THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING PLASTIC DEFORMATION OF NONMETALLIC PHASES WITHIN DUCTILE METALS Robert J. Warrick A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan Department of Chemical and Metallurgical Engineering 1963 June, 1963 IP-626

To my wife

ACKNOWLEDGEMENTS It is my pleasure to acknowledge a number of individuals and organizations for their indispensable assistance during the course of this investigation. I am particularly grateful to Professor Lawrence H. Van Vlack who served as Chairman of the Doctoral Committee. He stimulated interest in the topic, provided encouragement throughout the experimental work, and was always available for helpful discussions. Invaluable aid was received from the Doctoral Committee Members. Professor Robert M. Haythornthwaite made many helpful suggestions for test and stress analysis methods. The interest, discerning discussions and advice of Professors James M. Freeman, Maurice J. Sinnott, and Lars Thomassen are appreciated, as are those of Professor Co T. Yang who served on the Doctoral Committee while still at the University of Michigan. I should like to thank the faculty of the Department of Chemical and Metallurgical Engineering who provided helpful discussions and aid during the course of my graduate work and the departmental shop personnel who helped with equipment and supplies. Valuable discussions and assistance with experimental programs were obtained from a number of my fellow graduate students. In particular, I wish to thank Mro John W. Moore, Mr. Robert M, Rusnak, Mr. William E. Sutar, and Mro John Wo Upp for their help with metallographic work and routine measurements. The financial support for the thesis was provided by the National Science Foundation. The Wheelabrator Corporation and the General Electric Company supported part of my graduate work through fellowships. iii

I wish to acknowledge Miss Dolores Gillies, who typed the first draft of the thesis, and the Industry Program of the College of Engineering, which prepared the dissertation in its final form. iv

TABLE OF CONTENTS ACKNOWLEDGEMENTS................................................ LIST OF TABLES................................................... LIST OF FIGURES............................................ ABSTRACT............. o.......................................... CHAPTER I INTRODUCTION........................................... II LITERATURE REVIEW..................................... Ao Deformation of Nonmetallic Materials............... 1. Slip and Fracture Systems...................... 2. Strength of Ionic Solids...................... 3. Plastic Behavior of Ionic Solids............... ao Yield Stress............................... b. Surface Effects............................ 4o Fracture of Ionic Solids...................... a. Surface Effects........................... b. Plastic Flow Effects....................... c. Stress State Effects....................... Bo Deformation of Included Nonmetallic Phases......... C. Hardness and Yield Strength....................... Do Extrusion.......................................... 1o Extrusion Methods.............................. 2. Extrusion Pressure Curves...................... 3. Flow Patterns During Extrusion................ 4. Extrusion Theorieso.......o......oo o..o o o o o ao Homogeneous Deformation................. o o bo Plane Strain Extrusion Without Friction.... c. Plane Strain Extrusion with Friction....... do Slip Line Theories........................ e. Axially Symmetric Extrusion................ f. Stress Determination from Gridded Specimens.........00..................... go Empirical Formulas for Extrusion Pressure.. Page iii ix x xv 1 3 3 5 6 6 6 9 9 10 12 13 14 17 19 20 20 23 25 26 26 28 29 32 34 34 v

TABLE OF CONTENTS (CONT'D) Page 5. Factors Controlling Ram Pressure and Stress States............................... 35 a. Per Cent Reduction........................ 35 bo Die Angle................................ 35 c. Sample Variations........................ 36 d. Temperature.............................. 36 e. Extrusion Speed............................ 37 III EXPERIMENTAL PROCEDURE............................... 38 Ao Sample Preparation............................. 38 1o Extrusion Samples............................. 39 ao Sample Materials.......................... 39 b. Powder Metallurgical Treatment............. 39 c. As-sintered Sample Properties.............. 45 20 Strain Hardening Study Samples............... 45 30 Inclusion Hardness Samples................... 45 4o NaCl Hardness Samples......................., 45 50 Compressive Yield Strength Samples..........0 47 B. Testing Techniqueso............................. 47 lo Method of Inclusion Deformation.o.............o 47 a. Selection of the Deformation Method....... 47 bo Extrusion Equipment........................ 51 2. Metallographic Techniques...................... 54 30 Hardness Measurements......................... 54 a. Room Temperature Measurements............. 54 bo Elevated Temperature Measurements....... 55 40 Yield Strength Determinations................ 58 5o Sample Density Determinations,................. 58 IV ANALYSIS OF INCLUSION DEFORMATION..................... 62 Ao Strain Analysis of Deformation.................... 62 lo L/W Measurements.............................. 62 20 The "Mean L/W Value"......................... 64 vi

TABLE OF CONTENTS (CONT'D) Page 3. L/W Correction for Density.................00 64 4. L/W Correction for As-sintered L/W Ratio....... 68 Bo Stress Analysis of Deformation..................... 70 1. Extrusion Theory Selection.................... 71 2o Application of Shield's Analysis.............. 72 35 Ram Pressure as a Stress Index................. 73 V EXPERIMENTAL RESULTS.......o..o....................... 76 A. Strain Hardening Patterns......................... o 76 B. Flow Patterns o................................... 79 C. Calculated Stress Distribution..................... 82 D. Plastic Deformation and Fracture................. 82 E. Effect of Extrusion Parameters on Inclusion Deformation............................... 89 lo Effect of Die Angle........................... 91 2. Effect of Per Cent Reduction in Area......o... 97 35 Effect of Strain Rate.......................... 97 4, Effect of Temperature.......................... o108 5. Effect of Per Cent Nonmetallic Material........ 116 60 Effect of Nonmetallic Inclusion Size.......... 116 7. Effect of Inclusion Type.. O................... 126 8. Effect of Matrix Type.......................... 132 VI DISCUSSION OF RESULTS o............................... 139 Ao Strain Hardening Patterns.......................... 139 Bo Flow Patterns........................1.......... 141 Co Calculated Stress Distributions.................... 144 Do Effects of Extrusion Parameters.................... 146 lo Die Angle..................................... 146 2o Per Cent Reduction......................... 148 30 Strain Rate....o......................... 149 4o Temperature................................... 150 5o Per Cent Nonmetallic Material.................. 153 60 Nonmetallic Inclusion Size.................... 154 70 Inclusion Type............................... 156 8. Matrix Type.............o o................... 157 Eo Plastic Deformation and Fracture.............. 158 vii

TABLE OF CONTENTS (CONT'D) Page VII CONCLUSIONS..o..........o..........................O.. 166 A. Extrusion......................................... 166 B. Inclusion Deformation....o.oo................... 167 APPENDICES...................................................... 168 A MEASURED VS. TRUE L/W RATIO FOR AN ELLIPSOIDAL INCLUSION.................................... 168 B MODIFICATION OF SHIELD'S STRESS ANALYSIS.............. 171 C EXPERIMENTAL RAM PRESSURE AND "MEAN L/W VALUE" DATA,.. 176 D MATRIX AND INCLUSION HARDNESS VALUESo o................ 177 E HARDNESS VALUE EXTRAPOLATION......................... 178 BIBLIOGRAPHY.......................................... 180 viii

LIST OF TABLES Table Page I Extrusion Sample Types and Deformation Parameters Studied.................................... 40 II Heat Treatment Data for Powder Purification........... 40 III Pressing and Sintering Treatments..................... 44 IV As-Sintered Extrusion Sample Properties............... 46 V Ionic Single Crystal Heat Treatment Data............... 46 VI Extrusion Die Geometries.5............................ 53 ix

LIST OF FIGURES Figure Page 1 Direct and Indirect Extrusion Equipment............... 21 2 Ram Pressure versus Ram Displacement.................. 22 3 Schematic Flow Patterns for Extrusion................ 24 4 Dimensions for Extruding a Cylindrical Body........... 27 5 Slip Line Field for 50 Per Cent Reduction in Thickness and a 180~ Die.................... 31 6 Dimensions for the Axially Symmetric Extrusion of a Cylindrical Body............................... 33 7 The Effect of Pressing Variables on Sample Properties........................................... 42 8 Microstructures of As-Sintered Samples................ 43 9 Pressing Load versus Matrix Hardness................. 48 10 Stress Components Acting on a Material Element for Extrusion, Drawing, and Rolling................... 50 11 Extrusion Equipment.................................. 52 12 Typical Matrix Hardness Traverse...................... 56 13 Strain Hardening Pattern for a 60~ - 75 Per Cent Reduction Die.oo................................... 57 14 Hot Hardness Testing Equipment........................ 59 15 Compression Testing Jig............................... 60 16 Quantities Measured as an Indication of Strain........ 63 17 Typical Cumulative L/W Curve......................... 65 18 Typical L/W Traverse......................... 66 19 Schematic Figures Used for the L/W Correction for As-Sintered L/W Ratio............................ 69 20 Compressive Yield Strength versus Matrix Hardness for 90 Cu - 10 NaCl Samples.......................... 74 x

LIST OF FIGURES (CONT'D) Figure Page 21 Strain Hardening Patterns for 75 Per Cent Reduction Dies o,,,....... o..........................., 77 22 Strain Hardening Patterns............................. 78 23 Microstructures Illustrating Inclusion Flow between the Die Lips,............................. 80 24 Microstructures Illustrating Inclusion Deformation in an Extruded Sample...................o o......... 81 25 Calculated Stress Distribution for a 90 Pb - 10 NaC1 Sample Extruded through a 90~ - 75 Per Cent Reduction Die........................o......................... 83 26 Calculated Stress Distribution for a 90 Cu - 10 NaCl Sample Extruded through a 90~ - 75 Per Cent Reduction D ie o. o o o.. o o o o........................................ o 84 27 Calculated Stress Distribution for a 90 Cu - 10 NaCl Sample Extruded through a 60~ - 75 Per Cent Reduction Dieo o. o oo o... o. e..... o............................... 85 28 Calculated Stress Distribution for a 90 Cu - 10 NaC1 Sample Extruded through a 135~ - 75 Per Cent Reduction Die o..o............. o o..........................o 86 29 Calculated Stress Distribution for a 90 Cu - 10 NaCI Sample Extruded through a 90~ - 75 Per Cent Reduction Die........ o.................... o... o. o. o oo oo 87 30 Microstructures Illustrating Plastically Deformed and Fractured Inclusionso.......o................. o. 88 31 Radiographs Illustrating the Effect of Inclusion Orientation on Inclusion Deformation................. 90 32 L/W Traverses for Die Angle Study Samples.............. 92 33 Inclusion Deformation versus Included Die Angle....... 93 34 Ram Pressure versus Included Die Angle.................. 94 35 Inclusion Deformation - Ram Pressure Relationship for Die Angle Study Samples......................... 95 xi

LIST OF FIGURES (CONT'D) Figure Page 36 Microstructures Illustrating the Effect of Die Angle on Inclusion Deformation o...................... 96 37 L/W Traverses for Per Cent Reduction in Area Study Samples o o o................... o..................... 98 38 Inclusion Deformation versus Per Cent Reduction in Area,.................................................... 99 39 Ram Pressure versus Per Cent Reduction in Area,......... 100 40 Inclusion Deformation - Ram Pressure Relationship for Per Cent Reduction in Area Samples................. 101 41 Microstructures Illustrating the Effect of Per Cent Reduction in Area on Inclusion Deformation.............. 102 42 L/W Traverses for Strain Rate Study Samples............. 103 43 Inclusion Deformation versus Ram Velocity............... 104 44 Ram Pressure versus Ram Velocity...................... 105 45 Inclusion Deformation - Ram Pressure Relationship for Strain Rate Study Samples.......................... 106 46 Microstructures Illustrating the Effect of Strain Rate on Inclusion Deformation.......................o o 107 47 L/W Traverses for Temperature Study Samples............. 109 48 Inclusion Deformation versus Temperature............... 110 49 Ram Pressure versus Temperature...................... 111 50 Matrix Hardness versus Temperature...................... 112 51 Inclusion Hardness versus Temperature................... 113 52 Hot Hardness of Annealed Copper and NaC1.........,..... 114 53 Microstructures Illustrating the Effect of Temperature on Inclusion Deformation......................... 115 54 L/W Traverses for Per Cent Nonmetallic Material Study Samples........................................... 117 xii

LIST OF FIGURES (CONT'D) Figure Page 55 Inclusion Deformation versus Per Cent NaCl............. 118 56 Ram Pressure versus Per Cent NaC1..........,......... 119 57 Inclusion Deformation - Ram Pressure Relationship for Per Cent NaCl Study Samples...................... 120 58 Microstructures Illustrating the Effect of Per Cent NaCl on Inclusion Deformation........................ 121 59 L/W Traverses for Inclusion Size Study Samples......... 122 60 Inclusion Deformation versus Mean Inclusion Diameter.... 123 61 Ram Pressure versus Mean Inclusion Diameter...o........ 124 62 Microstructures Illustrating the Effect of Inclusion Size on Inclusion Deformation. o......................... 125 63 L/W Traverses for Inclusion Type Study Samples.,....... 127 64 Inclusion Deformation versus Inclusion Hardness......... 128 65 Ram Pressure versus Inclusion Hardness................. 129 66 Inclusion Deformation - Ram Pressure Relationship for Inclusion Type Study Samples.....................o o. 130 67 Microstructures Illustrating the Effect of Inclusion Type on Inclusion Deformation o....o................oo.. 131 68 L/W Traverses for Matrix Type Study Samples,.........,. 133 69 Inclusion Deformation versus Matrix Hardness,........... 134 70 Ram Pressure versus Matrix Hardness.................. 135 71 Inclusion Deformation - Ram Pressure Relationship for Matrix Type Study Samples o...................... 136 72 Microstructures Illustrating the Effect of Matrix Type on Inclusion Deformation................ o........o.. 137 73 The Effect of Metal/Inclusion Hardness Ratio on Inclusion Deformation and Type of Inclusion Deformation Behavioro..................................... x...... 160 xiii

LIST OF FIGURES (CONT'D) Figure Page 74 The Effect of Metal/Inclusion Hardness Ratio on Inclusion Deformation for Temperature Study Samples..... 162 75 Section through an Ellipsoidal Inclusion................ 169 xiv

ABSTRACT The plastic deformation of nonmetallic inclusions in a ductile. metal matrix was studied. Primary emphasis was on the effects of extrusion parameters and sample variations on the fracture and flow of the included nonmetallic phase. Secondary emphasis was on strain hardening and flow patterns during extrusion, effect of extrusion parameters and sample variations on extrusion pressure, and properties of two phase materials. The experimental materials included face-centered cubic metals (Pb, Al, Ag, Cu, and brass) and inclusions with the NaCl structure (KC1, NaCl, NaF, and LiF). Samples were prepared using standard powder metallurgical techniques. Extrusion was used as the deformation method since it provides stress states the same as, or similar to, those found in the common metal forming operations, i.e., extrusion, drawing, and rolling. Consequently, the results should apply for these forming operations. After extrusion, the samples were sectioned on a symmetry plane parallel to the extrusion direction. Inclusion length/width ratios were measured in this plane as an index of the amount of inclusion deformation while inclusion and matrix hardnesses were used as an index of the relative properties of the phases. A modification of Shield's stress analysis for plastic flow in a converging connical channel was used to calculate stress distributions. However, accurate stress distributions were not achieved because of an inability to accurately evaluate the stress required to change the direction xv

of material flow at the die entry and exit sections. As a result, ram pressure was used as an index of the stress levels in the samples. The ram pressure required for extrusion can be reduced by (1) use of more effective lubricants, (2) proper deformation equipment design, (3) decreasing the sample reduction, (4) decreasing the extrusion velocity, (5) increasing the extrusion temperature, and (6) decreasing the sample yield strength. NaCl-type ionic inclusions were found to have high ductilities when subjected to the confined stress conditions that accompany extrusion. As the stress levels are increased, three types of inclusion behavior occur: (1) rigid inclusions, (2) brittle-ductile inclusions (fracture and deformation occur), and (3) ductile inclusions. For a given sample, the minimum stress acting on the inclusions is limited by the relative metal/inclusion hardness ratio. The amount of inclusion deformation can be decreased by the following process changes: (1) use of more effective lubricants, (2) proper deformation equipment design, (3) decreasing the sample reduction, (4) decreasing the strain rate, and (5) proper operating temperature selection. The amount of inclusion deformation can be controlled by the following material variations: (1) changing the relative yield strengths of the phases, and (2) changing the relative amounts of the phases. Inclusion size does not have a major effect on the amount of inclusion deformation. xvi

CHAPTER I INTRODUCTION This investigation is concerned with the fracture and plastic deformation of NaCl-type nonmetallic inclusions in a face-centered cubic metal matrix. The topic is of importance since the deformation, or fracture and dispersion, of nonmetallic inclusions during forming operations can have a marked effect on the properties of a metal. In addition, increased usage of two phase materials, such as cermets, has created interest in the way these materials i.eform under load. As a hypothesis, it was suggested that the fracture and plastic deformation characteristics of an included ionic solid would be governed by the stress states acting on the inclusions. These stress states are expected to be influenced by both deformation process variables and sample parameters. For example, as sample reductions or metal/inclusion hardness ratios change, the stress levels acting on the inclusions would be expected to change. This ch.ige in stress levels would be expected to influence inclusion fracture and deformation behavior. A systematic investigation of the effects of process and sample variations was undertaken to determine the parameters affecting fracture and flow characteristics of an included ionic solid. While primary thesis emphasis was on the fracture and flow of an included nonmetallic phase, supplemental information was obtained in the areas of strain hardening and flow patterns during extrusion, effects of extrusion parameters and sample variations on extrusion pressure, and properties of two phase materials. -1

-2 A study of sulfide inclusions in steel ) stimulated the author's interest in this topic. Consequently, experimental materials were selected to match the face-centered cubic structure of austenite and the NaCl structure of MnS. Materials were selected to provide model systems allowing a broad range of sample variations; and to permit study at low deformation temperatures. The extensive literature on the properties of NaCl-type ionic solids makes them advantageous for study. Extrusion was used as the deformation method since it permits the study of a broad range of deformation parameters. In addition, extrusion provides stress states which are the same as, or similar to, those found in common metal forming operations; i.e., extrusion, drawing, and rolling. The effects of the following process and sample variations were studied: 1. 2. 3. 4. 5. 6. 7. 8. Die angle. Per cent reduction in area. Strain rate. Temperature. Per cent nonmetallic material. Nonmetallic inclusion size. Nonmetallic inclusion type. Matrix type,

CHAPTER II LITERATURE REVIEW This literature review presents material which is important to the study and understanding of the thesis topics. Pertinent references will be considered under the following headings: A, Deformation of Nonmetallic Materials, B. Deformation of Included Nonmetallic Materials, C, Hardness and Yield Strength for Two Phase Systems. Do Extrusion. A. Deformation of Nonmetallic Materials Ionic crystals have a special place in the history of crystal plasticity since they were among the first to be investigated systematicallyo Because of their ionic bonds, ceramic materials generally have higher stability, melting temperature, hardness, and resistance to chemical alteration than metals or organic materialso(2) Consequently, they are interesting materials for many engineering applications. However, lack of sufficient ductility limits their application in many caseso Ionic materials were long thought to be inherently brittle. However, in 1867, Reusch(3) studied the deformation of rock salt crystals and reported glide on (110) planes in the [110] directions. In 1903, Coblentz(4) reported that NaC1 crystals can be plastically bent if their surfaces are first wetted and in 1909, Milch(5) reported that sticks of rock salt could be bent very readily after heating. Since these early experiments with NaCl investigators have demonstrated the ductility of -3

-4 many alkali halides, sulfides, and oxides.(6-19) Plastic deformation has been observed in ceramic phases having a wide range of crystal structures: for example, NaCl structure (MnS, MgO, NaCl, KC1, LiF),(6,7) CsC1 structure (CsBr),(7) CaF2 structure (Cu2S,(8) CaF2(9,10)), hexagonal structure (A120l,(11,12) Sio2,(13) CaC03(14)), and diamond cubic structure (diamond). (5) The greatest ductility has been observed in ionic materials having the NaCl structure. Because only ionic materials with the NaCl structure were used in this thesis, and since the preponderance of the literature is devoted to such materials, most of the review will be devoted to NaCl type materials. The deformation characteristics of ionic solids will be considered under the following headings: 1. Slip and Fracture Systems. 2. Strength. 3. Plastic Deformation. 4, Fracture. The fracture stress for ionic materials is generally lower than the yield stress. Consequently, factors which control fracture behavior will be considered in detail. For a more general discussion of the mechanical properties of ionic solids, the reader is referred to the classic work of Schmid and Boas (1936)(16) and the more recent reviews by Laurent (1958),(17) Pask (1961),(18) and Gilman (1961).(19)

lo Slip and Fracture Systems Plastic deformation of ionic crystals occurs by the movement of dislocations through the lattice, The primary slip system for KC1l NaCi, NaF, and LiF is (110) [110] while the secondary slip system is (100) [110]o(20=24) Buerger(22) and Tammann and Salge(25) report slip on the (111) [110] system for NaCl at elevated temperatureso The probability of (100) [110] slip increases with increasing temperature and increasing ion polarizabilityo PbS and PbTe, which have high polarizability, use the (100) [110] glide system at room temperature (22,23526) For all active glide planes, the [110] glide direction requires the least displacement for structure restorationo In addition, like ions do not pass over each other with this glide direction. In many crystals, the slip planes are those of widest spacing (densest packing). These are the (100) planes for NaCl type materialso Huntington, Dickey, and Thompson(27) made detailed calculations of the dislocation energies for glide in NaClo Their calculations show that screw dislocations have a definite preference for (110) planes rather than (100) planeso Their results were not as clear for the preference of (110) planes over (100) planes for edge dislocations. Ionic crystals cleave along (100) planes with secondary cleavage along (110) planeso(l9) For cleavage with no accompanying plastic deformation or step formation, the energy required to form new surface represents the surface energy of the material. Surface energies calculated by Van Zeggeren and Benson(28) using the Born Mayer hard sphere model predict the observed cleavage behavioro

-6 2. Strength of Ionic Solids In a perfect crystal, dislocation nucleation by the action of stress alone would be expected to occur with stresses of approximately G/r4no(19) Searsj(29) Gyulaij(30) and Hulse(31) have shown that "whiskers" of LiFP NaCl, and MgO can withstand very high stresses without the occurance of plastic flowo Stokes(32) has shown that bulk MgO crystals can withstand stresses within 1/10 of their theoretical strengtho He removed "fresh' surface dislocations by chemical polishing and "grown in" dislocation sources (inclusions and precipitate particles) by annealingo 30 Plastic Behavior of Ionic Solids The plastic behavior of ionic solids will be discussed under the following headingso ao Yield Stresso b, Surface Effects, ao Yield Stress-The yield stresses of ionic crystals are determined primarily by the stresses required to nucleate and more dislocations in them, The effect of dislocation source has been demonstrated by Stokes(32) for MgO crystalso "Fresh" dislocations, introduced by cleavage or mechanical contact, move and multiply to generate slip bands at approximately 39000 psio The crystal can be quite ductile under these conditions When the "fresh" dislocations are removed by chemical polishing, the crystals have to depend on "grown-in" dislocations which are considered to be associated with precipitate particles -:r. inclusionso Chemically polished crystals deform elastically for stresses up to > 70,000 psio Crystals which rely on "grown-in" dislocation sources or

-7 homogeneous dislocation nucleation often fracture at, or soon after, yieldo This is probably a result of crack nucleation at intersecting slip bands, (See section A-4-b of this chapter,) When "fresh" dislocations are present in a crystal, the yield stress is determined by the stress required to move dislocations in the crystal. This was demonstrated by Johnston and Gilman(33) who showed that the macroscopic yield stresses of crystals are linearly proportional to the stresses required to cause motion of "fresh" dislocationso Consequently, when fresh sources are present, the yield stress is determined by the resistance of the crystal to dislocation motion, Ender,(34) Schoenfeld,(35) and Metag(36) show the effect of various solute additions on the yield strength of NaClo Luhman and Gorum(37) and Gorum, Luhman, and Pask(38) made the same type of studies for AgCl and MgO respectively. In all cases, solute additions increase the yield strength. The rate of strain hardening was observed to increase with increasing solute additions. Divalent ion additions to NaCl and AgC1 have a much greater effect than monovalent ions which may be explained in part by charged dislocations as discussed later in this section. For a given material and temperature, Phillips(39) reports that an increase in the strain rate results in an increase in the critical resolved shear stresso He reports that the change in the rate of strain hardening with log strain rate [d(da/de)/(d log e)] is independent of temperature for NaCl, LiF, and MgO. Strain hardening is essentially linear for ionic crystals, as can be observed from the stress-strain curves presented by Luhman and Gorum(37) (AgCl), and Gorum, Parker, and Pask(7) (KC1, KBr, MgO)o

-8 In general, a temperature increase would increase the ease of dislocation motion and thus decrease the yield strength. For NaCi, LiF, and MgO, Phillips(39) reports the following relationships c - aoe- (1) where ac is the critical resolved shear stress, ao is a stress constant, K is a constant (for constant strain rate), and T is the absolute temperature. A break in the log ac vSo absolute temperature plot occurs at approximately 300~K for NaCl, 420~K for LiF, and 440~K for MgOo Above these characteristic temperatures, there is a decrease in the rate of change of log ac/T. Pratt(4) points out that dislocations in an ionic crystal should be charged (have an excess of jogs of one sign)o Eshelby, et al.(41) suggest that vacancies formed in an ionic crystal will form a charge cloud around a dislocation which will lock the dislocation in place. They point out that the addition of divalent impurities to NaCl would cause maxima and minima to be expected in the yield strength vso temperature plot. Maxima and minima were observed by Ekstein(42) and Eshelby, et alo(41) The dislocation locking effect resulting from divalent impurities could account for the previously noted greater yield strength increase for divalent vso monovalent solute additions. Westbrook(43) reports that the yield stress is given approximately by the following equation for NaCl type ionic solids Y.S. = Hv/55 (2)

-9 where Y.S. is the yield stress in Kg/mm2 and Hv is the Vickers hardness in Kg/mm2. Gilman(l7) shows that there is a linear relationship between elastic modulus and microhardness for NaCl type ionic solids. The linear yield stress-hardness relationship would be expected since the theoretical stress required to move a dislocation is proportional to the elastic modulus (44) bo Surface Effects-The surface conditions of ionic crystals often limit plastic behavior by causing premature fracture, as will be discussed in the next section. However, surface conditions do not have a strong effect on the fundamental event in plasticity: the mobility of dislocations.(l9) Vaughan and Davisson(45) show that the stress required to move dislocations in a given NaCl crystal is independent of exposure to air, water, oil, and air aging. The surface is important as a source of "fresh" dislocations as pointed out in the previous section and as discussed by Stokes. (32) Because the number of dislocation sources in the surface control the distribution of glide, they also control the rate of strain hardening. As the number of interfering glide bands increases, the rate of strain hardening increases. The influence of the surface on the rate of strain hardening of MgO was demonstrated by Stokes.(32) 4, Fracture of Ionic Solids Fracture of ionic solids often occurs at stress levels lower than those required for appreciable plastic flow. In this section the factors influencing fracture will be considered under the following

-10 headings: a. Surface Effects. bo Plastic Flow Effects. c. Stress State Effectso ao Surface Effects-For all NaCI type ionic solids, ductility is limited by the presence of surface micro-crackso These micro-cracks can be introduced by conventional sample preparation techniques such as cleavage or sawing, grinding, and wet polishing.(46-48) Stokes, Johnston, and Li(47) have recently shown that surface defects can readily be introduced into polished ionic crystals by conventional handling techniqueso Clarke and Sambell(45) report that chemical polishing will remove cleavage micro-cracks and enhance the ductility of MgO crystals 5 to 10 times. Considering the susceptability of ionic crystals to surface damage and the marked effect of this damage on their ductility, it is not surprising that much conflicting data exist in the literatureo Special surface effects have been observed in the water soluble ionic salts such as KC1, KBr, and NaCl.(7) Freshly cleaved and/or water polished samples of these materials are ductileo(49) However, authors(49-51) have reported a loss of ductility with aging in air. The air aging effects are less or nonexistent for LiF and MgO.(7) Murray has shown that air aging raises the ductile-brittle transition temperature for NaCl and MgO.(52) Gorum, Parker, and Pask(7) report that the air embrittlement is caused by oxygen and nitrogen. Aerts and Dekeyser(50) report that nitrogen diffuses into NaCli pins the dislocations, hardens the material, and leads to premature failure. Class, Machlin, and Murray(51) have found that the surface of NaCl reacts with ozone in the atmosphere to form NaCl03.

-11 Three different mechanisms have been proposed for the air embrittlement phenomenon: 1. For a crystal containing flaws, gas absorption will result in a lowering of the critical stress for crack propagation according to the Griffith equation. This would cause a deterioration of the mechanical properties. (47) 2. A surface layer is formed which is difficult for the dislocations to penetrate.(7'55) The dislocations pile up against this surface layer and nucleate a crack. 3. The NaC103 layer forms a brittle coherent surface film which can start cracks in the underlying crystal.(51) A fourth mechanism might also be considered. The gases tend to pin "fresh" dislocations at the surface, decreasing the number of active slip systems. This would lead to fracture at interacting slip planes as discussed in the next section. Stokes, Johnston, and Li(47) have demonstrated that surface precipitates formed on improperly dried NaCl crystals lead to premature failure. They have also shown that air embrittlement is slowed down markedly if the crystals are stored in a desicator. Moisture in the air could cause local salt solution and reprecipitation and thus speed up the embrittlement phenomenon. The role of gases in air embrittlement phenomena is an area deserving more study for clarification. The major effect of chemical polishing is probably the removal of surface defects, such as microcracks.

-12 bo Plastic Flow Effects-Cracks can be nucleated in ionic crystals by the pile-up of dislocations against an obstacle,(54i58) Three types of crack nucleation mechanisms have been observed in MgO. Stokes, Johnston, and Li(54,55) report crack formation associated with the Stroh mechanism.(59) In their compression samples, dislocations were observed to pile up at a kink band and nucleate a crack. Washburn, Gorum, and Parker(56) observed crack nucleation associated with the Cottrell mechanism (60) They observed this type of crack formation in tension samples. Dislocation pile up on two intersecting (110) slip bands leads to crack nucleation along a (100) plane. Keh, Li, and Chou(57) proposed a third type of mechanism to account for (110) crack formation around hardness indentations in LiF and MgO. For their mechanism, an immobile dislocation is formed in a (112) plane having the vector l/2a[110]. Dislocations generated on the two intersecting slip planes (011) and (101) will pile up against this immobile dislocation and initiate a crack in the (110) plane. The significance of these mechanisms and observations is that they show cracks can be nucleated as a result of plastic flow. These crack nuclei can lead to catastrophic failureo Melankholin and Regel(61) studied crack formation and propagation in NaCl. They observed that cracks often stop in regions of local plastic hardening. This observation is substantiated by the later observations of Stokes, Johnston, and Lio (8) Stokes, Johnston, and Li(58) state that the amount of plastic deformation preceding cleavage fracture depends on the density and distribution of slip sources at the onset of plastic flow. Their reasoning

-13 followso A cleavage crack is nucleated at the end of a (110) slip band by the coalescence of edge dislocations, which are held up by an orthogonal intersecting (110) slip bando The crack nucleus propagates parallel to the (110) plane until halted by another (110) slip band parallel to the original (110) slip band. The (110) slip band prevents the crack from propagating further in the [1101 direction, but it is free to develop parallel to the [001] line of intersection. For fracture, the slit transfers to the (100) plane normal to the applied stresso The effectiveness of the adjacent (110) band in halting and stabilizing the crack nucleus is a function of its distance from the [001] line of nucleation. This distance determines the crack velocity when it hits the slip band and the stabilized crack length, which in turn determines its associated stress concentration. If all slip bands are parallel, high plastic deformation can occuro If a high number of intersection slip bands are present, the cracks are stabilized and high ductility occurs. This mechanism effectively explains the high ductility observed for crystals whose surfaces were sprinkled with hard powderso Co Stress State Effects-The ability of an ionic material to be plastically deformed is dependent on the stress state. Bridgman(62) has shown that the introduction of large hydrostatic pressures markedly increases the ductility of metals. Bridgman(63) has also shown that the strength and ductility of NaCl and A1203 are increased by the application of hydrostatic pressureo Hydrostatic pressures plus superimposed shear stresses would also account for the plastic deformation occurring naturally in rock strata. (13,64)

-14 Scheil(8) states that MnS inclusions within a steel deform plastically when the steel is deformed while inclusions fracture in a polished surfaceo He suggests that cleavability is suppressed by imbedding the inclusions in a metal matrix. Pask(18) states that hydrostatic pressures tend to eliminate pure tensile stresses, allowing the development of sufficiently high shear stresses to cause plastic deformation. Bo Deformation of Included Nonmetallic Phases The type, size, shape, and distribution of nonmetallic inclusions can have a marked effect on the properties of a metal. In freemachining steels, for example, sulfur is intentionally added to improve machinabilityo(65) Knowlton(66) and Boulger, et al (67) have observed that steels containing globular sulfide inclusions had higher machinability indiceso Inclusions have a detrimental effect on fatigue properties (68-74) ductility,(75-77) and impact strength (78) Cummings, et al(68) and Ransom(72) show that inclusion size and shape have a marked effect on fatigue properties while Gagnebin(75) and Ogawa, et alo(77) show that globular, randomly distributed inclusions are a requirement for high ductility. The size, shape and distribution of inclusions in metals can be controlled to some extent by forming operations. Pickering(79) suggests that inclusion fracture during deformation is necessary to disperse larger detrimental inclusions into smaller, less harmful particles, Proper selection of deformation parameters which favor spherical sulfide inclusions is important to producers of free-machining steelso

-15 Consequently, parameters affecting inclusion deformation and/or fracture and dispersion during metal forming operations are importanto No systematic study has been made of the effects of deformation parameters on the deformation and fracture of included nonmetallic phases. The material that follows will illustrate the types of studies that have been madeo In 1935, Scheil and Schnell(80) showed qualitatively that different types of inclusions in steel behave differently during deformationo Dehlinger(81) observed the deformation of FeS, NiS, Cu2S, Ag2S, and Cu2Se inclusions in their base metal. The inclusions were found to deform readily. Dehlinger(81) suggested that all sulfides, selenides, and telurides should be expected to deform readily in their base metal. For MnS, Cu2S, and Ag2S in their base metals, Scheil(8) reported the sulfide length/width (L/W) ratio is approximately equal to the square root of the metal L/W ratio. Van Vlack(82) studied the effects of composition and mechanical working on the L/W ratio of inclusions in resulphurized steels. He found that fracture and diminuition in the size of sulfide inclusions becomes pronounced on cold workingo Deoxidizers, such as silicon and aluminum, were found to increase the plasticity of sulfide inclusions during hot rolling. More recently, Van Vlack, et al.(l) studied the effects of steel composition and temperature on the equilibrium shape, and type of sulfide and sulfide-oxide inclusionso MnS deformation was known to increase with increasing silicon content in the steelo(82) A siliceous liquid was found to envelop the MnS inclusions with higher silicon levels in the

-16 steel. It was postulated that this liquid would facilitate the development of hydrostatic pressures on the MnS inclusions, MnS deformation would be facilitated by hydrostatic pressures rather than single shear stresses. Scheil(8) and Pickering(79) studied the effects of inclusion size on deformation. Scheil(8) reports the L/W ratios of Cu2S inclusions in Cu were statistically independent of their size. Pickering(79) states that small inclusions deform less readily than large oneso Pickering also states that larger inclusions fracture earlier. The probability of fracture increases with increasing deformation, Scheil(8) and Pickering(79) studied the effects of temperature on the deformation of inclusions in steel. For a 0.253 S steel, Scheil(8) found no change in sulfide deformation in the temperature range from -80 to 1250~Co However, the scatter in his data was largeo For silicate inclusions, he found low deformation at 700~Co At 900~C, the amount of inclusion deformation approaches that of the steel. Only slight deformation increases were observed above 900~Co Except for a 100~C difference in the brittle-ductile transition, Pickering's(79) silicate curve has the same shape. Pickering(79) found the silicates were brittle below 1000~Co He found that iron oxide inclusions become increasingly plastic as the rolling temperature decreases from 1150 to 800~Co Scheil and Sibert(83) made a statistical study of the extent to which needle-like brittle inclusions, such as FeAl and Si, in Al are broken on rollingo FeAl was found to be more brittle than Si, except at 400~0 where they were about the sameo

-17 Puttick(84) studied the deformation and fracture of pearlite. He observed plastic deformation of cementite lamellae in pearlite. The following sequence of events was found during deformation: lo Fine slip in ferrite near pearlite grainso 2, Large shear deformation of ferrite. 30 Eventual fracture of ferrite and pearlite in regions of heavy deformationo Gurland(85) and Nishimatsu and Gurland(86) studied the deformation of two-phase alloys of tungsten carbide and cobalt. WC (94RA) is considered to be a rigid, brittle inclusion in the Cobalt (42RA) matrix. The samples deform by plastic flow in the matrix, At stress levels approaching those required for sample fracture, fracture of WC grains is observed. In a 50 WC - 50 Co sample, slip lines were observed to pass through a few WC grains. Thermal history can create stress concentrations in the sample. Consequently, slip is observed with lower loads as the WC content increaseso Co Hardness and Yield Strength In extrusion theories, ram pressure is linearly proportional to the average yield strength of the sampleso Thus, the experimental data yield indirect evidence on the effects of sample variations on the average yield strength. A brief discussion of trends which can be predicted from the literature will be presented in this section. Indentation hardness is influenced by yield strength and rate of strain hardening. Hill(87) states that the indentation force per unit area is three times the yield stress for a spherical indentoro

-18 Westbrook(43) has shown a linear hardness-yield strength relationship for face-centered cubic metals and NaCl-type ionic solids. Hardness and yield strength will be considered simultaneously, since a factor which increases hardness will also increase average yield strengtho In general, hardesses and yield strengthsof two-phase materials are structure sensitiveo These properties are influenced by the size, shape, and distribution of the second phaseo Generally, the yield strength would not be expected to follow a simple law of mixtures rule sample = aa + bfb (3) where a is the yield strength and f is the volume fraction of the phase indicated by the subscripts. Edelson and Baldwin(88) have demonstrated that the second phase must be harder than the matrix for hardening to occuro Edelson and Baldwin(88) and Geisler(89) demonstrated that the second phase must be coherent with the matrix for the occurrence of major yield strength increases. Particle strenghtening would not be expected when the included phase is weaker than the matrix, Edelson and Baldwin(88) and Geisler(89) suggest that the yield strength increase is a function of the mean free path between particleso For Cu-Cr and Cu-Fe systems, where the second phase is coherent, the yield strength increases for mean free paths below approximately 12701, For WC-Co alloys, where. the phases are not coherent, the yield strength does not increase significantly until the mean free path is below 0lk(86) For noncoherent phases, particle strengthening would not be expected unless the mean free path was very small.

-19 Koppenaal and Parikh(90) and Keeler(91) show that the stress required to initiate slip in the matrix is largely independent of the amount of a harder, included phase. This stress is the stress required to start dislocation motion in the weakest phaseo Since the phases have different expansion coefficients, thermal history can create residual stress fields in the sample which can modify the applied stress required to initiate yieldo This effect is minimized as the inclusion spacing is increased. Koppenaal and Parikh(90) and Keeler(91) show data where the second phase has a marked effect on the rate of strain hardening of the matrix. As the amount of a hard second phase increases, matrix strain hardening increases rapidlyo Thus, for moderate strains, the yield strength. is influenced significantly by the amount of the second phaseo To date, no general theory exists for predicting the yield strength of two-phase samples for a broad range of sample variations. Do Extrusion The extrusion literature will be considered under the following headings~ 1o Extrusion Methods. 2. Extrusion Pressure Curves. 30 Flow Patterns During Extrusiono 4, Extrusion Theories. 5. Factors Controlling Ram Pressure and Stress Stateo For more general reviews, the reader is referred to Pearson and Parkin's book on extrusion, 92) Bishop's recent review on the theory of extrusion,(93) and Wistreich's review on wire drawing.(94)

-20 1o Extrusion Methods Extrusion can be accomplished by either the direct or the indirect method (Figure 1)o In direct extrusion, the billet is pushed down the chamber and through the die. For indirect extrusion, the billet is stationary while the die moves. Lower ram pressures are required for indirect extrusion since sliding friction between the billet and the chamber wall does not have to be overcome. However, the reduced ram cross-sectional area for indirect extrusion often limits the application of this method. There are a number of extrusion variables which can be readily studied, i.eo 1o Per cent reduction in billet areao 2o Die angleo.o Ram velocityo 4o Temperature of extrusion. 5. Coefficients of friction along the die and chamber wallso 60 Sample typeo 2. Extrusion Pressure Curves Ram pressure-ram displacement curves are shown in Figure 2, The ram displacement at point RP corresponds to the ram contacting the die lipo A hollow conical "pipe" forms at the ram end of the billeto This hollow "pipe" collapses readily when it enters the die resulting in the dip at the end of the ram pressure-ram travel curveo The linear decrease in ram pressure with increasing ram travel represents a decrease in the contact area between the billet and the chamber wall. As

-21 CLOSURE BLOCK DIRECT EXTRUSION INDIRECT EXTRUSION (o) (b) Figure 1. Direct and Indirect Extrusion Equipment.

200 I 180 160 180 W 140 - 135o CL Vl2~~~~ ~ zoo60~~~~ - / /<~~~ ~ /~RP 1j 120 80n,- ~~~~~~~~~900 rcn II00 W 600 80 60 40 20 0 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 RAM DISPLACEMENT, INCHES Figure 2. Ram Pressure versus Ram Displacement. 75 per cent reduction dies, having the indicated die angle, were used. I ro I\) I

-23 - the frictional forces associated with this area decrease, the ram pressure decreaseso 3o Flow Patterns During Extrusion The literature on flow patterns during extrusion is summarized by Pearson and Parkins (92) Sachs and Eisbein(95) show the effects of die cone angle on the flow distribution for tin sampleso The flow patterns were obtained by observing the deformation of a grid inscribed on a plane of symmetry. As the die cone angle decreases, the amount of grid distortion decreases. The amount of grid distortion increases as the sample surface is approached. With small die angles, there is a fairly close approximation to uniform deformation by simple elongation. For large die angles and strain-hardening materials, dead metal zones are observed at the die-container junctiono(96) Under these conditions, the sample forms its own die surfaceo A number of extrusion theories assume that the material in the die flows radially toward the virtual apex of the die (Point 0 in Figure 3(a)), Examination of the pictures of gridded samples presented by Sachs and Eisbein(95) indicate that this is a reasonably good approximation for small die angleso However, for large die angles the lines become curved as shown schematically in Figure 3(b). The use of lubricants decreases the shear distortion near the sample surface (92)

FLOW,-~/a~L I N E LI NES 0~ I / a) Flow pattern assumed by a number of extrusion theories / t/ X FLOW ~/ / ~~~~~LI NES b) Flow pattern for large die angles. This figures i based on results presented by Sachs and Eisbein(95) for gridded tin sample. Figure 3. Schematic Flow Patterns for Extrusion.

-25 4o Extrusion Theories An ideal extrusion theory would allow the prediction of the optimum combination of process variables for minimizing ram pressure and maximizing product qualityo However, a rigorous solution to the apparently simple problem of the extrusion of a cylindrical billet into a cylindrical rod is not known for the process variables, even under highly idealized conditions (93) The extrusion theories presented in the literature are for "simplified" cases of the practical problemso Solutions to actual problems are limited by the complexity of the mathematics involved and in some cases, by knowledge of the metallurgy involved. The solutions to many of these "simplified" problems are difficult to handle. They do represent a real advance toward the understanding of the extrusion process. Most extrusion theories simplify the problem by considering isotropic, incompressible, non-strain hardening materials, They do not consider the elastic stress components present in the material, nor in general, the stress required to change the direction of material flow at the die entrance and exit sectionso Many of the solutions are for the plane strain rather than the more common axially symmetric caseo For plane strain, the stress and velocity equations are hyperbolic equations and can thus be treated by known mathematical methods. The balance of this section will be devoted to various analytical and empirical extrusion theorieso Since ram pressure is used as an indication of stress state in the thesis, ram pressure-process variable relationships will be presented for the various theorieso

-26 ao Homogeneous Deformation-For homogeneous deformation only the work required to change the shape of the material is considered, This derivation does not consider the work required for redundant deformation or for overcoming frictional forces. In 1874, Fink(97) deduced the work required to deform a plastic material from observations on the rolling of metalso Siebel and Fangemeier(98) applied the same approach to calculate the ram pressure (RP) required for extruding a cylindrical rod from a cylindrical billet. They obtained the equation R P= -o ) (4) where ao is the tensile yield strength of the material, Ai is the initial cross-sectional area, and Af is the final cross-sectional area. Observed extrusion pressures are approximately 45 per cent greater than those calculated from Equation (4). bo Plane Strain Extrusion Without Friction-Hoffman and Sachs(99) present a derivation of extrusion stresses for the frictionless case. The stress components are indicated in Figure 4(a) where aR and oa are the principal stresses. It is a plane strain solution. The pertinent equations are 7Q O = O' O ( O(5) c =(i- 1!,^ ( D I (6) RP C- lo=)^ r2 ~ (7)

-27 I N J \ I,^\ \ \ / /.\,/ Df I — oR D I I Ri I I I I I N\ N N N / N / / / / / / a) Plane strain extrusion 9\ without friction. b) Plane strain extrusion with friction. Figure 4. Dimensions for Extruding a Cylindrical Body.

-28 Note that the principal stress components are independent of the distance of an element from the center line. This would not be expected to be the case. Equation (7) is the same as Equation (4). c. Plane Strain Extrusion with Friction-Hoffman and Sachs(99) present a derivation of stress fields for the extrusion of a cylindrical billet with friction between the billet and the die wall. A plane strain analysis is used to determine the stress components shown in Figure 4(b). The resulting equations are - 4L'JtU2%: C 26 (8) CTO B Co ( D+) CT'o 8 L w^ - o UD -e B, (DW _ D) (9)) RP _ i 8 + D 8 (10) where i B (= p is the coefficient of friction, and axf represents the pressure applied to the rod leaving the dieo In general, axf would be set equal to zero and the last term in the Equations (8) through (11) would drop out. Treco(lOO) has recently extended this analysis to allow for friction at the chamber wall and at the die exit. The problem in applying the above equations lies in determining the value of the coefficient of friction. The coefficients of friction

-29 are often empirical constants which are used to make the equations fit experimental data. Treco(100) has shown that a reasonable estimate of the coefficient of friction for his extended analysis will lead to maximum calculated ram pressure values within seven per cent of those obtained experimentally. do Slip Line Theories-Luders lines develop in annealed mild steel which is deformed slightly beyond the yield point. The Luders lines are lines of maximum shear stress along which there is relative movement of the adjacent material. The Luders lines intersect each other at 90~. The slip line deformation theory is based on the development of Luders lines. For a given deformation process, a slip line field is assumed which is composed of the so-called a and a lines which are orthogonal. Stress and velocity equations are then developed for the assumed slip line field. The solutions are developed for plane strain since the mathematics of the situation become too complex when a third strain direction is introducedo The laws governing the properties of slip line fields were developed by Hencky(101) (stress equations) and Geiringer(l02) (velocity equations). Methods of application of slip line theory are discussed by Hill(87) and by Prager and Hodge.(l03) Hill(104) first applied slip line theory to extrusion in 1948. Johnson(105-107) has applied the method to direct extrusion for a variety of die cone angles, reductions, and coefficients of friction. For all slip line solutions, the ram pressure is equal to the average tensile yield strength times a function which considers the die

-30 geometry and frictional conditions, As an example, for 50 per cent reduction in diameter and a square die, Hill(104) proposes the slip line field shown in Figure 5o The ram pressure is given by the following equation RP - k (\0t ) (12) where k = ao/ for the Mises yield condition or ao/2 for the Tresca yield conditiono Purchase and Tupper(l08) have shown that there is good agreement between the experimental and calculated flow pattern for an inscribed square grido The theoretical solution for 50 per cent reduction in thickness and a square die was compared with the flow pattern obtained for gridded lead using a square die and 4667 per cent reduction. Johnson(109) conducted a series of experiments using different materials, reductions, and friction conditions in an examination of extrusion pressureo His conclusions give a fair summary of slip line theory. They are listed below: 1o Lubricated extrusions are accompanied by coefficients of friction which are not smallo 2. In the case of lubricated slip over the die face, which occurs for die semi-angles up to about 600, pressures can be reasonably well predicted if a high value of p. is assumedo 3o With large angle or square dies, pressures will be predicted to better than about 15 per cent by the curves

-51 DEAD METAL DEAD METAL Figure 5. Slip Line Field for 50 Per Cent Reduction in Thickness and a 180~ Die.

-32 associated with the dead metal zone hypothesis and a rough container wallo 4o The method of allowing for strain hardening by using an averaged value of the yield stress is quite satisfactory for practical purposeso 5o Distorted grids, as predicted by the thoery, though not exact certainly provide very good first approximations. Dodeja and Johnson(ll0) suggest that the values k and ao be replaced by the average values k and ao for a strain hardening material. These average values could be obtained from a true stressstrain diagram a0 would represent the average value of true stress over the strain range of interesto In conclusion (4) above, Johnson shows that the average value works quite well for computing ram pressureo However, ao would not yield an accurate value for the stress state at various points in the materialo Stresses near the die exit would be higher than the values calculated using aoo The converse would be true near the entrance section of the dieo e. Axially Symmetric Extrusion-Shield(1ll) presents an analysis for the axisymmetric case of flow through a converging conical channelo This stress analysis method will be considered in detail in the thesiso The directions of the nonzero principal stress components are given in Figure 6. The ram pressure and the principal stress components are given by the following equations for the von Mises yield condition RP=C 90 (13)

-33 - 9ff 0~ Zf zi Figure 6. Dimensions for the Axially Symmetric Extrusion of a Cylindrical Body.

-34 _" -o_ _ (14) " = q- = cop 0(R) - 3 - (15) n = f7 - ^o -i (16) where T is (T = f(c)) obtained from the equation rC'(qJ) C. 7:'t P t a ~ TF ) =0C (17) and (R) -C eR ^ (18) A and C are constants of integration to be determined from the boundary conditions. The mathematical complexity of the "simplified" axisymmetric model considered here is readily apparento f. Stress Determination from Gridded Specimens-Thomsen and coworkers(ll2-116) have experimentally determined stress and velocity fields for extrusion, A grid is inscribed on a plane of symmetry through the sample and the sample is then extruded incrementally. After each deformation increment, the grid shape is photographedo The velocity field is found directly by analysis of the deformed grids while the stress field is determined from the experimentally determined velocity fieldo g. Empirical Formulas for Extrusion Pressure-Johnson(l07) states that ram pressure curves calculated from plane strain theory can be expressed empirically in the form R P = a + b 9e j (19)

-35 where r is the reduction and a and b are constants. Since r-= L_ - (20) Ae Equation (19) can also be stated as PP = + b Ioe (i) (21) Bishop(93) summarizes values of a and b to be used for various frictional conditions and reductions. For lead, Johnson(ll7) found a = 0.8 and b = 1.5. For lead, tin, and super-pure aluminum, he demonstrates that the use of these constants in Equation (17) leads to values of RP/2k which are within seven per cent of experimental RP/2k values. Reductions between 75 and 93.8 per cent were used with square dies. 5. Factors Controlling Ram Pressure and Stress States In this section, the effects of extrusion variables on ram pressure will be considered. A variable which increases the ram pressure will increase the stress levels in the sample. a. Per Cent Reduction-As shown in the previous section, experimental data would be expected to fit Equation (21)o Consequently, a linear relationship would be expected between ram pressure and loge(Ai/Af). bo Die Angle-The flow patterns presented by Sachs and Eisbein(95) show that more uniform deformation is achieved as the ie angle decreases. With more uniform deformation, the amount of redundant work would decrease, and the extrusion pressure would decrease. However, for a given

-36 reduction, as the die angle decreases, the area of contact between the billet and die wall increases. As this area increases, the ram pressure would be expected to increase for a fixed coefficient of friction. For a fixed die angle, the contact area increases with increasing reductions. These counteracting effects explain the die cone angle effect discrepancies which are reported in the literatureo For a fixed reduction, there is some optimum die angle. As the reduction increases, the optimum angle would be expected to increase. Johnson(l06) shows this ram pressure minimum using plane strain theoryo For a reduction of 20 per cent and a coefficient of friction of 0o20, the ram pressure minimum occurs for a die semi-angle of 17~o c. Sample Variations-For the extrusion theories, changes in the sample material are accounted for by yield strength changeso The theories presume a linear relationship between ram pressure and average yield strength. The results obtained with empirical formulas indicate that this is a good first assumptiono d. Temperature-For the various extrusion theories, temperature effects would be accounted for in the yield strength term. As the temperature increases, the yield strength and ram pressure would be expected to decrease. Sehishokin(ll8) presents the following empirical formula for use where temperature is the only variable R P = A e (22) where A and Xf are constants and T is the extrusion temperature This formula would only be expected to be valid for the hot working

-57 temperature range. For Equation (20) to be valid, the following yield strength relationship would have to be valid cO = B e (23) where B and X are constants and T is the temperature. e. Extrusion Speed-Pearson and Parkins(92) present experimental data for the effects of extrusion speed on ram pressure. Cadmium and lead were the experimental materials. The following equations fit the experimental data V = b(RP)9 (24) or Io9 (RP)= ( oeV+ o eC) (25) where V is the extrusion speed and a, b and c are constants. As the extrusion speed increases, the ram pressure would be expected to increase.

CHAPTER III EXPERIMENTAL PROCEDURE discussed discussed described following The following areas were investigated experimentally: 1. The effects of extrusion parameters and sample variations on inclusion deformation. 2. Strain hardening of copper during extrusion. 30 Hardnesses of pure inclusion materialso 4. Matrix and inclusion hardnesses in as-sintered and asextruded samples. 5. Hot hardness of sodium chlorideo 6. Compressive yield strength as a function of matrix hardnesso Experimental procedures used for the various studies will be under the headings: A. Sample Preparationo B. Test Methods. The procedures used for inclusion deformation analysis will be in Chapter IV. The methods used to treat hardness data are in this chapter. A. Sample Preparation Sample preparation techniques will be considered under the headings, which describe the sample type: 1. Extrusion Samples. 2, Strain Hardening Samples 30 Inclusion Hardness Samples. -38

-39 4. NaCl Hardness Samples. 5. Compressive Yield Strength Samples. 1. Extrusion Samples Standard powder metallurgical techniques were used to prepare the extrusion samples. a. Sample Materials-Face-centered cubic metals (Pb, Al, Ag, Cu, and 65-35 brass) were used as matrix materials and ionic solids having the NaCl structure (KC1, NaCl, LiF, and NaF) were used as inclusions. The sample types and deformation parameters studied are listed in Table I. High purity aluminum and brass powders were used while all other materials were reagent grade. The as-received KC1, NaF, and LiF powders were too fine for inclusion use. To produce coarser particles, each inclusion material was held at a temperature 50~F above its melting point for 1/2 hour in graphite crucibles and then furnace cooled. This procedure produced coarse-grained ingots having a grain diameter of approximately 1/4 to 3/8 inch. Each ingot was crushed and screened to produce inclusions in the desired size range. b. Powder Metallurgical Treatment-Granular silver and lead shot, having a size range of -40 +120 mesh, were used. All other metals were -120 mesh. The nonmetallic inclusions were -60 +80 mesh for all but the particle size effect samples. Surface oxides were removed from the lead, copper, and silver by a hydrogen anneal. The temperature and times used are listed in Table II.

-40 TABLE I EXTRUSION SAMPLE TYPES AND DEFORMATION PARAMETERS STUDIED - - - Type Sample Cu - NaCl Ag - NaCl Al - NaCl Brass - NaCl Pb - NaCl Cu Cu - KC1 Cu - LiF Cu - NaF NaCI Deformation Parameters Studied Die cone angle Per cent reduction in area Per cent nonmetallic material Inclusion particle size Strain rate Temperature Type of nonmetallic material Matrix type Matrix type Matrix type Matrix type Matrix type Temperature Per cent nonmetallic material Type of nonmetallic material Type of nonmetallic material Type of nonmetallic material Per cent nonmetallic material Temperature TABLE II HEAT TREATMENT DATA FOR POWDER PURIFICATION - =i= -- - - = - = = — = —- ,~r- -= -- - = —- - -L3 Metal Ag Cu Pb Temperature, ~F 1000 1000 500 Time, hrs. 1/2' 1/2 1/2

-41 A magnetic device was used so that the metals could be inserted in the furnace, annealed, and cooled to room temperature without being removed from the hydrogen atmosphereo The brass and aluminum powders were not annealed in hydrogen since the H2/H20 ratio was not high enough to remove the oxide layer from aluminum and brass powders dezincify during annealingo The large metal-nonmetallic phase density difference leads to phase separation during mechanical mixing. To avoid this problem, hand mixing, which results in more uniform phase distribution, was used. The samples were pressed in a single action, 3/8 inch diameter die. After the initial press, the samples were repressed from the opposite endo The two step pressing operation maximizes sample densification (Figure 7(a)) and yields samples which shrink uniformly during sinteringo The nonmetallic inclusions were flattened during the pressing operation, as shown in Figures 7(b) and 80 The inclusion length/width (L/W) ratio was used as an index of the amount of flattening where the length is measured in the direction of load application. Ideally, the L/W ratio would equal one in the unextruded sampleo The effect of pressing load on sample density is illustrated in Figure 7(c)o Above some optimum load, the samples grow during sinteringo The pressing loads listed in Table III were selected to minimize inclusion deformation and maximize sample densification. The samples were pressed at the same load prior to each sinter. The maximum sintering temperature is limited by the melting point of the nonmetallic phase for all but the Pb-NaCl and Al-NaC1

-42 I 8C 80 z w 0o _J o - 60 w O 0 I o 4C z Ur 0 aw 20 1 -- PRESSED FROM ONE END -- PRESSED FROM BOTH ENDS 2000 4000 6000 8000 PRESSING LOAD, POUNDS 10000 12000 14000 a) Effect of density. pressing method on sample Copper samples were used. 0.81 0 3. I 0.6 I I I I I I I I I I 0.4 0.21 b) 0 2000 4000 6000 8000 10000 12000 PRESSING LOAD, LBS. Pressing load vs. inclusion L/W ratio. Cu Per Cent NaCl samples were used. A single ing and sintering operation was used. 10 0, I I II I I - 10 press - z 0 0o I O z -) w a. PRESSING LOAD X 10 3 POUNDS c) Pressing load vs. per cent of theoretical density. Copper samples were used. Figure 7. The Effect of Pressing Variables on Sample Properties.

-435 p.. 11111 *I _m U. I I I9 Im zr ~~~~~~~~~~~~~'I( 1 S madaL a) Sample: 90 Pb - 10 NaC. L/W Ratio: 0.937. r r A' O. <'.; b) Sample: 90 Al - 10 NaCl. L/W Ratio: 0.714. __ 3+~_:LC, a _~-e~-~ _ 1p 4 -U a,4~. I. 4 _.,6 NW ammumw _.. c~~l~'-~4 4 4 c) Sample: 90 Cu - 10 LiF. L/W Ratio: 0.620. d) Sample: 90 L/W Ratio: Cu - 10 NaCl. 0.536. Figure 8. Microstructures of As-Sintered Samples. 37.5X.

-44 TABLE III PRESSING AND SINTERING TREATMENTS Sample Type Pressing Load, lbs* Sintering Temp. ~F(+10~F) Ag - NaCl 4,000 1150 Al - NaCl 4,000 1150 Brass - NaCl 6,000 1430 Cu 6,000 1430 Cu - KC1 6,000 1380 Cu - LiF 6,000 1430 Cu - NaCl 6,000 1430 Cu - NaF 6,000 1430 NaCl** 6,000 1430 Pb - NaC1 4,000575......................5.5 * Load on a 07375 inch diameter ramo * The NaCl sample received a single pressing and six hr. sintering treatment, samples since liquid inclusions were found to sweat from the samples. Since densification is controlled by the sintering of the metallic phase, the sintering temperatures were generally lower than the ideal sintering. temperature, Pure coppery for example, was found to sinter more rapidly at 1650~F than at 1430~Fo Three pressing and sintering operations were used for each sample. With the temperature and load limitations, three pressing and sintering operations were necessary to obtain satisfactory densificationo

The times for the three sinters were six hours, three hours, and three hours respectively. The sintering temperatures are listed in Table III. The samples were sintered and cooled in a hydrogen atmosphere. c. As-Sintered Sample Properties-The densities andinclusion L/W ratiosfor the various as-sintered sample types are presented in Table IVo 2, Strain Hardening Study Samples Strain hardening patterns were determined on annealed, high purity copper. The copper rods were machined to be a slip fit in the extrusion assembly and the sample length was approximately 0. 4 inches. The samples were annealed in hydrogen for one hour at 1430~F (+100) before and after machining. 3o Inclusion Hardness Samples Single crystals of KC1, NaF, and LiF were used for hardness samples. The crystals were cleaved along (100) crystal faces from the furnace-cooled melts described in section A-l-a of this chapter. After cleaving, the single crystals were annealed in hydrogen for the temperatures and times listed in Table Vo 4o NaCl Hardness Samples Room and elevated temperature NaC1 hardness measurements were made on optical quality single crystals purchased from the Harshaw Chemical Companyo Prior to testing, the crystals were annealed in hydrogen for the temperatures and times listed in Table V.

-46 TABLE IV AS-SINTERED EXTRUSION SAMPLE PROPERTIES Per cent of Sample Type Theoretical Density L/W Ratio Brass - NaC 90.8 0,55533 Cu - NaCl 94.7 05536 Cu - NaF 92.9 0o620 Cu - LiF 93o3 0o666 Cu - KC1 9308 0o461 Al - NaCl 96 5 0.714 Ag - NaCl 9609 o 745 Pb - NaCl 100.0 0.937 Cu 948 NaCl 9301 TABLE V IONIC SINGLE CRYSTAL HEAT TREATMENT DATA Annealing Temperature Annealing Time Sample ~F (+10OF) Hrs. KC1 1380 2 NaCl 1430 2 NaF 1430 2 LiF 1430 2

5o Compressive Yield Strength Samples Compressive yield strength was determined as a function of matrix hardness using 90 Cu - 10 NaCl extrusion type samples (Section A-1 of this chapter). The samples were hardened after the final sinter by repressing the sample from both ends in the original pressing dieo The pressing load-sample hardness relationship is shown in Figure 9. following B. Testing Techniques The experimental testing procedure will be described under the headings: 1. Method of Inclusion Deformationo 20 Metallographic Techniques, 53, Hardness Measurements. 40 Yield Strength Determinationso 5. Sample Density Determinationso 1L Method of Inclusion Deformation a, Selection of the Deformation Method~Plastic flow in a deformation study can either be "constrained" or "unconstrainedo " "Unconstained" plastic flow occurs with methods such as tensile testing under hydrostatic pressure, as used by Bridgman; (62) or tube testing under combined tension and, torsion, as used by Taylor and Quinney.(119) "Unconstrained" test methods are useful for determining the yield strength of a materialo However9 continuing changes in sample geometry make these method s unsuitable for deformations involving large plastic strainso "Constrained" flow tests, such as extrusion, involve high frictional forces which are often difficult to evaluateo However, they

-48 I QC) C) LUI x 20000 RAM LOAD, POUNDS Figure 9. Pressing Load versus Matrix Hardness. 90 Cu - 10 NaCl samples were used.

provide constant strain patterns throughout the test. The latter factor makes "constrained" methods more suitable for tests involving large plastic strains. Extrusion was selected as the test methodo A number of factors influenced this choice: 1. Extrusion theory is better developed than rolling theory. 2. Ram pressures can be easily measured. 3. Dies can be changed to provide a wide range of stress distributions in the sample. 4o One die can be used for a number of testso Consequently, the effect of individual parameters, such as temperature and sample type, can be readily determined. 5o Extrusion provides stress distributions the same as, or similar to, those found in the metal forming processes of major interest; i.e., extrusion, drawing, and rolling. The stresses acting on a material element are somewhat similar for extrusion, drawing, and rolling. The similarities are great enough that inclusion deformation results for extrusion would be expected to apply to the other forming operationso Consider the stresses on the elements shown in Figure 10o The roll shape can be considered as a conical die with a continually turning tangento For sheets with a width to thickness ratio greater than 10/1, the sheet does not deform significantly in the Z directiono Consequently, the stress ao would be compressive in the interior of the sheeto The constraint offered by the rolls would make a compressiveo For extrusion and drawing, the die constraint would make the stresses ac and a0

-50 - /m__ t a) Extrusion and drawing b) Rolling Figure 10. Stress Components Acting on a Material Element for Extrusion, Drawing, and Rolling.

-51 compressiveo As a result, a material element in rolling, drawing, and extrusion is subjected to compressive stresses in two directions. For all three processes, aR is compressive for some values of R and tensile for otherso The tensile and compressive regions depend on the nature of the process and the use of front and back tensions. For extrusion, aR is primarily compressive while the converse is true for drawingo The presence of tensile and compressive values of aR in rolling can be demonstrated by the following analysis. The sheet velocity increases during rolling since the area is being reduced. At some point (neutral point), the roll velocity equals the sheet velocity. From the roll entrance to the neutral point, the roll travels faster than the sheeto This velocity difference introduces tensile stress in the R directiono Between the neutral point and the exit, the rolls move slower than the sheet, introducing compressive forces in the R directiono bo Extrusion Equipment-A series of extrusion dies were used to permit studies of the effects of per cent reduction in area and included die angleo The geometries of these dies are listed in Table VIo Extrusions were run at subzero, room, and elevated temperatures. The basic extrusion equipment shown in Figure 11 was used for all runs, For subzero runs, the extrusion equipment was immersed in a dry icealcohol bath, The extrusion temperature was assumed to be the sublimation temperature of dry ice (-108.6~F)o For elevated temperature tests, the experimental equipment was inserted in a tube furnace with chromelalumel thermocouples placed at the locations indicated in Figure 11.

-52 -RAM EXTENSION:ONTAINER ASBESTOS (ELEVATED TEMPERATURES ONLY) IASE Figure 11. Extrusion Equipment.

-53 The reported extrusion temperature is the average temperature for the two internal couples. TABLE EXTRUSION DIE VI GEOMETRIES -- Die No. 1 2 3 4 5 6 7 Initial Diameter, (inches) 0.375 0.375 0.375 0.375 0.375 0.375 0.375 Final Diameter, (inches) 0.1885 0.1930 0.1885 0.1887 0.3118 0.2500 0.1276 Actual Reduction in Area, 74.73 73 51 74.73 74.68 30.87 55,55 88.45 Nominal Reduction in Area, 75.00 75~00 75.00 75.00 30.56 55.55 88.89 Included Die Angle (degrees) 60 90 135 180 90 90 90 A Baldwin-Southwark tensile machine (60,000 lb. maximum load) was used in compression to extrude the samples. Prior to each run, the table speed (and thus the ram velocity) was calibrated. A ram velocity of 0.10 inches per minute was used for all but the strain rate study samples. A Baldwin-Microformer-Recorder was used to record ram load as a function of time. The ram load-time curve and the ram velocity calibration were used to construct ram pressure-ram displacement curves, All die parts in contact with the sample were coated with a stearic acid-graphite lubricant. Ether was used as the lubricant vehicle.

-54 20 Metallographic Techniques Data measurements were made along a symmetry plane parallel to the extrusion direction, The final polished section was less than + 0.0015 inches from the centerline of the extrusion. The tolerance was generally less than + 0,0005 incheso The inclusion materials are highly water solubleo Consequently, ethyl alcohol was used instead of water for wet grinding all samples containing inclusionso With this exception, standard metallographic techniques were usedo Diamond paste was used for the final polishing. 35 Hardness Measurements a, Room Temperature Measurements-A standard 136~ diamond pyramid indentor was used with a Tukon microhardness testing instrument for all room temperature hardness measurements, A 10 gram load was used for inclusion hardness measurements in extruded samples while a 500 gram load was used for all other room temperature measurementso Hardness values obtained with a 10 gram load were 1/3 less than those obtained with a 500 gram load for the range from 10 to 210 DPH, Since the 500 gram measurements check with the standard hardness block supplied with the Tukon tester, all 10 gram DPH measurements were corrected. by a factor of 1o5o Inclusion hardnesses in extruded samples were measured within 0.25 mm of the sample centerline, Annealed inclusion hardnesses were measured on (100) faces of single crystals. In both cases, the reported results are the average of six measurementso

-55 To determine matrix hardnesses, three traverses were made perpendicular to the sample centerline. Hardness readings were taken in mirror image positions across the sample centerline with 0o50 mm or less between each indentation in the traverseo The average of the six hardness values from equivalent positions was plotted as a function of sample position (Figure 12). When a single hardness value is reported for a sample, it is the average of the hardness values across the sample. Matrix/ inclusion hardness ratios were calculated for hardnesses at the center of the samples. Strain hardening patterns were determined on partially extruded copper sampleso Hardness traverses were run perpendicular to the extrusion directiono These hardness traverses were never more than 1 mm apart and in regions where the hardness values were changing rapidly, they were 0o5 mm or less apart. The same is true for the distance between indentations in each traverseo The data were made symmetric by averaging the mirror image hardness values in each traverseo Hardness vso position plots were drawn for lines parallel and perpendicular to the extrusion directiono From these plots, lines of constant hardness were drawn as shown in Figure 13o The experimental constant hardness points are shown in the right half of the figure while smooth curves for the constant hardness profiles are shown on the left. Since the limit of accuracy for the hardness values is approximately two per cent, the curves shown in the right half of the figure are not justifiedo bo Elevated Temperature Measurements-Elevated temperature hardness measurements were made on (100) faces of NaCl single crystals. The

-56 140..... I.............I'I I 0 120 - - 1~~~ — l00o 1I a. 0 cn LLJ z Qa: Q cl I 2 80 60 40 20 0 0 SURFACE 0.2 0.4 0.6 0.8 1.0 CENTER NORMALIZED POSITION Figure 12. Typical Matrix Hardness Traverse.

-57 Figure 13. Strain Hardening Pattern for a 60~ - 75 Per Cent Reduction Die.

-58 hot hardness device shown in Figure 14 was used with a Vickers hardness testero The applied load was calibrated by comparing room temperature hardness indentations made with the hot hardness arrangement with those made on the same NaCi crystal with the Tukon testero A 500 gram load was used with the Tukon hardness machine to obtain the calibration factoro The temperature control was + 10~Fo The tests were run under a positive helium pressure equivalent to lo5 inches of mercuryo The indentations were measured at room temperature using the Filar eyepiece on the Tukon testero 4, Yield Strength Determinations The tensile machine and load recording system used for the extrusion studies (Section B-l-b of this chapter) were also used to determine compressive yield strength datao Parallel pressing surfaces were insured by the use of the compression jig shown in Figure 15o A stearic acid and graphite lubricant was applied to the ends of the sampleo The reported yield strengths were determined by the 0,1 per cent offset methodo 5~ Sample Dns Deteminations Sample density calculations were based on the weight and dimensions of as sintered samples which were weighed to + 0o0005 grams and measured to + 0o0002 incheso Ailowing for scatter between samples. the minimum accuracy for the per cent of theoretical density figures is + 05 per cent, The following formula was used to calculate the theoretical density of two phase samples 1gT X FM 7, -I^ (26)

RADIATION SHIELD SAMPLE POSITIONING SCREW 1360 DIAMOND INDENTER:SISTANCE WOUND FURNACE )ILS:FRACTORY TUBE -OUTLET TO VACUUM PUMP AND INERT GAS SUPPLY Figure 14. Hot Hardness Testing Equipment.

-60 Figure 15. Compression Testing Jig.

-61where PT is the theoretical density, PM is the metal density, PI is the inclusion density, fM is the metal volume fraction, and fI is the inclusion volume fraction.

CHAPTER IV ANALYSIS OF INCLUSION DEFORMATION The techniques used for the analysis of inclusion deformation are sufficiently involved to merit a separate chapter. They will be described under the following headings: (A) Strain Analysis of Deformation and (B) Stress Analysis of Deformation. Ao Strain Analysis of Deformation Analytic techniques were used for strain measurements will be considered under the headings: 1. L/W Measurements. 2. The'"ean L/W Value " 3. L/W Correction for Density. 4o L/W Correction for As-sintered L/W Ratio. 1o L/W Measurements Strain measurements were made along a symmetry plane parallel to the extrusion direction, The inclusion length/width ratios are used as a measure of the strain (Figure 16). If the inclusions are ellipsoidal, the L/W ratio in the plane of measurement represents the true inclusion L/W ratio as demonstrated in Appendix A. Inclusion L/W ratios were measured using the Filar eyepiece of a Tukon hardness testero The micrometer dials on the Tukon are used to determine the position of each inclusion in the sample. The measured L/W ratio is accurate to within + 1% of the actual L/W value in the polished surface, To eliminate end effects, inclusions were not measured near either end of the sampleo -62

Figure 16. Quantities Measured as an Indication of Strain.

-64 The extruded inclusions were measured in a series of longitudinal bands 0.30 mmo wide with the exception of the 30.9% reduction sample. Band widths of 050 mm were used for this sample and all as-sintered samples. To make the data symmetric, bands equidistant from the sample centerline were considered as pairs. An equal number of L/W values (10-15) was taken from each band in the pair. These L/W values are presented in a cumulative plot, as shown in Figure 17 and the 50% point of the cumulative curve is taken as the L/W ratio of the band pairo The L/W ratio of each band pair is plotted as a function of normalized position across the extruded sample, as shown in Figure 18. 2o The "Mean L/W Value" To illustrate the effects of various parameters on inclusion deformation, a single strain figure would be desirable for each sample. The "mean L/W value" was calculated for this purpose where the "mean L/W value" of a sample is given by the quantity At(L/W')A, (27) At where (L/W)i is the L/W ratio of band i, Ai is the cross-sectional area of band in and At is the total cross-sectional area of the sample. This quantity serves as a relative index for the amount of nonmetallic deformation occurring in a sampleo 3. L/W Correction for Density Most of the samples have less than theoretical density prior to extrusion. With the high stresses developed, some densification would be expected during extrusion and if it occurs, the resulting L/W value

-65 z L) _40 Iol 0 10 20 30 40 MEASURED INCLUSION L/W RATIO MEASURED INCLUSION L/W RATIO Figure 17. Typical Cumulative L/W Curve.

-66 30 25 20 0 3 5 -IJ 10 - - 10 5 0 -I I I I0 0.2 0.4 0.6 0.8 1.0 SURFACE CENTER NORMALIZED POSITION Figure 18. Typical L/W Traverse.

-67 for the bulk sample will be less than the L/W value for a sample having theoretical densityo The initial and final sample weights (Wt) will be the sameo Thus, (W+t) = (Wt), (28) where the subscripts refer to the states before (o) and after (f) extrusiono In terms of density (p), sample length (L) and diameter (W), Equation (28) becomes Lw o o = L W; = (29) or, solving for Lf, L = Lo W~ p (30) For no density change (po = Pf), the final sample length (Lf) would be L Lo \A/_ (31) For the bulk sample., Wf is fixed by the die opening. Consequently, dividing Equation (31) by Equation (30) we have (32) L~ po The corrected L/W value (LU/Wf) given by JLIf' L_ 4 Cf- ^ ) ((33) where (Lf/Wf) is the measured L/W value. The maximum L/W correction for densification would result when Pf is the theoretical density (PT) Consequently, the maximum correction

-68 for the bulk sample is given by /L o n itU2r OVL _)PT (34) (W PCoR~fCTED E )(W E ESU O The correction for the bulk sample is assumed to apply to the inclusionso Since theoretical density would not be expected during extrusion, Equation (3p1) probably yields the maximum correction neededo Unless specifically indicated, this correction factor will not be applied to the data presented in the thesiso Maximum corrected L/W values are less than 10% greater than the measured valueso 4o L/W Correction for As-sintered L/W Ratio The inclusions in the as-sintered samples have a L/W ratio less than one and should thus be corrected for an equivalent original L/W ratio of one~ Consider the model shown in Figure 19(a)o The shaded area in the as-sintered inclusion has a L/W ratio of one,while the extruded L/W ratio for the shaded area is Lf/6o The quantity Lf/b is the corrected L/W ratio. The ratio of the width of the shaded area to the total inclusion width (Lo/Wo) would be the same before and after extrusiono Therefore, b is given by c VJ ( Lfo) (35) Wo and the corrected L/W ratio is L (36f @\A )corRRcC-rr U < (36)'4/v

-69 Lf Before extrusion After extrusion a) Inclusion deformation near the center of the sample. Lf Wo Before extrusion After extrusion b) Inclusion deformation near the surface of the sample. Figure 19. Schematic Figures Used for the L/W Correction for As-Sintered L/W Ratio.

-70 The corrected L/W ratio is simply the measured L/W ratio for the extruded sample divided by the L/W ratio for the as-sintered sample. Sachs and Eisbein's(95) results for grid distortion during extrusion indicate that Equation (36) would give a valid correction for inclusions near the center of the sample. As the sample surface is approached, however, an originally rectangular region is deformed as shown in Figure 19(b), Higher shear stresses near the sample surface cause the change in the flow pattern. For the case shown in Figure 19(b), the corrected L/W ratio is P/& where 6 is given by Equation (35)~ Since P < Lf, the corrected L/W value for this case is given by (\ < (\(3 WV CoRREC-ED S o (Lo Since Lo/Wo is less than one, the correction given by Equation (36) is too large as the sample surface is approached, In general, measured L/W values are presented in this thesis. The measured data accurately indicate trends where the sample type is held constant. It should be realized, however, that corrected L/W values will be somewhat greater than the measured valueso Where data for several sample types are considered, a curve for the center L/W values will be presented. These center L/W values will be corrected for the original L/W ratio by use of Equation (36)o B. Stress Analysis of Deformation Stress analyses were considered to see if the stress levels acting on the inclusions could be determined. If the inclusion stress state could, be determined, a stress-strain correlation could be attempted.

1o Extrusion Theory Selection The extrusion stress analyses presented in the literature are listed below: lo Homogeneous deformationo 2o Plane strain extrusion without frictiono 3o Plane strain extrusion with friction. 4, Slip line theories. 5o Axially symmetric extrusion. 60 Stress determination from gridded samples. These methods will be referred to by number in the ensuing discussion. The extrusions theories associated with these methods are discussed in Chapter IIo The experimental data available for use with the extrusion theories are: 1. Die angleo 2o Per cent reductiono 3o Sample yield strength. 4o Ram pressureo 5. Strain hardening patternso The extrusion theory selected should be able to predict stress levels on the basis of these boundary conditions. The coefficient of friction between the billet and the die wall is unknown, Method six was eliminated since (1) incremental deformation would introduce extraneous variables for elevated temperature studies and since (2) it is unduly cumbersome for the number of samples to be

-72 consideredo The other extrusion theories are developed for isotropic, rigid-plastic, incompressible, non-strain hardening materialso Consequently, consideration of these factors would not lead to the selection of one theory over another, The following reasoning was used in selecting an extrusion theory from methods one through fiveo Methods one and two were eliminated (1) since they do not consider frictional forces and (2) since ram pressures predicted from them are significantly low. Method three was eliminated since (1) the stress components are independent of the angle cp (Figure 4(b)), which would not be expected, and (2) the analysis is for the plane strain case. Shield's(lll) axially symmetric analysis (method f -) was selected over the slip line theory (method 4) since it can be readily programmed for use with a computer, 2o Application of Shield's Analysis Shield(11l) applied his analysis for plastic flow in a converging conical channel to the case of drawingo The mathematic modification of his analysis for extrusion is presented in Appendix Bo The method and assumptions used in applying the modified analysis are presented in this sectiono Shield's(111) analysis assumes an isotropic, non-strain hardening material and does not account for the stress required to change the direction of material flow at the die entry and exit sectionso For the modified analysis, the samples are assumed to behave isotropically in the bulko Since the relative amount of stress required to change the direction of material flow decreases as the sample reduction increases, this

-73 stress was assumed to be a negligible portion of the stress required for extrusion (ram pressure). Since strain hardening during extrusion can not be neglected, hardness measurements were used to determine strain hardening patterns for the various die geometries (Figure 13). The compressive yield strength of 90 Cu - 10 NaCi samples was determined as a function of matrix hardness (Figure 20). Based on these two types of data, yield strength can be determined as a function of position in the extrusion. The yield strength-sample position data can not be readily analyzed mathematically. Consequently, an average yield strength value was used for the calculations. The stress profiles plotted in the results section are for the stress component divided by the average yield strengtho The actual stress could be obtained by multiplying the plotted values by the actual yield stress for the position being considered. However, this correction was not made since the stress profiles were not believed to be accurate. The ram pressure value used for the calculations was the extrapolated ram pressure (Chapter II - Section D-2). The extrapolation was made to eliminate the effects of sliding wall friction between the billet and chambero 3~ Ram Pressure as a Stress Index Since an accurate determination of the stress components was not obtained, ram pressure was used as a stress level index. As the ram pressure increases, the magnitude of the stress components increases.

a- 40/ II,rJ 30 /-I LLJ 20 0 i 0 20 40 60 80 100 120 140 160 MATRIX HARDNESS, D P H Figure 20. Compressive Yield Strength versus Matrix Hardness for 90 Cu - 10 NaC Samples.

-75However, for a fixed die and coefficient of friction, the stress distribution would be unchanged. Consequently, ram pressure is a good index of the relative stress levels acting on the inclusions.

CHAPTER V EXPERIMENTAL RESULTS The experimental results will be discussed under the following headings: Ao Strain Hardening Patterns. B. Flow Patternso Co Calculated Stress Distributions. Do Plastic Deformation and Fracture. Eo Effects of Parameters on Inclusion Deformation. A. Strain Hardening Patterns Strain hardening patterns for various reductions and included die angles are presented in Figures 21 and 22. The lines of constant hardness were determined for low oxygen copper, Rates of strain hardening can be estimated from the distance between the various constant hardness lineso The right side of each figure shows lines through the experimentally determined points while the left side of each figure has the lines smoothed out to show the more probable constant hardness profiles. The following observations can be made from the strain harden ing patterns: 1, As the die angle increases, (a) the strain hardening patterns become more complex, (b) regions of softer material begin to occur near the die-container junction, and (c) the maximum hardness increases in the region of the die lipso 2o Most of the strain hardening occurs before the sample enters the die, 3o Some strain hardening occurs near the container walls. -76

-77 a) 60~ die b) 135~ die c) 180~ die Figure 21. Strain Hardening Patterns for 75 Per Cent Reduction Dies.

-78 a) 55.6 per cent reduction die b) 75 per cent reduction die Figure 22. Strain Hardening Patterns. 90~ dies were used.

-79 Bo Flow Patterns Photomicrographs illustrating inclusion deformation between the die lips are presented in Figure 23. These results are for 90 Cu - 10 NaCl samples which were extruded through a 90~ included angle -75 per cent reduction die. The following observations can be made on the basis of these photomicrographs: 1, The inclusions are flattened somewhat before they enter the dieo (The L/W ratio in the as-sintered samples was 0o536e) 2o Most of the deformation occurs in the region of a line normal to the die wall and originating at the die lips (line AB)o 30 Lines drawn parallel to the long axes of the inclusions are not directed toward the virtual apex of the cone formed by extending the die lipso Inclusion shapes in an extruded 90 Al - 10 NaCl sample are shown in Figure 24, A 180~ included angle -75 per cent reduction die was usedo A longitudinal sample along a symmetry plane is shown in Figure 24(a). As the sample surface is approached, the inclusion L/W ratio is observed to increase. Numerical data illustrating L/W profiles across various samples will be presented in section E of this chaptero A section normal to the extrusion direction is shown in Figure 24(b)o As the sample is traversed from surface to center. the inclusion length in the circumferential direction does not change appreciably,

j(D H. CD HP CD ^ O (D O CD 0 C D c'l(D c O" CD w HH. U* C O c o H' 0 c+ o H O a t o 1-' 0 PD =' PP0 -' r) 0 CD ct0 X I F O H C) rHo'ICO 0 cn D -l

If_ lw.."'. ark =~~~~~~~~~~~~~I he e.xtrs.io airection. a) Section para _f~~~~ijt~~~~l~ -~l I I ~~...-............., / *..... I* l l SU, A |#,f'.I'~P.S-.~BB~Bggg~R ~C6 B Nab 444>~~~~~~~~~~Y$W~, ~~a "iBIe B e rstrr " I~ a 14e QBoo p (I) L, Q klF Q b) &. irectione inormal to the eotr usion orecttio' sec in nofma,,ionDe,tio Stratils0 Inc MicrOStructcures ilsrtn nlso eomt~ in a Extrucded Sample. in an F,)Ctl r

-82Co Calculated Stress Distribution Stress distributions calculated from the modified stress analysis are presented in Figures 25 through 29. The yield strength values used for the calculations are listed with each figure. Figure 25 shows the results obtained for the extrusion of a 90 Pb - 10 NaCl sample through a 90~ - 75 per cent reduction die. These results are significant since this sample does not strain harden during extrusiono Figure 26 shows the results obtained for a 90 Cu - 10 NaC1 sample extruded through the same die where the average value of the initial and final yield strengths was used in the calculations. The results obtained from these two calculations are almost identical. The effect of die angle on stress distribution can be observed in Figures 26 through 28. An average yield strength value was used for these calculations. As the die angle increases, the difference between the surface and center values of the stress components increases. The effect of using the final value of the yield strength for the calculations is shown in Figure 29. For all of the calculations, the shear stress reaches oQ/\3 well inside the samples. Do Plastic Deformation and Fracture Plastically deformed and fractured inclusions are shown in Figure 30. Typical plastically deformed inclusions are shown in Figure 30(a)o Regions where local sections of the inclusion have slipped over each other can be observed. When fracture occurs during deformation, the inclusions have the appearance shown in Figure 30(b)o Crystallographic

6 0 X 6 -4 T6 o a) Stress distribution at the die entrance -6 -4 2 0 2 4 4 6 o~- a6 o-o o'o o' R To 1I 1, I I ~ -6 -4 -2 0 2 4 6 0 ao 0 To T R a o -6 -4 -2 0 2 4 6 a' R To b) Stress distribution at the die exit Figure 25. Calculated Stress Distribution for a 90 Pb - 10 NaCl Sample Extruded through a 90~ - 75 Per Cent Reduction Die. 5o = 1,245 psi.

-6 -4 -2 0 2 4 0o so 6 0 I -6 -4 T R J o a) Stress distribution at the die entrance 6 0 I -6 -4 TR To I aI -6 -4 -2 0 2 4 wo o8 2 4 6 T R __ b) Stress distribution at the die exit 0R To Figure 26. Calculated Stress Distribution for a 90 Cu - 10 NaCl Sample Extruded through a 90~ - 75 Per Cent Reduction Die. aO = 27,150 psi.

0 -6 -4 -2 0 2 4 6 - 6 -4 - 2 0 0(To Co Co TR 6 So a) Stress distribution at the die entrance 0 2 4 6 (R 0s 0 I TR ( 0 I ao \jn -6 -4 -2 0 2 4 6 0 0 0o o -6 -4 -2 0 2 4 6 OR 0c b) Stress distribution at the die exit Figure 27. Calculated Stress Distribution for a 90 Cu - 10 NaCl Extruded through a 60~ - 75 Per Cent Reduction Die. ~o = 26,980 psi. Sample

XI -8 -6 -4 -2 0 2 4 -8 -6 -4 -2 0 2 4 0 0-l Co Co0 TR CR o 0r a) Stress distribution at the die entrance l 0 I TR ( Go -4 -2 0 2 4 6 8 Co 00,0 -4 -2 0 2 4 6 8 OR Go b) Stress distribution at the die exit Figure 28. Calculated Stress Distribution for a 90 Cu - 10 NaCl Sample Extruded through a 135~ - 75 Per Cent Reduction Die. Uo = 25,950 psi.

0 I To -6 -4 -2 0 2 4 6 -6 -4 -2 07 0% 9 0 0 Co- Co o 2 4 6 (OR 0 a) Stress distribution at the die entrance, 6 0 I -6 -4 TR0o0 I CO) I -6 -4 -2 0 2 4 0 0 Co Co -2 0 2 4 6 0o b) Stress distribution at the die exit Figure 29. Calculated Stress Distribution for a 90 Cu - NaC1 Sample Extruded through a 90~ - 75 Per Cent Reduction Die. ao = 44,700 psi.

-88 a) Plastic deformation of inclusions in a 90 Al - 10 Cu sample. 37.5X. A5.A.Aw b) Inclusion fracture during extrusion in a 90 Cu - 10 LiF sample. 37.5X. Figure 30. Microstructures Illustrating Plastically Deformed and Fractured Inclusions. All samples were extruded through a 90~ -75% reduction die.

-89 fracture planes are observed, metal is occasionally forced between the fractured pieces of the inclusion, and some inclusion dispersion occurs. Internal cracks in the inclusions are rapidly etched by the alcohol used in polishing. Consequently, cracks can be readily observed when they occur. It was felt desirable to investigate the effect of initial inclusion orientation on the final shape of the inclusion. To do this, cubic NaCl single crystals with known orientations were placed at the center of copper samples. The inclusions were oriented so that a [111], [110], or [100] inclusion direction was in the extrusion direction. Radiographs of the samples before and after room temperature extrusion are shown in Figure 31. The sample having the [100] orientation was not fully extruded and it cracked near the die entrance on removal from the die. The original inclusion orientation is preserved during extrusion for the [111] and [110] samples while the [100] sample tends to rotate so that a [110] direction lines up with the extrusion direction. The amount of deformation for the [111] and [110] samples is about the same. E. Effects of Extrusion Parameters on Inclusion Deformation In this section, the experimental data obtained as a function of various extrusion parameters will be presented graphically. The experimental ram pressure and L/W data are listed in tabular form in Appendix C while the experimental hardness data are listed in tabular form in Appendix D. The following plots are used to illustrate the

-90-..... #t *1 [110] a) Before extrusion. [100o] [111] 9. * U, i * I IA r.% I [100o [110] [111] b) After extrusion. Figure 31. Radiographs Illustrating the Effect of Inclusion Orientation in Inclusion Deformation. The black dots were caused by lead shot in the clay used to position the samples.

effects of the parameters: 1. L/W ratio vs. variable. 2. Z(L/W)iAi vs. variable. At 3. Ram pressure vs. variable. i 4. E(L/W)iAi vs. ram pressure. At In general, the fourth type of curve is a consequence of the factors governing the second and third types of curves. 1. Effect of Die Angle (Data and figures: 75 per cent reduction-variable angle dies, 90 Cu - 10 NaCl samples, Figures 32 through 36.) As the die cone angle increases, the following effects are observed: 1. There is a rapid increase in the L/W ratio near the surface of the sample, The L/W ratio near the center of the samples does not change significantly (Figure 32). 2. The amount of inclusion deformation increases (Figure 33). 3 The ram pressure increases (Figure 34). The bulk L/W value for the sample is approximately 8/1. With the L/W correction for density and original L/W ratio, the inclusion L/W ratios are above 8/1 for all positions and die cone angles. Consequently, the NaCl inclusions are the more ductile phase.

-92 1 80 1 60 140 120 100 0 180~ 80.J 60 135 0 40 90~ 20 - ^H- 600 0 0 0.2 0.4 0.6 0.8 1.0 SURFACE CENTER NORMALIZED POSITION Figure 32. L/W Traverses for Die Angle Study Samples.

-93 - 80 | 60 1800 <: % - 4- 1350 - <J 20 900 60~ 0 - I I i 0 1 2 3 4 DIE CONE ANGLE,RADIANS Figure 33. Inclusion Deformation versus Die Angle.

-94 1350 - 120 / (n a. 9. Id 60 " LIO Cr o) 80 Co 0 0 LG an t 0 4 0.. 1E 1- I AS0 1 2 3 4 DIE CONE ANGLE, RADIANS Figure 34. Ram Pressure versus Included Die Angle.

-95 80 60 - 3 <40 -J 20_ 0 0 40 80 120 160 RAM PRESSURE, KPSI Figure 35. Inclusion Deformation - Ram Pressure Relationship for Die Angle Study Samples.

-96 1" L *''..,,^ - * **. mm__ -'4- __ 4I; I, JI ~~~~~~~~~~~~~~~~~ ~~~-, L- ~~~~~~~~ 44 mmmmppp- - - - . -92IMMINK.- —. I "I WA- - , a) Die angle: 60~. b) Die angle: 90~ W.. —._! - -. ftw-..~~~~~~~~~~~~~~~~~~~~~~~~~ ts v*.6~ CL~I. - y~~"Mumo. _ —I~g%.kl -01 O - ~s~~~~~~~Z ~~~~m"ft",lL M"1eLI fo 14110 a I _ f l h m m m jjjjj j.. A=._ *d) Die -l ^ 80o. - d) Die angle: 180~. c) Die angle: 135~. Figure 6..... —' "cr'ussructures Illustrating the Effect of Die Angle on Inclusion Deformation. Samples: 90 Cu - 10 NaC1 Die: 75% reduction. 37.Sx.

-97 2, Effect of Per Cent Reduction in Area (Data and figures: 90~ - variable reduction dies, 90 Cu - 10 NaC1 samples, Figures 37 through 41) As the per cent reduction in area increases, the following effects are observed: 1o The inclusion L/W ratio increases for all positions across the sample (Figure 37)o 2, Both the amount of sample deformation and the amount of inclusion deformation increase. The curves for inclusion deformation and bulk sample deformation have the same general shape (Figure 38)o 30 The NaCl inclusions undergo more deformation than the matrix (Figure 38), 4. The ram pressure increases (Figure 39(a))o 5. The plot of ram pressure vso log (Ai/Af) is linear (Figure 39(b)). 3. Effect of Strain Rate (Data and figures: 90~ - 75 per cent reduction dies, 90 Cu - 10 NaCl samples, Figures 42 through 46) As the strain rate (ram velocity) increases, the following effects are observed: o1 The amount of NaCl deformation increases (Figures 42 and 43). 2o Ram pressure increases (Figure 44(a)). 3. Log (ram pressure)is a linear function of log (ram velocity) (Figure 44(b))o

60 0 < 30I 55.6 % 0 - 0.2 0.48 0 SURFACE NORMALIZED pOSITION igure 7 L/W for Per Cent Reduction Figure 57.in Trea Stuey SampleS i~n Area Study amls

-99 -4 20 40 60 PEoucr^ R 8O 100 PERCENT REDUCTION IN AREA Figure 38. nclusion Deformation versus Per Cent Re duction in Area.

-100 U) CL 120 LJ D UJ r 80 40 - - 0 0 20 40 60 80 100 PERCENT REDUCTION IN AREA a) Ram pressure vs. per cent reduction in area. 20c 88.4% 160 - 0/ 120 - U) U) W 80 Cr 55.6 % 40 3Q9% o / I I I I 0 0.5 1.0 1.5 2.0 2.5 Loge ( A ) b) Ram pressure vs. loge(Ai/Af) Figure 39. Ram Pressure versus Per Cent Reduction in Area.

-101 60 _ 50 88.4% 40. 30..J 20 I 0 i^ 55.6% / ^30.9 05.% 0 0 40 80 120 160 200 RAM PRESSURE, KPSI Figure 40. Inclusion Deformation - Ram Pressure Relationship for Per Cent Reduction in Area Samples.

"rrr, b,. - b) Reduction in area: 55.6. a) Reduction in area 30.9. Amr- - *~~~~~~~~~~~~ *.~~~~~%p~ I.rrI 106 o-' "I..~ __~adS~ I ~ 1~~Mbt _ _ _~~~~~ —m ---— 1'I6-"IIPI"-~IPI r............. 4w..-~~~~~la ~~iic~~~sF: w -- a~- ~' I cp~e ~~ ___ _ -- J r. OO *' 0 ----- - - - c) Reduction in area 73.5. d) Reduction e 41. ~rost^t IllustratingIon in ara 88.5. Figure. Microstructures Iustrt iffect f Per Cent Reduction in Area on Inclusion Deformation. Samples: 90 Cu - 10 NaC_. Dies: 90~ 37.5X.

.103 30 25 0.10 A 1 0.05 20 F<:[ I - - 5 - 0 -— II I-I 0 0.2 0.4 0.6 0.8 1.0 SURFACE CENTER NORMALIZED POSITION Figure 42. L/W Traverses for Strain Rate Study Samples.

-104 15 -J I 2. — - o 0 ----- ----— II I I, 0.2.4.6.8 1.0 RAM VELOCITY, I P M Figure 43. Inclusion Deformation versus Ram Velocity.

-105 I I I loo0 aLJ wt V) a. <E 0: 80 60H 40k 20 0 L 0.2.4.6.8 RAM VELOCITY, I PM 1.0 a) Ram pressure vs. ram velocity 11.60 w (I) U) W 0. a, 0 -Ji 11.58 - 11.56 - 0 11.54 11.52 F b) 0 0.5 1.0 1.5 2.0 2.5 3.0 Loge ( RAM VELOCITY) ) Loge (ram pressure) vs. loge (ram velocity) Figure 44. Ram Pressure versus Ram Velocity.

20 II I I I'p 15 e: I -W 10 I 0 To I 5 - 0 0 20 40 60 80 100 120 RAM PRESSURE, KPSI Figure 45. Inclusion Deformation - Ram Pressure Relationship for Strain Rate Study Samples.

-107 ~.i ~.. ~......' ~,.,. *' ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. ~ ~ ~..'i', ~ -.,': r~. - t,.` ~,. ~.'.'.''' i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~..~. -~~~~~~~~~~~~~~'''''''..~.,.,,.,,;,...''~'~~~~~~~~~~~ri _. ~ ~ ~ q_,.-...... ---- --—:,.,_,_,... ,, XrlPSIQ3QIII C'rr'" 41"1"x -"-"'fsnna,.-I %, xr. ~" a) Ram...... - — Ibm- -. - I lw.~~~~~~~~~~~ velocity: 0.05 ipm. b) Ram velocity: 0. I0 ipra. I r~~~~~~ 1Cllllr I IIC -. A o. - w -,~W~-.'...._.,,....-....l.L.~~~~~~~~~~~~~~~~~~~~~~-...." C) Ram~~~~~~~~~~~~~~~~~~~ )Rmvelocity: 0 30 i'mn d)Ram p,1 Figure 46. -. v-uIlzy: 0. 71 ipm. MiCrostructures Illustrating the Efrect Of Strain Rate on Inclusion Deformation. Samples. 90 Cu - 10 NaC. Die. 90 7% reduction. 37.5X.

4. Effect of Temperature (Data and figures: 90~ - 75 per cent reduction dies, copper, NaC1, and 90 Cu - 10 NaCl samples, Figures 47 through 53.) As an indication of the relative property changes of the individual phases, ram pressure and hardness curves are presented for copper and NaCl samples. Hardness values presented in Figures 50 and 51 were measured at room temperature after the samples were extruded at the indicated temperatures. The inclusion hardness values shown in Figure 51 were measured near the sample centerline. The hardness values described in Figure 52 were measured at temperature on fully annealed materials. The copper and NaCl used for the hot hardness measurements were from a different source than the extrusion materials. The following temperature effects were observed: 1. The shape of the L/W profile changes above 200~F. Above this temperature, the L/W curves do not tend to level off as the sample surface is approached (Figure 47). 2. A minimum in the amount of inclusion deformation occurs at approximately 300~F (Figures 47 and 48). More inclusions are fractured in the region of this minimum (200 to 500~F) than at higher or lower temperatures. Above 300~F, inclusion deformation increases with increasing temperature. 3. The minimum in the inclusion deformation curve corresponds to a maximum in the inclusion hardness curve (Figures 48 and 51).

60 I | i 60 50 -50 - 40 40 1316 OF 0 \ 1339~F 0 30- - 30 < 919 F <: nc- -108 ~F.1200 I0 - 0 -497 0000I 0 0 0.2.4.6.8 1.0 0.2.4.6.8 I. SURFACE CENTER SURFACE CENTE NORMALIZED POSITION NORMALIZED POSITION a) Traverses for temperatures below 400~F b) Traverses for temperature above 400~F Figure 47. L/W Traverses for Temperature Study Samples. I 0 RD 0 R

30.25 20 15 10 I I I I I I H 0 I 5 - 0. I I -200 0 200 400 600 800 1000 1 TEMPERATURE, OF Inclusion Deformation versus Temperature. 1200 1400 Figure 48.

160 - 120 (') 0n Y X CL~~~Cu w: 80- t)X Cu -NoCI —(C) (L 40 -,0 I -200 0 200 400 600 800 1000 1 TEMPERATURE, OF Figure 49. Ram Pressure versus Temperature The NaC1 curves is after Moore. (121) 200 1400

160 120 I Q 0_ CI I) w 80 40 I - A ~NaCI 0 - 200 0 200 400 600 800 1000 1200 1400 TEM PERATURE,~F Figure 50. Matrix Hardness versus Temperature. The NaC1 curve is after Moore.(121) The hardness values were measured at room temperature after extrusion at the indicated temperatures.

60 50 I 0_ U) r) z C, -J 0 Z) 401 30 - HI 20 - I 0 0 -2( Il I I I i )O 0 200 400 600 800 1000 1200 1400 TEMPERATURE,~F Figure 51. Inclusion Hardness versus Temperature. The hardness values were measured at room temperature after extrusion at the indicated temperatures.

40 I 30o Cu I 0 w) z Q I 20F I iI I o NoCI 0 I 0 200 400 600 800 1000 1200 1400 TEMPERATURE, Figure 52. Hot Hardness of Annealed Copper and NaC1. copper curvesis after Chao.(122) The

I.!4ow' -00N - -rllr eFIIC'Ir "Irr' ur*,t ~9 T"r"hiaUb, b%91F~ p B~ " ~rlPgqllr~ — VW,""~ a~ *~~~~~~~~~~* 11. -"oa 1 t,.jrrrrrsi:;I ~ - b ~emprat - re L AN-. 1 -t re, - 30 - t) Tempera -. -eper08 a) ~emperature. F. — mosoop",dr -999..~~~~~~rl Ir ~LI, W. ) -- 00"Op._ _ *~~~~~~~~~___d 111:.-..Mmmm w __oow. —.. jr ~,.- o WrfY.. _ t A Tempera ture 1316F C) Tempera ~ o~F' th Effect o ~ -eNeature ture - e. 0 lsC1. tion. samp?emp MicrostN ~eformaion. 7l. oIJnCI i 5nc reduction Die: 900 Figre 53

-116 4o Ram pressure decreases with increasing temperature for copper, 90 Cu - 10 NaC1, and NaC1 samples (Figure 49). 5, The cold to hot working region change occurs between 700 and 1100~F for pure copper (Figures 49 and 50). 60 The copper in the 90 Cu - 10 NaCl sample strain hardens less than in the pure copper sample during extrusion (Figure 50). 7. For annealed Harshaw NaCl single crystals and high purity copper, hardness decreases smoothly with increasing temperature (Figure 52), 50 Effect of Per Cent Nonmetallic Material (Data and figures: 900 - 75 per cent reduction dies, copperNaCl samples, Figures 54 through 58) As the volume per cent NaCl increases, the following effects were observed: 1o The amount of inclusion deformation decreases (Figures 54 and 55)~ 2. Ram pressure decreases linearly (Figure 56). The NaCl inclusions deform more than the copper matrix for all compositions (Figure 55), 60 Effect of Nonmetallic Inclusion Size (Data and figures: 90~ - 75 per cent reduction dies, 90 Cu - 10 NaC1 samples, Figures 59 through 62) The particle size reported is the arithmetic mean for the screened salt fraction used to make up the samples. A "mean L/W value" vs. ram pressure plot was not presented since the data reduce to a point.

-117 0 0 _ I- 15 Oc l0 SURFACE 2.4.6.8.2.4.6.8 1.0 CENTER NORMALIZED POSITION Figure 54. L/W Traverses for Per Cent Nonmetallic Material Study Samples. Per cents of the NaC1 phase are indicated.

20 15 I 0 - C- BULK SAMPLE -J ~-P34 5..... I I I I 0 20 40 60 80 VOLUME PERCENT Na CI Figure 55. Inclusion Deformation versus Per Cent NaCl.

-119 120 100 8 U) a. LJ 60 U) fU) UJ\ aL 40 2 20 0 20 40 60 80 VOLUME PERCENT NaCI Figure 56. Ram Pressure versus Per Cent NaCl.

2 20i —--------------------------- 15 I 14 -J *-coo 5% 0 - 100 % i 5 - 1 As IM 0 0 20 40 60 80 100 120 RAM PRESSURE, KPSI Figure 57. Inclusion Deformation - Ram Pressure Relationship for Per Cent NaC1 Study Samples.

-121 ~~brr~~rrdoppl ~~~icsrrr'~~-I 90mpw li mu 1 w-_ r e L-l -J 4' -~i e~~~~~~c)~~~~ 2w — _ IJCi~~~_0 A _ I I -.m - - - - --- - ------- - -M a) Sample: 95 Cu - 5 NaCl. b) Sample: 80 Cu - 10 NaCl.'-". —-o _- -- ------ _ - - a~l~~3Lb-IZ-cGs --- ON — I -— _ -,........... -- - ~9 -0 c) Sample: 50 Cu - 50 NaCI. d) Sample: 100 NaC1. Figure 58. Microstructures Illustrating the Effect of Per Cent NaC1 on Inclusion Deformation. Die: 90~ - 75% reduction. 37.5X.

-122 30 0 155 < - cr- la l 5- 0 0 I i I 1 0.2.4.6.8 1.0 SURFACE CENTER NORMALIZED POSITION Figure 59. L/W Traverses for Inclusion Size Study Samples.

20 | lll'I I IO ro 5 0 0.1 0.2 0.3 MEAN INCLUSION DIAMETER, MM. Figure 60. Inclusion Deformation versus Mean Inclusion Diameter.

-124 120! i " -. ~ I1~0 0 0 I 00 - 0 80 cn C_ 0 W 60 cnr LU O. 40 cr 2 20 0 II I I I 0 O.1 0.2 0.3 MEAN INCLUSION DIAMETER, MM. Figure 61. Ram Pressure versus Mean Inclusion Diameter.

-125 - ( -r -3F ~- -i".:~~; 1 -~ -, -*c~1 - -mni~ — ~FC a. rr- p - ~ ~ ~ rZ — ~~iF ~7) cr"C 7- rr. o- -r -rN' rr ~ ~~~~.C -ttdraP~~-~LC2 UL*J*i*,'~ - o Ir; C;~"~,:^-*l**C~T~~-WOW:"~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~::~'~ ~i~~ ~ ~ ~ ~ ~~~~~~iij~i:j z:,~~:: j'~J: i;'j:J:;/?:iz/~::~::i:j::' /!......... kow. i:...::~::,::::! i:'::i!::i i:i: 1 i:i: ii:ii!:i~,:i:: i:::}'i:::: {:!, i::::::~i:::il B~:' ~':::~:~'' ~ _.::: i~:!~!.!:iiii':,'?:?':::.:::0 0 0:::! 1:::::;. "'.. —::-:::~~~~~~~~~'Iuloftma""law a) Mean inclusion diameter: 0.048 mm. b) Mean inclusion diameter: 0.081 mm................... r BqsllT~ig~rlgll~l~j~*;i - r'~: i, i iiii'i ~c':: ~.i~-.:'i:i ii/::i:,:!ij!i:?,~'~':J!.jijJ[['[':':~':~:~.'~,':'~"./,.j~j"~:"i.:'~::',':::::::..j[i i::.?:'jij:j j i i i,.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~: ~ c) Mean inclusion diameter: 0.163 mm. d) Mean inclusion diameter: 0.275 mm. Figure 62. Microstructures Illustrating the Effect of Inclusion Size on Inclusion Deformation. Samples: 90 Cu - 10 NaCl. Die: 90~ -75% reduction. 37.5X.

-126 The following effects were observed: 1. The L/W profiles across all of the samples are approximately the same (Figure 59). 2. The amount of inclusion deformation decreases very slightly with increasing inclusion size (Figure 60). 3. The ram pressure does not change with inclusion size (Figure 61)o A least squares fit of a straight line to the experimental data indicates a slope of 1000 psi/mm which is not a significant variation. 7o Effect of Inclusion Type (Data and figures: 90o - 75 per cent reduction dies, 90 per cent copper - balance KC1, NaCl, NaF, or LiF samples, Figures 63 through 67) Inclusion hardness was used as the index of the inclusion propertieso Both average and final inclusion hardness values are plottedo The average hardness values plotted are the average of the inclusion hardness values before and after extrusion while the final hardness is the inclusion hardness in the extruded sampleso The inclusion L/W ratio in the assintered samples varies with inclusion type. To eliminate this variable, the L/W ratio at the sample center was corrected for the as-sintered L/W ratio and plotted in Figure 64(a). However, the L/W correction does not change the results appreciablyo As inclusion hardness increases, the following effects can be observed: 1o The amount of inclusion deformation decreases (Figures 63 and 64). 2o The ram pressure increases (Figure 65).

-127 30 25 20 NaC " KCI 15 Na F I0 0.2.4.6.8 1.0 SURFACE CENTER NORMALIZED POSITION Figure 63. L/W Traverses for Inclusion Type Study Samples.

ILJ z wO 0o d OWQ< 0 20 I 0 0 ( I I I I K Cl F NoF 4 V<E G Li F.. I I I I NR G o l^^ — ^^ ) 40 80 120 160 200 INCLUSION HARDNESS, DPH a) Corrected center L/W ratio versus inclusion hardness. I r) oD 20 I _-j I O 4 I I I I KCI NoF NocI K Cl PFJaF Li F I I I I 0 0 40 80 120 160 INCLUSION HARDNESS,DPH L/W ratio" versus inclusion hardness. 200 b) Measured'?nean Figure 64. Inclusion Deformation versus Inclusion Hardness.

-129 140 i I 120 Li F NaCI KCI / AVERAGE FINAL 100 cn Q. 80 V) () cr 60 a. tr 40 20 0 I I I I I 0 40 80 120 160 200 240 INCLUSION HARDNESS, DPH Figure 65. Ram Pressure versus Inclusion Hardness.

50 40 30 -._ -J -- 20 H I 0 l KCI * NoCI 10o NoF * Li F 0 0 20 40 60 80 100 120 RAM PRESSURE, KPSI Figure 66. Inclusion Deformation - Ram Pressure Relationship for Inclusion Type Study Samples.

-131-:: II."......; -~ Os 8: i - - t~~~~1 ~~.. ----- - ~- -: ~L............. I IsJ~~~~~~~~~~~~~~~~~~~r~~~~~~~~~ I 4 -- ---- I iri"... I I. Mmmmmmmmmpw-. -'-~r I........... --- 11... 11.... Mm __ ~~ -'..,I. a) Sample: 90 Cu - 10 KC1. b s:,-:d.S _,.. _ ~,, _ _ =.........,. _. 0.. *. ~. *,. *. Sample: 90 Cu - 10 NaC1.' -" * s AL. - 0 I >s -. - +~- ~ c) Sample: 90 Cu - 10 NaF. d) Sample: 90 Cu - 10 LTi Figure 67. Microstructures Illustrating the Effect of Inclusion Type on Inclusion Deformation. Die: 90 -75 reduction. 37.5X. $

-132 KC1, NaCl, and NaF inclusions deformed more than the copper matrix (8/1) while the converse was true for LiF inclusions (Figure 64). Many of the inclusions fractured during extrusion of the 90 Cu - 10 LiF sample. 8. Effect of Matrix Type (Data and figures: 90~ - 75 per cent reduction dies, 10 per cent NaCl - balance Pb, Al, Ag, Cu, or 75-35 brass samples, Figures 68 through 72) Both average and final matrix hardness values were used as an index of the matrix properties. The average hardness values plotted are the average of the matrix hardness values before and after extrusion while the final hardness is the average matrix hardness in the extruded samples. The L/W values measured at the sample center were corrected for the L/W ratios in the as-sintered samples and they are presented in Figure 69(a). As the matrix hardness increases, the following effects were observed: 1. The amount of inclusion deformation increases (Figures 68 and 69). NaCl inclusions in copper deform more than in an aluminum or silver matrix. However, the correction for as-sintered L/W ratio must be made for this result to be obvious (Figures 69(a) and (b)). 2. After a rapid initial increase in the amount of inclusion deformation, the rate of increase becomes constant at a lower value (Figure 69).

-133 - 30 25 20 0 r 15 U IO- -j0 5 0 0.2 0.4 0.6 0.8 1.0 SURFACE CENTER NORMALIZED POSITION Figure 68. L/W Traverses for Matrix Type Study Samples.

20 0 o w < o- _J 0o a' Pb 0 L30r 40 80 120 160 200 MATRIX HARDNESS, DPH a) Corrected center L/W ratio versus matrix hardness 240 jBrass 20 Ag Al 3,.-J 4 I 0 0' C - I ) 40 80 120 160 200 MATRIX HARDNESS, DPH b) Measured "mean L/W ratio" versus matrix hardness Figure 69. Inclusion Deformation versus Matrix Hardness. 240

200 160 I/ 0: Cu Z) rr Ag Q- 80- -'~ Al 40 Pb 0 I I 0 40 80 120 160 200 240 MATRIX HARDNESS, DPH Figure 70. Ram Pressure versus Matrix Hardness.

-136 O 3: a~ H z clJ (t) 0 -J._ 50 40 30 20 I 80 120 RAM PRESSURE, KPSI Figure 71. Inclusion Deformation - Ram Pressure Relationship for Matrix Type Study Samples.

aoomma. A4 i m u U I f u n __-III ~I ~~lll~q%(~l~.-..I~~p lyn*~lilPLIIIPI^ ~CI _ ___ a) Sample: 90 Pb - 10 NaC1. b) Sample: 90 Al - 10 NaC1. qlww~a I -.am 111101111 I - jl~~~;icr"Ilo -Dou g --- RPM - -— ~l *Jlrrrc __c —JZClsr IMMMFF- -__P~-rr~.~sdpL~~r~ra rar c) Sample: 90 Ag - 10 NaC1. Figue 72. Microstruetures i) Sample: 90 Brass - 10 NaC1. Finuse.ios nuuresIllustrating the Effect of Matrix Type on Inclusion Deformation. Die: 90' -75% reduction. 37.5X.

-1383o Essentially no inclusion deformation occurs for the lead matrix. For the aluminum, silver, copper, and brass matrices, the amount of inclusion deformation is greater than the amount of matrix deformation (Figures 68 and 69). 4o Ram pressure increases (Figure 70).

CHAPTER VI DISCUSSION OF RESULTS The experimental results will be discussed under the following headings: A. Strain Hardening Patterns. B. Flow Patterns. C. Calculated Stress Distributions. D. Effects of Extrusion Parameters on Inclusion Deformation. E. Plastic Deformation and Fracture. Conclusions are listed at the end of each section. Ao Strain Hardening Patterns The strain hardening patterns shown in Figure 21 can be explained on the basis of the shear stress levels developed in the samples. For the 60~ and 90~ dies, the samples slide along the lubricated die surface while for the 135~ and 180~ dies, shear occurs through the samples. Consequently, as the die angle increases, the shear stress levels in the samples increase. This conclusion is consistent with the results of Sachs and Eisbein(95) and Johnson.(109) The former show that the amount of sample distortion near the sample surface increases with increasing die angles. Johnson(l09) states that lubricated slip over the die face occurs for included angles up to about 120~. As the die angle increases, (1) the strain hardening patterns become more complex, and (2) the hardness increases near the die lips. These changes are a consequence of increasing shear stress levels in the samples. -139

The softer regions near the die-container junction for the 135~ and 1800 dies are the result of shear occurring through the sampleo For these die angles, the samples essentially form their own die surface behind the die lips. Consequently, the amount of strain and strain hardening ahead of this internal surface is reducedo The region ahead of this internal shear surface is the so-called "dead" metal regiono The same over-all stress distribution would be expected for the same die angle, even with different reductions in areao This is evidenced by the similarities in the strain hardening patterns for the 55.5 and 75 per cent reduction dies (Figure 22). As would be expected for heavier reductions, the hardness values for corresponding normalized positions are greater for the 75 per cent reduction die. Strain hardening near the container wall is a consequence of sliding friction between the sample and the container. The hardness increase behind the die entrance is greater than the increase between the die lips. The annealed sample hardness was approximately 40 DPHo Observation of the strain hardening patterns indicates that appreciable hardening occurs even in regions where no macroscopic flow occurredo This result suggests that appreciable hardening can occur under the influence of high confining pressures without appreciable plastic flowo Since the samples fit the chamber snugly, the hardening is not caused by flow perpendicular to the extrusion direction, The observed hardening is not the result of a second phase on sample densification since high purity copper bar stock was used for the samples. It was found experimentally that 90 Cu - 10 NaCl compacts could be hardened by pressing in a closed die (Figure 9). The matrix hardness

-141 increased from 25 to 90 DPH under a pressure of 126,000 psi. These samples fit the pressing die snugly. However, in these samples the density was approximately 94 per cent of theoretical and a softer second phase was present. Consequently, some strain could be expected during pressing. The evidence is strong that copper canbehardened by the application of high hydrostatic pressures. However, further experimental work would be required to establish this as fact. Conclusions - (Strain hardening patterns): 1. As the die angle increases, the shear stress levels near the sample surface increase. 2. "Dead" metal zones are formed for 135~ and 180~ dies. Shear surfaces are formed within the sample. 3. Strain hardening patterns are quite complex. Consequently, it seems unlikely that simple strain hardening pattern equations can be developed for use with the current extrusion theories. 4. There is evidence that materials can be hardened by the application of high confining pressures. B. Flow Patterns The flattening of the inclusions behind the die entry is a consequence of nonhydrostatic behavior of the matrix. The stresses in the extrusion direction are greater than those normal to the die wall. For the same pressure, a weaker material would be expected to behave more hydrostatically. Consequently, the amount of inclusion flattening would be expected to decrease as the matrix yield strength decreases as observed for the data in Table IV.

-142 The other observations on inclusion flow are consistent with the flow patterns for gridded tin presented by Sachs and Eisbeino(95) For their approximately 90~ included angle die, the bulk of the deformation of a square element occurs in the region of a line normal to the die wall and originating at the die lipso Flow is not directed toward the apex of a cone formed by extending the die lips as is assumed for some extrusion theories. The increase in inclusion L/W ratio as the sample surface is approached results from the shear stress distribution. This will be discussed more fully in section D-l of this chaptero The stress component directions referred to in the ensuing discussion are presented in Figure 6, The inclusion length in a circumferential (G) direction was not observed to vary significantly across a cross section (Figure 24(b))o Thus, the inclusion length in a circumferential direction is approximately equal to the inclusion width at the sample center. This width can be measured in a symmetry plane parallel to the extrusion direction. Consequently, L/W measurements in a symmetry plane parallel to the extrusion direction are a good index of the amount of inclusion deformationo For Shield's(lll) stress analysis ap = ago For TRp < o/< 3 oR is more tensile (or less compressive) than ap and o00 The inclusion deformation at the sample center is a consequence of these stress componentso As the sample surface is approached, the samples thin rapidly in the cp direction under the influence of the shear stress TRop There is no shear stress component in a plane perpendicular to the 9 direction. Inclusion deformation is controlled by both the difference between aR and a( = ao and by TRcpo

-143 TRp increases as the sample surface is approached and results in a decrease in the difference between a( and aR. Where there are moderate to large shear stress values, the inclusion L/W ratio increases markedly as the sample surface is approached (section D-l of this chapter). Consequently, the shear stress level has a major effect on the amount of inclusion deformation. The amount of inclusion deformation can be minimized by minimizing TRIo Conclusions - (Flow patterns): 1. Observed inclusion flow patterns are consistent with the flow patterns observed for gridded tin by Sachs and Eisbeino(95) 2. Inclusion flow patterns for 90~ and larger angle dies indicate that flow is not directed toward the virtual apex of the cone formed by extending the die lips. Consequently, the assumption of this flow pattern in an extrusion theory for large angle dies would only serve as a rough first approximation. 3. Inclusion length in the circumferential direction does not appear to be a function of the position of the inclusion in the cross-section. 4. L/W ratios measured in a symmetry plane parallel to the extrusion direction are a good index of the amount of inclusion deformation. 5. Inclusion deformation is controlled by both TRp and the difference in aR and aC = Co.

-144 60 For large values of TRW- the shear stress can have a much greater effect on the amount of inclusion deformation than the a - aR differenceo 7. For a given sample type and reduction, the amount of inclusion deformation can be minimized by minimizing the shear stress (TR)o Co Calculated Stress Distributions Lubricated slip along the die face occurs for the 60~ and 90~ dies (section A of this chapter). The shear stress calculated from the modified analysis for these dies reached ao/v3 well inside the sample which indicates shear through the sample Consequently, the calculated stress distributions do not accurately describe the stress state in the sampleo For this reason, no attempt was made to correlate inclusion strain with the calculated stress componentso The failure of the calculations is a result of the failure to consider the contribution of the end effects to the ram pressureo An accurate method of determining the contribution of the end effects to ram pressure must be established before Shield's stress analysis can be applied to practical extrusion problemso This statement in no way effects the validity of Shield's stress analysiso The contribution of the end effects to ram pressure is large for a 90' - 75 per cent reduction dieo The final matrix yield strength was used. in the calculations for Figure 29. For the determination of the constant C, this yield strength overestimate is equivalent to a 38 per

-145 cent reduction in the ram pressure. As shown in Figure 29, this reduction does not reduce the shear stress levels enough. Figure 25 was determined for a nonstrain hardening matrix (Pb). Figure 26 was determined for a material with high strain hardening qualities (Cu), using the average of the initial and final yield strengths for the calculation of the constant C. The fact that these figures are almost identical indicates that use of the average sample yield strength for the calculation of C would yield satisfactory results for a strain hardening material. This value could be used to determine the value of each stress component divided by aoo The yield strength value at each position in the sample would have to be used to accurately evaluate the individual stress components. As the die angle increases, the size of the TRY = o/\3 region increaseso This is a consequence of both increased shear stresses in the sample and increased stress required to change the direction of material flow at the die entry and exit sections. Conclusions - (Calculated stress distributions): 1o For 75 per cent reduction dies, the stress required to change the direction of material flow at the die entry and exit sections is a moderately large portion of the ram pressure ( > 38 per cent for a 90~ - 75 per cent reduction die). 2, An accurate method of determining the contribution of the end effects to ram pressure must be established before Shield's stress analysis(lll) can be applied to practical extrusion problemso

-146 3. For Shield's stress analysis, the use of an average yield strength value for the calculation of the constant C would yield satisfactory results. Do Effects of Extrusion Parameters In this section, the effects of individual extrusion parameters will be considered under the following headings: 1o Die Angle 2, Per Cent Reduction. 3. Strain Rate. 4. Temperature 5. Per Cent Nonmetallic Materialo 60 Nonmetallic Inclusion Size. 7o Inclusion Type. 8. Matrix Type. Copper - NaCl samples were used to study a number of the extrusion parameters. In all cases, the NaCl inclusions deformed more than the copper matrix. The greater NaCl deformation results from the fact that NaCl has a lower hardness and yield strength than copper. 1o Die Angle The effect of die angle can be explained on the basis of the stress distribution in the sample as discussed in sections A and B of this chapter. Near the sample center, the shear stress level is low, Consequently, the amount of inclusion deformation depends on the a - aR difference. At the sample center, this difference is the same for all die angles and a fixed reduction in area. Thus, the same center L/W

-147 ratio would be expected for the samples as is observed in Figure 32. As the sample surface is approached, the shear stress levels increase with increasing die angles resulting in increased inclusion deformation, In deforming composite samples, the a - aR difference results in a minimum amount of inclusion deformation. This minimum amount of deformation is illustrated by the results for the 60~ die where the amount of inclusion deformation is approximately constant across the sample. As the shear stress levels increase, the amount of inclusion deformation rapidly exceeds this minimum valueo Consequently, inclusion deformation can be minimized by minimizing the shear stress levels in the sample. The increase in ram pressure with increasing die angle is a consequence of two factors. As the die angle increases, the change in direction of material flow at the die entry and exit sections increases. As the die angle increases, the shear stress level increases. Both factors increase the required extrusion pressure. Shear occurs through the sample for both the 135~ and 180~ dies. As a result, the rate of ram pressure change decreases between these two die angles. As seen in the foregoing discussion, a process change which reduces shear stress levels will markedly reduce inclusion deformation and ram pressure. Process changes which lower the shear stress level include: 1, Use of more effective lubricants. 2. Use of lower angle dies for extrusion and drawing operations.

-148 Conclusions - (Die angle): 1. Ram pressure and inclusion deformation can be minimized by reducing the shear stress levels in the sampleso Shear stress levels can be reduced by the use of more effective lubricants and proper deformation equipment designo 2o Per Cent Reduction The total sample strain and the stress level transmitted to the inclusions increase with increasing sample reductions. Thus, inclusion deformation would be expected to increase with increasing sample reductions as is observed in Figure 38. As the reduction increases, the amount of work increases resulting in the ram pressure data shown in Figure 39o The experimental ram pressure data fit an equation of the form RP = xo ge(AJ) -b (38) where a and. b are constantso The linear dependence of the ram pressure Ae on loge (A) is in agreement with results presented by Pearson and Parkins(92) and Johnsono(107) Since strain hardening increases with increasing reductions, a plot of RP/oo would have a slightly flatter slope than that presented in Figure 39(b)o The slopes of the inclusion deformation and ram pressure curves increase rapidly with increasing reductionso As a result, several reductions, rather than one large reduction, would reduce the ram pressure requirementso Several reductions would also be expected to reduce the amount

-149 of inclusion deformation. Pearson and Parkins(92) show that the ram pressure is a linear function of loge (Ai) for strain hardening and nonstrain hardening materials. Thus, the stress transmitted to the inclusions is a linear function of loge (A). As a result, less incluAf sion deformation might be expected if several reductions were used, This conclusion was not verified experimentally. Conclusions - (Per cent reduction): 1. Inclusion deformation and ram pressure can be reduced by reducing the amount of sample deformation, 2. There is evidence that several reductions, rather than one large reduction, might reduce the amount of inclusion deformation. However, this was not verified experimentally. Ai 3. Ram pressure is a linear function of loge (A~) 3. Strain Rate Inclusion deformation increases with increasing strain rate (Figure 43). Since the bulk sample deformation remains constant, the amount of matrix deformation decreases. This result suggests that the rate of inclusion strain hardening increases less rapidly than that of the matrix for increasing strain rates. The slopes of the strain hardening curve (da/de) increase significantly with increasing strain rates (E) for both metals (120) and NaCl type ionic solids.(39) Approximately a two-fold hardness increase occurred for pure NaC1 which was extruded through a 90~ - 75 per cent reduction die while a five-fold hardness increase occurred for pure copper. Thus, for the

-150 same strain, copper strain hardens much more rapidly than NaCl. Therefore, the quantity d(da/de)/di would be expected to increase more rapidly for copper than NaCli The increase in ram pressure with increasing strain rate is a result of the increase in the rate of strain hardening. A linear relationship exists between loge (RP) and loge (extrusion velocity) which is in agreement with the empirical relationship presented by Pearson and Parkins(92) (Equations (24) and (25) in Chapter II)o Conclusions - (Strain rate): 1o Ram pressure and inclusion deformation increase with increasing strain rateso 2. A linear relationship exists between loge (RP) and loge (extrusion velocity). 4. Temperature The shape of the L/W profiles changesabove 200~F (Figure 47), probably as a result of a change in the coefficient of friction at the sample surface. The stearic acid lubricant melts below 200~F and boils at 7210Fo Consequently, as the extrusion temperature increases, there is a range where the stearic acid portion of the lubricant is liquid while at higher temperatures, most of the lubrication is due to the graphiteo These changes in the character of the lubricant would be expected to have some effect on the amount of inclusion deformation and ram pressure. However, the results indicate that this effect is not large. The effect of the lubricant is minimized at the center of the sampleo As shown in Figure 47, the center inclusion L/W values have the same temperature dependency as the surface L/W values.

-151 On of the most interesting effects of temperature is the shape of the inclusion deformation-temperature curve (Figure 48). There is a minimum in this curve at approximately 300~Fo This minimum would not be expected on the basis of the ram pressure curves (Figure 49) or the hardness curves (Figures 50 and 52) for the individual phases. However, the final inclusion hardnesses show that a maximum inclusion hardness occurs at 300~F (Figure 51)o Consequently, the L/W minimum at approximately 300~F is the result of an increase in the inclusion hardness. At elevated temperatures, the inclusion hardness values would be below the values measured at room temperature. The L/W increase with temperature above 3000F is a result of decreasing inclusion hardness and yield strength values. In general, the amount of inclusion deformation increases as the metal/inclusion hardness ratio increases. This will be demonstrated in section E of this chapter. The exact reason for the intermediate temperature hardness increase is not known. It is not associated with pure NaCl, as indicated by the ram pressure curve (Figure 49) or the hardness curves (Figures 50 and 52). Consequently, it must be caused by an impurity picked up from the copper during sintering. To check this, Harshaw NaCl single crystals were heat treated under copper powder for 30 hours. After heat treatment, they were rapidly cooled to room temperature. The room temperature hardness was approximately 3 DPH greater than that shown in Figure 52. The hardness differential increased up to approximately 500~F and was time dependent. These results suggest that an impurity was picked up from the copper which leads to a precipitation hardening reaction. Precipitation hardening and overaging would be consistent with the hardness values shown in Figure 51. The

-152hardness increase observed for the Harshaw NaCi single crystals was not as large as that observed for the inclusions. However, the Harshaw crystals were much larger. Thus, the amount of impurity diffusing into the Harshaw crystals is probably much smallero For 90 Cu - 10 NaCl samples, the shape of the ram pressure curve is controlled primarily by the copper matrix. As temperature increases, the amount of strain hardening and hot hardness decrease and thus the yield strength and ram pressure decrease. This is also true for the pure copper and pure NaCl samples (Figure 49)o For the copper sample, the change in the ram pressure slope between 700~ and 1100~F is a result of recrystallizationo In contrast, the hardness curve for copper indicates that recrystallization is complete at 920 F (Figure 50). The difference in the recrystallization range between the ram pressure and hardness curves results from the time dependence of recrystallizationo The samples were at temperature for approximately five minutes after extrusion was finished, The ram pressure required to extrude the 90 Cu - 10 NaCl sample is less than that required to extrude the pure copper sampleo Since NaCl has a lower yield strength than copper, the addition of NaCl lowers the sample yield strength and the required extrusion pressure. The hot hardness curves for annealed, high purity copper and NaCl have the anticipated shape. As temperature increases, interatomic spacing increases, and dislocation motion is easier, As a result, the hot hardnesses of high purity Cu and NaCl decrease smoothly with increasing temperatureo The hot hardness data indicates that the intermediate temperature increase in yield strength which was reported for impure NaCl by Ekstein(42) and Eshelby, et al (41) does not occur for pure NaCl.

-153 Conclusions - (Temperature): 1. The amount of inclusion deformation is dependent on the inclusion hardness (yield strength). In general, the amount of inclusion deformation decreases with increasing inclusion hardness. 2. Ram pressure decreases with decreasing hardness and yield strength of the sample. Since yield strength generally decreases with increasing temperature, ram pressure generally decreases with increasing temperature. 3. The addition of a softer inclusion phase to a material decreases the sample yield strength and the pressure required for extrusion. 4. The hardnesses of annealed, high purity copper and NaC1 decrease smoothly with increasing temperature. 5. Per Cent Nonmetallic Material The shape of the L/W traverses are almost the same for all NaC1 volume fractions and thus it follows that the stress distributions are approximately the same. The stress level in a sample decreases with increasing NaCl volume fractions, as indicated by Figure 56. Thus, as the stress level transmitted to the inclusions decreases, the amount of inclusion deformation decreases (Figure 55). Copper strain hardens more than NaCl for equivalent strains. As the copper volume fraction increases, the amount of odpper;deformation increases and the hardness differential between the copper and the NaC1 increases. As the hardness differential increases, the stress transmitted

-154 to the NaC1 increases relative to the yield strength of NaCl. Thus, the curve for NaCi deformation curves upward as the NaCl volume fraction is reduced. If the inclusion material were more rigid than the matrix, the opposite effect would be expected. As the volume fraction of the harder inclusion material increased, the inclusion L/W ratio would increase. The linear decrease in ram pressure (Figure 56) indicates a linear decrease in the average sample yield strength. Since the ram pressure line runs between the ram pressures of the two pure components, the average yield strength of the two-phase system is given by TAMPLE = cW ~ Cu t * \ OC I c kx (39) where a is the average yield strength and f is the volume fraction of the indicated phases. The inclusions are relatively large and noncoherent. This linear relationship would not be expected for very small, hard inclusions. Conclusions - (Per cent nonmetallic material): 1. A decrease in the amount of the metal matrix reduces inclusion deformation when the inclusion is the softer of the two phases. The converse is true when the inclusions are harder than the metal matrix. 2. Ram pressure decreases linearly with decreasing average sample yield strengths. 6. Nonmetallic Inclusion Size Inclusion deformation was observed to decrease very slowly as inclusion size increased (Figure 60). As inclusion size increases,

-155 the amount of copper strain hardening near an inclusion would decrease slightly since NaC1 deforms more than the copper matrix. This results in a slight decrease in the stress acting on the inclusions as inclusion size increases and explains the observed data trend. For an inclusion which deforms less than the matrix, the opposite effect would be expected; i.eo, inclusion deformation would increase slightly for increasing inclusion sizes as observed experimentally by Pickering. (79) Scheil(80) reported no size effect for Cu2S in copper which is not inconsistent with the above interpretation for the scatter in his data. In any event, inclusion deformation does not change appreciably with inclusion size changes. The ram pressure was independent of inclusion size for inclusion diameters down to 48 microns (Figure 61). The mean free path for the 48 micron NaCl inclusions is approximately 300 microns. Edelson and Baldwin(88) state that the second phase must be harder than the matrix for particle strengthening to occur. Softer, noncoherent inclusions would not be expected to cause significant dislocation pile-ups and particle strengthening. For a hard inclusion phase, particle strengthening would be expected for low intraparticle mean free pathso The mean free path required for hardening would increase as the particle coherency increased. The fact that ram pressure does not change with particle size suggests that average sample yield strength is determined by the volume fractions of the sample phases. This result would not be expected for fine, coherent, harder second phases,

-156 Conclusions - (Nonmetallic inclusion size): 1. Inclusion deformation is almost independent of inclusion size. The results show a slow decrease in inclusion deformation with increasing inclusion size. 2. Ram pressure is independent of inclusion size. This result would not be expected for very fine, hard, coherent inclusions. 3. Average sample yield strength is determined by the volume fraction of the phases present. Again, this result would not be expected for very fine, hard, coherent inclusions. 7. Inclusion Type For a fixed matrix type, inclusion deformation decreases with increasing inclusion hardness values. This result would be expected since inclusion yield strength increases with increasing inclusion hardness. The final hardness values for the copper matrix range from 120.8 to 136.5 DPH. The results indicate that a harder inclusion (LiF) deforms less than the matrix (Figure 64) while the converse is true for softer inclusions (KC1, NaCl, NaF). These results are a consequence of the relative yield strengths of the two phases. The ram pressure required to extrude pure copper is 110,000 psi. The results presented in Figure 65 indicate that the addition of a second phase, which is harder than the matrix, increases the ram pressure while the converse is true for a softer second phase. Since ram pressure is linearly proportional to the average sample yield strength, the results

-157 show that the addition of a harder second phase increases the average yield strength of a material. The converse is true for softer inclusions. This result would be expected for two reasons: 1. The harder inclusions add to the yield strength on a volume fraction basis. 2o As the inclusion hardness increases, the amount of inclusion deformation decreases. Thus, for a fixed sample strain, the amount of matrix strain increases as the inclusion hardness increases and the amount of matrix strain hardening would increase. Conclusions - (Inclusion type): 1. For a fixed matrix type, inclusion deformation decreases with increasing inclusion hardness. 2. Inclusions which are harder than the matrix will deform less than the matrix. The converse is true for softer inclusions. 3. The addition of a harder second phase increases the average yield strength of a sample while the converse is true for the addition of a softer second phase. 4. Ram pressure increases as the average sample yield strength increases. 8. Matrix Type The stress levels acting on the inclusions increase as the matrix hardness increases. Consequently, the amount of inclusion deformation increases (Figure 69). This trend is not fully evident until the

-158 measured L/W ratios are corrected for the L/W ratios in the as-sintered samples. The lead matrix is softer than the NaCl inclusions while the converse is true for the other matrix materials studied. When the matrix is softer than the inclusions, the matrix undergoes more deformation than the inclusions (Figure 69). The converse is true for matrix materials which are harder than the inclusions. The sample yield strength increases with increasing matrix hardness and results in an increase in the average sample yield strength. Conclusions - (Matrix type): 1. Inclusion deformation increases as the matrix hardness increases. 2. When the matrix is softer than the inclusions, the matrix undergoes more deformation than the inclusions. The converse is true for matrix materials which are harder than the inclusions. 35 Ram pressure increases as the average yield strength of the sample increases. E. Plastic Deformation and Fracture The high ductility possible for NaCl type ionic solids is illustrated by the L/W results presented in section D of Chapter V. Inclusions which are softer than the matrix are more highly deformed than the matrix. Final NaCl L/W ratios of several hundred to one were obtained using a 180~ - 75 per cent reduction die. Consequently, with the proper stress state, NaCl-type ionic solids have high ductilities.

-159 The results for inclusion type and matrix type suggest that inclusion deformation is controlled, at least in part, by the relative hardnesses of the inclusion and matrix involved. To show the correlation more clearly, corrected center L/W ratios were plotted as a function of metal/inclusion hardness ratio (Figure 73). The center L/W ratio was used since a correction must be made for the varying L/W ratio in the as-sintered samples. The hardness of each phase, before and after extrusion, was averaged to obtain the values plotted. As indicated in Figure 73, three types of inclusion behavior are observed as the metal/inclusion hardness ratio increases. These types of inclusion behavior can be classified as follows: 1o Rigid behavior. Neither deformation nor fracture occur. 2o Brittle-ductile behavior. Both deformation and fracture occur. In this region, the amount of inclusion deformation increases and the amount of fracture decreases as the metal/inclusion hardness ratio increases. 3. Ductile behavior. The inclusions are highly deformed plastically and fracture is almost nonexistent. However, highly deformed inclusions do fracture occasionally. The three regions of inclusion behavior can be explained on the following basiso The inclusion hardness is directly related to the flow and fracture stress of the inclusions while stress is transmitted to the inclusions through the metal matrix. Consequently, as the matrix hardness increases, the stress level acting on the inclusions increases relative to the flow and fracture stress of the inclusions.

20 0 or -J LJ w Iz w 0 UJ:L 0 u 15 10 5 0 0 I 2 3 4 5 6 7 8 METAL HARDNESS INCLUSION HARDNESS Figure 73. The Effect of Metal/Inclusion Hardness Ratio on Inclusion Deformation and Type of Inclusion Deformation Behavior.

-161 For low metal/inclusion hardness ratios (M/I ratios), the stress transmitted to the inclusions is below the flow and fracture stress of the inclusions. Consequently, the inclusions behave as rigid bodies in the matrix. For intermediate M/I ratios, the stress acting on the inclusion exceeds both the fracture and yield strengths. The hydrostatic stress levels are not high enough to prevent the development of pure tensile stresses in the inclusions, and fracture occurs. As the M/I ratio increases in this region, the hydrostatic stress levels increase. Thus, inclusion deformation increases, and the fracture tendency decreases. For high M/I ratios, high hydrostatic stresses tend to prevent crack propagation. The stress levels are high relative to the yield strength of the sample and the inclusions deform readily. The results from the temperature study should show the same deformation behavior since the same die was used. Unfortunately, the phase hardness values are not known at temperature. The room temperature hardness values were extrapolated on the basis of the derivation presented in Appendix E and the results are presented in Figure 74. The extrapolation would be most valid for the -108, 80, and 1316~F points. These points follow the predicted behavior exactly. The precipitation hardening reaction shown in Figure 51 occurs in NaCl in the intermediate temperature range. Therefore, the slope of the loge (hardness) vs. temperature plot would be expected to change from the slope obtained for pure NaC1 in this temperature range. In addition, with an impurity picked up from the copper matrix, intermediate temperature NaC1 yield strength increases could be expected. These intermediate temperature increases are predicted by Eshelby, et al.(41) if divalent ions

20 0 In, I 3: z LJ 0 Q tLJ 0o I 0 0 1 2 3 4 5 6 78 METAL HARDNESS INCLUSION HARDNESS Figure 74. The Effect of Metal/Inclusion Hardness Ratio on Inclusion Deformation for Temperature Study Samples.

are present in the NaCl lattice. Both factors could increase the intermediate temperature inclusion hardness over the extrapolated values and improve the data fit. While all of the data from the temperature study do not fall on the L/W curve predicted from Figure 73, the general trends are the same. The degree of agreement is all that can be expected within the limitations of the hot hardness estimates. As the reduction in area increases, the shape of Figure 73 would be expected to change. The M/I ratio is indirect evidence of the stress level transmitted to the inclusions relative to the inclusion yield strength. Increasing the sample reduction increases the stress level acting on the inclusions which would not be evidenced by the M/I ratio in non-strain hardening materials. In these materials, an increase in sample reduction would have the same effect as an increase in the matrix hardness. The effect of other process variables on the shape of Figure 73 can be predicted on the basis of their effect on the stress levels transmitted to the inclusions. Figure 73 is significant for two reasons. It shows the effect of stress level on inclusion behavior and it shows the effect of relative metal and inclusion properties on inclusion behavior. Consequently, Figure 73 enables predictions to be made about inclusion behavior. Any deformation parameter which changes the metal/inclusion hardness ratio will change the inclusion deformation behavior as shown in Figure 73. For a fixed sample type, the major parameter effecting the M/I ratio is temperature. Less significant changes can be obtained by varying the strain rate for strain hardening materials (see section D-3 of this chapter).

-164 The following discussion applies for dilute inclusion concentrations. Sample deformation will not occur until the matrix yield strength is exceeded. Consequently, the initial M/I ratio for the sample [(M/I)s] is important. Types of inclusion deformation behavior shown in Figure 73 for M/I ratios lower than the (M/I)s ratio can not be obtained by increasing the sample reduction. However, the behavior range for higher M/I ranges can theoretically be reached by increasing the sample reduction, This is true for both strain hardening and non-strain hardening materials since, in general, the matrix strain hardens more rapidly than the inclusions. The same argument can be made for shear stress (TRw) levels in the sample. If the original (M/I)s value is in the deformation range, increasing the shear stress level will increase the amount of inclusion deformation. However, even when the shear stress level is reduced to zero, rigid or ductile-brittle behavior would not be expected. Consequently, for initially ductile inclusions, rigid behavior or fracture and dispersion could only be obtained by decreasing the M/I ratio for the sample. In general, the shape of Figure 73 is at least qualitatively consistent with the results reported in the literature. (See section B of Chapter IIo) Pickering(79) reported that iron oxide inclusions become increasingly plastic as the rolling temperature decreases from 1150 to 800~C. The author is not familiar with hot hardness data which would prove that this decrease is consistent with Figure 73. Conclusions - (Plastic deformation and fracture): 1o With the proper stress state, NaCl-type ionic solids have high ductilities.

-165 2. As the stress level transmitted to the inclusions increases, three types of inclusion deformation behavior are encountered. a. Rigid inclusions. No inclusion deformation occurs. b. Brittle-ductile inclusions. Both inclusion fracture and deformation occur. c. Ductile inclusions. High inclusion deformations occur. The zones where the various types of deformation behavior occur can be related to the metal/inclusion hardness ratios. 3. Figure 73 can be used to qualitatively predict the effect of sample and process changes on inclusion deformation behavior. 4. The ability to change the type of inclusion behavior is limited in part by the metal/inclusion hardness ratio. Inclusion behavior types occurring for M/I ratios lower than the sample M/I ratio can not be achieved by means other than changing the sample M/I ratio. The converse of this statement is not true.

CHAPTER VII CONCLUSIONS The conclusions will be summarized under the following headings: (A) Extrusion, and (B) Inclusion Deformation. A more detailed presentation of the conclusions is given at the end of the indicated sections of Chapter VIo Ao Extrusion For 75 per cent reduction dies, the stress required to change the direction of material flow at the die entry and exit sections is a moderately large portion of the ram pressure. Consequently, an accurate method of determining the contribution of the end effects must be established before Shield's stress analysis can be applied to practical extrusion problems, (VI-C) The ram pressure required for extrusion can be reduced by: lo Reducing the shear stress levels at the material surface through use of more effective lubricants and proper deformation equipment design (VI-D-l)o 2o Decreasing the amount of sample reduction (VI-D-2)o 3. Decreasing the extrusion velocity (VI-D-3)o 4. Increasing the extrusion temperature (VI-D-4)o 5 Decreasing the yield strength of the material being extruded (VI-D-4, 5, 7, 8)o -166

-167 Bo Inclusion Deformation With the confined stress states which occur in extrusion, NaCl-type ionic solids can have high ductilities (VI-E)o As the stress levels increase, three types of inclusion deformation behavior occur: 1o Rigid inclusions (neither fracture nor deformation occur). 2. Brittle-ductile inclusions (inclusion deformation and fracture occur). 3. Ductile inclusions (high inclusion deformations occur). The minimum stress acting on an inclusion is governed by the metal/inclusion hardness ratio (VI-E)o The amount of inclusion deformation can be decreased by the following process and sample changes: 1. Decreasing the shear stress levels at the material surface by the use of more effective lubricants and proper deformation equipment design (VI-A, B, D-l)o 2. Decreasing the total sample reduction (VI-D-2)o 30 Decreasing the strain rate (VI-D-3). 4. Proper operating temperature selection (VI-D-4)o 5, Changing the relative yield strengths of the two phases (VI-D-7. 8, E). 60 Changing the relative amounts of the two phases (VI-D-5). Inclusion size does not have a major effect on the amount of inclusion deformation (VI-D-6).

APPENDICES APPENDIX A MEASURED VSo TRUE L/W RATIO FOR AN ELLIPSOIDAL INCLUSION If the inclusions have an ellipsoidal shape, the measured and true L/W ratios are the same. The equation of an ellipsoidal inclusion is X + 4 - 1 ( 0) where a, b, and c are constants equal to 1/2 of the major axis of the inclusion in the x, y, and z directions respectively (Figure 75). Since, c= L/2 (41) and b = w/z where L and W are the inclusion length and width, we can write (x + (Y) + = (42) L2 W2 Ca The section through the inclusion in the plane of measurement has -a constant value of z. Consequently, Equation (42) can be written as () -a E (43) LZ VJ2 C where E is a constant for the plane of measurement, The measured length of the inclusion (2m) is 2x for y = O. Consequently, an - -x = (44) and aL ~ = 2 y - b (4)) Dividing Equation (4-) by Equation (45), we have _ _ L (46) UJ, VW -168

-169 z x Figure 75. Section Through an Ellipsoidal Inclusion.

-170Consequently, for an ellipsoidal inclusion, the measured L/W value (Rm/cm) is the true L/W value.

APPENDIX B MODIFICATION OF SHIELD'S STRESS ANALYSIS The stress analysis used is a modification of Shield'sl analysis for plastic flow in a converging connical channel. In this appendix, the method is modified to apply to extrusion. This modification parallels that presented by Shield for wire drawing. The dimensions used for the analysis and the directions of the principal stress components are shown in Figure 6. The assumptions are the same as those presented by Shield, 1o General Equations For the von Mises yield condition, the stress components are given by the equations r =k-i (47) C - =- -k=i'3(T-) = k(R)-3k r (48) where?AR =- C lo~(R+A and k' = o/= (50) The quantity T(cp) is given by oT'f( +q ~ZcoItT T + 25/( =C G(51) -171

-172 A and C are constants of integration to be determined by the boundary conditionso At the die surface, ips - n k is a constant < 1. Thus, the boundary conditions on (52) where m T(Cp) are coci(O)=0 - (53) coefficient of friction at the die surface is unknown, m is Since the unknown o 2o Calculation of C The constant C is determined from the condition at the die entrance (at Zi). Here, the stress across the section is assumed to equal the extrapolated ram pressure (RP), The stresses at the die entrance and exit sections are given by G-)Z = - k C lo (Ze)+ k L 4) u-z.~F= - kC ioge (ZF4) k L (? (55) where L(_) =A+- iT3( l)'( + c^s 2 f)-3jrclf - 3s T o a? Subtracting (55) from (54), we have O-'z- -rz kC loe (Zh (56) (57)

-173 - Since the ram pressure is compressive, the mean extruding stress over the entrance section is given by - rrzto (- RP)=J i 2 Z a s ec d f 0 (58) For the mean stress over the exit section being 0, we have fj -z 2rr Z ta. T seccp?jT = 0 (59) The substitution of Equation (50) into Equation (58) leads to the result 1, - d, IAA - RP) [cz.;+ kC PR r Z. c, \ Ts c'-,pJ VI~ Z-f L" eC~ 9 L i~~~~ (60) Cancelling like terms and using Equation (59), we have tcx YN a (-R-P) = 2 C c,y5 J\ ~t,, cp sec'3 d (p I 0 (61) By noting that seco _ c (tO"v ) (62) J(P Equation (61) reduces to -(RP-)= kC C.1(0 2z (63) Substituting Equation (50) in Equation (63) and solving for C leads to the result F/f(RP) C = e(64) Equation (64) yields the value of the constant C to be used in solving Equation (51).

-174 3o Calculation of T(cp) Equation (51) can not be solved directly to determine T(p) o It was solved incrementally using a computero The computation was started with the known boundary condition T(O) = 0o Using Equation (51) in the following form, the slope of the T vso cp curve was determined for T = c =0. At = C-Tco. - _ 3 -T1)" Ac y (65) Multiplying the slope by a small Acp (Ap = cpj + 1 - cpj), the value of was determined at Tj + 1o This process was continued to obtain corresponding values of T and p. Whenever T was greater than 1, T was taken as 1o A value of T = 1 corresponds to shear through the sample, 4, Calculation of the Constant A The value of A must be known to solve Equations (48) and (49). This constant is determined from the fact that the mean stress at the die exit is equal to zero. From Equation (56), we have J tc^, Tc Secz yoI f 0 (66) 2TM Z a T tcn CD se c> d p = 0 (66) o azf is given by Equation (52) as = -kC\< loe z10 kZ; 4 1a 3 (12( os2?c)3ftd' -'3c, K1l (67) Substituting Equation (66) into (67) and solving for A leads to the result A [c \~ e J j/(t )(I+cos2p)+ 3ftrkp +=s i ta" p secs a d( It Ta r( p C8)

-175 Equation (65) was solved incrementally using a computer. following. Calculational Procedure The stress values calculated are all divided by aO. The sequence of operations was used: 1. Calculate C from Equation (64). 2. Determine T as a function cp using an incremental solution to Equation (51). 3. Determine the constant A from Equation (68) using an incremental solution. 4. Calculate TR -Jo from Equation (48). 5. Calculate aR/Uo, a0/-'o, and aG/o from Equations (48), (49), and (68).

APPENDIX C EXPERIMENTAL RAM PRESSURE AND "MEAN L/W VALUE" DATA Sample 9OCu-IONaC1 it T! if It t! I! it it 11 IT t! it It II,, I, I, 95Cu-5NaC1 9OCu-IONaC1 8OCu-2ONaC1 5OCu-5ONaC1 1OONaCI 9OCu-lONaC1 I, 11 it I! * Die 60~ -75o 90~ -75o L350 -75% 155~ -75o 180~ -755 90~-30.950 90~-55.5% 90~ -73.5 90 -88.5% 90 -75% ii it It 11 If 11 It,I IT tt It 11 I, if It 90~ -75o if It tt It Ram Pressure, Kpsi 89.o 104.0 126.0 130.0 25.1 65.0 104.0 173.6 102.0 o14.0 106.6 109.3 137.5 104.0 86.3 78.3 76.1 67.7 60.5 43.0 24.9 24.5 109.0 104.0 90.5 67.5 31.0 105.5 104.0 o00.5 100.0 102.8 103.5 Measured i Z (L/W)iAi At 6.5 15.4 38.2 55.5 4.5 6.6 15.4 43.3 14.8 15.4 17.2 18.1 15.0 15.4 10.5 6.7 8.3 153.5 18.2 20.3 24.6 16.6 15.4 12.5 9.9 8.0 15.1 15.5 14.o 15.6 16.0 14.7 Variable Die Angle. 11 11 ii tt tt Percent It 1i I1 Reduction It Ram Velocity(0.05ipm) " it ( O.lOipm) It " i (0.30ipm) (0.7lipm) Temperature (-108~F) (75 F) (2000F) (300~F) (395oF) (497~F) (710~F) (920~F) (11390F) (1316F) Percent NaCI II It II II Inclusn S (.275mm) Inclusion Size(0.275mm) 1" " (0.214mm) "f I" (O.163mm) It i" (0.137mm) " " I(0.081mm) "t "i (o.048mm) 90Cu-1OKC1 9OCu-IONaCI " 9OCu-1ONaF " 90Cu-1OLiF 9OPb-lONaCl 90~-75% 90Al-1ONaCl " 9OAg-1ONaCl 9OCu-IONaCI " 90 Brass-lONaCi 95.0 o14.0 105.0 115.0 4.9 50.3 79.3 lo4.o 186.5 147.7 110.0 95.0 78.0 80.9 51.4 34.0 17.6 15.4 10.1 3.6 1.0 16.6 18.1 15.4 22.2 11 it 11 I! ii 11 Inclusion Type It II It II ii t1 Matrix Type It II It II ii It Temperature(-108~F) "t (820F),I (307~F) (50o5F) (705~F),, (9100F) ", (11150F) lOOCu I! 1i 1i I! 11 ni It II II It ItI II *The included die angle and per cent reduction in area are listed.

-177 Sample 90Pb-lONaCl 90Al-1ONaCi 90Ag-O1NaCl 90Cu-lONaCl 90 Brass-lONaCl APPENDIX D MATRIX AND INCLUSION HARDNESS VALUES Mat r i x Hardn e s s Before Extrusion After Extrusic 4.0 4.0 30.8 60.1 36.8 97.0 40.5 120.8 61.1 210.2 Inc lusion Har dn ess Annealed Single After Dn Crystals Extrusion 17.4 26.6 17.4 45.8 17.4 31.4 17.4 30.0 17.4 42.2 90Cu-10KC1 90Cu-O1NaF 90Cu-1OLiF 90Cu-10NaCl(-108~F) 90Cu-lONaCl(75~F) 90Cu-1ONaCl(200~F) 90Cu-lONaCl(300~F) 90Cu-10NaCl(497~F) 90Cu-lONaCl(710~F) 90Cu-10NaCl(920~F) 90Cu-10NaCl(1139~F) 90Cu-lONaCl(1316~F) 42.2 40.5 42.0 40.5 40.5 40.5 40.5 40.5 40.5 40.5 4o.5 40.5 4o.5 42.0 42.0 42.0 42.0 42.0 42.0 42.0 42.0 42.0 113.0 132.0 136.5 154.0 120.8 110.0 101.0 89.0 63.0 54.4 55.9 56.5 10.9 58.2 116.2 17.4 17.4 17.4 17.4 17.4 17.4 17.4 17.4 17.4 21.3 109.7 193.3 31.4 30.0 42.6 52.7 48.5 37.4 34.4 32.3 30.6 lOOCu lOOCu lOOCu 1OOCu lOOCu 1OOCu lOOCu (-108~F) (82~F) (3070F) (550~F) (706~F) (910~F) (llL6~F) 141.5 131.0 125.0 117.0 105.5 63.1 62.7

APPENDIX E HARDNESS VALUE EXTRAPOLATION For the temperature effect study, it would be desirable to know the hardnesses of the sample phases at temperature. Room temperature hardness values were known for the NaCl inclusions and the copper matrix after extrusion at elevated temperatures. In addition, the hardness-temperature relationship was known for each pure phase. Based on this information, a method was sought to extrapolate the room temperature hardness data to elevated temperatures. The extrapolation method used will be presented in this section. As a first approximation, the hardness was assumed to obey the following relationship - BT H - Ae (69) where H is the hardness, T is the absolute temperature, and A and B are constants. Differentiating this equation leads to the result J H -H B (7) ___-R _(70) Equation (70) shows that the rate of hardness decrease with increasing temperature is a function of the hardness. Between room temperature and an elevated temperature, the average value of (AH/AT) was known for the pure annealed phaseso Based on Equation (70), we can write ( 1 T = 2L) H B) (71) \d T/ LT /p Hp Bp where p represents the pure, annealed phase and u represents the -178

-179 phase where only the room temperature hardness is known, Assuming Bu = Bp, we can write /H _ H, (72) (T/ oT H Using the value of (H/AT) calculated from Equation (72), the elevated temperature hardness can be obtained from the following equation AH\ (Te-To) = (H -H) (73) where the subscripts e and o denote the property of interest at the temperature of interest and room temperature, respectively. In general, Bu would not equal Bp. The difference between these two quantities would be expected to increase as precipitation hardening occured in the NaCl inclusions. Consequently, the extrapolation for NaCl hardness values would get worse in the intermediate temperature range. At best, this extrapolation method would only be expected to yield an approximation to the elevated temperature hardness values.

BIBLIOGRAPHY 1o Van Vlack, L. Ho, 0. Ko Riegger, R, Jo Warrick, and J. M, Dahl, "Sulfide Inclusions in Steel," Trans. AIME, 221, 220-228, (1961). 2, Van Vlack, Lo Ho, Elements of Materials Science, Addison-Wesley Publishing Co, Inc., Reading, Massachusetts, (1959) 3. Reusch, Eo, "On a Special Kind of Deformation in Rock-Salt and Calcite," Poggo Ann,, 132, 441, (1867)o After ref. 19o 4. Coblentz, W. Wo, "Note on the Bending of Rock-Salt," Physo Revo, 16, 389, (1903). 5. Milch, Lo. "The Increase of Plasticity of Crystals with Rise of Temperature," Neu, Jahrbo Min. Geolo Pal,, 60-72, (1909). 60 Parker, Eo Ro, J. Ao Pask, Jo Washburn, Ao Eo Gorum, and Wo Luhman, "Ductile Ceramics - A High Temperature Possibility," J. Metals, 10, 351-353, (1958) 7. Gorum, Ao Eo, E. Ro Parker, and J. Ao Pask, "Effect of Surface Conditions on the Room Temperature Ductility of Ionic Crystals," Jo Amo Ceramo SoC., 41, 161-164, (1958), 80 Scheil, Eo, "Statistical Structure Investigations. Io," Zeitschrift fur Metallkunde, 27, 199-209, (1935)o 9. Phillips, Jro, Wo Lo, "Deformation and Fracture Processes in Calcium Fluoride Single Crystals," J. Am, Ceram. Soc., 44, 499-506, (1.961) 10. Burn, Ro and Go T. Murray, "Plasticity and Dislocation Etch Pits in CaF2," J. Amo Ceram. Soc., 45, 251-252, (1962). 11. Wachtman, Jr., J. B., and L, H. Maxwell, "Plastic Deformation of Ceramic-Oxide Single Crystals," J. Amo Ceram. Soco, 37, 291-292, (1954). 12, Wachtman, Jr., Jo Bo, and L, H, Maxwell, "Strength of Single Crystal Saphire and Ruby as a Function of Temperature and Orientation," Jo Am. Ceramo Soco, 42, 432-433,(1959)o 13o Bailey, S. W,. Ro Ao Bell, and Co Jo Peng, "Plastic Deformation of Quartz in Nature, "Geological Society of America Bulletin, 69, 1443-1466, (1958)o 14, Turner, Fo J., "Experimental Deformation of Calcite Crystals," Geological Society of America, Bulletin, 65, 883-933, (1954)o 15. Howes, Vo Ro, and So Tolanski, "Pressure Crack-Figures on Diamond Faceso, o The Octahedral Face," Proc. Royal Soco, (London) A230, 287-293,(1955), also" IIo The Dodecahedral and Cubic Faces," ibid, 294-301o -180

-181 16o Schmid, E., and Wo Boas, Kristallplastizitaet, Springer, Berlin (1936). English translation, Plasticity of Crystals, F. A. Hughes and Co., Ltd., London, (1950). 17. Laurent, Jo F., "Plastic Deformation in Ionic Crystals," Le Journal de Physique et le Radium, 19, 602-611, (1958). 18. Pask, J, A. "Mechanical Properties of Ceramic Materials," 288318 of Mechanical Behavior of Materials at Elevated Temperatures, J. E Dorn Ed.,., York, 1 61, Dorn, Ed., McGraw-Hill Book Co., Inc, New York, (1961). 19, Gilman, J. Jo, "Mechanical Behavior of Ionic Crystals," 146-199 of Progress in Ceramic Science, Jo Eo Burke, Ed., Pergamon Press, New York, (1961). 20. Dommerich, So, "Festigkeitseigenschaften bewasserter Salzkristalls. VIo Richtungsabhangigkeit der Streckgrenze gleichmassig abgeloster Steinsalzstabvhen," Zeito Physik, 190, 189-196, (1934) 21. Pratt, P. Lo, "Similar Glide Processes in Ionic and Metallic Crystals," Acta Met., 1, 103-104, (1953). 22. Buerger, M. J., "Translation-Gliding in Crystals of the NaCl Structural Type," Amer. Mineralogist, 15, 174-187, (1935). 23. Gilman, J. J., "Plastic Anisotropy of LiF and Other Rock-Salt Type Crystals," Acta Met., 7, 608-613,(1959). 24. Shubin, G. V., "The Question of Brittleness and Plastic Deformations of NaCl," Izvest. Vysshikh Ucheb. Zavedenii, Fiz., No. 1, 98-103, (1961), 25. Tammann, G., and W. Salge, "Octahedral Glide of NaCl at High Temperatureso Decrease in the Yield Point with Rising Temperature," Neues Jahrbo Mineral, Geol. Palaon, suppl. 57, 117, (1927). 26. Rachinger, W. Ao, "Glide in Lead Telluride," Acta Met., 4, 647-649, (1956). 27, Huntington, Ho Ro, Jo Eo Dickey, and R. Thompson, "Dislocation Energies in NaCl," Phys. Rev, 100, 1117-1128, (1955). 28, Van Zeggeren, F., and G. C. Benson, "Calculations of the Surface Energies of Alkali Halide Crystals," Jo Chem. Phys., 26, 1077-1082, (1957) 29. Sears, G. Wo, "Strength of LiF Whiskers," J. Chem. Phys. Solids, 6, 300, (1958)o 30. Gyulai, Z. Z. "Festigkeits- und Plastizitatseigenschaften von NaClNadelkristallen," Zeit. Physik, 138, 317-321, (1954).

31. Hulse, Co 0O, "Formation and Strength of Magnesia Whiskers," J. Am. Ceram. Soco, 44, 572-575, (1961)o 32o Stokes, Ro J,, "Dislocation Sources and the Strength of Magnesium Oxide Single Crystals," Transo AIME, 224, 1227-1237, (1962)0 33. Johnston, Wo Go and J. Jo Gilman, "Dislocation Velocities, Dislocation Densities, and Plastic Flow in Lithium Fluoride Crystals," Jo Applied Physics, 30, 129-144, (1959)o 34, Edner, Ao, "Einfluss von Fremdzusatzen auf die Kohasionsgrenzen und die ultramikroskopische Solbildung Synthetischer Steinsalzkristalle, " Zeit, Physik, 73.9 623-648, (1932). 35. Schoenfeld, Ho, "Einfluss von Fremdzusatzen auf die Kohasionsgrenzen und die ultramikroskopische Solbildung Synthetischer Steinsalzkristalle," Zeito Physik, 75, 442-467, (1932). 36. Metag, Wo, "Einfluss von Fremdzusatzen auf die Kohasionsgrenzen und die ultramikroskopische Solbildung Synthetischer Steinsalzkristalle, Zeito Physik, 78, 363-374, (1932)o 37. Luhman, Wo J. and A, Eo Gorum, "The Effect of Impurities on the Strength of Silver Chloride," Acta MetO, 7, 685-687, (1959). 38, Gorum, Ao Eo, Wo Jo Luhman, and Jo Ao Pask, "Effect of Impurities on the Heat Treatment and Ductility of MgO," J. Am. Ceram. Soc., 43, 241-245, (1960)o 39o Phillips, Jro, Wo Lo, "Effect of Strain Rate and Temperature on the Stress-Strain Characteristics of NaC1, LiF, and MgO Single Crystals," Transo AIME, 224, 434-436, (1.962) o 40o Pratt, Po L, "Point Defects and Mechanical Properties of Ionic Crystals," Inst. of Metals Monograph and Report Series, 99, (1957)o 41. Eshelby, Jo Do, Co W, Ao Newey, Po Lo Pratt, and Ao Bo Lidiard, "Charged Dislocations and the Strength of Ionic Crystals," Philo Mago, 3, 75-89, (1958). 42, Ekstein, H,, "Torsion of NaCl," Unpublished work, quoted by Schmid, Eo, and W. Boas, Plasticity of Crystals, Fo Ao Hughes and Co., Ltd,, London, (1950). 43. Westbrook, J. H,, "Flow in Rock-Salt Structures," WADC Tech. Report, 58-304, ASTIA Document AD 15565, General Electric Co., Research Lab., (1958), After ref. 18. 44. Cottrell, Ao H., Dislocations and Plastic Flow in Crystals, Oxford Press, London, (1953).

-183 45. Vaughan, W. H. and J. W. Davisson, "Surface Mobility of Dislocations and the Joffe Effect," Report of NRL Progress, (April, 1959). After refo 19. 46. Clarke, F. J. P. and R, A. J. Sambell, "Micro-cracks and their Relation to Flow and Fracture in Single Crystals of Magnesium Oxide," Phil. Mag., 5, 697-707, (1960). 47. Stokes, R. J., T. L. Johnston, and Co H. Li, "Evironmental Effects on the Mechanical Properties of Ionic Solids with Particular Reference to the Joffe Effect," Trans. AIME, 218, 655-662, (1960). 48. Kronberg, Mo L., and J. E, May, "Ductility of Magnesium Oxide Single Crystals," J. Am. Ceram, Soc., 43, 438, (1960)o 49. Lad, R. A., C. A. Stearns, and Mo G. DelDuca, "Factors Affecting the Plasticity of Ionic Crystals," Acta Met,, 6, 610-611, (1958), 50. Aerts, Eo and W. Dekeyser, "Gases in Rock Salt and the Joffe Effect," Acta Meto, 4, 557-558, (1956), 51. Class, W,, Eo So Machlin, and G.To Murray, "NaCl Embrittlement by Surface Compound Formation, AIME Trans., 221, 769-775, (1961). 52o Murray, Go To, "Brittle-Ductile Transition Temperatures in Ionic Crystals," J, Am. Ceram. Soc., 42, 330-334, (1960)o 53. Bassett, Go A., "The Plasticity of Alkali Halide Crystals," Acta Met., 7, 754-755, (1959). 54. Stokes, R. J., T. L. Johnston, and C, H. Li, "Crack Formation in MgO Single Crystals," Phil. Mag., 3, 718-725, (1958), 550 Stokes, R. J., T. L. Johnston, and C. H. Li, "Further Observations of Stroh cracks in Magnesium Oxide Single Crystals," Phil. Mag., 4, 137-138, (1959). 56. Washburn, J., A. Eo Gorum, and Eo R. Parker, "Causes of Cleavage Fracture in Ductile Materials," Trans. AIME, 215, 230-237, (1959)o 57. Keh, A. S., J. C, Mo Li, and Y. T. Chou, "Cracks due to the Pilingup of Dislocations on Two Intersecting Slip Planes in MgO Crystals," Acta Met., 7, 694-696, (1959). 58. Stokes, Ro J., To Lo Johnston, and C. Ho Li, "Effect of Slip Distribution on the Fracture Behavior of MgO Single Crystals," Phil. Mag., 6, 9-24, (1961). 59. Stroh, A. N., Proc. Roy. Soc., A, 223, 404, (1954); ibid., 232 548, (1955); Advance Physo, 6, 418, (1957).

-184 60. Cottrell, Ao H., Theory of Brittle Fracture in Steels and Similar Metals," Trans. AIME, 212, 192-203, (1958). 61. Melankholin, N. Mo, and Vo R. Regel, "Investigation of the Destruction Process of NaC1," Soviet Phys., JETP., 2, 696-702, (1956). 62. Bridgman, P. Wo, Studies in Large Plastic Flow and Fracture, McGrawHill Book Co,, Inc, New York, (1952). 63. Bridgman, Po Wo, "The Effect of Hydrostatic Pressureon the Fracture of Brittle Substances," Jo Appl. Phys., 18, 246-258, (1947), 64. Griggs, Do, "Experimental Flow of Rocks under Conditions Favoring Recrystallization, Geological Society of America, Bulletin, 51, 1001-1022, (1940)o 65. "Metallurgical Factors Affecting the Machinability of Steel," Metals Handbook, 1948 Edition, 369-371, The American Society for Metals, Cleveland, Ohio, 66. Knowlton, H. B., "Machinability of Wrought Metals," Machining of Steels, American Society for Metals, (1938). 67. Boulger, Fo Wo, Ho Ao Moorhead, and T. Mo Garvey, "Superior Machinability of MX Explained," Iron Age, 167 90-95, (1951) 68. Cummings, Ho No, F, B. Stulen, and Wo C. Schulte, "Relation of Inclusions to the Fatigue Properties of SAE 4340 Steel," Trans. ASM, 49, 482-510, (1957). 69. Epremian, E. and R. Fo Mehl, "Investigation of Statistical Nature of Fatigue Properties," National Advisory Committee for Aeronautics, Tech. note 2719, (June, 1952). 70. Frith, P, H,9 "Fatigue Tests on Rolled Alloy Steels made in Electric and Open-hearth Furnaces," I.S.I.. Spec, Rep. No. 50, (1954). 71. Gadd, Eo Ro, "Fatigue from the Metallurgist's Viewpoint," J. Royal Aeronautical Society, 57, 565-580, (1953). 72, Ransom, J. To, "The Effect of Inclusions on the Fatigue Strength of SAE 4340 Steel," Transo ASM, 46, 1254-1263, (1954)o 73. Stulen, Fo B,, "On The Statistical Nature of Fatigue," American Society for Testing Materials Symposium on Statistical Nature of Fatigue, STP No. 121 23, (1952)o 74o Stewart, Wo C. and Wo Lo Williams, "Effects of Inclusions on the Endurance Properties of Steels," Journ. Am, Soco Naval Engrs,, 60, 475-504, (1948)o

-185 75. Gagnebin, A, Po, "Selenium Additions to Cast Steel-Influence on Sulfide Inclusions and Ductility," American Foundryman, 12, 43-52, (1947)o 76, "Nonmetallic Inclusions in Steel," Metals Handbook, 1948 Edition, 445-452, The American Society for Metals, Cleveland, Ohio. 77. Ogawa, S. Y., T. B, King, and No J. Grant, "Deformation and Fracture Characteristics of Fe-S, Fe-S-O, and Fe-S-Mn Alloys at High Strain Rates and Temperatures," Trans. AIME, 224, 12-18, (1962). 78. Kinzel, Ao Bo and Walter Crafts, "Inclusions and their Effect on the Impact Strength of Steel," Iron Age, 127, 781-783,(1931) 79. Pickering, F, B,, "Some Effects of Mechanical Working on the Deformation of Nonmetallic Inclusions," J. Iron and Steel Inst., 189, part 2, 148-159, (1958). 80. Scheil, E. and R. Schnell, "Mitt. Forsch Ver. Stahlwerke demnachst," after Scheil, E., "Statistische Gefugeuntersuchungen I," Zeitschrift fur Metallkunde, 27, 199-209, (1935). 81. Dehlinger, U., Zeitschrift fur Metallkunde, 26, 227, (1934) After ref. 8. 82. Van Vlack, L. H., "Correlation of Machinability with Inclusion Characteristics in Resulphurized Bessemer Steels," Trans. ASM, 4_, 741-753, (1953). 83. Scheil, E. and W. Sibert, "Statistical Researches on Structure. IV. The Deformation of Inclusions of Iron Alumnide and Silicon Crystals in Aluminum on Rolling," Zeitschrift fur Metallkunde, 32, 288-289, (1940). 84. Puttick, K. E., "The Structure, Deformation, and Fracture of Pearlite, Part II: Deformation and Structure," Jo Iron and Steel Inst., 185, 167-176, (1957). 850 Gurland, J., "Metallographic Study of the Deformation of a Particle Strengthened Alloy, WC-Co," Division of Engineering Report, Brown University, Providence, R. I., Nonr 562(19)/o1, (May, 1958). 86. Nishimatsu, C. and Jo Gurland, "Experimental Survey of the Deformation of a Hard-Ductile Two-Phase Alloy System, WC-Co," Division of Engineering Report, Brown University, Providence, Ro Io, Nonr 562(19)/2o, (Sept., 1958). Also, Trans. ASM, 52, 469-484, (1960). 87. Hill, R., The Mathematical Theory of Plasticity, Oxford Press, London, (1950)o

-186 88, Edelson, Bo Io, and Wo Mo Baldwin, Jr., "The Effect of Second Phases on the Mechanical Properties of Alloys," Trans. ASM, 55, 230-250, (1962). 890 Geisler, A., "Precipitation from Solid Solutions of Metals," Phase Transformations in Solids, John Wiley and Sons, New York, (1951). 90O Koppenaal, To J., and No M, Parikh, "Microstraining in FiberReinforced Siver," Trans. AIME, 224, 1173-1176, (1962)o 91. Keeler, J. Ho, "Tensile Properties of Zr-Cr Alloys-Particle Strengthening Effects," ASM Transactions, 48, 825-842, (1956)o 92. Pearson, Co Eo, and Ro No Parkins, The Extrusion of Metals, 2nd Edition, John Wiley and Sons, Inc., New York, (1960). 93O Bishop, Jo Fo W., "The Theory of Extrusion," Metallurgical Reviews, 2, 361-390, (1957)o 94, Wistreich, Jo Go, "Fundamentals of Wire Drawing," Metallurgical Reviews, 3, 97-142, (1958)o 95, Sachs, Go and Wo Eisbein, "Power Consumption and Mechanism of Flow in the Extrusion Process," Mitto Material, 16, 67-96, (1931)o 96. Johnson, W,, BoIoSoR.Ao Report, MW/E/55/54. After ref. 92. 97o Fink, Co, Zo Bergwes, Prevsz., 22, 200, (1874). After refo 92. 98. Siebel, Eo and Eo Fangemeier, "Researches on Power Consumption in the Extrusion and Punching of Metals, Mitt, Ko Wo Inst. fur Eisenforschung, 13, 29-43, (1931)o 99. Hoffman, 0. and Go Sachs, Introduction to the Theory of Plasticity for Engineers, McGraw-Hill Book Company, Inco, New York (1953)o 100o Treco, Ro M,, "Theoretical and Experimental Analysis of the Extrusion Process for Metals," Transo ASM, 55, 697-718, (1962)o It 101. Hencky, H., "Uber einige statisch bestimmte taelle des Gleichgewichts in plastische Gleichgewicht," Zeit, angew, Math, und Mecho, 3, 241-251, (1923)o 102, Geiringer, Ho, Proc. Third International Congress for Applied Mechanics, 2, Stockholm, (1930) 103. Prager, W. and P. Go Hodge, Jro, Theory of Perfectly Plastic Solids, John Wiley and Sons, Inco, New York, (1951)o

-187 104. Hill, Ro, "A Theoretical Analysis of the Stresses and Strains in Extrusion and Piercing," Jo Iron Steel Inst. 158, 177-185, (February, 1948). 105. Johnson, W., "Extrusion Through Wedge Shaped Dies, Part I," J. Mech, Physo Solids, 3, 218-223, (1955)o 106. Johnson, W., "Extrusion Through Wedge Shaped Dies, Part II," J. Mech. Phys. Solids, 3, 224-230, (1955). 107. Johnson, W., "Extrusion Through Square Dies of Large Reduction," Jo Mech. Physo Solids, 4, 191-198, (1956)o 108. Purchase, No W. and S. J. Tupper, "Experiments with a Laboratory Extrusion Apparatus Under Conditions of Plane Strain, " J. Mech Phys. Solids, 1, 277-283, (1952-3) 109. Johnson, W,, "Experiments in Plane-Strain Extrusion," J. Mech, Phys, Solids, 4, 269-282, (1956)o 110o Dodeja, Lo C. and W, Johnson, "Experiments in Plane-Strain Extrusion," J. Mech. Physo Solids, 5, 269-282, (1957). 11 Shield, Ro To, "Plastic Flow in a Converging Conical Channel," Jo Mecho Physo Solids, 3, 246-258, (1955). 112, Frisch, J. and E. G. Thomsen, "An Experimental Study of Metal Extrusions at Various Strain Rates," Trans. ASME, 76, 599-606, (1954). 113. Thomsen, E, Go, Co Yang, and J. B, Bierbower, "An Experimental Investigation of the Mechanics of Plastic Deformation of Metals," University of California Publications in Engineering, 5, University of California Press, Berkeley, (1954), 114, Thomsen, Eo Go, and J, Frisch, "Stresses and Strains in Cold Extruding 2S-0 Aluminum," Trans. ASME, 77, 1343-1353, (1955). 115. Thomsen, E. G., "Plasticity Equations and Their Application to Working of Metals in the Work Hardening Range," Transo ASME, 78, 407-412, (1956). 116, Jordan, To Fo, and E, G. Thomsen, "Comparison of an Unsymmetric Slip-Line Solution in Extrusion with Experiment," J. Mech. Phys. Solids, 4, 184-190, (1956)o 117o Johnson, Wo, "The Pressure for the Cold Extrusion of Lubricated Rod Through Square Dies of Moderate Reduction at Slow Speeds," Jo Inst. Metals, 85, 403-408, (1956-57)o

-188118o Schishokin, W. Po, "The Extrusion Properties of Metals and Alloys at Different Temperatures," Zhur. Priklado Khimii, i, 663, (1929)o After ref. 92. 119o Taylor, Go Io and Ho Quinney, "Combined Torsion and Tension of Thin Tubes, "Philo Trans. Roy. Soco A, 230, 323-362, (1931)o 120o Barrett, Co S., Structure of Metals, McGraw-Hill Book Co Inc, New York, (1952). 121o Moore, Jo W., Personal Communication. 122. Chao, Ho C., Personal Communication.