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Deformation characteristics of w-zn composites. Bala, Sathish Rao 1971

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DEFORMATION CHARACTERISTICS OF W-Zn COMPOSITES by SATHISH RAO BALA B.Sc, University of Mysore, INDIA, (1965). B.E., Indian Institute of Science, Bangalore, INDIA, (1969). A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of Metallurgy We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1971 In presenting this thesis in partial. fu1filment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or'publication of this thesis for financial gain shall not be allowed without my written perm issibn. Department of Metallurgy The University of British Columbia Vancouver 8, Canada Date December 17, 1971 ABSTRACT The deformation characteristics of continuous tungsten fibre-reinforced zinc composites have been investigated. Composites with a single crystal matrix containing up to 4.5 volume per cent of tungsten fibres were studied. The stress-strain curves of W-Zn composites showed positive deviations from the "rule of mixture" predictions. Theoretical work attributes the positive deviations to matrix hardening due to either one of the phenomena: (a) the difference in the lateral contractions of the fibre and the matrix; (b) the pile up of dislocations in the matrix at the matrix-fibre interface. In the present work the positive deviations in the elastic-plastic region of the stress-strain curves of the composites have been attributed to both (a) and (b). The positive deviations in the ultimate tensile strengths of the composites have been attributed to (b). . Composites containing up to 0.08 volume per cent of the tungsten wires deformed even after the fracture of the fibres. Dissolution of the matrix of these deformed composites showed that multiple necking had occurred in the fibres fractured to 1-5 mms length. Composites containing greater than 0.08 volume per cent of tung sten fibres fractured by cleaving through the basal plane of the matrix. iii No fibre fracture inside the matrix was seen except at the fracture end of the composite. Multiple necking of the fibres near the fractured end has been seen only in those composites which have deformed more than the free fibres tested individually. ACKNOWLEDGEMENT The author gratefully acknowledges the helpful discussions with his research director, Professor E. Teghtsoonian, and with Dr. Ainul Akhtar. He wishes to thank the members of the faculty and fellow graduate students of the Department of Metallurgy for their continued support and interest in this work. Financial assistance was received in the form of an assistant-ship under National Research Council of Canada grant number A-2452, and is gratefully acknowledged. iv TABLE OF CONTENTS Page I. INTRODUCTION 1 A. General ,1 B. Previous Work 2 C. Scope 9 II. EXPERIMENTAL PROCEDURE. 4 11 A. Selection of the Composite System 1B. Bonding Characteristics Between Tungsten and Zinc ... 12 C. Fabrication of W-Zn Composites. 5 D. Method of Growing Oriented Single Crystals of Zinc and Zinc Matrix 16 E. Preparation of W-Zn Composites with 0.0015 inch Diameter Tungsten Wires 8 F. Tensile Tests 9 G. Control Experiment to Examine the Behaviour of W-Zn Composite in Stage II 20 H. Volume Fraction of Fibres 22. III. RESULTS ..: 23 A. Metallographic Observations 2A.l. Slip Lines and Twinning in Deformed Zinc Single Crystals 23 A.2. Microscopic Observations of the as made CompositeA.3«v. Fractographic Observations 26 A.4. • Observations on Fibres Deformed to Fracture ... 30 A. 5. Electron-Probe Analysis of the Fibre-Matrix Interface in a W-Zn Composite 30 B. Tensile Properties 32 B. l.'' True Stress-True Strain Curve for W-Wires (Polycrystalline) . . . . ' 32 B.2. True Stress-True Strain Curve for Pure Zinc Crystal and Tungsten-Zinc Composites. ...... 37 B.3. Derived Stress-Strain Curves for the Matrices of the Composites 49 B.4. Resolved Shear Stress-Shear Strain Curves for Zinc Single Crystal and Tungsten-Zinc Composites 51 v vi Page IV. DISCUSSION. 61 A. Scatter in the Experimental Results 61 B. Cause for the Greater Elongation of W-Zn Composites Compared to the Free Tungsten Wire 63 C. Discrepancy Between the Experimental Strengths of W-Zn Composites and Values Predicted According to "Rule of Mixtures" 6D. Stage II of the Stress-Strain Curves of W-Zn Composites 67 (a) Hill 6(b) Tanaka et al 70 E. Stage II of the Derived Stress-Strain Curves of the Matrix Corresponding to the Second Stage of the Compo site Stress-Strain Curve 73 (a) Kelly et al 7(b) Neumann et al 5 F. Resolved Shear Stress-Shear Strain Curves of the Composites 80 V. SUMMARY AND CONCLUSIONS 82 VI. SUGGESTION FOR FUTURE WORK. 83 LIST OF FIGURES Figure Page 1 Schematic Diagram of the True Stress-Strain Curve of a Composite Showing Four Stages of Deformation 5 2a Cross-Section of a W-Al Composite, 10% Sodium Hydroxide Etch, X23 13 2b Cross-Section of a Stainless Steel-Al Composite, 10% Sodium Hydroxide Etch, X23. 13 3a Cross-Section of a W-Zn Composite, Zinc Etch, X23 14 3b Cross-Section of a Stainless Steel-Zn Composite, Zinc Etch X23 14 Schematic Diagram of the Liquid Metal Infiltration Set Up . 17 5 Schematic Diagram of the Load-Elongation Curve of a W-Zn Composite Obtained in the Control Experiment 21 6 Slip Lines and Twin Markings on the Surface of a Deformed Zinc Crystal, X130 24 7 Distribution of Fibres in a W-Zn Composite Containing (a) 100 Tungsten Wires, (b) 1486 Tungsten Wires, X12. ... 25 8a Cross-Section of a W-Zn Composite Showing Oxide Rings and Twins, X125 27 8b Longitudinal-Section of a W-Zn Composite Showing Discontin uity in the Oxide-Layers, X110. ..... 27 9 Twin Markings on the Fractured Surface of W-Zn Composite Containing (a) 100 Tungsten Wires, (b) 1486 Tungsten Wires, X275. . 28 10 Scanning:Electron Micrograph of the Fractured Surface of a W-Zn Composite Containing 743 Tungsten Wires, X715. .... 29 11a A Portion of the Extracted Tungsten Wire Fragments Obtained by Dissolving the Matrix of a Fractured W-Zn Composite Con taining 6 Tungsten Wires, X12 31 lib Multiple-Necking in the Fragments, X130 31 vii viii Figure Page 12 Composition-Distance Curve Obtained From Electron-Probe Analysis 33 13 True Stress-Strain Curve of a Tungsten Wire ......... 35 14a True Stress-Strain Curves of Zinc Crystal and W-Zn Composites. 41 14b True Stress-Strain Curves of W-Zn Composites 42 15 U.T.S. Vs. Vf% Plots of W-Zn Composites 46 16 Yield Stress Vs. Vf% Plots of W-Zn Composites 47 17 ^c Vs. V,% Plots of W-Zn Composites 48 de 18 Derived Matrix Stress-Strain Curves of W-Zn Composites and True Stress-Strain Curve of Zinc Crystal 50 II , Q da * -r-^- Vs. V£% Plots of the Matrix of W-Zn Composites 52 d £ t 20a Resolved Shear Stress-Shear Strain Curves of a Zinc Crystal and W-Zn Composites 54 20b Resolved Shear Stress-Shear Strain Curves of W-Zn Composites 55 21 C.R.S.S. Vs. Vf% Plots of W-Zn Composites 59 22 Slope of the Elastic-Plastic Region of the T -y Curves Vs. Vf% Plots of W-Zn Composites 60 23 Dislocation Pile Up Model 77 LIST OF TABLES Table Page 1 Fabrication Techniques, and Composite Systems Fabricated by These Techniques. 2 2 Composite Groups and the Appropriate Composite Systems . . 4 3 Elastic Constants for Tungsten Wire 36 4 Elastic Constants for Zinc Crystal 39 5a Tensile Properties Obtained From the True Stress-Strain Curves of Tungsten Wire, Zinc Single Crystals and W-Zn Composites 44 5b Tensile Properties Obtained From the True Stress-Strain Curves of W-Zn Composites 45 6a Tensile Properties Obtained From the Resolved Shear Stress-Shear Strain Curves of Pure Zinc Crystals and W-Zn Composites . 56 6b Tensile Properties Obtained From the Resolved Shear Stress-Shear Strain Curves of W-Zn Composites . 57 7 Estimates of the Scatter in the Experimental Results ... 62 8a Comparison of the Experimental and Theoretical Values. . . 65 8b Comparison of the Experimental and Theoretical Values. . . 66 9 D Values Obtained for W-Zn Composites According to Neumann et at 81 ix I. INTRODUCTION A. General: In recent years much, attention has been given to the possibil ity of increasing the strength of weaker materials by reinforcing them with stronger materials. Very good strength characteristics are found in materials which contain a large volume fraction of a hard second phase uniformly distributed through the major phase. These types of materials can be obtained by phase decomposition, by mechanical mixing and sinter ing (powder metallurgical means), and by internal oxidation. The strength properties of these materials entirely depend on the properties of the hard phase present in the weak matrix. Investiga tions on these materials led to the conclusion that a matrix containing uniformly dispersed needle shaped hard particles, with a good bonding of the harder phase to the matrix, would give maximum strength. Recent investigators have considered combinations of strong fibrous materials with relatively weak binder materials in.order to obtain strong composites. This has led to the fabrication of composite materials with mechanical properties superior to those of the bulk materials from which they are derived. Various techniques have been developed for incor-n ' porating strong fibres in the weaker matrices and many theories predicting the tensile properties have been developed. 1 2 B. Previous Work: Much of th.e past research, has been concerned with, fabricating a good composite. Fabrication methods can be broadly classified"*" as 'direct', and 'indirect'. Direct fabrication of fibre-reinforced metals can be carried out either by the growth, and arrangement of the fibres, using a controlled phase transformation, or by production and alignment by working in the solid state. Direct fabrication is usually done in a single operation. 2 3 CuA^-Al eutectic composites, Fe2B-Fe eutectic composites, 3 4 Ni^Si-Ni eutectic composites, Al^Ni-Al eutectic composites, Cu-Cr eutec tic composites"'" and Cu^Ca-Cu"'" composites are some examples of composites obtained by direct fabrication methods. The indirect method of fabrication involves two steps. The fibres are first obtained separately and are subsequently incorporated in the matrix to form the composite structure. Reinforcements have been made using either continuous or discontinuous fibres. Different approaches to the fabrication of composites, and the systems fabricated by these methods are given in Table 1. TABLE 1. FABRICATION TECHNIQUES, AND COMPOSITE SYSTEMS FABRICATED BY THESE TECHNIQUES 5 6 From Alexander et al. ' FABRICATION TECHNIQUE FIBRE-MATRIX SYSTEM Liquid Metal Infiltration B-Mg, W-Cu, Mo-Cu, Ta-Cu, B-Al, W-Ni, Steel-Ag, W-Ag. 3 Table 1 (Continued) FABRICATION TECHNIQUE FIBRE-MATRIX SYSTEM Hot Pressure Bonding B-Al, Be-Al, Stainless Steel-Al, SiG-Al, Coated B-Al, Si02-Al, B-Mg, SiC-Ti, Be-Ti, B-TI, Coated B-Ti. Cold Press and Sinter W-Ni, Mo-Ni, Mo-Ti, W-Ag. Plasma Spray B-Al, SiC-Al, Coated B-Al, W-W. High Energy Rate Forming B-Al, B-Ni, B-Ti, W-Al, W-Ni, SiC-Ti, SiC-Ni. Electro-deposition W-Ni, B-Ni, SiC-Ni, B-Al, SiC-Al, W-Cu. Chemical Vapour Deposition W-W, B-W, Be-Al. Extrusion and Rolling W-Ni, Mo-Ni, B-Al, Mo-Ti, B-Ti. The strong fibres used in the reinforcement of a weaker matrix-binder can be divided into three groups; whiskers, metallic wires and non-metallic wires (ceramic fibres). Composite materials can also be classified according to the fibres and matrix materials present in the composites. The different categories of composites, with, examples of composite systems that have been fabricated and whose deformation characteristics have been studied, are given in Table 2. 4 TABLE 2. COMPOSITE GROUPS AND THE APPROPRIATE COMPOSITE SYSTEMS FIBRE-MATRIX COMPOSITE SYSTEM INVESTIGATED: FIBRE-MATRIX Metal-metal W-Ag,7'8 W-Cu,9'10'11 Stainless Steel-A1 12,13,14,15 Cf. . _ 16 _ 10 Al, Steel-Cu, Mo-Cu, 17 18,19 fc Ta-Cu, B-Al, etc. Ceramic Cnon metal)-metal 20 21 22 C-Ni, Al203-Ni, Si3N4-Ag , Si02" 23 2 A oc Al, Glass-Al, A120 -Ti,"5 etc. Metal-non metal 26 26 Nichrome-Al_0 , Nichrome-Si09, 26 Stainless Steel - A1„0 , Stainless 26 27 Steel-Si02, Al-plastic, etc. Non metal-non metal 28 29 Glass-plastic, Carbon-polyester, 22 Si_N,-resin. 3 4 Most extensive studies have been done on the deformation charac teristics of composites containing continuous metallic fibres. 9 McDanels &t at. in their work on tungsten fibre-reinforced copper composites defined four stages of tensile deformation, Figure 1. I. Elastic deformation of fibre; elastic deformation of matrix. II. Elastic deformation of fibre; plastic deformation of matrix. III. Plastic deformation of fibre; plastic deformation of matrix. IV. Failure of fibre and matrix. (Continued on p. 6) True Strain Fig. 1. Schematic Diagram of the True Stress-strain Curve of a Composite Showing Four Stages of Deformation. 6 The tensile properties were discussed, making use of the "rule 30 of mixtures." The rule of mixtures is given by the elementary decomposi tion formula: f - clfl + C2f2' Cl + C2 - 1 (1) f, f^ and f are composite, phase 1, and phase 2 averages of any function f(x,y). c^ and are phase concentrations (by volume fraction) of phases 1 and 2 respectively. For a volume fraction of fibres greater than a certain ^minimum, value (to be discussed later) the ultimate tensile strength of a continuous fibre composite, a , is given by: cu o = CT. V, + a 'V , V_ + V = 1 (2) cu fu f mm f m Oj = the ultimate tensile strength of the fibre; CT ' = ,,the stress in the free matrix at the ultimate strain of the free fibre; = volume fraction of the fibre in the compsite; Vm = volume fraction of the matrix in the composite. , i; Expression (2) assumes that the .matrix strain e , the m fibre strain e.-^, and the composite strain are equal for a given load. 9 McDanels et al. generalized equation C2) to allow prediction of the stress in a composite, CT^, at any value of strain: CT = cr£V£ + CT V , V + V = 1 C3) c f f • m JD. m f where the a values represent stresses at any particular value of strain taken from the stress-strain curves of the components in the condition in which they exist in the composite. 7 On the basis of equal elastic strains in Stage I, expression (3) can be written as: E , = E V + E V (4) cl rf mm where E^j. = Young's modulus of the composite in Stage I, E^ = Young's modulus of the free fibre, E = Young's modulus of the free matrix, m A similar expression was also written for Stage II of the composite 9 stress-strain curve. Stage II was found to be linear by McDanels et al. der The slope of the second stage, -—, or the "secondary modulus", E , of a de c -LJ-composite is given by: da da E = -—• = E.V_ + (-r-Si) V (5) c II de f f de m da where -;— is the slope of stress-strain curve of the free matrix at a de given strain. In expression (5) E T is the dominating factor and hence da . the variation of ( ^. ) has negligible effect on \ If the fibres are to produce a material stronger than the work-hardened matrix alone, the strength of the composite must exceed the ulti mate tensile strength of the free matrix, a , i.e. mu a =a'. Vx + a Cl-V,) • > CT C6) cu f.u f m f mu Expression C6) sets a critical given by: . . a . . . - . a _ a mu ID. , mu V_ crit. = p f CT- - a ' a^ fu m fu 8 i for a and a,. >> cr (J). mu f u m Only if the volume fraction of fibres exceeds crit. will fibre strengthening occur. A-lso, if the fibres all break in one cross-section the composite will fail unless the ductile matrix can support the load. The maximum stress that the matrix can support is a . The failure of all the fibres results in mu immediate failure of the composite only if, a = cr. V, + cr '(l-V.) > cr (1-V,) (8) cu fu f m f mu f Expression (8) defines a minimum volume fraction, V^min.,"'" which must be exceeded if the strength of the composite is to be given by expression (2). Hence: i a - a TT . mu m ,n>. V,mm, = ; »- (9) f a-. + a - a ru mu m 9 These predictions were found,by McDanels et al. , to agree with ex periments on W-Cu composites, within the limits of the scatter observed in the experimental values. Howard"*"^ made investigations on steel wire-reinforced copper com posites. Values of the tensile strengths predicted according to expression (2) were found to be lower than the experimental values. Hence, he proposed an expression for ultimate tensile strength by modifying expression C2). In his expression an extra term, which is a function of the fibre diameter and hardness of the fibre, was added. Discrepancies between the experimental tensile strength and the predicted values according to expression (2) have been found by many other 9 10 6 8 31 Investigators ' ' ' working with, different composite systems. 32 Cooper obtained values of ultimate tensile strengths for differ ent orientations of the longitudinal direction of the tungsten fibres with the specimen axis of the W-Cu composites. The ultimate tensile strengths were found to decrease with increasing angle of orientation of the fibres. 13 Jackson et al. working with 50 volume per cent stainless steel-Al composites found that the ultimate tensile strengths increased up to 20° mis-orientation of the fibre axis with the composite axis and then decreased as the misorientation further increased. Kelly et.al.^ investigated the deformation characteristics of W-Cu composites. In their work the copper matrix was incidentally a single crystal. The ultimate tensile strengths and the Young's moduli in Stage I were found to agree with the predicted values according to the rule of mix tures within the limits of scatter in the experimental values. The predicted Young's modulus in Stage II and the stress values were found to be lower than the experimental values. Kelly et al.^ using expression (3) and the experimental stress-strain curves of tungsten wire and W-Cu composite derived the matrix stress-strain curve. The derived matrix stress-strain curve did not coincide with the stress-strain curves ob-11 33 tained for a free copper single crystal. Kelly et al., Tanaka et al., 34 and Neumann et al. gave explanations for this discrepancy and predicted . the slope of the Stage II of the derived matrix stress-strain curve. C. Scope: Kelly et al.^~ investigated the deformation characteristics of W-Cu composites. Their liquid-infiltration method of fabricating the W-Cu 10 composites incidentally also gave essentially a single crystal matrix. For the. purpose of the present Investigation it was felt that deformation studies on a different composite system with a single crystal matrix of specific orientation migh-t yield useful information regarding the work hardening behaviour of composites. Finding a suitable system, and fabricating composites with var ious volume percents of fibre reinforcement, were of specific interest. II. EXPERIMENTAL PROCEDURE A. Selection of the Composite System: Tungsten and stainless steel wires were chosen as reinforcing materials. Aluminum, magnesium and zinc were the possible matrix mater ials. It was desired that the reinforcing material and the matrix mater ial should not have any mutual solubility. Also, the matrix should be easily grown into a single crystal in the presence of reinforcing mater ial. Preliminary experiments were done by melting the matrix mater ial in a graphite crucible. The reinforcing material, in the form of wires, was kept vertically inside the molten matrix material. The matrix material was cooled in a controlled way to prevent pipe formation. This was done by lowering the crucible on to a metal block kept below the ver-tical tube furnace. When the crucible was lowered on to the metal block, the upper-half was kept inside the furnace. The cooling furnace acted as a "hot top." When the composite casting was cold it was removed from the graphite crucible. Using this method, W-Al, stainless st'eel-Al, W-Mg, stainless steel-Mg, W-Zn and stainless steel-Zn composites, with polycrystalline matrices, were obtained. A modified Bridgman technique was used to attempt to grow single crystal matrices in these composites. Growing a single crystal matrix of magnesium was not possible. The aluminum matrix grew into a single crystal in W-Al system but not in the stainless steel-Al system. In both cases alloy formation between aluminum 11 12 and the fibres occurred; Figures 2a and 2b. Zinc gave single crystal matrices in both systems W-Zn and stain less steel-Zn. Alloy formation between zinc and stainless steel was ob served. However, alloy regions around the fibres, as seen in the stainless steel-Zn composites, were never seen in the W-Zn composites. Figures 3a and 3b show cross-sections of W-Zn and stainless steel-Zn composites. From the preliminary experiments it was thus found that the W-Zn system was the most suitable system for experimentation. B. Bonding Characteristics Between Tungsten and Zinc: Bonding between the reinforcing material and the matrix mater ial is an important aspect of composite properties and hence an effort was made to determine the wetting characteristics between tungsten and zinc using a sessile drop experiment. The experiment was unsuccessful. Evaporation of zinc at or near the melting point due to its high vapour pressure was the principal problem. Hence, an attempt was made to deter mine directly the bond shear strength between tungsten and zinc, using the fibre "pull out" experiment. In the "pull out" experiment, a single tungsten wire was pulled out of the zinc matrix using an Instron machine. The "pull out" experi ment was also unsuccessful in that reproducible results could not be obtained, and the scatter in results was extremely large. (Continued on p. 15) 13 Fig. 2b. Cross-section of a Stainless Steel-Al Composite, 10% Sodium Hydroxide Etch, X23. Fig. 3b. Cross-section of a Stainless Steel-Zn Composite, Zinc Etch, X23. 15 C. Fabrication of W-Zn Composites: Initially, liquid metal infiltration of bare W-wires was used to obtain W-Zn composites. On sectioning the composites thus obtained, it was found that all the tungsten wires were together and there was no trace of zinc between the wires. Hence this method of fabrication was considered unsuitable. Liquid metal infiltration of tungsten wires with electrodeposited zinc coatings was found to be the most successful method of fabrication of W-Zn composites. For electrodeposition of zinc on W-wires the acid elec trolyte used was:"^ ZnS04 • H20 240 gms/litre Na2S0^ 40 gms/litre ZnCl2 10 gms/litre H3B°3 5 gms/litre Distilled water 1 litre pH 3-4 Temp. 30°C. Anodes were obtained by hot rolling 99.99% pure zinc blocks, at 36 150°C. into 10 cm x 27 cm x 0.25 cm plates. Tungsten wire, 0.01 inch in diameter, was wound on 27 cm x 13 cm plastic frames. The W-wire was cleaned using an HF + HNO^ acid solution. Tungsten wire wound on the frame acted as the cathode. An average wire diameter was determined, using a travelling microscope, prior to the deposi tion. Zinc was deposited onto the wires at a cathode current density of 75 2 amps/ft . The thickness of the deposit was adjusted as needed. The deposited 16 wire was then immersed in a solution of 50% HNO^ acid to get a shiny sur face, rinsed in distilled water and dried by natural evaporation over night . Zinc-coated tungsten wires were cut into 25-cm long pieces which were bundled together. A bundle was put inside an aquadag (colloidal sus pension of graphite In water) coated pyrex tube of 6 mm inner diameter. The number of coated wires bundled depended on the volume fraction of fibres desired in the composite. The bundle was kept in the middle of the pyrex tube. The tube was constricted near both ends of the fibre bundle. One end of the tube was immersed in molten zinc (99.999% pure) and the other end was connected to suction. The bundle was kept hot at 250 -300°C. by means of an induction heating coil. When the bundle was hot t. the molten metal was infiltrated into the bundle. Induction heating was switched off as the molten metal completely covered the bundle. The in duction coil then acted as a cooling coil. Figure 4 shows a schematic diagram of the liquid metal infiltration set-up. W-Zn composites thus ob tained had a polycrystalline zinc matrix. D. Method of Growing Oriented Single Crystals of Zinc and Zinc Matrix: Randomly oriented zinc crystals were grown using 99.999% pure zinc supplied by COMINCO (B.C. Trail, Canada). The modified Bridgman method was used torgrow 6 inch long x 1 1/8 inch diameter zinc crystals. The maximum temperature in the furnace was about 450°C. The growth rate was 3 cms/hour. (Continued on p. 18) 17 o o o o o o o > to suction constriction aquadag coated pyrex tube zinc coated sten wires induction heater molten zinc lite crucible resistance furnace Fig. 4. Schematic Diagram of the Liquid Metal Infiltration Set Up. 18 Seed crystals of circular cross-sections with basal plane orientation of 45° to the long axis were cut from the bulk crystals using the spark erosion method. Basal plane cleavage to a depth of 1-2 turns, due to spark damage, was seen in the seed crystals. Polycrystalline zinc rods of 3/8 inch diameter x 12 inches long were obtained by swaging 5/8 inch diameter pure zinc castings at 200°C. The seed crystal and the cleaned polycrystalline zinc rod were put inside an aquadag coated pyrex tube and vacuum sealed. The polycrys talline rod was then grown into an oriented single crystal using the modi fied Bridgman technique. The maximum temperature in the vertical crystal growing furnace was 450 - 10°C. The growth rate of the crystal was 2 cms/hr. The crystal was recovered from the pyrex tube by dissolving away the tube in 52% HF acid. The same procedure was followed to grow oriented single crystal matrices of zinc in W-Zn composites. In this case, pyrex tubes with 6 mm inner diameter were used. Also, polycrystalline zinc rod was replaced by the W-Zn composite. Oriented free crystals and composite crystals were cleaned using 50% HN0.J ac^' The orientations of the crystals were checked by the back-reflection Laue X-ray method. E. Preparation of W-Zn Composites With 0.0015 Inch Diameter Tungsten Wires: Tensile tests carried out on W-Zn composites containing 0.01 inch diameter W-wires showed a 3-6 mm length "pull out" of the tungsten wires on the fractured surface. This is because the load bearing capacity of the fibre is greater than the load bearing capacity of the fibre-metal 19 interface. The smaller the diameter of the fibre, the smaller is the load bearing capacity of the fibre. The use of fibres with smaller diameter minimizes fibre pull out. Hence 0.0015 inch diameter tungsten wires supplied by SYLVANIA CChem. and Met. Div., Towanda, Pa.) were subsequently used for reinforcing. The procedure adopted to prepare W-Zn composites of single crys tal matrix with 0.0015 inch diameter tungsten wires was the same as that with 0.01 inch tungsten wires. For the fabrication of composites containing 6, 12, 25, 100 and 186 wires, the electrodeposited wires were bundled together with thin strips of pure zinc prior to liquid metal infiltration. The composites prepared contained 6, 12, 25, 100, 186, 372, 743 and 1486 tungsten wires, of average diameter 0.0329 mm, in a cross-section 2 of 26-28 mm . The orientations of the zinc matrices involved were Y o 35-45° and XQ = 35-49°. XQ is the angle between the slip plane (0001) and the tensile axis of the specimen. Xq is the angle between the most favour able slip direction [1120] and the tensile axis of the specimen. Back reflection Laue X-ray pictures showed the absence of low angle boundaries in the matrix of the W-Zn composites. F. Tensile Tests: Pure zinc crystal and W-Zn composite test specimens, 3 inches long, were annealed at 400°C. for one hour. Test specimens of 0.0015 inch dia meter tungsten wire were annealed at 450°C. for one day. The test speci mens were then tested in an Instron tensile testing machine at a cross-20 head speed of 0.02 inch per minute on a 2 inch gauge length.. Load-elonga tion curves were recorded on an x-y recorder at suitable chart speeds. The wire and the composite specimens were mounted on the machine using split grips. The pure zinc crystal specimens were mounted on the machine using brass holders soldered to the specimens. For wire specimens, elongation was recorded according to the cross-head movement. An extensometer was not used because the W-wire was not strong enough to support It. In the case of pure zinc crystals, elongation was measured accord ing to the movement of cross-head. . In the earlier tests, the elongation of the composites was recorded according to the cross-head movement of the Instron machine. In later exper iments elongations were recorded according to the strain gauge extensometer. Also, all the tests with composite specimens were repeated using a 1 inch gauge length extensometer to measure elongation. The use of the strain gauge clearly showed four stages of deformation in the load-elongation curves as described earlier. G. Control Experiment to Examine the Behaviour of W-Zn Composite in Stage II: Control experiments were conducted to check whether the composites in stage II are plastic or elastic. W-Zn composite specimens were stressed t. into stage II and then unloaded. The initial part of the unloading curve was parallel to the elastic-elastic line, demonstrating that the matrix in stage II behaves plastically. A curve obtained from the control experiment is shown schematically in Figure 5. CContinued on p. 22) Elongation Fig. 5. Schematic Diagram of the Load-elongation Curve of a W-Zn Composite Obtained in the Control Experi ment. 22 H. Volume Fraction of Fibres: The volume fraction of fibres, in each composite, was calculated from the fibre diameter, the number of fibres and the diameter of the composite. III. RESULTS A. Metallographlc Observations: A.l. Slip Lines and Twinning in Deformed Zinc Single Crystals: An attempt was made to observe the slip and twin markings on the surface of the fractured zinc crystal. Figure 6 shows the slip lines and twins on the surface of a crystal. No secondary slip lines are seen. Twinning might be responsible for the serrations in the later stages of the load-elongation curves. Serrations were also observed in those com posites which behaved like a single crystal, in the final stages of defor mation. Since the surfaces of the composites were not good enough for optical examination, no photographs were taken. A.2. Microscopic Observations of the as made Composite: Composite specimens were spark-cut for metallographlc examina tion. The cross-sections were ground and polished using the diamond polish ing wheel. The polished cross-sections were viewed under the microscope to see the fibre distributions. It has been stated"'" that poor wettability be tween the fibers and the matrix would develop voids inside the composites. Microscopic examination did not show any voids in the composites. Typical macrographs of the cross-sections for two composites are given in Figures 7a and 7b. In some cross-sections oxide rings were seen encircling the tung sten fibres. This can be accounted for by the inability to use any sort of protective atmosphere while heating the electrodeposited tungsten wires prior to infiltration. The continuity of the oxide rings was investigated. (Continued on p. 26) 23 Fig. 6. Slip Lines and Twin Markings on the Surface of a Deformed Zinc Crystal, X130. 2 5 (a) (b) Fig. 7. Distribution of Fibres in a W-Zn Composite Containing (a) 100 Tungsten Wires, (b) 1486 Tungsten Wires, X12. 26 The polished specimen surface was etched with modified Gilman's solution: 320 gms Cr03 20 gms Na2S04 1000 mis H20 Figure 8a shows discontinuities in the oxide rings. It should be noted that all the fibres did not have the oxide rings encircling them. Longi tudinal sections of the composites were polished and examined. The oxide layer is discontinuous as shown in Figure 8b. Etching of the longitudinal section was avoided because the increased etching rate in the direction of the oxide layer itself tended to produce continuity. Incidentally the twins formed during grinding and polishing oper ations showed the continuity of the crystal matrix, by running parallel, inside and outside the oxide rings (Figure 8a). A.3. Fractographic Observations: Tungsten-zinc composites containing up to 0.08 volume per cent fibres fractured essentially the same as the pure zinc single crystal. Composites containing higher than 0.08 volume per cent fibres fractured by cleaving through the basal planes. The cleaved surface due to composite fracture was observed under the microscope. Twins in large numbers are seen in the matrix in the vicinity of the fibres as shown in Figures 9a and 9b. The twin density increased with the volume fraction of the fibres in the matrix. Also, fractographic observations were made using the scanning electron microscope. A typical scanning electron micrograph is shown in Figure 10. Stepsccan be seen connecting two fibres in the micrograph. (Continued on p. 30) 27 Fig. 8a. Cross-section of a W-Zn Composite Showing Oxide-rings and Twins, X125. Fig. 8b. Longitudinal-Section of a W-Zn Composite Showing Discontinuity in the Oxide-layers, X110. I Fig. 9. Twin Markings on the Fractured Surface of W-Zn Composite Containing (a) 100 Tungsten Wires, (b) 1486 Tungsten Wires, X275. 29 Fig. 10. Scanning Electron Micrograph of the Fractured Surface of a W-Zn Composite Containing 743 Tungsten Wires, X715. 30 A.4. Observations on Fibres Deformed to Fracture: Free tungsten fibres deformed to fracture showed a single necking at the fracture end. To investigate deformation behaviour of the fibres in side the matrix, some of the composite specimens deformed to fracture were etched to extract the fibres from the matrix using 50% nitric acid. One of the specimens which deformed essentially as a pure zinc crystal, was treated as described above. Small lengths of fractured fibres were recovered. The length of the bits varied from 1 to 5 mms. When these bits were viewed using transmitted light under the optical microscope they exhibited multiple necking profiles on the surface. The multiple neck ings were present near both fractured ends of the fibres. Figures 11a and lib show the length of the bits and the multiple necking. The other specimens, which fractured by cleaving through the basal plane, left behind continuous fibres on etching off the matrix. This shows that the fibres did not fracture inside the matrix prior to the specimen fracture. Also multiple necking was seen in the fibres near the fractured ends. The multiple necking was seen only in those composites which deformed greater than the free tungsten fibre tested. A.5. Electron-Probe Analysis of the Fibre-Matrix Interface in a W-Zn Composite: 37 It has been stated that alloy formation between fibres and the matrix at the interface deteriorates the properties of the fibre and hence of the composite. In the present W-Zn system, it is stated that the mutual 37 solubilities between the components of the system is essentially nil be-38 low 1350°C. To confirm this, electron-probe analysis of a W-Zn composite was carried out. (Continued on p. 32) 31 Fig. 11a. A Portion of the Extracted Tungsten Wire Fragments Obtained by Dissolving the Matrix of a Fractured W-Zn Composite Con taining 6 Tungsten Wires, x 12. Fig. lib. Multiple Necking in the Fragments, X 130, 32 A polished cross-section of the W-Zn composite was continuously scanned, in and out of a single fibre, along the diametric axis of the fibre. The scanned path-composition curve was recorded on an x-y recorder. The scanned path-composition curve obtained showed, with a resolving capa city of the electron-probe equal to one micron, no mutual solubility between tungsten and zinc. The scanned path-composition curve obtained for the W-Zn system is given in Figure 12. B. Tensile Properties: B.l. True Stress-True Strain Curve for Tungsten Wires (Polycrystalline): Tensile properties like stress, strain and elastic constants are of great importance in theoretical predictions. Also, the tensile properties of the materials vary with the fabrication methods by which they are obtained. Hence, an average true stress-true strain curve was plotted for 0.033 mm dia meter tungsten wire. This curve is obtained from the load-elongation curves of the tungsten wires using the relations: °f = i1 (1 + r-> <10) o o and ef = In CI + ~) (11) o where = the true stress; e^ = the true strain; .1 = the original gauge length; P = the tensile load (Continued on p. 34) if) o Q. £ o o Zn ' w Zn 1 1 ,0* 1 i Scanned-path (Distance) Fig. 12. Composition-distance Curve Obtained from Electron-probe Analysis, CJ to 34 and Al = the total elongation at P according to the cross-head movement of the Instron tensile testing machine. Figure 13 gives the average true stress-true strain curve for polycrystalline tungsten wire. This curve is an average of the load-elongation curves obtained for three wire specimens. All the specimens fractured at a strain of 2% and the results were reproducible. From the stress-strain curve of tungsten wire the yield stress °"f » the Young's modulus E^, and the ultimate tensile strength are obtained. These values are given in Table 5a. 3. Yield stress is defined to be the stress at which the stress-strain curve first deviates from linearity. Young's modulus is the slope of the linear portion of the stress-strain curve. The ultimate tensile strength is defined to be the maximum stress in the stress-strain curve. The reason for obtaining elongation according to the cross-head movement is that the wire was too thin to support an extensometer. The value of E^ obtained from the stress-strain curve is comparable to the value obtained by Kelly et al.^ Since mechanical properties vary accord ing to the manufacturing method the available data are not used except for the Poisson's ratio v^. The E^ value obtained; experimentally and value taken from 39 Lowrie and Gonas were used to obtain the shear modulus G^, the bulk mod ulus K^ and the plane strain bulk modulus k^ of the tungsten wires. The relations used are: Gf = 2(1 !%) <*2) v Kf • ~ 3(1 - 2vf) (13) and (Continued on p. 36) 0 I 2 3 True Strain, e, %. Fig. 13. True Stress-Strain Curve of a Tungsten Wire. 36 kfP • zci-r^i a4) assuming a homogeneous isotropic solid. The experimental and calculated values of elastic constants for tungsten wires are tabulated along with the published values of Lowrie 39 11 et al. and Kelly et at. , in Table 3. TABLE 3 ELASTIC CONSTANTS FOR TUNGSTEN WIRE PRESENT CALCULATIONS LOWRIE et at. KELLY et- al. Ef 3.6 X in7 A' 2 10 gms /mm 4 .1 X io7 , 2 gms/mm 6 2 10 u-w: (36.4*5)10 gms/mm 20 y-w: (38.1±2)106 gms/mm2 Gf 1.4 X in7 / 2 10 gms/mm 1 .6 X io7 / 2 gms /mm Kf 2.7 X in7 / 2 10 gms/mm 3 .1 X / 2 gms/mm vf 0. ,28 3.2 X in7 / 2 10 gms/mm The values of the elastic constants given in Table 3 were used in the theoretical calculations. 37 B.2. True Stress-True Strain Curve for Pure Zinc Crystal and Tungsten-Zinc Composites: With, a view to examining the deformation characteristics, and hence correlating the experimental results with the theoretically pre dicted results, true stress-true strain curves were drawn for pure zinc crystals and tungsten-zinc composites. True stress-true strain curves are more appropriate than engineering stress-strain curves even though at low strains the difference between the two is negligibly small. In the beginning, tensile tests were performed without using the extensometer strain gauge. The elongation was recorded according to the cross-head movement. Later, experiments were conducted using an extensometer for elongation measurement. The load-elongation curves ob tained from these two methods for similar materials show that the elastic elongation recorded according to the cross-head movement is much larger than the one recorded by using the extensometer. Also, the elastic elong ation recorded using an extensometer is too small on the chart to make any precise calculation of the strain or the elastic modulus of the material. Hence the Young's moduli of the pure zinc crystal and of W-Zn composites were made use of in calculating the appropriate elastic elongations. The Young's modulus of zinc crystals was obtained using the elas tic compliances. The Young's modulus of zinc varies according to the orien tation of the crystallographic axis-c with the tensile axis. In particular, the Young's modulus of a zinc crystal rod, the length of which makes an 40 angle 0 with the- c-axis of the h.c.p. lattice is given by: 38 iT = CSll+S33-2S13-S44)cos4e + C2Si3+S44-2Sll)cos20 + Sn <"> m where S^, ^i3> ^33 anc* ^44 are the elastic compliances. The values of the five elastic compliances for zinc taken from Wert et al.^ for calculation purposes are: Su = 8.38 x 10~13 cm2/dyne -13 2 = x 10 cm /dyne -13 2 S^^ = -7.31 x 10 cm /dyne S33 = 28.3 x 10"13 cm2/dyne and S 4 = 26.1 x 10~13 cm2/dyne. All the specimens, either zinc single crystals or composites with a zinc single crystal matrix, had their orientation of the basal plane with the tensile axis, XQ> lying between 37° and 44°. An average value of XQ = 40° and hence 0 = 50° was used in the calculation of E . m Also the rigidity modulus G , the bulk modulus K and the Poisson's m m ratio v^, for zinc crystals with Xq = 40°, were calculated using the re-lations: G~ = S44 +ICS11~S12)_1/2S44] C1_COs20) + 2CS11+S33-2S13-S44)cos20(l-cos20) m (16) K~ " S33 + 2CS11 + S12> + 6S13 C17m s13 and v = — -— (approximately) (18) m S33 39 The values of the elastic constants obtained for the zinc crystal Cx = 40°) are given in Table 4. TABLE 4 ELASTIC CONSTANTS FOR ZINC CRYSTAL E m 9.5 x io6 / 2 gms/mm G m 2.8 x io6 / 2 . gms/mm K m 6.0 x io6 / 2 gms/mm V m 0.26 These values of the elastic constants of a zinc crystal are use ful in the theoretical predictions involved. The Young's modulus of a composite, which is defined to be the Young's modulus in stage I, is obtained using expression (4). True stress-true strain curves were obtained from load-elongation curves of pure zinc crystals using the relations: *m = f-CL + eT) (19) o em = In (1 + eT) (20and Al Al + Al . Al e = _1 = _£ P = JP_ + _JE. on . T 1 1 A E 1 L ; o o o m o where 40 cr m = the true stress z m = the true strain 6T = the total engineering strain A o = the original area of cross-section 1 o = the original gauge length. p = the tensile load T = total elongation at P Al e = elastic elongation at P Al P = plastic elongation at P. The elastic strain at P was obtained using the relation A1e P (22) 1 A E o o m Similarly, true stress-true strain curves were obtained from the load-elongation curves of composites using the relations given for pure zinc crystals and replacing CT , e and E by cr , e and E ,. respectively. mm m.cc cl For composites Alg .^corresponds to the elastic-elastic elongation and Al^ = A1T-Ale. Figures 14a and 14b show true stress-true strain curves for specimens X-7, C-16, C-14, C-25, C-20, C-27, C-40, C-9 and c-55. x and C represent crystal and composite specimens respectively. Similar curves were also obtained for other specimens. The curves obtained for W-Zn com-9 posites show four stages of deformation as observed by McDanels et al. in W-Cu composites. (Continued on p. 43) 41 Fig. 14a. True Stress-strain Curves of Zinc Crystal and W-Zn Composites. 42 Fig. 14b. True Stress-strain Curves of W-Zn Composites. 43 Curves obtained for the composites containing up to 0.08 volume per cent tungsten fibres were similar to those of pure zinc crystals. Ser rations were found in the final stages of the load-elongation curves of pure zinc crystals and W-Zn composites containing up to 0.08 volume per cent fibres. These serrations are probably due to twinning. Serrations were also found in stage IV of the load elongation curves of the W-Zn com posites containing up to 0.08 volume per cent fibres. These serrations are due to fibre fracture. Serrations due to fibre fractures were found until the final stages of deformation. Serrations were not seen in the load-elongation curves of the composites containing more than 0.08 volume per cent of fibres. Since the curves for all the specimens tested are not given, the 1 important physical and tensile properties obtained from the curves of all the specimens tested are given in Tables 5a and 5b. The variations in ultimate tensile strengths of W-Zn composites, acu> with volume fractions of the tungsten fibres, V^%, present are shown in Figure 15. The ultimate tensile strengths of the pure zinc single crys tals, o" » are also plotted in this figure as points corresponding to zero volume per cent. The yield strengths of the W-Zn composites, a , are plotted cy against V^/d in Figure 16. The yield strengths of pure zinc crystals, o"my> are also plotted in this figure. da Finally, the slope of the second stage, c , of the true stress-de true strain curves of W-Zn composites were measured. These values are plotted against V^% in Figure 17. (Continued on p. 49) TABLE 5a TENSILE PROPERTIES OBTAINED FROM THE TRUE STRESS-STRAIN CURVES OF TUNGSTEN WIRE, ZINC SINGLE CRYSTALS, AND W-Zn COMPOSITES N 0 YIELD da STRESS AT 0.05% U.T.S. STRAIN *_c „ - STRESS -—• OFFSET FROM STAGE II AT U.T.S, SP. NO. WIRES V ,2 d£, 2 .2 , 2 gms/mm gms/mm gms/mm gms/mm % W-wire 100 23 x 104 — — 36 x 104 2.29 X-6 0 0 62 3520 104.67 X-7 0 0 59 — — 3500 116.64 C-7 6 0.023 82 2.8 X < 240 520 14.44 C-8 6 0.020 63 3.1 X 210 1500 57.20 C-16 6 0.018 56 1.2 X 103 190 3400 99.50 C-46 6 0.018 94 9.6 X 103 220 4600 138.03 C-13 12 0.036 67 2.7 X K 230 1300 51.90 C-14 12 0.037 63 3.0 X 104 280 3800 102.00 C-17 25 0.080 85 2.9 X A 260 350 2.62 C-18 25 0.081 97 4.0 X 104 390 940 48 C-19 25 0.080 98 4.9 X 10t 390 480 1.97 C-2 3 25 0.076 65 6.3 X 460 3200 104.83 C-25 25 0.078 66 7.7 X io4 430 3800 115.24 C-20 100 0.322 100 1.8 X 105 1100 1500 2.25 C-21 100 0.324 110 2.2 X 105 1000 1500 1.53 C-26 100 0.303 100 1.6 X 105 1100 1600 1.63 C-53 100 0.298 64 1.4 X io5 940 1100 1.04 TABLE 5b TENSILE PROPERTIES OBTAINED FROM THE TRUE STRESS-STRAIN CURVES OF W-Zn COMPOSITES SP.. NO. NO. OF WIRES Vf% YIELD STRESS , 2 gms/mm da c de gms/mm STRESS AT 0.05% OFFSET FROM STAGE II gms/mm2 U.T.S. , 2 gms/mm STRAIN AT U.T.S, % C-27 " 186 0.553 120 3.4 X, 10 1700 2600 2.59 C-29 186 0.558 130 2.4 X 10* 1700 2300 1.22 C-30 186 0.574 . 130 3.8 X 105 1600 2400 1.03 C-31 186 0.562 140 3.9 X 105 1800 2300 0.86 C-32 186 0.565 150 3.9 X 105 1700 2500 1.61 C-51 186 0.587 130 2.9 X 105 1900 2400 1.53 C-36 372 1.144 210 7.7 X 105 3500 5200 2.07 C-38 372 1.163 190 8.2 X 105 3700 4800 0.91 C-39 372 1.146 190 8.2 X 105 3300 4700 1.07 C-40 372 1.110 190 6.3 X 105 3300 4800 2.08 C-41 372 1.089 210 7.0 X 105 3600 4800 1.52 C-3 743 2.298 260 1.1 X 106 6300 8900 2.07 C-9 743 2.357 260 1.4 X 106 7000 10400 2.14 C-10 743 2.321 250 1.2 X 106 6800 9700 1.57 C-11 743 2.276 250 1.3 X 106 7000 9100 1.01 C-58 743 2.357 260 1.3 X io6 7400 10000 1.96 C-42 1484 4.568 330 1.9 X 106 13000 19400 2.39 C-55 1486 4.583 330 2.2 X 106 14000 19500 3.26 -O-49 da c a and -— values are found to.increase with, increasing value cy de of a values are also found to increase with, increasing value of f cu • • VJ. except at lower At lower values of V.,%, a values are found to f f f cu be constant as increases. B.3. Derived Stress-Strain Curves for the Matrices of the Gomposites: In most previous work,expression (4) has been found to be in good agreement with, the experimental values. In the present work it was not possible to find experimentally and hence to check the agreement with the expression. According to expression (3) if there is no interaction between the fibre and the matrix, the a value should follow the stress-strain m curve of the free matrix. To find the actual stresses in the matrix of II the composites, a^ values were derived from the experimental values of 11 a , V,., V and a_. The relation used is: c f m f " vH' Vm + Vf = 1 (23) m m This involves the assumption that e_ = e = e , and that the stress-strain i f m c curve of the fibres obtained individually, remains the same in the com posite. Expression (23) is the same as expression (3) except for a ", where m II a is the actual stress in the matrix of the composite. The derived stress-strain curves for the matrices of the composites, obtained using expression (23) are given in Figure 18. The stress-strain curve of pure zinc crystals is also given in the same figure. The derived stress-strain curves of the matrices in the W-Zn com posites exhibit linear portions corresponding to stage I and II of the (Continued on p. 51) 50 0-5 I True Strain Fig. 18. Derived Matrix Stress-strain Curves of W-Zn Composites and the True Stress-strain Curve of Zinc Crystal. 51 da stress-strain curves of the W-Zn composites. In expression (5) ~j^T~ is the slope of the stress-strain curve of the free matrix at a particular strain. This is based on the assumption that there is no interaction be tween the fibre and the matrix. For the actual matrix of the W-Zn com-dam " posites the slope j£" corresponding to stage II is obtained using the relation: da " E TT ~ E.V, m This assumes that e, = e = e and E, remains the same in the composite. f m c f r II da -2 values were calculated using the experimental values of E. de , it °" r ell da E^, V^ and expression (24). —— values are plotted against V^% in Figure 19. B.4. Resolved Shear Stress-Shear Strain Curves for Zinc Single Crystal and Tungsten-Zinc Composites: When working with single crystals or composites with a single cry stal matrix, it is sometimes more appropriate to present the deformation curves in terms of resolved shear stress-shear strain curves. Kelly and 42 Nicholson have found that single crystals containing particles of a hard second phase deform not by slip in a single glide system, but by slip, in many intersecting systems. This leads to a stress-strain curve inde pendent of orientation. In the present work, the matrix of the W-Zn composite was a single crystal. At room temperature, only one slip plane (0001) is operative in (Continued on p. 53) CM e e E o» m O 5 " da I"— Experimental -Kelly et al. (Theo.) Fig. 19, m de 1 2 vf % 3 Vs. V % Plots of the Matrix of W-Zn Composites, ho 53 h.c.p. zinc single crystalsat the orientation involved. Hence, it is assumed that the matrix in W-Zn composites deforms by slip on the basal plane (0001). Even though there are three [1120] slip directions, the most favourable one is assumed to be operative. Based on these assumptions, the resolved shear stress-shear strain curves were obtained for W-Zn composites and pure zinc crystals. A 3 The relations used to obtain the resolved shear stress-shear strain curves from the load-elongation curves are: T = l~sin xo r -s/(f")2 - sin2V (25) o o and Y = -r1 (*/T-)2 - sin2X - cosX ) (26) sin x » 1 ° ° o o P is the tensile load. 1 is the gauge length after deformation, i.e., at P. A and 1 are the initial area of cross section and gauge length respec-o o tively. XQ is the angle between the specimen axis (tensile axis) and slip plane [(0001)] . X is the angle between the specimen axis (tensile axis) and the most favourable slip direction ([1120]) . T and y are resolved shear stress and shear strain respectively. The resolved shear stress^-shear strain curves, for some of the specimens are given in Figures 20a and 20b. Similar curves were obtained for the other specimens. Physical properties, and the important mechanical properties from the resolved shear stress-shear strain curves are given in Tables 6a and 6b. (Continued on p. 58) CM* 0 ? -—-J—-. , L L , J L 8 » I » I 0 50 100 150 200 250 300 350 400 450 500 Resolved Shear Strain, y% %. Fig. 20a. Resolved Shear Stress-Shear Strain Curves of a Zinc Crystal and W-Zn Composites. ui 0 I 2 3 4 5 6 7 Resolved Shear Strain, Fig. 20b. Resolved Shear Stress-Shear Strain Curves of W-Zn Composites. TABLE 6a TENSILE PROPERTIES OBTAINED FROM THE RESOLVED SHEAR STRESS-SHEAR STRAIN CURVES OF PURE ZINC CRYSTALS AND W-Zn COMPOSITES SP. NO. NO. OF WIRES AREA OF C/S 2 mm Vf% C.R.S.S. T SLOPE OF THE ELASTIC-PLASTIC REGION 2 gms/mm gms/mm FLOW STRESS AT 0.05% OFFSET FROM ELASTIC-PLASTIC REGION gms/mm2 MAX. FLOW STRESS , 2 gms/mm STRAIN AT MAX. FLOW STRESS % X-6 0 41.29 0 40 44 31 — 770 318 X-7 0 41.60 0 40 47 26 — — 680 380 C-7 6 22.18 0.023 41 45 38 7.3 X 10 88 230 31 C-8 6 25.99 0.020 40 42 30 . 6.8 X 103 . 100 500 140 C-16 6 28.13 0.018 42 46 26 3.3 X 103 73 810 286 C-46 6 28.03 0.018 44 47 45 2.7 X 103 93 800 464 C-13 12 28.51 0.036 41 46 31 5.3 X io3 :98 440 126 C-14 12 27.90 0.037 41 47 29 7.3 X 103 110 880 302 C-17 25 26.73 0.080 41 45 39 6.7 X 103 110 160 5.6 C-18 25 26.16 0.081 41 45 45 8.4 X io3 180 340 113 C-19 25 26.74 0.080 41 45 46 10.2 X io3 180 230 4.2 C-23 25 27.88 0.076 40 45 30 12.8 X io3 210 710 320 C-25 25 27.45 0.078 40 45 30 14.9 X 103 200 760 370 C-20 100 26.39 0.322 40 47 45 34.9 X 103 420 670 5.1 C-21 100 26.25 0.324 40 47 47 39.3 X 420 660 3.5 C-26 100 28.08 0.303 41 46 46 35.5 X io3 470 710 3.5 C-53 100 28.56 0.298 38 40 30 31.8 X 103 400 520 2.2 Os TABLE 6b TENSILE PROPERTIES OBTAINED FROM THE RESOLVED SHEAR STRESS-SHEAR STRAIN CURVES OF W-Zn COMPOSITES SP. NO. NO. OF WIRES AREA OF C/S 2 mm V % C«R*S«S* o Tc X 2 Q gms/mm SLOPE OF THE ELASTIC-PLASTIC REGION , 2 gms/mm FLOW STRESS AT 0.05% OFFSET FROM ELASTIC-PLASTIC REGION , 2 gms/mm MAX. FLOW STRESS , 2 gms/mm STRAIN AT MAX. FLOW STRESS % C-27 186 28.62 0.553 40 45 53 C-29 186 28.36 0.558 40 45 58 C-30 186 27.57 0.574 42 46 61 C-31 186 28.15 0.562 42 46 64 C-32 186 28.02 0.565 42 46 68 C-51 186 26.99 0.587 38 40 63 C-36 372 27.67 1.144 40 42 102 C-38 372 27.23 1.163 40 42 92 C-39 372 27.63 1.146 37 39 88 C-40 372 28.51 1.110 37 39 89 C-41 372 28.55 1.089 37 39 96 C-3 743 27.51 2.298 37 40 122 C-9 743 26.83 2.357 42 46 122 C-10 743 27.24 2.321 40 44 116 C-ll 743 27.78 2.276 38 42 112 C-58 743 26.82 2.357 40 44 121 C-42 1484 27.65 4.568 39 40 158 C-55 1486 27.59 4.583 39 40 155 71.7 X 10 790 1200 5.7 50.6 X 750 1000 2.6 78.0 X io3 790 1100 2.2 91.8 X 103 750 1100 1.8 92.5 X 103 770 1200 3.2 59.1 X io3 910 1100 3.2 20.9 X 104 1500 2500 4.2 21.9 X 104 1600 2300 1.8 20.9 X 104 1400 2200 2.2 15.2 X 104 1300 2200 4.4 16.8 X 104 1500 2200 3.2 26.8 X 2600 4100 4.3 33.5 X 104 3000 4900 4.4 28.9 X 104 3000 4500 3.2 33.3 X 2600 4100 2.0 30.0 X io4 3200 4600 4.0 56.4 X 104 5900 9300 4.6 67.7 X io4 5900 9100 6.7 58 Critical resolved shear stress (C.R.S.S.) T is defined to be c the shear stress at which the load-elongation curve first deviates from linearity. Tc values are plotted against in Figure 21. Also, the slope of the elastic-plastic region Is plotted against in Figure 22. The plots were made to see the effects of fibre reinforcement on the work hardening behaviour. Fig. 21. C.R.S.S. Vs. V % Plots of W-Zn Composites. vO • 2UJUJ/UJ6 gOlx '|0|d / SA i jo uoi69j Dj|SD|d-0!|SD|3 jo adois IV. DISCUSSION A. Scatter in tKe Experimental Results: The scatter in the stress-strain curves of pure zinc crystals is due to conditions of growth, different impurities present and the low angle boundaries present in the crystal. The small scatter in the stress-strain curves of tungsten wires is due to the non-uniformity of the surface and residual stresses present in the wires. The scatter in the stress-strain curves of W-Zn composites, contain ing the same volume fractions of tungsten fibres, may be due to: (a) poor bonding between some fibres and the • : 20 zinc matrix (b) misorientation of the fibres with the . 13, 32 specimen axis (c) presence of zinc oxide (d) presence of low angle boundaries in the single crystal matrix of zinc Ce) non uniform distribution of the tungsten fibres in the matrix. No quantitative measurements are possible to estimate the contri-r butions to the scatter from each one of the above factors. An estimation of the maximum scatter in the experimental results was made. The estimates are given in Table 7. 61 TABLE. 7 ESTIMATES OF THE SCATTER IN THE EXPERIMENTAL RESULTS MEAN MEAN VALUE PERCENTAGE SCATTER V,% gms/mm2 (Maximum) cr 2.316 . 9700 +8 cu a 1.126 200 ±6 cy d°c 4.575 2.05 x 106 ±8 de cr 0 3510 ±0.3 mu dgm 4.575 4.3 x 105 ±39 de C.R.S.S. 1.126 95 ±7 Slope of the elastic-plastic 4.575 62 x 10 ±9 region of x-y plot 63 B. Cause for the Greater Elongation of W-Zn Composites Compared to the Free Tungsten Wire: Some of the W-Zn composites tested exhibited elongations greater than the elongation of a tungsten wire tested separately. The composites containing a volume fraction of the fibres less than V^min cannot fail since the ductile zinc matrix supports the load even after the fibres have fractured. In the present work a value of 0.0095 is obtained for V^min., using expression (9). The extra elongation observed in W—Zn composites containing a 44 volume fraction of fibres greater than V^ min. is due to multiple necking of the tungsten wires in the composites. Multiple necking in the fibres 44 was also observed in W-brass composites. The multiple necking of tungsten wires in a brass matrix was found to result from local strain hardening of the brass matrix in the vicinity of each neck enabling the matrix to control composite deformation locally. C. Discrepancy Between the Experimental Strengths of W-Zn Composites and Values Predicted According to "Rule of Mixtures": (i) In the present work it was assumed that expression (4) holds good for stage I. Hence, the value of V^ crit. was calculated using ex pression (7). The value of V^ crtt. is found to be 0.0099. It can he seen from Figure 16 that yield stress varies linearly with V^% above and below V^ crit. (ii) Excluding stage I of the stress-strain curves of W-Zn com posites, the stress-strain curves predicted according to expression (3) 64 did not agree with, the experimentally obtained stress-strain curves for W-Zn composites. The higher values of the stresses found in the experi mental curves can be attributed to: Ca) hardening of the matrix due to the different effective Poisson's ratio of the two constituents of the .„ 11,30,45 composite, Cb) hardening of the matrix by dislocation pile-ups caused by blocking of the motion of the dislocations in the matrix by the fibres"^'"^ Hardening due to Ca) and Cb) can account for stage II deformation. In stage III hardening due to Ca) becomes negligibly small since both con stituents of the composite deform plastically and Poisson's ratio for each constituent becomes the same. Hence, in stage III hardening in the matrix can only be due to Cb). This continues until fracture. The values of ultimate tensile strengths of W-Zn composite obtained according to expression (2) are found to be lower than the experimental values, (Figure 15, and Tables 8a and 8b). These differences in the pre dicted values and the experimental values are accounted for by matrix harden ing due to dislocation pile ups. Also there may be a small effect due to oxide present. Figure 15 shows that the value of crit. predicted (0.99%) accord ing to expression C7) is higher than the experimental value of crit. CO.825%) This also can be accounted for by matrix hardening. The extent of matrix hardening depends on the volume fraction of the tungsten fibres present in the matrix. This can be seen from Figure 18 in CContinued on p. 67) TABLE 8a COMPARISON OF THE EXPERIMENTAL AND THEORETICAL VALUES (W-Zn Composites) SP. NO. Vf% da de gms/mm Exp. 2 da Theo. da Theo. de de Eqn. (33) Eqn. (39) gms/mm2 gms/mm2 U.T.S. Exp. 2 gms/mm U.T.S. Theo. Eqn. (22 gms/mm da m de Exp. gms/mm da m Theo. de Eqn. (40 gms /mm' X-6 0 — X-7 0 — — — C-7 0.023 2.8 X < 8.7 X 103 8.4 X io3 C-8 0.020 3.1 X 7.4 X 103 7.2 X io3. C-16 0.018 1.2 X 103 6.9 X 10l 6.64 x 10^ C-46 0.018 9.6 X 103 6.9 X 103 6.66 x 10J C-13 0.036 2.7 X K 1.4 X 1.3 X < C-14 0.037 3.0 X io4 1.4 X 104 1.3 X 104 C-17 0.080 2.9 X 3.0 X K 2.9 X C-18 0.081 4.0 X K 3.1 X 104 3.0 X K C-19 0.080 4.9 X 104 3.0 X 104 2.9 X iot C-23 0.076 6.3 X 104 2.9 X K 2.8 X iot C-25 0.078 7.7 X 104 2.9 X 104 2.8 X 104 C-20 0.322 1.8 X 105 1.2 X io5_ 1.2 X 105 C-21 0.324 2.2 X 105 1.2 X 1.2 X 10S C-26 0.303 1.6 X 105 1.1 X 105 1.1 X 105 C-53 0.298 1.4 X io5 1.1 X 105 1.1 X 105 3520 3500 a = 88 m. m = 88 520 170 2.0 X < 4.3 X 1500 160 2.4 X io4 3.7 X 3400 150 5.5 X 103 3.4 X 4600 150 3.1 X 103 3.4 X 1300 220 1.4 X K 6.7 X 3800 220 1.7 X io4 6.9 X 350 370 4.4 X 104 1.5 X 940 380 1.1 X 104 1.5 X .480 370 2.1 X 104 1.5 X 3200 360 3.6 X 104 1.4 X 3800 370 4.9 X 104 1.5 X 1500 1200 6.8 X 10J 6.0 X 1500 1200 1.0 X 104 6.1 X 1600 1200 5.2 X 10t 5.7 X 1100 1200 3.2 X 104 5.6 X 10' 10, io; io; x io; io-ui TABLE 8b COMPARISON OF THE EXPERIMENTAL AND THEORETICAL VALUES (W-Zn Composites) SP. NO. V % da de gms/mm Exp. 2 da c de Theo. Eqn. (33) gms/mm2 da c de Theo. Eqn. (39) 2 gms/mm U.T.S. Exp. U.T.S. Theo, Eqn. (2) 2 gms/mm gms/mm da m de Exp. gms/mm da m de Theo. Eqn. (40) gms/mm2 C-27 0.553 3.4 X 10* 2.1 X 10 2.1 X C-28 0.558 2.4 X 10* 2.1 X 10* 2.1 X C-30 0.574 3.8 X 10^ 2.2 X 10* 2.1 X C-31 0.562 3.9 X 105 2.1 X 105 2.1 X C-32 0.565 3.9 X 10* 2.1 X 10* 2.1 X C-51 0.587 2.9 X 10* 2.2 X 10* 2.2 X C-36 1.144 7.7 X 10* 4.3 X 10* 4.4 X C-38 1.163 8.2 X 105 4.4 X 10* 4.4 X C-39 1.146 8.2 X 10^ 4.3 X 105 4.4 X C-40 1.110 6.3 X 10* 4.2 X 10* 4.2 X C-41 1.089 7.0 X 10* 4.1 X 10* 4.1 X C-3 2.298 1.1 X 106 8.7 X 10* 9.1 X C-9 2.357 1.4 X 106 8.9 X 10* 9.53 X C-10 2.321 1.2 X 106 8.8 X 10* 9.2 X C-11 2.276 1.3 X 106 8.6 X 105 9.0 X C-58 2.357 1.3 X io6 8.9 X 10* 9.3 X C-42 4.568 1.9 X K 1.7 X 106 1.9 X C-55 4.583 2.2 X io6 1.7 X 106 1.9 X x io: io; io; io; x 10" x io; x io; io-io: x io; x 10" 10 w-fibre 100 2600 2100 1.4 X 10 1.0 X 2300 2100 3.9 X 105 1.0 X 2400 2100 1.7 X 10* 1.1 X 2300 2100 1.9 X 105 1.1 X 2500 2100 1.9 X 1.1 X 2400 2200 8.1 X 104 1.1 X 5200 4200 3.7 X 105 2.1 X 4800 4200 4.1 X 10* 2.2 X 4700 4200 4.2 X 10* 2.1 X 4800 4100 2.3 X 10^ 2.1 X 4800 4000 3.1 X 10* 2.0 X 8900 8300 2.4 X 10* 4.2 X 10400 8500 5.3 X 10* 4.3 X 9700 8400 3.7 X 10* 4.2 X 9100 8200 4.8 X 10* 4.2 X 10000 8500 2.7 X 10* 4.3 X 19400 16400 2.7 X 10* 8.1 X 19500 16400 6.0 X 10* 8.1 X 6 x 10* 3.6 x 10* x 10 x 10 x 10, 10, 10L x 10, 10 10 10 ON ON 67 which the derived matrix stress-strain curves do not follow the stress-strain curve of the pure zinc crystal.' D. Stage II of the Stress-Strain Curves of W-Zn Composites: 30 Hill gave an upper and lower bound for the slope of the second stage of a composite. Tanaka et al.^ predicted the slope of the second stage and this was found to be in good agreement with the experimental re sults of Kelly et al.^~ The theoretical and the experimental values are compared and discussed. (a) Hill:30 In Hill's derivation, a single fibre with circular section embedded in a matching circular cylindrical shell of matrix was considered as a rudimentary composite. It was supposed that the composite cylinder was subjected to uniform lateral pressure P and to axial mean ten sion T acting through rigid constraints keeping the ends plane. By stress analysis and by frequent use of "law of mixtures," Hill derived an expression which is given by: 4V V (vf - v )2 E _ - V.E - V E = * m * - (27) cl f f m m Vj. .V He ' k. G mp fp m where k and k. are the plane strain bulk moduli for lateral dilatation, mp tp • •• ' without longitudinal extension, of the matrix and the fibre respectively. Hill obtained the bounds on E^ using elastic extremum principles 46 of potential energy and Reuss and Voigt estimates: 68 The Voigt treatment of reinforcement assumes that the strain throughout the mixture is uniform. Hence he gave an expression, ^Vf+Vm (28) for a polycrystal mixture, where is the overall Young's modulus of the mixture and E^ is the Young's modulus of the polycrystalline matrix. Reuss, also for a polycrystal mixture, assumed that the stress is uniform throughout the matrix and gave the expression, R f m where E is the overall modulus of the mixture. Neither assumption is correct: the Voigt stresses are such that the tractions at phase boundaries would not be in equilibrium, while the implied Reuss strains are such that inclusions (fibres) and matrix could not remain bonded. From these two estimates it is shown that ^ < [E - VfEf + VmEm] < Ev (30) The equality is only valid when = v . On the basis of similar arguments Hill obtained bounds on the plane strain bulk modulus, kc> of the composite for different transverse rigidities of the fibre and matrix.This in turn gives the bounds for E^, for elastic behaviour of the compound cylinder, as: 4V,V (v -v )2 . 4V V.(v.-v)2 v£ m I m ~ < E - VfE. - V E < * m I m (31) V, V -\ — cl f f m m — V, V . ... (-1— 4. m + r t 4- m I 1 s, (k + k. + G 'ClT" + kT G. } mp fp m mp fp f 69 for G- > G , where G. and G are the rigidity moduli of the fibre and r — m r m the matrix respectively. k and k,. are the plane strain bulk moduli of fibre and matrix mp f p respectively. These bounds are derived for the composite when matrix and fibre are both elastic. For inelastic behaviour, i.e., when the weaker phase (matrix) does not harden and the stronger phase (fibre) remains linearly elastic, v becomes equal to -jL k becomes K and E _ becomes the slope of the m ^ 2mp , m • , cl der . do" c c elastic-plastic region (stage II) . can also be denoted by E T_. de d£ J ell da Hill obtained an expression for ~r~~ of the elastic-plastic (in elastic) region of the composite. The expression is: de _f_ m 1_ K kc G m f p m da The bounds on — were also derived and are given by: de b J VrV (1 - 2vr)2 da. V£V (1 - 2vr)2 _XJ2 L_ < —~ - v E < 1— (33) Vc V . -de f f - V£ V • ^ J f , m , 1_ _f m • 1_ K k. G K k. Gc m rp m m fp f for G > G , also V + V, = 1. l — m m r da —— values calculated for W-Zn composites using the equality at Hill's upper bound in expression (33) are plotted in Figure 17. The experi-dac mental values of — obtained for W-Zn composites are higher than the values 70 predicted (Tables 8a and 8b). This discrepancy is accounted for by the assumptions made in deri ving the expression (33). Expression (33) is derived on the basis of dif ferent contractions of the two phases and gradual work hardening of the matrix. Experimental results show that matrix hardening is not gradual in the composite. Also, lateral constraint is not the only reason for work hardening of the matrix material. There is also hardening due to dis location pile ups in the matrix. Cb) Tanaka et al.:^~> Tanaka et al. predicted the slope of the second stage of a composite stress-strain curve. Their prediction was essentially in good agreement with the experimental values of Kelly et al.^ Their prediction was also checked in the present work. Tanaka et al. assumed that the matrix deforms plastically and that dislocations are present in contact with the interfaces. Homogeneous plastic deformation takes place by the operation of several slip systems. An expression was developed for the Gibb's free energy of a fibre-reinforced composite as a function of a plastic strain of the matrix. Thermodynamic stability led to an equilibrium relation between the applied stress a and the plastic strain in the matrix c . The linear c P 33 relation is given by: .... a . . AE.V.e . _ o , m f p , CTc " Cl - BVf) + Cl - BVf) C34) where cr^ is the yield stress of the composite, 71 3C1 - 2v,)E + 2(1 + v )E, . E, A _ t m m f ( f A (1 + v,)(l - 2v.)E + Cl + v )E, 4E } LJB) r r m m r m Cl - 2v,)E + E, , T, _ i m r . . ,1 (1 + v,)(l - 2v_.)E +(1 + v )E. u + VmME ; (36) ffm m f m v/ where A is a coefficient connected with the elastic energy- change. B is a coefficient connected with the external potential energy change. In a tensile test the recorded total strain e is the sum of the elastic strain and the plastic strain. For a composite e is given by: 1 £ = £P(1 - v+ Is- C37) c Ec is the overall Young's modulus of the composite (same as ^cj)• From ex pressions (34) and (37) , the measured rate of hardening in stage II is given by: da A V_.E de' ' .x-2 _,AV,E us; r(l-BVf) —~——] c Expression.(34) was derived on the assumption that low volume fractions of the fibres were present in the composite and that the stress field of any fibre did not interact with that of another. This involves ignoring the disturbance of the internal stress around a fihre by another fibre, which should affect not only the elastic energy but also the inter action energy with applied stress. The effect of the disturbance of the internal stress by other fibres is considered on the assumption that a fibre is surrounded by a phase which has elastic constants determined by the over all elastic constants of the composite. 72 Hence the hardening rate for a general volume fraction is given by: da AVi dr • —SLXiHr C39) fle 1(1 - B V_r + A V.] c r c t where Ac and B^ are calculated from A and B replacing Iexpressions (35) and (36)] E and v by E and v respectively. As a crude approximation m m E and v can be obtained by the "rule of mixtures." c c da values were calculated using expression (39). Calculated values are plotted in Figure 17. Also, calculated values are compared with the experimental values in Tables 8a and 8b. The experimental values are da c higher than the predicted values of -j—— . The composite containing 4.5 volume per cent of fibres was found to give a slope which agreed with the predicted value within the limits of experimental scatter. This may be due to mere coincidence, or the prediction may be in good agreement with the experimental values of W-Zn composites containing more than 4.5 volume per cent of fibres. It is also worth men tioning that Tanaka's predictions did not agree with experimental values obtained for W-Cu composites containing less than 10 volume per cent fibres. Predicted values were lower than the experimental values of W-Cu composites. In the present work disagreement can be accounted for by inhomo-geneous deformation, and by work hardening of the matrix due to dislocation pile ups. 73 E. Stage II of the Derived Stress-Strain Curves of the Matrix Correspond ing to the Second Stage of the Composite Stress-Strain Curve: Kelly"'"''" tried to explain . the work hardening behaviour of the matrix especially in the second stage of the composite. In his work on W-Cu composites, stress-strain curves for the matrix were derived from „ • 1 do the composite stress-strain curves. The slope of the second stage, . m de of the derived stress-strain curve was predicted and compared. The pre-diection was made on the basis of different lateral constraints of the constituents of the composite. n 34 dCT Neumann et al. predicted on the basis of dislocation de pile ups and hence predicted a value for the slip band spacing for W-Cu . ; . composites. 11 3 A The predictions of Kelly et al. and Neumann et al. are con sidered in the present work. (a) Kelly et al.^: The expressions given for the Young's moduli E^-j. and E^^. on the basis of the rule of mixtures, and on the assump tion of equal strains are true only if the Poisson's ratios are equal, i.e., = v^. When they differ, a lateral stress arises which is proportional to 2 2 v - v. . When fibre and matrix are elastic the value of Iv - v.I ; is 1 m f1 1 m f1 small and hence the effect of lateral stress on E _ is small. When the cl matrix yields plastically, the effective v becomes 0.5 since there is m 2 little work hardening, and the value of - | becomes much larger. This may affect the stress-strain curves of composites. Kelly et al.., following Hill's ideas and Love's47 solution of the problem of a tube under internal pressure, tried to explain the effect of 74 lateral constraints and derived an expression for the slope of the second linear portion of the derived stress-strain curve of the matrix correspond ing to stage IX of the composite. The model used was a composite cylinder, similar to that of Hill, with a stiffer phase surrounding the weaker phase. Considering the weaker phase to be a fluid with Poisson's ratio 0.5 and the bulk modulus of the matrix, and the stiffer phase to have the elastic properties of the rein-da forcing material, he derived the expression for which is given by: de der " V (1 - 2v )2 IT" V V , ' Vf +Vm = 1 WO) K k, G, m fp f do This expression is the same as that of Hill for —. II deda The values calculated for tungsten-zinc composites using the above expression are plotted against V, in Figure 19, and are tabulated in dam" Tables 8a and 8b. The calculated values of -z are lower than the experi-de mental values, even though the predicted values give an upper limit to the slopes. This sort of discrepancy was also observed by Kelly et al.^ and by 48 Stuhrke, Kelly et al. gave an explanation for the discrepancy by suggest ing that the matrix does not yield completely; i.e., a portion of the matrix remains elastic during the second stage. But no experimental evidence for this sort of behaviour is reported. This seems to be unrealistic because the matrix cannot remain elastic when higher stresses are involved. Predicted II da values of show that difference in the lateral constraints of the con-de. stituents is not the only cause for matrix hardening. The other contribution 75 may be due to dislocation, pile ups. The presence of dislocation pile ups can be qualitatively supported: Ca) Dislocation pile ups in the matrix would produce stress concentrations in the fibres, and hence fibres would appear weaker in the composite than when tested individually. This is obvious since most of the composite specimens fractured at lower elongations than that of the fibres tested individually (Tables 5a and 5b). (b) The large stresses in the matrix decrease as soon as the yield strain of the fibres tested alone is reached. This can also be seen in Figure 18. (c) The large stresses in the matrix should cause yielding of the fibres at a lower strain than when tested individually. It can be seen from Figure 14b that all the composites have yielded in stage II at strains lower than that of the fibre tested individually. 12 Pinnel et al. made observations on stainless steel-Al composites using transmission electron microscopy. Specimens deformed to strain -3 levels <_ 8 x 10 showed that the matrix deformed uniformly and the dislo cation densities and configurations were independent of distance from the matrix-fibre interface. Operation of multiple slip systems may be the reason for the absence of dislocation pile ups. (b) Neumann et al.^^: Neumann et dl. explained the work-harden-11 ing rate observed for the matrix by Kelly et al. on the basis of a system of parallel dislocation pile ups between the fibres. A cylindrical specimen 76 with, axial fibres was considered. The fibre direction was the y-direction. The assumed dislocation model is shown in Figure 23. Under tension in the y-directlon edge dislocations infinitely long in the z-direction pile up on slip planes of spacing against boundaries atx = ± — , where a is the fibre spacing. 49 - -Read calculated x-- in the coordinate system (x, y), for an in finite row of dislocations spaced at regular intervals. When the disloca tion wall is at x = 0 the x— value is given by: xy J , Gf y = 0) = ®L Re-{2*L— - ! ct8 f> 5 £- x (i-) 2y2(l-v)Dy gi.2.^ »y \ d (41) D (1 + 1) where d = —^ ; G, b and v are the shear modulus, Burgers vector and n yjl Poisson's ratio of the matrix respectively; Re is the real part of the func-tion inside brackets; and x(—) is a function of —. y y A continuous distribution of dislocation walls was considered. Then the sum of the Burgers vector of all the dislocations that lie on the slip planes between the positions x and x + dx is [b D(x)dxJ. D(x) is a distribution function of x. The shear stress on the slip plane at a point x1 = x with y = 0 is: a. xr (x, y = 0) = / DCx') |-x CX-^') dx' (42) -a y y i - sfl -a a x (x, o) is the^ shear stress in the interval — and — . In equilibrium . n V2 v/2 i — i — i a o* the total stress x (x, o) + =0 for all |x| _< —. a Is the applied ~\/2 tensile stress in the matrix In the y-direction. (Continued on p. 78) 77 fibre fibre I " VIQU" Fig. 23. Dislocation Pile-up Model. 78 The average plastic strain.e in the y direction for |x| < due to the formation of the pile ups is given By: a_ b ^ e = f-J" / x' D(P) dx' (43) y* -a. An Investigation of the^roperties of T"*"(X, O) and e. • leads to the following expression: e = oa • f(S-) C44) y where f (—) is a function of — , and can Be obtained by numerical compu-y y tatlon of the integral expression for e^. However was approximated analytically using the case of an isolated pile up given By LeiBfried."^ For D'» a, e. is given by: £P " 4 rTeT (45) y £ is the longitudinal total strain due to the stress a in the matrix far X. EL outside of the pile up, also equal to e. On the other hand for D « a, the expression for e reduces to: y P p T The simple interpolation function which can satisfy the extremum of £p is given by: I = e-T II - exp - (J' g-) j C46) y 34 Neumann et.al. gave an expression for the derived matrix stress it cr^ , which is given by: ^"^T' a) = Era r [e - Z(<r"i f-)J C47) m T matrix T p m D - "a y where e (& —) is the interpolation function given earlier, p m D y • On substituting the Interpolation function for e. in the ex it pression for a . it can be written: m am"C£T> a) - Ematrix V (1 " ^~^} <*8> y On differentiating this expression the following expression can be written: II da = E ' {l - [l - exp(- y-£-)']} : (49) de^ matrix . 4 Dv In this expression k'may be geometrically related"'"'" to the fibre diameter 0 by: a = 0( /(—T—)- 1 ) C50) V 2^ V da assuming a hexagonal fibre arrangement, gives the slope of the derived ET da stress-strain curve in the second stage. The expression for -j gives an T analytical expression with as the only adjustable parameter. Neumann et al. found a 9% error in the interpolation function given for e^. After a slight modification of this interpolation function on the basis of numerical computation, the error was estimated to be 3%. da " Using the modified interpolation function he calculated the value of d&T -for different values of to obtain agreement with the experimental values of Kelly et al.^ = lOp Is found to satisfy the experimental slopes obtained for tungsten-copper composites of different . In the present work the unmodified expression (49) was used to . .da- " calculate the slope -j^—by giving different values to D . It was found T. 80 that a single value of could not give the experimental slopes for different of the tungsten-zinc composites. The different values of D obtained for different V, are tabulated along with the average experi-7 dam" mental values of —— in Table 9. The D values which can be calculated from de y slip band spacing, require experimental confirmation. In the present work the surfaces of the composite specimens were not good enough for the direct measurement of slip band spacing. The sur faces could not be improved by any sort of increasedvpolishing because this exposed the fibres on the surface of the composite. It should be noted that the predicted values of may be higher than the actual value. The hardening of the matrix is caused by two phenomena taking place in the matrix: (a) hardening due to difference in lateral constraints of the constitutents; (b) hardening due to dislocation pile ups. F. Resolved Shear Stress-Strain Curves of the Composites: These are given only to show that stress-strain curves of the composites can be represented in the form of resolved shear stress-shear strain curves. C.R.S.S. varies linearly above and below crit. Cfound theore tically) . The slope of the elastic-plastic region is found to vary linearly with V,.. TABLE 9 n VALUES OBTAINED FOR W-Zn COMPOSITES 7 ACCORDING TO NEUMANN et al. AVERAGE Vf% AVERAGE da m de EXP. 2 gms/mm D VALUES y MICRONS 0.020 1.3 X io4 260 0.0365 1.6 X 104 196 0.079 2.3 X io4 141 0.312 4.1 X io4 76 0.566 1.4 X 105 771 1.130 3.4 X io5 62 2.320 3.8 X io5 42 4.580 4.3 X io5 29 V. SUMMARY AND CONCLUSIONS 1. W-Zn. composites show four stages of tensile deformation. 2. In stage I, since no precise measurement of the strain was possible, the "rule of mixtures" prediction was assumed to be true for calculating strain. 3. The experimental values of the slope of stage II were appreciably higher than predicted values. The discrepancy was attri buted to matrix hardening by: (a) different lateral constraints of the fibre and the matrix; (b) dislocation pile ups in the matrix near the fibres. 4. The experimental values of the ultimate tensile strength of the W-Zn composites were found to be higher than the values predicted according to "rule of mixtures". This was attributed to the matrix hardening due to dislocation pile ups in the matrix at the W-Zn inter face. 5. The experimental value of V, crit. was found to he 0.825%. 82 VI. SUGGESTIONS FOR FUTURE WORK Several lines of investigation can be suggested from the dis cussion of the present work. These include: (a) Further development of the experimental tech nique to obtain good surfaces of W-Zn composites for the measurement of the slip band spacings in the matrix. (b) Observations on the existence of dislocation pile ups using an etch pit technique. (c) A study of the deformation characteristics of W-Zn composites containing greater than 4.5 volume per cent tungsten fibres. 83 BIBLIOGRAPHY 1. Kelly, A., Davies, G. J., Met. Rev.' 103 1, (1965). 2. Pattnaik, A. and Lawley, A., Met. 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