m5: 00%: :53 w A _ 9V . 1mm” . m. y E m m Mm, m. Fm 31m. € eéiz farthebegr 8 . a WWW MEGPRQAR S? if 5. a I w. v u . an. .9; 9 furl” .13.. :21, km Y‘Jffl" Michigan State University This is to certify that the thesis entitled THE EFFECT OF BIAXIAL LOADING ON THE CRITICAL RESOLVED SHEAR STRESS OF ZINC SINGLE CRYSTALS presented by Jerry Allen Barendreght has been accepted towards fulfillment of the requirements for Ph. D. degree in Mechanical Engineering ABSTRACT THE EFFECT OF BIAXIAI.IOADING ON THE CRITICAL RESOLVED SHEAR STRESS OF ZINC SINGLE CRYSTALS BY Jerry Allen Barendreght The macroscopic deformation of {0001} <2ll0> type slip in hexagonal close-packed zinc single crystals is investigated by sub- jecting the crystals to a uniform biaxial state of stress. This was accomplished by loading flat tensile specimens in both the axial and transverse direction. The transverse load was applied with specially designed rubber grips. This allowed the effects of the crystal orientation and the resolved normal stresses on the active slip system to be uncoupled. In this study the crystal orientation was held constant and the resolved normal stresses at yield were varied by varying the biaxial stress ratio at yield. The design of the loading configuration that resulted in the largest region of uniform biaxial stress was verified by extensive elasticity and photoelasticity investigations. The elasticity in- vestigation established the geometrical limits for the flat tensile Specimen by modelling the specimen as a finite rectangular beam loaded transversely along it's sides. Fourier analysis was utilized in solving for the stresses in the beam. The photoelasticity investiga- tion with models tested at these limits verified the elasticity solution, established the geometry of the rubber grips and determined Jerry Allen Barendreght the limit that was experimentally practical. A series of uniaxial single crystal tests established the critical resolved shear stress to be 70 1:2 grams/mmz. The results of the biaxial shear stress experiments showed that the critical resolved shear stress of zinc single crystals decreases as the resolved normal stresses acting on the active slip system increases. These results were for {0001} <2Il0> type slip when the angle between the slip plane and the tensile axis and the angle between the slip direction and the tensile axis are both 45 degrees. THE EFFECT OF BIAXIAL LOADING ON THE CRITICAL RESOLVED SHEAR STRESS OF ZINC SINGLE CRYSTALS BY Jerry Allen Barendreght A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering 1971 ACKNOWLEDGEMENTS The author wishes to express his appreciation to Dr. W.N. Sharpe, Jr. for his guidance and Support throughout the course of this study. He appreciates the guidance of the other members of his graduate committee, Dr. R.T. Hinkle, Dr. H.1n WOmochel, and Dr. G.L. Cloud. The expert assistance of Dr. KAN. Subramanian with reSpect to crystal growth and preparation is gratefully acknowledged. The assistance of Dr. G.L. Cloud in regards to suggestions for the photoelasticity setup is deeply appreciated. Also, the helpful discussions and assistance of Dr. Raw. Little with the elasticity solution and Dr. J.L. Lubkin's computer results used to verify the author's computer program are gratefully appreciated. The author is thankful for the patient encouragement of his wife, Sharon. Finally, Special thanks to Mr. J.W. Hoffman and the Division of Engineering Research, College of Engineering for continuous support. ii Chapter I II III IV TABLE OF CONTENTS Acknowledgements List of Tables List of Figures INTRODUCTION 0......OOOOOOOOOOOOOOIOOOOOOOOOOOOOOO mADmG CWFIGURATION OOOOOOOOOOOOOOO A. Elasticity Investigation ..................... 1. Theory ................................... 2. Results ................................ B. Photoelasticity Investigation ................ 1. Specimen Details ......................... 2. Specimen Preparation ..................... 3. Testing Procedure ........................ 4. Experimental Results ..................... C. Discussion of Results ........................ SHEAR STRESS EXPERIMENTS C$c>w > mNchIm OOOOOOOOO0.00.00.00.00000000000000000CO THE BIBLIOGRAPHY APENDIX 00......0.0...0.0.0.0000...OOOOOCOOOOOOOO . Crystal Growth ............................... . Specimen Preparation ......................... . Testing Procedure ............................ . Shear Stress Results ......................... GeneraIReferenCe 00......OOIQOOOOOOOOOOOOCOOO Computer programs 1. Calculation of stress distribution in a finite rectangular beam ......... 2. Calculation of principal Stress difference distribution in photo- elastic models ....................... 3. Calculation of resolved shear Stress and resolved shear strain ............ iii Page ii iv 11 13 13 16 23 26 29 35 36 38 44 47 48 56 6O 89 91 93 94 97 99 Table LIST OF TABIES Results of Loading Photoelastic Specimen A in Transverse Direction Results of loading Photoelastic Specimen B in Transverse Direction Results of Loading Photoelastic Specimen B Biaxially Theoretical Shear Stress Distribution for Transverse Loading - H/C = 0.5 Sumary of Data for Shear Stress Experiments iv Page 39 4O 41 42 85 Figure 10 LIST OF FIGURES Coordinates for calculating resolved shear stresses Biaxial loading of a flat tensile Specimen Coordinate transformation and stress definition Resolved normal stress as a function of or ientat ion (¢) Finite rectangular beam for elasticity model Stress distribution in the test section of a finite rectangular beam for x1/H = 0.0 and H/L << 1 a. 011/011 vs. H/C b. 012/0111 vs. H/C I c. 022/011 vs. H/C I d. (022 - 011)/011 vs. H/C Stress distribution in the test section of a finite rectangular beam for X 1/H = 0.5 and H/L << 1 a. (022 - 011)/011 vs. H/C I Stress distribution in the test section of a finite rectangular beam for X 1/H = 0.75 and H/L‘<< 1 a. (022 - 011)/011 vs. H/C . b. 012/011 vs. H/C Stress distribution in the test section of a finite rectangular beam for X 1/H = 1.0 and H/L << 1 a. (022-011)/all vs. H/C Circular polariSCOpe (schematic) Page 14 17 18 18 19 20 20 21 21 22 22 25 Figure 11 12 13 14 15 l6 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Fractional fringe order determination Loading frame for photoelasticity investigation Load cell and load cell calibration curves Photoelasticity Specimen details Bondability index of common elastomers Rubber grip details a. Rubber grip cutting jig b. Gluing jig for photoelastic specimens Effect of rubber grip slits on the Stress distribution in a finite rectangular beam. Slits cut to within 1/8 inch of bonded interface Calibration curve for photoelasticity modelling material Zinc specimen details Crucible design Specimen packed in crucible (schematic) Crystal growing furnace (schematic) Specimen holders a. Holder A b. Holder B Wire saw Gluing jig for zinc specimens Aluminum grip details Biaxial testing machine Effects of improper alignment and no prestress on biaxial loading of Specimen Typical nominal stress-strain curve Photomicrograph of slip in Specimen's test section vi Page 27 27 28 30 31 32 33 33 34 37 45 49 50 51 52 54 S4 55 57 59 61 62 Figure 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 Nominal stress-strain curves Nominal stress- Nominal stress- Resolved shear stress-strain Resolved shear stress-Strain Resolved shear stress-strain Resolved shear Stress-strain strain curve strain curve Nominal stress-strain curve Nominal stress- Nominal stress- Nominal Stress- and 15 Nominal stress- Resolved shear stress-strain Resolved Shear Resolved shear Resolved shear Resolved shear Resolved shear Resolved Shear Resolved Shear Resolved shear Resolved Shear stress to To strain curve strain curve strain curve strain curve stress-strain stress-strain stress-strain stress-strain stress-strain stress-strain stress—strain stress-strain stress to T at yield vii 0 Test No. 1 and 18 Test No. 4 Test No. 5 curve Test No. 1 curve Test No. 4 curve Test No. 5 curve Test No. 8 Test No. 9 and 10 Test No. 11 TeSt No. 12 Test No. 13, 14, Test No. 16 and 17 curve Test No. 9 curve Test No. 10 curve Test No. 11 curve Test No. 12 curve Test No. 13 curve Test No. 14 curve Test No. 15 curve Test No. 16 curve Test No. 17 - resolved normal Page 63 65 66 69 71 72 73 74 75 76 77 78 79 8O 81 82 83 84 88 I INTRODUCTION The usual testing procedure in the investigation of the mechanical properties of single crystals is uniaxial tension or compression. It has been shown that, when a uniaxial stress is applied to a specimen and macroscopic yield occurs, the resolved shear stress acting on a slip system is constant and is independent of the crystal orientation with reSpect to the tensile axis of the specimen (1, 2). However, there have been cases where the critical resolved shear stress has been found to vary (3, 4, 5, 6). All of these experiments have been in uniaxial tension or compression, where a variation in the crystal orientation is accompanied by a variation in the resolved normal stresses on the slip plane. The concept of critical resolved shear stress in single crys- tals is a well known concept first investigated by Schmid (1). Schmid showed that the yield stress of hexagonal metals (cadmium, zinc, and magnesium) varied greatly with orientation. Later, Schmid, Boas et 81., (2) showed that when the tensile stress at was converted to a resolved Shear stress using, T = Ct sin x cos Y , the resulting shear stress at yield was constant for each metal. This constancy of yield stress is usually referred to as Schmid's law. Y is the angle between the uniaxial Stress axis and the slip direction, and ¢ is the angle between the uniaxial stress axis and the slip plane normal. See Figure 1. T is the shear stress on the slip plane and the Shear stress at yield iS referred to as the critical resolved Shear stress (CRSS). The CRSS of many metals was examined further by different investigators who found differences concerning the constancy of the CRSS (or To) for all orientations of the standard stereographic triangle. Opinion on the orientation-dependence of now is *0 divided, partly because of difficulties inherent in the accurate measurement of To. Often it is very difficult to determine when plastic flow commences; and T0 is quite structure-sensitive, being affected markedly by trace impurities or dislocations introduced in handling (7). Rosi and Mathewson (8) investigated high-purity aluminum single crystals for the change in the CRSS with temperature and found that the Schmid law was obeyed. They used this uniformity of behavior as an indication that their method for producing and preparing tensile- test Specimens resulted in structurally-uniform single crystals. Rose (9), studying plastic properties of copper crystals, found that the Schmid law was confirmed (with the possible exceptions of the [100] and [111] orientations). The investigation of Penn, Hibbard and Leppers (10) on axial extension of alpha brass (70/30) single crystals Show good agreement with the critical resolved shear stress law. More recently, Hassen (3) expressed the opinion that at "'0 a given temperature is independent of orientation in nickel; although, some experimental Scatter was observed in his experiments. On the other hand, Andrade and Aboau (4), and Diehl (5) found that, in Slip direction Figure 1. Coordinates for calculating resolved Shear stresses. copper, orientations near the center of the stereographic triangle gave nearly constant values of but this result was no longer To, true for orientations approaching the boundaries of the triangle where the operation of other slip systems becomes more likely. Further evidence that FCC single crystals do not obey the Schmid law was demonstrated by Lficke and Lange (6) using 99.5 and 99.99 percent pure aluminum Single crystals. In Maddin's and Chen's review of "Glide in Face Centered Cubic Metals," they stated that "Similar results (to Lficke and Lange) for high purity copper Single crystals have been reported by Cupp and Chalmers" (11). Barrett (12) reported that an increase of hydrostatic pressure increased the flow stress for nickel and aluminum during plastic deformation; earlier, he had stated that the normal stress from hydrostatic pressure up to 40 atm. had no effect on the CRSS (13). Hull, Byron and Noble (14) reported that tantalum and Silicon- iron single crystals having orientations between [110] and [111] along the edge of the unit triangle obey the Schmid law when deformed in tension. Tungsten single crystals, however, do not obey such a law in this region and exhibit a change in Slip system which cannot be accounted for by geometrical considerations alone. Also, their results for compression tests cannot be explained in terms of a Simple Schmid law. HCP metals have agreed with the Schmid law more closely than have ECG and FCC metals. Jillson (15) found that zinc in uniaxial tension was highly consistent with the Schmid law for slip along the basal plane; Similar results were obtained by Burke and Hibbard (16). This constancy of CRSS also has been demonstrated for cadmium and magnesium. Since theories for predicting the stress-strain curve of a polycrystalline aggregate use as a basis for predicting the yield “‘0 stress, it is evident that the dependence of T upon complex stresses 0 needs to be understood. Honeycombe (17) reviewed those theories which led generally to the determination of the mean orientation factor m for the relationship a = RT, between the tensile stress and the resolved shear stress. Sachs (18) and Taylor (19) found values for m of 2.238 and 3.06 respectively, and BishOp and Hill (20) confirmed that the approximate value for m is about 3.1. When these values for m were calculated, the Schmid law was always assumed valid. The validity of this law has been shown only for uniaxial tension in HCP single crystals but not for the complex state of stress actually occurring in crystals in the aggregate. All of the investigators mentioned above have studied the mechanical properties of metals subjected to uniaxial stress fields. This type of testing leaves the normal stress and orientation effects coupled. The purpose of this experimental investigation is to test the hypothesis that macroscopic yield in single crystals is determined only by the shear stress on the active Slip system and is independent of the resolved normal stresses. This hypothesis is, of course, based on the assumption that all other variables Such as dislocation density, impurities, oxide films, temperature and etc. can be held constant. In order to test the hypothesis, one needs to separate the effect of the two above mentioned variables, namely, the crystal orientation and the resolved normal stresses. This separation of variables was accomplished by testing single crystals of zinc in a uniform biaxial state of stress. The uniform biaxial stress field was imposed on an ordinary flat tensile specimen by gluing rubber grips to the edges as shown in Figure 2. The relation between stresses on the slip plane and a biaxial load may be understood from Figure 3. The crystals were oriented such that the slip plane normal, the slip direction, and the tensile axis were all in the same plane. For this Special orientation, the X'X'XS coordinate system is simply a rotation of 9 degrees about 1 2 the X3 axis. Xi is perpendicular to the slip plane and X' is 2 colinear with the slip direction and Y = x = 90 - ¢. The stresses in the unprimed axis are transformed to the primed axis by ' = °ij 0’11- “js °rs ' (2) The first subscript indicates the direction of the normal to the plane on which the stress is considered; the second subscript denotes the direction of the stress itself. Noting that Y = 90 - ¢, the stress components on the slip system (primed coordinates) as a func- tion of ¢ are: all = all sin ¢ + 022 cos ¢ , (3) 02.2 = 011 COS (D + 022 Sin 95 , (4) Oiz = (022 - 01].) Sin ¢ COS Q 9 (5) for °21 = “’31 = c’13 = °33 = °23 = °32 = 0 ° (6) Slits in rubber *Pl _.l. [-1/3" Rubber Grips Bonded Interface P Figure 2. Biaxial loading of flat tensile specimen. Slip direction, x5, 552 “\\ X2, 022 Y Xi, 011’ Normal "¢ / 9 I. X 1’ c’11 Slip plane Figure 3. Coordinate transformation and Stress definition. ¢ - degree Figure 4. Resolved normal stress as a function of orientation (¢). These formulae may be written in terms of the ratios of normal stresses on the Slip plane to the shear stress on the slip plane as: a /o 0 /o = tan + cot ¢ (7) o /o . . 11 22 l a /o = _ cot ¢ + _ tan ¢ . (8) 22 12 1 011/022 1 011/022 These equations are plotted in Figure 4 with the stress ratio °11/°22 as a parameter. The parameter all/022 introduced by the biaxial load permits independent variation of the resolved normal stresses and the crystal orientation. The effect of the resolved normal stress on the CRSS of single crystals can be investigated by holding ¢ constant while the resolved normal stress is varied by varying all/o On the 22' other hand, if one wants to Study the effect of crystal orientation on the CRSS, all and 052 at yield can be made constant while the orientation is varied by adjusting the biaxial load. However, the Specimen geometry does not permit complete variation of the normal stresses since only tensile forces are applied. The curves for 011/022 = 0 are thus lower boundaries on the stress ratios. In this research program, ¢ was held constant and all and 052 at yield were varied by varying all/022' An elasticity solution coupled with a photoelasticity in- vestigation established the design of the loading configuration. The elasticity solution fixed the limits on the width of Specimen (2H) to length of loading (2C) ratio (H/C). These geometrical limits were such that 80 percent of the central portion (test section) of the Specimen was subjected to a uniform biaxial stress field. 10 The photoelasticity investigation verified that the rubber grips generated a uniform normal surface traction at the bonded interface. These rubber grips were slit in order to reduce the Poisson effect of the rubber and minimize the shear stress at the bonded interface between the rubber grip and the Specimen. If this shear stress is reduced to zero, the stress applied to the specimen boundary is effectively a pure normal stress. The zinc single crystals used in the shear stress experiments were grown by the modified Bridgman (21) technique. There was no attempt to obtain the lowest value for the critical resolved shear. stress, but the emphasis was on generating single crystal Specimens that would yield consistently. The crystal orientation was Such that I 11 “22' The results obtained from a series of uniaxial tests showed ¢ = W was equal to 45 degrees; therefore, 0 equals reproducible yielding of the zinc Single crystals. The results of the biaxial Shear stress experiments indicated that when the orienta- tion was held constant, and the normal Stresses acting on the basal slip system at yield increased, the resolved shear Stress at yield . . ' decreased. By varying 011/022 at yield from 0.0 to 0.62, all/T0 was varied from 1 to 3. This variation in the resolved normal stresses resulted in a decrease of oiZ/T from 1.0 to 0.7 for 0 I all/TO greater than 2. II IDADING CONFIGURATION The objective of this section is to describe the design of a loading and specimen configuration that will allow one to test anisotropic structures and materials under biaxial loading. There are several ways one can obtain a biaxial State of stress. A thin walled cylinder, a large flat plate, a biaxially loaded square or a flat tensile specimen may be used. The following must be considered in determining the loading configuration; goal of research program, type and structure of material needed to obtain this goal, size of loads required, and how state of Stress can be verified. In this research program, the goal required the variation of the normal stress at yield on the active slip system of zinc single crystals. Zinc Single crystals have a low yield strength; therefore, the load requirement will be low. The use of a thin walled cylinder was eliminated, because the axes of anistropy are continuously varying with reSpect to the principal stress axis. A large flat plate was eliminated by the inability to grow a sufficiently large single crystal of preferred orientation. A preliminary photoelasticity investiga- tion of a thin square loaded biaxially showed a region of uniform biaxial stress too small to be useful. The final choice was a flat tensile Specimen loaded in the transverse as well as the axial direction. This was the most favorable choice for the following reasons: it could be grown in the existing crystal-growing furnace 11 12 to the size required for the shear stress experiments, preliminary photoelasticity investigation indicated that it had a large region of uniform biaxial Stress, and the state of stress at the interface of the grips and Specimen could be verified by comparing an isotropic elasticity solution with a photoelasticity solution. The design and geometry of the specimen and rubber grips required careful investigation to insure that a large region of uniform biaxial Stress would be available for observation. Therefore, an elasticity and photoelasticity solution were undertaken to verify the state of stress. The Specimen was modelled as a finite rectangular beam 2L long and 2H wide with its sides Subjected to a uniform normal surface traction over a length 2C. The elasticity solution was in the form of a single Fourier series. The results of this solution established the limits of the geometry parameter, H/C, which would produce the largest region of uniform biaxial stress. The flat tensile photoelastic specimens were made to fall within these limits. The grips used to apply the transverse load to the flat tensile Specimen were designed such that they would generate a uniform normal stress at the bonded interface yet not reinforce the sides of the Specimen appreciably. The details of the elasticity and photoelasticity investigations are discussed at length in section A and B below. The results of the elasticity and photoelasticity investigation verified that the rubber grips could be used to generate a uniform normal stress at the bonded interface between the rubber grips and the Specimen. The biaxial stress difference (022 - all) for a region that covered approximately 80 percent of the test section 13 was equal to 10215 percent of the applied transverse stress oil For this entire region, the photoelasticity and elasticity solutions were within i_2 percent of each other. Therefore, rubber grips slit at 1/8 inch interval and to within 0.035 inch of the bonded interface can be used to generate the uniform normal surface traction needed to carry out the shear stress experiments. A. Elasticity Solution The purpose of the elasticity investigation was to determine the specimen geometry that would insure the largest region of uni- form biaxial stress when loaded in both the axial and transverse direction. 1. Theory The solution of the stress distribution in the Specimen's test section was considered in two parts, and the results were superimposed. The first part was that of a simple tensile specimen subjected to and axial load. The solution to this part iS _ I I — 022, where a 22 = PZ/AZ’ A = 2H: and t is the Spec1men °22 2 thickness. For the second part, the Specimen was modelled as a thin finite rectangular beam with a width (2H) to length (2L) ratio (H/L) << 1 which was subjected to a uniform normal stress, 011’ in the X direction aS shown in Figure 5. Fourier analysis was 1 utilized to determine the stress distribution in the finite rectan- gular beam (22). The boundary conditions are: l4 W/l/lfl Test sect ion Figure 5. Finite rectangular beam for elasticity model. 15 I o , -1,< x2 < -C and C < X2 < L, 012 = 0, x2=iL’ °22=°’ 012:0' Since the body forces are zero, the following form of the Airy stress function was considered, 2 m DOX2 @ = n2100s anx2[A Cosh anxl +-DBnX1 Slnh enxlj +--E——- . (9) Parforming the necessary operations on equation (9), applying the boundary conditions to solve for the constants A and D, and select- ing 3 equal to nn/L, one can arrive at the following solution n for the stresses in the beam. I . . . m 2011 . CosthX1(BnH CosthH-iSthnH)-BnX181nthH81nthX1 all: 2 SlanC “=1 anL BnH + Sinh gnu Cosh gnu oils °(COS BnXZ) 'f L (10) I . . . a = 2 2011 Sing Cl:(BnXISinhanx1+ZCosthX1)SinhanH-Coshenxl(BnHCosthH+SinhenHil 22 “=1 BnL n BnH +-Sinh_BfiH Cosh BnH J -(Cos BnX2E} (11) I . . . . a g a; 2011 Sina C[(BnX1CosthX1-l-SinthX1)S inthH-S inthX1(BnHCosthH-PS lnthH)] 12 “=1 anL n BnH +-Sinh BnH Cosh BnH -(Sin an2)} . (12) Equations (10), (11) and (12) were evaluated by summing the first 200 terms of the series to obtain the desired convergences of the solution (23). By dividing both sides of the equations (10), (11) and (12) by 011’ the solution was obtained in dimensionless form. 16 To determine the effects of Specimen geometry on the Stress distribu- tion, C was held constant, and H was varied such that the ratio of the width of the Specimen to the length of loading (H/C) varied from 0.05 to 1.5. 2. Results The results are plotted with the stress ratios on the ordinate and the geometric parameter (H/C) on the abscissa. Each family of curves, for each stress ratio, was plotted for a particular Xl/H distance along the X1 axis with Xz/C as a parameter as shown in Figures 6 through 9. With the solution plotted in this form, the stress distribution can be determined for any H/C ratio for which H/L << 1. Figure 6a through 6d shows the following stress distribu- tion in the Specimen's test section for XI/H equal to 0.0: a, 011/011:1 vs. H/C; b, 022/011 vs. H/C; c, 012/0?1 vs. H/C; d, (022 - 011)/oi1 vs. H/C. Since the resolved shear Stress is related to the stress difference (022 - all), the Specimen geometry that yields the largest region of uniform biaxial stress difference along with 012 = 0 is the Specimen geometry suited for the testing of the zinc single crystals. Therefore, only the Stress difference and Shear Stress results are shown in Figures 7 through 9, for Xl/H equal to 0.5, 0.75 and 1.0, respectively. In examining the stress difference curves, it is observed that Specimens with an H/C ratio equal to 0.25 have the largest region of uniform biaxial Stress dif- ference, i.e. 022 - 011 = (1.02 : 0.02)aIl1 over at least 80 percent of the specimen's test section. For H/C equal to 0.5, 022 - all is equal to (1.02 i;0.05)a§1 over 80 percent of the test section. As 17 .H vv s\m was 0.0 u m\Hx sow Emma Smaswcmuoou muflcflm m mo cofiuoom umou use aw eoflusewuumflp mmouum .o ouswwm u\m .m> Hwo\aao .m o\m ¢.H N H o.H w o. a N. W - u q 1 u - 00H IIIIIIIIIIIIIIII w.o 0.0 «.0 N.o N 0.0 u o\ x on oo 0 .I. I / D II I ow . OOH ONH 18 A.v.ucoov o madman U; :3... .. 0:1 1: N H o4 m. a. s. N. o . q a . a a . ON. 4 m S: m> SHESo .n Ex .3 N; o; m. o. a. N. . - . . q . . . . 3 In Z / D II o .I. 19 A.p.ucoov o ouswwm o\m q.H N.H o.H w. \0 \‘T. N O OOH lllllllllllllllll m.o o.o ca 8 ) D Z Z . 0 TL TL( ow 1: ID I TL . c/o ooa Qua 20 110 {- I j l l .2 .4 .6 .8 BIG 3. (022 011)/011 vs. H/C 5 1- h. ”>- -—-- . ’ \‘~~\X2/C 3 0.4 0 p» " 4i-.=: 0.0 .\° 0.6 I '\ H -5 b HI-O \ON \ 0.8 67 -10 #- ——._ _——~~ 1.0 l l 4 .2 .4 .6 .8 H/C I b. 012/011 vs. H/C Figure 7. Stress distribution in the test section of a finite rectangular beam for Xl/H B 0.5 and H/L‘<< 1. 21 110— lOOh- 92 = 0.0 t 0.4 F‘ 0.6 Hot-l 90 .- 0.8 \ AH H O N ON v 80 1- i! J, J, l l o .2 .4 .6 .8 H/C a (o c )/oI vs H/C 22 11 ll 5 r- ” e 0 O H: '5 t' \ .\ 0.8 3 v N 67-20 \ 1 O 0 .2 .4 .6 .8 H/C I b. Figure 8. Stress distribution in test section of a finite rectangular beam for X1/H = 0.75 and H/L << 1. 22 110 — 100 .\° I :1 H J? 90 AH H o = 0.0 I N N o " 80 0.4 50 0.8 0.6 1.0 1 1 I I _ 2 .4 6 .8 H/C I a (022 011)/011 vs H/C 5 - xz/c = 0.0,. ,1.0. 0 h. o\° I bag: -5 _ \ N 67 -10 .. IL J It 1 .2 .4 .6 .8 H/C I b. 012/011 vs. H/C Figure 9. Stress distribution in the test section of a finite rectangular beam for XI/H = 1.0 and H/L << 1. 23 H/C is increased above 0.5, the size of the uniform stress difference region steadily decreases. Therefore, for equal to °22 ' °11 (1.02 j:0.05)oi1 over 80 percent of the central portion of the test section; the extreme limits of H/C were established to fall between 0.25 and 0.50. These limits allow one to subject a Specimen to the largest region of uniform biaxial stress. Based on these results, the photoelastic specimens were made such that the stress distribution could be verified at the upper and lower limits of H/C. B. Photoelasticity Investigation The purpose of the photoelasticity investigation was to design rubber grips such that they would generate a uniform transverse normal Stress at the bonded interface and to verify the Specimen geometry suggested by the elasticity solution. Also, the lower experimental limit of H/C was to be established. The above were verified by demonstrating that the size and shape of the uniform biaxial Stress region was the same as the elasticity solution for flat tensile specimens subjected to both transverse and biaxial loads. Normal incidence was used to determine the principal stress difference (aII - OI) for all the tests. An oblique incidence Study was attempted; but due to the large stress difference and the small value of OI, it was impossible to separate the stresses. This impossibility was verified by a calculation which showed that, for a one percent change in the oblique incidence fringe value, the principal stresses would have to change as much as 33 percent. Also, the elasticity solution showed a maximum change in the stress levels along principal axis of rotation of not more than 6 percent which 24 would require the detection of less than 0.2 degrees rotation of the analyzer. This was less than the error band of the Tardy compensation method used to measure the fractional fringe orders. Uniaxial tests with and without rubber grips were run to calibrate the modelling material and Show that the rubber grips didn't reinforce the Specimens, and photoelastic specimens were subjected to transverse and biaxial loads to design the rubber grips and to verify the elasticity solution. The following equipment was used to test the models: a circular polariscope; a photodiode which was mounted on a X-Y scanner to sense the intensity of the light transmitted through the model; and a simple testing machine consisting of an 8 to l lever system to apply the longitudinal load, and a pulley System to apply the transverse load. The circular polariscope shown schematically in Figure 10 was equipped with a mercury light source. The camera shown to the right of the analyzer was used to project the image of the specimen onto the plane of the photodiode. The X4Y Scanner allowed point by point determination of the fringe values for the entire test section. The output signal from the photodiode was observed on an oscilloscope. Then the fringe value was determined by rotating the analyzer to obtain the angle Ym for which the intensity I was a minimum. The fringe value (N) was calculated by using the relation- ship, N = n + ym/180, where n is the value of the dark-field isochromatic fringe that is moved to the point of interest by a clock- wise rotation of the analyzer. The error in determining the fringe value (24) was reduced to a minimum by reading the value of y both before and after the minimum intensity, where the noise level was the 25 L Optical axis ‘——- X-Y s canner Photodiode output to oscilloscope +__ Camera Field lens k/4-plate f...- / y . / / Model 1 )(flt-plate A»- Polarizer *— Diffusser +—— Filter (1 = 570) t Focus lens * -o——— Hg light source Figure 10. Circular polariscope (schematic). 26 lowest, see Figure 11. Then the following equation was used to determine Ym’ Ym = (ya + vb)/2 where Ya and Yb are the angular values of y before and after the minimum intensity, reSpectively, required to produce the same intensity I Even 0' though the time required to record all the data was considerable, the value of the first point determined at the start of the test didn't vary by more than :13 percent when checked at the end of the test. When the datum points had been taken, the principal stress difference was determined by all - OI calibration constant for the material. = KN, where K is the The lever arm friction of the testing machine, see Figure 12, was such that when a 3 gram load was applied, a shift in the fringe value could be detected. The load cell and its calibration curve are Shown in Figure 13. The load cell was used to measure F2 for the uniaxial tests and F1 for the transverse and biaxial tests. P2 for the biaxial tests was calculated by a static moment equation derived for the lever arm system. The load cell was capable of detecting 10 gram change in the load. Hence, the error in determining the values of the input loads was less than i 0.5 percent. The comparison of the unaxial results indicated that the rubber grips didn't reinforce the specimen. The transverse and biaxial test results were compared with the elasticity solution and found to agree to within 1 2 percent. 1. Specimen Details The Specimens were milled out of 1/8 inch thick sheets of photoelastic material, PSM-l, purchased from Photolastic Corporation. 27 Error band Intensity - I Ym = (Va + Yb)/2 . N = n + vm/180. Figure 11. Fractional fringe order determination. Output to type N l ' strain ' indicator K\\ P2 Load cell Model Figure 12. Loading frame for photoelasticity investigation. 28 mo>uao :oHumunHHmo OH m m n :8 53 6cm :8 e63 mueoemuoaxm mmouom umosm .m mocoEwuoaxm owummHSOBOLm .< new m>uno :oHumhanmo HHou quH mucoewhomxm mmouum woman you nmuHo>u nowwfiaae< zpso comomuomamo .ma spam-a awn u AHo u HHbv monopommfin mmouum Hmafloewpm oqu oHN omH 0mg ONH om 00 on 4 . 1 . . q . a o 03 u .1 605 ofimmaoBofiv TEE new o>uou COwumpanmo lw.o +3 $6 I HH 23 + 88 u AHo - 3 1390 N - anteA 38u113 38 The results of loading Specimen A in the transverse direction are tabulated in Table 1. The solutions agree along the center line of the specimen. But the experimental solution away from the center- 1ine of the specimen fluctuates above and below the theoretical results, because the specimens test section is too narrow for the 1/8 inch thickness of the specimen. Therefore, this Specimen geometry is experimentally impractical for use in the shear stress experiments. The results of loading specimen B in the transverse direction are tabulated in Table 2, and the results of loading Specimen B biaxially are tabulated in Table 3. Here the solutions agree through- out the entire test section. Table 4 is the elasticity solution of 012/011 for sepcimen B loaded transversely. This indicates that the shear Stress throughout the entire central portion of the test section is less than 1’5 percent of 011' Since the solutions agreed where the shear Stress was zero, one can safely say that the elasticity and photoelasticity solutions agree to within 1 2 percent for the size and shape of the uniform biaxial stress field, and that the rubber grips did generate a uniform normal surface traction at the bonded interface for H/C equal to 0.5. Also, by comparing the trans- verse and the biaxial results of Specimen B, it is observed that even when the rubber grips are subjected to a load the specimen test section isn't reinforced. C. Discussion of Results The loading configuration design was established to be a flat tensile Specimen subjected to both axial and transverse loads. The transverse loads were applied by adhering Slitted rubber grips to 39 U\ N 0.0 0.0 0.0 0.0 0.0 0 q d d 1 d 1 1 q d. d 41 ii 00.0 00.0 00.0 A.6000000.0 00.0 00.0 00.0 00.0 00.0 0000 00.0 00.4 A.0000000.m 00.» 0000 0000 N000 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 0000 00 ..0 00 ..0 00..0 00..0 0000 00.. 0 0000 00..0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 0000 0000 0000 00.m 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 0000 00.m 05.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 500000009 omuo>memua C0 4 coEHanm o0ummHoOuosm w50pmoa mo muapmom .H oHan omN.o oom.o om~.o mum.o H/IX 4O c0000000m mmpo>msmufl SH m ome0ooam UHummHoouosm mewomoq mo muHSmom 05 x 0.0 0.0 0.0 0.0 0.0 0 ll 1 q q a q u q q 1 a q d d]! 00.0 00.0 00.0 5.6000000.0 50.0 50.0 00.0 00.0 00.0 00.0 00 ..0 00..0 5.00000 00..0 50 ..0 50.. 0 00+. 0 00.. 0 00..0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 q H ‘ G G d d d 00.0 00.0 0000 00.0 00.0 00.0 00.0 00.0 00.0 00.0 05.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 05.0 00.0 00.0 50.0 00.0 00.0 0q00 00.0 00.0 00.0 00.0 00.0 00.0 00.0 50.0 00.0 00.0 q u T q d a J H a 50.0 05.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 I d u q q q a l a 00.0 50.0 00.0 00.0 50.0 00.0 00.0 00.0 00.0 .N mHQmH omN.o oom.o omn.o mmw.o H/IX 41 05 x 0.0 0.0 0.0 0.0 0.0 0 q q d a q q u 4 d d a q d 1‘ 00.0 00.0 00.0 5.0000000.0 50.0 50.0 00.0 00.0 00.0 00.0 00.0 50.0 A.0000000.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 I u d d d d 4 d 1 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 05.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 I J ‘1 d q 1C1 G 1 I 00.0 05.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 50.0 00.0 00.0 d d d d 1 d I d d 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 00.0 0000 00.» 50.x 00.m 00.» 0000 0000 0000 00.» zaamemHm m coewomam owummHoOuosm wC0mmoq mo muaamom .m manna omN.o oom.o Om~.o mum.o H/IX 42 U\ x o.H o.o o.o o.o N.o o :1 I q q q q a q a a q 1 d u a 1. q u q u . 1. a q . o o.o o.o o.o . o.o o.o o.o o.o o.o o.o I I1 I 4 q I q I I \U-Noo oo.o: mo.Ou No.01 Ho.o No.0 No.0 Ho.o Ho.o o.o I Q I I 1 ‘ I I ILomoo MH.Ou OH.01 no.0: No.0 mo.o mo.o No.0 Ho.o o.o I I I I I I H I .1 mNeo BH.Ou NH.O- o.o mo.o «o.o no.0 mo.o o.o o.o I I 1 I d I I I I OOOH o.o o.o o.o o.o o.o o.o o.o o.o o.o m.o u U\m n wcwomoa mmum>msmuH you sewuanwuumwn mmouum Hmmnm Hmuwumuomnfi .o mHan 43 the edge of the Specimen's test section. An elasticity solution coupled with a photoelasticity investiga- tion verified this loading configuration. The elasticity solution suggested that the geometrical limit (H/C) of the Specimen should fall between 0.25 and 0.50 for a uniform biaxial stress region that covered approximately 80 percent of the Specimen's test section. The biaxial stress difference in this uniform biaxial stress region was equal to 102 i 5 percent of the input stress oll’ i.e. (022 - 011) = (1'02.i 0°05)°Il' The photoelasticity investigation established the geometry of the rubber grips to be 1/8 X l X 4 inches with slits at 1/8 inch intervals which were Slit to within 0.035 inch of the bonded interface. Photoelastic models were made to Specifications that would yield models for testing at the upper and lower limits of H/C. When the results of the photoelastic models subjected to trans- verse and biaxial loads with the rubber grips were compared with the elasticity solution, it was observed that the solutions at the upper limit of H/C agreed to within i;2 percent throughout the entire region of uniform biaxial stress, but the lower limit proved to be experi- mentally impractical. Therefore, the dimensions of the zinc Specimen were made to the same Specification as the photoelastic models with H/C equal to 0.5 except for minor changes which enable the zinc single crystals to be grown. The details of these changes are covered in section A of Chapter 111. Also, since the uniform biaxial stress region symmetrically covered 80 percent of the Specimen's test section, the clip gage length for the clip gage to be used in the shear stress experiments was established at 0.8 inch and located symmetrically about the X1 axis along the centerline of the Specimen. III SHEAR STRESS EXPERIMENTS The elasticity solution and the photoelasticity investigation suggest that the zinc specimen for the shear Stress experiments should have a geometric parameter, H/C, of 0.5 to obtain the largest region of uniform biaxial stress. The flat tensile specimen shown in Figure 20 meets these geometric requirements and still falls within the Specimen size limit (maximum width of 3/4 inch) of the crystal growing furnace to be discussed in section A below. The purpose of the shear stress experiments was to subject zinc single crystals to a uniform biaxial state of stress in order to determine the effect of the resolved normal stress on the critical resolved Shear stress. This was accomplished by loading the single crystals with rubber grips in the manner described in the previous chapter. Since the crystals of an aggregate are subjected to a very complex stress field, the response of the crystals to biaxial loading may lead to a better understanding of the behavior of aggregates. The Specimen orientation was Such that g and Y were both 45 degrees. This choice of orientation was based on the following four reasons: (1) Jillson had the closest agreement with theory for this orientation, i.e. for a Schmid Factor (cos ¢ cos Y) equal to 0.5; (2) One can easily determine the yield point for this orienta- tion, because the easy-glide region of the Stress-strain curve is nearly flat; (3) The yield strength is the lowest Possible. there- fore, one can vary all/O22 at yield over a larger range of values 44 stock specimen [Ll-1‘ Seed 7 __L 01. 45 1.1. a. Stock Specimen and seed Figure 20. Zinc Specimen __J L b. Final Specimen Cleaved basal c. Seed details plane 46 than would be possible for a higher yield strength; and (4) Twinning won't occur for this orientation (29, 30, 31). The approach in the crystal growth was to produce Single crystals as nearly identical as possible, but not necessarily with a minimum To. This was Sucessfully accomplished and the details of crystal growth and handling procedure are given in section A and B below. The shear stress experiments were carried out in uniaxial and biaxial tension. The uniaxial tension tests established the con- sistency of the CRSS, the negligible effect on the Slip mechanism of gluing the rubber grips to the sides of the crystals, and the uniformity of the surface traction generated by the rubber grips. Specimens with and without rubber grips were tested with the load applied parallel to the tensile axis. To test the uniformity of the surface traction at the bonded interface, additional specimens were loaded in tension in the transverse direction and polished and etched to determine the uniformity of the Slip throughout the test section. The result of the etching verified the existence of uniform slip. The biaxial tension tests required simultaneous application of P1 and P2. This was accomplished by a balance and pulley system coupled with an Instron. The details of the testing procedures are discussed in full in section C of the present chapter. The results of the shear stress experiments Show that for the uniaxial test T varied between 68 and 72 grams/mmz. The results 0 of the biaxial loading are plotted in Figure 53 where the resolved Shear Stress ratio at yield (OIZ‘y/TO) is plotted as a function of the resolved normal Stress to critical resolved shear stress ratio 47 (Olliy/TO) at yield. One can see that the experimental results fall well below the theoretical curve for all‘y/TO greater than 2. In conclusion, one could safely say that the resolved normal stress on the basal slip system does effect the CRSS of zinc Single crystals. For a more detailed account of the results, see section D of the present chapter. A. Crystal Growth The specimen stock was machined from 1/4 inch thick strips sawed out of hexagonal ingots of high purity zinc (99.99+) purchased from Mattiessen and Hegeler Zinc Company. The specimen stock in Figure 20a was made long enough to yield the desired specimen Size shown in Figure 20b. It was necessary to round all corners to pre- vent the nucleation of crystals of different orientation at these corners. The lower end (stem) of the Specimen stock was machined at a 45 degree angle with respect to the tensile axis. Also, the seed was cleaved on the basal plane by chilling it in liquid nitrogen (31) which made it possible to obtain the exact orientation of the seed. The cleaved surface of the seed was then welded to the 45 degree surface of the stem with an acetylene torch using ammonium chloride as a flux. Both the Specimen Stock and seed were cleaned in dilute hydrochloric acid before and after welding. Next, the orientation of the seed with respect to the stock was checked by Laue' back-reflection. The stem was bent when necessary to obtain the correct orientation. Then the seed and seed end of the Specimen were polished for 5 minutes in polishing solutions developed by Vreeland et a1. (32) to remove dislocations introduced during handling 48 and by sharp corners at the weld joint. The seed end of the specimen was then placed in the slot of the lower thermal block (TL) shown in Figure 21. The slot of the upper thermal block (TU) was placed over the top of the specimen, and the alumina powder mixture was packed around the Specimen through the Openings in T The thermocouples used to record the axial U' temperature gradient were positioned inside the crucible at the upper and lower thermal blocks as shown in Figure 22. This made it possible to control the solidus-liquidus interface at 3/8 1 1/8 inch from the bottom of the seed. The packed crucible was then SuSpended by alumel wire such that the lower end of the seed was 2 inches below the top of the lower furnace as Shown in Figure 23. The furnace was preheated to the temperature necessary to melt the entire specimen except for the lower 3/8 inch of the seed. Then the soak period was set of 3 1/2 hours. This soak was required to obtain a uniform temperature distribution throughout the entire Specimen. At the end of the soak period, the furnace was program cooled such that the crystal growth rate was 1 mm/min.; and at the same time, a constant axial temperature gradient was maintained. A typical axial temperature gradient, as recorded by the thermocouples in the crucible during soak, is as follows: Upper thermocouple......4900C, Lower thermocouple.....412°C. B. Crystal Preparation During all the specimen preparation, the Specimens were trans- ported by means of the holders Shown in Figure 24. Holder A was 49 Thermocouple tie down holes I 0%: 3/4 I l '4 _..| |.._0.13, I 0.87 3/8 in. Dia. L o—1/8 Upper thermal block - T 0.87 Hole for lower TC +1 |-0.13 85: n—7/8—u U \\\\\\\\\\\\\\\\ «J: S Ii ~«-l \\\\\\\ I, lower thermal block - TL Crucible half Figure 21. Crucible design 50 Upper thermocouple (Tau) a A. ‘ L 's. ’J'o; . 0.0; :r. ‘6 \J (a. . o ‘ o e ‘D.‘ ' \..° Alumina Powder :o:b.’ c'e' :cu- ‘} ‘Lz‘.’ -. ~ .0“ .- . 0’ I‘d. a'.: . O ':.. .0 ’43. a. \ ‘. ' 0 Q . ‘.'\-.' ’::’; s‘..§ .: {5.} .I... .' ..O..~\ ' 7. . a C . ’ “g .; N" =' .,- .‘ ‘0 's‘ml.‘ ~ ‘ o . O o '.|. Crucible I\\;.wfiz\\\P\\\‘\\5I\\§\\\:\\\f\\‘ ‘\g:\\:\\:§\:\<:\\_ ;I\;>\:\q 0—-To recorder Lower thermocouple (TCL) Specimen Seed Figure 22. Specimen packed in crucible (schematic) 51 Alumel wire TCL TCU 'II I Z: '5 'v :5 I5 r A . _ \ \ \ \1 ' w”’——-Tu I \ : ""“ TC co t l .§§§hi: n ro :l TC differential 'I I- II - L g. \\ \ \J 1’ l” 10—— Insulator I f fl\\\\m _ . i :: TC differential “$02: ” ‘ "l<:l| Specimen :ilL-__..I T L . Cruc1ble L I — Zirco tube pd tFlow of dry Nitrogen Figure 23. Crystal growing furnace (schematic) 52 b. Holder A 8. Holder 3 Figure 24. Specimen holders 53 designed to hold the Specimen during the X-ray of the Specimen and the cutting of the specimen to length. Holder B was designed such that the remaining specimen preparation could be completed without the Specimen ever being removed from the holder or handled directly. Both holders were made of plexiglas, and the specimens were held firmly in position with neoprene rubber pads. This handling pre- caution was taken to keep the specimen's dislocation density from increasing. Also, the holders protected the test section from accidental damage. The crystal specimen was cleaned in concentrated hydrochloric acid and visually examined. Next the crystal was placed in holder A and x-rayed to insure that the crystal was a Single crystal and of the correct orientation. The crystal and holder A were then posi- tioned on the wire saw table as shown in Figure 25, and the specimen was cut to an overall length of 4 inches. The Specimen and aluminum grips (shown in Figure 27) were cleaned with 50 percent hydrochloric acid, and the rubber grips were cleaned with acetone. Next the Specimen was placed in holder B, and then fastened to the gluing jig as Shown in Figure 26. The gluing jig was designed such that all the grips could be aligned and then adhered to the Specimen without re- moving the Specimen or holder B from the jig. Following the gluing of the aluminum and rubber grips to the specimen, the clip gage tabs were positioned and glued to the Specimen with the aid of a traveling microscope. After the tabs had been cured in the furnace, the entire Specimen was chemically polished for two minutes with solutions suggested by Vreeland et a1. Then, the specimen was thoroughly dried with a hot air dryer, and the Specimen dimensions were measured. 54 Figure 25. Wire saw Figure 26. Gluing jig for zinc specimens 55 ‘ 1 tn I I I . I 0 I I q J— ——————————— 4. ~ T“““‘““T I L. .......... 0 In—I A if *— l/L-N 1/16 in. Dia. Grove 1’” I--- ~-- 13/16 ———~I I- L I Q's-I Figure 27. Aluminum grip details “b -‘Eh-db 1 5/16 56 Chemlok 305 two part epoxy based adhesive was used for adher- ing all the grips and clip gage tabs to the specimen. In all cases, the bonded joints were cured in a furnace for 8 hours at 1200F to obtain the maximum bonding strength. The clip gage tabs were 1/16 Diameter X 3/32 eyelets purchased from United Shoe Division Machinery. It is believed that the consistent growth procedures, the careful handling of the specimen, and the careful Specimen prepara- tion as described above were the reasons for the consistent results obtained in the shear stress experiments. C. Testing Procedure The prepared specimen and holder B were loaded into the test- Ing machine, and all the grips were pinned to their appropriate load- ing links before the holder was removed. Then the clip gage was attached, and the specimen was preloaded with 200 grams. The sensitivity of the clip gage was such that one could sense a 5 X 10-5 in/in change in the gage length. Immediately following the preload, the subsequent loading was started with as little delay as possible. The strain rate in the Specimen test section was 27 X 10-.4 percent per second. In all the tests P was applied with the Instron and 2 P with the pulley System shown in Figure 28. This pulley system was actuated by filling a 20 gallon bucket with water at a constant rate. P was measured with the load cell Shown in Figure 13, and l the continuous P vs. time curve was recorded on an X4Y recorder. 1 The test equipment was calibrated just prior to running each test. For the uniaxial tests, the load P2 was applied with a constant crosshead speed of 0.02 cm/min. The specimens with the 57 Front view . Back view Biaxial testing machine Figure 28. 58 rubber grips glued to the edge of the test section were preloaded in the P1 direction with 180 grams, before the axial load was applied. This kept the rubber grips aligned with the specimen so that there wasn't any bending Stress applied to the test section. The P2 vs. time curve was recorded on the Instron chart recorder, and P2 vs. percent strain curve was continuously recorded on a X4Y recorder. The uniaxial transverse load was applied in the P1 direction with the pulley system such that the strain rate in the test section was the same as the axial loading. The P1 vs. time curve was recorded on an X4Y recorder, and the P1 vs. percent Strain curve was recorded on a second XJY recorder. The biaxial loading of the specimens required careful align- ment of the Instron heads and the pulley system. It also required both axial and transverse preloading of the specimen. The effects of improper alignment and no preload on the simultaneous loading of the specimen are shown in Figure 29. A polycrystalline Specimen was subjected to a biaxial load. Curve A shows a typical P2 vs. time curve for prOper prestress and alignment as recorded on the Instron chart recorder. Curve B is for improper alignment with correct pre- stress. Curve C is for no prestress with prOper alignment. As can be seen from the above results, it is necessary to carefully align the equipment and properly prestress the Specimen to insure the simultaneous loading of the Specimen. It should be emphasized that the preloads were well below the load required for yielding. Careful control of the loading was required to prevent negative straining (i.e. a negative elastic response caused by apply- ing P1 to fast.) and to prevent premature plastic deformation in 59 A B C :[_—W— .54 H L- _’ Load vs. Time Curves A - PrOper alignment and prestress B - Prestress and improper g) __ a l1gnment .01 C - Proper alignment and m no prestress _ l J- 0.25 0.5 Time - min. Figure 29. Effects of improper alignment and no prestress on biaxial loading of Specimen. 60 the test section before P1 was fully applied. If P1 was applied too fast, the resulting negative strain has to be recovered before a positive resolved shear stress would reSult. Also, for large values of P , creep might occur. If P 1 was applied too slow, the test 1 section would be plastically deformed by the axial load before P 1 was fully applied. But when the loads are applied simultaneously such that all < 022 as shown in Figure 30, the creep doesn't occur, there is no negative strain to recover. The Single crystals were chemically polished and etched with Vreeland's solutions after they had been tested. Then they were examined on a Bausch and Lomb Research Metallograph. The etch pits revealed that the specimens yielded uniformly on the basal slip system throughout the entire uniform biaxial stress region of the test section. Figure 31 is a typical photomicrograph of uniform basal slip observed in the test section. D. Shear Stress Results All the single crystals with ¢ = Y = 45 degrees were deformed under uniaxial and biaxial tension by {0001} <2IIO> type Slip. The deformation was stopped at one percent or less to save the Specimens. The results of all the tests were plotted with the stress on the ordinate, and the strain on the abscissa. The nominal stress-strain curve for the four uniaxial tests are plotted in Figures 32 through 34, and the resolved shear stress- strain curves are plotted in Figures 35 to 38. Test Numbers 2, 3, and 6 were run primarily for the purpose of testing the equipment. The crystals used were not good crystals; that is, the orientation .coE0oodw mo wc0om0a Hm0xm0o you o>uso oE0uImmonum Hmc0eoc Hmo0ozfi 61 .om musw0m .c0E I oEHH oqa ONH ooH om ow oq ON 0 1 a q I q d d d O #0 b J On I ooH J onH NNO Ill/ll .J ooN .l Omm 62 Figure 31. Photomicrograph of uniform slip in Specimen test section. .oH can H .oz 00de 0050.30 :0muuwammmuum HmEEoz .Nm 005w:— 0 J 0 00.10 1 000 w." nun/1119.18 - 220 l on." 2. l ooN I. omN 64 a .02 0009 o>aso ewmuumnmmouum 0000562 .00 0.0000 N I w 0. 5. 0. 0. 0. 0. N. 0. 0 0 0 0 q 0 q q 0 0 600 u 0. u 0 I 00 D Z Z 1 000 . on J 9 m 0/ 7y I 000 .- I 005. omm 65 m .02 000B o>0so e0maumnmmouum Hmc0eoz .qm oasw0m N I 0 m. N. 0. m.. a. m. N. 0. 0 01 I 0 0 A 0 0 0 0 00“ I0 1.00 000 0mg 00N omN mm/meJS - ZZo 2 T - grams/mm 120 F. 105'- 75 .— 60 i- 45 - 30L 15 ' 66 y @ = Y 45 O 4 1 1 l L 1 1 1 I 1 L 0 .l .2 .3 . .5 .6 v - Z Resolved shear stress-strain curve Test N0. 1 Figure 35. 67 4 80 p, 70 —- 6O .0 50 40 g F \ E m LI 60 I t. 30 " 20 .- lO .— ¢=Y=45 0 1 1 1 1L J. L, I l 0 l v - Z Figure 36. Resolved shear stress-strain curve Test No. 4 80 r- 70I' 60 .. 50 40 L- 2 301. T - gram/mm 20 Figure 37. l l 1. l I l l v - Z Resolved shear stress-strain curve Test No. 5 80 {r- 70 #- 60 - 50 . 40p 2 r - gram/mm 30 5. 20 10 Figure 38. Resolved shear 69 y - Z stress-strain 46 *6 II curve Test No. 18 70 wasn't correct or the dislocation density wasn't the same as the other crystals. Therefore, the results of these tests were not in- cluded. Test No. 1 and 4 were Specimens having rubber grips adhered to the test section. The load was applied in the P2 direction after the Specimen was preloaded in the P1 direction. Test No. 5 was loaded in the axial direction without rubber grips glued to the test section. Test No. 18 was loaded transversely by applying the load with the pulley system. This test was run last because of the possibility of damaging the testing equipment. The test was ended abruptly, when it started to yield, to prevent catastrophic yielding due to the high energy stored in the extended rubber grips. The uniaxial results tabulated in Table 5 indicate reproducible yielding; i.e., To equal to 70‘: 2 gram/mmz. The results also indicate that the rubber grips didn't effect the yielding of the crystals. These results verify that all the single crystals had approximately the same dislocation density and oxide film coating before running the tests. Therefore, since the specimens for the biaxial Shear stress experiments were grown and prepared the same as the uniaxial Specimens; the variation in the resolved shear stress at yield would primarily be due to the normal Stresses. The nominal stress-strain curves and the resolved shear stress-strain curves for the biaxial tests are plotted in Figures 39 to 43 and Figures 44 to 52, respectively. The yield point for all the curves was determined by the three methods described below. These methods were used to establish the error due to the inter- pretation of the results. The lowest possible value for yield was taken as the point where the resolved shear Stress-strain curve showed oH new a .oz umoe mezzo. gamumummouum HmEEoz .mm meow?“ 71 .26 .\. n w . d _ q _ _ o Om.» n- ? u S We: 1 2: In I e u D. o i 0...: a . co NND w... / m / m llmi m I. SUN L 0mm Z 72 HH .oz ummH Q>HDU Cw whuml mmmhum H QCHEOZ .oq muowfim oq me .20 Ha o on O U 2: B U D. D Z Z o2 . co 1 B m / m m o8 0mm 73 Na .02 ummm. o>uso chuumnmmopum HQCwEoz .Hq muswfim N I w h. e. m. a. m. N. H. o d 1 q . T q a o x) qu u.- m’ u 6 1 cm :0 .lo I 1 02 m. D. 0 Z Z . co 1 02 u m l w m 7v .. 08 So ‘1 S L 0mm 74 mg new «a .ma umoh mo>poo samuumnmmouum Hmcfleoz .mq acumen N I w .56 a a . o o? u a u e 1 on I‘ll! HMO III!) 0 I \ VI I A .. 2: e u D. D z 7v . I o2 a.” e m \ / m Z 1 all Nuolly o8 2 3 L 2 omN 75 NH can 0H .02 ummh o\o| m mo>uzo samuom-mmocom HmcHeoz .mq muswam So ca OOH omH oo~ 0mm oom mm/melfi - ZZo pug Ito 80 70 60 50 2 40 w - gram/mm 30 20 10 Figure 76 .2 .4 .6 y - Z Resolved shear stress-strain curve Test No. 9 l .0 80 70 60 50 2 40 c - gram/mm 3O 20 10 Figure 77 L L 1 L l '01 O .1 .2 .3 v - Z 45. Resolved Shear stress-strain curve Test No. 10 78 80,. 7O - soi- 2 4O _ 7 - gram/mm 30.. 20 _. 10- 45 -.l 0 .2 .4 .6 y - Z Figure 46. Resolved shear Stress-strain curve Test No. 11 ll 80 7O 6O 50 2 40 7 - gram/mm 30 20 10 Figure 79 47. Resolved Shear stress-strain curve Test No. 12 12 50 " 30$- 2 20 L ~ - gram/mm 10 - 80 13 -10 I— -20 b Figure 48. Resolved shear stress-strain curve Test No. 13 80 70 6O 50 2 40 w - gram/mm 30 20 10 Figure 81 14 49. Resolved shear Stress-strain Test No. 14 80 70 60 SO 2 40 w - gram/mm 30 20 10 Figure 82 y - Z 50. Resolved shear stress-strain curve Test No. 15 15 l. 2 r - gram/mm 83 80 r 70 i- 60 ‘ 50 _ 16 40 *- 30 P- 20 F' 10 P (b = Y = 450 O b J I I A -.1 O .1 2 .3 y - Z Figure 51. Resolved shear stress-Strain curve Test No. 16 t w vr I FE'IW- 2 T - gram/mm 80 70 60 50 4O 30 20 10 Figure 84 -.l 0 .1 .2 y - Z 52. Resolved shear stress-Strain curve Test No. 17 17 85 Table 5. Summary of Test Specie Orientation Type Area Area Gage O' ‘ No. men Y of A1 A2 Length 11 y NO' (deg.) (deg. Tes‘ 2 2 2 (mm ) (mm ) (mm) (g/mm ) 1 16 45 45 Uniaxial 80.5 39.6 20.32 0.0 4 6 45 45 " 79.4 39.2 20.32 0 5 3 45 45 " - 36.3 20.32 0 9 2 45 45 Biaxial 81.2 39.6 20.29 29.7 10 11 45 45 " 80.0 40.0 20.11 53.6 11 12 45 46 " 79.9 38.4 20.33 36.2 12 14 45 45 " 80.4 39.4 20.31 66.7 13* 17 45 45 ” 81.3 40.0 20.97 98.9 14 18 45 45 " 81.2 39.4 20.32 131.7 15 19 45 45 " 80.4 39.1 20.32 101.5 16* 20 45 45 " 81.3 40.0 20.32 160.5 17 15 45 45 " 81.2 40.0 20.32 138.1 18 8 45 46 Trans- 79.0 39.3 10.57 147.0 verse * Failure before yield o\y - Value of stress at yield (0.02% offset) 86 Shear Stress Experiment Data Ozz‘y °11‘y “il‘y °12‘y Oil‘y °11‘y °12‘y 81 82 2 Ozz‘y 2 2 olZ‘y T0 T0 (kg/min.) (kg/min.) (8/mm ) (g/mm ) (g/mm ) 145.5 0.0 70.5 70. 1.00 1.01 1.01 - 50. - 0 68.0 68. 1.00 0.97 0.97 - 21. - 0 70.0 70. 1.00 1.00 1.00 - 21. 169.0 0.18 99.0 70. 1.41 1.41 1.00 16. 24. 198.2 0.27 125.6 72. 1.75 1.79 1.03 15. 21. 181.2 0.20 108.6 72. 1.50 1.55 1.035 21. 25. 202.7 0.33 134.7 68. 1.98 1.92 0.97 20. 26. 218.8 0.45 158.8 59. 2.64 2.26 0.86 30. 22. 229.7 0.57 180.7 49. 3.69 2.58 0.70 31. 22. 223.5 0.45 162.5 61. 2.66 2.32 0.87 30. 27. 262.5 0.62 211.5 51. 4.14 3.02 0.73 25. 28. 243.0 0.57 190.6 52. 3.67 2.72 0.74 27. 21. - - 72 72 1 1.03 1.03 31. - 87 the first deviation from linearity. The intermediate value of CRSS was determined by the 0.02 percent offset method and the maximum value by a 0.1 percent offset. The tabulated results listed in Table 5 for the biaxial tests are all based on the 0.02 percent offset method. These results show that the resolved normal stresses varied from 70 to 212 gram/mm2 by varying a at yield 0 to 0.62. 11/°22 This caused the resolved shear Stress to drop from 70 to 49 gram/mmz. To better illustrate the effect of the resolved normal stress on the CRSS for {0001} <2110> type Slip of zinc single crystals, the results of the uniaxial and biaxial tension tests were plotted as shown in Figure 53. The resolved shear stress at yield (giz‘y) and resolved normal stress at yield (ail‘y) were both normalized to To, where is the CRSS for the uniaxial tests. The maximum 70 error bar indicates the maximum possible error in the interpretation of the data as described above. When all‘y/TO is increased above a value of approximately 2, the experimental results begin to fall below the theoretical curve. For Oil‘y/TO equal to 3, there is a 30 percent drop in ch‘y' 88 .oaowz on 0e co mmmuum HmEuo: mo>~0mouu 0 16 E On mmweum umosm om>~0mmm o % .\ .Hmo mantra O Nee\666w N H.oa a 6363» accuse musaamm % assume smo.o - .Nwo uouuw Eszxwz H3 .mm muawfim oe o D T.I 7P. .A m. u. 0 0.4 IV CONCLUSION The mechanical behavior of polycrystalline materials has been tested extensively in a biaxial state of stress, but there has been' little attention given to the effects of a complex State of stress on the mechanical behavior of metal single crystals. The effect of a complex stress field on a single crystal is important because aggregate theories use the value of the critical resolved shear stress determined by uniaxial testing to predict the stress-Strain curve of aggregates, while the crystals within these aggregates are actually subjected to a complex stress field. In previous investiga- tion, the orientation dependence of Single crystals has been examined by subjecting single crystals to a uniaxial state of stress. When the crystal orientation was varied, the resolved normal stresses at yield on the active slip system were also varied. This coupling of the resolved normal stresses and the crystal orientation effects was separated by subjecting single crystals to a uniform biaxial state of stress. The main theme of this experimental investigation was to test the hypothesis that macroscopic yield in single crystals is determined only by the shear stress on the active slip system and is independent of the resolved normal stresses. The following conclusions have been drawn from the present research program: 89 90 To obtain a uniform biaxial Stress region in a flat tensile specimen over approximately 80 percent of the test section, the width to uniform transverse loading length ratio (H/C) should fall between 0.25 and 0.50. These limits were established by an extensive elasticity investigation. Slitted rubber grips can be used to produce a uniform stress field at the surface of the sample. This is verified by the photoelasticity investigation for isotropic materials and by the uniform Slip lines for anisotropic zinc crystals. The grips were made of a 1/8 inch thick neoprene rubber with slits at 1/8 inch intervals. These slits were cut to within 0.035 inch of the bonded interface. This was necessary in order to eliminate the shear stress at the bonded interface which arises as a result of the Poisson effect. When tested biaxially, zinc single crystals show that the resolved shear stress for yield decreases with increasing resolved normal stress. The crystal orientation was held constant with ¢ = Y = 45 degrees and the resolved normal stress was varied from 70 to 212. This variation in the resolved normal stress resulted in a 30 percent drop in the resolved shear Stress. Therefore, one could conclude that macrosc0pic yield of zinc single crystals is not only determined by the shear Stress on the active basal slip system, but is also dependent on the normal stresses acting on that slip System when ¢ = Y = 45 degrees for {0001} (2110) type slip. THE BIB LIOGRAHIY l. 11. 12. 13. 14. 15. 16. 17. 18. 19. THE B IB LI OGRAPHY E. Schmid, Proc. Internat. Congr. of Appl. Mech. Delft, 342 (1924). E. Schmid and W. 8088, Kristallplastizitat, Springer Verlag, Berlin, Original German edition (1935). English translation published by F.A. Hughes and Co., (1950). P. Haasen, Phil. Mag., 3, 284 (1958). W.N. Andrade, D.C. and D.A. Aboau, Proc. Roy. Soc., A240, 304 (1957). J.Z.F. Diehl, Metallkunde, 41, 331 (1956). K. lficke and H. Lange, Z. Metallkunde, 43, 55 (1952). RJW.K. Honeycombe, Progr. Mat. Sci., 2, 95 (1961). F.D. R081 and C.H. Mathewson, Trans. AIME J. of Metala,188, 1159 (1950). F.D. Rosi, Trans. AIME J. of Metals, 200, 1009 (1954). RJW. Fenn, W.R. Hibbard, Jr. and H.A. Lepper, Jr., Trans. AIME, 188, 175 (1950). R. Madin and N.K. Chen, Prog. Metal Phys., 5, 53 (1954). C.S. Barrett, An International Conference held at Lake Placid by Fisher, Johnston, Thomson, Vreeland, 238 (1956). C.S. Barrett, Structure of Metals, McGraw-Hill, 346 (1952). D. Hull, J.F. Byron and F.W. Noble, Can. J. Phys., 45, 1091 (1967). D.C. Jillson, Trans., AIME J. of Metals, £999 1129 (1950). E.C. Burke and W.R. Hibbard, Trans. AIME, 124, 295 (1952). RJW.K. Honeycombe, The Plastic Deformation of Metals, St. Martin's Press, New York (1968). G. Sachs, Z.D. Ver. deut. Ing., Z_, 734 (1928). G.I. Taylor, J. Inst. Metals, 62, 307 (1928). 91 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. F.F.W. Bishop and R. Hill, Phil. Mag., 42, 414, 1498 (1951). C.K. Chyung, "0n the Improvement of Lattice Perfection in High PUrity Zinc Crystals Grown from the Melt," M.S. Thesis, Michigan State University, (1962). Private communication - RJW. Little and notes for forthcoming book. Private communication - J.L. Lubkin. G.L. Cloud, "Improvement in Use of Photometric Methods for Measurement of Birefringence", Expr. Mech., 8, 138 (1968). C.H. Paterson, "Special Adhesives for Rubber Bonding", Symposium on Adhesives for Structural Applications, 59 (1962). W.M. DeCrease, Rubber Age, 81, 1013 (1960). J.W. Gallagher, Adhesive Age, 22, Jan. (1968). J.A. Svigelj - Supplied samples of Chemlok 305 adhesive - Hughson Chemical Company, Eire, Pannsylvania. C.H. Chyung, "Nucleation of Deformation Twins in Zinc Bicrystals," Ph.D. Thesis, Michigan State University, (1965). E.A. Anderson and D.C. Jillson, J. Met., 1191, Sept. (1953). R.F. Miller, Trans. AIME, 122, 173 (1937). .._.."' GENERAL REFERENCE RAW.K. Honeycombe, The Plastic Deformation of Metals, St. Martin's Press, New York, 1968. E. Schmid and W. Boas, Kristallplastizitat, Springer Verlag, Berlin, Original German edition (1935). English translation published by F.A. Hughes and Co., 1950. C.S. Barrett, Structure of Metals, McGraw-Hill Book Company, Inc., New York, Toronto, London, 1952. R.C. Dove and P.H. Adams, Experimental Stress Analysis and Motion Measurement, Charles E. Merrill Books, Inc., Columbus, Ohio, 1964. J.W. Dally and W.F. Riley, Experimental Stress Analysis, McCraw- Hill Book Company, New York, 1965. R.F.S. Hearmon, An Introduction to Applied Anisotropic Elasticipy. Oxford University Press, 1961. 93 APPENDIX Computer Programs m:z_hzcu om .cu .x.1.x>Cm .cu.x.a.>xw .ou.x.w.xxm >au.x.o.>>m w>z.Hux om cc mx2.~uw om cc exxusau>a exxmomom~¢~.muma ~+>zuw>z H+xzuwxz >z\Iu>I xz\4xuxx ~\\m.mHu.u Q.xm.m.mfiu.n I.xm.mom~u.u ._x.xm.mom.nm.u 0.xm.NIH.h2.x2.a.4x.leu .Nemuz. aqwm H Numuz mudaz .mwmmwm th Z— cwzzsm qzawh we .\2 n x422 6m~x th u ozcgq m20~m~>~m 4<30m uC .02 u >2 .mqulx th 0204< m2©~mw>~m 4 wuqmmnm ml» #4 zcmhuth owugaad th m~ a eqwaq zomhuawh>m ux>om cz< >xm .>>m .xxm mmmmwmhw wIH mom mw>qom zCm.AHH.Hm.>xm.AHH.Hm.>>m .Agfieam.xxm zcumZmz—c .Emmn umasw66uomu muwaww a Ca cowusnwuumwv mmouum mo cowuwH30Hwo .H 95 mZZHHZCU OCH Ax.w.xxm I .y.n.>>mnax.1.x>cm xmm*.>me>xmm.%au + Ax.o.>xmu .x.w.>xm xmu*.>>mn.x.w.>>m xmu*.xImsquu>xu >1ma>2malrmu>Im+>Iu*>za.*IImn>xmu A>Iu*.m + >1m4>2m.*11m uxmu >Iu*zmvxuu>Iu A>zm.Imu>Im z>*.HIx.u>2m w>z.an ocH cc >Iazmu2> xIazmHZX QU\umuau IIuaxxm+Izmucu musuuuum zzm\a*.muuu .uzm.z_mumu IIm+ IIuarzmuqu .Izm.ImuIIm .Izmvruuxxu IazmuIZm uazmuuzm maazuzm memomH¢H.m*zu22m x .¢~. mom czw kaw egqu H oh cw wzzahzco .NIH. h<2d0u .mm.aaz. mama: Afic.m~uc~\vx.mmwahm xxl>>.xm\\. hqszm .mx2.~uo.~¥.w.x>Cm. gem.aaz. mk_az 1.0.m~¢o~\.\.mmwmhm I >x.xm\\. hqszu .mx2.~uw..x.o. >xm. .mm.maz. wh_m3 «Ac.m~mc~\.\.mmwabm I xx.xm\\v hqszu Amx2.~uo..x.o. xxm. 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