)VIESI_J RETURNING MATERIALS: P1ace in book drop to Llanmuss remove this checkout from ‘1-Inzyl-IL. your record. FINES wil] be charged if book is returned after the date stamped below. DEFDRHQTIDN STUDIES ON ALPHA-BETA BRASS Bv Nitin Sudhakar Kulkarni A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Metallurgy, Mechanics and Materials Science 1984 ABSTRACT DEFORMATION STUDIES ON ALPHA-BETA BRASS By Nitin Sudhakar Kulkarni The deformation structure in polycrystalline specimens of two-phase aloha-beta brass was analyzed by Transmission Electron Microscopy. after deforming the specimens to two different strain levels (5 and 8 percents). In specimens that were deformed to 5 percent strain level, the dislocations were confined to single slip bands and no cross-slip was observed. 0n the other hand. specimens that were deformed to 8 percent strain level revealed complex tangled dislocations and deformation on intersecting slip planes. Burgers vector analysis of “Dislocation-Pairs". observed in the alpha phase,suggests that they may be screw dislocations that have double cross-slipped onto parallel slip planes. The stress-strain relationships and stress-strain distributions in the two-phase alpha-beta brass were also studied by using a continuum mechanics approach with the use of Finite Element Method Analysis. This analysis was performed bv using the ANSYS computer program. ACKNOWLEDGEMENTS I would like to take this opportunity to sincerely thank my adVisor, Dr. K. N. Subramanian, but for whose guidance and encouragement this endeavour would not have ended successfully. I also take this opportunity to extend my most sincere gratitude to Dr. N. Altiero without whose advice and guidance, the Finite Element Analysis could not have been carried out. His help is greatly appreciated. A special thanks to my colleagues, 5. Shekhar and Tong C. Lee, for having helped me during the TEM investigations. This work could not have been completed without the help of all the aforementioned. List of Figures. List of Tables TABLE OF CONTENTS I I INTRODU01‘10N I I I I I I I I I I I I I I II. HISTORICAL BACKGROUND. . . . . . . . . . Il-A. II-B. II-C. Copper-Zinc System. . . . . . . . a) D) General Alpha-Beta Brass Two-Phase Material Investigations a) b) c) General Investigations Using Model Systems Plastic Deformation and Slip Finite Element Method . . . . . . III. EXPERIMENTAL PROCEDURE . . . . . . . . . III-A. III-B. Transmission Electron Microscopic Study of the Deformation Structure Finite Element Analysis. . . . . IV. RESULTS AND DISCUSSION . . . . . . . . . IV-A. IV-Bo Transmission Electron Microscopic Analysis of Deformation Structure in a) b) c) d) e) Alpha-Beta Brass . . . . . . . Overall Deformation Structure Burgers Vector Analysis of Dislocations Near the Boundary Analysis of “Dislocation-Pairs“ Grain Boundary Dislocations In-sztu Deformation Studies in a High Voltage Electron Microscope Finite Element Method Analysis of Stress-Strain Behavior in Alpha-Beta Brass. . . . . . . . . a) Stress-Strain Relationships for Different Volume Percents iii ix 3o 36 42 S7 57 S7 V1. b) c) d) of Beta Phase Stress-Strain Relationships for Different Grain Size Distributions Stress-Strain Distributions Across the Alpha-Beta Interface Comparison of Experimental Stress-Strain Curve with the FEM Calculated Curve CONCLUS I 0N8 I I I I I I I I I I I I I I I I V-A. TransmiSSion Electron Microscopy Studies . . . . . . . . . V-B. Finite Element Method Analysis . . . REFERENCES iv .110 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure LIST OF FIGURES The Copper-Zinc Phase Diagram. (Raf: 15). I I I I I I I I I I I I I I I Schematic of a bicrystal consisting of crystals ‘A’ and ‘8’. Loading is along the Z-direction. . . . . . . . . . . . . Schematic of ‘Jet-Polishing’ unit. . . . Schematic of the final electropolishing unit. . . . . . . . . . FEM mesh for alpha-beta brass containing 20 volume percent beta phase (beta phase is shaded dark): a) Fine grain distribution. b) Medium grain distribution. c) Coarse grain distribution. . . . . . FEM mesh for alpha-beta brass containing 40 volume percent beta phase (beta phase is shaded dark): a) Fine grain distribution. b) Medium grain distribution. c) Coarse grain distribution. . . . . . FEM mesh for alpha-beta brass containing 60 volume percent beta phase (beta phase is shaded dark): a) Fine grain distribution. b) Medium grain distribution. c) Coarse grain distribution. . . . . . FEM input stress-strain curve of ‘Alpha’ and ‘Beta’ brasses . . . . . . . FEM mesh, for alpha-beta brass containing 20 volume percent beta phase (shaded dark), used for calculation of the stress-strain distributions, across the alpha beta interface . . . . . . . . Deformation Structure in the alpha phase of the alpha-beta brass deformed to 5 percent strain level. . . . . . . . Deformation Structure in the alpha phase of the alpha-beta brass deformed to 5 percent strain level. . . . . . . . UT 14 38 40 47 49 m m 60 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 13. 14. 17. 18. 20. 21. Deformation Structure in the alpha phase of the alpha-beta brass deformed to 5 percent strain level. . . . . . . Deformation Structure in the alpha phase of the alpha-beta brass deformed to 8 percent strain level. . . . . . . Deformation Structure in the alpha phase of the alpha-beta brass deformed to 8 percent strain level. . . . . . . Deformation Structure in the alpha phase of the alpha-beta brass deformed to 8 percent strain level. . . . . . . Deformation structure in the alpha phase of the alpha beta brass near the boundary deformed to: a) 5 percent strain level. b) 8 percent strain level. c) 8 percent strain level . . . . . . Dislocations near the boundary in the alpha phase of the alpha-beta brass with different diffraction vectors: a) 73‘: [T113 b) ‘9‘: [0023 c) ‘5’: czogi d)‘§=[1111............. Dislocations near the boundary in the alpha phase of the alpha-beta brass with different diffraction vectors: a) 73: [T11] b) ‘9‘: [00g] c) ‘9‘: [2023 d)‘g‘=[ozoi............. Dislocations near the boundary in the alpha phase of the alpha-beta brass with different diffraction vectors: a) ‘9’: [T113 b) ‘g‘= [00233 c) ‘9‘: [20133 m-g"=[1113............. "Dislocation-Pairs" observed in the alpha phase of the alpha-beta brass. . Boundary dislocations observed in the alpha phase of the alpha-beta brass. . vi 61 63 64 66 71 74 78 84 BS Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure n. 2b. 27. 29. 31. Boundary dislocations observed in the alpha phase of the alpha-beta brass. . . Still photographs of dislocations in motion in the alpha phase of the alpha-beta brass during in-sita deformation in an HVEM . . . . . . . . . Still photographs of dislocations in motion in the alpha phase of the alpha-beta brass during in-sita deformation in an HVEM . . . . . . . . . FEM calculated curves of alpha-beta brass for 20, 40 and 60 volume percents of beta phase. . . . . . . . . . . . . . FEM calculated stress-strain curves of alpha-beta brass containing 20 volume percent beta phase for fine (I) and coarse (II) grain size distribution. . . FEM calculated stress-strain curves of alpha-beta brass containing 40 volume percent beta phase for fine (I) and coarse (II) grain size distribution. . . FEM calculated stress-strain curves of alpha-beta brass containing 60 volume percent beta phase for fine (I) and coarse (II) grain size distribution. . . FEM calculated stress distribution plot for alpha-beta brass contaiing 20 volume percent beta phase, in the loading Y-direction. . . . . . . . . . . FEM calculated stress distribution plot for alpha-beta brass contaiing 20 volume percent beta phase, in the transverse X-direction . . . . . . . . . FEM generated stress contour distribution plot in the 60 element mesh for alpha-beta brass containing 20 volume percent beta phase, in the loading Y-direction. . . . . . . . . . . FEM generated stress contour distribution plot in the b0 element mesh for alpha-beta brass containing 20 volume percent beta phase, in the transverse X-direction . . . . . . . . . vii 91 .101 .102 Figure Figure T FEM calculated strain distribution plot for aloha-beta brass containing 20 volume percent beta phase . . . . . . . .103 Comparison between the experimentally determined stress-strain curve of bOCu-4OZn brass and the FEM calculated curve of alpha-beta brass containing 20 volume percent beta phase. . .105 viii Table Table Table Table Table Table Table Table LIST OF TABLES Typical Mechanical Properties of MUNTZ METAL (Ref. 16). . . . . . . . . Typical Metallurgical Data of MUNTZ METAL (Ref. 16). . . . . . . . . Electropolishing data for "Dishing Technique". . . . . . . . . . Electropolishing Data for Final Step of Thin Foil Preparation . . . . . . . Results of the Burgers Vector AnaIVSis of Dislocations in Figure 17. Results of the Burgers Vector AnaIVSis of Dislocations in Figure 18. Results of the Burgers Vector Analysis of Dislocations in Figure 19. Yield Stress Values Obtained by Using the 0.2 Percent Offset Method from Curves in Figure 25. . . . . . . . . . ix I. INTRODUCTION Two-phase materials find wide ranging engineering applications. Various steels, two-phase titanium-aluminum alloys, copper-zinc alloys, copper-tin alloys and copper-aluminum alloys are some of the most widely used two-phase alloys. The mechanical properties of these two-phase materials will differ with changes in the properties of individual phases, the distribution and volume fractions of each phase. Other metallurgical factors that affect the mechanical properties of such two-phase materials are crystal structure of each phase. crystallographic orientation of the phases in relation to each other, strain-hardening in each phase, strain-rate effects and the dislocation behavior and slip in each phase. A polycrystalline aggregate of single-phase material consists of grains which are of the same crystal structure. In such single-phase materials, different regions with different crystallographic orientations are separated by grain boundaries. In a two-phase alloy, each phase has a different chemical composition,which quite often possesses a different crystal structure. The boundary between the two phases in a two-phase material is called a phase boundary which is analogous to a grain boundary of a polycrystalline single-phase material. In order to understand the basic mechanical behavior of a single-phase material, a single crystal or a series or parallel type bicrystal is commonly used. In a series bicrystal the grain boundary is oriented normal to the loading 3X15. whereas in the parallel type bicrystal, the grain boundary is oriented parallel to the loading axis. The ideal unit to study the behavior of a two-phase material is a two-phase bicrystal or a multicrystal (1,2,3). Two-phase bicrystal incorporates a single crystal of one phase joined to a single crystal of the other phase. In a multicrystal. a Single crystal of one phase is JDIDEU to a few large grains of the other phase (3). A duplex crystal in which the two phases have a definite relative crystallographic orientation relationship is also of interest (3). Hingwe at al. (1,2,3) and Nilsen et al. (4.5.6) have studied the mechanical properties of bicrystals of alpha-beta brass at room temperature under uniaxial tension. at various strain-rates. Kezri-Yazdan et a1. (7.8.?) have studied the effect of temperature on mechanical behavior of such specimens at elevated temperatures. All the above mentioned types of crystals are only "Model Systems“. They help to better understand basic deformation characteristics of a single or two-phase material. However. the engineering materials in practical use are polycrystalline and their overall mechanical behaVior arises as a result of the basic considerations suggested earlier, although the overall picture will be more complicated. 3 The aim of this investigation is to study the deformation behavior of polycrystalline two-phase alpha-beta brass. The present study is divided into two parts: a) Transmission Electron Microscopic (TEM) study of the deformation structure in alpha-beta brass. b) Finite Element Method (FEM) analysis of elasto-plastic stress-strain relationships in alpha-beta brass. The present TEM study mainly concentrates upon the analysis of the deformation structure in the alpha phase of the alpha-beta brass. The results of this present work will be compared with the TEM studies on 70 Cu-30 Zn alpha brass performed by Karashima (10,11). The FEM analysis of alpha-beta brass is based on FEM studies performed by Jinoch et a1. (12) and Ankem et a1. (13.14) on alpha-beta Ti-BMn alloy. Alpha-beta brass is similar to a large number of two-phase alloys of engineering importance, and it is hoped that the results of this study may be applicable to various other two-phase alloy systems. II. HISTORICAL BACKGROUND 11-A- Qgeeec:;i99_§xstsm a) Gggggal; The term ‘BRASS’ is applied to a class of alloys that consist essentially of copper and zinc. They are a very important class of commercial alloys. Their popularity is mainly due to their high corrosion resistance, good machinability, and the varying mechanical properties obtainable by changes in the composition. The Copper-Zinc phase diagram (15) is presented in Figure 1. It can be observed from this diagram that the face centered cubic (F.C.C.) ‘Alpha’ phase can accommodate up to about 38 weight percent zinc in solid solution. The alpha phase is fairly strong and possesses excellent ductility. Many commercial brasses are made of this solid solution and they possess good formability. 70 Cu-30 Zn Cartridge Brass is one of the many alloys that is typically alpha brass. As observed from this diagram, addition of more than approximately 38 weight percent Zn results in the appearance of a second solid solution. This phase is called ‘Beta’ and it differs from the alpha phase in many ways. It has a cesium-chloride (CsCl) structure and is harder and stronger but much less ductile than the alpha phase. The beta phase undergoes a modification in the arrangement of solute atoms in the crystal lattice between a temperature range of 454-468 degrees Celsius (15). The high temperature stable phase is called ‘Beta’ (15). It has on the average a body centered cubic (B.C.C.) A2 type Figure l. The Copper-Zinc Phase Diagram. (Ref: 15) IEIGNT PEI CENT ZINC 40 50 60 (o 20 so 10 no so "00 1 1 l 1 l \ \ .000 \\ 900 a g 31.9 '3" (32.5) c u :00 \ l\\ 9700 1 I. 9170!? N J g mug!) \ \ 5 . :600 O? o___. C t \ \ ' \ \ i". . ”500 I 468‘ (3313: ‘23 “'2 . . '1 [disk “‘9’ ‘ u ‘2" :I!‘ ‘°° ‘ 912; M, Y\ (97) 4 (Zn) 300 /34.1 \ \ : i ' | [(35.2) .‘ ' : : ',| : zoo ,1 . ‘ 7 . . : o u I I I , '. ' 99.69 100 - .1 (99.1) o 10 20 QO so . so 70 no so 100 c: Home PEI cut zmc :- 6 crystal structure. On cooling below the above mentioned temperature range, the solute atoms assume an ordered cubic 82 type of structure. This low temperature form is termed as ‘Beta-Prime’. The brasses varying in composition from 38 to 46 weight percent Zn, contain both alpha and beta phases. These are termed as the two-phase alpha-beta brasses. This family of brass is the main focus of this study. Brasses containing approximately 4b to 50 weight percent Zn contain the beta phase only. The brass alloys in this comp051tion range do not find much commercial application as they are hard and brittle. With the zinc content approximately greater than 50 weight percent, a very hard brittle ‘Gamma’ phase appears. Alloys containing the gamma phase are useless in engineering applications. b) Algba:§gta_§§§§§; Brass containing b0 weight percent Cu, 40 weight percent Zn is representative of this class. and is commonly called as ‘MUNTZ METAL’. Muntz metal finds extensive use in the manufacture of pipes. tubes, and general purpose castings (16). This alloy has relatively poor cold drawing and forming properties as compared to other copper alloys, but has excellent hot working properties. It is the strongest of the commercially used Cu-Zn alloys. The typical mechanical properties of this alloy are listed in Table 1. The typical metallurgical properties of this alloy are listed in Table 2 (16). TABLE 1 Ixeigel_megbaoigal_Ecgeecti§§-Qi_dyNTZ NEIAL (Ref- lél Yield strength: 15 x 10: - 38 x 10: KPa Tensile strength: 37 x 10 — 51 x 10 KPa Shear strength: 28 x 10 KPa Percent Elongation: 10 - 52% Modulus of Elasticity: 1.03 x 107 KPa Modulus of Rigidity: 3.86 x 10 KPa Note: 1 psi = 6.895 KPa Ixeisel-metellucgisel miscestcsctucgi Ecxstel-§icuggg;g; Liquigus_lsmeecatucei TABLE 2 -9s2é-gi-EQBIZ "ETAL (Ref- 191 Two Phase Alpha and Beta. a) Alpha phase: A1 type face-centered cubic. b) Beta phase: (1) High temperature beta phase has A2 type body-centered cubic structure. (2) The low temperature beta prime phase has B2 type Cesium-Chloride (CsCl) structure. Beta phase appears dark when etched with ferric chloride. and golden yellow when etched with ammonium chloride. 117B°K §elidu§-lsmeecatuce_8eoge; 698 -B73°K BQE:!QCEIOQ-I§@Q§E§EB ce_Beogei 89B -1073°K II-B- I!Q:Ebe§e-datgciel-lexe§tigetiso§ a) Ggggcgl; Unckel (17) was one of the first to investigate deformation in polycrystalline two-phase materials. In his investigations, alloys of copper containing 6 weight percent iron, leaded brass, alpha-beta brass, complex brass (60 Cu-40 Zn brass with additions of Al, Fe, Mn, Sn, and Ni) and alpha-delta bronze were deformed by cold-rolling. The particle size was measured before and after rolling. For materials in which the second phase was softer than the matrix, like leaded brass, the amount of reduction undergone by the particles on rolling was greater than the reduction of the specimen as a whole. In all the other alloys, the inclusions were harder than the matrix, and the second phase particle deformed less than the matrix. Unckel (17) reported that as the amount of deformation is increased, the deformation of the matrix and particles tended to become more and more homogeneous, i.e. the hard particles deformed more and the soft particles deformed less. This can be attributed to the work-hardening of the softer matrix phase. Chap and Van Vlack (18) confirmed Unckel’s conclusions that the extent of the deformation of the second phase depends upon the relative hardness of the particles of second phase as compared to that of the matrix. The first detailed metallographic study of two-phase materials was undertaken by Honeycombe and Boas (19). They studied deformation behavior of alpha-beta brass and 10 observed that during deformation, slip first appeared in alpha grains which became heavily deformed before slip was observed in the beta grains. When a heavily deformed specimen was repolished and restrained, slip lines started again in the alpha phase. Slip lines occasionally crossed the alpha-beta phase boundary. The orientation relationship between the alpha and the beta phase for this to occur was {110} in beta parallel to {111} in alpha as well as (111) in beta parallel to <110> in alpha. Honeycombe and Boas (19) also noted that more deformation occurred in the region of the beta phase near an alpha-beta boundary than in the interior of the beta grain. Clarebrough (20) subsequently studied silver-magnesium alloys which consisted of soft silver rich grains of F.C.C. alpha phase and grains of hard beta phase having a CsCl structure. These alloys deformed in a manner similar to two-phase alpha-beta brass. Studies indicated that below 30 volume percent of beta phase. the alpha phase deformed more than the beta phase. When the volume fraction of the beta phase exceeded 30 volume percent, the extent of deformation in both the phases were the same and equal to that of the alloy as a whole. Clarebrough and Perger (21) observed similar behavior in alloys of copper and zinc. Honeycombe and Boas (19) have shown that. in alpha-beta brass. preferential deformation of the softer alpha phase can occur. Mechanical properties of ductile materials containing either soft or hard particles have also been 11 studied by Gurland (22) using different silver alloys. From his studies Gurland concluded that the type of second phase particle is not an important parameter for overall deformation characteristics so long as it does not deform during deformation of the alloy. He also suggested that, the effect of the strong second phase particle on strengthening is due to the restriction of plastic deformation of the softer matrix by the particles (22). Slip is initiated in the matrix by the stress concentrations caused by the presence of hard particles during loading. Margolin, Farrar and Greenfield (23) have studied the deformation of alpha-beta titanium alloys. In these alloys the alpha phase was distributed either as equiaxed alpha, in an aged beta matrix or as widmanstatten plates and grain boundary alpha in an aged beta matrix. For specimen with approximately same value of yield stress, the fracture strength of equiaxed alpha structure was found to be higher than that of the widmanstatten structure of the same grain size. Correspondingly, the reduction of area was higher for specimens with the equiaxed structure as compared to those with widmanstfitten structure. All the studies that have been discussed so far have been carried out using bulk polycrystalline two-phase materials (17-23). To understand the process of deformation in two-phase materials, the role of the phase boundary during the deformation needs to be investigated. Investigations using Model Systems of two-phase materials 12 are reviewed in the following section. b) lozestigetion§-Q§iog_dggsl-§xstemsi Model Systems such as bicrystals and duplex crystals have been used to understand the basic mechanisms of deformation in two-phase materials. To study the mechanical properties and deformation mechanisms of two-phase materials, the number of variables must be minimized. The ideal fundamental unit for studying the mechanical behavior of two-phase material would be a single crystal of one phase joined to a single crystal of another phase. Such a unit is called a two-phase bicrystal (1). A duplex crystal has been defined as an oriented crystallographic unit consisting of two phases with a definite crystallographic orientation relationship with each other (3). Eberhardt et a1. (24) have developed a method for producing two-phase alpha-beta brass bicrystals by diffusion bonding. The interface boundaries of these bicrystals are neither sharp nor stable. Bicrystals produced by such a method are in a metastable state. since they are obtained by ice-water quenching (25). Takasugi, Izumi and Fat-Halla (25-28) tested such diffusion bonded isoaxial specimens of alpha-beta brass at specific strain-rates, at various temperatures. Under such loading conditions, the strains in the alpha and beta phases are the same, and the boundary does not experience any stress. Hingwe and Subramanian (1) have developed a technique for producing bicrystals and duplex crystals of alpha-beta brass by using specialized 13 heat-treatment schedules. The phase boundaries in bicrystals grown by them are extremely sharp ( < lam thick ) and are stable. similar to that in actual two-phase materials. Studies on alpha-beta brass by Hingwe and; Subramanian (2) employed a series type bicrystal where the phase boundary was normal to the tensile axis. Under such loading, alpha phase, beta phase and the boundary experience the same stress due to the external load. Under any general state of stress, macroscopic and microscopic compatibility conditions must be satisfied at the grain boundary (3). For an isoaxial, symmetric bicrystal, consisting of crystal ‘A’ and crystal ‘B’ (see Figure 2). with the planar boundary normal to the Y-axis. the macroscopic compatibility conditions that must be satisfied are: [$351538 ; EzEJfiEzEJa ,- E.;J.=E.J. (E1) where,€ and 6&2 are strain components and )X ’ 6!! subscripts ‘A’ and ‘8’ represent the regions in the crystal ‘A’ and ‘B’, respectively. In addition, the strain in the loading direction in crystal ‘A’, [:2ng , should be equal to the strain in crystal ‘B’, [625] B’ and these two in turn should be equal to the total strain,‘€ of the bicrystal. z'z’ These compatibility relations can become incompatible in an anisotropic crystal. Compatibility conditions are more complicated in a polycrystalline aggregate. 14 Figure 2. Schematic of a bicrystal consisting of crystals 'A’ and ‘B’. Loading is along the Z-direction. E on «D F 15 Hingwe and Subramanian (2) deformed bicrystals of alpha-beta brass in uniaxial tension to study the initiation of plastic deformation and its propagation across the phase boundary. The effectiveness of the phase boundary as a barrier for slip propagation depends upon the relative crystallographic orientation of the two phases. Slip was observed in the beta phase only after secondary slip and cross-slip took place in the alpha phase. This may be due to: 1) The difference in the shear moduli of the alpha and beta phase or. 2) Difference in the Burgers vector of slip dislocation in the two phases, or 3) Difference in the number of available slip systems (2). Nilsen and Subramanian (5) have carried out tensile-tests on bicrystal specimens of alpha-beta brass at various strain-rates, and for specimens having two different types of boundary geometries, corrugated and flat. According to them, the corrugated boundary was more effective in blocking the propagation of slip from alpha to beta phase. All specimens tested were found to be strain-rate sensitive. At low strain-rates, both boundaries were ineffective barriers, and coarse slip was observed in the alpha phase. Fine slip was observed at high strain-rates in the alpha phase. Slip propagation in the beta phase requires the movement and multiplication of superlattice 16 dislocations. These superlattice dislocations are composed of two perfect dislocations jOiflEd to one another by an antiphase boundary. Since the dislocations are connected to one another by an antiphase boundary, they must move as a single unit. As a result, their motion is difficult and requires a high stress. As any slip line crosses from the alpha phase into the beta phase, it will proceed only to a limited extent. This extent depends upon the level of stress concentration created in the alpha phase near the phase boundary. The effect of this stress concentration in the beta phase diminishes in direct proportion to the distance from the phase boundary. When the slip line reaches a position at which the stress concentration is insufficient to continue the propagation. it will stop. Nilsen and Subramanian (4) have also carried out investigations on bicrystals of alpha-beta brass consisting of oriented and equiaxed duplex boundaries. Equiaxed boundaries were found to be more effective in blocking the propagation of deformation from alpha to beta phase, than the oriented boundary. Nilsen and Subramanian (4,5) concluded that wider slip bands formed in alpha phase produced a more effective stress concentration so as to activate slip in beta phase across the boundary. Fine or multiple slip formed in alpha phase at high strain-rates, did not produce effective stress concentrations to cause slip across the boundary. High strain-rate, produces fine slip in alpha phase by activating many dislocation sources. Interaction of fine slip with the boundary causes multiple slip which immobilizes a large fraction of the dislocations. Kezri-Yazdan and Subramanian (29) have studied the role of phase boundary orientation relative to the tensile axis in bicrystals of alpha-beta brass. Under such a geometry, the phase boundary experiences both shear and normal stresses during loading. It was found that in the bicrystal specimens having inclined boundaries, the interaction of slip in the alpha phase with the phase boundary was not the motivating force for creating deformation in the beta phase. In these specimens, the slip in the beta phase normally occurred on its own. This may be due to the shear stresses acting on the inclined boundaries. Fat-Halla et al. have studied operative slip systems in diffusion bonded alpha-beta brass two-phase bicrystals at 150°K. Their studies showed that the primary (111)[I01J system was operative in the alpha phase and W3 [111] system was operative in the beta phase, where $5 is the actually observed slip plane in the beta phase lying between [:01] and [II2J. Fat-Halla et a1. (30) have also studied the elastic incompatibility of isoaxial bicrystals of alpha-beta brass during deformation. Kawazoe et a1. (31) have also performed fatigue tests on diffusion bonded two-phase bicrystals of alpha-beta brass. The fatigue tests were performed at 30 Hz in tension and compression, under a constant stress amplitude of about 18 1100 MPa. Predominant slip in the alpha phase occurred on the primary (111) plane, while that in beta phase occurred on planes apart from (T01). rather close to (211). Interface studies by Takasugi and Izumi (32) on bicrystals of alpha-beta brass led them to believe that the phenomenon of interface sliding was strongly dependent upon the nature of the interface. the orientation relationships, the shear direction and the microstructure of the interface. Margolin and co-workers (33-38) have published a series of papers on the deformation behavior of beta brass using beta brass bicrystals. Chuang and Margolin (33) studied the stress-strain relationships of beta brass bicrystals. Three isoaxial bicrystals and one non-isoaxial bicrystal of beta brass, oriented to have the grain boundaries parallel to the axis of loading were used in their studies to investigate the stress-strain relationships. It was found that grain boundary strengthening was greater in non-isoaxial bicrystals of beta brass. Lee and Margolin (34) studied the stress-strain behavior of beta brass using bicrystals and tricrystals of beta brass. Their studies showed that the flow-stress in the bicrystal specimens was higher than the single-crystal flow-stress at the same strain level, whereas the tricrystal flow stresses were only slightly higher than those of bicrystal specimens at the same strain level. Secondary stresses in tricrystals had a slower decay rate than in bicrystals. Based on this observation, 19 Lee et al. have suggested that the secondary stresses would spread across a grain more rapidly in a polycrystalline material, where the constraints are the highest (34). The average grain boundary stress in tricrystals was found to be smaller than the corresponding stress for bicrystals despite the increased constraint. Lee and Margolin (35) have also studied the effective Young’s modulus in isoaxial bicrystals and tricrystals of beta brass. Studies showed that the slope of the elastic portion of the stress-strain curve of bicrystals was markedly higher than that of its component crystals. The increase in slope of tricrystals was only slightly higher than the bicrystal specimen. These changes in slopes arise due to the elastic shear incompatibility at the grain boundary. The apparent elastic modulus of some bicrystal grain boundaries was found to be higher than that of the beta grain interior. Margolin et al. (36) showed that, if examined separately. the grain boundary and the grain interior possess separate and different stress-strain relationships. Yaouchi and Margolin (37) have also studied the grain boundary contribution to the Bauschinger effect in beta brass bicrystals. Their studies revealed that at similar strain levels in the grain boundary and the grain interior, the former unloads faster due to its higher modulus than the latter. Thus Yaguchi et a1. (37) suggested that at overall zero stress level, the grain boundary experiences compression and the grain interior 20 experiences tension. After prior tensile straining, if the specimen is compressed, the grain boundary will yield first (37). Yaguchi and Margolin (38), from their studies, have shown that the flow in the twinning direction of the bicrystal required a higher stress than flow in the anti-twinning direction, and prior slip in compression raised the stress required for flow in tension. c) Blast;c-2219cmatieo_aod-§lie figggcgl; Plastic deformation of a crystalline material causes rapid dislocation multiplication with increasing plastic strain. 1n the early stages of deformation, the dislocation movement tends to be confined to a single set of parallel slip planes (6). On further deformation, slip occurs in other slip systems, and different dislocations moving in different slip systems interact. This dislocation interaction depends upon many factors such as temperature, crystal structure of the material, strain, strain-rate, inclusions or precipitates, and the nature of stacking faults (6). During the plastic deformation of a polycrystalline material, the dislocations must overcome the resistance resulting from obstacles such as impurity atoms, point defects, other dislocations and grain boundaries. Dislocations tend to pile-up at the grain boundaries (6). Because of this dislocation pile-up, a back stress is created which eventually may stop the dislocation source from operating (6). In the region where the leading dislocation is stopped at a boundary, a stress 21 concentration will occur. If the stress concentration is high enough, the slip proceeds across the boundary and moves on a coherent slip plane in the adjacent region. Various mechanisms have been proposed to explain the continuation of slip through the boundary into the adjacent grain (6). One possibility is that the high stress concentration preceding the leading dislocation in the slip band causes the dislocations to cross the boundary and move into the adjacent grain (39). These newly created dislocations, then under the combined action of the applied stresses and the stresses due to dislocation pile-ups at the boundary continue to glide in that region. Another theory predicts that the leading dislocations in the pile-up produce a stress concentration at the grain boundary. and this stress concentration causes some of the boundary dislocations to move into the adjacent grain (40). Another theory predicts that the stress concentration, produced by the dislocation pile-up at the boundary, activates dislocation sources present in the nearby region of the adjacent grain (41). The decay in the intensity of slip at the boundary could according to Margolin et a1. (40) imply that there is "elastic opposition" to the operation of slip system as the slip approaches the boundary. Hence, cross-slip according to them, becomes more favorable near the boundary (40). Resistance to plastic deformation as a result of elastic restraints of the adjacent grain and constraints at the boundary is termed as ‘elastic oppo51tion’ by Margolin et a1. (40). Elastic opposition to the operation of slip in one grain by the neighboring grain has been observed by Margolin et a1. (40). He states that although the elastic interaction assists the nucleation of slip, plastic incompatibility opposes it. At a distance from the boundary, plastic incompatibility does not have a significant effect on the operation of slip. But as slip approaches the boundary, the compatibility effects become more prominent, causing decay of the slip intensity and the occurence of cross-slip. Blast;s-Qeigcmetieo_§ebéyioc-Qi-elpne-§ca§§i Alpha brass is a single-phase substitutional solid solution of zinc in copper. The zinc atoms. in alpha brass, occupy random substitutional atomic positions in a F.C.C. crystal lattice structure. Alpha brass usually deforms along close packed {111) planes in the <1T0> directions. Alpha brass is stronger than copper and the increase in strength of alpha brass. as compared to copper. may be attributed to the solid solution hardening caused by the addition of zinc atoms to copper (3). Presence of zinc affects both the stress necessary to initiate plastic deformation and subsequent work-hardening (3). The critical shear stress in alpha brass crystals increases linearly with the zinc concentration. Ardley and Cottrell (42) have studied the yielding phenomenon in alpha brass crystals at room 23 temperature. Well marked yield points have been observed in Virgin and strain-aged specimen. Further. both the upper yield stress and the yield drop, increase markedly with increasing zinc content (42). When an F.C.C. alpha brass single-crystal is deformed in uniaxial tension, the plot of the resolved stress as a function of the shear strain exhibits three stages of deformation, i.e. Stages 1. II and Ill. As the zinc concentration is increased, Stage I (easy-glide region) becomes longer, Stage II (linear-hardening region) increases in range. and Stage III starts at a higher stress level. At the end of the easy-glide. the shear stress in the primary slip system. has been found to be the controlling factor in initiating secondary slip. It is assumed that easy-glide ends when the stress concentration around the clusters of dislocations on the primary slip plane is sufficiently large to initiate slip on another slip plane. In case of alpha brass. higher stress concentration is required to initiate secondary slip as compared to other F.C.C. materials. This means that bigger clusters of dislocations are required on the primary slip planes: this can be achieved by a long easy-glide. Stacking fault energy is one of the parameters that is important in determining whether cross-slip is possible. Lowering the stacking fault energy widens the separation between the partial dislocations. Widening the separation of partial dislocations makes cross-slip difficult. 24 however, sometimes pile-ups of dislocations in the primary slip plane lead to the activation of nearby sources on the cross-slip planes. Hence cross-slip in alpha brass can take place in spite of its low staking fault energy (43). Karashima (10) has studied the dislocation structure in deformed alpha brass using Transmission Electron Microscopy. He observed that at low strains (1-5 percent) dislocations piled-up against the boundary and at obstacles within a crystal. It was also observed that with increaSing degree of deformation (5-10 percent), dislocations from two or three slip systems interacted to form dislocation tangles and kinks. A prominent characteristic of alpha brass is that most of the dislocations introduced during cold-working are confined to the primary slip planes. This behavior is quite different from that found in metals having a high stacking fault energy like beta brass (10). Karashima’s studies showed that slip progresses through grain boundaries and twin boundaries, with a clear indication that slip was continuous across the boundary into the neighboring grain. Slip can be induced into the neighboring grain with the aid of stress concentration from the pile-ups of dislocations against the grain boundary (10). Such a pile-up can activate dislocation sources present in the neighboring grains or make the grain boundary dislocation sources operative. Mitchell and Thornton (44) have investigated the role of secondary slip during deformation of copper and 25 alpha brass single crystals. Secondary slip is likely to be important during Stage 11 of deformation. They have found that with increaSing Zinc content the amount of secondary slip decreases (44). Elestis_Qs£Qcmation_§ebaxiec_9i_§sta-§csssi Beta brass is an intermediate solid solution of the Cu-Zn system. The ordered beta prime phase has a cesium-chloride structure, which is equivalent to two interconnected simple cubic unit cells; one of copper and the other zinc. This type of structure is referred to as the B2 type superlattice. Beta brass normally deforms by slip on {110} planes and in <1I1> directions and in general is a material having high stacking fault energy. In the ordered state, the dislocation behavior of the beta prime phase is different from that of the disordered B.C.C. beta phase. A slip vector displacement causes an atom in one layer to move from one position to a new position at a unit distance. This produces a change in the local arrangement of the atoms on the slip plane, thereby creating an antiphase boundary (6). This boundary has a specific energy, depending upon the degree of order in the lattice. This disorder produced by the slip process, can be removed by a second dislocation, which restores the original atomic arrangement. Thus a perfect dislocation in an ordered lattice consists of two ordinary dislocations on the slip plane joined by an antiphase domain boundary (45). This is similar to an extended dislocation consisting of two 26 partial dislocations joined by a stacking fault (45). Since single unit dislocation produces an antiphase domain boundary. it is energetically favorable for it to be associated with a second unit dislocation which removes the high energy fault. This pair of dislocations are called superlattice dislocations, and move on the same slip plane in the crystal. A high stress, however, is required to move these superlattice dislocations (46). Two distinct structural characteristics, namely slip bands and deformation bands, have been observed in deformed beta brass (47). Slip lines in beta brass are less prominent than those in F.C.C. metals (47). Both temperature and mode of deformation have been found to affect the occurence and spacing of deformation bands (47). Because of the highly anisotropic nature of beta brass, the electron microscope contrast theory based on isotropic invisibility criteria fails, except for pure edge or pure screw dislocations (48). Head et a1. (48) have computed the theoretical image profiles of dislocations according to the two-beam dynamical theory, and considering the full anisotropic strain fields of the dislocations (48). Comparing the character of the theoretically predicted and experimentally observed dislocation images, they have concluded that the majority of dislocations in beta brass are screw dislocations of type a<111> gliding on {T10}. Some of these a<111> dislocations split into two a/2<111) dislocations. Other dislocations are of a<010> type glisSile on {001}. Elastic properties of cubic crystals are governed by values of three elastic stiffness constants (CTI , C12 and C44 ) and the elastic compliances (S11 , S12 and S44 ) (49). The displacement field around a dislocation, and the electron diffraction contrast from the dislocation, involve only ratios of these constants: A: 2C44_ : 2 _S_1_1:_S_L2_ . Two of the most characteristic TEM dislocation contrast effects observed in beta brass by Head et a1. were: 1) {111} dislocations showing double images when using either a £110] or a [112] reflection perpendicular to the dislocation line, and 2) Markedly dotted appearance of dislocation images in the [110] reflection. Observation of paired-dislocations of Burgers vector of a/2<111> has also been reported by Head et al. (48). It has been suggested that these like dislocations of Burgers vector a/2<111> together may constitute a superlattice dislocation of a<111> separated by a small region of disordered material. Head et a1. (48) have reported that the slip system <010> {101} is also operative in some cases in beta brass. Dislocations of Burgers vector a{100> were found to be glissile on {100}. 30 11-0- Eipite_§lemept_netbgg An important part of the present investigation involves using a continuum mechanism approach with the help of Finite Element Method (FEM) analysis to analyze the elasto—plastic deformation behavior of alpha-beta brass. FEM analysis was used in this present investigation to determine the stress-strain levels in the alpha and beta phases and in regions near the boundary during loading. FEM is an approximate numerical method which can be effectively used for solv1ng various problems of stress analysis in engineering (12). The solution of a continuum mechanics problem is approximated by polynomials on simple areas called ”Finite Elements“. The area of the problem is diVioed into these finite elements. The values of the solution at the nodes of the finite element mesh. are determined by solving a large system of linear algebraic equations, assembled into a matrix operation. These equations represent physical quantities or parameters speCific to the given problem. The division of the total area into elements is fairly arbitrary and each element may have different material properties. linear or non-linear. With the necessary input data, FEM can be used to study mechanical behaVior of multi-phase alloys, provided the stress-strain behavior of the indiViqual phases are known or can be approximated. Zienkiewiez (53) has given a thorough treatment of the theory and application of FEM. The use of FEM to investigate metallurgical problems 31 has been described by Swedlow and Smelser (54) as an “opening in the traditional wall between mechanicists and metallurgists." One of the first so called “openings in the wall" was made by Fischmeister et a1. (55). They used FEM to study plastic deformation of two-phase materials. They applied FEM analysis to plain-carbon (0.21 percent carbon) steel and calculated the stress-strain curves for different hardness ratios of the ferrite and martensite phases. They studied deformation in the indiViOual phases by uSinq a two-dimensional “plate“ model Of the ferrite-martensite microstructure. This model was obtained by subdividing each phase region of the microstructure into triangular elements. The deformation of all the phase regions was calculated for a plane strain and linear work-hardening condition in both phases. Both phases were assigned identical values of Young’s modulus and work-hardening rates. Sundstrfim (56) studied the Ielastic-plastic behavior of tungsten carbide-cooalt (WC-Co) composite using FEM. The aim of his investigation was to ascertain whether FEM could be applied to predict elastic-plastic behaVior of a two-phase material. His investigations showed that continuum mechanics applied to a two-dimensional model of real microstructure comprising of 20-30 grains gives a reasonable agreement with the experimental stress-strain curves at low stress levels. However. he concluded that the plastic strain distribution calculated by FEM is very inhomogenious at the microscale. 32 Secondly, FEM cannot give a very high resolution on the microscale of stress and strain fields in the phases. The reason for it, according to Sundstrfim, lies in the computational limitations leading to a relatively coarse sub-diVision of the model into elements. Mean values of the field quantities over the critical region, however, gives information about the interaction between phases. According to Swedlow and Smelser (54), FEM analysis tends to be “overstiff". It generates low strains for a given stress level or excessive stress at a given strain level. From the theory of elasticity it is known that there is a Similarity in the solutions of two and three-dimensional mechanics problems with equivalent geometry. This resemblance can also be found in the plastic range if one considers the two-dimensional problem under plane strain condition (12). Physically, the assumption of plane strain means that there is a strong constraint to the plastic flow perpendicular to the plane of the two—dimensional plate. Therefore, it is assumed that the two-dimensional model gives a fair description of the real three-dimensional structure. All materials are anisotropic on the grain size level. FEM assumes isotropy for all the phases. However, if there is a large difference in the elastic moduli values between the phases, the influence of anisotropy of the phases is then diminished, thus justifying the assumptions of isotropy. Also, the FEM model is not the truest 53 representation of the volume fractions and the distribution of phases in the bulk material. Limitations in the calculation capaCity determine the number of elements available for the model. Karlsson and Sundstrbm (57) used FEM to study inhomogeneiety in plastic deformation of two-phase steels. In their investigation, the FEM analysis was performed on the ferrite and martensite microstructure of an alloyed low-carbon steel using constant strain elements. Their investigations revealed that a very inhomogenious strain distribution exists in two-phase structures. The inhomogeneity is strongly dependent upon the hardness ratio of the two phases. Maximum strains were found to appear away from the interfaces. Margolin at 31. (12.13.14) published a series of three papers on deformation studies on alpha-beta Ti- 8Mn alloy using FEM. The first of these studies, performed by Jinoch. Ankem and Margolin (12). analyzed stress-strain curves of alpha-beta Ti- 8Mn alloy uSinq stress-strain curves and approx1mate volume fractions of each phase. Stress distributions in the two phases were numerically calculated in their studies using an IBM 360/65 computer by means of a finite element NASTRAN program developed by NASA. They used a uniform mesh of 392 triangular elements for the FEM analysis. Sub-areas of alpha phase were located by projecting a photo-micrograph onto a graph of the size of the entire mesh. The alpha particle shapes were idealized to make calculations simpler while 34 maintaining volume fractions of alpha phase constant at 17 percent. Their investigations revealed that the FEM calculated stress-strain curve lies below the experimental one. The reason for this is attributed to the fact that the beta grains and the alpha particle sizes in the Ti -8Mn alloy were smaller than the single phase alpha and beta grain sizes in the alloy whose stress-strain curves were used in the FEM calculations. Secondly, the role of the interface could not be discarded. or ignored, in a two-phase alloy. Jinocn et a1. (12) also investigated stress distribution in the alpha-beta phases of the fi -8Mn alloy. Stress gradients were found to be present in both phases. In the beta phase, stresses were higher near the interface. and in some instances, this was found to be true in the alpha phase as well. For stresses below those producing plastic strain. the strain in alpha phase were higher than in the beta phase. Ankem and Marquin (13,14) have presented a two-part analysis of FEM calculations of stress-strain behavior of alpha-beta Ti- BMn alloys. Part I dealt with the determination of stress-strain relations and Part II with the stress-strain distributions. They used the NASTRAN computer program (developed by NASA) to calculate the effect of particle size, matrix and volume fractions on the. stress-strain relations of alpha-beta ri- 8Mn alloys. For a given volume fraction, the calculated stress-strain curve was higher for a fine grain size than for a coarse grain 35 size. When the stronger beta phase was the matrix, stress-strain curves reached higher stress values. They also concluded that the difference in the calculated and experimental curves diminished at higher strains. Their investigations did not consider anisotropy of alpha and beta phases either in their elastic or plastic state. Also, no distinction was made between slip behavior in _eguiaxed and widmanstatten alpha phase. Investigations also revealed that the average strains in the alpha phase were higher than the average strains in the beta phase. Values 0f strains were higher in the alpha phase than in the beta phase. The maximum value of strain in the alpha phase appeared at the center bf the particle away from the constraints imposed by the beta phase. Maximum strains in the beta phase appeared at the alpha-beta interface. The longitudinal stresses, in the tensile direction, were higher in beta phase. In this work, the above mentioned investigations by Margolin and associates have been used as a baSis for the FEM analysis of alpha-beta brass (12,13,14). The theoretical and experimental procedure used is described in detail in the following chapter. III. EXPERIMENTAL PROCEDURE III-A- Iceosmissien Electron Hicresseeis_§tsdx of the Deformation Structure Large grained polycrystalline specimens of alpha-beta brass were used for the present study to aid in the TEM observations. The large grained specimens used in this study were prepared by using the following procedure. The available stock was rolled down to a thickness of about 1mm using a cold-rolling mill. Tensile specimens of 15mm gauge length and 7mm width were cut from them. These tensile specimens were then annealed for i-hour at 425 degrees Celsius under a charcoal cover to relieve any prior stresses induced in the material. The annealed tensile speCimens were then subjected to a strain-annealing treatment to obtain a large grain size of about 0.5-1.0mm for this study. The annealed specimens were first deformed in tension to a critical strain of 8 percent, using a micro-tenSile testing device attached to an Instron machine. These strained specimens were then heat-treated at 825 degrees Celsius for 72-hours in evacuated quartz tubes to obtain the large grain size. These large grained heat-treated specimens of alpha-beta brass were then subsequently strained to the order of 5 percent and 8 percent plastic strains, using the micro-tensile testing device mentioned earlier. The strain levels were chosen in order to induce definite amounts of plastic strain in the specimens. These deformed specimens were then chemically 36 37 thinned down to 0.20mm using a solution of 60 volume percent distilled water and 40 volume percent nitric acid. The thinned speCimens were washed with methanol. Three millimeter diameter discs were then cut out from the gauge length of the thinned speCimens, using a very sharp pair of scissors to minimize the damage from the cutting operation. The central portion of the 3mm discs were made transparent to an electron beam by uSing the following two-step polishing process. The first step, the conditions for which are given in Table 3, is known as the ‘Dishing Technique’ (58). It was used to produce a preferential polishing in the central region of the specimen using a funnel-shaped apparatus shown in Figure 3. After polishing the 3mm specimen for abut one-half minute, the specimen was turned over and polished on the other side as well. The ‘Dishing Technique’ or ‘Jet-Polishing’ created a shiny disc-shaped concave surface in the central part of the specimen. The main aim of produCing concave-shaped surfaces on both sides of the specimen was to produce conditions suitable for causing perforation in the central region of the speCimen. during the second step of the polishing process. In the second step. the ‘dished’ specimen was held between Circular platinum loops as shown in Figure 4 and electro-polished under conditions described in Table 4. The cathode during this electro-polishing step was a cylindrical stainless-steel sheet with slots cut out of it. 38 Figure 3. Schematic of ‘Jet-Polishing’ unit. (by courtesy of S. Shekhar, MMM, Michigan State University). polishingsss solution To power ‘‘‘‘‘ "T‘T‘\§ supply Stand Specwmer .__ vTo power supply Platinum ' Petridish wire ; L2, I Y7 with a hole Reservoir / I I I l I I I l _-_ __._----..- Up FIGURE 3. 39 TABLE 3 ELEEECQQQLLEOIDQ_Q§§§_iQC_:QL§DLQQ_I§EDOIQQ§:s CompoSition of Electrolyte Voltage Temperature 4 parts of Methyl Alcohol 50 volts 243 °k plus 1 part of Nitric Acid (d.c.) (by courtesy of S Shekhar. MMM, Michigan State University) 40 Figure 4. Schematic of the final electropolishing unit. (by courtesy of S. Shekhar, MMM, Michigan State University). To cathode._ Fr——*rlo anode \ L -- - ..... «< ------- .. - Electrolyte Stainless steel sheet ”*_+" T' 5,. with slots [WE , cut out _. ”j,,s»—Platinum wire a ‘w’rj,,ssrs" loops Specimen FIGURE 4. 41 TABLE 4 Elsetceeelisnip9-Dsts-tec-Eipsl_§tep 9i_Ibio_EQil-Eceeacstieos Composition of Electrolyte Voltage Temperature 4 parts of Methyl Alcohol 8 volts 243 K plus 1 part of Nitric Ac1d (d.c.) (by courtesy of S Shekhar, MMM, Michigan State University) 42 A pointed light source was used to determine when perforation occurred in the central portion of the spec1men. The final electro-polishing time was about 1-3 minutes, and as soon as the hole was formed in the center of the specimen. the speCimen was removed from the electrolyte and washed thoroughly in running methanol for at least 7-10 minutes. These large grained specimens of alpha-beta brass were then examined in a conventional Transmission Electron Microscope, operated at 100KV. III‘B- tidite_Elemeot_edslxsi§ The numerical calculations for the FEM analysis were carried out on the PRIME 750 computer system by means of the ANSYS Engineering Analysis Computer Program developed by Swanson Analysis Systems, Inc. The earlier part of this section deals with the important features of the ANSYS Computer Program. A description of the procedure adopted to analyze the stress-strain relationships and stress-strain distributions in alpha-beta brass uSing the ANSYS program is presented in the latter part of this section. The analysis capabilities of the ANSYS Computer Program (59) include: static and dynamic analysis, elastic, plastic, creep and swelling, buckling and deflection, study and transient state heat transfer, and fluid and current flow. The program is capable of analyzing two- and three-dimensional frame structures, piping systems, two-dimensional plane and axisymmetric solids, 45 three-dimensional solids. flat plates. shell structure and other non-linear type problems. The ANSYS Computer Program uses a ”WAVE FRONT“ (or FRONTAL) direct solution method (59). It can give a highly accurate result in a minimum amount of computer time. The number of equations that are active after any element has been processed during the solution is called the "wave-front“ at the point. The present problem at hand is solved uSing three phases: 1) Pre-proceSSing phase, 2) Solution phase. and 3) Post-processing phase. The ANSYS ore-processor routine contains powerful mesh-generation capabilities, as well as, capabilities to define the material properties, material constants, and loads. The solution routine formulates the element matrices. performs the overall matrix triangularization and also computes the displacement, stress and other field parameters. The post-processing routines can plot geometries, stress contours, temperature contours, stress-strain curves, time-history graphs. The ANSYS element library (59) offers 95 different element types to carry out a wide range of engineering analyses. For the stress-strain analysis of alpha-beta brass, element type 42, termed as STIF 42, was used (59). STIF 42 is used for problems involving two-dimensional isoparametric problems. ‘STIF 42’ (59) is used for two-dimensional modeling of 44 solid structures. The element can be used as a biaxial plane element (plane stress or plane strain) or as an axisymmetric ring element. The element is defined by three or four nodal points having two degrees of freedom at each node, i.e. translations in nodal X- or Y-directions. The STIF 42 element (59) has plastic, creep, swelling, stress stiffening, and large rotation capabilities. The solution printout from a full execution run using the ANSYS program (59) consists basically of two parts: 1) Nodal solution, and 2) Element solution. The Nodal solution gives the displacements at each node of the finite element mesh. The element solution gives the normal stresses in each element, the three principal stresses. plastic and elastic strains in each element and a large amount of other additional stress-strain data. For plastic analysis, the non-linear properties of the material have to be defined (59). All materials are assumed to be isotropic. ANSYS uses the Initial Stress Approach (59), also described as the Residual Strain Method for analyzing plasticity effects. This procedure defines a reference elastic material stiffness with corresponding reference elastic strains. Any departures from linearity are treated as initial strains. The Initial Stress Approach uses an Iterative Solution Technique with a constant triangularized stiffness matrix and a changing load vector (59). Yielding is based on the Von-Mises 45 yielding criterion. The triangularized matrix is automatically reused for each iteration. Six different types of loading behaviors can be selected. For the present analysis, the material behavior under loading was assumed to be non-linear elastic. This loading behavior is conservative. i.e. non-path dependent. It is useful for materials undergoing only loading, and no unloading. The materials under this loading behavior show no hysterysis effect. To generate a non-linear stress-strain relationship. the Young’s modulus. the Poisson’s ratio. five strains in the non-linear region of the stress-strain curve, and the corresponding stresses for the five strains defined earlier, have to be specified (59). The ANSYS program assumes the stress-strain curves to be a function of temperature, i.e. the stress-strain curves are different at different temperature levels. If the temperature factor is constant, then one has to define the same stress-strain curve as if they were obtained at two different temperatures. Linear interpolation is used between the points specified on a curve and between curves of different temperatures. This part of the present section describes the experimental procedure used to analyze the stress-strain relationships in alpha-beta brass. A uniform ‘mesh’ of 60 triangular elements was prepared for the analysis. The FEM analysis was performed for three different volume percents 46 of beta phase and three different grain sizes as shown in Figures 5, 6. and 7. The beta particles were distributed uniformly in the mesh while maintaining the volume fraction constant at the appropriate level. The shapes of the particles incorporated in the mesh were not the same as those in the actual speCimen. This difference could influence the calculated stress-strain behaVior. Meshes in Figures 5, 6. and 7 were used for the calculation of the overall stress-strain curves. For calculations of the stress-strain distributions, the beta particles in the mesh were incorporated into a single particle while maintaining the volume fraction constant. The stress-strain curve for alpha brass was based on the stress-strain curve of brass containing 70 weight percent copper and 30 weight percent zinc. The stress-strain curve for beta brass was extrapolated from the available values of Young’s modulus, yield stress and the corresponding strain value at the yield stress level (60). The input stress-strain curves are shown in Figure 8. 1n the meshes shown in Figure 5, 6, 7, node ‘J’ was conSidered fixed, and all the other nodes along 38 could move in the x-direction only. All the other nodes in the mesh could move in both x- and Y-directions, except the nodes along CD. which had the same Y-displacement at any stage of deformation. For generating the stress-strain curves, the applied stresses were known for each loading condition. The corresponding strains were calculated from 47 Figure 5. FEM mesh for alpha-beta brass containing 20 volume percent beta phase (beta phase is shaded dark): a) Fine grain distribution. b) Medium grain distribution. 48 Figure 5 (cont’d.). c) Coarse grain distribution. FIGURE 5 (cont'd.) 49 Figure 6. FEM mesh for alpha-beta brass containing 40 volume percent beta phase (beta phase is shaded dark): a) Fine grain distribution. b) Medium grain distribution. a a“! "a e x "r ‘fu \n 0 Figure 6 (cont’d.). c) Coarse grain distribution. FIGURE 6 (cont'd.) 51 Figure 7. FEM mesh for alpha-beta brass containing 60 volume percent beta phase (beta phase is shaded dark): a) Fine grain distribution. b) Medium grain distribution. 52 Figure 7 (cont’d.). c) Coarse grain distribution. 53 Figure 8. FEM input stress-strain curve of ‘Alpha’ and ‘Beta’ brasses. 600 KBHDSS STRESSlMPB-l 400 "”,,2_ // a-Bross zoo /////// Younos Moouws IlMPa) Ea iLOxIJ E - 15-15 _ 5 YIELD STRESS: a - no Mh ~ 152 MPa 0 T B T T o 4 5 a STRAIN. PERCENT FIGURE 8 . 54 the common displacements of the nodes on the line CD and the initial length AC, i.e. values of strain 6y. corresponding to the prescribed values of stress 0", were calculated as a weighted average of load displacements in the nodes along CD divided by AC. Stress-strain distribution plots were made by using meshes like the one presented in Figure 9. As mentioned earlier. all the beta particles in the mesh were incorporated into a single particle, keeping the volume fraction constant. Stresses on adjacent triangular elements, i.e. elements sharing a square in the mesh, were averaged to obtain stresses at the center of each square on line 88. Strains were calculated from the displacements in the Y-direction of two successive nodes, one of which lies below the line GE, and the other above the line 36, shown in Figure 9. Accordingly, seven strain values and six stress values were calculated for each case. Meshes in Figures 5, 6, and 7 could be considered to be a two-dimensional representation of a three-dimensional cylindrical test specimen. The meshes correspond to the upper half of the longitudinal section through the center of the specimen (51). Thus AB is the diameter of the tensile specimen and CD is the end of the upper half of the gauge length. It has been suggested that this two-dimensional representation can reasonably represent a three-dimensional behavior because the field equations in both cases have the same format (12). Stress-strain curves 55 Figure 9. FEM mesh, for alpha-beta brass containing 20 volume percent beta phase (shaded dark), used for calculation of the stress-strain distributions, across the alpha beta interface. FIGURE 9 . 56 were calculated by FEM for three cases. i.e. 20. 40 and 60 volume percent of beta phase. For each volume percent of beta phase. three different grain sizes were analyzed as shown in meshes in Figures 5, 6. and 7. Stress-strain distribution plots for the selected volume percents of beta phase and the three different grain sizes were also calculated. The detailed results of the FEM analysis are presented in the following chapter. IV. RESULTS AND DISCUSSION lV-A. 1§d_eoelysis_et-tbe_9eiecmetieo_§tcestuce ip-elpbe:§ete-§cess A detailed TEM analySis of the deformation structure in the two-phase alpha-beta brass could be diVided into the following categories: 1) The analysis of the deformation structure in the alpha phase and the beta phase. 2) The deformation structure hear the alpha-alpha and the beta-beta grain boundaries. and 3) The deformation structure near the alpha-beta phase boundary. This study mainly focused on the analysis of the deformation structure in the alpha phase of the two-phase alpha-beta brass in the interior of the grain and hear the alpha-alpha grain boundary. The results of this analysis are compared to prior investigations carried out on single phase 70Cu-302n alpha brass by Karashima (10.11) to check whether the deformation structure in the alpha phase of the two-phase alpha-beta brass is similar to the deformation structure in a single-phase alpha brass. a) Overall Deformation Structure —-———-—--_--——--—-—u.-— 60Cu-402n brass were deformed to two different strain levels. The two strain levels chosen were 5 and 8 percents. The first part of this subsection discusses the 57 58 deformation structure in specimens deformed to 5 percent strain level. The deformation structure in specimens strained to 8 percent strain level will be discussed in a later part of this subsection. In the second part of this subsection, observations relating to dislocation interaction with the boundary will be presented. The deformation structure in the alpha phase, in specimens deformed to 5 percent strain level is presented in Figures 10, 11 and 12. It can be seen from Figures 10 and 11 that the dislocations in specimens deformed to 5 percent strain level were confined to definite single slip planes. No cross-slip was observed in specimens strained to this strain level. Dislocation tangling and complex dislocation structures were absent under these conditions. Another feature observed was pinning of the dislocations in some cases, as can be seen from Figure 12. With increased degree of deformation to 8 percent strain level dislocations present in two or three slip systems interacted an caused dislocation tangles. This resulted in complex dislocation structures such as those presented in Figures 13. 14 and 15. In specimens that were deformed to 8 percent strain, slip on intersecting slip planes was also observed. Qislesetieo-lotecection With ths_§euodec11 Observation of the deformation structure in specimens that were deformed to 5 percent strain level presented in Figure 16(a), revealed dislocation pile-ups at the boundary. Absence of 59 Figure 10. Deformation Structure in the alpha phase of the alpha-beta brass deformed to 5 percent strain level. FIGURE 10. Figure 11. Deformation Structure in the alpha phase of the alpha-beta brass deformed to 5 percent strain level. FIGURE. 11. 61 Figure 12. Deformation Structure in the alpha phase of the alpha-beta brass deformed to 5 percent strain level. GURE 12. F4 is 62 Figure 13. Deformation Structure in the alpha phase of the alpha-beta brass deformed to 8 percent strain level. FIGURE 13. 63 Figure 14. Deformation Structure in the alpha phase of the alpha-beta brass deformed to 8 percent strain level. FIGURE 14. 64 Figure 15. Deformation Structure in the alpha phase of the alpha-beta brass deformed to 8 percent strain level. FIGURE 15 . 65 progress of slip through the boundary suggested that this strain level was not suffiCieht enough to cause the dislocations to overcome the resistance to slip propagation provided by the boundary. However, in specimens that were deformed to 8 percent strain level dislocations piled-up at the boundary. initiated slip in the adjacent grain. This can be seen very clearly in Figure 16(b) and 16(c). Three models have been proposed to explain the initiation of deformation in the adiacent drain due to interaction of slip in one grain with the grain boundary. The stress concentration created due to the pile-up at the boundary either, 1) Forces the leading dislocations of the pile-up to move into the adjacent grain (39). or 2) Forces the boundary dislocation sources to multiply and cause dislocation motion into the adjacent brain (40), or 3) Activates some of the dislocation sources in the adjacent grain (41). Slip nucleation at the grain boundary has been observed in Si-Fe, beta-Ti, Cu. Beta brass, alpha-Ti, Ni and 304 stainless steel (40). Margolin et a1. (40) have studied grain boundary nucleation of slip in beta Ti- 8Mh alloy. According to Nabarro (41). a slip band cannot cross from one grain to another without irregularity unless: 1) The slip direction is the same in both Figure 16. Deformation structure in the alpha phase of the alpha beta brass near the boundary deformed to: a) 5 percent strain level. FIGURE 16 . 67 Figure 16 (cont’d.). b) and c) 8 percent strain level. , ‘x 5133. a _‘ f. .4" %/‘ 0.15M") FIGURE 16 (cont'd.) 68 the grains. and 2) The chaice of the crystal plane for slip is not critical in the adjOinihq grain. According to Nabarro (41), when the slip directions are different in the adjacent grains, it is very difficult for a dislocation to pass from one grain into the adjacent grain as the stress needed to drive the dislocation through the grain would be of the order of the theoretical shear strength of the material. Hence. as a result of the difficulty in forc1ng a dislocation from one grain to another, Cottrell (41) suggested that. slip does not have to spread from one grain into the neighboring grain. Instead. the stress concentration created by the dislocation pile-up at the boundary may induce sources of dislocations. in the neighboring grains. to become active (41). In this case, the dislocations in the adjacent grain could be initiated from a source which is some distance away from the boundary. Such a feature can be observed in Figure 16(c). wherein the dislocations are seen to be present not very close to the boundary, but at a distance away from the boundary. Karashima (10) has performed similar TEM analyses on 70Cu-302n alpha brass to study the deformation structure in this material. The results of his study can be summarized as follows: 1) ln specimens that were strained from 1-5 percent dislocations were confined to slip planes, whereas 69 specimens that were strained from 5-10 percent revealed complex tangled dislocation structures. 2) Dislocation arrays piling up against the boundaries without slip propagation in the adjacent grain were observed at small strain (1-5 percent). On the other hand, continuation of slip across the boundaries was observed at higher strain levels (5-10 percent). According to Karashima (10), the dislocation arrangements in a material, vary according to the degree of working. Thus, the work-hardening mechanisms during the various stages of deformation in turn are affected by the deformation structure in the material (10). The work-hardening, during the early stages of deformation, is mainly based upon the back stresses created due to the piling up of dislocations at the boundary (10). However, the formation of complex tangled dislocations and vacancy jogs is the main factor for the work-hardening in the later stages of deformation. Formation of such complex sessile dislocations is a prominent characteristic observed during deformation of alpha brass. However, in materials with a high stacking fault energy like beta brass or aluminum (10), most of the dislocations introduced during cold-working are confined to single slip planes, and thus formation of complex tangled dislocation structure is not a very common feature in such materials. On comparing the results of the present study with the 7O investigations on single phase 7OCu-3OZn alpha brass by Karashima (10), one can notice a close similarity between the deformation behaVior in the alpha phase of the two-phase alpha-beta brass and the single phase alpha brass. b) autos:s-yes:9c-apelxsi§_ei_91§lesetloos peac_tbe_§Qupdacy Burgers vector analysis of dislocations near the boundary was performed to determine the character of dislocations in their relaxed configuration. The analysis also prov1des details of the active slip plane and the active slip direction. A set of dislocations in the alpha phase near a boundary with different diffraction vectors are presented in Figure 17. There are two types bf dislocations in this set. marked as ‘A’ and ‘8’. Burgers vector anaIVSIS of the dislocations showed that both types of dislocations ‘A’ and ‘B’ were of mixed character. Dislocations of type ‘A’ have a Burgers vector of aIZLITUJ and that of type ‘B’ have a Burgers vector of a/2E01I]. The results bf this analysis are tabulated in Table 5. Another set of dislocations present in the alpha phase near a boundary with different diffraction vectors is presented in Figure 18. This set of dislocations, also has two types of dislocations marked as ‘C’ and ‘D’. Burgers vector analysis showed that the dislocations of type ‘C’ are mixed in nature. whereas that of type ‘D’ are screw in nature. The Burgers vector of dislocations of type ‘C’ is a/2E1IOJ, 71 Figure 17. Dislocations near the boundary in the alpha phase of the alpha-beta brass with different diffraction vectors: a) [111] b) (002] .5. 9 FIGURE 17 . Figure 17 (cont’d.). c) d) om! 72 [2023 [111] FIGURE 17 (cont'd.) .i 73 TABLE 5 BessIts_9i-tbe_§ucgscs-2estec finalisis_gi_91slecetigos In Figure 17- Type of Burgers Slip Dislocations Vector Plane Direction Character A £1T03 (111) [21?] Mixed B [01?) (111) [T103 Mixed 74 Figure 18. Dislocations near the boundary in the alpha phase of the alpha-beta brass with different diffraction vegtors: a) 5 [111] o) 6 L002] FIGURE 18 . Figure 18 (cont’d. ) .~ (2) d) 0ND. 75 [2053 c0201 FIGURE 18 (cont'd.) 76 TABLE 6 825!l£§-Qi-§D§_§QC9§£§-!§E£QC fiflél¥§l§-Qi-Ql§LQEé£LQD§-lD-ElQQ£§_l§; Tvpe of Burgers Slip D1slocatlons Vector Plane Dlrect1on Character c [130] (111) [10?] Mixed D L10?) (111) L10?) Screw 77 whereas that of type ‘D’ is a/2ElOTJ. The results of this analysis are tabulated in Table 6. Another set of dislocations in the alpha phase near a boundary are seen in Figure 19. Dislocations that occur as pairs (hence+orth referred to as "Dislocation—Pairs“) are clearly ViSible in these figures. These pairs are indicated by arrows. Dislocations in this set. as marked as ‘E’. are screw in nature with a Burgers vector of a/ZEIOTJ. Table 7 gives the details of the Burgers vector analVSis of these dislocations. The "Dislocation-Pairs" observed in Figure 1? are discussed in more detail in the following subsection. All the specimens used for the Burgers vector analysis were deformed to a 5 percent strain level. c) Boélxsi§_9i_:Qi§ngétiQn:Eéicsi An important observation made during the TEM analySis was the presence of "Dislocation-Pairs“ in the alpha phase of the alpha-beta brass. These "Dislocation-Pairs" were observed to occur successively on a slip plane. Karashima (11) in his investigations on alpha brass observed Similar dislocation pairs in deformed alpha brass. He discounted the possibility that they could be double-images of dislocations produced due to the TEM contrast effects. His analysis is based on the observation that some o+ the dislocations produce single dislocation images. whereas their neighbors possesSing the same 78 Figure 19. Dislocations near the boundary in the alpha phase.of the alpha-beta brass with different diffraction vectors: a) 6 [T113 b) 6 L002] FIGURE 19 . Figure 19 (cont’d.9. c) d) 79 .11?) FIGURE 19 (cont'd.) 80 TABLE 7 Beault§-9i_tbe_§gcggc§-!e§tgc enslxsi§_9i_9islgcatigu§-iu-Eiguce-12; Type of Burgers Slip Dislocations Vector Plane Direction Character E [10?] (111) [1073 Screw 81 character appear as "double images". Such an observation cannot be explained on the basis of TEM contrast effects. Karashima (11) has considered two possibilities to explain the occurance of these "Dislocation-Pairs”. One suggestion is that these pairs are superlattice dislocations, however, as he has pointed out, presence of superlattice dislocations or long-range order has not been reported in alpha brass (11). On this basis, the double images of dislocations observed in the alpha phase of the alpha-beta brass cannot be due to superlattice formation. Another suggestion by Karashima is that these are unit dislocations moving in pairs: the second one pushing the first in order to overcome short-range order hardening. Seeger (61) has originally proposed such a theory for Ag-Au binary alloys. Karashima considered this hypothesis to be a possible explanation for explaining the occurrence of "Dislocation-Pairs" because short-range order hardening has been reported in alpha brass (11). In the present investigation, Burgers vector analysis was performed on such paired dislocations observed in Figure 19. This analysis showed that these ”Dislocation-Pairs” possessed screw character. So they may be dislocations that have cross-slipped onto parallel slip planes due to dislocation pile-up stresses that arise at the boundary region. The results of the Burgers vector analysis performed on such "Dislocation-Pairs" are tabulated Table 7 (refer to the previous subsection). It can be observed from Figure 19 that the dislocations occur in pairs only near a boundary, 82 whereas, the dislocations present in the same slip plane at a distance away from the boundary are single. Based on this observation, the formation of "double-images” could be explained in the following manner. As dislocations approach a boundary, their movement is inhibited by the back stresses created due to dislocation pile-up at the boundary. To overcome this resistance created due to the pile-up of dislocations. the screw dislocations double cross-slip onto parallel slip planes. In the relaxed state they will arrange themselves into a minimum energy configuration (62). These cross-slipped screw dislocations will appear as pairs when observed in the TEM under certain tilt conditions. The above mentioned hypothesis could also explain why these dislocations occur in pairs only near the boundary, and not away from it. At locations away from the boundary, the dislocations do not have to cross-slip due to the absence of the pile-up stresses. Hingwe et a1. (2) and Nilsen et a1. (4) have observed similar cross-slipping phenomena in the alpha phase of bicrystals of alpha-beta brass. Optical and Scanning Electron Microscopy analysis performed during their investigations revealed that slip lines in the alpha phase, when approaching the alpha-beta phase boundary result in cross-slip or multiple-slip to avoid crossing the phase boundary. Subramanian (62) has proposed that screw dislocations, after cross-slipping onto parallel planes, arrange themselves in a minimum energy configuration in the relaxed state. The "Dislocation-Pairs" have also been 83 observed to occur in random fashion among tangled dislocations as can be seen in Figure 20 (indicated by arrows). The hypothesis that these dislocation pairs are screw dislocations that have cross-slipped onto parallel planes could also explain the occurence of these pairs among tangled dislocations, because tangled sessile dislocations hinder the motion of dislocations in a fashion similar to the resistance offered to the dislocation movement by grain boundaries. Further studies need to be performed to clarify this proposed model for the occurence of such "Dislocation-Fairs". d) Etaia_§sungacx_2islgséti9u§ The boundary dislocations observed in the specimens of alpha-beta brass are seen in Figures 21 and 22. These dislocations, as can be noted from the figure are not dislocations in a slip trace, but are present in a boundary. Close examination of diffraction patterns on both sides of the boundary, revealed a slight angular rotation between the diffraction patterns, indicating that these dislocations are present in a low-angle tilt boundary. 2) Iuz§its-2s£9:matisn-§tugiss in a High YQLE§Q§-§lEESEQfl-fliSEQ§EQE§-L§!§EL Apart from TEM studies of the deformation structure in a conventional 100KV Electron Microscope, preliminary investigations of in-sita deformation of micro-tensile 84 Figure 20. "Dislocation-Pairs" observed in the alpha phase of the alpha-beta brass. 20. FIGURE 85 Figure 21. Boundary dislocations observed in the alpha phase of the alpha-beta brass. . rm mmDOHrn. 86 r1 r‘. Figure se. Boundary dislocations observed in the alpha phase of the alpha-beta brass. . mm 3:54“; 87 specimen of alpha-beta brass in an HVEM were carried out. Figures 23 and 24 are those of dislocations that were in motion during zn-sica deformation in an HVEM. These still photographs were taken from a video-tape on which the dislocation motion was recorded. Dislocations moving in a slip trace are seen in Figure 23 (marked as ‘F’ on the photograph). Dislocation pile-ups (marked as ‘6’) and dislocations moving on intersecting slip planes (marked as ‘H’) during in-sztu deformation in an HVEM are presented in Figure 24. IV-B- E§d_eoalxsis-Qi_§tcss§:§tcain_§sbaxigc Qi_eleba:§sta_§:ass The results of the present FEM analysis could be divided into the following categories: 1) Analysis of the stress-strain relationships for different volume percents of beta phase, 2) Analysis of the stress-strain relationships for different grain sizes of alpha-beta phases, 5) Analysis of the stress-strain distributions across the alpha-beta interface, and 4) Comparison of experimental and FEM calculated stress-strain curves. a) §tcssszstcaiu-leatigusbins-i9c_Qiiiscsut_!Qlues Percents of Beta Phase FEM calculated stress-strain curves of alpha-beta brass for three different volume percentages of beta phase (20, 40 and 60) are illustrated in Figure 25. As can be observed 88 Figure 23. Still photographs of dislocations in motion in the alpha phase of the alpha-beta brass during 2n-szta deformation in an HVEM. FIGURE 23. 89 Figure 24. Still photographs of dislocations in motion in the alpha phase of the alpha-beta brass during in-sita deformation in an HVEM. FIGURE 24 . 90 from this figure the stress-strain curve for 60 volume percent beta phase attains higher stress levels for similar strain levels, as compared to the stress-strain curve for 40 volume percent beta phase (in the plastic region). Similarly, the stress-strain curve for 40 volume percent beta phase, in turn, reaches higher stress levels for a given strain level. when compared to the stress-strain curve for 20 volume percent beta phase (in the plastic region). The grain Sizes and the phase distributions of beta phase used in the meshes for the calculation of these stress-strain curves are shown in Figures 5(a), 6(a) and lie). HS can be observed, all three curves deViate from linearity at a Stress level of about 110-150 MPa. Although the exact yield stress level of each curve is difficult to determine, an approximate estimate of the value could be obtained by using the 0.2 percent offset method for Yield p01nt determination. The yield stress values from the three stress-strain curves, Obtained by using the "Offset Method“ are tabulated in fable 8. These values compare very well with the published handbook values of yield stress for alpha-beta brass (refer to Table 1). b) Stress-Strain Relationships_fg:_giffggggt _ -----~-._---—.—-- ~‘--—.—--- Stress-strain curves of alpha-beta brass containing 20, 40 and 80 volume percent of beta phase, for different grain sizes were calculated by using FEM analv51s. The stress-strain curves of alpha-beta brass containing 20 91 Figure 25. FEM calculated curves of alpha-beta brass for 20, 40 and 60 volume percents of beta phase. BCXD STRESS(MPBJ ZCXD \\ HDC) 691 UN SIZE~ E VOLUME FERCENT BETA PHASE: I 60% 7 H 40% I“ +- 20% O I I T 1 1 C) C12 C14 C16 ()8 LC) 12 STRAIN.PERCENT FIGURE 25. 92 TABLE 8 Yield_§t:s§§_!alus§-thaiueg-bx-using_tbs_Qiz_Escssnt Qi:set-flstb9g-ttQm_§ut¥ss-in_Eiguts-2§s Volume Percent Yield Curve Beta Phase Stress (MPa) I 20 160 ll 40 175 III 60 210 93 volume percent beta phase for two different grain sizes are presented in Figure 26. It can be seen from the figure that as the grain size gets smaller, the stress-strain curve tends to shift upwards, i.e. for equal strain levels, the stress-strain curve for finer grain size attains higher stress levels than the stress-strain curve for a coarser grain size (in the plastic region). The stress-strain curves at two different grain sizes for alpha-beta brass containing 40 volume percent of beta phase, are presented in Figure 27, and the stress-strain curves at two different Grain Sizes for alpha-beta brass containing 50 volume percent beta phase are presented in Figure 28. In both these figures, it can be opserved that the stress-strain curve for a finer grain size reaches higher stress levels (for a given strain) than the stress-strain curve for a coarser grain size (in the plastic range). One possible explanation for this behaVior according to Jinoch et a1. (12) is that the flow stress, at a given strain level, for a finer particle configuration is higher than for a coarse particle configuration. This is because the strain difference between the alpha and the beta phases is smaller for a finer particle configuration (12). When the particle configuration is coarse, the strain in the softer alpha phase is mucn greater than the strain in the harder beta phase. Whereas, when the particle configuration is finer, the difference in strain levels between the alpha and the beta phases is not so large. Hence, as a result, the 94 Figure 2b. FEM calculated stress-strain curves of alpha-beta brass containing 20 volume percent beta phase for fine (I) and coarse (II) grain size distribution. 300 STRESS I MPG-1 200 \ av 100 I .4 O. 6 I O. STRAIN. PERCENT FIGURE 26. 1.4 95 Figure 27. FEM calculated stress-strain curves of aloha-beta brass containing 40 volume percent beta phase for fine (I) and coarse (II) grain size distribution. 300 ,I/‘/ /_l smessmpm / M / 200 /// 100 cunve 653.2? ‘ 1 J- I -L. 0 T I r . I 0 0.6 o. 1.0 12 1, STRAIN . PERCENT FIGURE 27. Figure 28. FEM calculated stress-strain curves o+ alpha-beta brass containing 60 volume percent beta phase for fine (I) and coarse (ll) grain size distribution. 300 STRESS ( MP5} \\ g 200 100 / GRAIN CURVE SIZE C) I: STRAIN.PERCENT FIGUREAZB. 97 stiffer beta phase undergoes considerably larger strain. Since the beta phase is stiffer. larger strains in beta induce the stress levels to reach a higher value. This,in turn. results in the stress-strain curve for a finer particle configuration to reach higher stress levels (51) than the curve for a coarser particle configuration. The relative grain sizes and the phase distributions used in the meshes to calculate the stress-strain curves are presented in Figures 5. b and 7. c) Stress—Strain Distributions Across the 6l2D§:§sté_lnteciac§ Stress distributions in the alpha-beta brass containing 20 volume percent beta phase. in each element across the line SE in Figure 9, are plotted in Figures 29 and 30. The stress distributions across the phases are calculated in the loading Y-direction and in the transverse X-direction. The stress distributions have been calculated for three different externally applied stress levels, i.e. 207. 241 and 276 "Pa for distributions in the Y-direction, and 172. 207 and 241 "Pa, for distributions in the X-direction. Large differences in the stress levels in both the X- and Y-directions are evident from Figure 29 and 30. For stress distributions in the loading Y-direction, the stresses in the beta phase were found to be considerably higher than the stresses in the alpha phase, as can be observed in Figure 29, for any particular level of externally applied stress. Stress distribution in the 98 Figure 29. FEM calculated stress distribution plot for alpha-beta brass contaiing 20 volume percent beta phase, in the loading Y-direction. 450 18 APPLIEDS'RESS LEVEL a A 207 MPa x 241 MPa .. 276 MPa STRESSIHPal ' " . . GRAIN SIZE“: 300 : : d .\ A A /' .4 \\. .// _‘ ‘\.\‘ ‘//‘ ~\\‘A A’/// 150 «f c: A ‘f *7 f d: s O I I I I I I I I I I I I I I I I I I I I I I I I O 3 6 12 15 DISTANZBpm) FIGURE 29. 99 Figure 30. FEM calculated stress distribution plot for alpha-beta brass containing 20 volume percent beta phase, in the transverse X-direction. +5.0 APPLIED STRESS LEvEL _‘ A 172 MPa _ o 207 MPa STRESSIWBI x 241 MPa _ GRMN wzfl Q1;: O ‘_ A * \z I ‘1 ‘\ .‘—-4 C ‘ _ x———+———x ./// - :\\\\\\\A A///:///. -5-0 \\\3 // -1CLO ./ / I I I I ' I I I I Ij I I I I T I Ij I I I I I T O 3 6 9 12 15 18 DISTMCEtpm) FIGURE 50. TOO transverse x-directions also showed steep gradients across the phases. Stresses were found to be generally compressive in nature in the transverse X-direction at the applied stress levels. as can be observed in Figure 30. The difference in the stress levels between the alpha and the beta phase tended to increase at higher externally applied stress levels. At stress levels of 172 MPa and below. the stresses in the transverse X-direction in the beta phase. were found to be tenSile in nature. whereas the stresses in the alpha phase were compreSSive. This is probably due to a lower Young’s modulus value of the beta phase as compared to the alpha phase. As a result of this. beta phase expands more along the Y-direction as compared to the alpha phase. This in turn will require the beta phase to contract more in the X-direction. The alpha phase resists this and may cause tenSile stresses in the beta phase. At higher stress levels. the beta phase undergoes plastic deformation and experiences compression like the alpha phase. The stress contour distributions in the 60 element mesh used for the FEM analysis are presented in Figure 31 and 32. The stress levels at different locations on the mesh are marked. The stress contour distribution for stresses in the loading Y-direction at an externally applied stress level of 207 MPa is presented in Figure 31. The stress contour distribution for stresses in the transverse x-direction at the same external stress level is 101 Figure 31. FEM generated stress contour distribution plot in the 60 element mesh for alpha-beta brass containing 20 volume percent beta phase, in the loading Y-direction. (The stress levels marked are in MPa). FIGURE 31. 102 -rr- 5 Figure . FEM generated stress contour distribution plot in the 60 element mesh for alpha-beta brass containing 20 volume percent beta phase. in the transverse X-direction. (The stress levels marked are in MPa). [.- t 103 Figure 33. FEM calculated strain distribution plot for alphafbeta brass containing 20 volume percent beta phase. STRAIN . PERCENT RI GRAHJSIZG L5 ’ 1.0 ’//,, X o 207 MPa X 05 T O J J! := :I I I I 3 I I I T' ”III T'T I I I T T O I I I I 3 I I I b 9 12 15 13 DIST/Imam) FIGURE 33. 104 presented in Figure 32. The stress contour distributions shown in Figures 31 and 32 are plotted for alpha-beta brass containing 20 volume percent beta phase for the grain size and phase distributions shown in Figure 5(a). The strain distributions across the phases on line 53 (see Figure 9) is plotted in Figure 33. As can be observed from this figure, the strains in the beta phase were generally higher than the strains in alpha phase at the externally applied stress levels. Strains in the beta phase were higher near the alpha-beta interface as compared to the strains in the beta grain interior. The strain gradients calculated in this analysis, however, were not as steep as those observed by Jinoch et a1. (12) in their FEM investigations of Ti- 8Mn alloy. This difference in the steepness of the strain gradients may be due to the larger difference in the yield stress values of the alpha and beta Ti- 8Mn input curves. as compared to the difference in the yield stress values of the alpha and beta brass input curves used during the present FEM analysis. d) 9909a:1290_Q£-§§Qecimental-Stce§§:§tcain-§ucxe uith the FEM Calculatsg_§ucxs§ A comparison of the FEM calculated stress-strain curve of alpha-beta brass (containing 20 volume percent beta phase: same as in the 60Cu-402n brass) with the experimental one is made in Figure 34. As can be seen from the figure. the FEM calculated stress-strain curve is at a higher level than the experimental curve. One possible 105 Figure 34. Comparison between the experimentally determined stress-strain curve of 60Cu-402n brass and the FEM calculated curve of alpha-beta brass containing 20 volume percent beta phase. 400- STRESS ( MPaJ // 2CHD- I-FEM Ii—EXPERIMENTAL I FEM CURVE__ i 1 GHAHISIZE ;\q/; 0 1 '2 3 4 5 6 7 STRAIN.PERCENT FIGURE 34 . 106 explanation is the following: The FEM input curve for the alpha brass used during the analysis was extrapolated by performing a tensile test on a 7OCu-302n tensile specimen and incorporating the published handbook values of the Young’s modulus and the yield stress into the curve. The FEM input curve for beta brass was extrapolated from published values of Young’s modulus and yield stress (b0). lf these input curves themselves reached higher levels than the experimentally predicted curves for these materials, then the FEM calculated curve of alpha-beta brass would reach higher levels than the experimental stress-strain curve. This is because the FEM calculated curve is calculated from the FEM input curves of the alpha and beta brass. The modulus of elasticity of the FEM calculated curve is about 1.04 x 108 KPa which agrees well with the published handbook value of 1.03 x 108 KPa (see Table 1). The yield strength value of the FEM and the experimental stress-strain curve is about 170 MPa. This value agrees well with the published handbook values of 150-380 MPa (see Table l). The grain-size and the phase distribution in the mesh used for the FEM calculation of the stress-strain curve is presented in Figure 5(a). V. CONCLUSIONS V-A - I§D_§§Qgi.§§ 1) The dislocations in polycrystalline specimens deformed to 5 percent strain level, were confined to definite single slip bands. No cross-slip was observed in specimens deformed to this strain level. On the other hand, in specimens that were deformed to 8 percent strain. dislocations moving in intersecting slip planes tangled with each other forming complex dislocation structures. 2) Extensive dislocation pile-ups were observed at the boundary in specimens strained to 5 percent, and as well as the 8 percent strain level. In speCimens deformed to 8 percent strain level, dislocations piled-up at the boundary initiated slip in the adjacent grain. 3) "Dislocation-Pairs" were observed in the alpha-phase of the deformed alpha-beta brass. The "Dislocation-Pairs“ were observed to occur in pairs near a boundary and not away from it. Burgers vector analysis of such pairs of dislocations indicates that these are SCREW dislocations that may have cross-slipped onto parallel planes and present as pairs in a relaxed configuration under the absence of any applied stress. 107 V-B e 108 Egfliiflélxgi‘é 1) 4) Stress-strain curves obtained for alpha-beta brass containing 60 volume percent beta phase tends to be higher than the one for 40 volume percent beta phase in the plastic range. The stress-strain curve of alpha-beta brass containing 40 volume percent beta phase is in turn higher than the stress-strain curve for a distribution of 20 volume percent beta phase in the plastic range. For a given volume fraction of beta phase, the FEM calculated stress-strain curves with a finer beta phase distribution tend to be higher than those with a coarser beta phase distribution. Stress distributions both in the loading Y-direction and the transverse X-direction show gradients. Stresses in the beta phase were found to be higher than the stresses in the alpha phase in the loading Y-direction. The stresses in the transverse X-direction were generally compressive. However, below the externally applied stress level of 172 MPa, the stresses in the transverse x-direction in beta phase were tensile, whereas those in alpha were compressive. Strains in the beta phase were found to be higher than the strains in the alpha phase at the externally applied stress levels. The strains in 5) 109 the beta phase were generally higher near the alpha-beta interface than in the beta grain interior. The FEM calculated stress-strain curve attained higher stress levels, for similar strain levels as compared to an experimentally calculated stress-strain curve. The values of the yield stresses on both the curves are in good agreement. The Young’s modulus value of the FEM calculated curve agrees well with the experimentally predicted data. 10. 11. VI. REFERENCES A. Hingwe and K.N. Subramanian. ”On Growth of Two-Phase Bicrystals of Alpha-Beta Brass." J. Crystal Growth, 2;, 287, (1974). A. Hingwe and K.N. Subramanian. ”Deformation of Duplex Crystals and Two-Phase Bicrystals of Alpha-Beta Brass.“ J. Mater. Sci., 19, 183, (1975). A. Hingwe. "Duplex-Crystals and Two-Phase Bicrystals of Alpha-Beta Brass -- Growth and Mechanical Properties. Ph.D. Thesis.“ Michigan State University. (1973). C. Nilsen and K.N. Subramanian. I'Deformation of Alpha-Beta Brass with Oriented and Equiaxed Duplex Geometries." phy. stat. 501., 22, 281, (1983). C. Nilsen and K.N. Subramanian. "Role of Strain Rate and Phase Boundary Geometry on Deformation Behavior of Two-Phase Bicrystals of Alpha-Beta Brass." J. Mater. Sc1., 12, 788. (1984). C. Nilsen. "Mechanical Properties of Alpha-Beta Brass -- Studies on Model Systems, Ph.D. Thesis.“ Michigan State UniverSity. (1973). A. Kezri-YaZdan and K. N. Subramanian. "Elevated Temperature Deformation BehaVior of Alpha-Beta Brass Bicrystals, Part 1 -- Ordered Beta." J. Mater. Sci., (in Press). A. Kezri—Yazdan and K. N. Subramanian. “Elevated Temperature Deformation Behavior of Alpha-Beta Brass Bicrystals. Part 11 -- Disordered Beta.“ J. Mater. Sc1.. (in Press). A. Kezri-Yazdan and K. N. Subramanian. "Elevated Temperature Deformation Behavior of Alpha-Beta Brass Bicrystals, Part 111 -- Supporting Studies." J. Mater. Sc1.. (in Press). 8. Karashima. "Electron Microscope Studies of Deformed Alpha Brass. Part I -- General Arrangement of Dislocations." Trans. J.1.M.. g, 134, (1961). S. Karashima. l‘Dislocation Pairs Observed in Alpha Brass." Trans. J.l.M., §, 48, (1962). 110 14. 15. lo. 18. 19. N 1:. 1.1 y... I N M 23. 111 J. Jinoch, S. Ankem, and H. Margolin. “Calculation of Stress-Strain Distribution for an Alpha-Beta Ti-BMn Alloy." Mat. Sci. Eng., 34, 203, (1978). S. Ankem and H. Margolin. "Finite Element Method Calculations of Stress-Strain Behavior of Alpha-Beta Ti-Mn Alloys: Part I. Stress-Strain Relations." Metall. Trans. A., ,39, 595, (1982). S. Ankem and H. Margolin. "Finite Element Method Calculations of Stress-Strain Behavior of Alpha-Beta Ti-Mn Alloys: Part 11. Stress and Strain Distributions.” Metall. Trans. A., 13a, 603, (1982). M. Hansen and K. Anderko. Qgggtigggigg_gi_§;ggcx fillgyg, McGraw-Hill, New York, 650, (1958). Metals Handbook- E:gescties-ang_§slectign-gi matscial§i_xglums_11_§9itiee-§. L- Taylor (50-). American Soc1etv of Metals. Ohio, 1021, (1961). H. Unckel. J. lnst. Metals, 91, 171, (1937), (as cited in reference 6). H. Chao and L. Van Vlack. "Deformation of Oriented MnS IncluSions on Low-Carbon Steel." Trans. Met. Soc. A.I.M.E., 233, 1277, (1965). R.N.K. Honeycombe and w. Boas. Aust. J. Sci. Res., [A], 1, 70, (1948). L.M. Clarebrough and G. Perger. Aust. J. Sci. Res., [A], 3, 72, (1950). L.M. Clarebrough and G. Perger. Aust. J. Sci. Res., 5, 114, (1952), (as cited in reference 6). J. Gurland. “Comparison of Dispersion Hardening in Four Silver-Base Alloys of Equivalent Compositions." Trans. Met. Soc. A.I.M.E., 221, 407, (1961). H. Margolin, P. Farrar, and M. Greenfield. The Jaffe and N. Promisel (Eds.). Pergamon Press, New York, 795, (1970). A. Eberhardt, M. Suerey, and B. Baudelet. "Plane Interface Boundary Sliding in Alpha-Beta Brass.“ Scripta Metall., 9, 1231, (1975). 25. 2b. 27. 29. 30. 35. 36. 112 T. Takasugi, O. Izumi, and N. Fat-Halla. "Activated Slip Systems During Yielding of Alpha-Beta Brass Two-Phase Bicrystals." J. Mater. Sci., 13, 2013, (1978). T. Takasugi, O. lzumi, and N. Fat-Halla. “Plastic Deformation and Fracture Behaviors of Alpha-Beta Brass Two-Phase Bicrystals.” Acta Metall., 2Q, 1453, (1978). T. Takasugi, O. Izumi. and N. Fat-Halla. “Dynamic Observation of Fracture Phenomenon in Alpha-Beta Brass Two-Phase Bicrystals“. J. Mater. Sci., 23, 2462, (1978). T. Takasugi, O. lzumi, N. Fat-Halla. “Operative Slip Systems in Alpha-Beta Brass Two-Phase Bicrystals at 150K." J. Mater. Sci., ,3, 1651, (1979). A. Kezri-Yazdan and K.N. Subramanian. "Role of Phase Boundary Orientation Relative to the Tensile Axis in Deformation of Two-Phase Bicrystals." (To Be Published). N. Fat-Halla. T. Takasugi, and O. Izumi. "On Elastic Incompatibility During the Deformation of Alpha-Beta Brass Two-Phase Bicrystals.“ J. Mater. Sci., 15, 3071. (1980). H. Kawazoe, T. Takasugi, and O. lzumi. "Fatigue of Alpha-Beta Brass Two-Phase Bicrystals." Acta Metall., 22, 1253, (1980). T. Takasugi and O. Izumi. "Interface Sliding in Alpha-Beta Brass Two-Phase Bicrystals." Acta Metall., 22, 465, (1980). Y. Chuang and H. Margolin. “Beta Brass Stress-Strain Relations.“ Metallurg. Trans., 4, 1905, (1973). T. Lee and H. Margolin. “Beta Brass Bicrystal and Tricrystal Stress-Strain Behavior." Metallurg. Trans., 2A, 157, (1977). T. Lee and H. Margolin. “Effective Young’s Modulus in Isoaxial Bicrystals and Tricrystals of Beta Brass." Scripta Metall., ;;, 713, (1977). H. Margolin, F. Hazareh, and H. Yaguchi. "Grain Boundary Contribution to the Bauschinger Effect." Scripta Metall., 22, 1141, (1978). 113 37. H. Yaguchi and H. Margolin. "Grain Boundary Contribution to the Bauschinger Effect in Beta Brass Bicrystals.“ Scripta Metall., ,4, 627, (1980). 38. H. Yaguchi and H. Margolin. "Uniaxial Tension and Compression and Cyclic Stress-Strain Behavior of Beta Brass." Scripta Metall., ;4, 631, (1980). 39. G. OgilVie. J. Inst. Met., (81), 49, (1952-53), (as cited in reference 6). 40. H. Margolin, K. Hashimoto, and H. Yaguchi. "Grain Boundary Nucleation of Slip.“ Scripta Metall., 15, 181, (1981). 41- A- Cottrell- Qislgsatiens-ang_filastic Flow 19 Qcygtaig, Claredon Press, Oxford, 107, (1953). 42. G. Ardley and A. Cottrell. Proc. R. Soc., A222, 328, (1953), (as Cieted in refernce 6). 43. l. Fourie. "The Life History of Individual Slip Lines During the Plastic Deformation of Alpha Brass Single Crystals." Acta Metall., g, 88, (1960). 44. T. Mitchell and P. Thornton. Phil Mag., lg, 315, (1964), (as cited in reference 6). 45. M. Marcinkowski, N. Brown, and R. Fisher. "Dislocations configuration in AuCugand AuCu Type Superlattices." Acta Metall., 2, 129, (1961). 46. N. Brown. Phil. Mag., 4, 693, (1959), (as cited in reference 6). 47. A. Greninger. Trans. A.l.M.E., A22, 369, (1938), (as cited in reference 6). 48 A. Head, M. Loretto, and P. Humble. "Identification of Burgers Vector and Unstable Directions of Dislocations in Beta Brass.“ phy. stat. sol., 29, 521, (1967). 49. P. Humble. "A comparison of Electron Microscope Images in Dislocations based on Isotropic and Anisotropic Elasticity Theory." phy. stat. sol., 21, 733, (1967). 50. A. Head. "Unstable Dislocations in Anisotropic Crystals." phy. stat. sol., $2, 185, (1967). 51. R. Dewitt and J. Koehler. Phy. Rew., ;;6, 1113, (1959), (as cited in reference 50). 55. 56. 57. SB. 59. 60. 61. 62. 114 A. Head, M. Loretto, and P. Humble. "lnfluence of Large Elastic Anisotropy on the Determination of Burgers Vector of Dislocations in Beta Brass by Electron Microscopy.“ phy. stat. sol., 29, 505, (1967). O.C. Zienkiewiez. Einitg Elgmggg_Method in Eggggggcggg_§g;gggg§, McGraw-Hill, New York, (1971). J. Swedlow and R. Smelser. “Discussion on Calculation of Stress-Strain Curve and Stress-Strain Distributions for an Alpha-Beta Ti-8Mn Alloy." Mat. Sci. Eng., 39, 139, (1979). H. Fischmeister, J. Hjalmered, B. Karlsson, G. Lindon, and B- SundstrOM- 101:9-10tecnatioual anieceucs_90 Stcggggh_gf_flgtegi§;§_§nd Alloxg, Institute of Metals and Iron and Steel Institute, Cambridge and London, A, 621, (1973). B. Sundstrom. "Elastic-Plastic BehaVior of WC-Co Analyzed by Continuum Mechanics.” Mat. Sci. Eng., 22, 265, (1973). B. Karlsson and B. Sundstrom. "lnhomogeneity in Plastic Deformation of Two-Phase Steel.” Mat. Sci. Eng., 161, (1974). P. Hirsch, A. Howie, R. Nicholson, D. Pashley, and M. Whelan- Electcgn_niccgscogy of Tbin-§cxstal§z Butterworths, London, 36, (1965). G. DeSalvo and J. Swanson. AN§!§_§gglgggclgg BOQLXEl5_§¥§£§@§_:Z-Q§§E§-Défluéi;_li Swanson Analysis Systems Inc., Houston, PA. 0- H- Westbrook- nesbapical-EcQQsttie§-Qi lDEECQ§£é£LE_QQ@EQEOQ§; donn-Wilev and 8005. 12. (1959). 9- Seeqer- Qislecatiens-§ng_flecnanicel_flcgegctiss_ei 95y§g§;§, John-Wiley and Sons, New York, 389, (1957). K. N. Subramanian. "Development of Deformation Bands." phy. stat. sol., 24, 683, (1982).