THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING AN INVESTIGATION OF THE ANISOTROPY OF TENSILE AND COMPRESSIVE PROPERTIES OF COMMERCIALLY PURE ALUMINUM PRESTRAINED UNDER SIMPLE DIRECT STRESS Alimullah Khan A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Science in the University of Michigan Department of Mechanical Engineering 1963 January, 1964 IP-656

Doctoral Committee: Professor Joseph Datsko, Chairman Professor Robert M. Haythornthwaite Assistant Professor Alexander Henkin Professor Robert C. Juvinall Professor Clarence A, Siebert ii

ACKNOWLEDGMENTS The author wishes to acknowledge his appreciation and indebtedness to everybody who aided in this investigation, particularly: Professor Joseph Datsko, Chairman of the Doctoral Committee, except for whose encouragement, guidance, advice and help in all possible ways, it would have taken a great deal longer to complete this investigation than it has. Professors Robert Mo Haythornthwaite, Robert Co Juvinall, Clarence A. Siebert and Alexander Henkin, members of the Doctoral Committee for their advice and assistanceO The Agency for International Development and the Government -of East Pakistan, for financial support to meet his expenses here, in Pakistan and for transportation and travel. Professor Gordon Jo Van Wylen, Chairman, Mechanical Engineering Department and the General Electric Company for financial support for -the research, The staff of the Mechanical Engineering Machine Tools Laboratory for their assistance in preparing specimens of very exacting qualityo The Civil Engineering and Engineering Mechanics Departments for allowing the author to use their laboratory facilitiesO The members of the staff of the Industry Program of the College of Engineering, the University of Michigan, for their assistance in preparation and reproduction of this dissertation0 The friends who helped to keep the morale high during the long period of experimental worko Last, but by no means the least, the author's wife, Mrs, Quazi Jahan Ara Khan, who shouldered all the family responsibilities for more than two years, which made the endeavor possible4 iii

TABLE OF CONTENTS Page ACKNOWLEDGMENTS................................................ LIST OF TABLES............................................. LIST OF FIGURES................................................ CHAPTER I. INTRODUCTION...................................... iii vii viii 1 IIo THEORETICAL CONSIDERATIONS....................... General Remarks................................ Yield Conditions.................................. Strain Hardening...........................o.. Stress-Strain Relations.......................... Anisotropic Yield and Flow..............o........ Physical Theory............................... III. REVIEW OF LITERATURE................................ General Remarks................................ Theoretical Investigations........................ Experimental Investigations.................. IVo EXPERIMENTAL PROGRAM................................ A. Selection of Material.................. Reasons for Selecting Aluminum.............. Fabrication Procedure................o.... B. Program Outline......................... C. Experimental Procedure................. o. 8 8 10 12 14 17 17 20 20 25 32 45 45 45 46 47 49 Processing of Material........... Designation of Prestrain......... Annealing....................... Prestraining.................. Strain Rate.................. Tension Test Specimens........... 0 0 0 00 0 0 0 0 * 0a 0 0 9 0 0 0f 0 0 0 0 0 00 0 0 0 0 0 00 0 0 49 50 53 54 55 58 58 59 59 61 62 Compression Test Specimens. Plane Strain Specimens..... Designation of Tests....... Fabrication............... Testing Procedure.......... ~~~~~,~~O)Qb ~ ~ 0 a 0: 0 0 0 0 0 * 0 "' v

TABLE OF CONTENTS CONT'D Page V. DISCUSSION OF RESULTS................................. 65 Introduction.o.....o....................... 65 Definitions............................... o.......... 65 Material.....,................................... 66 Prestrain.................................. o 66 Strain Recording.....o.............................. 67 Tension Test and Correction for Triaxiality of Stress in the Neck........,., o................67 Compression Test and Correction for Friction......... 70 Plane Strain Test.O.............................. 71 Stress-Strain Curves........................ 71 General Remarks...............o....................... 72 Stress-Strain Curves for Annealed Material......... 78 Stress-Strain Relations for Tension Tests After Tensile Prestrains for Individual Prestrain Stateso 81 Stress-Strain Relations in Compression After Tensile Prestrain for Individual Prestrain States.......... 83 Tensile Relations After Compressive Prestrain for Individual Prestrain States..................o 83 Compression Tests After Compressive Prestrains for Individual Prestrain States....................... 87 Comparison Between Tension and Compression Tests for Prestrains of Nominally Equal Magnitudes..... 87 Tension Tests After Tensile Prestrains............... 99 Compression Tests After Tensile Prestrains......... 101 Tension Tests After Compressive Prestrains........... 104 Compression Tests After Compressive Prestrains........ 104 0.2% Offset Yield Stress..................... o.... 107 Average Flow Stress.............................. 112 Flow Stress at c = 0.6o......................... 0 113 Yield Stress and Strain Hardening Exponent......... 114 Poisson's Ratioo,...... o..........................o 115 Residual Ductility........... o............ 129 Limitations of Present Investigation,................. 136 VI. CONCLUSIONS......................................... 139 APPENDICES.... o...o..0 0 0 0. 0 0., 0 0.. o o. o............. o 143 BIBLIOGRAPHY..o........ oo............o.. o................... l48 vi

LIST OF TABLES Table Page III-I Designation of Aluminum Alloys....................... 23 III-II Strength of 24ST and 24SRT Aluminum Alloys........... 40 IV-I Composition of 1100-0 Aluminum....................... 47 IV-II Prestrain Designations............................... 5 V-I Flow Stress in ksi................................... 73 V-II Plastic Constants for Annealed 1100-0 Aluminum....... 81 V-III 0.2% Offset Yield Stress and Difference from 2%z 0............................................. 109 V-IV Average Flow Stress and % Difference from az(av)..... 110 V-V Flow Stress at e = 0.6 and % Difference from a(.6)z. 11l V-VI(a) Flow Constants for Tension Tests in the X Direction.. 116 V-VI(b) Flow Constants for Tension Tests in the Y Direction.. 117 V-VI(c) Flow Constants for Tension Tests in the Z Direction.. 118 V-VI(d) Flow Constants for Compression Tests in the X Direction.. 0.... 0 0................... 119 V-VI(e) Flow Constants for Compression Tests in the Y Direction............................................ 120 V-VI(f) Flow Constants for Compression Tests in the Z Direction,.......................................... 121 V-VII Plastic Poisson's Ratio........................ 122 V-VIII Yield Stress in Plane Strain Tests................... 123 V-IX Residual Ductility in Tension Tests Subsequent to Prestraining..............................o.e..... oo 134 vii

LIST OF FIGURES Figure Page II-1 Tresca Criterion for Yield on the al - a2 Plane...... 10 II-2 Von Mises Criterion for Yield on the a - a2 Plane0.. 11 IV-1 Axes of Orthotropy................................... 51 IV-2 Specimen for Compressive Prestraining................ 52 IV-3 Specimen for Tensile Prestraining.................... 52 IV-4 Large Grips for Tensile Prestraining............... 56 IV-5 Set-up for Prestraining in Tension............... 57 IV-6 Tension Specimen................................. 57 IV-7 Small Grips for Tension Test............... 57 IV-8 Compression Specimens............................ 60 IV-9 Plane Strain Specimen........................O...O 60 IV-10 Adapter for Plane Strain Test..................... 64 IV-11 Set-up for Plane Strain Test........................ 64 IV-12- Electromechanical Clip-on Gage....................... 64 V-1 Comparison of Data Obtained by Electromechanical Clip-on Gage and by SR-4 Gage.............0.........o 79 V-2 a-e Diagram for Annealed 1100-0 Aluminum in Tension in the X, Y and Z Directions..................... 79 V-3:-e Diagram for Annealed 1100-0 Aluminum in Compression in the X, Y and Z Directions.......... Oo 79 V-4 a-e Diagram for Annealed 1100-0 Aluminum in Tension and in Compression in the X Direction................ 80 V-5 a-e Diagram for Annealed 1100-0 Aluminum in Tension and in Compression in the Y Direction.....o...,,,.. 80 viii

LIST OF FIGURES CONT'D Figure V-6 Page ar- Diagram for Annealed 1100-0 Aluminum in Tension and in Compression in the Z Direction................ 80 V-7 V-8 V-9 V-10 V-ll V-12 V-13 Prestrain State Til. a-c Diagram for Tension Test in the X, Prestrain in the X, Prestrain in the X, Prestrain in the X, Prestrain in the X, Prestrain in the X, Prestrain in the X, Prestrain in the X, Prestrain in the X, Prestrain in the X, Prestrain in the X, Y and State Y and State Y and State Y and State Y and State Y and State Y and State Y and State Y and State Y and State Y and Z Directions........................ T15. a — Diagram for Tension Test Z Directions........................ T20. a-e Diagram for Tension Test Z Directions........................ T25. a-E Diagram for Tension Test Z Directions.............o. Tll. a-e Diagram for Compression Z Directions....................... T15. a-c Diagram for Compression Z Directions......................... T20. a-e Diagram for Compression Z Directions........................ T25. a-E Diagram for Compression Z Directions...o.................... C2. a-c Diagram for Tension Z Directions........................ C35 a-c Diagram for Tension Z Directions................... C4o a-c Diagram for Tension Z Directions.......................... C5. cr- Diagram for Tension Z Directions.................. C6. o-c Diagram for Tension Z Directions....................... 82 82 82 82 84 84 84 84 85 85 85 85 85 V-15 v-16 v-17 V-18 Prestrain State in the X, Y and V-19 Prestrain State in the X, Y and ix

LIST OF FIGURES CONT'D Figure V-20 Page Prestrain State C7. a-c Diagram for Tension in the X, Y and Z Directions....................... V-21 Prestrain in the X, V-22 Prestrain in the X, V-23 Prestrain in the X, V-24 Prestrain in the X, V-25 Prestrain in the X, V-26 Prestrain in the X, V-27 Prestrain in the X, V-28 Prestrain in the X, V-29 Prestrain in the X, V-30 Prestrain in the X, V-31 Prestrain in the X, V-32 Prestrain in the X, V-33 Prestrain in the X, V-34 Prestrain in the X, State Y and State Y and State Y and State Y and State Y and State Y and State Y and State Y and State Y and State Y and State Y and State Y and State Y and State Y and C8, a-c Diagram for Tension Z Directions........................ C9. c-e Diagram for Tension Z Directionso...................0 C10. a-e Diagram for Tension Z Directions................... Cll. a-e Diagram for Tension Z Directions...................... C2, ar- Diagram for Compression Z Directions............... C3. c-e Diagram for Compression Z Directions........................ C4. c-c Diagram for Compression Z Directions......................... C5. a-e Diagram for Compression Z Directions...................... C6, ac- Diagram for Compression Z Directions......................... C7. a-e Diagram for Compression Z Directions,........................ C8. a-e Diagram for Compression Z Directions o....................... C90. a- Diagram for Compression Z Directions...................... o ClOo cx-~ Diagram for Compression Z Directions......................... Cl1. a-e Diagram for Compression Z Directions........................ 86 86 86 86 86 88 88 88 88 88 89 89 89 89 89 X

LIST OF FIGURES CONT'D Figure V-355 v-36 V-37 V-38 V-39 v-40 v-41 V-42 v-43 V-44 Page Prestrain States Tll and and Compression in The X Prestrain States T15 and and Compression in the X Prestrain States T20 and and Compression in the X Prestrain States T25 and and Compression in the X C2, a-c Diagram for Tension Direction.................. C3o ca- Diagram for Tension Direction................... C40, o-E Diagram for Tension Direction................... C5. ar- Diagram for Tension Direction.................. Diagram for Tension Direction................... Diagram for Tension Direction.................. Diagram for Tension Direction............... o. Diagram for Tension Direction................... 91 91 91 91 92 92 92 Prestrain State and Compression Prestrain State and Compression Prestrain State and Compression Prestrain State and Compression Prestrain State and Compression Prestrain State C6. a-e in the X C7. Cain the X C8. a-E in the X C9.o ain the X ClO. a.- Diagram for Tension in the X Direction................... Cll. o-e Diagram for Tension 92 92 and Compression in the X Direction.................. v-46 V-47 V-48 V-49 Prestrain States Tll and and Compression in the Y Prestrain States T15 and and Compression in the Y Prestrain States T20 and and Compression in the Y Prestrain States T25 and and Compression in the Y Prestrain State C6. a-e and Compression in the Y C2. a-E Diagram for Tension Direction.................. C3. a-E Diagram for Tension Direction................ C4o ca- Diagram for Tension Direction................... C5. a-e Diagram for Tension Direction.................. Diagram for Tension Direction................... 93 93 93 93 xi

LIST OF FIGURES CONT'D Figure Page V-50 Prestrain State C7. a-e Diagram for Tension and Compression in the Y Direction................... 95 V-51 Prestrain State C8. a-c Diagram for Tension and Compression in the Y Direction................. 95 V-52 Prestrain State C9. a-e Diagram for Tension and Compression in the Y Direction.................. 95 V-53 Prestrain State C10, a-e Diagram for Tension and Compression in the Y Direction............ 95 V-54 Prestrain State Cll. a-e Diagram for Tension and Compression in the Y Directiono.................. 95 V-55 Prestrain States Til and C2. a-E Diagram for Tension and Compression in the Z Direction................... 97 V-56 Prestrain States Tl5 and C3. cr- Diagram for Tension and Compression in the Z Direction.................. 97 V-57 Prestrain States T20 and C4. a-c Diagram for Tension and Compression in the Z DirectionO................. 97 V-58 Prestrain States T25 and C5, A-e Diagram for Tension and Compression in the Z Direction................ 97 V-59 Prestrain State C6, a-e Diagram for Tension and Compression in the Z Directiono............... 98 V-6o Prestrain State C7. a-e Diagram for Tension and Compression in the Z Directiono.................. 98 V-61 Prestrain State C8. a-e Diagram for Tension and Compression in the Z Direction,............... 98 V-62 Prestrain State ClOo a-e Diagram for Tension and Compression in the Z Direction,.................. 98 V-63 Prestrain State C11. ar- Diagram for Tension and Compression in the Z Direction,............... 98 V-64 a-e Diagram. Tension Tests in the X Direction After Tensile Prestrain.................. 100 xii

LIST OF FIGURES CONT'D Figure Page V-65 c-e Diagram. Tension Tests in the Y Direction After Tensile Prestrain............................. 100 V-66 a-e Diagram. Tension Tests in the Z Direction After Tensile Prestrain o.......... 00............0.0 100 V-67 a-e Diagram. Tension Tests in the Z Direction as Continuation of Tensile Prestrain.............. 102 V-68 a-e Diagram. Compression Tests in the X Direction After Tensile Prestrain............................ 103 V-69 a-e Diagram. Compression Tests in the Y Direction After Tensile Prestrain............................ 103 V-70 a-c Diagram. Compression Tests in the Z Direction After Tensile Prestrain............................. 103 V-71 a-e Diagramo Tension Tests in the X Direction After Compressive Prestrain...................... 105 V-72 a-e Diagram. Tension Tests in the Y Direction After Compressive Prestrain........................ 105 V-73 o-e Diagram. Tension Tests in the Z Direction After Compressive Prestrain..o...............o o..... 105 V-74 a-e Diagramo Compression Tests in the X Direction After Compressive Prestrain...................... ^ 106 V-75 a-e Diagram. Compression Tests in the Y Direction After Compressive Prestrain......................... 106 V-76 a-e Diagram. Compression Tests in the Z Direction After Compressive Prestrain........................ 106 V-77 c-e Diagramo Compression Tests in the Z Direction as Continuation of Compressive Prestrain............ 108 V-78 Variation of Plastic Poisson's Ratio with Prestraino. 126 V-79 Yield Stress in Tension After Tensile Prestrain...... 127 V-80 Yield Stress in Compression After Tensile Prestrain.. 127 xiii

LIST OF FIGURES CONT'D Figure Page V-81 Yield Stress in Tension After Compressive Prestrain.. 127 V-82 Yield Stress in Compression After Compressive Prestrain...................................... 127 V-83 Yield Stress for Plane Strain Tests After Prestraining in Tension....... *........... 128 V-84 Yield Stress for Plane Strain Tests After Prestraining in Compression..................128 V-85 Yield Surfaces for 1100 Aluminum Alloy After Tensile Prestrains...............o.....o.................... 130 V-86 Yield Surfaces for 1100 Aluminum Alloy After Compressive Prestrains.......... o......... o...... 131 V-87 Residual Ductility for Tension Test in the X Direction.......................50.................. 133 V-88 Residual Ductility for Tension Test in the Y Direction......o................................. 133 V-89 Residual Ductility for Tension Test in the Z Direction.. o o................. o....o............0o o 133 V-90 Hardness Traverse Along the Diameter of a Cold Worked High Strength Steel Cylinder................. 137 A-1 Co-ordinate Directions on the Tc Plane................ 147 A-2 Construction of the Yield Surface................... 147 xiv

CHAPTER I INTRODUCTION One of the many demands the contemporary engineers have to meet is to design and manufacture structures and components under conditions of service which needs utmost ingenuity and extensive knowledge of the properties of materials. In recent years these demands have grown more exacting; particularly in the space field where weight is so critical. To meet them a number of exotic materials have been developed and highly sophisticated analysis are called foro To make the best use of the available materials and the methods of analysis, the most important prerequisite is the detailed knowledge of the behavior of materials, in general, and also for particular materials. Unfortunately, our knowledge of the properties of materials as functions of processing history is not yet adequately known The mathematical theory of plasticity has been very well developed during the past century within the limits of the assumptions on which it is based. For the purpose of mathematical tractability, the materials are idealized; they are usually assumed to be isotropic in the annealed state, and either rigid-ideal-plastic or elasticideal-plastic in their plastic properties. Theoretical investigations have also been made for ideally strain hardening materials, although to a much restricted scale. On the other hand, investigations on the behavior of anisotropic materials are very few indeed. -1

The limitations of the present day theories of plasticity is not so much the lack of availability of technical literature as the limits of application of the theories, particularly so far as mechanical engineering practice is concerned, The theories have been developed primarily to satisfy the needs of structural engineering; to predict the behavior of structural members where a few percent of plastic strain causes the structure to become useless or the geometry changes so much that the stress distribution changes substantially from that at the initiation of yield. In the analyses which takes strain hardening into consideration, stress and strain are assumed to be linearly related. For real materials, this can be approximately true for small strains only. Hence these analyses are also limited to rather small strains. But very large strains are quite common in mechanical engineering practice. Processes like extrusion,drawing, etc, involve large strains. As another very important application, materials are cold worked to enhance their strength. Cold working also improves the machinability of soft annealed materials like low carbon steels, aluminum, copper,. etco The situation becomes much more complicated when cold worked material is used to fabricate machine components, particularly when fabrication involves cold straining in a direction different from that the material had undergone during previous cold work. This kind of application is common in sheet metal work. It is necessary to know the property o the material in the cold worked condition to be able to ensure that a certain fabrication procedure is possible

without undue waste in terms of scrap and machine capacity, It is also necessary to be able to predict the properties of the material after fabrication to ensure proper functioning of the component in usage. Since Bauschinger noticed in 1881 that when a material is loaded beyond its elastic limit, unloaded and reloaded in the opposite direction, the subsequent elastic limit is lower than that of the original material, it has been known that plastic deformation causes an otherwise isotropic material to become anisotropic. It can also be reasoned that since plastic deformation is primarily distortional, it is likely that a material, after plastic straining, should acquire mechanical properties which are different in different directions. The purpose of mechanical engineering design is to take advantage of the properties of materials to produce sound machines and machine components at minimum costo This, of course, presupposes that the properties of all engineering materials are known in all possible detail. Unfortunately, this is not quite soo Design stresses are usually based, from safety considerations, on the lowest strength expected of the material. But, at least in certain applications, where weight of the components concerned is critical, advantage can and should be taken of the variation of the strength of a material. Or, if it is known that a material is weaker in a particular direction, precautions can be taken to ensure safe functioning of the components made of it, Although it is well known that cold working produces anisotropy of mechanical properties, systematic data regarding the directionality of these properties of cold worked materials is very scarce, and our

-4 knowledge about them is very incomplete and inadequate. In view of these facts it was deemed profitable to systematically explore, document and compare the directionality of some of the mechanical properties of cold worked material, The material chosen for this investigation was commercially pure aluminum. The reasons for selecting this particular material is discussed in Chapter IV. The material was prestrained to various degrees in tension and in compression and the following investigations of properties were made, (a) Determination of the stress-strain relations. The relationship between stresses and strains in the plastic range were investigated for three orthotropic directions, both in tension and in compression. Torsion tests were not included since the material was anisotropic, and if subjected to plastic torsion, the principal strains would not stay in coincidence with the orthotropic directions and an analysis would be rather untractableo (b) Construction of yield surfaceso Yield surfaces were constructed from yield data obtained from tension, compression and plane strain tests and measurement of Poisson's ratios. (c) Residual ductility was determined from measured fracture cross-section for tension tests in all the orthotropic directions and correlated to prestrainso A few definitions may be examined here for use later on. Engineering stress is defined as the load divided by the initial cross-section and engineering strain as deformation divided by the

initial length. So long as the deformation and change in geometry is small, engineering stress and strain can be used to describe material behavior. But for large strains and consequent change in geometry, engineering stress and strain looses its significance and meaning, The logical thing to do then is to use natural stress and strain. Natural stress is defined as load divided by instantaneous area, and natural strain as the summation of instantaneous strains based on instantaneous length, rather than the original length. Assuming constancy of volume, this can also be expressed in terms of instantaneous cross-section. These concepts had been suggested and used as long ago as 1907 by Ludwik. Since the present investigation deals primarily with large strains, all stresses and strains mentioned hereafter are natural stresses and strains and mentioned simply as stress and strain for the sake of brevity, unless specifically mentioned otherwiseo In the case of tension tests, after the test specimen starts necking, the stress distribution at the neck is rather complex, and Bridgmans semi-empirical correction has been applied to correct for the triaxiality of stresses. Hence, whenever applicable, stress mentioned in case of tension tests always means corrected true stress. Since aluminum does not show a sharp yield point, an artificial yield point had to be defined. The usual definition of yield stress as the stress at 0.2% plastic strain was not satisfactory in all cases, and a modification was necessary. This has been discussed in detail in Chapter V.

-6 Residual ductility has been defined as the true strain at fracture of a specimen fabricated from prestrained material~ In literature, ductility has often been defined as reduction in area in percento Although the two definitions are equivalent, the later definition as the theoretical upper limit of 100%, whereas the first definition does not have any such upper numerical limito Also, since all the strains have been measured as true strains, this is logical to measure the ductility also in the same termso Perhaps the most important reason for measuring ductility in terms of natural strain is the fact that natural strains are additive, When a certain amount of ductility has been used up by working a material, the residual ductility can be found by direct subtraction of the used up ductility from the expected ductilityo This is not possible when percent elongation or reduction of area is used as a measure of ductilityo In this investigation the term "direction" means one of the three axes of orthotropy denoted by'the symbols X, Y and Zo The designation of these directions have been explained in Chapter IV. All of the testing has been done with reference to these three directions onlyo Two kinds of tests can be made in a particular direction, a tension test and a compression testo They are considered to be opposite in "sense" with respect to each other~ When a subsequent test is such that the strains produced by the test is a continuation of the previous strain in a certain direction, the sense of the subsequent test is said to be the same as that of previous strain.

-7 Originally the term "Bauschinger effect" was used to indicate the reduction in the elastic limit on reversal of loading after plastic yielding by the first loading. However, this term has been used subsequently in a much wider senseo In the present investigation the Bauschinger effect is defined as the difference between the flow stresses for testing in tension and in compression in a certain direction, In engineering practice the percent elongation at fracture has been widely used as a material property~ Unfortunately, this quantity varies with many geometrical factors like gage length, size, taper, etc., and hence can hardly be called a fundamental property of the material. Hence, no attempt was made to collect data on percent elongation, Fracture strength or stress defined as fracture load divided by the fracture cross-section is widely reported as a property of ferrous materialso In the present investigation and also in previous tests it has been found that aluminum test specimens do not break into two until the cross-section becomes very small and the load comes down to few pounds onlyo Also, fracture actually starts within the neck and just before the specimen breaks into two pieces, the load is supported by only a fraction of the apparent cross-sectiono Obviously, in such a case the fracture stress means very little and was not calculatedo

CHAPTER II THEORETICAL CONSIDERATIONS This chapter deals with the outlines of some of the theories of plasticity concerning the yield and flow of metals. No attempt has been made to describe the theories with rigorous mathematics, or to lay down all the assumptions involvedo The theories are described only in so far as it is required to build a physical concept of the mechanism of yield and flow of metals0 Some of them will be dealt with in more detail in the chapter "Review of Literature." Empirical stress-strain relations for uniaxial loading are also discussed. General Remarks The theory of plasticity assumes the existence of a yield surface which is defined as a surface in stress space such that any stress condition within the space enclosed by the surface produces elastic. deformation only; and when the stress is removed, the strains also vanisho Plastic flow can occur only when the stress state is on the yield surfaceo In general, four mechanical models of materials are used in plasticity analysis- (1) Rigid-ideal plastic; (2) Elastic-ideal plastic, (3) Elastic-strain hardening and (4) rigid-strain hardeningo Although there is no material which is perfectly rigid under any condition of stress, if the plastic strains are large compared to the -8

elastic strains, the rigid-plastic approximation may be very goodo On the other hand, when elastic and plastic strains are of the same order of magnitude, it is more logical to assume the material to be elasticideally plastic rather than rigid-plastic. When a material is considered as ideally plastic, it is assumed that plastic straining does not change the properties of the material, and that the strain rates at yielding are of arbitrary magnitude. The yield surface remains stationary in stress space for this type of materialo For a previously cold worked material subjected to small plastic strains as encountered in structural applications, these assumptions may represent actual behavior to a close approximationo However, for larger strains, the assumption of a stationary yield surface is not realized in practice Nearly all metals have their recrystallization temperature above the ambient temperature, and become stronger with increasing straino Hence, the yield surface does not remain stationary0 The material is then called strain hardenable In the theory of plasticity it is assumed that for uniaxial loading increments of stress and strain are linearly relatedo This can be approximately true for strains of a few percent in magnitude for certain metalso As a material becomes stronger with increasing strain, the yield surface is no longer stationary in stress space, and has to move or change in shape or size to accommodate the increase in strengtho

Yield Conditions The two most widely used yield conditions for homogeneous isotropic materials are as follows: (a) Tresca Criteriono This theory postulates that yielding occurs when the maximum shear stress in the material concerned reaches a certain maximum value. This can be expressed mathematically by the equation~ {(s2 - S3) - 4k }{(S5 S1)2 4k ( - S2) - 4k2} = 0 (IIx-l) where S1, S2, and S3 are the stress deviators referred to principal axes and k is the yield strength of the material in shearo The surface in stress space is a regular hexagonal prism of infinite length, its axis being equally inclined to the three. principaL.stress axes and each side having a width ~ ko The trace of the prism in a plane con3 taining two stress axes is a hexagon as shown in Figure II-1. 62 Figure I-l. Tresca Criterion for Yield in a1-a2 Plane.

-11 (b) Von Mises Criterion. In this theory it is assumed that the material yields when the shear strain energy reaches a maximum value. The same condition is reached when octahedral shear stress reaches a maximum value. Mathematically it can be expressed as: 2 2 2 2 sl + + S3 - 2k = 0 (II-2) The surface in stress space is a right circular cylinder of infinite length and radius 12ko The axis of the cylinder being inclined equally to the three stress axes. The trace of the cylinder on a plane containing two stress axes is an ellipse as shown in Figure II-2. a Yield stress for uniaxial loading Figure I.o2. Von Mises Criterion for Yield in a1 - a2 Plane. Other Theories of Yield Criteria Several other criteria have been suggested at different times to describe the yield phenomenon. Well known among them are maximum

-12 principle stress, maximum strain and maximum strain energy criteria. Recently Haythornthwait Has suggested a maximum reduced stress criterion for yielding. This is intended primarily for use as bounding yield condition when the available amount of experimental data is limited. This postulates that yielding occurs when the largest stress deviator reaches a certain maximum magnitude. Strain Hardening( 3) When a strain hardenable material is subjected to a state of stress not included within the yield surface, plastic strains result and the strength of the material changes in such a way that the yield surface translates, expands, or distorts until the stress is contained on the yield surface. There are several theories which attempt to predict the motion of the yield surface. Unfortunately none of them seem to represent the actual process in general, although they produce good qualitative approximations in certain cases. Several common theories are explained below. (a) Isotropic Hardening. This theory postulates that the shape of the yield surface remains the same, and all the dimensions are increased by a constant factor. In other words, if F - f(ai ) = 0 represents the original yield surface, the final yield surface can be represented by F kf(ij) = 0 where k is a constant. Unfortunately this theory cannot accommodate the well known Bauschinger effect. Also (4) it has been shown mathematically by Drucker(4) that isotropic strain hardening is not physically permissible, although it is convenient to (5) use. Tension-torsion tests by Marin and Wiseman(5) corroborates the statement.

-13 (b) Kinematic Hardening. This rule, as developed by Prager,'7 assumes that the yield surface is rigid and does not rotate, but translates in a direction normal to the surface at the loading point. This is a mechanical model which accounts for the Bauschinger effect, but for large strains it becomes increasingly less valid since the yield surface is known to become. larger with increasing plastic deformation. Methemati2 cally, if the initial yield surface is represented by f(aij) = k = const, the yield surface after plastic deformation is given by f(c i -i) = k2 (II-3) where.j. represents the translation of the yield surface. (c) Piecewise Linear Theory of Hardening. P. G. Hodge(9 developed a more general theory which can take into account any combination of isotropic hardening and Bauschinger effect. This theory assumes that the yield surface is made up of a finite number of linear loading functions of the stress, and that the plastic stress-strain curve can be approximated by a single straight line. Elastic strains are neglected. The loading functions are assumed to be able to translate parallel to themselves and independent of each other, but not so much so that one of them may vanish. This was found valid when the load point stayed in contact with one side of the yield surface, but not in general cases. A theory based on an infinite number of loading surfaces was (45) discussed by Sanders. The fundamental feature of this theory is the treatment of a curved yield surface as the limiting case of plane loading surfaces when the number of such surfaces tend to infinity.

(d) Slip Theory of Hardening. Batdorf and Budiansky(1011) developed a theory of plasticity based on slip in crystals. The primary deformation of the yield surface according to this theory is to change the shape of the surface near the loading point by producing or rotating planes to include the subsequent loading point. In conclusion it may be stated that experimental evidence(2) indicates the actual behavior of a material under an arbitrary state of stresses above the elastic limit to be exceedingly complicated; and even if a completely accurate theory could be devised, it would probably be too cumbersome to be of any use(5) Therefore, it is still necessary to investigate the yield behavior of cold worked materials for engineering design purposes. Stress-Strain Relations For materials like mild steel that exhibit a sharp yield points it is possible to approximate the plastic stress-strain relations for small strains under uniaxial load by a linear function. Unfortunately this is not true for most materials, particulary at large strains. Various empirical relations have been suggested by investigators. The first attempts to find a relationship for the plastic extension of metals was at the beginning of this century. As noted by Hill(l), as early as 1909 Ludwik suggested a power relation for annealed material given by a = a + bem where a, b and m are experimental constants. In the opinion of Hill, if ~ does not exceed 0.2, then, for low values of e, this relation

would produce an xunderestimated, and at high values of cj an. overestimated value of a, since in his opinion, the exponential law does not provide for sufficiently rapid variation in stresso In 1948-49 Voce and Palm suggested a more complicated relation given by a a + (b-c) (le ) (IIX4) where a, b, and c are experimental constants. They report it gives better results for certain metals. Unfortunately, this is a rather awkward relation for convenient use However, Zaitsev(13) has shown that Ludwik's power law can be used as a good approximation if the stresses are corrected for triaxiality after necking starts in a tensile testo For the material used in the present investigation a plot of the stress-strain curve on logarithmic co-ordinates was a straight line to a close approximation in the range of strain of our interesto This suggests that within that range the stress-strain relation can be expressed as,(14l15,l6) a = em (1-5) where a and m are experimental constantso o When a material is strained to a true strain of say, ~p, in a certain geometrical direction of the material, unloaded, and then loaded again, then except for a small anelastic effect near the new elastic limit, the stress-strain curve follows the extrapolation of the original curve, since the second loading is just a continuation of the first one. Hence, if we wish to express the stress-strain relation of a material that has been cold worked to a true strain

-16 of ep, it is obviously best to refer it to the annealed condition and hence, if we denote the subsequent current strain by e, the relation is given byO a = ao(p + ) (II-6) where the constants ao and m are the same as in Equation (II-5). This is valid only for straining the material in the same direction and sense as the prestraino Unfortunately such a simple relation cannot be deduced when the direction or the sense of straining is different from that of the prestrain, and any such relation has to be found experimentally. From Equation (II-5) we have log a = log or + m log C Hence, if the stress-strain curve is plotted on logarithmic co-ordinates, the intercept of the curve at e = 1 gives the value ao, and m is the slope of the curve which is constanto To find the slope of the curve for prestrained material strained in the same direction as the prestrain, we have, from Equation (II-6), In a = in o + m n (p + ) (11-7) Let y = in a x = ln Then e = ex From Equation (II-7) y = in ao + m in (ep + eX)

or dy = m e dx. Ep + ex dy d(log a) e or - = d(l = m — (11-8) dx d(log e) Hence, it is apparent that the stress strain curve for previously cold worked material on logarithmic co-ordinates will have an increasing slope. In practice, it is possible to draw an approximate straight line for small strains~ From Equations (II-7) and (II-8) it is also apparent that the slope of such an approximate line will be smaller with increasing value of ep, but its intercept at e = 1 will be higher. Similar general comments can be made for straining in directions other than that of prestraining, but the relation shown in Equation (II-8) is not expected to hold true. Anistropic Yield and Flow A yield criterion of material with orthotropic anisotropy (17) in strength has been suggested by Hillo7) Similar relations from different assumptions have been stated by Lo Ro Jackson et al(18) and by J Eo Dorn(10) and applied to the plastic flow of anisotropic sheet metalso Since these will be discussed in Chapter III, "Review of Literature," they are not presented here Physical Theory In the theory of plasticity, the material is treated as a continuum, although in reality it is not. All metals consist of atoms

-18 arranged geometrically according to the structure of their crystals. A polycrystalline material, as used in the present investigation, is composed of a large number of crystals with grain boundaries between them. The grain boundaries are areas of misfit and the atoms there are not properly lined up with the geometrical patterns of the grains they separate. Although in the present investigation the material is treated as a continuum, a few comments about the physical nature of strain hardening should be made here. 6 Undeformed annealed metallic crystals contain about 10 to 8 10 dislocations per sqg cm. arranged in a random spatial networko On the application of a stress, some of the network elements can act as Frank-Read sources (dislocation generators), since their end points are more or less fixedo Also, often other structures in the crystals act as sourceso The mutual elastic interaction of dislocations increases with the dislocation density. In a highly cold worked material the density 11 12 may be 10 to 10 dislocations per sqo cmo This is considered to be the basis of work hardeningo () Another effect is the mutual encounter of nearly parallel dislocations moving on different intersecting slip planes, resulting in a sessile dislocation which forms a barrier to other dislocations. A Frank-Read source on a slip plane where a barrier occurs can emit only a limited number of dislocation loops even under increasing shear stress because the loops cannot move further from the source than the

-19 barrier permits and will pile up against it. Long range elastic stresses are associated with such dislocation pileups.. Large stresses also occur on other slip planes in the neighborhood of a pileup and these limit the dislocation mobility on other slip systems(21) and hence plastic deformation is resisted. Another important factor in strain hardening is the mutual intersection of non-parallel dislocations. This usually results in the formation of jogs and subsequent formation of point defects. A dislocation loop expanding through a forest of obstacles is likely to have an irregular shape.(22) Under certain conditions the jogs may be rendered immobile due to interaction with other obstacles. The loop then leaves behind vacancies or interstitials. With larger plastic deformation local tangles gradually develop into a network of subgrain boundarieso Some very interesting electron micrographs have been presented by Snowden(23) showing dislocation patterns of cold worked aluminum crystals. More references to similar work have been made in Chapter III.

CHAPTER III REVIEW OF LITERATURE General Remarks Recent literature on anisotropic behavior of materials has been reviewed in this chapter. Existing literature in this field can be roughly subdivided in three broad groups as follows: (1) Those which present a theory of anisotropic strength and flow of material and try to check the theory by experiment; (2) Those which try to present a physical model of anisotropic behavior; (3) Those which investigate the actual behavior of materials The present investigation wills by and large, fall in the third group. However, to draw significant conclusions, it is necessary to keep in mind and also to refer to the findings of the first group of studies, and hence. short reviews of some works which are properly placed in the first group has been included here. The first few reviews are from this group. Reviews of articles from the third group follows those of the first. Concise discussions of some of the articles of the third group that have direct bearing on the present investigation are presented. Space restrictions have forced others only to be mentioned. -20

-21 From the second group, a few references have only been mentioned. It has long been recognized that plastic strain is essentially a distortional (245) rather than a volumetric change. Plastic strain is therefore intrinsically directional. In cold working a material its mechanical properties change in response to this directional strain. Therefore, it cannot be postulated that an initially isotropic material will remain so after plastic strain. If the principal axes of stress remain unaltered and coincident with the axes of orthotropy while the strains develop, the principal axes of strain should also remain coincident with those of the stress, but the response of the material to stresses in those directions will not necessarily be the same; in fact they will be different in general, and this gives rise to anisotropy of (26,27) mechanical properties. ) Although the relations between stresses and strains for an annealed and presumably isotropic metal can be expressed as a function independent of direction, the stress-strain curves of a slightly coldworked metal have been found to vary considerably with the direction of testing.(8) Three types of anisotropy have been listed by Klinger and Sachs(28 29,30): (a) Crystallographic anisotropy which results from preferred orientation or texture in the crystal structure of the metal. This develops gradually in an initially apparently isotropic material with the increase in cold work.

(b) The second kind of anisotropy is due to mechanical fibring and elongation of inclusions, impurities and secondary phases due to processing. The probable effect of this kind of anisotropy is the variation of fracture characteristics (25.32) (c) The third kind of anisotropy has been called anelastic anisotropy which is supposedly due to residual stresses set up due (29) to cold working. 9) Anistropy in mechanical strength which causes earing during drawing is generally objectionable; but it has been said that the difference in properties in directions perpendicular and parallel to the sheet can be an asset in good drawability and be beneficial to press performance, Most of the experimental investigations concerning anisotropy of mechanical properties has been concerned with the behavior of rolled sheet or drawn tubes and rods. Theoretical investigations were made on the basis of certain basic assumptions. Anisotropic behavior of materials were predicted according to these theories. The predicted behaviors were then compared with the behaviors of actual materials to assess the validity of the theories. A number of investigators have used aluminum alloys for their tests and all of them used the old designations to identify them. In the following review the designations used by them has been retained. The table below is included as a ready reference showing the old and the new aluminum alloy designations.

-23 TABLE III-I DESIGNATION OF ALUMINUM ALLOYS Old Number New Number 2S* 1100 11S 2011 14S 2014 17S 2017 24S 2024 3S 3003 32S 4032 52S 5052 61s 6061 75S 7075 *The'S' designates a wrought material. Theoretical Investigations Yield criterion of an anisotropic material. Hill postulated that the yield criterion for an anisotropic material should be of the form 2f~c~..) 2 2 2 2f(cij) = F(Oy-%) + G(Cz-ax) + H(ax-ay) 2 2 2) + 2LT + 2MT + 2NT 1 yz zx xy where a and T's are the direct and shear stresses referred to the axes of orthotropy and F, G, H, L, M and N are parameters characteristic of the current state of anisotropy. It was assumed that no Bauschinger effect was present. It was shown that if U, V, and W denotes the yield stresses in the principal directions of anisotropy and R, S and

-24 T the yield' stresses in shear with respect to the principal axes, then, 2F 1 V2 W2 U2 2G = -+ - W2 U2 V2 1 1 1 2H = +U2 V2 W2 2L = 1 R2 2M. _ 1 s (III-2) 1 and 2N = - T For an isotropic material U = V = W and R = S = T and U = 3Ro Hence F = G H= 2 2U 1 3 1 L = M= N= 2-= (III-3) 2R- 2 U2 and Equation (III-1) reduced to Von Mises criterion for yielding for isotropic materials. The strain increment relations are given by dex = dX [H(ax - cy) + G(ax- az)] dey= dX [F(oy az) + H((y - ax)] dez = dX [G(z - x) + F(az - y)] dyyz = dXIr dy7zx = daMT and dxy = dTxy (III-4)

-25 where c and y's are shear strains referred to the axes of orthotropy and X a rate constant. Jackson, et al. ) described anisotropy in terms of several parameters describing deviation from isotropy and plastic Poisson's ratios. They assumed that strain increments can be expressed as: dex = d (a ax byxay Czxz) de dE: (-ap. a, +ba b c[ta) y = xy x y z ) and de = - (-az a - ab + cz ) z xz x yz y z (III-5) where a = "Effective stress" F; 32 2 = [ _~ -_ —3 —[- Kz Kxz (Cx - y) + K2 z (rz - x) - 2(Kz + Kx + KyzKx) - y z ( + Kyz(y - c z) ] de = "Effective strain" axdex + oydey + azdez =a K yz K xz Cy z z de y aax de aax when a = a = y z when c = a 0 x z when a = a = 0 Y when ay = az = 0 z

-26 dz ^ = ba, when a = = o yz by zba etc. a, b and c are measures of deviation of the material from isotropy and a + b + c = 3. x y and z are axes of orthotropic symmetry. In an isotropic material, a = b = c = 1 and xy = dz... - They used rimmed annealed sheet steel, aluminum killed annealed and temper rolled sheet steel and "Cor-ten" sheet steel for their tests. Straining was done by direct tensile tests and circular and elliptical bulging by hydraulic pressure. True effective stresses were plotted against true effective strains assuming isotropic symmetry and again assuming orthotropic symmetry. In the first case tensile and bulge tests showed wide divergence. In the second case about half the tests showed good agreement. Assuming that the components of strain rates are linear functions of stresses, Dornl ) formulated six plastic strain rate equations as followsC = A AAT + A T + A T + AA4 + A+ A1TT xx 11 xx 12 yy 15zz l4txy 15 yz l6zx Cxy A41Txx A42Ty 4 4 zz A44Txy A45Tyz + A46Tzx etc. (III-6) *Composition: 0.10 max. C, 0.1 - 0.3 Mn, 0.1 - 0.2 P, 0.5 - 1.0 Si, 0.3 - 0.5 Cu, 0.5 - 1.5 Cr., balance Fe. Small amount of Ni may be added.

-27 This is noted that in a general case Akl / Alk and hence the 36 coefficients may be different from one another. However, for planer plastic flow, as in sheet metals and assuming symmetry about y-z plane, the equations reduce to: xx ~yy xy xy Allxx + A12Tyy A21Txx + A22Tyy A44xy and (III-7) if x and y are orthotropic axes and Tz =0. Assuming constant anisotropy, relations similar to that of Jackson, et al.(8) were found. This theory was experimentally verified by Hazlett et al. It was shown that Equation (III-7) can be expressed as: dCxx - {llaxx *+ 12ayy} deyy d {21axx + 22ayy} and de = {44axy (III-8) where de = Strain increment on a fiber instantaneously in the XX cross rolling direction. dey = Strain increment on a fiber instantaneously in the yy rolling direction. de = Shear strain increment between fibers instantaneously xy in the cross-rolling and rolling directions respectively. *x and y are the transverse and the rolling directions respectively, and the z direction is normal to the x and y directions.

-28 x = Normal stress in the cross rolling direction on a xx plane whose normal is in the cross rolling direction. a = Normal stress in the rolling direction of a YY plane whose normal is in the rolling direction. = Shear stress in Y direction on a plane whose xy normal is in the X-direction. a.. = Coefficients of anisotropy (i = 1 to 6, j = 1 to 6) a = 4aX(~lLaoxx + ~'12ayy)+ oyy(c2laxx + C22a^yy) + a44axy and = /c22(dx ( O12 + Cl2,)dcxxde + all (dEy2) (dE)2 and X~ = Il44 V112 -0 11221 2442 Observations on the deformation of mild steel plates were in excellent agreement with the theory. However, the ratios C21/522 and 021/all were -0o48 for this material, while for an isotropic material it would be 0.5 which means that the material was rather close to isotropy to begin with and hence the agreement may not be significant. For 3S-0 aluminum alloy for which the coefficients were all = 1.0l; 12 = -o383; 2 = -0.34 and 22 0o957 agreement between theory and observation was not very good. The theory (39) was applied to data reported by Stang et al.( for 24S-T3 alloy.. The coefficients were 11 = 1, a = -0o436, -0.375 and a2 0.939. 11 12 21 -2 Again, the theoretical and experimental values were not in precise agreemento Deviations were noted with 61S-T6 alloy also,

-29 A rather different theory was proposed by Fisher ) This is based on the concept of slip in metal crystals. Slip bands are thought of as regions of relative motion between two plastically undisturbed portions of a crystal. During the slip process the slip bands are unable to support a shearing stress in the direction of slip. After slip has occurred, however, the bands very rapidly regain their ability to withstand shearing stresses. During slip the elastic strain energy near the slip band is relieved and a smaller amount of strain energy is concentrated near the perimeter of the slip band, unless the band terminates at a free surface of the metal. The plastic stress-strain relationships were derived assuming this slip mechanism of plastic flow. The essential features of this theory are as follows: (1) The definition of an effective stress proportional to the square root of the strain energy per unit volume released by slip in the neighborhood of a slip band. (2) The definition of an effective strain such that the integral of effective stress with respect to effective strain gives the work done in deforming a unit volume of material. (3) The assumption that the effective strain (or the work done per unit deformed volume) is a satisfactory measure of deformation. (4) A demonstration that there exists a functional relationship between the effective strain and the effective stress required to maintain a constant strain rate at a constant temperature.

-30 (5) A demonstration that the plastic strain increments are proportional to the difference between (a) the elastic strains before slip and (b) the elastic strains in the slip region after slip. Principal axes of stresses and strains were assumed to coincide with the axes of orthotropic symmetry. It was also assumed that after slip, the stresses in the slipped region are linear functions of applied principal stresses a1, a2 and a3 1= 1ll1a + 133 a2 = 21a1 22a2 + 23a3 5 = G + a2 +V a 3 311 32 2 333 T23 = ~4151 +'42a2 *'43a3 f31= 51a1 + Y52a2 + QU53U3 and T = O(a + (III-9) ~and T12 = C61gl + C62a2 + C'6303( From these assumptions, it was shown that the effective stress can be defined as: a = E [( —a2) + (l+m)(ac2a ) + (o3 a ) ] And the effective strain increment can be defined as: de = k (de + ad2 + ad3 3) 2 2 2 Pdl! + dC2 +.de3 2 r[(BE_-dC2)2 + (l+m)(de2-dE3)2 + (de -f1) 2j

-31 where 4( 22 - 1l1)(l - Cll - 2ca2) (1 + 53l)(l - 11) and 1 + a11 2a22 1 - ca11 Tests on an aluminum bar, 10 feet long and 6 inches in diameter showed that the predictions of this theory deviated from the experimental value by about 4% compared to about 9% deviation for distortion energy theory, Batdorf and Budiansky(10 11) proposed a theory of plasticity based on slip in the crystals of polycrystalline aggregates. The basic assumptions are as follows: (1) A metal that is isotropic in the gross mechanical sense can be sufficiently approximated theoretically by an aggregate of randomly oriented crystals regarded identical with each other except for orientation, (2) The grain is the smallest structural unitO (3) Local variation of stress within a grain can be neglected. (4) The grains deform by slipo The theory was successfully applied in the case of biaxial, loading, but its application to tension followed by partial unloading and then to torsion was found unsatisfactory. One consequence of the postulates of this theory is the possibility of production of a corner on a otherwise smooth yield surface~ Naghdi, et al. )(41) surface, Naghdi, et al. tested anisotropic tubes of 24S-T4

-52 aluminum alloy subjected to variable ratios of tension and torsiono Their findings suggest existence of such yield corners. Linearity of the increment of the plastic strain in the increment of stress could not be ascertained. Ashkenazi(42) formulated anisotropy of yield strength of materials assuming that the yield strength can be treated as a fourth order tensor, and developed mathematical relations which would permit the determination of the whole field of resistance in the material if its strength in several basic directions are experimentally determined, Experimental Investigations Perhaps the earliest experimental investigation in the anisotropy of yield properties of cold strained material was by Professor Bauschinger(43) of Munich. He showed that when a material is loaded beyond its elastic limit, unloaded and reloaded in the opposite sense, the second elastic limit is found to be lower than that of the virgin material, Also, he concluded that the higher the stress in the first loading, the lower is the elastic limit for subsequent loading in opposite senseO This effect is known as the Bauschinger effect and was first recognized in 1881, Subsequently it was found that not only was the elastic limit lowered, but the material was weakened for subsequent loading in the opposite direction after prestraining, and the stress strain curve for the subsequent test was below that of the virgin material, (44) Schwartzbart, et al. studied the Bauschinger effect in copper and brass prestrained in tension and tested in compression. Their findings are as follows:

33 (a) The Bauschinger effect develops at very small prestrains, less than ep = 0o005, and remains essentially constant up to a prestrain of Ep = 065. (b) The Bauschinger effect disappears after a strain in the subsequent compression test exceeded e = 0,01o (c) Tensile prestrain weakens the material in subsequent compression. The degree of weakening increases with prestrain up to about p = 0.30, above which the effect is constant. (d) The shape of the stress-strain curve in compression after tensile prestrain is a function of the prestraino For prestrains below Ep = 0o15, the form of the curve is similar to that for an annealed specimen. Above this prestrain, the curve exhibits a region of low slope immediately after initial yielding. The slope of this initial part, in turn, decreased with increasing prestrain until it becomes practically horizontal at a prestrain of e = 0.24. This behavior is similar to the "yield point effect" in mild steel. Once the initial region of low slope was exceeded the slope of the compression curve was essentially parallel to the virgin compression curveO They introduced a correction factor for compression to take into account the texture produced by cold work in such a way that ratio of flow stress in tension and in compression is a constant at any strain level. The validity of such correction seems questionable.

Naghdi, et alo investigated the change of the yield surface of 24S-T4 aluminum alloy tubes under a program of tension and torsion where the tensile stresses were always less than the yield stress, They reported the following observations: (a) The initial as well as subsequent yield surfaces in tension-torsion plane are convex; (b) The initial yield surface is essentially symmetric about the tension axis; (c) While the yield stress in tension is unaltered, the yield surfaces in the neighborhood of the torsion axis display a pronounced Bauschinger effect which gradually vanishes as the curve approaches the tension axis; (d) The effect of work hardening rapidly diminishes as the tension axis is approached but is accompanied by a gradual linearization of the subsequent yield surfaces in the first quadrant, producing a region of high curvature about the torsion axis, This is reminiscent of corners predicted by the slip theory of plasticity (10) This may also be viewed in terms of incremental theory based on an infinite set (45) of plane loading surfaces (45 (e) The initial yield surface is almost identical with von Mises type yield surface. Cunningham, et al (46) reported variation in the distortion energy ellipse for different stages of straining. Biaxial tension

-535 tests with variable stress ratio for 14S-T4 aluminum alloy was made by Marin and Hu(47) which showed that neither distortion energy, nor maximum shear criterions were adequateo Similar tests on 14S-T6 aluminum alloy and cold drawn mild steel(8) and 24ST aluminum alloy(49) showed similar resultso (57) Based on Hill's theory of plasticity( for anisotropic materials, Hu(50) derived plastic stress-strain increment relations for strain hardening materials, He also discussed the influence of anisotropy on the plastic behavior of metals in a state of plane stress or plane strain. It was found that the pressure carrying capacity of a thick walled cylinder under internal pressure can be increased by decreasing the axial strength. For conventional biaxial stresses, the method of interpretation of test results were found valid for biaxial tension tests of anisotropic materials, but not valid for tension-torsion tests. Bridgman(51) investigated the variation of yield and flow stresses in the longitudinal and the transverse directions of steels cold worked to large strains under high superimposed hydraulic pressure. His investigations can be divided into the following sections: (a) Simple Tension After Prestraining in Tensiono Specimens were pulled in tension under different hydrostatic pressures and. then pulled again under atmospheric pressureo It was possible to pull to a much higher strain under pressure than was possible under atmospheric pressure, and in spite of that fact there was residual ductility left for straining at atmospheric pressureo The stress-strain points on

-36 the second pulling at atmospheric pressure fall on a single line irrespective of the pressure or the strain of the first pulling. The stress-strain curve for pulling under pressure lies higher than the stress-strain curve for atmospheric pressure; the difference being greater with higher pressure, (b) Simple Tension After Prestraining in Compression. The material was strained in compression to a true strain of about 1.25 under hydrostatic pressure. Tension specimens cut in logitudinal and transverse directions were tested. It was found that the strength of the material increased in both directions compared to the virgin material, but hardening.was not equal in two directions. The transverse direction was found to be much stronger in tension than the longitudinal direction. The strain hardening rate was lower and so also the residual ductility. (c) Simple Tension After Prestraining in Two Dimensional Compression. The material was prestrained in compression in such a way that the strain in one transverse direction was zero. The prestrain was about 0.4, The material was found anisotropic after prestraining9 although the strength was higher in all directions compared to the virgin metal. The tensile strength was lowest in the direction of compressive prestrain, but the residual ductility at fracture was highest. The mechanical properties in the other two directions did not show any such consistency. (d) Compression After Tension Under Pressure. Compression tests were made on specimens taken from the necked region and tested

-37 in the same direction. For small prestrains, reduction in the compressive strength was noticed. (e) Simple Compression After Prestraining in Compression. The material was found anisotropic after prestraining to different strains. Strengths in the transverse directions were found lower than in the longitudinal direction, although higher than that of the virgin material. (f) Simple Compression After Prestraining in Two Dimensional Compressiono Cr-V steel and stainless steel were used for this testO Strain in one direction was held at zero. Subsequent compression in the direction of zero strain showed slightly lesser strength than the original compressive stress in the direction of compression. Moreover slight strain-softening was noticed during the subsequent testo Mehringer and McGregor(52) investigated the variation of stress-strain properties of cold rolled low carbon steel, They rolled steel bars to different reductions in thickness, machined tensile specimens cut out at different angles with the direction of rolling and plotted true stress-strain diagrams. However, they did not consider the triaxiality of stresses(53555) at the neck, and true stress was defined simply as load divided by instantaneous area at the neck, and hence the stresses obtained after necking were too high. The material used was isotropic in the annealed state so far as the strength was concerned, but was anisotropic in strains which was manifested by the ellipticity of the test specimens at strains beyond necking. They observed that the cross sections remained circular during uniform strains, and became elliptical only after

necking started, They also investigated the effect of cold working by tensile pull on true stress-strain curves. Mehringer and McGregor made the observations as below, The "modulus of strain hardening" m = =a/e is not greatly affected up to a reduction of 10o% It decreases for higher reductions, and the greatest decrease is for the 90 inclination to the rolling direction. Increased reductions by rolling also reduce the true uniform strain at the maximum load, and the true fracture strainso No appreciable increase in the true fracture stress takes place until about 30% reduction is reached, except for the specimens at 90 to the rolling direction. True stress at maximum load is increased by large reductions by rolling. The net effect of stretching up to 15o on the true stress-true strain properties is equivalent to the same reduction by cold rolling on specimens cut out at 00 to the rolling (and stretching) direction. However, for specimens cut at 900 to the rolling direction, 10% reduction by rolling is equivalent to a 15% reduction by stretching. Up to 10% reduction by rolling, the true stress at maximum load remains constant for all directions. After that, it increases rapidly. True stress at fracture remains practically constant till about 35% reduction for

-39 specimens at 00 and 45 to the rolling direction and then increases slightly. For specimens at 90, it remains constant to about 15% and then increases a little more rapidly than those at 0 and 45~ The modulus m decreases with increasing reduction by rolling, although the scatter is rather large, Vickers hardness values increases rapidly with small reductions in rolling. The rate of increase of hardness decreases until a reduction of about 45% is reached. Thereafter, it increases uniformly. All true stress-true strain diagrams were plotted in Cartesian co-ordinates, (28) Klinger and Sachs investigated the anisotropic behavior of 24ST and 24SRT aluminum alloy sheet, During fabrication, 24ST sheet is a stretched, and/or rolled by about 1% while 24SRT is rolled by 5 to 6% to increase its strength. They found that the tensile strength is higher than the compressive strength when tested parallel to the grain. When tested perpendicular to the grain, the strengths are of intermediate value, the compressive strength being higher than the tensile strength. If no cold work is applied, the strengths are almost identical in both directionso Table III-II shows some of the data obtained by them. It was also found that after prestraining by rolling, the transverse stress-strain curve in tension is similar to the longitudinal stress-strain curve in compression. Apparently, rolling acts, in this respect, similar to stretching by tension.

For any given plastic strain, the stress decreases first approximately linearly with the angle between rolling direction and testing direction to about 50 9 and then decreases at a much slower rate, to 90~o The total difference increases rapidly with decreasing strain, the transverse strength being about 76% of longitudinal strength in tension, Anisotropy in compression is considerably smaller than in tension, Cold rolled 24ST aluminum plates were found to have definite degree of preferred orientation. For annealed material, the yield strength varied only slightly, but the plastic Poisson's ratio showed large deviation from isotropyo TABLE III-II STRENGTH OF 24ST AND 24SRT ALUMINUM ALLOYS 0.2% Offset Yield Flow Stress in ksi Stress in ksi at 1% Total Strain Tension Compression Tension Compression 24ST Parallel to Grain 51,5 42.0 5530 8o 0 Perpo to Grain 43o5 4.7 5 48o5 51 o 24SRT Parallel to Grain 630o 53~0 64o5 58,0 Perp, to Grain 55.5 58.5 5905 61o0 Ebert56 investigated the transverse and longitudinal properties of steel bars, cold drawn through carbide dies of 80 half angle. Most specimens were roller straightened and others were left unstraightened.

He found that the steels respond to the strengthening effect of cold work as regards the tensile and yield strengths in the same relative manner in both the transverse and longitudinal directionso The shape of the trend curves for the yield and tensile strengths as functions of the amount of cold work are not as consistent for the transverse tests as they are for longitudinal tests, the major difference being the absence of the last rapid increase in strength. He observed two apparent anomalies. First, the leveling off of the "strength trend curve" for 1040 steel at reductions above 25%, and perhaps even a slight decrease with increasing cold workO Second, the slight decrease in the tensile strength of 1060 steel at a very small amount of cold work, The tensile strength and yield strength values are somewhat lower for the transverse specimens compared to those for the longitud(4) inal ones. This difference was quite small when stress relieved values were compared. Two steels, 1040 and 4615 showed high initial directionality in ductility which increased with further cold working. Ebert concluded from his tests that cold working by drawing produces little directionality in the tensile strength, marked directionality in the yield stress, which can be removed by stress relieving and a limited directionality in the ductility. Rosalsky(57) used 3/4 inch 2S-0 aluminum plate to investigate the effect of prestraining by different stress states on yield and fracture properties. Tensile prestrains were produced by tension and by cross compression, and compressive prestrains by direct

-42 compression and by bulging. The material was tested in tension in different directions. No stress-strain curves were plotted. His conclusions were that the fracture properties of 2S-0 aluminum depend not only upon the nature and magnitude of the prestrain, but also upon the stress state under which a given prestrain is effected. For tensile prestrains of the same magnitude (after a critical amount of prestrain) a stress state of biaxial compression resulted in a higher retained ductility than did a stress state of simple tension. However, for tensile prestrains of the same magnitude, a stress state of simple tension resulted in the same retained tensile ductility as did a stress state of cross compression. For compressive prestrains of the same magnitude, a stress state of simple compression resulted in a higher tensile fracture stress in subsequent testing, than such prestrain obtained by biaxial tension. The shapes of the curves obtained for the tensile fracture stress versus prestrain differed considerably when the stress state for prestraining was varied, No significant changes in the tensile yield strength versus prestrain curves were obtained by variations in the prestraining stress state. For the tests, the material was annealed at 650 for 30 minutes. From the yield data it seems unlikely that annealing was complete. Burns and Heyer(58) investigated the anisotropy of strength and plastic properties of low carbon sheet steel. They used flat specimens in an attempt to measure variations in plastic strains.

-43 Since the slip line fields for flat and for round specimens are very different from each other, the correlations found may be doubtful. It has been recommended by Rittenhouse and Picklesimer(8) that only round specimens should be used for such studieso Stowell and Pride(27) deduced a large number of semi-empirical relations between Poisson's ratios in the plastic range and presented stress-strain curves both in tension and in compression. Tests were carried out on aluminum blocks and stainless steel sheets. Correlation appears to be goodo Gerard and Wildhorn(59) reported a set of test data for the variation of Poisson's ratio in the yield region of the stress-strain curve for the aluminum alloys 14S-T6, 24S-T4 and 75S-T6. Miller(60) reported experimental stress-strain data for tension and compression of different thicknesses of 75S-T6 aluminum alloy sheets which showed small anisotropy of strength at small plastic strains, but the stress-strain diagrams almost coinciding for elastic strains and large plastic strains, Anisotropic behavior of Zircaloy-2 (61) which is an important material for reactor applications, was investigated by Mehano He analyzed the stress-strain characteristics under combined stresses according to Hills theory for anisotropic flow, although it has not been shown whether Hillss postulates are satisfied. Some of the data seem to be good. A very extensive investigation on the effects of fabrication on the anisotropic behavior of Zircaloy was made by Rittenhouse and Picklesimer, (6263)

A few investigations on the dislocation patterns produced by cold working are mentioned below. No Fo Mott(64) investigated the phenomenon of piling up of dislocations against a barrier, theoretically as well as experimentally, Wilsdorf and Schmitz(65) presented electron micrographs of glide dislocations and dislocation tangles in 99~9994 pure aluminum single crystals oriented for single slip for deformations up to 1%. Tomlinson(66) deformed magnesium, cobalt, copper, nickel, aluminum, brass and iron 0.8 to 25% by rolling and by tension and prepared electron micrographs of dislocation tangles and subgrain boundarieso Segall and Partridge 67) investigated the dislocation arrangements in deformed aluminum polycrystals.

CHAPTER IV EXPERIMENTAL PROGRAM From the preceding chapters, viZo Theoretical Considerations and Review of Literature, it is seen that although a large amount of work has been done on the anisotropic behavior of metals under different conditions of plastic straining our knowledge about the same is far from completeo Most of the investigations about the anisotropic behavior of prestrained materials have been limited to small prestrains onlyo However, as noted in the Introduction, there are a variety of processes like extrusion, drawing, etc. which are common in mechanical engineering practice and involve very large plastic strainso Experimental data regarding directional variation of mechanical properties developed by such large strains is very scarce. In an attempt to explore this field the following experimental program was drawn up. This chapter on experimental program has been divided into three sections and discussed as below: A. Selection of Material; Be Program Outline; C. Experimental Procedure. Ao Selection of Material Reasons for Selecting Aluminum As mentioned in Chapter I, commercially pure aluminum was selected as the material for the present investigationo The reasons -45

for this selection are as follows: (a) Being a single phase solid solution, it would be possible to extend the conclusions arrived at to a wide variety of materials; (b) It is a very widely used engineering material, is readily available and is inexpensive; (c) Blocks big enough to yield tensile specimens can be prestrained without too great difficulty. (d) Being a fcc solid solution, it has a large number of slip planes and was expected to exhibit reasonable isotropy in the annealed condition. Fabrication Procedure One bar of 1100-F aluminum, 4 inches in diameter and 12 feet long was purchased as supplied by retailerso The chemical composition of the material is given in Table IV-Io The fabrication procedure used to manufacture this bar is as follows: The metal is cast as a 14" x 14" square ingot. hot rolled to 6-3/8" x 6-3/8" bloom stock, cooled to room temperature and scalped to 6" x 6" square billet. The billet is hot rolled to finish size, cooled to room temperature, and finally roll straightened. It has been stated that the cold work in processing is limited to the small amount involved in roll straightening. As such, it was assumed that the grains would be randomly oriented, Unfortunately, it did not corroborate to later experimental findings. The material was

-47 TABLE IV-I COMPOSITION OF 1100-0 ALUMINUM Required Element Minimum Maximum Actual Aluminum 99.00 --- 99007 Iron ----- --- 57 Silicon ----- ---.12 Iron + Silicon ----- 1.00.69 Copper ----- 0.20.18 Manganese -----. 05.008 Zinc ---- O.O.00 Others Each ----- 0.05.03 Total. —-- 0 15 o06 found to be anisotropic at least as far as the strains are concerned even in the annealed state. B. Program Outline For the: purpose of investigation of the directional variation of the mechanical properties, it was decided to proceed as detailed below: (a) The material was to be machined to produce prestrain specimens for tensile and for compressive prestrains,

-48 After machining, the prestrain specimens were to be annealed and then strained to different prestrain states in tension and in compression at the lowest practicable strain rateo (b) The axes of orthotropy were to be found by measurement of diameters of the prestrained materialo (c) Blanks were to be cut from the prestrained material with reference to the three orthotropic diredtions to make specimens for the tests mentioned below in (d)o (d) Three kinds of tests were to be made, vizo, (.i) Tension tests; (ii) Compression tests; (iii) Plane Strain. testso For tension and for compression tests, an average of two tests were to be taken for each caseo Test specimens were to be machined out of the blanks cut from the prestrain specimenso (e) True stress-true strain diagrams were to be plotted -from the data obtained,from the tension and compression tests and flow constants determinedo The curves were to be drawn.in different combinations to assess the differences between the curveso (f) Yield surfaces were to be constructed with the help of the yield data obtained from the three kinds of tests and measured values of plastic Poisson's ratios.

-49 (g) Fracture cross-sections were to be measured and residual ductilities in tension were to be calculated and plotted against prestrainso Co Experimental Procedure This section is devoted to descriptions of the processing of the material, equipment used, specimens, and methods of testingo Processing of Material The material was purchased as a 4 inch diameter bar in the 1100-F conditiono An outline of prior processing has been mentioned in a preceding section of this chaptero It was anticipated from this processing history that the material should be in an almost annealed condition with very small amount of cold work in it and also that it should be isotropic; at least there should be an axial symmetry of properties. Unfortunately, that was not the caseo A hardness traverse of a cross-section showed variations from 57 RE to 67 RH from the center to the periphery, which is equivalent to true strains from 0.05 to 0.15 according to previous investigations by the authoro Also it was found that when an annealed specimen was strained along an axis parallel to the axis of the bar, transverse strains were not uniform, but a circular section became approximately ellipticalo The directions for largest and smallest strains remained always constant irrespective of the location of the specimen in the baro Also, when a specimen taken at right angles to the bar axis was strained, one of the principal axes was always found parallel to the bar axis. Hence, it was deduced

-50 that the material was not isotropic, but was orthotropic in nature, one of the axes of orthotropy being parallel to the axis of the bare Designation of Prestrain Now, since it was recognized that the material was anisotropic, it was necessary to designate the three principal directions of orthotropy by some meanso The axial direction was called the vZ' direction, the direction of largest transverse strain when the material was stressed uniaxially along the Z direction was designated as the'XT direction and that of smallest transverse strain was called the TY' direction. They are shown in Figure IV-lo On all specimens tested, at least two of these directions were marked on the specimens at all stages of fabricationo The material was always prestrained in the VZ directiono The material was tested in the annealed condition, 10 conditions of varying degrees of compressive prestrain, and 4 conditions of tensile prestraino The coding system and their repsective prestrains are given in Table IV-IIo It was possible to prestrain in compression to a true strain of 1.953, but in tension only to 002450 The limiting factor in compressive prestrain was the tallest specimen of 4 inch diameter that could be compressed without causing bucklingo A specimen 4 inches in diameter and 7inches long was successfully compressed, but an 8 inch long specimen buckled and had to be discarded. On the other hand, in tensile prestrain the limiting factor was the end of uniform elongation and incipient necking. Figures IV-2 and IV-3 show the approximate

-51 Y x Y z I I x 7 Figure IV-1. Axes of Orthotropy.

-52 4" i r I I.' O 1 so NNC I Figure IV-2. Specimen for Compressive Prestraining. Figure IV-3. Specimen for Tensile Prestraining.

-53 TABLE IV-II PRESTRAIN DESIGNATIONS* A Zero prestrain C2 Compressive prestrain 0o.06 C3 Compressive prestrain 0o1494 C4 Compressive prestrain 0o204 C5 Compressive prestrain 0.244 C6 Compressive prestrain 0.300 C7 Compressive prestrain 0.455 C8 Compressive prestrain 0o673 C9 Compressive prestrain 1953 C10 Compressive prestrain lo010 Cll Compressive prestrain 1.490 Tll Tensile prestrain 0o1098 T15 Tensile prestrain. 0152 T20 Tensile prestrain 0.206 T25 Tensile prestrain 0.245 * Designation of tests are described in a later section of this chapter. dimensions of the specimens before prestraining, The length of the compression specimens were such that after prestraining the length was reduced to approximately 2 inches except for specimens C9 and Cll, for which the final lengths were 0.633 inch and 1576 inches respectivelyo Annealing The specimens for prestraining were machined, and then annealed. It was first expected that one hour at 900~F would suffice to bring the material to its softest condition, but in practice it was found that annealing was not complete in. one hour; presumably since the surface was machined bright and very little radiant heat was absorbed by the specimens. Therefore the annealing time had to be increased to three hours for the compressive prestrain specimens and to two hours and thirty minutes for the tensile prestrain specimenso They were allowed, to

-54 cool in the furnace which took about 20 hours to reach 100 F from the annealing temperature of 9000Fo After annealing the Rockwell H hardness was found to be about 30-32 which is about the softest condition that this material can be brought tOo A slightly different procedure had to be followed for testing at zero prestraino Since it would be very difficult to machine test specimens from annealed material, the specimens were made first and then annealed for 35 minutes at heat and cooled in the furnaceo Prestraining The compressive prestraining was done in a 400,0001lb. Tinius Olsen mechanical type testing machine between two hardened and ground steel plates0 Some of the specimens had to be remachined to reduce the diameter before prestraining could be completed0 A commercial preparation of MoS2 was used to lubricate the ends of the blocks in order to reduce the friction and avoid barrellingo This lubricant worked admirably well; so much so that after 10% reduction in length the variation of diameter from the ends to the middle of a block was never more than a few thousandths of an incho In case of the longer specimens, the lengths were reduced by 00200 inch by each loading, unloaded and reloadedo The ends were lubricated after every reduction of length of 0o600 incho For shorter specimens ends were lubricated more frequently, Tensile prestraining was done in a 120,000 lbo Riehle mechanical type testing machine. To prestrain such large specimens

-55 special adapters had to be made which are shown in Figure IV4,. There was hardly any variation in diameter at all over the middle 6 inches of the strained specimeno This section was used for testingo Figure IV-5 shows one tensile prestrain specimen mounted on the testing machine for prestrainingo Strain Rate It is well known that stresses are functions not only of strains, but also of strain rateso However, stresses are of little concern while prestraining since the variable concerned is strain and not stresso Still, to eliminate any possibility of introducing strain rate as a variable, the rate was kept as lov as practical. Compressive prestraining was done with a machine head speed of about 0O050 inch per minute, which is equivalent to about 0O00042 7/n per second at the last stages of compressiono It is considered.slow enough to give practically static condition of straining(o68)'In case of the C9 specimen, where the final length was less than 2 inches, a slower head speed was used, For tensile prestraining the strain rate was about OoO001,"/ per second. No increase in temperature was detectable to touch during the straining operation. After prestraining, smaller test specimens were cut out of the blocks and tested. Three kinds of tests were made, vizo, (i) Tension test (ii) Compression test (iii) Plane strain test

-56 PULL ROD 3/4" BOLT PRESTRAIN SPECIMEN Figure IV-4. Large Grips for Tensile Prestraining.

-57 Figure IV-5 Set-up for Prestrain. ing in Tension. I Figure IV-6. Tension Specimen. Figure IV -7 Small Grips for Tension Test.

-58 Two identical specimens of the same orientation were used for each of the tension and compression tests. Tension Test Specimens A sketch of a tension test specimen is shown in Figure IV-6. Lengths of the specimens varied according to the size of the block from which the specimens were taken. It was convenient to have the specimens as long as possible to facilitate measurements, and hence no attempt was made to standardize the lengths. The tolerance on the diameter was +.005 inch and no attempt was made to make it any closer. However, the variation in diameter of a single specimen was kept within 0.0002 incho Special grips had to be made for testing the specimens. They are shown in Figure IV-70 Three-quarter inch gage length was used. A small step was machined on the specimens to prevent the specimen from necking within the grips. Compression Test Specimens Originally it was intended to make the compression test specimens of the same diameter as the tension test specimens, but it was found impracticable to test specimens of so small a diameter and hence the diameter had to be increased, Some compression specimens of about 0.400 inch diameter were tested. Then it was found that with increasing prestrain, certain compression specimens behaved very much like an ideal elastic-plastic material and if the length was substantially larger than the diameter, localized plastic deformation would start and render the specimen useless. Hence, it was

-59 attempted to make the compression specimens as large in diameter as possible from 1/2 inch square blankso As such, the diameters varied from o400 inch to 0435 inch and lengths from o400 inch to 0450 inch. With specimens of so small lengths it was found difficult to measure the small plastic strains accurately. To circumvent this difficulty, two types of specimens were usedo The longer ones were used to measure small strains and shorter ones were used for large strainso They are shown in Figure IV-80 Plane Strain Specimens Plane strain specimens are shown in Figure IV-9. To obtain a condition of plane strain, the width was made large compared to the length and thickness0 The flanges of large section effectively eliminated any contraction of the width. Theoretically, the length. should have been large compared to the thickness) but it was found impracticable to make very thin specimens without introducing substantial strain during manufacture, Designation of Tests Every test had to be given a code designation for referenceo The first letter and numeral denotes the prestrain state. The next letter is. T C or S indicating tension, compression or plane strain test respectively. In case of tension or compression tests) the next letter is X, Y or Z denoting the axis of the specimen with respect to the axes of orthotropyo In case of plane strain specimens, after the letter S, there are two letters from among X, Y and Zo The first letter indicates the direction of zero strain, and the second

-60 -A0.4"d - I 0.4"dl — I - I(a) Short Compression Specimen. s 11 I (b) Long Compression Specimen. Figure IV-8. Compression Specimens. --- i.8" -0.4"H T I 0.120 of Figure IV-9. Plane Strain Specimen.

letter shows the direction in which the specimen has been pulledo When more than one specimen of a kind is tested, they are numbered, and the number is placed at the end of the designation code. As an example, C8TY2 refers to a tension specimen whose axis is parallel to the Y direction and the prestrain state of the block from which it has been machined is C8o Fabrication After prestraining, the major and minor axes were marked. The blocks were first sliced with a band saw to produce blanks from which test specimens were to be machined. To prevent any overheating with consequent change in strength state of the material, the blocks were cooled in ice prior to slicing and were frequently cooled during cutting too. Tension and compression test specimens were machined on a lathe with a copious supply of kerosine to produce a good surface finish and also to keep the specimens coolo It was essential to keep the thickness of the material disturbed during cutting operation to a minimum, For this purpose, the tools were ground and honed, to a sharp edge0 The tool signature was 10, 30, 8, 8, 20, 0O 0 in accordance with the ASA standards0 During the finishing cuts, the feed was 0o0012 inch per revo and the speed was about 60 fpmo The plane strain specimens were prepared by milling with sharp end cutting cutterso Copious amount of kerosine was used as a coolanto

-62 Testing Procedure All of the testing was done on a 60,000 lb. hydraulic drive testing machine. (a) Tension Tests. A clip-on electromechanical strain gage and recorder was used for most of the tension tests. A load-extension diagram was obtained until the maximum load was attained, at which time the gage was taken off and instantaneous cross sections at the neck was recorded along with instantaneous loads. Measuring the instantaneous cross section posed certain problems. The material being anisotropic in transverse strain, a circular section did not remain circular, but became elliptical. Hence, it became necessary to record both maximum and minimum diameters, and the area was calculated assuming that the cross section remained elliptical, which is true to a close approximation. Measurements were made with a pointed micrometer until the surface became too rough to make accurate measurements possible. However, the specimens were strained until fracture and the cross section at fracture was noted. (b) Compression Tests. As mentioned before, there were two kinds of compression specimens. The longer ones were for small strains and the shorter ones were for large strains. For small strains an electromechanical clip-on strain gage with Baldwin-Southwork stress-strain recorder was used to obtain a load-deflection curve from which the stress strain diagram was calculated up to a strain of about 0.02. Shorter specimens were compressed between two ground flats and the ends were lubricated

between each compression to reduce friction. After every compression the load was noted, the specimen taken out and measured with a micrometer. (c) Plane Strain Tests. In the case of the plane strain tests, only the yield stress is required, The stress-strain diagram was not plotted, and only the yield load was noted from the load-deflection diagram, To test these specimens special adapters were designed and made. The deformation of the specimens was picked up by an electromechanical clip-on gage. Figure IV-10 shows the adapters used and Figure I-11 shows the adapter and an electromechanical clip-on gage mounted on the testing machine together with a servomotoro Two electromechanical clip-on gages are shown in Figure IV-12.

Figure IV-10. Adapter for Plane Strain Test. Adapter Figure IV-11t. Set-up for Plane Strain Test. Electromechanical Clip-on Gage Figure IV-12. Electromechanical Clip-on Gage.

-65 CHAPTER V DISCUSSION OF RESULTS Introduction The results of the present investigation as obtained by tests outlined in Chapter IV are presented and discussed in this chapter. They are also compared with the findings of previous investigators where possibleo The few references made in this chapter are primarily for the purpose of analyzing the data obtained and comparing with the present work. The stress-strain relations for different tests are presented in the form of tables and also as curves on logarithmic co-ordinates for comparison. Yield and ductility data are also given both as tables and diagramso Definitions Before starting the discussion of the results obtained in the present investigation, it would be of advantage to restate the meaning of some terms used in this chapter. The term "flow stress" has been used in the literature to mean different things in different places. Whenever this term is used hereafter, it means the stress level as a function of strain under a given condition of straining. The term "strain" always means the natural strain measured from the beginning of a particular test, irrespective of the prestrain or prior history of the material. By "stress" is meant the true stress,

-66 corrected by the use of the Bridgman correction factor whenever applicableo Stress and strain are denoted by a and e respectively. Material Since the material was found to be orthotropic as regards strains, it was expected to be orthotropic in flow properties, too. Hence, it was necessary to test it in the annealed condition as well as in the prestrained conditions in all the three directions. Since two specimens of each kind were tested in tension and compression, for each prestrain state, six tension specimens, six long compression specimens and six short compression specimens were tested. It was not found feasible to repeat plane strain tests since it took more material to make these specimens than was available. Prestrain Compressive prestrain is expressed as Y, 0 p = ln f where io and if are original and final lengths of the block respectively, Elastic strain is neglected. Tensile prestrain is expressed as ef Ep = in where io and if are gage lengths before and after strainingo Diametral strains are also expressed in a similar way. In compression, the ratio of the two transverse strains were found to be 1.47 and that in tension lo59. In accordance with the above definition, prestrains are always

-67 numerically positiveo This has been done to avoid carrying along the positive and negative signs, Strain Recording In most of the tension and compression tests an electromechanical strain gage was usedo For a few tension tests SR-4-electric resistance strain gages were used as a check. Figure V-1 shows a comparison of data obtained by the two methods for testing two C9TY specimenso The largest strains recorded by the electromechanical and by the SR-4 electrical strain gages were 0O017 and 0o015 respectively, the maximum load having occurred at those strainso It is apparent from the diagram that the agreement is reasonably good with very little scattero Tension Test and Correction for Triaxiality of Stress in the Neck Specimens with low prestrain when tested in tension in the Z direction showed substantial uniform strain before necking started, as is expected in such caseso During the uniform straining the crosssection did not remain circular, but became elliptical, which is in contradiction to the observations of Mehringer and McGregor(52) in testing low carbon steels, With larger prestrains and while testing in the X and Y directions, the uniform elongation was small and necking started earlyo In addition, the section became elliptical0 The stress distribution becomes complex after the initiation of necking, and hence reasonably accurate true stress-true strain diagrams can be plotted

-68 for small strains only unless a correction for the triaxiality of stresses at the neck is resorted too Three semi-empirical approaches for such corrections are availableo The correction factor arrived at by Siebel(69) and by Davidenkov and Spiridonova() are identical, although derived in different wayso The relationship for the correction factor derived by Bridgman(53) is rather different from the other as shown latero In all the approaches, the material is assumed to be isotropic, and hence a circular section is assumed to remain circularo To apply the correction, it is necessary to measure the radius of curvature of the specimen at the neck on a plane containing the axis of the specimen, and also the radius at the necko The correction factors are given as follows: Bridgman's Correction factor Kb = (1 + n b (1 + l) In (1+ ) a 2R SiebelYs Correction factor Ks = 1 + — 4R where a = minimum radius; and R = radius of curvature at the necko In the present investigation, the material being anisotropic, the necked down section became elliptical and hence the radii of curvature were different in different directionso Hence, even if it were possible to measure the radii, it would not have been possible to use these corrections without modification~

-69 However, there have been attempts to relate the ratio a/R with the current strain state, so as to eliminate the necessity for the measurement of the radii, which is a rather cumbersome process. Kula and Larson (70) and Nunes(71) investigated the relationship between the ratio a/R and strain, and suggest that to a good approximation the ratio a/R can be replaced by the strain after necking starts. Since the necking begins at the maximum load, the following substitution can be made: a = ( e-m) where e = current strain; and Em = strain at maximum loado Assuming this substitution is valid for the material under consideration, it now becomes possible to apply corrections to the test data, using either Bridgman's or Siebel's correction factoro The choice seems to be rather immaterial since for e - em = 1. 0, the SiebelPs correction factor is only about 3% less than the Bridgmanqs correction factor, and the difference is much smaller for lesser values of C - em Considering the uncertainties of the assumptions involved in the derivation of these factors, in the substitution for a/R values and the experimental errors involved, there seems to be hardly any reason to prefer one to the othero This could be important if data are collected to larger strains, but the largest strain recorded in the present investigation was about o,0o For many specimens, the largest strain was only 0.6. The largest strain readings had to be restricted as the necked surface became rough and after a certain stage accurate measurements could no longer be made.

-70 It was decided to use the Bridgman correction factor since this has been used in many more investigations than the other. Compression Test and Correction for Friction As mentioned in Chapter IV, two kinds of specimens had to be used for compression test; one kind for small strains, and the other for larger strains where elastic strains can be neglected, Hence, it is possible to take the specimen out from the machine to measure its reduction in length. To reduce friction, the ends of the specimens were lubricated with a commercial preparation of MoS2 which worked very well, even though friction cannot possibly be entirely eliminated. According to Alexander(72) a coefficient of friction of 0.0056 will cause an error of about 1,4% for a diameter to height ratio of 5, To estimate the coefficient of friction encountered while using MoS2 as a lubricant, a near accident during the compressive prestraining may be mentionedo A cylinder of about 4-1/2 inches in diameter was being compressed with a spherically seated compression head, using MoS2 as a lubricant, Slight misalignment of the piece caused it to slip and it shot out of the machineo The two faces had become non-parallel, and the maximum difference in length on a diameter was found to be only 050 inch, Therefore, the friction angle was tan1 10, ioeo the friction coefficient was only about 0,00560 The error due to so small friction is negligible and hence no correction is required.

Plain Strain Test The plane strain test is a biaxial stress test, As explained in Chapter IV, the direction of zero strain of such a test specimen SUY is X and it is loaded in the Y directiono Therefore, the stress on a Y plane is given by the load divided by the cross-sectional area and that on a X plane is given by xx = -1xy yy where xy is the plastic Poisson's ratio in the X direction for uniaxial loading in Y direction, Stress-Strain Curves Since the primary interest in the present investigation is in the plastic stress-strain relations, not much emphasis is given to the elastic range of the stress-strain diagram. Although a few tension specimens were tested with the help of SR-4 post-yield strain gages, most of the work was done with an electromechanical clip-on gageo The primary reason for using electromechanical clip-on gages is the ease with which they can be handled to give good results cheaply -4 and quicklyo While at very small strains of the order of 10-5 or 104 ino/in. SR-4 gages are more reliable, at larger strains, the accuracy of both type of gages are comparableo Furthermore, it is rather difficult to properly fix the SR-4 gages on specimens of very small diameters as used in this investigation, The stress-strain curves are plotted on log-log co-ordinates, rather than in linear Cartesian co-ordinates. If plotted in Cartesian

-72 co-ordinates, most of the curves look very similar and the differences are not clearly visibleo At large strains, all the curves become so flat that hardly any strain hardening is noticeableo On the other hand, at small strain, differences in strain at any stress level become obliterated. and two curves plotted together for comparison become too close to comparisono Also, since the stress-strain relation for an annealed material can approximately be expressed as a power relation, the curve is a straight line on logarithmic co-ordinates, while no such visual effect is possible when plotted in Cartesian co-ordinateso Flow stresses at different strain levels for the tests are given in Table V-Io This gives the average of results obtained from two tests of each kindo The tension test data has been corrected. with the help of modified Bridgman correction factor and the expression for the corrected true stress now is: L a- A (1 + -— ) in (1 + 2m) E "m 2 where L is the load, and A the instantaneous cross-sectional areas General Remarks The results of various tension and compression tests for different prestrain states are presented in different combinations for comparison purposes in Figures V?2 to V-77o A few general remarks may be made before the comparative discussion.

TABLE V-I FLOW STRESS IN KSI Annealed Material: Sp = 0.000 Strain TX TY TZ.002.003.004.005.007.01.02 3.5 4.0 4.5 4.9 5.5 6.3 8.2 3.3 3.7 4.1 4.5 5.1 6.0 7.8 3.5 4.0 4.4 4.8 5.4 6.3 8.4.03.05.07 9.4 10.9 11.8 9.0 10.4 11.1 9.7 11.1 11.7 0.1.15.2.25.3.35.4 12.5 13.5 14.4 15.2 15.9 16.5 17.1 12.0 13.5 14.5 15.5 16.4 17.0 17.7 12.9 14.4 15.5 16.4 17.2 17.9 18.5.45 17.6 18.3 19.1.5.6 18.2 19.0 18.9 19.8 19.6 20.5.7 19.8 20.7 21.4.8 20.4 21.5 22.3.9 1.0 21.1 21.8 22.3 23.0 22.9 23.5 Cx CY CZ 3.5 4.0 4.4 4.8 5.5 6.3 8.3 9.3 10.7 11.7 12.6 13.8 14.8 15.4 16.5 17.1 17.7 18.4 18.8 19.8 20.7 21.5 22.3 23.0 3.6 4.1 4.5 4.9 5.5 6.3 8.3 9.3 10.6 11.6 12.7 14.0 15.1 16.0 16.7 17.4 18.0 18.5 19.1 20.0 20.7 21.5 22.1 22.7 4.1 4.5 5.0 5.4 6.1 6.9 8.7 9.6 10.9 11.8 13.0 14.4 15.4 16.3 16.9 17.5 18.3 18.7 19.3 20.2 21.0 21.7 22.4 23.0 TX 10.5 11.2 11.5 11.8 12.0 12.2 12.4 12.5 TY 11.4 12.0 12.4 12.5 12.5 12.6 12.7 12.8 TZ 12.2 12.3 12.4 12.5 12.6 12.8 13.4 13.7 Prestrain State TEl; Ep = 0.1098 12.5 12.8 13.1 13.4 13.9 14.4 14.7 15.2 15.6 15.9 16.3 17.1 17.7 12.8 12.9 13.2 13.3 14.0 14.5 15.1 15.4 16.0 16.5 16.9 17.4 18.2 19.1 14.2 14.5 15.2 16.0 16.5 17.1 17.7 18.3 18.8 19.3 19.8 20.6 12.5 cx CY CZ 11.2 12.1 12.5 12.6 12.9 13.2 13.6 13.8 14.1 12.2 12.6 12.9 13.0 13.1 13.3 13.5 13.7 14.1 9.5 10.2 10.7 11.0 11.5 11.9 12.2 12.3 12.4 14.5 14.8 15.5 16.2 16.8 17.4 17.9 18.4 14.4 14.8 15.5 16.2 16.8 17.4 17.9 18.4 12.5 12.8 13.6 14.6 15.5 16.4 17.0 17.6 18.9 19.4 20.1'20.9 21.5 22.2 22.9 18.7 19.3 20.0 20.7 21.5 22.3 23.0 18.2 18.7 19.7 20.5 21.2 21.8 22.4 GI rresran Mate TLJ p = 0.15C Tt 10.9 11.9 12.4 12.6 13.0 13.2 13.4 13.4 TY 12.2 12.9 13.2 13.4 13.4 13.5 13.7 13.9 TZ 12.6 12.8 -12.9 12.9 13.0 13.2 13.5 13.8 13.4 13.5 13.8 14.1 14.5 14.8 15.3 15.7 16.0 13.9 14.0 14.2 14.8 15.4 15.7 16.3 16.7 17.0 14.4 14.6 15.4 16.0 16.7 17.3 17.7 18.4 18.7 16.6 17.0 17.8 18.7 17.5 17.9 18.7 19.2 19.6 20.5 21.3 21.9 22.6 23.4 Cx 13.0 CY 13.1 CZ 10.7 13.4 13.5 13.7 13.8 14.0 14.4 14.5 13.7 14.0 14.0 14.0 14.1 14.2 14.4 11.5 12.1 12.4 12.7 13.0 13.3 13.3 14.7 14.9 15.3 15.9 16.5 17.1 17.6 18.2 18.7 14.7 15.2 15.5 16.2 16.5 17.4 17.8 18.3. 18.6 13.4 13.4 13.5 14.1 14.8 15.6 16.5 17.2 17.8 19.3 19.7 20.5 21.4 22.1 22.7 23.4 19.1 19.5 20.4 21.2 22.0 22.7 23.5 18.4 19.0 20.0 20.8 21.5 22.3 23.0 Tt 11.5 12.7 13.5 13.8 14.2 14.3 14.5 14.6 14.6 TY 12.7 13.9 14.3 14.4 14.5 14.6 14.9 15.1 15.1 TZ 13.6 13.9 14.0 14.2 14.3 14.4 14.7 15.0 15.5 14.7 15.2 15.7 Prestrain State T20; ~p -.206 14.9 15.2 15.6 15.8 16.3 15.4 15.9 16.2 16.6 17.2 16.6 16.9 17.2 17.6 17.6 18.0 18.4 18.9 18.3 19.3 19.8 20.7 21.3 22.1 16.3 16.9 17.5 18.0 18.5 19.0 19.5 19.9 20.4 22.7 23.5 24.3 Cx 13.3 14.3 14.6 14.7 14.9 15.0 15.4 CI 14.1 14.7 15.1 15.2 15.3 15.4 15.5 CZ 10.9 12.0 12.6 13.2 13.6 14.o 14.4 15.5 15.8 16.0 16.3 16.8 17.4 18.0 18.5 19.0 19.5 19.9 20.4 21.3 21.9 22.7 23.4 24.1 15.6 15.8 16.0 16.4 16.8 17.4 17.9 18.4 18.8 19.4 19.7 20.2 21.0 21.6 22.4 23.1 23.7 14.4 14.4 14.1 14.0 14.4 14.8 15.6 16.4 17.1 17.8 18.4 18.9 19.9 20.8 21.6 22.4 23.0

TX 12.3 13.5 14.o0 14.4 14.6 14.8 14.9 15.0 Tr 13.4 14.4 14.6 15. 1.0 55.2 15.4 15.4 TZ 14.8 15.2 15.3 15.3 15.4 15.5 16.0 16.2 Cx 14.4 15.1 15.5 15.6 15.9 16.0 16.2 16.4 C 14.1 15.4 15.7 15.8 15.9 i6.0 16.2 16.2 CZ 10.4 11.7 12.5 13.0 13.6 14.2 14.6 14.7 15.1 15.4 16.5 16.5 16.3 14.8 15.2 15.5 16.8 16.6 16.5 14.7 TABLE V-I COET'D Prestrain State T25; ep = 0.245 15.2 15.6 15.9 16.3 16.6 15.8 16.1 16.6 17.0 17.4 17.1 17.6 18.3 18.8 19.3 17.0 17.5 18.0 18.5 18.9 16.6 17.2 17.6 18.2 18.6 14.6 14.9 15.3 15.8 16.5 17.0 17.9 19.7 19.4 19.1 17.3 17.4 18.3 20.2 19.9 19.5 17.8 17.7 18.8 20.6 20.3 19.9 18.4 18.1 19.2 21.0 20.7 20.3 18.9 18.7 20.0 21.7 21.5 21.0 19.8 19.4 22.6 23.4 22.2 21.7 20.7 22.8 22.4 21.5 23.5 23.1 22.4 24.1 23.7 23.0 Strain.002.003.004.005.007.01.02.03 TI 12.1 12.5 12.6 12.7 12.9 13.1 13.4 13.5 TI 12.5 12.9 13.1 13.2 13.3 13.4 13.6 13.7 IZ 9.2 10.0 10.4 10.6 11.2 11.5 11.8 12.0 CX 10.5 11.5 11.8 12.2 12.5 12.7 13.0 13.3 CT 12.0 12.6 13.0 13.3 13.4 13.5 13.9 14.1 CZ 12.4 12.6 12.8 12.9 13.1 13.4 13.7 14.0.05 13.7 14.0 12.1 13.5 14.4 14.4.07 14.1 14.1 12.3 13.6 14.5 14.9 Prestrain State C2j Ep = 0.106 0.1.15.2.25.3 14.3 14.9 15.3 15.7 16.2 14.4 15.0 15.6 16.2 16.8 12.5 13.1 13.7 14 15.1 14.1 1.6.4 15.9 165 4.8 15.5 16.0 16.6 17.3 15.5 16.5 17.3 17.9 18.5.35 16.5 17.3 15.7 17.1 17.7 19.0.4 17.0 17.7 16.3 17.6 18.4 19.5.45 17.4 18.1 16.9 18.1 18.9 19.9.5 17.7 18.4 17.5 18.5 19.5 20.4.6 18.6 19.2 18.5 19.5 20.4 21.2.7 19.5 20.0 19.2 20.4 21.4 21.8.8 20.2 20.7 20.1 21.2 22.2 22.5.9 1.0 p.~ 20.8 21.6 21.9 22.9 23.1 22.6 23.7 23.6 TI 12.9 13.2 13.3 13.4 13.5 13.7 14.0 14.2 TY 13.0 13.6 14.2 14.3 14.3 14.4 14.6 14.7 TZ 9.8 10.8 11.5 12.0 12.4 12.6 13.1 13.3 14.4 15.0 13.5 14.5 15.1 13.6 Prestrain State C3j q = 0.1494 14.6 15.0 15.4 15.8 16.4 15.1 15.5 15.9 16.5 17.1 13.7 14.1 14.9 15.1 15.7 16.8 17.6 16.3 17.1 18.0 16.9 17.7 18.5 17.5 18.1 19.0 17.9 18.9 19.8 18.8 19.4 20.4 19.7 20.0 20.5 21.0 20.6 21.5 22.1 CX 12.5 13.3 13.8 14.o 14. 14.5 14.7 14.9 CT 12.9 13.5 13.9 14.3 14.5 14.6 14.8 14.9 CZ 13.2 13.4 13.5 13.7 13.9 1a.1 14.5 14.7 15.1 15.1 15.3 15.3 15.3 15.6 15.4 15.9 16.4 16.9 17.5 15.6 6.216. 7 17.2 17.7 16.1 16.8 17.5 18.2 18.7 18.0 18.2 19.2 18.6 18.6 19.7 19.0 19.1 20.2 19.5 19.6 20.6 20.4 20.6 21.5 21.2 21.4 22.3 21.8 22.0 23.0 22.5 22.5 23.6 23.3 23.2 24.3

TABE V-OI OCT'D Strain.T.TZ.002.003.004.005.007.01.02.03.05 13.3 15.8 14.1 14.3 14.4 14.4 14.5 14.5 14.8 14.3 15.1 15.5 15.6 15.7 15.8 16.0 16.2 16.3 10.1 1.1 11.6 12.2 12.6 13.1 13.5 13.5 13.5 Preetrain State C4 qp = 0.204.07.1.15.2.25.3 15.1 15.3 15.9 16.2 16.5 17.0 16.3 16.7 17.1 17.6 18.1 18.7 13.5 13.8 14.2 14.8 15.2 15.6.35.4.45.=.6.7.8.9 17.3 17.8 18.1 18.5 19.3 19.9 20.5 21.0 19.0 19.5 19.9 20.3 21.0 21.7 16.0 16.9 17.4 17.9 18.8 19.5 20.4 1.0 ex cr cz 12.7 13.5 13.8 14.2 14.5 14.7 15.2 15.3 19.3 14.4 15.3 15.6 15.9 16.0 16.2 16.3 16.4 16.5 14.2 14.5 14.6 14.7 14.9 15.1 15.5 15.7 15.9 15.5 15.6 16.2 16.6 17.1 17.5 18.0 18.5 19.0 19.5 20.4 21.3 22.0 22.6 23.4 16.6 16.7 17.2 17.6 18.1 18.5 19.0 19.4 19.8 20.2 21.0 21.6 22.4 22.9 16.3 16-7 17.4 18.0 18.5 19.0 19.5 20.0 20.4 20.8 21.5 22.3 23.0 23.7 24.5 Strain TX TI TZ.002.003.004.005.00-.01 14.0 14.6 14.9 15.0 15.1 15.2 15.2 15.8 16.2 16.4 16.5 16.6 11.1 12.1 12.6 31.0 13.5 13.9.02.03.05 15.4 15.5 15.6 16.7 16.7 16.8 14.5 14.6 14.6 Prestraln State 05; 9 = 0.244.07.1.15.2.25.3 15.8 16.1 16.5 16.8 17.2 17.4 16.8 17.0 1f.4 17.8 18.2 18.5 14.7 14.8 15.1 15.5 15.9 16.4.35.4.45 17.9 18.4 18.7 19.0 19.4 19.8 16.8 17.3 17.9.5 19.1 20.2 18.4.6.7.8 19.9 20.4 21.0 20.9 21.6 22.0 19.2 20.1 20.7.9 1.0 21.7 21.3 x 12.7 14.0 14.7 15.2 15.6 15.9 Cy 12.6 14.4 15.3 15.6 16.0 16.1 CZ 14.5 15.0 15.3 15.4 15.5 15.6 Ts 14.5 15.3 15.5 15.7 15.9 16.0 T 14.9 16.4 16.9 17.0 17.1 17.2 TZ 11.4 12.7 13.5 14.0 14.6 15.3 Cx 14.7 15.9 16.3 16.5 16.9 17.1 13.9 15.3 15.9 16.3 16.6 16.9 CZ 15.3 16.0 16.4 16.5 16.6 16.8 16.2 16.4 16.4 16.6 16.0 16.3 16.3 16.5 17.2 17.3 15.6 15.8 17.3 16.9 17.0 17.2 17.1 17.2 16.6 16;8 16.9 17.1 17.6 18.0 18.4 16.9 17.0 17.3 17.5 18.0 18.5 18.9 16.5 16.6 17.o 17.6 18.4 18.8 19.4 Prestrain State06; _p = 0.30 16.6 16.6 16.7 17.0 17.5 17.8 18.2 17.3 17.3 17.6 18.0 18.5 18.8 19.2 16.0 16.1 16.2 16. 6.3 6 17.1 17.5 16.8 16.8 17.0 17.5 17.9 18.3 18.6.17.4 17.5 17.6 18.0 18.4 18.7 19.0 17.4 17.6 18.0 18.5 19.0 19.5 20.0 18.7 19.1 19.5 19.9 20.5 21.4 22.2 22.8 23.5 19.3 19.7 20.1 20.5 21.3 22.0 22.6 23.4 24.0 19.8 20.2 20.6 21.0 21.7 22.5 23.1 23.8 24.4 I 18.7 19.0 19.4 19.7 20.3 20.9 19.6 20.0 20.3 20.7 21.4 22.0 18.1 18.7 19.1 19.7 20.7 21.5 22.2 19.0 19.4 19.6 20.0 20.6 21.3 22.0 22.5 23.4 19. k 19.7 20.1 20.4 21.1 21.9 22.5 23.1 23.7 20.5 21.0 21.4 21.7 22.5 23.2 23.8 24.5 25.1 TX 15.3 16.7 17.0 17.0 17.1 17.2 T 16.6 18.2 18.5 18.6 18.7 18.8 Z 12.5 13.9 14.6 15.2 15.9 16.5 Cx 14.8 16.4 17.0 17.5 18.1 18.3 CT 15.9 17.5 18.3 18.6 19.0 19.3 CZ 16.5 17.5 17.7 17.9 18.0 18.2 17.4 17.5 18.9 19.0 17.2 17.3 18.5 18.5 19.4 19.2 18.5 18.7 Prestrain State CT; Ep = 0.455 17.6 17.6 17.7 17.9 18.2 18.5 18.7 19.0 19.0 19.1 19.3 19.7 2.1 20.5 17.3 17.3 17.4 17.6 17.9 18.4 18.6 18.5 18.5 18.6 1.9 19.1 19.4 19.6 19.1 19.0 19.0 19.3 19.5 19.7 20.1 18.9 19.0 19.3 19.7 20.1 20.5 20.9 19.1 19.6 19.8 20.1 20.7 21.2 20.8 21.3 21.7 22.0 22.8 23.6 24.1 24.5 19.0 19.4 19.7 20.2 21.1 22.0 22.9 23.9 19.9 20.3 20.6 21.0 21.6 22.4 23.3 23.7 24.4 20.4 20.6 20.9 21.3 21.9 22.5 23.0 23.6 24.1 21.2 21.5 21.9 22.2 22.8 23.4 24.0 24.5 25.1

TABI V-I CCT'D TXC 16.0 17.5 18.0 TIr 17.2 19.0 19.6 TZ 12.2 14.2 15.2 Ce 14.1 16.4 17.5 Cy 16.5 18.7 19-7 CZ 17.2 18.6 19.0 18.0 18.3 18.5 18.8 19.0 19.9 20.3 20.5 20.6 20.6 15.7 16.6 17.4 18.3 18.5 18.0 18.8 19.3 19.5 19.4 20.2 20.4 20.5 20.5 20.5 19.2 19.5 19.6 20.0 20.1 Prestrain State C8; gp 0.673 19.2 19.3 19.3 19.5 19.7 20.0 20.3 207 21.0 20.7 20.8 20.8 20.9 213 21.5 22.0 22.5 22.8 18.8 19.1 19.2 19.2 19. 19.7 20.2 20.4 20.7 19.4 19.4 19.4 19.5 19.6 19.9 20.2 20.5 20.7 20.6 20.6 20.6 20;6 20.8 21.1 21.4 21.6 21.9 20.4 20.5 20.8 21.2 21.5 21.9 22. 22.5 22.7 21.3 21.6 22.3 22.7 23.2 23.6 24.3 21.2 21.6 22.4 23.5 21.0 21.4 22.0 22.6 23.3 23.9 24.6 22.2 22.5 23.0 23.6 24.4 24.9 25.5 23.0 23.3 23.9 24.3 24.8 25.3 26.0 Bt 17.7 21.2 22.7 23.6 24.4 24.9 25.2 25.2 25.2 Tr 20.3 22.9 24.0 24.5 25.2 25.7 25.9 26.2 26.2 Prestrain State C9; Ep = 1.953 25.2 25.2 25.5 25.7 26.0 26.2 26.2 26.3 26.5 26.8 26.9 27.1 26.4 26.6 26.9 27.1 27.7 27.4 27.8 28.1 28.3 29.1 29.8 30.4 cx 16.0 19.0 20.8 CT 16.5 20.2 22.5 CZ TE 16.5 18.3 18.8 T 1t7.6 20.1 20.7 Z 133.6 15.5 16.7 cx 15.9 18.1 19.1 Cy 15.2 18.0 19.5 CZ 17.9 20.1 20.7!3 17.5 19.9 20.9 TI 18.3 21.2 22.3 3 3 14.6 16.6 18.0 21.7 23.1 24.0 25.2 25.7 23.6 25.0 26.0 26.8 27.1 26.4 26.5 26.5 26.7 26.9 19.2 19.5 19.8 20.1 20.2 21.0 21.4 21.5 21.8 21.9 17.5 18.5 19.4 20.4 20.5 19.5 20.1 20.5 21.2 21.4 20.5 21.5 22.2 22.6 22.9 21.2 21.7 22.0 22.2 22.5 21.3 2.7 22.0 22.4 22.5 22.7 23.3 52.6 24.0 24.2 18.9 20.0 2.1 22.5 22.8 25.7 25.4 25.1 24.8 24.7 24.7 24.8 24.9 24.9 27.0 26.8 26.5 26.1 26.0 25.9 25.9 25.9 25.9 27.0 27.2 27.3 27.5 27.6 27.8 28.0 28.3 28.5 Prestrain State C10; Ep = 1.01 20.3 20.3 20.3 20.7 21.0 21.3 21.6 21.9 22.3 22.0 22.1 22.1 22.3 22.5 22.8 23.1 23.4 23.8 20.6 20.7 20.7 20.8 20.8 21.1 21.4 21.7 22.0 21.5 21.5 21.4 21.3 21.4 21.5 21.7 22.0 22.2 23.3 23.2 22.8 22.6 22.6 22.7 22.9 23.0 23.3 22.9 23.1 23.2 23.5 23.9 24.2 24.4 24.7 24.9 Pre.train State C01;.p = 1.49 22.6 22.7 22.8 23.2 23.5 25.7 23.9 24.2 24.4 24.3 24.3 24.4 24.7 24.9 25.2 25.6 25.9 26.1 22.8 22.7 22.7 22.6 22.7 23.0 23.2 23.6 23.9 25.1 25.2 25.5 25.9 26.4 27.0 27.7 26.0 26.0 26.3 26.7 27.3 27.9 28.6 28.7 29.0 29.5 29.9 30.3 30.9 31.3 I a' 22.6 22.9 23.5 24.2 24.2 24.6 25.2 25.7 22.4 22.8 23.5 24.6 22.4 22.6 23.3 23.9 24.5 25.1 25.8 23.5 23.8 24.5 25.1 25.8 26.5 27.3 25.2 25.5 26.0 26.6 27.1 27.6 28.1 24.7 24.8 25.2 26.2 26.4 26.7 24.1 24.4 25.1 25.7 26.8 26.0 Cx 14.66 17.3 18.9 20.0 21.5 22.2 23.0 23.5 Cy 16.6 19.5 21.3 22.3 23.5 2.4 24.4 25.1 CZ 17.3 20.7 22.4 23.0 2.6 24.1 24.6 24.9 24.3 24.0 23.9 23.6 23.6 23.5 23.6 25.0 25.0 24.7 24.5 24.4 24.4 24.5 25.2 25.4 25.5 25.7 26.0 26.2 26.5 23.7 23.9 24.1 24.3 24.7 25.2 25.7 26.3 26.8 24.6 24.7 24.8 25.0 25.4 25.9 26.3 26.7 27.3 26.7 26.9 27.1 27.4 27.7 28.1 28.5 28.9 29.2

Common materials do not show any sharp yield point A transition zone connects the elastic part of the curve with the plastic part. It appears that the random variations in this transition zone are rather large. When the straining in a subsequent test is of the same sign, either positive or negative, and direction as the prior. strain, this transition region is rather short and the curve has a small radius of curvature, On the other hand when they are of different signs, the transition range is much larger and the transition from elastic to plastic range is very gradual. The stress-strain relationship is rather widely variable for small strains, eOg, e < OoO.1 The reason may be the different amounts of residual stresses present in the finished specimenso By the time the material has undergone a strain of about OOl, it is believed all the grains have yielded in the direction of loading and hence the stress-strain diagram is reliableo Hence, for all subsequent discussions in this chapter, when any comparison is.made between two curves, it is always assumed that the strains are greater than OoOl unless specifically mentioned0 A true strain of OoOl is equivalent to 1% cold worko Prestraining increases the strength of the material in all directions, in tension and as well as in compressiono In general, the differences in flow stress are only of the order of 1 ksi, or about 4-5%, except when the direction of stresses is reversedo In most cases, the difference between two tests is less than 200 psi, but in a few cases it has been as high as 600 psio Hence, a difference in flow stress of 1 ksi is possibly significant, but a difference

-78 of a few hundred psi is not. Whenever it is mentioned that a certain difference is "slight" or "small", it is understood that the difference is less than 500 psio Differences shown as percents are rounded figures onlyo Stress-Strain Curve for Annealed Material Figures V-2 and V-3 show the plastic stress-strain relations for testing in the X, Y and Z directions in tension and in compression respectively0 It is seen that the curves are not straight lines on log-log co-ordinates in the whole plastic rangeo In other words, the relation a ='aocm is not valid for constant m over the range investigated. However, this relation holds good from about c = 0ol to c lo0o Fortunately, we are primarily interested in this region only, since the smallest prestrains in tension and in compression are 0o1098 and 0o106 respectivelyo Although the general relationship is valid for each of the tests the flow constants a0 and m Vary from test to test as shown in Table V-IIo It is noticed from Figure V-2 that the flow stress in tension in the Z direction is about 1 ksi higher than that in the X directiono The flow stress in the Y direction is the same as that in the X direction at e = 1o5 and is intermediate between those in the X and Y directionso It is seen in Figure;. V-V that the flow stresses fOr the compression tests are very close to each othero Figures V-4, V-5 and V-6 show the comparison between tension and compression tests in the X, Y and Z directions respectivelyo Although there are slight differences between the tensile and compressive

-79 z Co w k 30 o, 33 I0 Figure V-1 8 x__ __ __~ _ _ __ __ __ __ Comparison of Data Obtained by Electromechanical Clip-on Gage and by SR-4 Gage 3 ----- 10' 2 a 4 6 7 10' 2 STRAIN IN IN/IN J 4 5 7 10-' U) W Co Im z Co Co U) 30 /__ Figure V-2 9 _ II ------- y. o.- Diagram for Arnnealed 11000 Aluminumin Tonsin in the, Y and Z Directions. a I - -- Diagram for Annealed 1100-0 Aluminum a l l in ComprTssion in the X, Y and Z Directions. I0 -.0005.001.002.005.01.02.05 0.1 0.2 0.8 I.C STRAIN IN IN/ IN D

-80 z nc Iw I Figure V-5 a-e Diagram for Annealed 1100-0 Aluminum 4 -____________^^ ^ ^'_________ _______________ _ in Tension and in Compression in the v -_JC ^ Y Direction. 0 ________ 1SC0 X ---- ---------------- I I ----------— m —-.. I I _ _ X. ___ c _______ __; Figure V-6 /4 _ C _.. _..-..._ -----.._-.-. Diagram for Annealed 1100-0 Aluminum in Tension and in Compression in the._ ____f__ _________-_ -- Z Direction. _.I... ^.. 6. z 0 ag Id A.000 001,00O.005.01.02.05.0.1 0.2. 1.O0 STRAIN IN IN/IN

flow stresses for each of the three directions tested, they are within the range of experimental error and random variations. TABLE VII PLASTIC CONSTANTS FOR ANNEALED 1100-0 ALUMINUM (Average of two tests of each kind) Kind of Test Direction a in ksi m Tension X 21.8 0,260 Tension Y 23,0 0.280 Tension Z 2354 0.262 Compression X 23.0 0o275 Compression Y 22.7 0.255 Compression Z 23.0 0.250 Stress-Strain Relations for Tension Tests After Tensile Prestrain for Individual Prestrain States Figures V-7 to V-10 show the stress-strain relations for tension tests in X, Y and Z directions for different tensile prestrain states. It appears that the flow stresses in the X and Y directions are more or less the same and are lower than that in the Z direction by 0 to 3 ksi, ioeo, by about 0 to 20%o The differences between the flow stresses in the different directions is least at about e = 0.01 and continuously increases with increasing strain. During uniaxial tensile loading, the transverse strains are compressive. Consequently, a tensile specimen cut out in the X or Y direction has compressive prestrain, and so the differences in the flow stresses may be due to the "Bauschinger Effect."

-82 ca x z 0n 0 I(0 Iz G$ eW U$ 30 20 - 10 ------- -____ _ —-- - Figure V-7 Prestrain State T11 7 -/ a-e Diagram for Tension in the X, Y and Z Directions. 30 40- _ _,,, I 04 < = - XFigure V-8 7 - _____ ____ Prestrain State T15?7' a- e Diagram for Tension in theX, Y and Z Directions. so t IsI 10 Figure V-9? _ r _ __ ____ ______ - Prestrain State T20 a-if Diagram for Tension in the X, Y and Z Directions. to I - 10 80 -- ---------------------------- Figure V-10 7 -, Prestrain State T25h cr-e Diagram for Tension in the /X, Y and Z Directions................ 4 /........ ____________________________________ 0) z I0 ag.0006.001.00 00.01.02.05 0.1 0.2 STRAIN IN IN/IN 0.5 I10

-83 Stress-Strain Relations in Compression After Tensile Prestrain for Individual Prestrain States Figures V-ll to V-14 compare the stress-strain relations in compression in the X, Y and Z directions for tensile prestrainso Here the flow stresses in the X and Y directions are identical. The flow stress in the Z direction is 1 to 3 ksi9 or about 5 to 20% lower than in the other two directions up to a strain of about 0.2o The differences gradually decrease after that until the flow stresses are about the same for all directions at e = loOo Since the flow stresses in the X and Y directions are higher than in the Z direction, it supports the findings in the previous section on tensile loading. Here again is the evidence of a Bauschinger effect of an equal magnitude to that in the previous section. Tensile Relations After Compressive Prestrain for Individual Prestrain States Figures V-15 to V-24 presents comparisons between the tensile stress-strain curves in the three directions for compressive prestrainso It is found that the curves for the Z direction are always lower than those for the X and Y directionso For the prestrain state p = 0o106, the flow stress in the X direction is slightly higher than that in the Y direction. Except for this, the flow stress in the Y direction is the highest and that in the X direction is intermediate between those in the Y and. Z directions. The differences however, become smaller with increasing prestrains. As was true in the two previous cases of tensile prestrain a Bauschinger type effect is evident here. for compressive

z co U) Un'a Ir0 U) z U) IU) z U) U) w U) w In eU) z U) WO o0 20 10 -- --— ^'^*^ ________ ______-______ —------ -------- __________ _ _______ _________ _________ ___ —I 5 Figure V-11 7 Prestrain State TlU a-e Diagram for Compression in the X, Y and Z Directions. 30 20 2I\w _ - ~ ________ __~_ ----- IL0 - _______ _______Figure V-12 Prestrain State Ti~ 7I~~~~~~~~~~ ~a-c Diagram for Compression in the X, Y and Z Directions. _ _ _ _ _ _ _ _ I _ _ L _.0005.001.002.005.01.02 STRAIN IN IN/IN.05 0.1 0.2 0.5 1.0

-85 w z w 0 w on 0 ah I0 0) Is 10 7 8 4 Figure V-16 ________SCA^'~~~~~~ ________ ________ Prestrain State C3?7 - c_ - Diagram for Tension in the f^"/~~~~~~~~~~~~~ ~~~~~X, Y and Z Directions. E I I 30 to 0 10 0 0 u,. 1 u T0 I-? 0) (/) on z,a aan 30 7 - -- - ---- - ------ - a-e Diagram for Tension in the X, Y and Z Directions. 4 - --------------.0005.001.002.005.01.02 STRAIN IN IN/IN.05 0.1 0.2 0. 1.0

-86 z i Ia hi C I I Figure V-20 Prestrain State C7 ~ -- ~~ >y - ------ -------- ----- ------ --------- ------ ola-a Diagram for Tension in the X, Y and Z Directions. so Figure V-21 Prestrain State C8 t7_ _ -___ ~ /~ ------------ ------------ - ae- Diagram for Tension in the X, Y and Z Directions. B$ —-- - I a U Wa to ---- -- --- ------- 10 - igure V-23 Preatrain State C10 // - - r - ----------------------- a o -4 Diagram for Tension in the X, Y and Z Directions. 4~ __ ___ — zi.000.001.001.000.01.01.05 0.1 0. 0.5 1.0 STRAIN IN IN/IN

-87 prestrain in the Z direction. Tensile strains are induced in the X and Y directions. Consequently, tensile specimens cut out in the X and Y directions should. be expected to have higher flow stresses in the Z direction. This is borne outo However, the magnitude of the Bauschinger type effect appears to decrease with increasing prestrain. Compression Tests After Compressive Prestrains for Individual Prestrain States Stress-strain diagrams for compression tests after compressive prestrains are shown in Figures V-25 to V-340 The flow stresses in the X direction are found, in general, to be lower than those in the other directions, although the differences are rather small. Flow stresses in the Y and Z directions are almost identical for strains up to 0,1 and for higher strains the flow stress in the Y direction is, in generals lower than that in the Z direction. The data presented. in the curves from these tests are consistant with all the previous tests, Here again a Bauschinger type effect is evident, although the effect is small. During compressive prestrain, the strains in the X and Y directions are tensileo Specimens cut out with their axes in these directions, on subsequent compressive loading, have flow stresses that are lower than those in the Z direction, This trend is identical in all the above discussed cases. Comparison Between Tension and Compression Tests for Prestrains of Nominally Equal Magnitude The prestrains had been planned in such a way that the first four compressive prestrain states have approximately the same numerical values of prestrains as the tensile prestrain stateso This made it

-88 I a a a ce iI (A m (, co U IC 0n Im a Co Ito II tO u\~~~~~~~~~~~~~~~~~ /y^^^ ~~~Figure V-25 Prestrain State C2 c-E Diagram for Compression in " ^~~~~~~~~~~~~~~/ ~~~~the X, Y and Z Directions. t o l\ i - ^ - _ _ _ _ _ _ _ _ _ _ _ _F i g u r e v - 2 6'I - _ ^ ________ ______________ _______ ______ _______ _______ __ _ _Prestrain State C3 _ a-e Diagram for Compression in,~~~~~~~~~~~~~~/...~~the X, Y and Z Directions. to mc - 10 Figure V-27 * XX ______f__________ ______.. -~ _ Prestrain State C4 ~~~/y'~~~~~~~' ff~~~~~~a-e Diagram for Compression in./z~~~~~~~~~~~~~~ y ~~~~the X, Y and Z Directions. 30 -- --. —---,- 10 Figure V-28 ____ _ I I I I Prestrain State C5 a-ce Diagram for Compression in I ~/^ _ __ ____ ____ ____________________ l l ~~~~ the X, Y and Z Directions. 0 I I Figure V-29 Prestrain State c6 c-e Diagram for Compression in the X, Y and Z Directions. 4 | Y ______ I __ I O (a 400M.001.0t.000.01.02 STRAIN IN IN/IN.06 0.1 0. OS 1.0

-89 1 1 1 r - III I I.jto as s 10 l0.to z i 10 li 4 90 20 B.o * 10 eo as hi 50 8 20 as I 5I0 o E I- _ S 51 * 10 0 0o a) S zn Figure V-30 Prestrain State C7 u —c Diagram for Compression in - the X, Y and Z Directions. --------------------------------------------------- \ ____I _ I I I.R ~ ~ L _ _ _. I.__ L _ ___I_ I I__ _ _ I. I I _ _I I I - X _/ Figure V-32 Prestrain State C9 a-e Diagram for Compression inthe X, Y and Z Directions. I I.0005.001.002.000.01.02 STRAIN IN IN/IN.0 0.1 0.2 05 1.0

90-o possible to compare the stress-strain curves of the material prestrained to approximately the same amount in tension and in compression. Figures V-35 to V-38 show the results of tension and compression tests in the X direction for both tensile and compressive prestrains of nominally the same magnitude. It appears that for compressive prestrain the tension and compression curves are very close to each other, tension curve being slightly lowero For tensile prestrain, the curves for tension tests are 1 to 3 ksi, or 5 to 20% lower than that in compression. Also, curves for compression tests in the X direction are about the same whether the material is prestrained in tension, or in compression, but for tension tests the flow stress is lower by 1/2 to 2 ksi, or up to 10% for tensile prestrain than for compressive prestrain. Figures V-39 to V-44 show the comparison between tension and compression tests in the X direction for compressive prestrains greater than 0.25. For strains more than 0.1, the flow stresses are practically the same, and between e = 0.01 and 0.1, the material seems to be slightly stronger in compression. Figures V-45 to V-48 show the stress-strain curves in tension and in compression in the Y direction for e < 0.25, prestrained both in tension and in compression. Observations here are similar to those for the X direction. At strains greater than 0.005, the curves for the compression tests for both kinds of prestrain and that for the tension after compressive prestrain are all very close to each other, while the curves for the tension tests in the Y direction after tensile prestrain are 0 to 2 ksi, i.e., up to 15% lower than the other three.

-91 z to w c U) I as I 0 oD 30,o C2CX F _igure V-35 Prestrain States Tll and C2 ////_ - - cs Diagram for Tension and Compression in the X Direction. $0 C - 1T C3TXI T15TX I0 /y^0 ^ rigure V"56 Prestrain States T1? and C0 /I o,- - ---------------------- Diagr am for Tension and - Conpression in the X Direction..3i I avi I I I - I I IX~II c Ky~~ IT2 X z aI0 n 10 w 5't w.. ^ -'A - A Figure V-37 Prestrain States T20 and C4 ca- Diagram for Tension and Compression in the X Direction.'I I.0006.001.002.005.01.02.05 0.1 0.2 0. 1.0 STRAIN IN IN/IN

-92 rn z wi I0) 0 hi Iz Un 0 n I0 Is CGTX Figure V-39 ____ _ Prestrain State c6 c-e Diagram for Tension and Compression in the X Direction,0 804C~~~~1- - ---- ------ ---- ---- ------ ---- ----- ---— Figure V-40 _____ _____ 1 Prestrain State C7 c -e Diagram for Tension and 8:/~~~ 5 | { j j/~~~~~~~~ Compression in the X Direction. 4 // ___ _____________ _________________ I _________ (a z h. 0 a: U) z h, 0 I (0 0: 30 COTX-, -, - -- = -=== -- cr-e Diagram for Tension and Compression in the X Direction. 30 /C OCX o > / C 10 TX9Figure C TX Is Figure V-43 ~~~__10 ____ ____________ _____________________________ | Prestrain State C10 c-e Diagram for Tension and Compression in the X Direction. 30 10 - ~.~ --- /+llcx ] C 1 TX X Figure V-43 0 _ ____ _______ __________| - ____ _ _ _ _ _ _ _ _ Prestrain State C10 /a- Diagram for Tension and Compression in the X Direction. 7 ^~~.000o.001.002.005.01.02 STRAIN IN IN/IN.06 0.1 0.2 OA 1.0

-93 1so I a I0 0 z a I C IID U 0 an 0 I0 ro _-_ _ _ t _- 1 Figure V-45 Prestrain States T11 and 2 _?.~~~~~~//ff"~~~~~~~~ a-es Diagram for Tension and Compression in the Y Direction.:'- -- I III I I r _ r_____T15_C3___________ ICL L - I I I I I j @ I /S l l l l g 0 Ec-s Diagrai for Tension and ^_ _ _ _ ________ Compression in the Y Direction. 51.| | OSTYT r S t +20TY j 1 425~~.TY 0 /^r~~~~~~~~~~~ ~~~Figure V-46 I __ ___.... I Prestrain States T15 and C _ |?]/// l l l l l l |^~ ~ ~ ~ ~ ~ ~ ~ c a-e Diagram for Tension and Compression in the Y Direction. I I C4 C 1 4CY 4 _ _ _ Figure V'-47?_. y ___ _____ ____ ___ _____ ___ _Prestrain States T20 and C3 _?/^~~~~~~~~ // o~~~~~~~~~a-( Diagram for Tension and Compression in the Y Direction. 40~ ~~~ Z^. ___ __________________________________ _Figure V-48 Prestrain States T25 and C5'a-e Diagram for Tension and Compression in the Y Direction. b~....ii.....000.001.00.006.01.02 STRAIN IN IN/IN.06 0.1 O.2 0.5 1.0

-94 Figures V-49 to V-54 show the comparison between tension and compression tests for compressive prestrains greater than 0.25. The flow stresses are practically identical except for prestrain states e > 0~4 and strains greater than 0.2, in which case, the p compression curve is 0 to 1-1/2 ksi, i.e., up to 8% lower than the tension curve. An interesting phenomenon was observed during compression tests in the X and the Y directions. In the X direction, for prestrains p > 0.45, the flow stress reaches a maximum, and then decreases slightly with increasing strain before beginning to increase monotonically. This strain-softening is only a fraction of a ksi at lower prestrains, increasing to about 1 ksi at p = lo95. This phenomenon was observed by Bridgman(51) and by Ebert,(56) but it seems that they did not put any importance to them or dismissed as experimental error. This effect has been consistently encountered in the present investigation, and it seems tobearegular phenomenono As a possible explanation of this phenomenon, it is recalled that when the material is prestrained in compression in the Z direction, strains in the X and Y direction are tensile, while the subsequent testing is compressive. It is likely that while prestraining, certain crystallographic planes were active for slip and dislocations were pinned down by different obstacles while moving on those planes. During subsequent testing, some of them, although they were sessile in one plane, found mobility possible in other planes which became active on change of direction of strain, and hence a drop in flow stress.

-95 II z - 7 20'U 0, 20 z 0) 10 /) L. C7TY__'C7TY Figure V-50 Prestrain State C7 U 7 cra-e Diagram for Tension and Compression in the Y Direction. CSTY IO Figure V-51 Prestrain State C8 7 - -c Diagram for Tension and Compression in the Y Direction. /___ ___ ____ ___ ___ _____ ___ ___I I________ z z 0 0 a: Az Un 0'U i Iz CA - I I. 1 1I 1 1 L I ] CIITY z 0 10 U) 0 7 u) If Figure V-54 Prestrain State Cll a-e Diagram for Tension and - Compression in the Y Direction. I I 5, I I I I I I I I I 4I I I 1I I.0008.001.002.005.01.02 STRAIN IN IN/IN.05 0.1 0.2 0.5

-96 The flow stress increases until a maxima is reached between e = 0o01 and 00050 It then decreases to a minima at about c = 0,15 to 0o25, before it starts increasing monotonicallyo The difference between flow stresses in tension and in compression in the X and Y directions is always rather small, but that in the Z direction is appreciable Figures V-55 to V-58 show the stress-strain curves in tension and in compression in the Z direction for both tensile and compressive prestrains of the same nominal magnitudes for p <.25o It is apparent that the flow stress is 5 to 20% lower when the direction of testing is reversed0 The effects are similar for both tensile and compressive prestrains, although probably the reduction in flow stress is slightly lesser for tensile prestrains than for compressive prestrainso From Figures V-55 to V-63 it is apparent that the Bauschinger effect is much more pronounced for stresses near the elastic limit than for stresses associated with, say, 0.% offseto From Figure V-55, the elastic limit is about 32% lower when measured in a sense opposite to the preload, but for a 0.4 offset, the yield strength is only about 15% lower in the opposite senseo In some cases this value is only 10% lower o Figures V-59 to V-63 show the comparison between tension and compression tests in the Z direction after compressive prestrains p > 0.25. The flow stress in tension is always less than that in compression. The minimum difference of about 8% occurs at a strain of about 0.02 and the maximum between 0.2 to 0.40 The maximum difference is about 3 ksi or about 15%o The difference tends to become

-97 as z Cft U) 0 z 0) U) w w 0)'a I 0 U) w w I 130 - _ _T Z_ Figure V-56 Prestrain States T15 and C5;~7 - -t —---- --- o-e Diagram for Tension and Compression in the Z Direction../:t - /___ _____ ____ _ _________ __T_ Prestrain States T20 and C4:7 - / — e Diagram for Tension and ^y'/^^E~ K ] [ j j j ~~~~~~~~ Cceipression in the Z Direction. C51rz C5 —Z —T- - Z Figure V-57 /^^~ ~~~~~~~~~~~~~~~~ ^ _____ _______Prestrain States T20 and C54 7 / >_ ------ ------- ------- -------' a-e Diagram for Tension and 0Compression in the Z Direction. Y J J — ~ // __CST__ ___________.0005.001.002.005.01.02 STRAIN IN IN/IN.05 0.1 0. 0.5 1.0

-98 z a hi oc IIa z 0n 0 I0 a i II I IL — - (4-C8TZ —-- Figure V-61 _______ ______ /~____~__ __~_____.Prestrain State C8? / / ------- ----- ~ ffIII - Diagram for Tension and Compression in the Z Direction.,/ /.'- I [ CIOCZ to ^~~~~~~~~100~~~~ 0<-~ _:Figure V-62 Prbetrain State C8 *; e S —------------------------- - cye-e Diagram for Tension a nd Ccmnression in the Z Direction. m] - II ___ ___ ^^^^cz^^^3r==^Tz::::=CIICZ —C IIT I10 Figure V-63 _X _____ ___ ___ ______r__________ Prestrain State C10? / -------------— ~ <a-C Diagram for Tension and - /Comression in the Z Direction. 4 [ j f Ie 0.0001.001.003.005.01.01 STRAIN IN IN/IN.05 0.1 0.3 0 1.0

-99 smaller at higher strain, but whether this reduction in difference is actual or is only apparent depends on the validity of the correction that has been applied to find the true stress at the necked down regiono Tension Tests After Tensile Prestrains Figure V-64 shows the stress-strain relations in tension in the X direction after tensile prestraino The curves show increasing flow stresses for increasing prestrains. However9 it is also seen that all the curves intersect the stress-strain curve for the X direction in the annealed stateo This means that although the material is strengthened in the X direction for lower strains, beyond a certain value of strain, the material is weaker than in initially annealed material. In other words, prestrain in the Z direction is not additive to strain in subsequent testing in the X direction, Also, it is apparent from this behavior that the concept of "effective strain" which is a positive definite quantity is not applicable, at least for large finite strainso Figure V-65 is similar to Figure V-64, except that the curves are for the Y direction, The same conclusions as above are reached from thiso Figure V-66 shows the results of tension test after tensile prestrain. both in the Z directiono Here the subsequent testing should be only a continuation of the prestraining process, and as expected, all the curves are above that for the annealed state To check that the tension tests are just a continuation of the prestrain

-100 I 0 cc m - 0 fi z a, 0) w c 0i (n to 80 - - --- I 0. A^-^ Figure V-64 y^^^ —I~~~~~ ______,0010~~~~~~~~~~~O~~ _____ ____ae Diagram.? _ - -------- Tension Tests in the X Direction' After Tensile Prestrain. 30 T20,._T T25, I T15 I0 Ay I Figure V-65 y7 - / _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _a-e D ia g ra m. Tension Tests in the Y Direction After Tensile Prestrains 4. 30... 1-T20 ------ a _ _ _ _ _ _ _ _ _ _ -TII O 10 Figure V-66 A ( I a-e Diagram. Tension Tests in the Z Direction After Tensile Prestrains 4.0005.001.002.005.01.02 STRAIN IN IN/IN.05 0.1 0.2 05 1.0

-101 process, stress against (p + e) for plastic ranges only were plotted in Figure V-670 The best straight line drawn gives a stress-strain relation as 0,262 a = 22.4 262 This is about 4.3% lower than that obtained by testing annealed specimens, and is within the range of experimental erroro Compression Tests After Tensile Prestrains Figures V-68 and V-69 show the results of compression tests in the X and Y directions respectively for tensile prestrains. Since the material underwent compressive strains in the X and Y directions during prestraining, it is expected that the flow stress would be greater than in the annealed state for all strainso This expectation is confirmed. Flow stresses increase with increasing prestrain as is expected, Figure V-70 shows the results of compression test in the Z direction after prestraining in tension. This set of curves are interesting in that although flow stress increases with increasing prestrain for strains up to about 0o25, they all converge at that strain to the flow stress value of annealed material, and for higher strains, the curves are practically coincident, irrespective of the prestrain valueo In other words, for any tensile prestrain, the material behaves like an annealed material for compressive strains greater than 0.25,

-102 z Co wn IZ: IJ 15 0 T II I A T 15 I- X T 20 + T25 10 I I I I I. I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.1 ep+e IN IN/IN IN TENSION Figure V-67. a-e Diagram. Tension Test in Z Direction as Continuation of Tensile Prestrain. 1.0

-103 z 0 0 w a t I 3I,1 a 30. - Bo 1A- I Figure V-69 ~~/~~^y~~~~~ A^^^'~~~~~~~ o-Ce Diagram. T -'' // - --------------------- ^ - ------------- Compression Test in the Y Direction After Tensile Prestrain. ~ ____ --- 4~~~~~~~~~~,. 10 10 I.0001.001.001.005.01.01 STRAIN IN IN/IN.05 0.1 0.3 0.1 1.0

-104 Tension Test After Compressive Prestrain Figures V-71 and V-72 show the stress-strain curves for tension in the X and Y directions after compressive prestraino By the same token as for Figures V-68 and V-69, it is expected that the curves should all lie above that for annealed material. However, the curve for annealed material crosses first two or three curves for low prestrain values below e = 10o. The cross over is slight and may be due to some experimental error or the factor used to correct for the triaxiality of stresses at the neck, Figure V-73 shows stress-strain curves for different prestrain states when tested in tension in the Z direction after prestraining in compression. They seem to be converging toward the curves for annealed material, but data could not be obtained for sufficiently large strains to verify ito Compression Test After Compressive Prestrain Figures V-74 and V-75 presents the results of compression tests in the X and Y directions respectively after compressive prestraining. In the same way as in Figures V-64 and V-65, they should cross the curve for annealed material0 It is not obvious from the diagrams, since sufficiently high strains could not be attained. The interesting phenomenon of strain softening has been discussed before and can be clearly seen in specimens C6 and C9o However, it is not so pronounced in all the specimenso Figure V-76 gives the comparison between stress-strain diagrams for compression tests in the Z direction after compressive

-105 z cn 0 z I0 on z a c I0 rA.0005.001.002.005.01.02.05 0.1 0.2 O0 1.0 STRAIN IN IN/IN

cI r c on ai ol Ia.0001 00 01 001S.00o.01.02.0 0.1 0.1 O6 1.O STRAIN IN IN/IN

-107 prestrains. As in the case of Figure V-66 the subsequent testing is only a continuation of the prestraining processo To check this, stress was plotted against (sp + ~) in Figure V-77, and it was found that the data points were indeed on an approximately straight line on log-log co-ordinates showing a stress-strain relation: = 22.9 00254 ksi which is almost identical to that for the annealed material and for the tensile prestrain specimens. It is rather difficult to compare quantitatively such a large number of stress-strain diagrams as are present in this investigation; particularly since each curve is an average of only two tests and, therefore, no statistical analysis for each kind of curve is possibleo Variations in flow stresses at low, intermediate and large strains are not similar and hence no single figure can adequately describe the difference between two curves. However, in an attempt to make the picture a little more clear Tables V-III, V-IV and V-V have been prepared, 0.2% Offset Yield Stress Table V-III shows the stresses at 0.2% plastic strain for the different tests and the percent difference between these stresses and the same for tension or compression test in the Z direction. according as whether the prestrain is tensile or compressiveo For tensile prestrains, it is observed that compared to the yield stress for the tension tests in the Z direction, the yield stress:

30 2 z (I) C') w Cl) w Icr. I — 20 H I I O 2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.5 2.0 Ep6E IN IN/IN IN COMPRESSION Figure V-77. a-e Diagram. Compression Test in Z Direction as Continuation of Compressive Prestrain.

-109 TABLE V-III 0.o2 OFFSET YIELD STRESS AND DIFFERENCE FROM a 2)z (a) Tensile Prestrains TX Tll ksi Diff in % ksi T15 Diff in % 11.3 -8 9 12 0 -63. 1301 -6.4 1306 -11o TY 1200 -302 13.0 1.6 1402 1.4 14.5 -502 TZ 12o4 0 12o8 0 14o0 0 1503 0 CX -0o8 1306 6.3 1405 306 1504 0.7 CY 1206 1.6 1309 8.6 1500 7.1 1506 2.0 CZ 10,2 -1707 11,6 -9,4 12,2 -12o 9 12o 0 1200 -20.6 ksi T20 Diff in % ksi T25 Diff in % (b) Compressive Prestrains ksi Diff in % ksi c3 Diff in % 1205 -0.8 1208 -5.2 1300 3.2 1400 5.7 10.0 -20.6 1100 -18.5 1105 -807 1305 0 1207 0.8 1307 1.5 12o6 0 13.5 0 4 ksi Diff 14.0 in % -3.4 ksi C5 Diff in % C6 ksi Diff in % ksi C7 Diff in % c8 ksi Diff in % 14.8 -2.0 15.4 -6.1 16.9 -4 5 18.0 -5~3 18.7 -10.1 153. 5.5 16.0 6.0 16.7 1.8 18.5 4.5 19.6 3.2 20.8 0 11.2 -22.8 12.53 -18.5 13.0 -20.7 1404 -18.6 14.6 -23.2 16.4 -21,2 13.6 -602 14.4 -4.6 16.1 -1.8 16.9 -14.5 17.1 -10.0 19.o -8.7 15.5 6.9 14.8 -2.0 15.6 -409 18.1 2.3 19.7 3.7 19.5 -6.3 14.5 0 ldol 16,4 o 17.7 o 19o0 o 2008 0 ksi C10 Diff in % ksi Cll Diff 21.0 in % -6.7 22.4 -0.4 17.7 -213. 18.7 -16.9 21.4 -4.9 22.5 0 ksi C9 Diff in % 23.0 -12,9 24.2 -8.3 20.9 -20.8 22.9 -1303 26.4 o

-110 TABLE V-IV AVERAGE FLOW STRESS AND % DIFFERENCE FRCM (av) (a) Tensile Prestrains TX l ksi Diff in % ksi T15 Diff in % ksi T20 Diff in % 13.1 -10.5 13.9 -6.7 15.0 -63. TY 13.5 -8.8 1404 -3.4 1505 -3.1 TZ 14o8 0 1409 o 16o0 0 CX 14o8 0 15o4 3.4 16.3 1.9 CY 14.8 0 15.4 3.4 1604 2.5 CZ 13.4 -9.5 14o0 -6,0 14.7 -8.0 ksi T25 Diff 15.4 in % -9.4 16.0 -5.9 17.0 0 1700 o 16.9 -0.6 15.0 -11.8 (b) Compressive Prestrains ksi C2 Diff in % 14.2 -702 14.5 -4.6 12.6 -17.1 14.o -709 14.9 -2.0 15.2 0 ksi Diiff 14.8 in % -6.9 c4 ksi c4 Diff in % C5 ksi Diff in % ksi C5 Diff in % ksi C7 Diff in % c8 ksi Diff in % 15.4 -702 16.1 -5.9 16.9 -6.1 17.8 -7.3 19.2 -6.8 15.4 -301 1608 1.2 17.3 1.2 17.5 -2.8 19.5 0.5 21.0 1.9 13.9 -12.6 14.0 -1507 14.9 -12.9 1601 -10.5 17.3 -10.0 18.6 -9.7 15.6 -1.9 15.8 -4.8 17.0 -0.6 17.4 -5.3 19.0 -1.0 19.5 -5.3 15.8 -0.6 17.0 2.4 17.2 0,6 17.7 -1.7 193. 0.5 2007 0.5 15.9 0 16.6 o 17.1 o 18.0 0 o 19.2 0 20.6 0 20,6 O ksi C10 Diff in % ksi Cll Diff in % 20,5 -10.9 22.2 -3.5 24.5 -5.2 203. -11.7 22.2 -11.5 21.2 -7.8 25.2 -7.6 22.9 -0.4 24.4 -2.8 2300 0 25.1 0 22.7 -9.6 ksi C9 Diff in % 2502 -7.4 26.2 -3.7 24.6 -9.6 26.1 -4.0 27.2 0

-111 TABLE 0.6 AND FLOW STRESS AT e = v-v % DIFFERENCE FROM (0o6)z. Prestrain (a) Tensile TX T ksi 17.1 Diff in % -17.0 ksi 17.8 T15 Diff in % -13.2 ksi 18.3 T20 Diff in % -14.1 5 ksi 18.7 5 Diff in % -13.8 TY 17.4 -15.5 18.7 -8.8 19.8 -700 20.0 -7.8 TZ 2006 0 20.5 0 213, 0 21,7 0 Cx 20.1 -2.4 20.5 210 21,5 09 21.5 -0.9 CY 20.0 -2.9 20.4 -0,5 21.0 -1.4 21.0 -3.2 CZ 1907 -4.4 20.0 -2.4 19.9 -6.6 19.8 -8.8 ksi C2 Diff in % ksi Diff in % c4 ksi Diff in % C5 ksi Diff in % ksi 6 Diff in % ksi C7 Diff in % ksi Diff in % ksi C10 Diff in % ksi Diff in % (b) 18.6 -12.53 18.9 -12.1 19.3 -10.2 19.9 -83. 20.5 -9.8 20.7 -9.2 22.3 -6.7 2305 -9.6 25.2 -9.0 27.7 -5.4 Compressive Prestrain 19.2 18.5 1 -9,4 -12.7 - 19.8 18.8 2 -7.9 -12.6 - 21.0 18.8 2C -2.3 -12.6 - 20.9 19.2 2C -3.7 -11.5 - 21.4 20,7 2C -4.9 -8.0 - 22.8 21.1 21 0 -7.5 -5 24.3 22.4 22 1.6 -6.3 -8 25.2 23.5 23 -3.1 -9.6 -10 26.7 25.1 24 -3.6 -9.4 -10 29.1 - 25 9.5 o 3.0 ).4 ~.l ).4. o1 ).5 0o5 ) o 5 ).6 3.4 L.6 ).3 o.0;~3 1.4.7.8.5.6 20.4 -3.8 20.6 -4.2 21.0 -2.3 21.3 -1.8 21.1 -6.2 21.9 -4.0 23.0 -3.8 24.5 -5.8 25.4 -8.3 26.3 21.2 0 21.5 0 21.5 0 21.7 0 22.5 0 22.8 0 23.9 0 26.0 0 27.7 0 29.5 C9 ksi Diff in % -1.4 -13 -10.9 o

-112 (a) In tension in the Y direction is the same; (b) In compression in the X and Y directions is about 5% higher; (c) In tension in the X direction is about 8% lower; (d) In compression in the Z direction is about 10 to 20% lower. For compressive prestrains, when compared with the yield stress for compression in the Z direction, it is found that (a) The yield stress in tension and in compression in the X direction are up to 13% and 20% lower respectively, (b) Except for the prestrain state C9 the tensile yield stress in the Y direction is slightly higher and the compressive yield stress in the Y direction is about the same as the yield stress in compression in the Z direction. (c) Yield stress in tension in the Z direction is about 22% lower than the compressive stress in the same direction. These observations are consistant with the previous remarks concerning the Bauschinger effect. Average Flow Stress Table V-IV shows the "average" flow stresses and the percent difference from that for strains of the same direction and sense as the prestrain. This average flow stress is the average of flow stresses at the following strains: 0o005, 0.007, 0.01, 0.02, 0.035 0o05, 0.07, 0.1, 0.2, 0.3 and 0.5 ~

-113 This average flow stress is used in an attempt to correct for any irregularities that may be present in the stress-strain curves. It is seen from this table that the comments on Table V-III are in general valid for Table V-IV also, except that the differences are smaller. Also, from Figures V-7 to V-63 it is seen that the variations in the flow stresses is the largest at small strains and it gradually diminishes with increasing strains to a minimum at about = Oo01o Flow Stress at e = 0.6 Table V-V is similar to Table V-III, except that the flow stresses are taken at C = 0.6. This is done to show the differences at larger strainso It is seen that the Bauschinger effect is smaller for the Z direction, and is not very consistant for the other directions. However, it may be recalled that the modified Bridgman correction factor has been used to find the flow stress at large strainso The uncertainties inherent in the correction may have affected the numerical results at c = 0.6. Concluding the discussion on Figures V-2 to V=77 it may be stated as a general comment that the Bauschinger effect, defined as the difference between the tensile and the compressive flow stresses, is present at all prestrain states. It is present not only in the direction of prior loading, but also in the transverse directionsO If the strain in the transverse direction is, say, tensile during prestraining, flow stress in this direction will be higher in tension than that in compression. Similarly for compressive strainso It also

-114 appears that, in general, the flow stresses for uniaxial testing in the Y direction, the direction of smaller transverse strains, is higher than those in the X direction, the direction of larger transverse strains. It was also noticed that, in general, for the range of strain of about 0.003 to about 005 the highest flow stresses occur in the Y direction. In fact, the flow stresses in the Y direction in this range of strain is higher than that in the Z direction even when the tests in these two directions are both of the same sense as that of the prestraino Yield Stress and Strain Hardening Exponent Cold worked materials do not, in general, show a sharp yield point. Therefore, it is necessary to define a yield strength (or stress) for the purpose of engineering application to design, A very common definition of yield strength is the flow stress at an arbitrary plastic strain, usually 0.2%. A similar definition will be used in the present investigation, with slight modification. It is well known that when a material is strained, unloaded, and loaded again, yielding occurs at a stress lower than that originally applied. A noticeable amount of plastic deformation occurs before the previous highest stress state is reached. Since in this investigation an attempt is made to relate the subsequent properties to the previous history, it is attempted to define yield stress in such a way that when the material is reloaded in the same direction and sense, the yield stress obtained is the same as the maximum stress applied during pre straining, From Figures V-67 and V-77 and Table V-I it is found that

-115 the flow stress at a strain of about 0,01 satisfies such conditionso Hence, the yield stress may now be defined as the flow stress at e = 0,01. The stress-strain diagrams for testing in the same direction as the prestrain has an increasing slope; but it is possible to approximate certain parts of the curves by straight lineso In all other cases, there appears to be a portion of the curve that is reasonably straight. The slope of the curve gives the strain hardening exponent in that range. Table V-VI gives the yield stress, slope m and the strength constant a for the different tests, and the range of strains over which the exponential form of relation is valid. The table does not include the properties of the annealed material, as this definition of yield stress is not identical to the published values of yield strength. Also, for the annealed material, plastic flow starts at m very low stress, and a relation of the form a = Oe~ is valid only for strains greater than e = 0.1, as mentioned before. The proportional limit has also been recorded, but the scatter is very large due to the difficulty of actually deciding where on the curve it is that proportionality ceases. Poisson's Ratio The plastic Poisson's Ratios for the tension tests are. shown in Table V-VII. In this table, a, means the ratio of strain in the a direction to that in the P direction when loaded uniaxially in the P direction. Since ig + 7j = -1 where c, D and y are three orthotropic directions, it is necessary to find only three independent Poisson's ratios. Since two complimentary Poisson's ratios cannot

TABLE. V-VI:(a) FLOW CONSTANTS FOR TENSION TESTS IN X-DIRECTION Yield Valid in the Range Stress 0o of Strain Prop. in ksi in ksi m from to Limit Remarks in ksi Tensile Prestrain Tll 12.2 13.7 00235 o0055 0,11 6.8 T15 13.2 14.6 0.0219.0055 0.15 6o0 T20 14.3 15.7 0,0203.0055 0.19 5.8 T25 14.8 16.2.o0182 o0055 0,18 4.9 Compressive Prestrain C2 13.1 15.5 0.0359.0025 0014 9.0 C3 13.7 15,9 0.0337.002 0.18 10o0 C4 14.4 16.0 0.0236 o0035 0 13 9.7 C5 15o2 16.8 0.0213 o 0035 O.14 10.0 c6 16.0 17.9 Oo0238.0035 0020 lO.9 C7 17.2 18.9 0.0200 o0035 0.23 9.6 C8 18.5 20.4 0.0202.0035 0.30 11.4 ClO 19.8 21.7 0.0192.oo6 0.25 12o3 Cll 22.0 24.1 0.0153 o007 0.25 12.6 C9 24.9 26.2 0.0091 o009 0.35 1309

-117 TABLE V-VI (b) FLOW CONSTANTS FOR TENSION TESTS IN Y-DIRECTION Yield Stress in ksi Tensile Prestrain Tll T15 T20 T25 12.6 13.5 14o6 15.4 d0 in ksi 1308 1.5.0 16.1 16.4 Valid in the Range of Strain from to m 00015 0.0197 000218 00213 o0035 0003.0035.0o45 0015 0012 0014 0015 Prop. Limit Remarks in ksi 7.0 9.0 8.0 6.3 Compressive Prestrain C2 C3 C4 C5 C6 C7 C8 ClO Cll C9 13.4 1404 15.8 16.6 17.2 18.8 20.5 21.5 23.6 25.7 15.00 16.o 17.5 18.o 18.4 20.0 21.7 230.6 25.6 28.1. 0.0252 0o0225 00021.4 0.0189 0,0158 0.121 0001.33 0o0197 0,0162 00 o1.48.0025.003.0035 oo04.0035.0035. oo6.0045.o07.009 0.09 00o.09 0.15 0011 0.1.8 0017 0.24 0.23 0.32 0.21 0.4 9.2 9.8 10.6 1201 11.01 1.2.0 12.6 12.9 14.6 15.5

-118 TABLE V-V (c) FLOW CONSTANTS FOR TENSION TESTS IN Z-DIRECTION Yield Valid in the Range Stress 0o of Strain Prop. in ksi in ksi m from to Limit Remarks in ksi Tensile Prestrain Tll 18 15.7.0393.0016 0o 6 10.0 T15 13.2 15.7.0342.0020 0.05 9.7 T20 14.4 16.7.0308.0020 0.05 11.2 T25 15.5 18.0.0303.0025 0.07 8.2 Compressive Prestrain C2 11.5 13.8.0396 00oo6 013 5.0 C3 12.6 15.2 o 0391.007 0.16 5.5 C4 13.1 14.8.0245.010 0.15 5.8 C5 13.9 15.9.0259.010 0.18 6.8 C6 15.3 17.1.0229.009 0.16 6.4 C7 16.5 18.4.0158.010 0.22 7.4 C8 17.4 20.1.0244.016 0.28 7.4 ClO 19.4 21.4 o0098.016 0.28 7.9 Cli 21.1 23 0.0032.018 0.25 9.8

-119 TABLE V-VI (d) FOR COMPRESSION TESTS IN X-DIRECTION FLOW CONSTANTS Yield Stress in ksi Tensile Prestrain Til T15 T20 T25 13.2 14.o 15.0 16.o 0 in ksi 16.1 16.2 17.1 17.4 Valid in the Range of Strain from to m.0438.0319.0279.0177 oo4.0025 0035 004 0.09 0.10 Oo10 0,09 Prop. Limit Remarks in ksi 7.9 9.5 10.2 10,3 Compressive Prestrain C2 c3 c4 C5 c6 C7 c8 C10 Cll C9 12.7 14.5 14.7 15.9 17.1 18.3 19.3 20.5 22.2 24,0 15,0 16.4 16,4 17,8 18.5 19.6 20.0 22,0 24,3 26,0.036.0282.0268.248 o,189.0149.0083. 0102.0067.0057 oo6 o005 oo6 o009 o004 oo6.008 v011.022 016 Oo10 0012 0013 0o15 0,25 0,25 0.27 0.33 0072 0.72 6.6 7.6 9.8 8.6 10.8 8.9 9.0 9.6 906 10.4 Inflected Inflected Inflected Inflected Inflected

-120 TABLE V-VI-(e) FOR COMPRESSION TESTS IN Y-DIRECTION FLOW CONSTANTS Yield Stress in ksi Tensile Prestrain Tll T15 T20 T25 13.3 1401 1504 16.o co in ksi 1503 15.5 16.4 16.8 0.0271 0.0189 0.0139 o00oo98 0.003 0.003 0.0035 0.0035 o.o6 00055 o.o7 0.10 Valid in the Range of Strain m from to Prop. Limit Remarks in ksi 9.1 9~3 10o 10.2 Compressive Prestrain C2 C3 c4 C5 c6 C7 c8 C10 Cll C9 13.5 14.6 16.2 16.1 16.9 19.3 20.5 22,2 24.4 2600 1506 1600 17.4 18.1 18.3 20.1 21.3 23.5 25.0 26.7 0.0298 0o.0045 0.09 8.4 0,0214 0o0167 0.0224 00015 0.012 00 oo84 0 0102 00003 00002 o0004 00004 00005.o008 0.005 0 009 0.011 0o011 ooo8 0.15 o.16 00o.14 0030 0030 0.45 0.55 0.75 9.0 1002 7.0 9~3 10,8 11.0 1004 10.8 1104 Inflected Inflected Inflected Inflected

-121 TABLE V-VI (f) FOR COMPRESSION TESTS IN Z-DIRECTION FLOW CONSTANTS Yield Stress in ksi 0 oi in ksi Valid in the Range of Strain from to Prop Limit Remarks m Tensile Prestrain Tll T15 T20 T25 11 9 13.0 1400 14.2 14.0 14,3 15.1 1504 0.0376 000206 o0184 0.0130 oo008 0.007 0.007 o 010 0012 o014 0020 o.18 5.02 7.4 7.0 6.2 Compressive Prestrain C2 C3 c4 C5 c6 C7 c8 C10 Cll 1304 14.1 1501 1506 16o8 1802 19,6 22.0 24,1 17,0 1706 1800 1802 1809 20.2 21,8 24.5 2700 oo0516 000015 0o08 oo0461 00385 0.0317 0.0259 0.0241 0.0230 0.0230 0.0230 0o0016 0,0017 0.0025 0.0030 000037, 0oo4o 0.0070 0.0080 0007 o.o8 0o11 0oll 0.13 o.16 0025 0.30 906 9.8 1004 11.0 12.0 1307 1300 15,0 16,2 C9265 28.0 0.0107 0~25

-122 TABLE V-VII PLASTIC POISSON'S RATIOS* xy Tensile Prestrain Tll T15 T20 T25 0.71 0,62 0.57 0.63 zy 0.29 0,38 0.43 0037 kyz 0.38 0040 0042 0.42 0062 o.6o Oo58 0.58 tzx 0050 0050 0052 o050 Ixy( calculated) 0o50 0050 0O48 0o50 0.620 o.600 0.580 0 580 Compressive Prestrain C2 C3 c4 C5 c6 C7 C8 C10 Cll c9 0.66 0.65 0.64 0062 o.60 0.55 0052 0052 0052 0.55 0034 0.35 0,36 0.38 0040 0o45 0o48 0O48 0O48 0o45 0.35 0039 0.395 0.39 0.35 0041 0042 0.37 0,39 o,65 0O48 o.61 0o43 o.605 0o42 o.61 0.375 o,65 0o45 o059 oo56 Oo58 o,56 0o63 0o59 0o61 o065 - 0o.58 0052 0.57 0. 58 0.625 0.55 0044 0044 0041 0.35 0042 0.619 o0665 o 675 Oo715 0 643 o.541 o.541 0.510 0o447 0.521 * The sign of Poisson's Ratios are always negativeo

-123 TABLE V-VIII YIELD STRESS IN PLANE STRAIN TESTS STRESS IN KSI Specimen SXY SZY SYZ SXZ SZX SYX Tensile Prestrain Tll 1407 14.1 14o4 15.8 13.5 13.5 T15 16.o0 148 14.7 18.1 14 5 14,4 T20 17.0 1661 1765 175 1506 15.5 T25 17.6 1605 1703 18.5 16.5 16.6 Compressive Prestrain C2 1503 14.5 1305 14.8 1400 14.8 C3 17.0 15o7 1407 15 0 15.5 16.1 c4 17.2 16.7 15.1 16.8 16 7 16.2 C5 18o2 18.o0 16.5 17.6 17.8 17.2 c6 19.1 18.4 17.4 17.4 18.0 18.4 C7 21.6 20.1 19.2 20.9 19.3 19.8 c8 23.1 21.6 19.7 21.0 20.7 2102 C10 25.5 24.2 22.7 24.9 23.6 22.3 Cli 27.0 25.7 24.1 24.9 25.0 24.7

-124 both be. exponential functions of prestrain, it is assumed that they are polynomial functions of prestraino For compressive prestrains, it was found by inspection of Table V-VII that the values of xz is approximately a constant of value 0o6~ Hence, it was necessary to find only two other Poisson's ratios, vizo Ixy and iX as functions of prestraino Similarly, it was found that for tensile prestrain, pzx = yx ==0~5~0 Hence, it was necessary to find only xy and -xzo However, the PoissonBs ratios must satisfy the relation Ixytyzpzx - -,= 1 (v-l) pyxPzylxz This relation has been derived in Appendix Ao For tensile prestrains, we have, [zy = xy'yz = ~"lJxz and ptzx = yx = Oo50 o Substituting these in Equation (V-l), we have x-y(&"ll xz) = FIxz(-laxy) or xy xz Similarly, for compressive prestrains, we have Izy = Hlxy Pzx = -lIyx =xz = -o6 Lyz = 1-'1xz = 0oo 40

-125 Substituting in Equation (V-l) we have, -0.4 xy(-l-yx) = -0.6 yx(-1-xy) or t4xy (1 + yx) = 1l5 iyx + 15 xytyx or xy(1- 0.5 0 yx) = 1.5 yx ~or n = 15 ~yx = 3 yx (V2) or - 1.5 ky Y (V-2) xy 1 - 0.5 [yx 2 - oyx The last column of Table V-VII shows the calculated values of axyo The calculated and experimental values of axy are plotted against prestrain in Figure V-78. Although the plot shows considerable scatter of data, a trend curve was drawn to relate ixy with tensile and compressive prestrains. Figures V-79 to V-82 show the yield stresses plotted against prestrain in tension and in compression on log-log co-ordinates. Table V-VIII shows the yield stresses obtained in the plain strain tests, and in Figures V-83 and V-84 they are shown plotted on log-log co-ordinates. The data from Figures V-79 to V-84 were used to draw the traces of the yield surfaces on the i- plane for different prestrain states. The procedure for this construction is shown in Appendix B. Figures V-85 and V-86 show the yield surfaces thus obtained for tensile and compressive prestrains respectively. The stress scale ~5 has been shown times the actual on the E plane so that the 42 values of stresses along the stress axes can be read directly off the diagram.

x 0 I — 4 U) z 0 U) IL U) 4 -i 0. X EXPERIMENTAL VALUE 0 CALCULATED VALUE 0.7 x 0.4 x' x _ _ _- _ _ x x,.,, 00.6 ~ ~ b` 4 0.4 0.5 0.41 I I 0.3 0.2 0. I 0 0.01 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 TENSILE COMPRESSIVE PRESTRAIN 1.5 2.0 Figure V-78. Variation of Plastic Poisson's Ratio with Prestrain.

I U) 0: - Q? X- DIRECTION _ Y- DIRECTION Z- DIRECTION X- DIRECTION - Y- DIRECTION ----- Z-DIRECTION y 30 z 25 -- - n -- = 20 - 15. I I I 0) 0) w IC. 0) an -J Sa 20 - 15 - 10 _ IO0 101 0.1 10 II Q15 0.2 PRESTRAIN IN Figure V-79. 0.3 IN/IN Yield Stress in Tension After Tensile Prestrain. 0.2 0.3 0.4 0.5 07 1.0 1.5 2.0 PRESTRAIN IN IN/IN V-81. Yield Stress in Tension After Compressiv Prestrain. Figure re I X- DIRECTION Y- DIRECTION Z- DIRECTION -Z z U) w Ir a.j w -7J 30 25 z X- DIRECTION - Y- DIRECTION Z-DIRECTION 201 15 10 I I I 0C 3. a) ahi w 201 15 n10 I I I I I I I I I 0.1 0.15 0.2 0.3 PRESTRAIN IN IN/IN Figure V-80. Yield Stress in Compression After Tensile Prestrain. 0.1 02 0.3 0.4 05 PRESTRAI IN Figure V-82. Yield Stress Prestrain. 0.7 1.0 15 2.0 IN/IN in Compression After Compressive

-128 a o 0: 0 3: 20 15 10 I I I I xz *4 % I I I I I. I 0.1 0.2 0.3 0.1 0.2 0.3 PRESTRAIN IN IN/IN 0.1 0L 0O Figure V-83. Yield Stress in Plane Strain Test After Prestraining in Tension. 30 25 20 a E I,I *2%.I I I I,, OQl o.a 04 0., 0.,7, 0. I 1'.1 G2 0.3 0.4 O 0. 0.7 1.0 L PRESTRAIN IN IN/IN s0k i 18s 201 SO:: — - 5 I1 0.1 0.2 0 04 0.5 0.7 1.0 1. PREITRAIN IN IN/IN Figure V-84. Yield Stress in Plane Strain Test After Prestraining in Compression.

-129 In Figure V-85 it is seen that with the increase of tensile prestrain the yield surface increases in size, ioe, strengthening occurs in all directions. The yield surface is not symmetrical about the normal to the it plane passing through the origino Since the material was prestrained in tension in the Z direction, yield stress in compression in the Z direction is lesser than that in tension in the same direction, and the geometrical centers of the traces of yield surfaces is offset in the direction of tension in the Z direction by about 1/2 to 1-1/2 ksio Since the shape of all the curves are closely similar, it appears that although a certain amount of kinemetric hardening occurred at the early stages of prestrainirg, at prestrains less than 01l, subsequent hardening is primarily isotropic. Figure V-86 shows yield surfaces for compressive prestrainso Similar comments as for Figure V-85 can be made in this case too, It is apparent from Figures V-86 and V-86 that the yield criterion for cold worked aluminum is not satisfactorily described either by Tresca criterion, or by von Mises criteriono However, a corner seems to develop gradually at the loading point with increasing compressive prestrains9 similar to what may be predicted by slip theory, but since the surfaces expand also, a combined effect of kinematic, isotropic and slip mechanisms of hardening seems likely at higher prestrainso Residual Ductility Of all the mechanical properties of materials, ductility shows the widest range of scatter, and the present investigation is

A o`111' su!T'esaJJ GT aTsuaG JsacV j OTTV umulrI 0TV 0 OTTo saOJ SonS PTJTa'P9-A a.MST x xtz x x 1X X \ x 01i X x 01 I S ~OZ X~ -o0T

-131 Figure V-86. Yield Surfaces for 1100 Aluminum Alloy After Compressive Prestrains.

no exceptiono The fracture strains in two transverse directions and the longitudinal strain at fracture for different prestrain states are tabulated in Table V-DXo Assuming that the ductility of an annealed material is constant, and since a tension test in the Z direction is merely a continuation of the previous tensile prestrain, it is expected that we should find p + d = Consto where ep is the prestrain in tension, and ed the residual ductility for tension test in the Z directiono If ductility is plotted against prestrain in Cartesian co-ordinates, this relation will appear as a straight lineo Therefore, although such a simple relation is not likely to be found for compressive prestrains, residual ductility for tension tests in X, Y and Z directions were plotted against prestrains in Cartesian co-ordinates in Figures V-87, V-88 and V-89 respectivelyo It is observed from these diagrams that the highest ductility in the X and Y directions are found in the annealed condition of the material0 For testing in the Y direction, ductility falls off rather rapidly with increase in prestrain of either kind, but the curve seems to level off for compressive prestrains above 0~4~ For testing in the X direction, ductility decreases almost uniformly for tension as well as in compression, the curve being steeper for tensile prestrainso For testing in the Z direction, the slope of the curve for tensile prestrain is.unity as is expectedo With increasing compressive

-135 S'- i. Ax _. I x 1.4 1.3 - m I.I x 1.0 0.2 00 02 0.4 0.6 0.8 1.0 15 TENSION COMPRESSION PRESTRAIN Figure V-87. Residual Ductility for Tension Test in the X Direction. 1.6 1 l.5 0 1.3 a.: 0 W 1.2 1.1 0 1.0 0 o0.0 0.2 0.4 o.6 0.8 1.0 1.5 TENSION COMPRESSION PRESTRAIN Figure V-88. Residual Ductility for Tension Test in the Y Direction... 1.7 - 1.6 -a a r 1.3 TENSION COMPRESSION PRESTRAIN Figure V-89. Residual Ductility for Tension Test in the Z Direction.

-154 TABLE V-IX RESIDUAL DUCTILITY IN TENSION TESTS SUBSEQUENT TO PRESTRAINING (Avo of Two Tests) Prestrain State Direction of Testing Cx Cy ~z Ex "y z Annealed X Y Z 1.530 1.092 0.959 1.009 1.466 o681 0.521 0.374 1.640 T 11 (ep = 0o1098) T 15 (Cp = 0152) T 20 (e = 0.206) T 25 (p = 0.245) C 2 (p = o.o106) x Y Z X Y Z X Y Z X Y Z X Y Z X Y Z X Y Z 1,558 o0891 0.925 1.476 1.005 o0830 1.433 0.869 0.857 1.351 o.856 0.774 1.570 o873 1.027 1.6o6 o850 0.947 10553 0.931 0.907 0o999 1.285 0 637 0.991 1.385 0.671 0.928 1,289 0.701 0,826 1.209 0o615 0.905 1 289 o.668 0.944 1.266 o.636 0.863 1.346 0.677 0.559 0.394 1.562 0.485 0.380 1.501 0.505 0 420 10558 0.525 0.353 1.389 0.665 0.416 1,695 0 o6'62 o.416 1.583 0 690 0. 415 1.584 c3 (cE = 0,1494) c 4 (p = 0.204

TABLE V-IX (CONT'D) Prestrain State Direction of Testing Ex Cy z c 5 (Ep = 0,244) c 6 (Ep = 0.300)) c 7 (E = 0o455) c8 (ep = 0.673) c9 (cp = 1953) C 10 (Ep = 1,01) C 11 (p = 1,49) x Y z x y z x z x Z X Y Z X Y Z X Y Z X Y Z X Y Z X Y Z 1.495 0.720 0.932 1,389 0 684 0.916 l. 4ol 0.739 0.838 1.105 0.704 0.849 0.950 0.547 1.290 0.592 o.829 1.130 0.523 0.790 0.782 1 182 o.650 0.735 1.120 0.590 0 618 1.174 0 421 o480 1 141 0O 489 O o480 1.053 0.554 1 147 0.492 0,470 1.074 0,526 0.713 0.462 1,582 0,654 o436 1 506 0,783 0.435 1.259 0o625 0.437 1o338' 00470 0 506 0.736 0,555 1o321 0.660 0,551 1.316 *, and e are the strains in the X, Y and Z directions respectively at fracture.

-136 prestrain the residual ductility is reduced and the curve levels off at prestrains of about 0.7. It is also noticed that the material has maximum ductility for tension tests in the annealed condition in the Z direction. This is as expected since commercial bars and plates, regardless of whether they are hot or cold rolled, have the inclusions elongated in the direction of rolling and consequently the ductility in the transverse direction is in general much lower than that in the longitudinal direction. Limitations of the Present Investigation. Certain assumptions, tacit or expressed, are always part of experimental investigations. Some such assumptions made for this investigation are critically examined below, (a) Continuity of Medium. This assumption is made in the treatment of certain branches of science, such as hydrodynamics. Even without going down to the atomic level, this assumption is not strictly valid for metals of commercial preparation and fabrication, since microdiscontinuities, cracks, inclusions, etc. are always present. They can easily be discerned under a microscope after polishing and etching. (b) Homogeneity. Metals in general are not microscopically homogeneous, but their elastic and plastic properties are usually averaged out over macroscopic dimensions, at least for the annealed material. But in the cold worked condition of a material even macro(73) scopic homogeneity is questionable. Figure V-90 shows hardness

-137 45 A,._ RC I l -'f I COLD WORKED 40 FACE 55 Rc 80 45 50 R 45 40 I~~~~~~~~~ Af I A1 -11 & I I~~~~~~~~~ I1III ~~ I~~~ 1/8 MACHINED OFF 1/4 MACHINED OFF Figure V-90. Hardness Traverse Along the Diameter of a Cold Worked High Strength Steel Cylinder.

-138 traverses across the diameter of a high strength alloy deformed by compression from an annealed stateo The traverses are for different depths from one end of the blocko The periodicity of the hardness values is obviouso Since the strength of a given material is directly related to its hardness, it is obvious that the deformation has not been uniform although barrelling was kept to a minimum by proper lubrication, (c) Time Dependenceo Although all the tests were carried out at very low speeds - about 00050 inch per minute head speed - the strain rate was not uniform since the deforming lengths were not the sameo The effect of strain rate, therefore, was small but not absent. Also, 1100-0 aluminum alloy is known to creep while sustaining a load. The effect of creep was, however, very small compared to the large strains to which the material was subjected and hence it was ignored,

-139 CHAPTER VI CONCLUSIONS From the previous analysis of the data obtained in this investigation the following conclusions can be made concerning the strain hardening characteristics of annealed, hot rolled commercially pure aluminum. These same conclusions, for the most part should be applicable to any single phase solid solution of a metal having a cubic latice structure. The conclusions are divided into four groups as below: Ao General Properties Determined by Processing History 1. In the annealed state such a material is approximately isotropic in strength properties, but not necessarily in strains. 2. The material is orthotropic and one of the axes of orthotropy being the axial direction. B. Expected Results and Conclusions of Previous Investigations Confirmed 1. Stress-strain relations for all states of prestrain and direction of testing can be expressed by an exponential type relation, a = a~o, over certain ranges of strain. 2. The value of the constant ao increases with increasing prestrain for any given kind of testing in a given direction. 3. The exponent m decreases with increasing prestrain.

4. Ductility in all the directions is a maximum at the annealed state of the material and decreases with increasing prestrain, both tensile and compressive. 5. Ductility in the transverse direction is lower than that in the axial direction, Co Conclusions in Variance with Current Concepts 1. Strains produced by loading consecutively in two directions are not additive. For large strains the concept of "effective strain" seems to be questionable, 2. Flow stress for testing in the opposite sense to that of the prestrain is always lower than that in the same sense and direction as the prestrain; the difference is about 5 to 20o% The Bauschinger effect, defined as reduction of flow stress on reversal of the sense of loading, is present for all prestrain states. 35 When stressed along the axial direction, the resultant strain hardening affects the flow properties in all the directions essentially the same way. In other words, the strain hardening is primarily isotropic, with a small kinematic component of 1 to 1-1/2 ksi superimposed on it, This is in contradiction to the kinematic hardening theory which does not seem to be valid for large prestrains as occurs in the present investigation,

-141 Do Further Observations and Conclusions 1. The range of strain over which the exponential type of relation is valid increases with increasing prestrain, 2. The direction of smaller transverse strain appears to have higher flow stress properties than the direction of larger transverse strain. In fact, it is found that the transverse yield strength in the direction of the smaller prestrain is up to 5% higher than the longitudinal yield strength in the direction and sense of the prestrain. This is rather different from what one would expect since the strains obtained in the two transverse directions are different in sense from that in the direction of loading. 3. When tested in axial compression after axial tensile prestrain, the stress-strain curves appear to converge to that of the annealed material for subsequent strains greater than 0.25, In other words, the effect of previous prestrain is removed after a certain reversed strain in this case, 4, (a) When tested in compression in a transverse direction after an axial compressive prestrain of about 0.4 the material shows a yield point phenomenon similar to that of annealed low carbon steels. The stress-strain curve reaches a maximum of stress, and then a slight strain softening is noticed.

(b) The amount of strain softening increases with increasing prestrain, being a fraction of a ksi at about C = 0.4 and about 1 ksi or about 4% for C = 2.0. 5~ The Bauschinger effect is present not only in the direction of prestrain, but also in the transverse directions, During prestraining, if the transverse strain in a certain direction is, say, tensile, then the flow stress in tension in that direction is higher than that in compressiono Similarly for compressive strainso 6, The greatest amount of directionality in flow stress is found at small plastic strains where the elastic and plastic strains are of comparable magnitudeo

APPENDIX A RELATIONSHIP BETWEEN POISSON'S RATIOS According to Hill's theory for anisotropic flow, plastic strain increments for multiaxial loading are given by Equation (III-l). When the principal directions of stresses coincide with the axes of orthotropy, the coefficients L, M and N do not appear in the equations, since T's and y's are all identically zero. For uniaxial loading in the X direction, (dex) = (dX)(H + G) cx (dEy) = -(dX) H ox and (dez) = -(dX) G ax Therefore (day) -H yx (H + G) (dz) = -G zx -- = — (dex) (H + G) where Iu is the plastic Poisson's ratio in the cs direction for loading in the 5 direction. Similarly, it can be shown that, -H xy (F + H) ~xx = -F y (F + H) = — G (G + F) -143

and Iyz -F (G + F) Hence, we have -H (F + H) -F -G (G + F) (H + G) yxy yzxzx yx zy xz = 1 -H (H + G) -F -G x -- x (F + H) (G + F)

APPENDIX B CONSTRUCTION OF YIELD SURFACE It is assumed, and to a close approximation it is true, that the yield criterion for metals is independent of superimposed hydrostatic stress of arbitrary magnitude. A consequence of this postulate is that the yield surface is a cylinder of infinite length, its generator being equally inclined to the axes of stress space. Hence, a plane equally inclined to the axes will be normal to the generator of the yield surface and the yield surface is completely defined by the trace of the yield surface on such a planeo Such a plane passing through the origin of the co-ordinate axes is known as the gt planeo The projection of the three stress axes on the n plane make 120~ with each other as shown in Figure A-lo It can be readily shown that the projection of a point in stress space with co-ordinates aaam agg and a7y is given by x = o - YY y 2= a Y a T2 6 on the t plane, x and y being two co-ordinates on the X plane as shown in Figure A-lo In the case of plane strain tests, if agl is the applied principal stress in the P direction, then the principal stress in the a direction for zero principal strain in that direction is P agp3og', the third principal stress being zero0 Wczg is the plastic Poissones ratio in the 0 direction for loading in the P direction,.145

Knowing the values of pi and all, the stress point can be shown on the t plane~ In practice it is more convenient to use a geometrical con\F2 struction as shown in Figure A-20 OB represents 2 ag5 and BA is BX drawn parallel to a -aais. The point X is such that -=I muLa~ ~OB'O The point X shows the projection of the point esenting the stress state f or the plane strain test on the n plane

-147 07 ASf Y x'PROJECTION OF STRESS STATE -- - 0. --- X aa yy Figure A-1. Co-ordinate Directions on 7 Plane. Cf B A ryrI, aa rr aa Figure A-2. Construction of Yield Surface.

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