model coefficients at the die wall, r2 = 0 and at the center line, r2 = 7 are not determined directly from the experimentally evaluated AP's, but since AP is equal to zero at the die wall and center line, to insure that the modeling deviations A P, are also equal to zero, the values of the m model coefficients A and B are extrapolated to zero at the die wall and center line, (See Figure 29 and Equations (70), (71), and (72).) With

72 these extrapolations on model coefficients A and B, the model deviation function AP, meets constraint Equation (67). The die wall value of o and B are determined by a linear extrapolation of the first two experimentally determined coefficients adjacent to the die wall. The other extrapolated values of the model coefficients at the die wall or the center line are set equal to the value of the coefficient adjacent to the die wall or center line, respectively. The resulting modeled deviations A P now meets the constraint equations and the extrapolated coeffim cients form a completed set. TABLE VI EXPERIMENTAL POINTS FOR DETERMINING MODEL COEFFICIENTS R = 0.276 Die Geometry: H = 0.724 y = 22.50 Material: 6061 aluminum Normalized Upstream Velocity: VA = 1.0 Lubrication: (See Lubrication, Chapter II) r2 APmax r1(APmax) APmin r1(APmin) rl (l/3APmax) r1(1/3 APmin) 0.468 0.100 -0.320 -0.0240 -2.733 0.980 -4. 033 1.003 0.160 -0.670 -0.0263 -2.983 0.698 -4.283 1.538 0.173 -0.980 -0.0243 -3.109 0.455 -4.409 2.072 0.140 -1. 220 -0.0221 -3.203 0.283 -4.503 2.607 0.077 -1.450 -0.0166 -3.210 0.120 -4.510

73 TABLE VII MODEL COEFFICIENTS R = 0.276 Die Geometry: H = 0.724 Y = 22.50 Material: 6061-0 aluminum Normalized Upstream Velocity: VA = 1.0 Lubrication: (See Lubrication, Chapter II) Experimental Points: (See Table VI) Skewed Symmetry Coefficients: c1 = (4/T)r2+1, c2 = (4/7r)r2+3 r2 A a c B b 3 0.468 0.155 1.513 -0.939 0.0407 1.421 -1. 945 1.003 0.222 1.421 -1.315 0.0387 1.430 -2.264 1.538 0.223 1.371 -1.614 0.0324 1.442 -2.467 2.072 0.172 1.338 -1.830 0.0271 1.465 -2.656 2.607 0.090 1.313 -2.033 0.0196 1.480 -2.712 An interpolative definition of the modeled real metal path line for any path line desired is defined using the complete set of model coefficients. Interpolation by piece-wise cubic splines is the method of approximation used to define the model coefficients between experimentally calculated values for any value of real metal path line, (See Cubic Spline Fit, Appendix D.) The resulting cubic splines are plotted in Figure 29. The interpolated model coefficients equal the experimentally

0.30 2.00 /b r4 0.r( 51~~~~~~~~~~~~~~~~~~~~~~~1 100 -( |: |I OQ EXPERIMENTALLY DETERMINED COEFFICIENTS a).i? / E X O EXTRAPOLATED COEFFICIENTS PQ \ I I ---------— PIECEWISE CUBIC SPLINE INTERPOLATION 0 o.a0 r2 000 0 7r/2 tr O 7r/2 DIE WALL CENTER LINE DIE WALL CENTER LINE r;_, -path lilne r2, path line C. P2 = r2 + A/(a(rl-a)2+ 1(r-a)) - B/(b(rl-P)2+ ec2(rl- )) -500 DIE GEOMETRY: Y = 22.50 H = 0,724 rR = 0.276 4 -3. 00 _ MATERIAL: 6061-0 ALUMINUM / NORMALIZE UPSTREAM VELOCITY: VA = 1.0 -2.00 LUBRICATION: (SEE LUBRICATION CHAPTER II) EXPERIMENTAL POINTS: (SEE TABLE VI) -1.00 SKEWED SYMMETRY COEFFICIENTS: C1= (4/I)r2+l, 0.00oo I I -r2 C2= (4/r) r2+3 0 Ir/2 DIE WALL CENTER LINE Figure 29 Model coefficients versus real metal path line

75 determined and extrapolated values of the model coefficients when the selected real metal path line coincides with one of the experimental path lines in Figure 9. The interpolated model coefficients are continuous and possess continuous first and second derivatives with respect to r2; the functions representing a perfect fluid are continuous and their derivatives of all order are continuous. The modeled deviation function A P, is continuous and its derivatives with respect to rl of all order m 1 are continuous, therefore, the modeled real metal path lines are continuous and the partial derivatives of this model are continuous at least through the second order. The modeled path lines meet the requirements for a continuum. The model coefficients and the first derivative of the model coefficients with respect to any value r2 are defined by a subroutine subprogram DCONS (See DCONS, Subprograms, Appendix D.) This subprogram returns values of the model coefficients and their derivatives when given a value of the real metal path line r2. With the model coefficients for a real metal path line, the functional subprogram RM, (See RM, subprograms, Appendix D.) returns the p2(r1 r2) value for the modeled real metal path line when r1 is given. The functional subprogram RM is the computer implementation of Equation (73). The subroutine subprogram DCONS and the functional subprogram RM are the digital computer methods employed to model the real metal path lines. MODELED REAL METAL VELOCITY FIELDS The Lagrangian model of the real metal flow field requires the velocity of the metal particles along their path lines but the velocity of a real metal particle is not directly observable with the split billet techneque. For steady flows, the path lines and stream lines are

76 coincident and when the path function equals a constant, a path line is defined. For a steady flow, the path function and stream function are identically equal and for incompressible, two dimensional and possibly rotational flows, the stream function defines the velocity field. In terms of the path function for the steadily flowing aluminum, the real metal velocity field has the form in the physical plane, Dr2 Dr2 k vk z k = 1 k zk 2 (76) The coordinate transformations defined for the z, w, s, p, q, and r planes are unique in each plane for the representative image of the die cavity. Therefore, Equation (65) implicitly defines the unique function, r2 r 2(p1 P2), and Equations (12), (16), and (21) implicitly define the unique functional relations, P1 = P1(Z1'Z2) and p2 P2(z1,z2). Now the partials in Equation (76) can have the forms Dr2 ar2 ap1 Dr2 P2 (77) = ~+ Dz2 aPl Dz2 DP2 aZ2 and 1r2 3r2 DPl Dr2 p2 (78) 1'l 1 2 z1Z

77 Let e =/ (79) and e = k3Xq, (vector cross product) (80) nq 3 q where k3 is the unit normal to the z plane. Then using the Cauchy Reimann conditions on the conformal relationship between z and p planes, together with Equations (77) through (80), Equation (76) becomes, Dr2 - ~ v-= I ( eq p enq) (81) qP2l nq Equation (81) is expressed in terms most closely related to the selected reference frame. When r2 is the modeled real metal path function, Equation (81) represents the modeled real metal velocity field. If the modeled real metal velocity field is to be evaluated as a function of position along the path line, the expression for q, Dr 2//p2, and ar2/3P1 must be expressed as functions of position along the modeled real metal path line. Upon selection of a modeled real metal path line r2, and position along this path line r1, DCONS together with RM returns the image of this real metal particle in the p plane. From this image in the p plane, the subroutine QF returns the velocity q, of a perfect fluid particle at this position. Since Equations (64) and (73), p1 = r1 (64) P2 = r2 + APm(rlr2) (73) implicitly define, r2 = r2(P1,P2)' the partials 3r2/3p1 and ar2/aP2, must be evaluated by the techniques

78 for partial differentiation of implicitly defined functions. The subroutine subprogram PDPRM (See PDPRM, Subprograms, Appendix D.) returns the values for these two partials, when r1, r2, the model coefficients, and the derivatives of the model coefficients are given. The subroutine subprogram DCONS that has been called to evaluate q, established the values of the model coefficients and their derivatives. From the results of DCONS, RM, QF, and PDPRM the functional subprogram VF (See VF, Subprograms, Appendix D.) returns the value for Equation (81). Equation (81) expresses the velocity of a real metal particle in terms of the deviation in the velocity of the real metal particle from the velocity of a perfect fluid particle at the same point. These deviations are the direct consequence of the deviations between the modeled real metal and the perfect fluid path lines and the possibly rotational character of the real metal velocity field at the specific point. Since the real metal path line has the implicit form, P2 = r2 + AP(r1 r2) (65) the deviations in velocity between a real metal particle and a perfect fluid particle at the same point are directly related to the Lagrangian measure of rotationality AP, selected for these steady flows. The images of an array of modeled real metal path lines in the modeled real metal velocity plane are shown in Figure 30. The velocity at a particular position along a path line is determined by the coordinate values of that modeled path line point in the modeled real metal velocity plane. An example for a particular path line point is shown in Figure 30, while the images of an array of modeled real metal path lines in the perfect fluid velocity plane are shown in Figure 31.

79 iv2 /r2 = 2.607,~~/ z-rr2 = 2.072 V2. r2 0. 468 A V1 D V1 - 1.000 - 1,381 = Semi-die angle: y = 22.5 Reduction in area: R = 0.276 Outlet to inlet Material: 6061-0 aluminum dimension ratio: H = 0.724 c = (4/rr)r+1 c2 (4/r2+3 Figure 30 Images of an array of modeled real metal path lines in the modeled velocity plane q2 A q D q' 1.000 1.381 - Semi-die angle: y = 22.50 Reduction in area: R = 0.276 Outlet to inlet Material: 6061-0 aluminum dimension ratio: H = 0.724 c = (4/7T)r2+1, c2 (4/')r2+3 Figure 31 Images of an array of modeled real metal path lines in the perfect fluid velocity plane

80 The differences in Figures 30 and 31 are the results of the modifying partials in Equation (81); the curves shown in Figure 30 are computer generated, (See Modeled Real Metal and Perfect Fluid Velocity Hodograph Plots, Appendix D.) The functions defining the modeled real metal velocity field are continuous; however, differentiation of functions such as A P which have experimental input tenc to magnify experimental errors. m Finite deformations for the modeled real metal flow field can be defined in terms of integrals of strain rates. Strain rate equations require differentiations of the velocity field. This information has included an additional magnification of experimental error with respect to AP. The time line method for determining finite deformations uses m a single integration of the velocity information and integration is a smoothing process. Finite deformations defined by the strain rate method are the result of smoothing operations on information that has a double magnification of experimental error with respect to APl when compared to finite deformations defined by the time line method. Constraint Equation (68) insures the uniqueness of the coordinate transformation between the p and r planes. Since 1= r1 (64) then r2 DP2/Dr2 = 1/ p now constrain Equation (68) can be reformulated, 0r," > 0. (8

81 Upon inspection of Equation (81), Equation (82) can be interpreted as constraining the velocity of a real metal particle to having a component of velocity in the same direction as a perfect fluid particle at the same point. The subroutine subprogram PDPRM that returns the partial derivatives of the modeled real metal path line is used to check to see that constraint Equation (82) is satisfied for the array of points used to define the modeled real metal deformation field in the next section. COMPUTER IMPLEMENTED MODEL OF REAL METAL DEFORMATION FIELDS Now the deformation field for the real metal can be modeled. The computer implementation for the deformation field of the modeled real metal is analogous to the computer generated deformation field of the perfect fluid. An incremental array of modeled real metal path lines is used to defined the time function. The time function is defined by integrating scalar Equation (31) along each modeled real metal path line of the array. The modeled real metal path lines are defined, r2(1) = 0 r2(2) = K1 (83) r2(I) = N(K2)+K where N = 1-2, and I = 3,..., 25 resulting when I = 25, r2(25) = T since VA = 1.0 for the real metal flow. The constants K1 and K2 are the same as in Equation (51). Therefore, in the limit up and down stream the modeled real metal path line array is coincident with the perfect fluid path line array defined by Equation (51). The index I identifies a particular modeled real metal path line.

82 The positions along the modeled real metal path lines are defined, r1(J) = r1 (I) + (K3) (84) where M = J-1 and J = 1, 2,..., 41 and the constant K3, is the same as in Equation (52). Index J identifies a position along a path line. When J = 1, r(l) = r{ (I) The index I in r{ (I) identifies the starting point along each path line as being uniquely determined for the modeled real metal path line. The r{ (I)'s are determined by the intersection of the time equaling zero line in the physical plane with the real metal path lines. The time equaling zero line for the model real metal is chosen to correspond to the experimentally selected line between circles numbered 199, 200,.. 209, 210 and circles numbered 217, 218,..., 227, 228 in Figure 9. This selected time line facilitates the comparison of the modeled real metal deformation field with both the real metal and the perfect fluid deformation fields in Figures 9 and 23 respectively. The perfect fluid points plotted in Figure 21 have an analog with respect to modeled real metal points when the points in the physical plane are defined, z(I,J) = z(r(I,J)). (85) Equation (85) indicates that the plotted points are defined for a modeled real metal. The path lines are represented by straight line segments between images of modeled real metal path line points in the physical plane. Equation (83) and (84) together with computer subprograms DCONS, RM, AND ZF define the modeled real metal path line points, (See Real Metal Deformation Field Modeling Computation and Plot, Appendix D )

83 The velocity of the modeled real metal particle at each point defined by Equation (85) is defined v(I,J) = v(r(I,J)). (86) As in Equations (83), (84), and (85), the index I identifies a path line and the index J indicates a position on a path line. v(I,J) is defined using the information needed to evaluate Equation (85) together with subprograms PDPRM and VF. An increment of time is defined, At = 1/2[(/I v(I,J) ) + (1/jv(I,J+l) )]X (87) [ z(I,J+l) - z(l,J)|] Starting with time equaling zero and summing the time increments along each incrementally modeled real metal path line, every point in the array z(I,J) is assigned a time t(I,J). The times t(I,J) are converted to integers, and the time lines z(I,K) are determined in a completely analogous manner to those for the perfect fluid. To implement the computer aided plots of the time lines and path lines of the modeled real metal in the format of those kinematic variables shown in Figure 9, every fourth modeled real metal path line and every other modeled real metal time..ine is plotted, (See Real Metal Deformation Field Modeling Computation and Plot, Appendix D.) The modeled real metal deformation field resulting form the computer aided plot is shown in Figure 32. This deformation field includes strain ellipses. From the time lines, velocity fields, and the path lines are determined the transformation coefficients for a given strain ellipse. That is to say that the strain ellipses are determined and plotted in a

84 completely analogous manner to those for the perfect fluid. This coliection of strain ellipses shown in Figure 32 is an incremental representation of the strain field for the modeled real metal. The kinematic variables for the steadily flowing modeled real metal, i.e., (1) the path lines, (2) the time lines, (3) the strain ellipses, are presented in Figure 32 in the same format as the experimentally determined kinematic variables of the flowing real metal shown in Figure 9. Both the mathematically modeled and the experimentally determined deformation fields are interpreted from a Lagrangian point of view. The entire kinematic effect of a real metal steadily flowing with friction is represented by analytical manifestations of the modeled deviations, A P m DOWNSTREAM Sermi-die angle: y = 22.50 Reduction in area: R = 0.276 Outlet to inlet Material: 6061-0 aluminum dimension ratio: H 0.724 c1 = (4/7)r2 +1 c2 = (4/T)r2+3 Figure 32 Modeled real metal deformation f ield

85 SLIP LINE GENERATED DEFORMATION FIELD Slip line field theory applies to non-homogeneous plane strain deformations of a rigid-perfectly plastic isotropic solid. A perfectly plastic solid has a constant yield strength so clearly this theory does not apply to the extrusion of the work hardening 6061-0 aluminum described in this dissertation. However, the deformation field of a perfectly plastic solid extruded through the die geometry defined in Figure 12 can be compared to the real metal deformation field shown in Figure 9. This comparison is analogous to the way the real metal is compared to a perfect fluid, and the deviations of the real metal path lines with respect to the solid path lines could be modeled. The kinematical analysis of the extrusion of a perfectly plastic solid is accomplished using slip line theory. Slip line analysis for the extrusion of the perfectly plas(43) tic solid is completed using the graphical cord method. The slip line field and velocity hodograph are shown in Figure 33. From the slip line field and the velocity hodograph, the path lines and time lines are constructed. The starting time line is selected to correspond to the starting time line of the perfect fluid and modeled real metal deformation fileds shown in Figures 23 and 32. Path lines and time lines for the perfectly plastic solid are shown in Figure 34 where the selected path line and time line increments correspond to the experimentally selected path line and time line increments in Figure 9. The path lines for the perfectly plastic solid shown in Figure 34 are continuous but do not have continuous first derivatives due to discontinuous changes in the velocity field. It can be proposed that the real metal path lines would be modeled by starting with the slip line field path lines and modeling the real

86 V/H VV/ DIE WALL \ \ Semi-die angle: y = 22.50 Outlet to inlet Reduction in area: R = 0.276 dimension ratio: H = 0.724 Figure 33 Slipline field and velocity hodograph DOWNSTREAM Semi-die angle: y = 22.50 Outlet to inlet Reduction in area: R = 0.276 dimension ratio: H = 0.724 Figure 34 Deformation field of a perfectly plastic solid

87 metal path line deviations relative to these lines. If this technique were used to model smoothly changing real metal deformation fields as exemplified by those shown in Figures 9, 10, and 11, at each location of discontinuous change in velocity there would have to be an offsetting change in the modeled deviations so that when slip line field path lines were taken together with the modeled deviations, the results would be smoothly changing. Modeling real metal path lines relative to perfectly plastic solid path lines can be seen to introduce additional constraints when compared to the modeling technique which employs a perfect fluid. The generalized slip line problem statement developed from kinetical considerations by Richmond(2) can include a realistic material model. The solution to this problem would result in smoothly changing path lines, however, the detailed deformation history of the material still would be required for a work hardening material model.

Chapter IV COMPARISON AND DISCUSSION OF MODELED DEFORMATION FIELDS AND EXPERIMENTAL RESULTS In Chapter III the real metal deformation field shown in Figure 9 is used as the basis for a mathematical model. In this chapter, this modeling method is applied to the real metal deformation fields shown in Figures 10 and 11. This method requires the comparison of the real metal path lines with the perfect fluid path lines and then the deviations between these path lines are modeled. These deviations are interpreted to be a Lagrangian measure of rotationality and from the modeled deviations, the modeled real metal path lines are defined. For this steady state extrusion process, the modeled real metal velocity field is defined in terms of the modeled real metal path lines. With the modeled velocities as a function of the modeled real metal path lines, the modeled real metal deformation field can now be determined from a Lagrangian point of view. The results are presented by making the following superimpositions for comparative purposes. The perfect fluid deformation field, the modeled real metal deformation field, and the slip line deformation field are superimposed on the real metal deformation field. COMPARISON OF THE PERFECT FLUID DEFORMATION FIELDS WITH THE REAL METAL DEFORMATION FIELDS The real metal deformation fields shown in Figures 10 and 11 are compared to corresponding perfect fluid deformation fields in Figures 35 and 36 respectively. As in Figure 24, the deviations between the real metal path lines and the perfect fluid path lines are regular. On traveling downstream the real metal path line first deviates toward the 88

89 die wall and then deviates toward the center line before coming coincident with the perfect fluid path line once again. \OQ Semi-die angle: y = 22.50 Reduction in area: R = 0.451 Outlet to inlet Material: 6061-0 aluminum dimension ratio: H = 0.549 Figure 35 Perfect fluid deformation field superimposed on the real metal deformation field. (Note that the real metal has numbered ellipses.) The real metal deformation fields are photographed using a copy camera with Kodak metallographic plates. The copy camera assures the orthogonality between the object photographed and the glass photographic plate; the latter assures a minimum of distortion during development. This photographic procedure is selected to minimize the distortions of

90 geometric relations on the photographic negative and subsequent projections. Semi —die angle: ~ = 450 Reduction in area: R = 0.449 Outlet to inlet Material: 6061-0 aluminum dimension ratio: H = 0.551 Figure 36 Perfect fluid deformation field superimposed on the real metal deformation field. (Note that the real metal has numbered ellipses.) To determine experimentally the value of the deviations indicated in Figures 35 and 36, the image of the real metal deformation field is projected on to an orthogonal curvilinear coordinate system, (See Orthogonal Curvilinear Coordinate System Computation and Plot, Appendix D.)

91 The orthogonal curvilinear coordinate system shown in Figure 37 is used for analyzing the real metal deformation field shown in Figure 11. The deviations between the real metal path lines and the perfect fluid path lines are determined experimentally. By using this curvilinear coordinate system, the Lagrangian measure of rotationality AP, is modeled in terms of the coordinates defined in the complex potential plane p Semi-die angle: = 450 Outlet to inlet Reduction in area: R = 0.449 dimension ratio: H = 0.551 F igure 37 Orthogonal curvilinear coordinate system The experimental points for determining the model coefficients for

92 field in Figure 10 are tabulated in Table VIII. The resulting coefficients A, a, a, B, b, and B from the nonlinear equation solving computer program are tabulated in Table IX, when the selected skewed symmetry terms have values, c1 = (2/T)r +1 and c2 (2/T)r +2. The modeled coefficients are only a function of the real metal path line r2. In Figure 38, cubic splines define the model coefficients for any path line. TABLE VIII EXPERIMENTAL POINTS FOR DETERMINING MODEL COEFFICIENTS R = 0.451 Die Geometry: H = 0.549 Y = 22.50 Material: 6061-0 aluminum Normalized Upstream Velocity: VA = 1.0 Lubrication: (See Lubrication, Chapter II) r2 APmax rl(APmax) APmin rl(APmin) r1(ll/3APmax) r1(1/3 APmin 0.487 0.115 -0.946 -0.0783 -4.869 0.363 -6.032 1.018 0.175 -1.340 -0.1035,-4.731 0. 056 -6.112 1.549 0.200 -1.594 -0.1035 -4.582 -0.154 -6.159 2.080 0.180 -1.785 -0.0810 -4.413 -0.388 -6.201 2.611 0.109 -1.931 -0.0450 -4.263 -0.622 -6.225

93 TABLE IX MODEL COEFFICIENTS R = 0.451 Die Geometry: H = 0.549 Y = 22.5~ Material: 6061-0 aluminum Normalized Upstream Velocity: VA = 1.0 Lubrication: (See Lubrication, Chapter II) Experimental Points: (See Table VIII) Skewed Symmetry Coefficients: cl = (2/7)r2+1, c2 = (2/T)r2+2 r2 A a c B b 0.487 0.187 1.538 -1.543 0,114 1.629 -4,296 1.018 0.264 1.422 -2.008 0.141 1.400 -4.070 1.549 0.283 1.376 -2.281 0.134 1.294 -3.868 2.080 0.248 1.398 -2.442 0.100 1.224 -3. 652 2.611 0.147 1.462 -2.541 0.054 1.184 -3.471

0 30 2.00' 0,20 c 1,00 o/. O0 EXPERIMENTALLY DETERMINED COEFFICIENTS 0,10 0.10 -/ ( \ | I ) EXTRAPOLATED COEFFICIENTS PIECEWISE CUBIC SPLINE INTERPOLATION 0,00 r2 0,00. I I I I, r 0 7r/2 r' 0 ir/2 DIE WALL., CENTER LINE DIE WALL -, -iath line CENTER LINE P2 r2 t A/(a(ri-a)2+6Cl(rl-a)) B/(b(rlp)2+eC2(r -500'r-t400 DIE GEOMETRY: Y= 22.5~'0 -3,00 - H = 0.549 R= 0.451 -2,00 MATERIAL: 6061-0 ALUMINUM NORMALIZE UPSTREAM VELOCITY: VA= 1,0 LUBRICATION: (SEE LUBRICATION, CHAPTER II) -1.00 EXPERIMENTAL POINTS: (SEE TABLE VIII) SKEWED SYMMETRY COEFFICIENTS: C1= (2/r)r 2+1, C 1 20.00. I!. = r2 C2 = (2/r)r2+2 0 7r/2 r DIE WALL r, path line CENTER LINE Figure 38 Model coefficients versus real metal path line, r2

95 MODELED REAL METAL VELOCITY FIELD Using Equation (76), the images of an array of modeled real metal path lines in the modeled real metal velocity plane are calculated and shown in Figure 39. The velocity at a particular position along a path iv2 r2 = 2.607 r2 = 2,072 = 1,538 f 1,003 V2 2252= 0,468 2 0-3.990 A V1 D V1 -1^~ ~ 1.000'= i,~~~~~~~~~~1.821 - Semi-die angle: Y = 22.5 Reduction in area: R = 0.451 Outlet to inlet Material: 6061-0 aluminum dimension ratio: H = 0.549 c = (4/Th)r2+1, c2 = (4/r)r2+3 Figure 39 Images of an array of modeled real metal path lines in the modeled real metal velocity plane v line is determined by the coordinate values of that point in this velocity plane as shown in Figure 39. The images of an array of modeled real metal path lines in the perfect fluid velocity plane are shown in Figure 40. The differences between Figures 39 and 40, and the differences between the path lines in Figure 35 are the result of the rotational character of the real metal flow field. The frictional effects and those metallic properties that contribute to this rotational character are kinemtaically modeled through the Lagrangian measure of rotationality, APR. m

96 iq2 22 _ 2,607 1.0.0 20.072 11000 r2- 1.00 Semi-die angle: Y = 22.55 Reduction in area: R = 0.451 Outlet to inlet Material: 6061-0 aluminum dimension ratio: H = 0.549 c (4/Tr)r2+1, c2 (4/)r2+3 Figure 40 Image of an array of modeled real metal path lines in the perfect fluid velocity plane q. COMPARISON OF PERFECT FLUID DEFORMATION FIELDS WITH MODELED REAL METAL DEFORMATION FIELDS From the modeled path lines and velocities, the modeled real metal deformation field is determined for the steadily flowing 6061-0 aluminum shown in Figures 9 and 10. These modeled real metal deformation fields are compared to corresponding perfect fluid deformation fields in Figures 41 and 42. As in Figures 24 and 35, when the perfect fluid and real metal deformation fields are compared, the deviations between the modeled real metal path lines and the perfect fluid path lines shown in Figures 41 and 42 are regular. On traveling downstream the modeled real metal path line first deviates toward the die wall and then deviates toward the center line before becoming coincident with the perfect fluid path line once again. This modeled path line behavior is similar to the real metal path line behavior. Additionally, deviations between the modeled

97 time lines and the perfect fluid time lines in Figures 41 and 42 are similar to the deviations between the real metal time lines and the perfect fluid time lines in Figures 24 and 35. DOWNSTREAM Modeled real metal Perfect fluid Semi-die angle: y =22.50 Reduction in area: R = 0.276 Outlet to inlet Material: 6061-0 aluminum dimension ratio: H = 0.724 Figure 41 Perfect fluid deformation field superimposed on the modeled real metal deformation field

98 DOWNSTREAM Modeled real metal Perfect fluid Semi-die angle: Y = 22.50 Reduction in area: R = 0.451 Outlet to inlet Material: 6061-0 aluminum dimension ratio: H = 0.549 Figure 42 Perfect fluid deformation field superimposed on the modeled real metal deformation field COMPARISON OF MODELED REAL METAL DEFORMATION FIELDS WITH REAL METAL DEFORMATION FIELDS The modeled real metal deformation fields are determined for the steadily flowing 6061-0 aluminum shown in Figure 9 and 10. These modeled deformation fields are compared to the corresponding real metal deformation fields in Figures 43 and 44. The experimental extrusion is conceived to be symmetric and two dimensional. Actually, the billets become somewhat thicker as indicated

99 by the foreshortening ratios in Table V. In Figure 43 the area between two sets of concentric ellipses is shaded. The smallest ellipse in each set is the experimentally deformed strain ellipses while the next larger is the modeled strain ellipse. The two modeled strain ellipses are constructed symmetrically about the center line. From the differences between the two shaded areas, the experimental extrusion is seen to be asymmetrical. Upon inspection, the extrusion dies were found to be slightly asymmetrical. The modeled real metal deformation fields shown in Figures 43 and 44 are in error. Besides foreshortening, the errors are of two types: 1. errors in displacemnets from the original time line 2. errors in the relative displacements or strains. Errors in displacements are differences in location between the modeled strain ellipses and the corresponding experimental strain ellipses. Errors in relative displacements are differences in shape and orientation between the modeled strain ellipse and the corresponding experimental strain ellipse irrespective of errors in location. The errors in displacements between symmetric modeled strain ellipses and the txperimental strain in Figure 43 graphically point out the asymi:-leur Jcal character of the real metal deformation field. To the author, this asymmetrical character is not obvious on first inspection of the flow field in Figure 9. Somewhat finer distinctions with respect to real metal flow fields appear to be possible when the real metal flow field is compared to a modeled flow.

100 DOWNSTREAM Modeled real metal Semi-die angle: y = 22.50 Reduction in area: R = 0.276 Outlet to inlet Material: 6061-0 aluminum dimension ratio: H = 0.724 Figure 43 Modeled real metal deformation field superimposed on the real metal deformation field. (Note that real metal has numbered ellipses.) The selected differential radius 6 of the neighborhood of points for which the strain is to be defined must be small with respect to the characteristic radii of curvature of both the time lines and path lines, if an ellipse is to accurately indicate the strain state. Additionally,

101 Modeled real metal Semi-die angle: Y = 22.50 Reduction in area: R = 0.451 Outlet to inlet Material: 6061-0 aluminum dimension ratio: H = 0.549 Figure 44 Modeled real metal deformation field superimposed on the real metal deformation field, (Note that real metal has numbered ellipses.) the differential radius 6, must be small compared to these characteristic radii of curvature for any displacement of the neighborhood of points if an ellipse is to accurately represent the changing strain for these displacements. The corners in the extrusion die walls which correspond to points B and C in Figure 13 are singularities. The radius of curvature for the path line at these corner points is zero. For path lines that pass close to these corner points the strain ellipses that result from neighborhoods defined by increasingly smaller differential

102 radii more accurately represent the strains in these areas. This effect is seen in the areas adjacent to the die wall corners in Figures 9, 10, and 11. COMPARISON OF SLIP LINE DEFORMATION FIELDS WITH REAL METAL DEFORMATION FIELDS The slip line modeled deformation fields, as determined for a perfectly olastic solid, are compared to the corresponding real metal deformation fields in Figures 45, 46, and 47. Semi-die angle: y = 22.5~ Reduction in area: R = 0.276 Outlet to inlet Miaterial: 6061-0 aluminum dimension ratio: H = 0.724 Figure 45 Slip line deformation field superimposed on the real metal deformation field

103 The flowing perfect fluid is described by Laplace's (an elliptic) partial differential equation, whereas the flowing perfectly plastic solid is described by a hyperbolic partial differential equation. The comparison of the real metal deformation field with that of both the perfect fluid and perfectly plastic solid indicates that the real metal flow field is described by neither an elliptic nor a hyperbolic partial differential equation. The hyperbolic equation of a flowing perfectly r% I g ^~ 1 O:'~ ~ ]: DOWNSTREAM Semi-die angle: y = 22.50 Reduction in area: R = 0.451 Outlet to inlet Material: 6061-0 aluminum dimension ratio: H = 0.549 Figure 46 Slip line deformation field superimposed on the real metal deformation field

104 plastic solid confines the deformations to within the slip line field, whereas the flowing perfect fluid as described by an elliptic equation is disturbed in the limit both up and down stream from the tapered die section. The real metal deformation zone can be seen to be greater than that described for a perfect plastic solid but less than that of a perfect fluid. DOWNSTREAM Semi-die angle: Y = 45 Reduction in area: R = 0.449 Outlet to inlet Material: 6060-0 aluminum dimension ratio: H = 0.551 Figure 47 Slip line deformation field superimposed on the real metal deformation field

105 PERTURBATIONS OF THE DEVELOPED MODEL FOR THE REAL METAL DEFORMATION FIELD The model of the real metal deformation field developed in Chapter III is created in an explicit manner; however, there are equally valid alternatives to that modeling method. This section deals with two perturbations or small changes to the modeling method of Chapter III. The first perturbated modeling method is based on alternate ways of extrapolating the model coefficients, other than A and B, to the die wall. (See page 72, Chapter III.) The second perturbated model is based on alternate functional relations for the skewed symmetry coefficients cl and c2. The five interior values of the model coefficients shown in Figure 48 are determined experimentally and have the same values as those coefficients shown in Figure 38. The values of the model coefficients in Figure 48 are extrapolated in the same manner as those model coefficients shown in Figure 38 except for the die wall values of a and B3. In modification 1, the die wall values of a and B in Figure 48 are defined to be equal to the first experimentally determined value adjacent to the die wall. Interpolation by piece wise cubic splines completes the definition of the model coefficients shown in Figure 48 so that any real metal path line can be modeled. In Figure 49, modification 1 of the modeled real metal deformation field is superimposed on the modeled real metal deformation field developed in Chapter III. The effect of the modified model coefficients is most readily seen in the deviations of the modeled time lines from the experimental time lines on traveling down stream through the die adjacent to the die walls.

0.30 2.00 44-. ~~~~~0)~~~~~~~~~~~~ ~J ~. <' r2 ~,- 0 ~ I I I O O, 2 0 O12 o e DIE GEOMETRY: = 22.5OEFFICIENTS H = 0724 I i 00 I R= 0.276 MATERIAL: LINE DIE W6061-0 AL, atUMINUMTER LINE -2.00 SKEWEO SYMMETRY COEFFICENTSY C= (4/22.5+1,'1 2 0 276 2 C= 4R2+3 0,00 -, 2~ DIE WALL r2IC path line CHAPTER III Figure 48 Modification 1, model coefficients versus real metal path line, r2

107 8Q88O ~DOWNSTREAM Modification 1. -— Model developed in Chapter III Semi-die angle: Y = 22.50 Reduction in area: R = 0.276 Outlet to inlet Material: 6061-0 aluninum dimension ratio: H = 0.724 Figure 49 Modeled deformation field based on Modification 1 superimposed on the modeled deformation field developed in Chapter III Modification 2 is created using new functions for the skewed symmetry coefficients. Table VIII, new model coefficients are calculated when c1 = (6/r)r2+1 and c2 = (6/) r2+5. The experimentally determined model coefficients are plotted in Figure 50. This model coefficient set is extrapolated in Figure 50 by the procedure of Chapter III. Interpolation by piecewise cubic splines completes the definition of the model coefficients. The modeled real metal deformation field based on these new coefficients is superimposed in Figure 51 on the modeled real metal deformation field developed in Chapter III. The effect of the new coefficients

0.30 2,00 O 0.20 o A 1,00 0 EXPERIMENTALLY DETERMINED COEFFICIENTS PQ 0.10 O EXTRAPOLATED COEFFICIENTS PIECEWISE CUBIC SPLINE INTERPOLATION 0.00 2 1- I I I, 0 7r/2 7r 0 io/2 DIE WALL t t CENTER LINE DIE WALL r., ll line CENTER LINE o P2 = 2+ A/(O(rl-a)2+eCl(rl-a)) - B/(b(rl-p)2+eC2(rl- )) -5.00 4J -4. 00 DIE GEOMETRY: Y= 22.50 H = 0.724 -3.00 P R = 0.276 c MATERIAL: 6061-0 ALUMINUM a NORMALIZE UPSTREAM VELOCITY: VA = 1.0 -2,00 LUBRICATiON: (SEE LUBRICATION, CHAPTER II) EXPERIMENTAL POINTS: (SEE TABLE VI) -1.00 SKEWED SYMMETRY COEFFICIENTS: C1 = (6/) r2+1, C2= (6/ir)r2+s5 0.00 0 7/2' DIE WALL r2, path line CENTER LINE Figure 50 Modification 2, model coefficients versus real metal path line, r2

109 is most readily seen in the deviation between the modeled time lines. These deviations manifest themselves at a greater distance from the die wall than those associated with modification 1. -- Modification 2. QOO Model developed in Chapter III Semi-die angle: y = 22.5 Reduction in area: DOWNSTREAM R = 0.276 Outlet to inlet dimension ratio: H = 0.724 Material: 6061-0 sluminum Figure 51 Modeled deformation field based on Modification 2 superimposed on the modeled deformation field developed in Chapter III. The two perturbations discussed in this section together with combinations of these changes, are available to the kinematic modeler of real metal deformation fields. The model used in Chapter III, compared to the modifications in Figures 49 and 51, is judged to be more effective. The perturbated techniques are presented to indicate ways of creating the alternate choices a kinematic modeler requires to assay goodness of fit.

110 POTENTIAL FLOW AND PLANE-STRAIN EXTRUSION Potential flow theory has been applied to plane-strain extrusion by Shabaik, Kobayashi, and Thomsen(44) and to axisymmetric extrusion by (45) Shabaik and Thomsen. In each case the flow field developed from the potential solution is compared to an actual flow as visualized by using a split billet technique (grids are used but no circles). The similarities and differences were noted. That the potentially derived flow field is irrotation in character as compared to the rotational character of the actual flow field was emphasized by Richmond and Devenpeck in their discussion of the Shabaik, Kobayashi, and Thomsen paper. The technique presented in this dissertation starts with the differences between the flow fields and models these differences in terms of a curvilinear coordinate system defined by the potential solution. The resulting modeled flow is rotational in character. CONCLUSIONS 1. The principal strains and directions are readily obtained experimental data and modeled computer output. 2. The modeled real metal deformation field provides a base for determination of experimentally related variations. 3. The perturbation techniques utilized in this study effectively enhance the goodness of fit. 4. The modeled measure of rotationality, AP, minimizes error magnification in determination of rotationality. 5. The Lagrangian measure of rotationality, AP, is influenced by frictional effects and metallic properties. 6. The real metal deformation zone is greater than that for a perfectly plastic solid, but less than that of a perfect fluid.

111 7. The Lagrangian kinematic model is applicable to continuously deformable materials without requiring a material model. FUTURE APPLICATIONS One of the strongest attributes of the techniques presented in this dissertation is the ability to amass large amounts of numerical information accompanying the kinematical relationships. Logically this leads to the employment of this method for obtaining a better understanding of the effects of strain history on product properties, perhaps resulting in improved material models. In like manner property distributions within extrusions can be determined which reasonably could lead to directly applicable correlations between strain histories and deformation related properties. Lubrication influence on extrusion and other processes is an area to be studied in hopes of putting numerical values on lubricants and frictional effects. The techniques presented are directly applicable. Further, the work of this thesis should be extended to other die geometries as well as axisymmetrical three-dimensional cases for extrusions, indirect extrusions and drawing.

APPENDIX A SELECTED PROPERTIES OF 6061-0 ALUMINUM The 6061-0 aluminum is received in the T6511 temper (extruded and stress-relieved stretched). The original bar stock of dimensions 1.0 inch by 2.5 inches in 12 foot lengths is completely annealed to bring it to the "0" temper condition. The measured properties are compared to typical properties as tabulated in the ASM Metals Handbook, Properties (15) and Selection, Volume 1 in Table X. TABLE X SELECTED MECHANICAL PROPERTIES 6061-0 ALUMINUM MEASURED TYPICAL Yield Strength 6,650 psi 8,000 psi Tensile Strength 18,100 psi 18,000 psi Elongation 31.2% 30% Reduction in Area 74.4% * See TABLE XI HARDNESS Measured Hardness: 63R average of 5 readings, range 1 R. 10 RE average of 4 readings, range 2 RE. Typical Hardness: 60 - 75 RH Rockwell Hardness 112

113 YIELD STRENGTH VERSUS PERCENT COLD WORK The 6061-0 aluminum is work hardenable. From information determined during a tensile test, the yield strength and percent cold work are tabulated in Table XI. The yield strength of 6061-0 aluminum doubles within 5 percent cold work. The cold work is defined in terms of the changing cross sectional area. TABLE XI YIELD STRENGTH VERSUS PERCENT COLD WORK Yield Strength Percent Cold Work 6780 psi 0.94 9180 psi 1.72 12600 psi 3.26 13850 psi 4.42 TENSILE BEHAVIOR The plastic strain data is determined on the run, i.e., the diameter data is taken while the test load is being applied. The modified stress state due to the hour glass shape of the tensile specimen.-fter necking is not included as a correction. (See Figure 52.) MODELED TENSILE BEHAVIOR The tensile behavior of this metal is modeled, GT = 29,500 (C).255 psi, (88) where o is the true stress and E is the true plastic strain.

114 ANISOTROPIC BEHAVIOR The aluminum behaves in an anisotropic manner during the tensile test. The originally round tensile specimen cross section becomes elliptical by the time of fracture. The minimum and maximum diameters at fracture are Dmin = 0.255 inches, and Dmax = 0.310 inches. An isotropic material would have had a circular cross section. The information used to create the true stress versus true strain graph in Figure 52 results from measurement taken on the diameter. 200 I 10 5 _ MATERIAL: 6051-0 ALUMINUM Of EXPERIMENTAL TRUE STRESS VS, 0 0 0 TRUE PLASTIC STRAIN ___b __o- = 29,500(E )0.255.2 I. I I I I I 0,01 0,02 0.05 0.10 0.20 0.50 1,00 e, TRUE PLASTIC STRAIN Figure 52 True stress versus true plastic strain

APPENDIX B RAM LOADS AND VELOCITIES DURING THE DIRECT COLD TWO DIMENSIONAL EXTRUSION PROCESS The ram loads and velocities during the direct cold two dimensional extrusion of 6061-0 aluminum are summarized in Figure 53. 90. _, C0 Q —Q- EXTRUSION III 0,0 1.0 2.0 3,0 RAM TRAVEL (INCHES) SEMI-DIE REDUCTION AVERAGE EXTRUSION ANGLE IN AREA VELOCITY I --- 450 0.449 0,352 IN/MIN II -0E —-- 22.50 0.451 0.351 IN/MIN III -- - 22.50 0.276 0.339 IN/MIN (i EXPERIMENTAL COMIENT NUMBER Figure 53 Ram loads and velocities during the direct cold two dimensional extrusion of 6061-0 aluminum Experimental comments: (79 In extrusion I at this point, the extrusion machine was disassembled and the split billet relubricated as shown in Figure 7 before resumption of the extrusion process. 115

116 Q In extrusion II at this point, the extrusion process was halted overnight and resumed the next morning. O The increasing rate of loading with ram travel was discovered to be a manifestation of the formation of fins. The fins are formed when the billet is extruded between the parting surfaces needed for the disassembly of the extrusion machine, (See Table XII. ). Q In extrusion II at this point, the maximum piston stroke was exceeded. A spacer block was added and the extrusion completed, KO The billet in extrusion II was 1.4% thinner than the billets used in the other two experimental extrusions. The more gradual initial increase in load with ram travel in this region is thought to be associated with the initial deformation of the billet within the container walls, 6. Extrusions I and II proceed in steps shown in Figure 53, In extrusion I, the standard deviation among the step velocities is 0.0204 inches per minute. In extrusion II, the standard deviation among the step velocities is 0.0733 inches per minute. Extrusion III was extruded continuously with load readings taken at equal time intervals. Within the accuracy established in extrusions I and II, the ram travel distance at each time interval is calculated from the average velocity in extrusion III, 7. The numbers next to the billet shown in Table XII were scribed onto the billets before extrusion. These numbers correspond to numbers on the extrusion machine, The parts of the extrusion machine were numbered to assure that the machine is assembled in the same way each time.

117 8. The fins were removed before the photographs in Figures 9, 10, and 11 were taken. TABLE XII FIN THICKNESSES Position 2 3 1 ~34 infin n fin L in Fin Thickness Semi-Die Reduction Fin Thickness 2 3 Angle in Area 1 4 0.007 0.011 450 0.449 0.005 0.01' 3 *** o n 0.009 22.50 0.451 n 0.009 n n 22.50 0.276 n n experimental, assuming thickness remains constant ** inches negligible

APPENDIX C THE SCHWARZ CHRISTOFFEL TRANSFORMATION FOR THE GENERAL QUADRILATERAL WITH TWO VERTICES EXTENDED, OF THE EXTENDED VERTICES, ONE CONTAINS A SOURCE AND THE OTHER CONTAINS AN EQUAL SINK. COMPLEX VARIABLE NOTATION THAT IS PECULAR TO THIS WORK The complex variable will be denoted by lower case Roman letters, p, q, q., x, y, z, where the real part and the imaginary part will be denoted by subscripts 1 and 2 respectively, for example, w = w(w1,w2), w = W1 + iw2, or w = r eiBw w where r=W2 21/2 r =(wl + w ) w 1 2 and -1 =w tan (w2/w ) A specific complex point will be denoted by a capital Roman letter, A, B,..,, L, M, N, where the real part and the imaginary part will be denoted by subscripts 1 and 2 respectively, for example, A = Al + iA2 or A = A(A1,A2). If we have one or several conformal transformations, the image of a specific point with respect to a variable can have the form when required, for example, 118

119 A =A + iA2, Z lz l 2z or z =A(Alz A ). which refers to the image of the point A in the z plane. The reasons for the subscripted complex notation are to economize the symbolization and to facilitate the use of two dimensional tensor or vector calculus when these techniques would seem useful. THE SCHWARZ CHRISTOFFEL TRANSFORMATION The general quadrilateral with the vertices extended is shown in Figure 54. The Schwarz Christoffel transformation, iz2 Z PLA[NF ANALYTIC REGION +7r r FOR CONFORMAL / A / TRANSFORMATIONS s,w,P,q \'c WHERE O h 1 B Z Figure 54 Physical plane z U-k -1) dz JR E (w-ak) ( dw t k=l in general maps the interior of a polygon onto the upper half plane, the w plane in this case. The ck term represents the interior angle of the

120 of the Kth vertice, while the ak term represents the image of the Kth vertice in the w plane. The transformation is completed by integration, n (oY(k -1) z Rlt kCw-ka Tr w + Tt (89) where the constant R can be used to control the relative size and orientation of the polygon in the z plane, and the constant Tt can be used to translate the polygon in the z plane. In general three vertices can be mapped arbitrarily whereas the images of the remaining vertices must be uniquely determined. For the quadrilateral in Figure 54 let,. = = 0 = C~ = -'Y+ ~, 1 A 0' 2 B C= =t =Y+-T, ~+ = = 0, 3 =C 4 D also let the images of the four vertices in the w plane have the following values, 1 A =0, a2 =B = b where O<b<l, a3 = C = 1, a= D = the extended point. The Schwarz Christoffel transformation for the quadrilateral in Figure 54, dz _t w-ff dw w w-bj (90) will be rationalized by the following procedure. Let m = Y/T, where O<m<l, When m is an irrational number, the rational fraction f/g will be used to approximate m as closely as desired. Before completing the rationalization of this derivative the constants R and b will be determined. t

121 Point A will be considered a source and point D will be considered an equal sink. The complex potential function P for the flow in the w half plane, Figure 55 is P = Q Log w, where Log w is the principal value of log w. iw2 W PLANE ANALYTIC REGION FOR CONFORMAL TRANSFORMATIONS, Z,S,P,q A B C W1 Figure 55 The Schwarz Christoffel upper half plane. Point D is extended. Let the real constant VA denote the velocity of the flow in the limit far from the offset to the left in the physical plane, Figure 54. Let q be the velocity where, q = complex conjugate d d dz dz and lim q = VA. Re (z)W- - oo Therefore the strength of the source on the left and of the sink on the right in the physical plane, is then

122 and then p = VALog w. Figure 56 and 57 show the p plane and the q plane respectively. iP2 P PLANE iVAr ANALYTIC REGION FOR CONFORMAL TRANSFORMATIONS, z,s,w,q VA LOGb B C P1 Figure 56 The complex potential plane, P2 equal to a constant is a streamline. Points A and D are extended. iq2 q PLANE ANALYTIC REGION FOR CONFORMAL TRANSFORMATIONS, Z,S,W,P B A D ~ —-' VA h Figure 57 The velocity plane, q. Point C is extended,

123 The complex velocity, dp Ap dw q dz dw dz V f/g A w (w-b / w Rt- (91) When w = 0, q = VA. Therefore, VA f/g VA= R (b) (92) t V When w approaches point D, q = V since Q V hV D h A DQ Therefore, from Equation (91), VA V h Rt or R = h, (93) and from Equation (92), g/f b =(h). (94) Now that constants Rt and b have been evaluated, the rationalization of the Schwarz Christoffel transformation will be completed. This mothod of evaluating Rt and b is suggested by Churchill.(46) The transformation from Equations (90), (93), and (94) dz _h w-l f/g dw w w-hg/f can be rationalized by defining a new complex variable f /w-hg/f f/g sW J

124 Now (w-hg/f )1 /g w-1 g/f sg = w-h w-1 so that hg/f s g W = (95) 1 - s and so that dz h 1- s\ (96) dw sf hg/f sg iS2 i/f h 1 S /? A D/// ANALYTIC REGION FOR CONFORMAL S PLANE TRANSFORMATIONS, Figure 58 The rationalizing plane, s, Point C is extended. Writing the Schwarz Christoffel transformation between the z plane and the s plane we get dz _ dz dw, (97) ds dw ds

125 where from Equation (95) dw d / hg/ f -s ds ds 1 dw g-l [(hg/ s) _ 1 ds gs 2 L-s g S(1 - sg)) _ (98) Combining Equations (96), (97), and (98) dz_ ghsg-f-1 1 dsg) (hg/f sg) and integrating?g-f-l d g-f-1ds z = ghJ gh g/f s + Tt.(1 - s) (h sg) From the algebra these fractions are expanded into partial fractions and integrated. Now that the algorithm for the Schwarz Christoffel transformation is complete, the reflection or symmetry principle (47)will allow the results to be interpreted as also applying to either physical system showr in Figure 59. If = /4 in Figure 54, z = h Log 1_s - i+ Log ( i -[L (5h+s) i Log( ih+s)] -(1-h)

126, CENTER LINE AND PLANE OF SYMMETRY a. Converging channel flow y // CENTER LINE AND PLANE OF SYMMETRY b. Split and separated channel flow Figure 59 Two additional equivalent physical configurations for the Schwarz Christoffel transformation

APPENDIX D COMPUTER AIDED ANALYSES AND PLOTTING PROGRAMS The subroutine and function subprograms collected in the first section of this appendix are used by the remaining programs. The presentation of these remaining programs is in the order in which they appeared in the text of this dissertation. Comment cards are used to describe the various sections of the longer functions, subroutines and programs. GENERAL PURPOSE SUBPROGRAMS Subprograms Brief Description Real Function RM Given rl, r2and the model coefficients from subroutine DCONS, RM returns the image of the real metal path line point in the potential plane. Real Function DT Given two path line points and the velocities at these points, DT returns the time increment based on the trapezoidal rule between these points. Complex Function RINT Given two vectors in complex forin, the times associated with these two vectors, and an intermediate value of time, RINT returns a vector in complex form that is a linear interpolation with respect to time between the two given vectors. Complex Function SF Given a point in the potential plane, SF returns a point in the rationalizing plane. 127

1.28 GENERAL PURPOSE SUBPROGRAMS, continued Complex Function ZF Given a point in the rationalizing plane, ZF returns a point in the physical plane. Complex Function QF Given a point in the rationalizing plane, QF returns a point that is a complex conjugate of the point in the perfect fluid velocity plane as defined in the text. Complex Function VF Given the image of rl, r2 in the perfect fluid velocity plane, and the partial derivatives from PDPRM, VF returns the modeled real metal velocity vector in complex form. Subroutine DGCONS Given r2, DGCONS returns the model coefficients and their derivatives with respect to r2. Subroutine DGCP2 Subroutine called by DGCONS for computational purposes, Subroutine PDPRM Given a modeled real metal point, PDPPM returns the partial derivatives of r2 with respect to P1 and P2 using the information from DCONS. Subroutine MINV Matrix inversion using Gauss-Jordan reduction with a maximum pivot strategy. This subroutine subprogram is only a slightly modified form of the program presented by Carnahan, Luther, and (4)a Wilkes, and is not listed.

129 GENERAL PURPOSE SUBPROGRAMS, continued Subroutine MATVEC Multiplies a matrix and a vector, This subroutine subprogram is only a slightly modified form of the subroutine presented (48) by Carnahan, Luther, and Wilkes, and is not listed, Subroutine NLSYS This subroutine solves a system of nonlinear equations, given a sufficiently accurate initial estimate of the solution and equations to be solved in Subroutine AUXFCN. This subroutine is the same as (49) that of Brown and Conti and is not listed. Subroutine AUXFCN This subroutine contains the equations to be solved by NLSYS and is listed in the Experimental Modeling and Coefficients Computation section.

130 GENERAL PURPOSE SUBPROGRAMS, continued REAL FIlJNTII)N RM*H(AA,HH,GXX) C... THIS FIINCTI IN KRETURNS RM, THE IMAGE OF A MODFLEF) RFAL MFTAL PATH C... LINE PU)INtT IN1 THE PO(TENTIAL PLANE, GIVEN THE MOI)ELFD PATH LINE C... POINT ANI) THE MODEL C)EFFICIFNTS. IMPL ICIT RFAL 8(A-H) DOUn(LE PRECISION AA,HH,PI,C1,C2 REAL "H8 (xx ( ) P I=3. 1415 92 6 (.1=2.5 C 2=CL+7 a1=(HH — GXX (3 ) 2 A2=-CL1(HH-GXX(3)) a3 = ( 2: -(C,-X ( 6 ) )**2 a3=(HH-(;xx(A)),:42 A4=(C2 ( rH-(,XX ( 6 ) ) IF (A2,?,1.7() A2=70 IF (A2.LT.-(U) A2=-70 IF (A4.( T. 70) A4=70 IF (A4.LT.-7()) A4=-70 HI =1. O)75 HH1=AlI)LOG(GXX (2 ) H I=H I IF (BHH1.LT.14.0) b1=GXX(2)e*A1 H2=DEXP (2 ) HB3=A3*DLOG(CXX (5)) h3 =H I IF (HB3.LT.174.0) B3=CXX(5)**A3 H4=DEXP ( A4 ) RM=GXX(I ) / ( 1+2 )-GXX ( 4 ) / (H3+ 4)+AA RF TUJRN F N(.) REAL FUJNCTION DTHI A,H,C, D) C.. HIS F.)NCTION RETIRNS THF TIMF INCREMENT DT USING THF TRAPEZO)IDAL C... RULE, GIVEN TWO PATH LINE POFINTS A AND H, AND THFIR RESPFCTIVE C... VELOCITIES C AND ). C MPL EX'; 1 6 A, H,C,) nT=(), 5( /LS/C i +/CO)Ahs DA)) *Cr)ABHSS (H-A) ETtJRN FNI) EN)) W$ t. r.****r*- **>- X*****...<****g c*~*e**t*=......... c*z t44t***c*$4

131 GENERAL PURPOSE SUBPROGRAMS, continued COMPLEX FUNCTION RI NT* 1 ( Z2 1,T2,T1, T,IT) C... THIS FUNCTION RETURNS RINT, A VECTOR IN COMPLEX NOTATION THAT IS A C... LINEAR INTERPOLATION WITH RESPECT TO TIME HETWEEN TWO GIVFN C... VECTORS. COMPLEX*16 Z1,Z2 DOUtBLE PRECISION TiTI,T2,DTT T= I T*DDT R INT=( T-T1)/( T-TT1 ) )*( Z2-Z 1 )+Z1 RETIORN END COMPLEX FUJNCTION SF*16( AA,HB,CC) C... THIS FUJNCTION RETURNS SF, A PI)INT IN THE RATIO(NALIZING PLANF, C... GIVEN A POINT IN THE PO)TENTIAL PLANF. ([OMPLEX'16 AH,AC,AA,CC,SSF I)OUBLE PRECISIOrN HH,BHH HBB=Bb*:H AB= ( AA- B bB) / ( AA-CC ) AC=CDSORT( AK) SSF=CDS(R T AC ) SF=CDS(T ( SF ) IF (AIMAG (SF).GT.O) SF=OC(3IJG, (SF) RE TtJRN F NI) COMPLFX FI)NCTION ZFJ1::l A(A, H,CC) ),;F F, A, F) C... THIS FIJUf'CTIf)Nl kFTIURNS 7f, A PlTINT!fNl JHF PHYSICAL PLANF ITVFN A C... POINT IN THF RATI)JtNAl _I7 ZI\jr, PLANF. IHF SCHWARP[7 C'HPII tSFFI_ C... TRAN SFrMA I NI IS FO )III FI TI) liF f ) I F 7 F. (S F- APP FNI X C.) ( FIMP L F X 1 A G, AC(, AI (,AC, A FC. A HC, AI GAi A(.,A) [ 1\ F, AI,,', I, A ) i) -, F,I FP C)1l /1tIAt- ~ FCA I I(,(A ( A(,(,= (-l- +A ) / (-F -A ) ii)(- =( I + ) / I ) A[ )=('!1 )t. I hS fAi'G )

132 GENERAL PURPOSE SUBPROGRAMS, continued (=(1)*AA ) / ( ),-a 6 f=(;1)L I'-, ( A —G ) IF (AI,( Af( ).-..Ah. R AL(LG).I.() AF= i)(, ln K ( AF A F (,= ( F a;:, + /. 6 ) / ( F A.: h F,- s a ) AF =C)L U(-,,( )A aGG'; = ( -{.zH'::HH+ a ) / ( -K:, ) G(; =C D L I; ( A C,; ) AHG= ( E:'H+ AA ) / ( E )E -AA) aH=CT)L- I(; ( AH ) Al 1(= ( BH + A ) / ( h,-AA ) AI =C()L(;,( A 16 ) IF (AIMA(A,(AIG).EO.O. ANO).RFAL(AIG,.LT.O) A I=r)CroNJG(AI) /F =H ( — EA*EAC-EF AD+AF ) + ( -E +A*AG,+F -AH-A ) +CC R E T IU N ENn COMPLEX FtIUNCTION OF (*6(AAK,CC ) C... THIS FUNCTION RETt)RNS OF, A POINT IN THF CrInlJIIATE PFRFFCT C... FLUtID VELOCITY PLANE GrIVEN A POINT IN, THE POTENTIAL PLANF,, (THIS C... IS THE COMPLEX CONJUGATE OF THE OF r)FFINFD IN THE TEXT.) COMPLFX*L6 AA )OUIBLE PRECISION BSCC (F=( HB/CC ) AA R ETURN END COMPLEX FUNCT ION VF*( 16,PDP,POP2) C... THIS FtUNCTION RFTIJRNS VF, THE MODELF[) REAL MFTAL VFLO(CITY VFCT'p" C... IN COMPLEX FO(RM, GIVEN THE PARTIAL nFRIVATIVES FRf1M PDPRM AND FHt C... IMAGE OF THE MODFLED REAL PATH LINE POINT IN THE PERFECT FLUtID C... VELO)CITY PLAN'F. COMPL E X 1 6 (,O E, PF, NPF UOD(HLF PRFCISIfIN VPF,RFPF-,SFPEF C O=DCONJG, ( 0 ) VPF=CAKS( C( O) t PF =CO/VPF REPF=)LE( REAL (EPF) ) SFPF=DHLF ( AIMAG(EPF) ) FNPF=DCMPLX(-SFPF,REPF) VF=VPF ( DP2FPF-PI)P 1FNPF ) RE TURN E NI)

133 GENERAL PURPOSE SUBPROGRAMS, continued SUHROIJTINE I)GCONS (AA,GXX,D)GXX) C... THIS SUBROUITINE RETURNS GXX(1,...,6), THE MODEL C(IFFFICIEN1-S AND) C... THEIR RESPECTIVE DERIVATIVES DGXX(1,...,6) WITH RESPECI T1 THEF C... MODELED REAL METAL PATH LINE. IMPLICIT REAL*8(A-H) DOUBLE PRECISION AA,PI,GC REAL*8 GXX(6),DPI(7),DGXX(6),GCC(h,7),X(6,7),Y(,7) INTEGER ID(6) COMMON GCC,X,Y,ID DO 1 I=1,7 1 DPI(I)=x(1,I) J=6 C... THE FOLLOWING CONDITIONAL STATEMENTS DETERMINE THE PROPER INTERVAL C... FOR THE SPLINE FITTED MODEL COEFICIENTS AND THE DERIVATIVES C.. WHICH ARE RETURNED BY DGCP2. IF (AA.GE.DPI(1).AND.AA.LT.DPI(2)) J=1 IF (AA.GE.DPI(2).AND.AA.LT.DPI(3)) J=2 IF (AA.GE.DPI(3).AND.AA.LT.DPI(4)) J=3 IF (AA.GE.DPI(4).AND.AA.LT.DPI(5)) J=4 IF (AA.GE.DPI(5).AND.AA.LT.DPI(6)) J=5 )O 2 K=1,6 CALL DGCP2 (K,J,AA,GC,DGC) GXX(K)=GC 2 DGXX(K)=DGC RETURN END ****************************************4*****4**** *4 4**4**4**4**4 4****4 SUBROUTINE DGCP2 (K,J,AA,GC,DGC) C... THIS SUBHROUTINE UISES THE INTERVAL DETERMINED BY DCU)NS T(l CALCULAiF C... THE SPLINE FITTED MODEL COEFFICIENTS AN[) THEIR DERIVATIVFS WIIH C... RESPECT TO THE REAL METAL PATH LINF, WHICH ARE RFTURNED TO DCRNS. IMPLICIT RFAL*8(A-H) REAL*8 GCC(6,7),S(7),X(,7,XX(7 ),Y(6,7),YY(7),H(7) INTEGER 1D(6) COMMON GCC,X,Y,ID M=ID(K) D)O 1 I=1~M XX(I)=X(K,I ) YY(I )=Y(K,I) 1 S(I)=GCC(K, I ) DO 2 II=1,6 2 IHHI I )=X ( 1,1+1)-x( 1, I ) L=J H=HH(L)

134 GENERAL PURPOSE SUBPROGRAMS, continued IGC=( S ( L *H )*(XX(L+)-A )* +((L+)/(2H) ): ( I A - X L ) ) *(AA -XX +( ]+1)/H-H*( (%(_+! +I)/6))*(AA-L))+ (YY(L/)/H-*(S(L)/a))"(XX(L+1)-AA} I,'GC=-(S(L)/(?*H))*( XX(L+])- AA)-*2+(S(L+ I)/(2';H))"(AA-XX(L))*-'~'~+( 1( L+ 1) /H-H' ( ( L+ /)/h ) ) -(YY L /H-H( S ( L )/6) ) RETtJRN F ND SUIHRU JI IN, E PI)PKM (hH, AA, XX, DGXX, P P 1i, PDP2) C... CIVFN THF CUiD)Rl)INATFS HIF A MHJf)DLFD) PFAL MFTAL PATH LINF POINT, C... THIS S(JhkO)tJTI NF kFTIJ)',S THEF PARTIAL )FR IVATIVFS (fF THE M(f))ELFD C... WEAL MFTAL PATH I INE WITTH PRFSPECT IfD P1 AN() P2 IN PFOPl AND PD)P2 (... RESPECT1IVELY. IMPLICIT R F AL' ( A-H, -7 ) W FAL* 8 (;XX ( ),I)GXX ( h) PI =3. 14159265 C 1=2. C2=C 1+7 C 3=0.0 A= — G X (3 ) A 1 =A*' 2 a2 =-C 1 A H=H-GCXX(6) A 3 = * * 2 A4=C2Th IF (A2.(.70) A2=7(0 IF (A2. L T.- ) A2=-7() IF (A4.(T.70) A4=7(0 IF 1A4.1lT.-70) A4=-7() I = 1. F) 7DT,B1 =A1?l) _((D CXX (2 )) hi =H I IF (KHK.I 1.1-74.(0) 4 Sl: CXX(2) A1 2 =f)FXP ( a2 ) K3=A3 I_)(7 (,XX ( 5) ) K3 =H I I F I H43. lT. 1 74.() 3=C, XX ( 5) A3 4=)F X P ( A4 ) )H I = 1. ()) 4 7 I'1 = K 1 + 5 2 I)l =2Dl) I (( i)l) 4'1 =IDH I IF ( I)l T. 1 4.0) D1 =H1+2? 1)2 =K3+H4 I)D2:2':I)L DIfG 1)2)!)2 =F)H I I F (!))2.LT. 174. O) 1)2=K3+H4 - =3 2 f)LG(D(,XX(I) )'; F 3:2 -':: ~: t~;)(, {GX ( 5 ) );: H -4=?2 4 k1 =r2: -( -(. -: A A+C 1.-,)gXX ( -3 ) )

135 GENERAL PURPOSE SUBPROGRAMS, continued F3 =A3: ( (,XX ( 5 ) A. ( A3-1 ) ) nf, XX ( 5 )-2*H B r)CL ) ( GXX ( 5 ) ):xH':-)C, X X ( 5 ) -4=H4( C 3H-C?)GXX ( 6 ) ) I iNE= —C 1.12+2 1 i*fr)L)C; ( ) )* A TWrI=C244+2 4H3I)L(G ( GCXX ( 5) ) H PDPl ( XX ( 1 ) ( NF ) /1**2-GXX ( 4 T ) ( W ) 02 2 ) / (1-( -)XX ( 1 ) i/ + 4)/1)2)-(GXX( 1) (Fl+F2)/(1 **2-(-XX(4)*(F3+F4)/F)2'*2)) PDP2=1/( l+)GXX( 1 )/Dl -GXX( 1{ )*(FlI+F2) ( l**2 )-F)GXX ( 4 )/D+ XX ( 4 )(F IF4)/(D2*-<2) ), ETl RN F Nn ***t *****t> *******$$*ee***4**e*e y*eXWt*8*Xqt e-

136 ORTHOGONAL CURVILINEAR COORDINATE COMPUTATION AND PLOT C... THIS PROGRAM TOGETHER WITH *PLOTSYS (SFF REFERENCE 42) COMPUJTES C... AND GENFRATES THE PLOT DESCRIPTIONS FOR THE ORTHOGONAL CUJRVILINFAR C... COORDINATE SYSTEM SHOWN IN THE RIGHT HAND SIDE OF FIGUtIR 12. IMPLICIT REAL*8(A-H) i)OUBLE PRECISION PI,VFL,TB COMPLFX*16 E,F,G,SF,ZF,7,GG,HH DIMENSION X(4500), Y(4500) DATA H/0.549/,VEL/1.0/,GG,/(-.70711)/,HH/(-.70711,-.70711)/, IE/(1.0,0.0)/,F/(O.O, 1.0)/ PI =3. 1415925 C=PI/12 TB=0.41421 TAAA=2.50 TBBB=10.50 TSS=1.095 TRN=2#(PI*TSS+TAAA) U=-((1-H)*PlI)/TB C... I IDENTIFIFS THE STREAMLINE,. )O 1 1=1,25 CO=-(0.00183H)*(1-25) IF (I.FO.1) CO=O.O A=(( I-1 )*PI)/24-CO C... J IDENTIFIES THE POTENTIAL LINE. DO 1 J=1,50 D=-6.870+(J- )*C C... ZF DEFINES THF POSITII)N OF A POINT (I,J) IN THE PHYSICAL PLANE. Z=ZF ( SF( CDEXP(I)CMPLX(), A )/VEL ),H.,E ),H, B, E,F GG, HH ) C... X AND Y ARF PO(SITI()NS IN THE ABSOLUITE REFERENCE FRAME USED BY C... *PL(ITSYS. X( ( I-1) bO+J )=AIMAG( Z )*TSS+TAAA I Y((I-1)*50+J)=RkAL(Z)*TSS+TBBH C... THE NFSTFI) LOOPS TERMINATED AT STATEMENT NUJMBER 2 CALCULATE THE C... MIRROR IMAGE NEEDED[) FOR A PLOT SYMMFTRICAL TO ITS CFNTER LINE. 1)O 2 K=1,25 lf) 2 J= I 1,50 X((K-1)O-50+1250+J)=-X((K-1)*5()+J)+TRN 2 Y ((K-1 )'50+1250+J)=Y( (K-1 )50+J) (... LOO()PS 3 ANl) 4 PLOfT STRPAMLINFS. 10)f 3 M= 125

137 ORTHOGONAL CURVILINEAR COORDINATE COMPUTATION AND PLOT, continued 3 (;ALL PLINF (X((M-l)*50+1),Y((M-1)*50+1),50,1,0,O,O) )0 4 L=1,24 4 CALL PLINE (X((L-1)*50+1251),Y((L-)*50+1251),50,11,,0,()) C... LOOPS 5 AND 6 PLOT POTENTIAL LINES. Dn 5 M=1,50 5 CALL PLINE (X(M),Y(M),25,50,0,0,0) DO 6 L=1,50 6 CALL PLINE (X(1250+L),Y(1250+L),25,50,OO,0) CALL PLTFND STO)P FND *444 **************4******4Ban4**44**4**

138 PERFECT FLUID DEFORMATION FIELD COMPUTATION AND PLOT C... THIS PROGRAM TOGETHER WITH *PLOTSYS (SEE RFFERENCE 42 COMPUTES C... AND GENERATES THE PLOT DESCRIPTIONS FOR THE SUPFRIMPOSED PERFECT C... FLUID DEFORMATION FIELD SHOWN IN FIGURE 35. IMPLICIT REAL*8(A-H) DOUJBLE PRECISION PI,VEL,TH,RM INTEGER ID(6),IT(25,41) REAL*8 GC(6,25),XX(6,7),YY(6,7),GX(6),DX(6), T25,41),DGC(6,25) 1,DS(25) COMPLEX*16 EF,G,SF,ZF,OF,GG,HH,P,W,S, Z(25,41),,VF,ZZ(2),VV(2), IZT(25,41),RINT,ZTT,VTT, V(25,41),VT(25,41),FRMENRM,ZEBAV,EBAV DIMENSIO)N X(2400), Y(2400), XXX(2400), YYY(2400), XS(25), YS(25), 1xXS(25), YYS(25) COMMON (7C,XXYY,ID DATA H/0.549/,VEL/1.0/,GG/(-0711,.,70711)/,HH/(-.70711,-.70711)/, lE/( 1.,O.O)/,F/(O.O,1.0)/ DO 1 I=1,6 1 ID(I)=25 WRITE (6,27) (ID(I),I=1,6) 2 READ (5,30) (OS(I),I=1,25) WRITE (6,29) (DS(I),I=1,25) PI=3.14159265 C (.942164) IS THE FORESHORTENING CORRECTION. O)DT=(PI/12)*(.942164) C=PI/12 TB=0.41421 TAAA=2.50 TBHH=10.50 TSS=1.095 TRN=2* (PI TSS+TAAA) SPI=SNGL(PI) C RO IS THF SCALED RADIUlS FOR THE U)NDEFORMED CIRCLES. RO=O.15*(SPI/1.2)*TSS/2. RADIAN=180./SPI C I.. I IDENTIFIES THE PERFECT FLUID PATH LINE. DO 3 I=1 25 CO=-(.001838)*(1-25) IF (I.FU.1) CO=O.O A=((I-1)*PI)/24-CO Z( 1 )=0 ZZ (2 =0 VV( 1 =() VV(2 )0 T (I,1) =() lT(I, )=O

139 PERFECT FLUID DEFORMATION FIELD COMPUTATION AND PLOT, continued C... J IDENTIFIES P(OSITIONS ALONG THE PERFECT FLUID PATH LINE,. D00 3 J=1,41 H=-((1-H)*PI /)/T C... DS(I) IDENTIFIES THE TIME EQUALS ZFROF TIME LINE. I)=DS( I)+(J-1 )*C C... ZF IDENTIFIES A POSITION OF A POINT IN THE PHYSICAL PLANE. Z(I,J)=ZF(SF(COEXP(DCMPLX(D,A)/VEL),H,F),H,NE,F,GG.,HH) C... X AND Y ARE POSITIONS IN THE ABSO)LTE REFERENCE FRAME UJSED BY C... *PLOTSYS. X( (I-1)441+J)=AIMAG(Z(I,J) )*TSS+TAAA Y((I-1)*41+J)=REAL(Z( I,J))*TSS+TB) H IF (I.EO.1.OR.I.EO.25) GO TO 3 (=OF(SF(CDXP(DCMPLX (D, A )/VEL),H,),VEL, H ) C... V(I,J) IS THE VELOCITY ASS()CIATED WITH EACH PERFECT FLtID PATH C... LINE POINT. V(I,J)=DCONJG(Q) ZZ( 1 )=ZZ (2) ZZ(2)=Z(I,J) VV( 1)=VV(2) VV(2)=V(I,J) IF (J.EO.1) GO TO 3... 1(II,J) IS THE VALUE OF THE TIME FUNCTION ALONG THE PERFFCT FLJIl) C... PATH LINE, WHICH IS DETERMINED BY SlIMMING THE OT'S. T( I,J)=T I,J-1)+DT(ZZ( 1),Z(2),VV(),VV(2) ) C... ( I,J) IS CO)NVERTEI) TO AN INTEGER. ATT=T( I,J) )/)[)lT T=SNGL ( ATT) IT(I,J)=IFIX( TT) 3 CONT I NtE C... THE MIRRKIR IMAGEF (OF THF PATH LINFS IS CALCIJLATED. 0)0 4 K=1,24 0I0) 4 J=1,41 X((K-1)-'41+ 1025+J)=-X( (K-I)),;41+J)+lwr, 4 Y((K-i):-'41+1()25+J)=Y( (KI-); 41+J) C.... IO)PS 5 ANO! h PLrIT THE PATH LINFS. I)0) 5 M=1,25,4 CALL PLINE (X((M-1)-: 41+1),Y((M-1)' 41+1),41,1,0,0,0) 0I[) 6 L=1,21,4 k CCALL PlINE (X((L-1); L 41+1(i2h),Y(( L-1 ),41+10?),41, 1,0,0,0) C... LOfJPS THRI)IIGH I) o)FTFTMMINF IF THFRF ARF 7FRFI, ONIF O1R MIIF TI1MF,... IINF PE)l"ITs HFl WF'I\ TWf) PA1H LINE P(lll[TS ANl[) ASS..1J ES AN FnIJAL

140 PERFECT FLUID DEFORMATION FIELD COMPUTATION AND PLOT, continued C... NU) HFR i-F TIME LINES FOR THE DEFORMATI ON FIEL)D. THF VELO)CITY C...AT EACH TIMF LINE PO)INT IS DFTERMINFD. DO 10 1=2,24 DO 10 J=2,41 N=J L = I M=J-1 IF ( I.E(.24.AND.J.O.41 ) GO TO 7 (;O TO 8 7 R=IT(I,J) R=R/2. KCH=IT( I,J)/2 KLAST=IT(I,J) IF (R.GT.KCH) KLAST=KLAST-1 8 CONT I N)F IF (IT(L,M).EO.IT (LN )) GO TO 10 ITT=IT(LN)-IT(LM) DO0 9 KL=1,ITT K=IT(L,M)+KL Z TT=R INT( Z (L, N) Z (L,), L, N), T ( L ),DDT, K) ZT(I,K)=ZTT VTT=RINT( V( L,N),V(L,M),T( LN),T(L,M ), DT,K) 9 VT ( I,K)=VTT 10 CONT I NtE WRITE (6,28) KLAST C... THF MAJOR AND) MINOR DIAMETERS AND THEIR ORIENTATIONS ARF C... DETERMINFD. I) 13 KR=1,2 DO 13 I =3,23,4 KLL=KLAST-1 l)O 13 K=1,KLL,2 hEE=COAHS(VT(I,K )) ERM=VT( I,K)/HEE FNKM=*FERM ZEBAV=ZT(I+1,K)-ZT(I-1,K) FHAV=ZEHAV/C)AHS( ZE6AV ) FPEE1 =DkFAL ( FRM*DCNJG( FAV) ) FPEF=U)ARCOS(FPEF1 ) FEE=PI/2-FPEE AEE=DTAN(FEF) AE 1=1 /HFF**2 AE2=-AFF* AF AE3=( HFF **2+AE-E'**2/HFF 2 ) AE4=-R( J4, 2 aAE 1 =( AE1+AF3 ) AAE2=DSJRT( (AF1)*2-4. ) E 1= ( AAF 1-AAF2 )/2 E2=( AAF 1+AA F2 )/ 2 f)MM2=S)RT ( l./ (El/(-AF4)) ) i)MN2=oS(T) T 1. / ( H2/ (-AF4) ) AR(;=?.AE2/ ( AFI-A3) I)ITJHF T= (I)ATAN ( AR) ) )HHFI A=f)UTHFI /2. I)T =1)TAN ( H)FTA ) f)NM=A 1 +2.:AF2*F)TT+AF 3 I)TT**2 i)CH 1 =-AF4/0)NM I)M(=DFCH * ( I. +)TT4*2 )

141 PERFECT FLUID DEFORMATION FIELD COMPUTATION AND PLOT, continued DCH=DSORT ( )M D ) C... THE MAJUR DIAMETER DIRFCTIO)N IS CHFCKFD AND CORRFCTED IF NEEDFD). IF (SNGL ( DCH).NE.SNGL(DMM2)) DHETA=PI/2.+DHETA DROT =DIMAG( ERM) DROT2=DARS I N( DROT1 ) ROT=SNGL (DRrlT2) TTHFT=SP I /2.-(SNGL(DHTA ) +RT) XO=AIMAG(ZT(,K) )*TSS+TAAA YO=REAL(ZT(I,K) )*TSS+TKKI XALPHA=TTHFT*RADI AN XAEE=SNGL (DMM2) XBEE=SNGL ( DMN2 ) C... WHEN KR FQUALS TWO, THE SET OF ELLIPSES IN THE MIRROR IMAGE IS C... PLOTTED. GO TO (12,11), KR 11 XO=-XO+TRN XALPHA=-XALPHA C... THE STRAIN ELLIPSES ARE PLOfTTED. 12 CALL PLIPS (XO,YO, XAEE,XHEE, XALPHA, O.0,360.0,O) 13 CONT INUIE C... THE POSITI()HNS f)F THE TIME LINE POINTS IN THE AHS(ILIJTE RFFERENCE C... FRAMF lJSF HY *PLOTSV S ARE DEF NF). I)O 14 L=,KLAST 0f) 14 M=2,24 XXX (IL-1)*23+M-1 )=AIMA(ZT(M,L) )*TSS+TAAA 14 YYY((L-1)*23+M-1 )=R AL(T I(M,L ) )*TSS+Th C... THE MIKkIfJ IMAG7E I)F THF TIME LINE POlINTS IS I)FTERMINEI). I)O 15 _=lI,KLAST 1)O 15 M=?,24 XXX ( L-I+KLAb1 )23+m-1 )=-XXX (( L-1);) 2'+M-1 ) +Tk 15 YYY ( ( L-1+KLST ) 2 3+M-1 ) =YYY (L-1)2 3+M-1 ) C...PLOT DESCkRI If)NS (IF THE TIMF LINFS Akf (GENERATE) HERE. DO 16 L=2,KLAST.2 16 CALL PLINF (XXX((L-l):235+1i),YYY((L-1);23+1),23,1,0,(,O) OI)[ 17 L=2,KLAST,2 17 CALL PLINF (XXX(3S-; IL-1+KLS T)+l ),YY (23*(L-14KLASf )+1),, 3,1,(,O,, I ) C. PL()T DFSC. RIPTIOJNS THF IJMF F(0llI S 7FkI) LFVFL CIJRVF AMF C...(E E!RHAT;J iFR-. i)O 1H I =2. 24 X S( I ) =A IAL ( / ( I, 1 ) ) TSS+1 AAA 8 YS ( I )=kAL (/ ( I 1 ) )::1'i+ 0)0 1 I=2,/4 XXS( I =-X%( I )+1PN 1' YYS( I ):= ( I ) (. AIL L. Pt iFIr ( X S I 2 ). Y S ( 2) 23, 1,() 0,) 0

142 PERFECT FLUID DEFORMATION FIELD COMPUTATION AND PLOT, continued CALL PLINE (XXS(2),YYS(),923, 1,,0,0) CALL PLIFNI) C... SELECTFD NUtMERICAL ()I)TPUT IS PRINTED. DO 20 J=1,41 20 WRITE (6,23) (T(I,J),I= 5,21,4) nO 21 J=1.,41 21 wRITE (6,24) (IT(I,J),I=3,23,2) WRITE (6,25) 00 22 L=1,KLAST 22 WRITE (6,26) IZT(M,L),M=5,21,4) GO TO 2 STOP 23 FORMAT (5(2X,E13.6)) 24 FORMAT (2X, 11 ( 2X, 12)) 25 FORMAT(2X,'THE LEVEL CURVES OF THF TIME FI)NC. FOLLOW') 26 FORMAT (2X,5(2X,E13.6)) 27 FORMAT(2X,D(1), I = 1,6 S',6(2X,12','') 28 FORMAT(2X,'KLAST IS',2X,12) 29 FORMAT (1H,6F13.6) 30 FORMAT (4(2X,E13.6)) END

143 PERFECT FLUID DEFORMATION FIELD COMPUTATION AND PLOT, continued Selected values of the time function, T(I,J), are listed below in the following format: T(5,1),T(9,1),T(13,1),T(17,1),T(21,1) T(5,41),T(9,41),T(13,41),T(17,41),T(21.41) 0.0 0.0 0.0 0.0 0.0 0.299 0.272 0.253 0.242 0.235 0.608 0.543 0.502 0.478 0.465 0.923 0.808 0.744 0.707 0.687 1.226 1.063 0.977 0.928 0.902 1.503 1.306 1.200 1.141 1.109 1.754 1.534 1.413 1.344 1.307 1.981 1.746 1.613 1.538 1.496 2.189 1.944 1.803 1.721 1.677 2.380 2.128 1.980 1.895 1.848 2.555 2.299 2.147 2.059 2.010 2.718 2.459 2.304 2.214 2.163 2.868 2.607 2.451 2.360 2.308 3.008 2.746 2.588 2.497 2.446 3.136 2.874 2.717 2.627 2.576 3.256 2.994 2.838 2.749 2.699 3.366 3.106 2.952 2.865 2.815 3.467 3.210 3.059 2.974 2.926 3.559 3.306 3.160 3.078 3.032 3.643 3.396 3.255 3.178 3.134 3.719 3.481 3.346 3.273 3.231 3.788 3.562 3.434 3.364 3.325 3.854 3.639 3.518 3.453 3.416 3.920 3.716 3.601 3.540 3.505 3.989 3.792 3.682 3.624 3.591 4.060 3.868 3.762 3.708 3.676 4.133 3.944 3.842 3.790 3.760 4.207 4.021 3.921 3.871 3.842 4.282 4.098 4.000 3.952 3.924 4.359 4.175 4.079 4.032 4.005 4.435 4.253 4.158 4.112 4.086 4.513 4.331 4.237 4. 192 4.166 4.590 4.410 4.316 4.271 4.246 4.668 4.488 4.395 4. 51 4.326 4.767 4.566 4.474 4.43()0 4.405 4.825 4.545 4.553 4.509 4.485 4.9Q4 4.724 4.632 4.588 4.564 4.982 4.02 4.711 4.667 4.643 5.061 4.,Ij 4.79() 4.746 4.72? 5. 139 4.96) 4. 69 4.825 4. 0()1 5.218 5.()39 4.947 4.904 4.8S0

144 PERFECT FLUID DEFORMATION FIELD COMPUTATION AND PLOT, continued Selected values of the time function after conversion to integers are listed below in the following format: IT(3,l),IT(5,1), IT(21,l),IT(23,1) IT(3,41),IT(5,41),..,IT(21,41),IT(23,41) 0 0 0 0 o 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 2 2? 2 2 2 1 1 1 1 1 4 3 3 3 3 3 2 2 2 2 2 5 4 4 4 4 3 3 3 3 3 3 6 6 5 5 5 4 4 4 4 4 4 7 7 6 6 5 5 5 5 5 5 5 8 8 7 7 6 6 6 6 6 6 6 9 8 8 7 7 7 7 6 6 6 6 10 9 9 8 8 8 7 7 7 7 7 11 10 9 9 8 8 8 8 8 8 8 11 11 10 9 9 9 9 8 8 8 8 12 11 11 10 10 9 9 9 9 9 9 12 12 11 11 10 10 10 10 10 9 9 13 12 12 11 11 11 10 10 10 10 10 13 13 12 12 11 11 11 11 11 10 10 14 13 13 12 12 11 11 11 11 11 11 14 14 13 13 12 12 12 12 11 11 11 15 14 13 1- 13 12 12 12 12 12 12 15 14 14 13 13 13 13 12 12 12 12 15 15 14 14 13 13 13 13 13 13 13 16 15 14 14 14 13 13 13 13 13 13 16 15 15 14 14 14 14 14 13 13 13 16 15 15 15 14 14 14 14 14 14 14 16 16 15 15 14 14 14 14 14 14 17 16 16 15 15 15 15 15 14 14 14 17 16 16 15 15 15 15 15 15 15 15 17 17 16 16 16 15 15 15 15 15 15 17 17 16 16 16 16 16 16 15 15 15 18 17 17 l 16 16 1h 16 16 16h 16 18 17 11 17 17 17 16 16 16 16 16 18 18 17 1t 17 17 17 16 16 16 16 19 18 iH 1 - 17 17 17 17 17 17 17 19 18 18 18 17 17 17 17 17 17 17 19 19 19 18 18 18 18 17 17 17 17 20 19 19 19 18 18 18 18 18 18 18 20 19 9 19 19 19 18 18 18 18 18 20 20 19 19 19 19 18 1 8 18 18 21 20 20 19 19 19 19 19 19 19 19 21 20 2() 2(0 19 19 1 19 19 19 19 19 21 21 20 20 20 20 19 19 19 19 19

145 PERFECT FLUID DEFORMATION FIELD COMPUTATION AND PLOT, continued The points in the physical plane, z(I,K) = Zl(I,K) + i z2(I,K) are on a time function level curve when K is constant, These points are listed below in the following format: z 2 (5,1), z1(9,1),z2 (9,1),z1(13,1),z2(13,1) Z1(17,1),z2(17,1),Z1(21,1),z2(21,1) z!(5,18),z2(5,18),z1(9,18),z2(9,18),z1(13,18),z2(13,18) z 1 (17,18),z2 (17,18), z 1(21,18), z (2118) -4.127 0.535 -4.110 1.093 -4.100 1.625 -4.100 2.139 -4.095 2.643 -3.900 0.553 -3.869 1.116 -3.847 1.645 -3.841 2.154 -3.831 2.651 -3.675 0.579 -3.626 1.144 -3.592 1.670 -3.577 2.171 -3.563 2.660 -3.451 0.614 -3.380 1.180 -3.332 1.699 -3.308 2.192 -3.289 2.670 -3.226 0.660 -3.129 1.223 -3.066 1.734 -3.033 2.215 -3.009 2.682 -2.994 0.719 -2.872 1.273 -2.793 1.773 -2.751 2.242 -2.721 2.696 -2.753 0.788 -2.607 1.330 -2.511 1.817 -2.459 2.272 -2.423 2.711 -2.501 0.864 -2.330 1.394 -2.219 1. R66 -2.155 2.305 -2.114 2.727 -2.236 0.948 -2.040 1.464 -1.912 1.920 -1.839 2.341 -1.791 2.745 -1.957 1.039 -1.736 1.539 -1.590 1.977 -1.507 2.379 -1.454 2.764 -1.664 1.135 -1.413 1.620 -1.250 2.037 -1.156 2.418 -1.098 2.783 - 1.35 1 1.2 39 - 1. 069 1. 705 -0. 8 8 6 2.099 -0.785?.457 -0.723 2.802 -1.016 1.350 -0.698 1.793 -0.496 2.158 -0.390 2.492 -0.327 2.818 -0.651 1.467 -0.289 1.875 -0.078 2.207 0.026 2.521 0.088 2. H32 -0.237 1.588 0.156 1.933 0.360 2.238 0.459 2.539 0.518 2.840 0.241 1.660 0.613 1.959 O.807 2.254 0.9(11.549 0.9 58 2.845 0.711 1. 67, 1.067 1.96H 1.256 2.261 1.347. S54 1.402 2.848 1.1^9 1.'81 1.518 1.913 1.705 2.265 1.745?.567 1.849 2.849

146 EXPERIMENTAL MODELING COEFFICIENTS COMPUTATION C... THIS PR(OGRAM SOLVES A SYSTFM OF 6 NONLINEAR EtQUATIONS FU)R THF 6 C... MODEL C(IFFICIENTS. REAL#8 X(6),Y(6), Z (6),CI,C), T2,PI COMMON Y,Z,C1 P I=3.14159265 I)0O 1 I=1,5 JJ=4'I+1 CO=(.06H72)*( 1.-(JJ)/25.) T2= ( P I/6 )* I-CO C1 =(6/P )*T2+1 READ (5,2) X(l),X(2),X(3),X(4),X(5),X(6) READ (5,2) Y (l), Y (2),Y(3),Y (4),Y(5),Y (6) READ (5,2) Z(1),Z (2 Z(3), Z(4),Z(5),7 (6) WRITE (6,2) X( 1),X(2), X (3),X(4),X(5),X(6) WRITE (6,2) Y (1),Y(2), Y (3),Y(4),Y( 5), Y (6) WRITE (6,2) Z(1), Z (2),Z(3),Z(4),Z(5), Z (6) N=6 NU M S I C, = 3 MAX I T=4() IPR INT= 1 C... SEF RFFFRENCE 49 FOR SLJHROIJT INE NLSYS. CALL NLSYS (N,NIJMSIG,,MAXIT,IPRINT,X) WRITE (6,3) WRITE (6,2) X( 1),X(2),X(3),X( (4),X(5),X6) WRITE (7,2) X (1 ), X (2)X(3),X(4) X (5),X(6) CONT I NIJF 2 FFORMAT (3(2X,F13.6)/3(2X,F13.6)) 3 FRMAT(X,'THF CALCUtLATFD Cl)FFFIFCCENTS AR' ) N F) C... THIS SUIJHr)tJT IN CU-JNTAINS THE h NC)NL InFAR E!)/ATIODlS To) HF SnLVFI). C.. INTFGFR Cr)NATANT KK WHICH CAN HAVE VALIIFS r)F I THRUUf)JH h C... F[ IERMINFS TH PRFKTINENT F'IU)ATIJIN. StHERf)l)TIN, AUJXFCNr (XX,FF,K6K) REALXH -F,XX(),YY(6),77 (l),C1C;2 C( r ()l YY,ZZ,CI I =KK (, =C l +4 IF ( I.F-(.b. E. I.-F. ) ) If l F=X ( 1 ) / ( XX ( ) - ( ( YY( I ) -XX( ) ) *2 ) +)F XP(-C (; (YY( I )-XX(X) )) )-XX 4) / ( X ( -): (Y (Y( I ) -XX (, )) — ) +f)- XP ( (2; ( YY ( I ) -XX (6 )) ) +17 ()

147 EXPERIMENTAL MODELING COEFFICIENTS COMPUTATION, continued (;( Tl( 2 1 FF=-XX ( 1 ) I( 2* ( YY ( -XX I 3 )*nLO( xx ( 2 ) )*xx ( 2 YY I -xx ( 3 ) * ll*F=-XXP(- (l*(YY( I )-XX(3) )) DLG)/(X X( 2) ),(X(2)- ( ( YY ( I )- XX) fXP(-Cl) ( 2(l )-XX(3) ) ) )*2+XX(4)*(2*(-YY( )-XX () ), )L(,(XX(5) )*XX(5)**( (YY( X X ( 6 ) ) *2 ) +C2*)EXP ( C2-Y ( -XXI6 XX ) ((YY ) ) ) ) / ( I )-XX ( 6 ) ) 2 ) 4FXP(C2 (YY( I )-XX (6) ) )*2+7Z ( I ) 2 EJRETURN -ND

148 EXPERIMENTAL FUNCTION DEFINITION USING A CUBIC SPLINE FIT C... THIS PRH()(GAM DETERMINES A CUJBIC SPLINF THROUGH THE FXTRAPYOLATEO C,... SETS ()F MODEL COEFFICIENTS. IMPLICIT KFAL8 (A-HO- ) INTEGFR ID(6) DIMENSII JN 5(50), A(50,50), Y(50), X(50), B(50), H(50), G0(7,7 ), XX 1(7) 0)0 1 1=1,6 1 ID( I )=7 DO 2 1=2,6 2 READ (5,19) (G[)(I,J),J=1,6) I)0 3 I=2,6 3 WRITF (6,19) (G)(I,J),dJ=1,h) P =3.14159265 XX 1)=0. I)O 4 I=2, 7 4 XX( I )=PI*( 1-1 )/-C( HHH=XX ( 2) / ( XX( 3 )-XX 2 ) ) C... FHE MODFlL ClFFFICIENTS ARE FXTRAPf)LATFO. G(D ( 1) =0 GO ( 1, 2 )=(;D( 2,2) D( 1, 3 ) =( (,)( 2 3 )-Gl) ( 3,3 ) HHH+[) ( 2,3 ) D(, 1,4)=O (;O( 1,5) =0U( 2,5 D( 1,6)= (()( 2, h' )-CD ( 3, 6) )HHH+G(?, 6) (;O ( 7, 1 ) =0 (; 0 7,2)= ([)(.2) (;D ( 7, 3 ) =(,) (, 3 ) G0( 7,4)= 0 (,0( 7,5 ) =-;( 6,5 ) (;[( 7,6 ) =f,, ) 0) 1]7 L=,6 N= 11) I ) WR I T (,2()) N n)0 5 1=1.N M= =I IF ( 11)() I.F ( ) M= 1+1 X(I )=XX(M) NR I f ((,10) (X(I), I =1,r ) IR IT- (h, ) (X( I).I= ) IF ( I I, IF I M. + Y( I )=G ) (, L ) oRIT (, ]) (Y (I I ), I ),Rk k 1 F (, 2)

149 EXPERIMENTAL FUNCTION DEFINITION USING A CUBIC SPLINE FIT, continued C... THE LIFAN F) I-FFFNNCE F() IAII()NS A F J)FTFNMINfE) WITH TWO) ADDITIO)NAL C... CONF)ITI(INS FlING SFT IJN THE PARTICtILAP M()DEL COEFFICIENT'S SFCOND C... )ERIVAIIVEVFS WITH RESPFCT TO THE PATH LINE AT THF POINTS ADJACENT C... TO THE ) IE wALL ANI) CFNTER LINE. DO 7 I=1,MAX 7 H( I )=X( 1+1 )-X( I) ( 1 )=( ( Y (3)-Y2))/H (2)-( Y (2)-Y () )/H( ) / (0. 5 ( H( 1) +H( 2) ) ) (N)=( YN )-Y N-1 ) )/H N-1)-(Y(N-1)-Y (N-2) ) /H ( N-2 ) ) / (.5 H(N-1)+H( IN-?2))) IF (L.EO(.2) b(1)=3*(Y(4)-2-Y(3)+Y(2))/H(3)**2 -)0 ( I=2, MAX A (1, 1)=(0. as(1,2)=1.!)00 9 J=-3,N 19 A( 1,J)=O. MAX2=N-2 I)O 10 J=I,MAX2 10( A(N,J)=O. A(N,MAX)=I. a. (N, N ) =(). f00 15 K=P,MAX K I=K-1 I =K K 2=K+ 1 O,0 15 J=],N IF (J.FO.K1 ) Gl TOI 11 IF (J.FO.K) (;n TO-) 12 IF (J.F(,K2). rK 0 Tft 13 (;O TOJ 14 11 A) I,J)=H( -1)/H I );0 TO() 15 1_ A( I. J)=?2' (H( I )+H (I-1) )/HK ) (0 TO) 1 5 13 A(I,d)=]. (7(GO T) 15 14 A( I, J)=(). 1 6 C)NT I NIJF wRITE (6,.23 0)() 16 I=l1, lh wRITE (I,21) I((I,J).J=1.N) FP S=(). F-1l9 (.... MI NV I nVJFR I S THF lN IFAk I) FFI NElCF NATP I X. CAL i M l 1V (N, FN,A,I)E'iEN) wK I T (6,24 ) )F - FPR M ATV-C o)F TEk,NIF'lF% THE Iltr\JKNf)WN SFCf)nOFD I)FRIVATIVFS [HAT D[)tFINE H I ISP IIl \1F 11 IitG THF- Il)PIIT 1)F MINl\/. A.L LA 1, \/F { h,,.S,!,IN) ^(KO T I ( ) ( I I =1 ) ()/ f -1, T Ir 1qi

150 EXPERIMENTAL FUNCTION DEFINITION USING A CUBIC SPLINE FIT, continued 1~ F 9~MAT (2X.F1 3., )) 2() F()kMAT(1H,,'(,l ISm,?X,T2//'THF X VFLT()l IS'//) 1 F- )KMAT (4 ( 2X. F 1 3.6) ) 2? FOKMAT(HO,'ITHF Y VFCT[fJR I%'/1H ) 23' FORMAT(IH()O,'HE LINEAK I)IFFE-FFNCF MATRIX IS'//) _4 FORMAT( IHO,'F)TER IS',2X,F13.6) 25 FORMAT(1HO, THE SFCONO) I)ERRIVATIVF VFCT(IR IS'//) F N L)

151 EXPERIMENTAL FUNCTION DEFINITION USING A CUBIC SPLINE FIT, continued The following output is needed to plot the spline fitted model coefficient A shown in Figure 48. The x vector contains values of experimentally selected real metal path lines, while the y vector contains values of model coefficient A for these selected path lines. The second derivative vector defines the cubic splines, and similar output is generated for the remaining model coefficients. N IS 7 THE X VECTOR IS 0.0 0.4868390 00 0.1017790 01 0.1548740 01 0.207969D 01 0.261064D 01 0.314159D0 01 THE Y VECTOR IS 0.0 0.1866770 00 0.2637a70 00 0.2833770 00 0.2478060D 00 0.147152D 00 0.0 THE LINEAR DIFFERENCE MATRIX IS 0.0 0. lO1000000 01 0.0 0.0 0.0 0.0 0.0 0.916919D 00 0.383384D 01 0.100000D 01 0.0 0.0 0.0 0.0 0.0 0. 100000D 01 0.400000D 01 01()00000 01 0.0 0.0 0.0 0.0 0.0 0.100000 01 0.4000000 01 0. 100000D 01 0.0 0.0 0.0 0.0 0.0 0.1000000D 01 0.4000001) 01 0.100000D 01 0.0 0.0 0.0 0.0 0.0 0.100000D 01 0.4000000 01 0.1000000 01 0.0 0,.0 0.0 0.0 0,0 0()100000 01 0,0 DETER IS ().513475D 02 THE SECOND DERIVATIVE VECTOR IS -0.8254381) 00 -0.468107D 00 -0.140464D 00 -0.194264D 00 -0.256497() 00 -0.1649400 00 -0.7338270-01

152 MODELED REAL METAL AND PERFECT FLUID VELOCITY HODOGRAPH PLOTS (..,. THIS PRlIGRAM TOGETHER WITH VPLOTSYS COlMPIUTES AND GENERATES THE C... PL!T DEFSCRIPTIONS FOR THF MODFLED RFAL METAL VELOCITIES AND THE C.. PERFECT FLUII) VELOCITIES AS A FUINCTI(IN OF INCREMENTAL STEPS DO)WN C.., THE MODELED REAL METAL PATH LINF,. IMPLICIT REAL*8(A-H) DOOBLE PRFCISION PI,VEL,TH INTEGER ID(6) REAL*8 GC(, 7),XX (6,7),YY(, 7),GX (), DGX (6 C OMPL FX 16 E, F G, SF, Z FF,, GG, HHP,W,, 7 F0,, V I)IMFNSIFON X( 1200), Y(1200), XXX(1200), YYY(I()0) COMMON GC,XX,YY,ID DATA H/O.724/,VEL/1.0/,GG/I(-.70711,.7071) /,HH/(-.70711,-.70711 )/ IF/(1.O,O.O)/,F/(O.O,1.D)/ DO 1 1=1,6 1 ID( I )=7 wRITE (6,9) ( 11) ),I=1],) )0O 2 I=1.6 READ (5,10) IXX(I,,J),,J=1,7) READ (5.10) (YY(I,J),J=1,7) 2 READ) (5,10) (GC(I,J) J=1,7) D O 3 1=1,6 3 WRITE (6,11) (XX(I.J),J=1,7) 1)0 4 I=1,6 4 WRITE (,1R1) (YY(I,J),J=1,7) )0O 5 I=1,6 )5 WRITE (6,11) (GC(I,J).J=1,7) T AAA=-2 TAA=TAAA T 3HH= 3 Th= 13 TSS=8 I =3.1415925) C... I I)ENTIFIFS [HF M)lj)EL-!) RFAL MEFTAL PATH LINF. O) 6 I=3,l1,? (L)-(.()68 /2):( 1.0-I/13. )(26/25. ) C... IC(,IN S oI-lEWrMINF-S HF- APPPI(JPPIATF FlO)FL C(JFFFICIFtN,1C ANI) 1HFIR C... I)R IVAI IVS. (;ALL )(7(,1CiN (A,GX, )GX) (... I 1i)FNT I IS A PSs I T IN IHN A PATH LI J\,F. I() 6 J=1,H1 IB=().4 1421 (.=P I /24

153 MODELED REAL METAL AND PERFECT FLUID VELOCITY HODOGRAPH PLOTS, continued 0)=-6. bo+ ( J-1 ) *C C... KM DETFRMINES THE POSITION OF THE MODELED REAL METAL PATH LINE C... POINT IN THE POTENTIAL PLANE. AA=RM ( A nD, GX ) C... OF DETERMINES THE PERFECT FLUID VELO(CITY. O=(F(SF(CDFXP ( DCMPLXD,AA )/VL ),H, ),VEL,H ) C... PDPRM DETERMINFS THE PARTIAL DERIVATIVES REOIQIRED TO CALCUILATE C... THE MODFLED REAL METAL VELOCITY. CALL PDPRM (D,A,GX,DGX,PDPI,PDP2) C... VF DETERMINES THE MODELED REAL METAL VELOCITY. V=VF ( O, P)P, PDP2 ) O= )C0INJG ( Q ) C... x AND Y ARF THF POSITIONS O(F IHE FRFF VECTnR TIP ASS()CIATED WITH C... MODELFD KREAL METAL VELIJCITIES IN THE ABSOLUTE RFFERFNCF FRAME C...',PL[)TSYS CAN UJSF. X((I-1) H1+J,))=REAL(V) TSS+TAAA Y((I-1)H 1+,J)=AIMA,(V)I*TSS+TBHB C... XXX AND YYY ARE POSITIONS ASSOCIATFF WITH THF PERFFCT FlllUID C... VELOCITIFS IN THE ABSOLt)TE RFFERENCF FRAME *PLOTSYS CAN IJSE. XXX ( I-1)*1+J ) =REAL( )*TSS+TAA 6 YYY((I-t )*H1+J)=AIMAG(Q)*TSS+TB C... THE PLOT DESCRIPTIONS ARE GENFRATE HfRE. CALL PFrNI) (h.0,3.0) CALL PENIM)N (10.0,3.0) CALL PFNIIP (0.0,0.0) 1)() 7 M=3,11,2 7 CALL PLINE (X((M-1)*81+1),Y((M-1)*81+1)+I)81,10,0,0) CALL PFNIJP (6.0, 13.0) CALL PFNf)N (10.0(),13.0) (.ALL PFni IP ( 0.(0,0.0) fO 8 L=3,11,2 8 CAIL PLINE (XXX((L-1)I1+1I),YYY((L-1)*81+1),Ri,1,O,0,0) C ALL PL1 FNI) 9 FrOR'A T (2 2 ))I, I=1,6', (2X, I = I 2,',')) I) -OkYMAT (4(2X,E13.6)) l F)H t4MAT (7(2X,F13. 6) FN[)

154 MODELED REAL METAL DEFORMATION FIELD COMPUTATION AND PLOT C... THIS PRIJGRAM T(IGETHER WITH *PLOTSYS CNMPtITES AND GENERATES THE C... PL)T DESCRIPTIUiNS FOR THE MODELED REAL METAL DEFORMATION FIELDS C... UJSING DASHED LINES. IMPLICIT REAL*H A-H) )0oJtLE PRECISION PI,VFL,T,R RM INTEGER IO(6),IT(25,41) REALH (';C(6,7),XX(6,7),YY(6,7),GX(6),DGX(6),T(25,41 ),)S(25) CDIMPLEX*16 E,F,G,SF,ZF,QF,G,HH,P,W,S,Z(25,41 ),O,VF,ZZ(2),VV(2), IZT(25,41 ),RINT,ZTTVT,VTTV(25,41),VT(25,41),ERM,ENRM,7EAV,FEAV I)IMENS I ()N X(2400), Y(2400), XXX(2400), YYY(2400), XS(25), YS(25), IxXS(25), YYS(25) COMMOUN (;C,XX,YY,ID DATA VFL/1.O/,CGG/(-.70711,.70711)/,HH/(-.70711,-.70711)/,E/( 1.,O. 10) /,F/ ( ().0, 1.0)/ I)O 1 1=1,6 1 I I)(I )=7 WRITE (6,33) (ID(I),I=1,6) 2 IO)t 3 I= 1,6 READ (5,35) (XX(I,J),J=1,7) READ (5,35) (YY(I,J),J=1,7) 3 READ (5,35) (GC(I,J),J=1,7) READ (5,27) H,ADDT READ (5,3H) (S(I), I=1,25) D)O 4 I=1,6 4 WRITE (6,36) (XX(I,J),J=1,7) [)O 5 I=1,6 5 WRITE (6,36) (YY(I,J),J=1,7) DO 6 I=1,6 6 WRITE (6,36) (GC(IJ),J=1,7) WR ITF (6,28) HADDT WRITF (6,37) (F)S(I),I=1,25) PI =3.14159265 C... AD[T I S THE Ff)RFSHo)RTENIING7 CO)RRECT I)N. )DT=(PI/12)'ADnDT C =PI /12 TB =0. 4 1 4 21 1 AAA=2,.(D TKHB= 1 0.50 TSS=1.1013 1RN=2' ( P ITSS+TAAA) SPI=SNGL(PI) C... Pl IS THEF SCALED RADIUIS FOR THE UNDFFFORMED CIRCLES. )=). 15:: ( SP I /1.2 )'TSS/2. At)I AN= 1 lH./SP I C.... I I[)FNTIFIES THF M[)D)LFF) RFAL MFTAL PATH LINE.

155 MODELED REAL METAL DEFORMATION FIELD COMPUTATION AND PLOT, continued DO 7 1=1.25 CO=-(0.001a38)*(I-25) IF (I.EO.1) CO=O.O A=(( I-1)*PI )/24-CO C... DC[)NS DETERMINES THE APPROPRIATE MODFL COEFFICIENTS AND THFIR C... DERIVATIVES. CALL DGCONS (A,GXDGX) ZZ( 1)=0 ZZ (2 )=0 VV( 1 )=0 VV(2) = T (I,1) =0 IT(I, 1) =O C... J IDENTIFIES THE POSITIONS ALONG THE MODFLED REAL METAL PATH LINE. )0 7 J=1,41 H=-( ( I-H)P I )/TH C... DS)(I) IDENTIFIES THE TIME EOIJALS ZEFR(J TIME LINE. )=1)5( I )+ J-1 )i*C C... RM DETERMINES THE POSITION OF THE M0))FLED RFAL MFTAL PATH LINE C... POINT IN THE P)OTENTIAL PLANF. AA=R( A, D,GX ) IF (I.Eo.1.OK.I.F0.25) AA=A C... ZF IDENTIFIES A POSITION (OF A POINT IN THE PHYSICAL PLANF. I( I,J)=ZF (SF (CDEXP DCMPLX( D, AA)/VEL ) H, F),H,h,FF,GG,HH) (... X AND Y AKF Pr)SITII)NS I\ THF A4KSOLIJTT E RFEREnICE FRAME UIJSE) BY C. *.: PL)T SY. ( (I-1 )o41+J )=AIMA( z( I,J) ): TSS+ TAa Y ( ( 1-1 ) - 41+,l ) = FAL ( Z ( I ~ J ) ):: TSS+ THhh IF (I. F(J. 1.. I.EO.25) C,1 I(1 7 C... c(F FEiTErfINFS THF P-PFFC{L FLUID1) VFTLUCITY. ()=U(F ( SE ( C i)F x P ( ICMP L X ( f), a a ) /V E L )I, H F ), \IF I, H ) C... PKRM U)FIFkM I NFS THF PAP)TIAL I)FRIVl)I\/IFS PF(U)IJRFI) TI) (CALCI(LATE (C... IHE M(JI)FLF)) EFAL MEtIAL VFLIUCITY. C(ALL pUI)PKM (Ij,A,CX,)C, X,P)PI,P1)P2) V... (1) I TH- VELUCIITY ASS)CIA'TE) IrI [- H ACH NI)FU)FLfC) PF:A MFITAL (.... a1H L Il'l PIRj\T. V( L, I) -\,/1(d PUP, 1P 2 )P ) /Z( 1 ) =// () /Z (?)=Z ( I,,) vV( V )=VV(2) vv(,)=vv(IJ) VV(/)=V(I,,))

156 MODELED REAL METAL DEFORMATION FIELD COMPUTATION AND PLOT, continued IF (J.EO.1) GO TO 7 C... T(IJ) IS THE VALUE OF THE TIME FU)NCTION ALONG THE MODELED REAL C... METAL PATH LINE, WHICH IS DETERMINED BY SUMMING THE DT'S. T(I,J)=T(I,J-1)+DT(ZZ(1I),ZZ(2),VV )VV(2)) C... T(I,J) IS CONVERTED TO AN INTEGER. ATT=T( I,J)/1)DT TT=SNGL ( ATT) IT(I,J)=IFIX(TT) 7 CONTINUE C... THE MIRROR IMAGE OF THE PATH LINES IS CALCULATED. f)O 8 K=1,24 UDO 8 J=1,41 X((K-1)*41+1025+J)=-X( (K-1)*41+J)+TRN 8 Y((K-1)*41+1025+J)=Y((K-i)*41+j) C... LOOPS 9 AND 10 PLOT THE PATH LINES. DO 9 M=1,25,4 9 CALL PDSHLN (X((M-1)*41+1),Y((M-1)*41+1),41,1,0.08,0) DO 10 L=1,21,4 10 CALL PDSHLN (X((L-1)*41+1026),Y((L-I)*41+1026),41,1,O.08,O) C... LOOPS THROJUGH 14 DETERMINE IF THERE ARE ZERO, ONE )OR MORE TIME C... LINE POINTS bETWEEN TWO PATH LINE POINTS AND ASSURES AN EQUAL C... NUMBER (iF TIME LINES FOR THE DEFORMATION FIELD. THF VELOCITY C... AT EACH TIME LINE PO)INT IS DFTERMINED. 0O0 14 I=2,24 0)O 14 J=2,41 N=J L=I M=J-1 IF (Il.E.24.AND.J.EO.41) GO TO 11 GO TO 12 11 R=IT(I,J) R=R/2. KCH=ITI I,J/2 KLAST= IT(I IJ) IF (R.GT.KCH) KLAST=KLAST-1 12 CONTINIJF IF (IT(L,M).FO. IT(LN)) GO T(1 14 ITT=IT(L,N)-IT(L.M) )DO 13 KL=1,ITT K=IT(L,M)+KL ZTT=RINTIZ(L,N),Z (L,M), T(L,N),T L,M), DDT, K) ZT(I,K)=ZTT VTT=RINT(V(L,N),V(L,M),T(L,N),T(L,M),DDT,K) 13 VT(I,K)=VTT 14 CONTINUJE WRITE (6,34) KLAST C... THE MAJ()R AND MINO)R DIAMETFRS AND THEIR O)RIENTATIONS ARF C... )ETFRMINFI).

157 MODELED REAL METAL DEFORMATION FIELD COMPUTATION AND PLOT, continued DO 17 KR=1,2 DO 17 1=3,23,4 KLL=KLAST-1 DO 17 K=1,KLL,2 BEE=CDABS(VT( I,K)) ERM=VT(I,K)/HEE ENRM=F*ERM ZEBAV=ZT(I+1,K)-ZT(I-1,K) EBAV=ZEHAV/CDABS(ZEBAV) FPEE1=DRFAL(ERM*DCONJG(EBAV)) FPEE=DARCOS(FPEEI) FEE=PI/2-FPEE AEE=DTAN( FEE) AE 1=1/EE**2 AE2=-AEE*AE1 AE3=( EE**2+AEE**2/BEE**2) AE4=-RO**2 AAE1=(AE1+AE3) AAE2=DSORT((AAE1)**2-4.) BE =(AAF 1-AAF2)/2 BE2=(AAE 1+AAE2)/2 oIMM2=DSORT(l./(BE1/(-AE4))) DMN2=DSORT(I./(BE2/(-AE4))) ARG=2.*AE2/(AE1-AE3) DUTHET=(DATAN(ARG)) DH ET A = D(ITHE T/2. DTT=DTAN( DHETA) DNM=AE1+2.*AE2*DTT+AE3*DTT**2 DCHI=-AE4/DNM DMD=DCHL*(1.+DTT**2) DCH=DSR T(DMO) C... THE MAJOR DIAMETER DIRECTION IS CHECKED AND CORRECTED IF NEEDED. IF (SNGL(DCH).NhE.SNGL(DMM2)) DHETA=PI/2.+DHETA DRJT 1=D IMAG( ERM) DRF)T2=DARSIN(DKOTl) ROT=SNGL (DROT2) TTHFT=SPI/2.-(SNGL(DHFTA)+ ROT) XO=AIMAC,(ZT(I,K))*TSS+TAAA YO=REAL(ZT(I,K) )*TSS+THHB XALPHA=TTHFT*RADl AN X AEF=SNGL ( MM2) XBEE=SNGL(D MN2) C... wHEN KR FOQUALS TO, THF SET OF FLLIPSFS IN THE MIRROR IJAGF IS C.. PLO(TTEI). (;n T(o (16,15), Kk 15 XO=-XO+TRN XALPHA=-XALPHA C... TH STRA I N LL I PSFS AF PLOrTTIFD. 16 CALL PFLIPS (XI),YO,XAEE,XHHEE,XALPHA,O.O,36(.O, O) ]? CO17 NT IJF. C... TH- PO.%lilftINS (iF THE TIME LInlF PrOI;IS IN THF AHS()LiJIF PRFFERFNCF C... FRaMF ISFI) HY,PLHITSYS ARF [FF InFO).

158 MODELED REAL METAL DEFORMATION FIELD COMPUTATION AND PLOT, continued DO 18 L=1,KLAST DO 18 M=2,24 XXX((L-1)*23+M-1)=AIMAGIZT(M,L))*TSS+TAAA 18 YYY((L-1)*23+M-1)=REAL(ZT(M,L))*TSS+TBBB C... THE MIRROR IMAGE OF THE TIME LINE POINTS IS DETERMINED. DO 19 L=1,KLAST DO 19 M=2,24 XXX((L-1+KLAST)*23+M-1)=-XXX((L-1)*23+M-1)+TRN 19 YYY((L-1+KLAST)*23+M-1)=YYY((L-1)*23+M-1) C... PLOT DESCRIPTIONS OF THE TIME LINES ARE GENERATED HERE. DO 20 L=2,KLAST,2 20 CALL PDSHLN (XXX((L-1)*23+1),YYY((L-1)*23+1),23,1,0.08,O) DO 21 L=2,KLAST,2 21 CALL PDSHLN (XXX(23*(L-1+KLAST)+1),YYY(23*(L- 1+KLAST)+l),23,1,0.08 1,0) C ~~~ PLOT DESCRIPTIONS OF THE TIME EQUOALS 7ERO LEVEL CURVE ARE C.. GENERATFI) HERE. 00 22 1=2,24 XS(I)=AIMAG(Z(I,1))*TSS+TAAA 22 YS(I)=REAL(Z( I,1) )*TSS+TbH DO 23 1=2,24 XXS(I)=-XS(I)+TRN 23 YYS(I)=YS(I) CALL PDSHLN (XS(2),YS(2),23,lt,O.08,) CALL PDSHLN (XXS(2),YYS(2),231,0O.OHO) CALL PLTFND C.. SELFCTFI) NIJMFRICAL OIUJTPUIT IS PRINTED). I)O 24 J=1,41 24 WRITF (6,29) (T(I,J),I=5,21,4) I)O 25 J=1l,41 25 WRITE (6,30) (IT(IJ),I=2.24) wRITE (6,31) DO 26 L=1,KLAST 26 wRITE (6,32) (ZT(M,L),M=5,21,4) (; TO 2 STO P 21 I nFORMAT (2X,E13.6,2X,F13.6) 28 FOKMAT(1X,/,'H IS',1X,E13.6,/,1X,''AfDT IS',IX,F13.6,/) 29 FORkMAT (5(2X,E13.6)) 3() FORMAT (2X,23(2X,12)) 31 FORMAT(?X,'THF LEVEL CIJRVFS F)F THE TIME FtJNC. FOLLOW') -3? FORMAT (2X,5(2X,E13.6)) 33 Ff)RMAT(2X,'If)(]), 1=1,h IS',6(2X,I2,',')) 34 FORMAT(2X,'KLAST IS',2X,I2) 35 F~nwMAT (4(2X,F13.6)) h FlORMAT ( (2X,F 13.) ) 3/ F[. /ORMAT ( 1H,F'13.) 38 -)R MAT (4(2X,F13. 6)) FNM)

159 MODELED REAL METAL DEFORMATION FIELD COMPUTATION AND PLOT, continued Selected values of the time function, T(I,J), used to determine the modeled real metal deformation field shown in Figure 42 are listed below in the following format: T(5,1),T(9,1),T(13,1),T(17,1),T(21,1) T(5,41),T(9,41),T(13,41),T(17,41),T(21.41) 0.0 0.0 0.0 0.0 0.0 0.271 0.258 0.254 0.248 0.247 0.552 0.518 0.506 0.494 0.490 0.841 0.775 0.7'? 0.734 0.727 1.120 1.023 0.989 0.966 0.95R 1.377 1.259 1.213 1.189 1.180 1.614 1.480 1.424 1.398 1.392 1.834 1.688 1.621 1.593 1.590 2.040 1.885 1.808 1.775 1.773 2.233 2.070 1.985 1.946 1.942 2.413 2.246 2.153 2.107 2.096 2.584 2.413 2.314 2.258 2.237 2.745 2.573 2.466 2.400 2.365 2.897 2.725 2.610 2.531 2.481 3.043 2. 69 2.745 2.652 2.588 3.181 3.003 2.869 2.763 2.690 3.311 3.126 2.983 2. 866 2.789 3.432 3.237 3.089 2.964 2.886 3.541 3.337 3.187 3.058 2.981 3.638 3.428 3.280 3.150 3.075 3.723 3.513 3.368 3.239 3.168 3.799 3.593 3.454 3.326 3.259 3.870 3.670 3.537 3.411 3.348 3.939 3.746 3.618 3.495 3.435 4.009 3.822 3.699 3.579 3.521 4,081 3. 8 9 7 3.778 3.661 3.606 4.154 3.974 3.858 3.743 3.689 4,229 4.050 3.937 3.824 3.772 4.305 4.128 4.016 3.904 3.R53 4.381 4.,205 4.095 3.985 3.934 4.458 4.283 4.174 4.065 4.015 4.535 4.361 4.253 4.144 4.095 4.613 4.439 4.332 4.224 4.175 4.691 4.51A 4.411 4.303 4.255 4. 769 4. 596 4.490 4.382 4. 334 4.848 4.675 4.569 4.462 4.414 4.926 4.753 4.648 4.541 4.493 5.005 4,.32 4.726 4.620 4.572 5. OR 3 4. ) 11 4.805 4. h699 4. h51 5.162 4.99() 4.884 4.77H 4.730 5.241 5.068 4. 963 4. H7 4. ()19

160 MODELED REAL METAL DEFORMATION FIELD COMPUTATION AND PLOT, continued The time function after conversion to integers used to determine the modeled real metal deformation field shown in Figure 42 is listed below in the following format: IT(3,1),IT(5,1),...,IT(21,1),IT(23,1) IT(3,41),IT(5,41),...,IT(21,41),IT(23,41) 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 3 3 3 3 3 3 3 2 2 2 2 4 4 4 4 4 4 3 3 3 3 3 5 5 5 5 4 4 4 4 4 4 4 6 6 6 5 5 5 5 5 5 5 5 7 7 7 6 6 6 6 6 6 6 6 8 8 7 7 7 7 7 7 7 7 7 9 9 8 8 8 8 7 7 7 7 7 10 9 9 9 8 8 8 8 8 8 8 10 10 10 9 9 9 9 9 9 9 9 11 11 10 10 10 9 9 9 9 9 9 12 11 11 11 10 10 10 10 10 10 10 12 12 11 11 11 11 10 10 10 10 10 13 12 12 12 11 11 11 11 11 10 10 13 13 13 12 12 12 11 11 11 11 11 14 13 13 13 12 12 12 12 11 11 11 14 14 13 13 13 12 12 12 12 12 12 15 14 14 13 13 13 13 12 12 12 12 15 15 14 14 13 13 13 13 12 12 12 15 15 14 14 14 14 13 13 13 13 13 16 15 15 14 14 14 14 13 13 13 13 16 15 15 15 14 14 14 14 14 13 13 16 16 15 15 15 14 14 14 14 14 14 16 16 16 15 15 15 15 14 14 14 14 17 16 16 16 15 15 15 15 15 14 14 17 17 16 16 16 15 15 15 15 15 15 17 17 17 1h 16 16 16 15 15 15 15 18 17 17 17 16 16 16 16 16 15 15 18 18 17 17 17 16 16 16 16 16 16 18 18 17 17 17 17 17 16 16 16 16 18 18 18 17 17 17 17 17 16 16 16 19 19 18 18 18 17 17 17 17 17 17 19 19 18 18 18 18 17 17 17 17 17 19 19 19 1 18 18 18 18 17 17 17 20 19 19 19 19 19 18 18 18 18 18 20 20 19 19 19 19 18 18 18 18 18 20 20 20 19 19 19 19 19 18 18 18 21 20 20 20 20 19 19 19 19 19 19 21 21 20 20 20 20 19 19 19 19 19

161 MODELED REAL METAL DEFORMATION FIELD COMPUTATION AND PLOT, continued The points in the physical plane, z(I,K) = zl(I,K) + i z2(I,K) are on a time function level curve when K is a constant, in Figure 42. These points are listed below in the following format: z1 1((59) z,1),1z2(5, 1M ),z2 l,z1(13,1),z2(13,1), z1(17,1),z2(17,1),z1(21,1),z2(21,1) z (5,18) 2(5,18),z(9,18) (9,18),z (13,18), ~i(518 ), 2 (,82'!( 1 2 z2(17,1l) z (17,18),z (21,18),z (21 18) -4.099 0.467 -4.090 1.009 -4.095 1.546 -4. 104 2.081 -4. 107 2.612 -3.846 0.476 -3.835 1.020 -3.843 1.555 -3.853 2.087 -3.858 2.615 -3.596 0.504 -3.580 1.046 -3.590 1.572 -3.601 2.097 -3.607 2.620 -3.349 0.554 -3.325 1.091 -3. 333 1.603 -3.345 2.115 -3.352 2.628 -3.103 0.626 -3.067 1.156 -3.070 1.654 -3. 08 3 2.145 -3.093 2.640 -2.854 0.715 -2.806 1.239 -2.798 1.725 -2.809 2.192 -2.823 2.662 -2.600 0.810 -2.538 1.329 -2.517 1.805 -2.523 2.255 -2.539 2.695 -2.341 0.908 -2.263 1.418 -2.228 1.887 -2.222 2.324 -2.232 2.737 -2.074 1.006 -1.982 1.509 -1.931 1.973 -1.906 2.402 -1.894 2.789 -1.798 1.108 -1.695 1.605 -1.627 2.068 -1.569 2.483 -1.510 2.837 -1.515 1.214 -1.404 1.708 -1.308 2.158 -1. 195 2.535 -1.083 2.846 -1.227 1.325 -1.100 1.809 -0.952 2.215 -0.778 2.538 -0.648 2. 835 -0.933 1.439 -0.759 1.888 -0.550 2.230 -0.340 2.530 -0. 20 2. H32 -0.620 1.548 -0.355 1.929 -0.116 2.234 0.101 2.534 0.208 2.83/ -0.252 1.636 0.097 1.948 0.329 2.244 0. 542.543 0.643 2.H42 0.2(03 1.674 0.558 1.961 0.77P 2.255 0.987 2. %51 1.085 2.846 0.6 /0 1. 1.0138. 1 3 1.98 1.227 2.261 1.433 2.555 1.531 2.848 1.127 1.681 1.464 1.972 1.677 2.264 1.881 2.557 1.978 2.849

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