A STUDY OF THE STRENGHT OF TUNGSTEN CARBIDE-COBALT ALLOYS FROM A FRACTURE MECHANICS VIEWPOINT THESIS 5012 THE 9mm 0? Ph. D MICHIGAN STATE UNIVERSITY ROY CARL LUETH 1972 "' t’ . "' A? I: L “ : ,' 4 I. 3 1' l ‘i Michigan State " University (ar‘t. This is to certify that the thesis entitled A STUDY OF THE STRENGTH OF TUNGSTEN CARBIDE - COBALT ALLOYS FROM A FRACTURE MECHANICS VIEWPOINT presented by ROY C. LUETH has been accepted towards fulfillment of the requirements for Ph.D. Metallurgy degree in / ./ / /,// _ 7 ( :lj’ /’7’ .7 /_// ,/ I. I / j"/ / /’/ / Major professor Date 1/ I 'AV / '4 V 0-169 ABSTRACT A STUDY OF THE STRENGTH OF 'IUNGSTEN CARBIDE-COBALT ALLOYS FROM A FRACTURE MECHANICS VIEWPOINT By Roy Carl Lueth The critical strain energy release rate and stress intensity parameter have been determined for nine tungsten carbide-Cobalt alloys by a wedge loaded double cantilever beam test. The nine alloys studied had uniform average grain sizes ranging from 1.5 to 8 microns, and cobalt Acontents from 3 to 15 wt. percent. The implications of these data have been discussed as they affect the theories of strength of cemented carbides. The critical strain energy release rate depends on the energy absorbed in the area adjacent to the tip of a metastable crack in the material. The energy absorbed at the tip of the crack depends on the amount of plastic work done on the binder contained in the plastic yield zone, as no flow was detected in the tungsten carbide Roy Carl Lueth particles. The amount of plastic work done on the binder in the yield zone is dependent on the volume of binder which plastically flows, which in turn is dependent on the binder film thickness, and the number of grains (and binder films) the plastic zone includes. As a result, the critical strain energy release rate generally increases as the binder film thickness increases. The mode of fracture of a tungsten carbide-cobalt alloy varies widely depending on several compositional and microstructural factors. If the plastic zone radius is less than one grain diameter, then the alloy will fail through the binder, regardless of the carbide grain size. If the plastic zone size is large enough to include several grains, then the failure mode will depend on the carbide grain strength; the larger grained alloys (weak grains) will fail through grain fracture, the smaller grained alloys will fail through ductile binder failure. Medium grain size alloys with large plastic zone sizes will fail with various amounts of the above modes, depending on cobalt content and grain size. The larger the cobalt content and the larger the grain size the more fracture will tend to be transgranular. Roy Carl Lueth The strength of these alloys in tension depends on the inherent flaw size, yield strength, and fracture toughness. The shape of the transverse rupture strength versus binder film thickness curve is a necessary result of these factors. The compressive strength of these alloys is dependent on carbide grain strength and dis- location density in the binder at the failure stress. The stress imposed on the grains of tungsten carbide-cobalt alloys has been found to be constant for a given grain size regardless of cobalt content, and as a result the compressive strength will rise as the binder film thick- ness goes down for a given grain size. A STUDY OF THE STRENGTH OF TUNGSTEN CARBIDE COBALT ALLOYS FROM A FRACTURE MECHANICS VIEWPOINT By Roy Carl Lueth A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Metallurgy, Mechanics and Materials Science 1972 © COpyright by ROY CARL LUETH 1972 ii ACKNOWLEDGMENTS The author would like to express his gratitude to Dr. A. J. Smith for his assistance during this work, and also to the staff of the Hard Matals Research section of the General Electric Metallurgical Products Department, especially Mr. William A. Powell who supplied the test material; and Dr. E. W. Goliber and Mr. T. E. Hale for many helpful and stimulating discussions. A special appreciation is expressed to G. F. Paluch who conducted most of the test work and R. J. Wills who did all of the microsc0py and photographic work, and to Dr. Robert Little who contributed a great deal more than technical assistance to this project. iii TABLE OF CONTENTS INTRODUCTION ------------------------------------ THEORY OF STRENGTH OF TUNGSTEN CARBIDE-COBALT ALLOYS --------------------------- SUBJECT RESEARCH ................................ EXPERIMENTAL PLAN ............................... COMPLIANCE CALIBRATION .......................... LOAD MEASURING SYSTEM --------------------------- DISPLACEMENT MEASURING SYSTEM ................... MEASUREMENT OF DATA AND TREATMENT ............... EFFECT OF MODULUS ON COMPLIANCE ................. TEST PROCEDURE .................................. RESULTS AND DISCUSSION ---; ...................... CONCLUSIONS ..................................... BIBLIOGRAPHY .................................... APPENDIX I THE DOUBLE CANTILEVER BEAM TEST -- APPENDIX 11 MATERIALS USED ................... APPENDIX III APPARATUS AND SETUP .............. APPENDIX IV FRACTOGRAPHS ..................... iv 12 43 44 46 46 47 48 51 54 58 83 85 87 Table II III IV LIST OF TABLES Strength of Tungsten Carbide-Cobalt Alloys Assuming Cobalt to Govern, No Size Effect, Carbide Infinitely Strong Physical PrOperty Data The Effect of Plastic Zone Size on Mode of Fracture Compressive Strength Data Material Structure Data Page 32 59 65 73 122 Figure LIST OF FIGURES System W-Co-C Section at l400°C (25500F) Quasi-Binary WC-Co Diagram Transverse Rupture Strength Versus Mean Free Path Effect of Test Procedure Upon Relationship Between Compressive Strength and Hardness for Various WC-Co Alloys Diagrammatic Representation of Thermal Stresses in Carbide and Cementing Phases of WC-Co Alloys Dislocation By-Pass Model for Calculating Orowan's Stress A Comparison of the Observed Yield Stresses in Shear of WC-Co Alloys with the Theoretical Values for the Dispersed Particle Strengthening Effects of Carbon Content on the Strength of WC 10% Co Alloy Having Mean Grain Size of 2.2 Microns Compressive Strength as a Function of Percent WC (Gurland) vi Page ll 17 20 22 28 30 35 39 Figure 10 11 12 13 Campliance Versus Crack Length Stress Strain Curve for 15% Co 1.5 Micron Grain Size Alloy in Compression Critical Strain Energy Release Rate Versus Binder Film Thickness for WC-Co Alloys Compressive Stress Versus the Fourth Root of Plastic Strain APPENDIX III Figure 1 2 Test Sample and Calibration Wedge Test Apparatus Apparatus for Calibration of Instrumented Wedge vii Page 50 57 61 69 126 127 128 LIST OF SYMBOLS - Microns - Surface Energy Normal Stress - Shear Stress ”dqssjz - Force A" J - DisPIacement 1 - Compliance p - Load G - Shear Modulus Medulus of ElaSticity~ I?! l GIc - Critical Strain Energy Release Rate KIc - Critical Stress Intensity Parameter Y’ - Poisson's Ratio 5 - Strain v - Crack Opening Displacement viii INTRODUCTION Tungsten carbide-cobalt alloys are liquid-phase sintered compacts consisting of 3 to 25% cobalt, the re- mainder being tungsten monocarbide. The cobalt appears as a thin binder film 0.1 to 1.0,“ thick between irregularly shaped tungsten monocarbide particles 1 to 5/{in diameter. This type of composite enables many of the best prOperties of each component to be realized-~the result being an extremely hard, strong material, the major disadvantage being the brittle or semibrittle nature of the fracture. The cobalt in these alloys, because of the-restraint which this phase sees, remains in an F.C.C. configuration at rbom temperature rather than in its equilibrium H.C.P. structure. It also contains percentages of carbon and tungsten in solution, the amounts depending on process variables. The tungsten monocarbide, which is an H.C.P. structure with the carbon arranged in the interstices, is virtually free of dissolved cobalt. These materials have the highest compression strength of any commercially available material. Because of their strength and their extremely good wear resistance, they are without equal as materials for rock drilling and similar 1 2 applications. These composites, because of their strength, hardness and relatively good toughness over a large tem- perature range, have found great use in other fields such as metalcutting and metalforming Operations. They have become and will continue to be the prime materials where this combination of properties of wear resistance, hard- ness and strength can best be exploited in industry. The fracture mechanisms are not well understood in these alloys. Such things as failure criteria, initiation points, and fracture paths have been the subject of much research; however, a large number of conflicting ideas are in evidence in the literature. If these mechanisms and criteria were better understood, the alloys could be used more effectively as this would come into play during the design stage of industrial equipment. Furthermore, the prOper existing grade best used in any Specific application could be better selected using the design data of existing equipment. Many theories have been advanced which prepose to elucidate the mechanisms of fracture in tungsten carbide- cobalt alloys-~these have met with varying success in attempting to fit experimental data. The main theories will be dealt with at some length. First, however, to 3 gain some awareness of the complex nature of these alloys, the process variables and how they can affect the results of any mechanical test, and thus any theory to explain these mechanical test results, will be explored. There are two main areas where processing variables can enter into the resultant mechanical pr0perties. First, consider the mechanical preparation of the alloys. Prior to sintering, the material is in a powder form. The preparation of this powder is critical to the final product. During processing of the raw powders contamination is the greatest danger. This contamination consists of foreign materials getting into the raw powders. Also contamination occurs in storage drums, pickup of iron, etc. After the raw powder has been prepared, a fugitive binder is added to facilitate compaction of the powder. This binder is usually paraffin and is mixed with the tungsten carbide-cobalt powder. This can be a critical step, as without careful control, the powder could be put into lumps which will bevery detrimental to the pressing Operation. Also this procedure creates heat and, due to the powder's low oxidation resistance, if the powder is exposed to air this can cause oxidation. The powder is then screened and granulated for better flow in the automatic presses. Here care again must be taken.to avoid contamination. The granules must also be of a certain "hardness"--if they are too "hard" they will not flow preperly and during pressing voids may result. The high cobalt materials are less sensitive to flaws during the mechanical handling processes and, in general, produce better compacts. It is harder to get a good compact with low cobalt grades. The finer the powder the poorer the flowing characteristics--shrinkage is greater and they are harder to compact, and thus more susceptible to voids. The fine powders sinter better as there is a greater surface area to volume ratio and thus a greater driving force. They do shrink more, however, due to pressing difficulties. The coarse powders during sintering will shrink approxi- mately 15%, the fine about 18%. It can be seen from this very brief discussion that getting a mechanically sound (free from macroflaws) product can be difficult and that the difficulties increase with alloys of lower cobalt content and finer grain size (finer tungsten carbide powders). Thus in comparing the failure characteristics of alloys of different compositions, it should be recog- nized that there may be a systematic variation in sound- ness of the material (freedom from flaws). 5 The other area which is critical for good mechanical properties is the chemical and metallurgical interactions during sintering. The development of a dense, strong material in the tungsten carbide-cobalt system is accom- plished via liquid phase sintering. In tungsten carbide- cobalt alloys, the eutectic reaction between tungsten carbide and cobalt forms the liquid as the temperature is raised; the position of this eutectic depends on such things as composition, free carbon, etc. This liquid penetrates between the tungsten carbide particles due to capillary action. The driving force for this effect is the reduction of surface free energy. The penetration of the liquid is dependent on the differences in free energy of the various surface interfaces, such as gas solid, liquid solid, solid solid. This can be expressed as the angle of contact of the liquid with the solid. The smaller the angle the larger the reduction in free energy by replacing a solid gas interface with a liquid solid interface. The angle of contact of a liquid solution of tungsten carbide and cobalt with solid tungsten carbide is essentially zero. This is a major factor in achieving up to 99.9 theoretical density. All other sintering mechanisms still operate, however, such as solution and reprecipitation, volume diffusion, etc., but they play a more or less secondary role in the densification. A much more complete discussion of sintering mechanisms and processes can be found in reference (1) (Schwartzkopf). There are several phenomena which may be observed in the final product which are detrimental to good mechanical prOperties. First consider several types of porosity. By convention the industry refers to porosity as being one of three types. The first type is termed "A" porosity and is observed as pores less than lO/Iin diameter. This type of porosity can have several causes. It can result from incomplete penetration of the liquid between the particles of carbide and can be alleviated by raising the sintering temperature, thereby reducing the contact angle of cobalt on tungsten carbide, thus increasing the activity of the liquid. The second, a so-called "B" type porosity, refers to holes larger than le. "B" type porosity may be a gross version of "A" type porosity. If caused by incomplete penetration of the liquid phase, it could be corrected by remilling to break down the aggregates or by increasing the sintering temperature. "B" type porosity may also be due to gas pockets formed at liquid phase sintering tempera- tures. Other causes are large oxide patches and pressing voids which have partially closed. The last type of porosity to be discussed is type "C". It is not really porosity but rather precipitated graphite, much like that which occurs in cast iron. This morphology appears when the free carbon is above 0.05%. For free carbon below 0.05%, it occurs in clusters on cooling, frequently out- lining the cobalt grain boundaries. Even with elimination of porosity in the sintering process, the final product may not be acceptable. Segre- gation Of the cobalt may appear. This is termed "binder laking", and appears in conjunction with rounded carbide grains. If we continue to "soak" the compact the mobility of the binder is increased and more tungsten carbide goes into solution. This allows the binder to flow prOperly, and laking will be eliminated. As the compact cools, the tungsten carbide will precipitate on existing grains and they will become more angular. When laking is eliminated and tungsten carbide crystal growth advanced, the properties are found to be in the Optimum range for that particular grade. Soaking for too long a time can cause excessive grain growth to the point Of being detrimental to prOperties. Further heating will cause density to dr0p off as bloating of the compact occurs due to residual gases and increased binder activity. Both of these can affect mechanical prOperties in a negative manner. Carbon control is another extremely important param- eter in the sintering Operation. As can be seen from a ternary isothermal section of the equilibrium phase diagram (Figure l), the system cobalt-tungsten carbide-carbon is quite complex containing several different phases. For tungsten carbide-cobalt alloys, the narrow region connect- ing tungsten carbide to cobalt is the working area of the diagram. The region Of equilibrium for tungsten carbide- cobalt is very narrow,and above this region (toward the C corner) free carbon will exist; with tungsten carbide- cobalt alloys,below this region several carbon deficient phases exist. The only one of interest to tungsten carbide- cobalt alloys is eta phase,as any one of the other three are observed only in rare occasions,and then only when the sintering process has gone far from the desirable one. Eta phase is CO3W3C and is a member of the M6C family. Eta phase grows in an uncontrolled manner at the expense of the binder phase and appears in macrOSCOpic prOportions. Eta phase does show good corrosion resistance but is extremely detrimental to mechanical prOperties. A vertical quasi- binary diagram taken through the tungsten carbide-cobalt 10 region is shown in Figure 2. The reactions occurring during sintering are discussed in detail by Schwartszpf, and will not be given here. To produce a quality carbide, precise carbon control must be exercised. From our prev- ious discussion it can be Observed that a deviation of I.06% carbon can have great consequences with respect to mechanical prOperties. This can also cause surface effects which will change the normal strength characteristics. Considering all of the above factors in any testing program, especially those which involve surface, strength such as transverse rupture testing, care must be taken not only to evaluate the quality of the general micro- structure but surface conditions as well. When comparing the fracture characteristics of alloys of different com- position the greater susceptibility Of some composition ranges to porosity, etc., must be carefully appraised and such matters as binder migration within the test specimen must be avoided. All of these factors must be considered in any explanation of testing phenomena. ll ooa 0o cm as Zow ou measummoV puma . m.m mcwocom Eoum wouOflpoum I III nuwoouum o>wmmoudaou .l m.mI.III I¢oH I’ll/l I, I m. 1.1:: I I I]: [OH IILMUU/II III/I /////// I I [ID I I III / l I/ I /I I/ I// II I I ‘ It \ v / x . // .x I / //\\\ | / . Odom a sound mo mquIIIIIIIIIIIIUVIII I l /, .l I: II _ F . _ . _ . com ooq 00m 000 00m oow (ISd €01) HIDNHHIS HAISSEHJWOD 40 however, shows that there is considerable dislocation movement and multiplication even in very low cobalt alloys at stress levels much below the failure level. Kreimer further states that even during what is considered brittle fracture there is always some evidence of at least micro- plasticity (Orowan's modification to the Griffith theory of brittle fracture). This plastic deformation should be found mostly in the cobalt phase and thus the more cobalt phase showing on the fracture surface the larger amount the amount of energy necessary to fracture the material. Kreimer states that the area of the fracture which is com- posed of cobalt is prOportional to the volume of the cobalt phase in the alloy and that the work of plastic deformation then is prOportional to the cobalt content (P = AC where P is plastic work, C is cobalt content and A is a constant). In view of the general findings of other investigators that the amount of tungsten carbide vs. cobalt fracture is strongly grain size dependent and may be dependent on other factors such as speed of crack, etc., this would seem to be a rather strong assumption. The impact strength of tungsten carbide-cobalt alloys is composed of three parts; namely, work of elastic deformation, work of plastic defor- Ination of the cobalt region adjacent to the fracture 41 surface, and the work expended in the formation of this surface a T ‘aE + P +7. Kreimer believes that the first and last terms are to be neglected and that only the work of plastic deformation is important in fracture. He also assumes that the critical crack length is independent of cobalt content and that the ultimate strength of tungsten carbide-cobalt alloys is determined by the stress necessary for crack prOpagation, not the stress necessary for initia- tion. With these assumptions, the Griffith/Orowan equation can be written 05 = AEC where A is a constant, E is the modulus of elasticity and C is cobalt content. He presents several plots of 0"‘(transverse rupture)2 vs. EC - all of which form fairly straight lines. This fact would be expected as the modulus of elasticity changes little in this range (left of the TR vs. Co curve) and it is well known that the TR is dependent on cobalt content and tungsten carbide grain size. The final form of the equation for strength which he presents is (7!- ABC + K 'where K is a constant which governs the grain size dependency of the strength. Many of these assumptions here are directly Opposite to those of Drucker and others. This theory then says that as cobalt content goes down, it takes less energy to prOpagate existing cracks and thus the material is less 42 strong. This should hold in all forms of loading; however, the data of Lueth and Hale contradicts this theory for the case of compressive strength, and this prOposal does not fit the available data in all respects. Many theories of strength of tungsten carbide-cobalt alloys have been put forward; however, as yet none explain the total range of experimental data, rather each attempts to explain one small portion of the data. All have some validity but further work must be done to sort out those ideas which can best be used to further understand the alloys, and thus be able to put them to better and more extensive use. There are other theories which try to explain the behavior of these alloys. It has been endeavored to review only those which have received the most complete acceptance, as many of the others have even more obvious flaws. SUBJECT RESEARCH Most of the theories of strength and behavior of tungsten carbide-cobalt alloys depend heavily on the initia- tion area--that is, tungsten carbide or cobalt on an inter- phase boundary and the propagation (which phase is trans- versed by the crack). MOst of the studies of fracture of these alloys have been done on transverse rupture bars or similar tests in which the test is immediately catastrophic. Here the fracture is extremely fast and may not reflect the true nature of the initial crack which must start and propagate initially very slowly. To prOperly apply the Griffith/Orowan theory to the material, no assumptions should be made about the fracture until data have been provided to substantiate these assumptions. The purpose of this research is to obtain KIc and GIc values for the tungsten carbide-cobalt alloy system and to examine the implication that these parameters have on the prOposed theories to explain the strength of cemented carbides. This was accomplished by a wedge loaded double cantilever beam.system. Pertinent information of the applicability and mechanics of this test can be found in Appendix I. 43 EXPERIMENTAL PLAN Nine alloys were used in this study with various cobalt contents and grain sizes. Nominal grain sizes and cobalt contents of these alloys are listed below. Cobalt Content NOminal Grain Size Allgy No. ,_ (Wt. %) Cu) 1 3 1.5 2 3 3 3 6 7-8 4 9 1.5 5 A 9 3 6 9 7-8 7 15 1.5 8 15 3 9 15 7-8 These alloys were hydrogen sintered at appropriate temperatures and the carbon contents in the median range for the alloy. Photomicrographs of the structure with porosity and soundness ratings can be found in Appendix II. The test bars were 3" x 1/8" x 1/2" with a 3/4" x .35 slot cut in one end and a 1/16" groove cut along the center the full length of the sample (see Figure 1, Appendix III). All grinding was done with a diamond grinding wheel of 44 45 100 grit and all grinding marks were parallel to the longitudinal axis of the specimen. The groove was neces- sary to prevent the crack from deviating from the center line of the specimen during prOpagation. The ungrooved side of the specimen was polished prior to testing to aid in observing the extent of the crack. Testing was done in a specially designed fixture (see Figure 2, Appendix III) which allowed the advance of a carbide wedge (via a micrometer screw) into the slot provided in the sample. The arms of the double cantilever beam could then be displaced and this diaplacement measured via attached transducers. The length of the crack was measured through the use of a travelling stage (to which the total fixture was attached) and a microscope mounted above the sample. The data gathered then consisted of arm displacement and crack length. In order that this informa- tion be converted to the desired KIc and GIc data the com- pliance calibration of the system at any given crack length must be determined. The method of obtaining compliance is discussed in the next section (see Appendix I for mathe- matical deve10pment). COMPLIANCE CALIBRATION The strain energy release rate G is equal to A2 ax G = (-———) ( ) 2A2b aZLS where b is the width of the fractured surface. In order to determine the strain energy release rate of any given material we must first determine the compliance A of the material in the configuration of the test. UH> A (Compliance) = This is essentially the inverse spring constant of the system. A must be determined for various crack lengths such that %A may be determined at any given crack length. The accuracy of the compliance can depend only on the accuracy of the measurement and since the compliance changes very little with crack length, the correctness and sensitivity of the measuring system is paramount. LOADVMEASURINC SYSTEM To measure the load on the sample a transducer was built which consisted of tungsten carbide cobalt wedge with a groove machined along the mid line of the wedge (see figure 1, appendix III). Strain gages were placed at the bottom of the groove. These gages produce a signal when the wedge was loaded at the tip. Since the wedge was displaced into the machined portion of the specimen to 46 47 produce increasing displacements, and thus loads, the point of contact of the specimen on the wedge transducer would change; therefore, the wedge must be calibrated along its usable portion. This was accomplished via a fixture which allowed the wedge to be loaded by an exterior mechanism at any point on the usable part of the wedge (see figure 3, appendix III). This total apparatus was then placed on an Instron load cell and data on load versus output from the wedge was taken as a function of loading position. The wedge strain gage signal was read through a BAM l amplifier and a Barber Coleman recorder. The output was found to be linear with respect to load at any given point of application on the wedge. The slope of the load versus wedge output data were plotted against position on the wedge and this was also found to be linear. The load could be read to within 1 .25 lbs over a range of 50 lbs, or about i .5% error. From this information, the known starting point and the displacement read off of the micrometer the load could be determined on the specimen during compliance calibration in the test fixture. DISPLACEMENT MEASURING SYSTEM The displacement was measured via a Daytronic variable inductance transducer and read on a Daytronic transducer amplifier meter (see figure 2, appendix III). This system has a summing capability for the two transducers such that lateral displacement of the specimen, if any, will not be 48 recorded as displacement. This system allows measurement of five micro inches; however, readings were taken routinely of .0002", thus error here could only be attributed to positioning of probe and initial calibration of the instru- ment. An error of less than .5% is expected. MEASUREMENT OF DATA AND TREATMENT Load versus displacement data were taken and several samples were taken at crack length increments of : .1" over the length of the specimens from 1" to 3". All these data were then plotted for each crack length and the compliance determined. The deflections were found to be linear with load as required. The compliance data from all samples and crack lengths were then plotted on a master curve (see figure 10) and as can be seen, all of these data correlated quite well. A regression analysis was performed on these data and equation of the form Y = AXB gave the best fit with an index of determination of .996. This result agrees well with theory. The displacement using the regular theory is if 6 2.6M From the regression analysis 3 = 90.8 x 10- L To obtain %% the first derivative was taken of this equation in the normal manner. This procedure eliminated the need to determine experimentally the slope of the master plot at any .point. With this information G for this material may be deter- Inined with only displacement and crack length measurements. 49 FIGURE 10 COMPLIANCE VERSUS CRACK LENGTH 50 ocoo mugs: Lanna ca auwcmu Russo Doom C r Dar—Arthur COCO ooom ¢©.HA ouoa x HR.mmm Se.mu QIOH x w.om mx< r< r<. 414:: ooom 0H AH NH 7‘01 x aouertdmoa 51 EFFECT OF MODULUS 0N COMPLIANCE Since the elastic modulus of the several grades of carbide which were used in the study differ, the effect of this difference must be considered. >J u p19 u mp< From Gillis and Gillman 2 Y = PL3 + kFL(2(l+v)) + CL F o 3E1 AE EI F 3 2 2 Y E = L + kFL(l+v) + CL F 0 '3I A ' I Observing that the right side of this equation depends only on sample configuration parameter and load except for a slight contribution due to change in D then YOEC = (f(sample parameters))P 1:. E c K: II f(sample parameters) Y E =—-9——= f P ACEC since F is the same for all grades then. f = ACEC = AMEM A A E O [:1 ll 52 ACEO A = M EM Therefore, SAC E = 3AM E 8L c a M 81M = SAC (E2) 3 3L EM G = A2 (BAC)(E£) = A2 3A0 EM M 2 EC 3L EM 2bA2 3L Ec 2bl (——) c 0 EM Comparison of Standard Theory (unslotted beam) vs. experiment (slotted beam) From simple beam theory Y_EL_3_ 0 El for L=l 2Y 3 _ o _ gL_ _ -6 A - —E— - 3EI - 57.71 x 10 From the more complex beam theory and using the results of Brown and Srawley k(l+v) + HE ) N 2‘! 3 o - _ L 3/2H ”15"“? “1*?” DJ 53 _ —6 3 3/u<.69) A - 57.71 x 10 L (1 + .5175 ) + (u)(2.183)(1.3) x 10“6 (.79) -6 3 5 2(2 183) x 10'6 A = 57.71 (1.5175) L + ' ° .79 for L=l x = 87.08 x 10’6 From compliance calibration A = 90.8 x 10"6 or about 3.5% difference This could possibly be accounted for in some warping of the beam due to the fact that the bottom is not flat (due to slot). It appears that the slot in the beam does not introduce serious deviation from standard double cantilever beam theory. TEST PROCEDURE The test samples were placed in the test fixture and the arms of the double cantilever beam displaced until a natural crack was initiated and arrested. The length of this precrack varied from 1/2" to 2" depending on the fracture toughness of the material. The total length of the crack was then measured (including the original diamond wheel notch) and the wedge withdrawn to close the natural crack. The diaplacement transducers were then zeroed prior to beginning the test. The arms of the wedge loaded double cantilever beam (hereafter referred to as WLDCB) were dis- placed until the crack began to move; data were then taken on total crack length and arm displacement. When ten read- ings had been taken in this manner the wedge was then with- drawn and the procedure repeated. This was done to elimi- nate any cumulative error which might result if all readings were taken in succession. Usually four to six sets of ten readings were taken on each sample. After completion of the test the fractured sample was examined to ascertain whether the crack had run off center into the thicker material on either side of the slot. If this had occurred those data points taken from the affected area were con- sidered invalid and discarded. The consequences of a 54 55 crack running off center would be to have the crack traversing a double thickness area (1/8" thick) and to make the arms of the beam unequal in thickness, thus negating the compliance calibration. Samples were also prepared to obtain other pertinent physical prOperty data such as transverse rupture strength, compressive strength, yield, and modulus of elasticity. The transverse rupture data were obtained from 1/4" x 1/4" bars broken in three point bending on 5/8" centers. The samples were ground using a 100 grit diamond wheel with the grinding marks running parallel to the longitudinal axis. The compression samples were prepared in a similar manner and were 1/4" x 1/4" x l" with the ends ground parallel to better than .0003". The test was run with 1 mil steel shims t0p and bottom of the sample (see refer- ence #7). The compression samples were also instrumented with two strain gages on Opposing sides and the strain data used in the modulus of elasticity calculations were the average of these two gages. A typical stress strain curve can be seen in Figure 11; the yield strength was evaluated by the .002% offset method. The arm displacement, crack length, and modulus of elasticity data were converted to 51c and KIc values using 56 FIGURE 11 STRESS STRAIN CURVE FOR 15% Co 1.5 MICRON GRAIN SIZE ALLOY IN COMPRESSION Stress x 103 psi 472 409 346 283 220 158 94 31 4 57 Ultimate Strength - 662,000 psi Preportional Limit - 158,000 psi Modulus of Elasticity - 83.1x106 psi 5 6 7 Strain x 10'4 FIGURE 11 10 11 58 a computer program based on the compliance calibration and calculations on pages 120-121 of Appendix I. All of these physical property data along with binder film thickness are compiled in Table II. RESULTS AND DISCUSSION The critical strain energy release rate GIc for cemented carbides bears a near linear relationship to binder film thickness for the low to medium binder film thicknesses with some leveling out at the higher values of this parameter (see Figure 12). The energy which is absorbed during fracture (the controlling factor in the critical strain energy release rate) is controlled by the manner in which the material fractures. The mode of fracture of the various alloys was studied by two techniques. First, motion pictures were taken I through the microsc0pe while the crack slowly prOpagated across the viewing field via the WLDCB method. Individual frames of this film were then printed and sequences of prOpagating cracks could then be studied. Magnifications of approximately lOOOX were achieved in this way. Secondly, the fractured surfaces were studied via transmission electron microfractography (see Appendix IV). Both of these tech- niques indicate that in the fine grained alloys (grain size 59 0.00 0.00 000.00 00.0 000.000 000.000 000.000 0-0 0 00. 0.00 0.00 000.0 00. 000.000 000.000 000.000 0-0.0 . 0 00. 0.00 0.00 0000.0 00. 000.000 000.000 000.000 0.0 00.0 0.00 0.00 000.00 00.0 000.000 000.000 000.000 0-0 00. 0.00 0.00 000.00 0.0 000.000 000.000 000.000 0-0.0 w 00. 00 0.00 000.0 00. 000.000 000.000 000.000 0.0 . 0.0 0.00 0.00 000.00 0.0 000.000 000.00 000.000 0-0 00. 0.00 0.00 000.00 00.0 000.000 000.000 000.000 0-0.0 00 00. 0.00 0.00 oom.00 0.0 coo.Noo ooo.0n0 ooo.o~0 0.0 «000008 :0 000 000 < 00030000 00 :0 000 0c0\mo0 00 000 uommwo 0000. 000 0000008 00 cu «mocxu0nh hu0u0umu0m museums: nonmemuam comm ammo0oa nuwcouum «mm 0000» camcouum «~00 cwmuu N .u3 8000 uuvc00 no 030300: zu0mcouc0 hwuocm 000000 u>0mmwuasou «>0mmoumsou «030030 mmuuum 00 0uu0u0uu ouws0u0= onuu>mcmuu 0mouuwm0 2 grain diameters failed in a combination of grain fracture and binder failure with a larger amount of grain fracture as the binder content is increased. The fine grained alloys largely resist grain fracture regardless of the plastic zone size and the large grained alloys with ry:>2 failed by grain fracture. The type of fracture which dominates in a given tungsten carbide-cobalt alloy is determined by several factors. As the binder at the tip of a crack yields, dislocation multiplication and migration occurs; if the zone over which this happens is large enough to include several tungsten carbide grains, then these 65 0050000 umUCHQ 0000050 0 0.0 0 0 0050000 000000 0000050 000 00000000000 0 0 0-0.0 0 0050000 000003 0000050 000 00000000000 0 0. 0.0 0 00500000 00000 0 0.00 0 0 00500000 00000 @500 £u03 0050000 umvc0o @000050 0 0.0 0-0.0 0 0050000 000:0o 0000050 0:0 00000000000 0 0.0 0.0 0 00500000 00000 0 0.00 0 00 o0500mm 000002 000000000000 5003 000000 000000 00 0050000 0 0.00 0-0.0 00 0050000 000:0o 0000050 0 0.00 0.0 00 00500000 mo 000: 00008000 0000006 0000006 :0 N .03 000900 00000 00 00 00 00 @000 00000 mmDHU5 microns may not be monocrystalline and would thus have planes of weakness. If the particles are not strong enough to withstand this stress the particles in the plastic zone will fracture followed by the ductile failure of the remaining binder ligaments. If the zone over which plastic deformation occurs is not large enough to include the area one grain diameter away from the tip of the crack the plastic deformation will be largely confined to the one binder film where the crack tip is located. If this is the case the amount of plastic work will be con- siderably less than for alloys where larger areas have yielded, and the critical strain energy release rate will be much lower. This would be the case for the 3% alloys and the fine grained 9% alloy. This dislocation migration and multiplication has been observed in tungsten carbide-cobalt alloys through internal friction studies by H. Doi, Y. Fujiwara and K. Miyake(13)- 67 In their investigation they studied the effects of plastic deformation in compression on both the amplitude dependent and amplitude independent components of internal friction. They established that the fourth root of the plastic strain is proportional to the magnitude of the stress above yield. = C /6 A0 = (O ‘ 0.00002) 1 p This was done using standard strain gage techniques. They also found that by normalizing the plastic strain with the volume % cobalt that the constant of proportionality is essentially independent of cobalt content or grain size. These relations have been found to hold in the present study (see figure 13) such that A0 = (G ’ 0.00002) ’ “2 l-f where f is the volume fraction of WC. Using the Granato Lucke theory of dislocation damping, Doi found the distance between pinning points of the dis- locations to be approximately .07“ microns at room temperature. This was also found to be independent of plastic strain, cobalt content, or grain size. They established that the relation between dislocation density and amount of plastic prestrain is: 68 FIGURE 13 COMPRESSIVE STRESS VERSUS THE FOURTH ROOT OF PLASTIC STRAIN 69 37 B 590 15%A 530 b N 0 157.36 410 -9% C 350 15% C Compressive Stress x 103 psi 290 230 170 .03 .12 .l. .20 V .24 FIGURE 13 70 where B is a damping constant. (B for cobalt has not been directly measured and could range from 10'"!4 to 10-1.) From this relationship and the relation between stress above yield and plastic strain then (A0)2 = (o - c 0002) C A02 = GEE pB 12] 2.7 x 10 or A02 = Kp where K is a constant. First consider the effects of these dislocations on the WC particles. Ansel and Lenel (Reference # in) show that the stress on a precipitate particle (WC grain) is T = no where o is the stress applied to the sample and n is the number of dislocations which are being forced against the particle. They then derive the relation that the number of dislocations is equal to 2A0 n=-—JS- 71 where o is the applied stress, A is the free distance between particles (binder film thickness) and b is the burgers vector of the dislocation. They then assume that the limiting stress which a particle can withstand is proportional to the modulus of the material; then if U; is the modulus of the carbide particle “C C7 C2 T fracture = for [\J >J O or is proportional to l;;:. Calculations show that this form of a relationship is invalid in either compression or tension for WC cobalt alloys. The relationship which Ansell and Lenel derive for the average number of dislocations around a particle with respect to the applied stress in this case is of doubtful validity, since the equation was derived for a two dimensional model of a single pile up crystal. If we use the results of Doi (Reference # 8), however, then the dislocation density, and thus the average number of dislocations on any particle, is prOportional to the stress above yield to the second power 01" A02 = Kp The stress on a WC particle is wc Coo = K2 A0 0 Q II where C and K are constants. 2 72 Since K is independent of cobalt content and grain size, 2 then 2 °wc {A0 CULT for all alloys. If we calculate this parameter from the compression data of the alloys used in this study, the stress on a WC particle at failure is found to be constant for a given grain size regardless of cobalt content (see table IV). 73 «no ooo.0me 000.00 0 00 000 000.000 ooo.~00 0 m 000 000.000 ooo.o00 m 0 0000 000.000 coo.000 m-m.~ 00 0o00 coo.0m0 ooo.~0~ 0-0.0 m 0000 coo.mmo 000.000 m-m.~ m 0000 000.000 ooo.000 0.0 00 0000 000.000 ooo.0- 0.0 m 0000 000.com ooo.o~q 0.0 m 03 02 w. 0038 00000000 00 000000 00 00000000 00000000 0000 00000 000000000000 000000 0>000000500 0000» 00080000 0>000000600 <0<0 00020000 0>000000200 >0 000 0.1. The error would then be 1% by neglecting this. At the instant that the crack begins tO propagate the surface energy Of the crack must be provided by the strain energy and the work done by the applied force. This is essentially the Griffith criterion. BS 3U _.._.=....__+_.. 8 8 3L That is, the rate at which surface energy is used per unit crack extension must equal the strain energy release rate plus the work done by the external forces. In the system tO be used here the loading is via a wedge,and thus the dis- placement Of the force is controlled and is taken as zero as the crack begins tO propagate. Thus, as _ 3g = 5L'— — a or S U 3EI 2 S = yLw U = 6 ’ .2L§ ° 31.. 2 3Ewt 2 do 3E %§ t3W a: S = wa - 3 2L 3 3E %— a: YL= 3 2L 1 3 2 2 §§=Y=—-a-g=3Ei'—§'t 6061.1 3L aL “L6 3U 3E t3 52 __.= 0:6 3L 8L5 This is the strain energy release rate. From this we can calculate the stress intensity parameter for this configura- tion _ 1/2 KI - (G E) /§ E t3/2 60 KI a 2 2 77L The derivation Of this parameter and its relation to G will now be derived,starting with the equilibrium and compatibility equations of plane extension elasticity. 30 3T 3T 30 —£+_fl=0 0 —).(_X.+——la:0 8x 3y ’ 3x 3y 2 92 and assuming an Airy type stress formulation 2 2 2 x 8y2 y 3x xy Bxay we obtain a biharmonic nature for O or Vu¢ = 0. Choosing ¢ = V, + Xv, + Yw3 will satisfy equation if each is harmonic, that is V W1 = 0 using z=x+iy and a function 7(2) and its derivatives 32 az =7;7=Z;—E=Z' NH 8 Q) N have harmonic real and imaginary parts if the function is analytic. if E = Re? + 11m? then V2(Re7) = V2(ImZ) = O This is a result Of the Cauchy-Riemann condition, that is 93 BRef 31m? 8 , 31m? = BReE - If we let this automatically satisfies equilibrium and compatibility. The resulting stresses are Q N ReZI - yImZ' x I = I 0y ReZI + yImZI Txy = - yReZI, Then any function z a sizl I £ Ore - 3r (r 86) - - (l—l) r [ClAcosel - C2Asinle + C3(A-2)cos(l-2)6 - Cu(A-2)sin(A-2)6] at e = 0 from Oee = O ; 02 = - CA from Ore = O ; lCl = - (A-2)C3 from the conditions at 6 = a -(A-2)C ——_—I——i sinxd + C3sin(A-2)d + (-cosAa + COS(A-2)G)Cu = O (A-2) = C3[sin(A-2)a - __I—_ sinla] + Cu[cos(A—2)a — cosla] O C3[(A-2)cos(A-2)d - (A-2)COSAG] - Cu[(l-2)sin(A—2)a - Asinxa] = O 106 By Cramer's rule to satisfy these homogeneous equations A must be the root of the transcendental equations obtained by expanding the determinate Of the coefficients and equating this to zero sina sin(A+l)d = i a (A+1)a Let A + 1 = Y sinyd = : ysind When a = 2n this corresponds to the case Of a crack with stress free surfaces and siny2n = O or Y = n/2 This yields the stress function ¢ = rY+1EClsin(Y+l)e + c cos(Y+l)6 + c sin(y-l)6 2 3 + Cucos(y-l)6] This will satisfy stress free edges along 6 = O 3 e = a for n = 1,2,3,ooo ¢ = P(n/2+1)[Clsin(%+l)6 + C cos(%+1)6 + C sin(%—l)e 2 3 + Cucos(%—l)6] 107 From the general definition of the stress function 2 ,=1__32_<2..1_2$ , , =__1_i_sL+_l_3_i r 2 2 r 36 re r area 2 36 r 36 r 2 06 = a 2 36 both O and I must vanish at e O and 6 = 2n O r9 from the above equation this can be done by requiring F(0) = O at 0 = 0,2“ and F"(6) = O at 0 = 0,2“ 01” n n n n [Clsin(§+l)e + C cos(§+l)e + C sin(§—l)6 + Cu(§—1)6] 2 3 at e = O [02 + CA] = O or Cu = - C and n n n n n n Cl(§+1)cos(§+l)e - C2(§+l)sin(§+1)e + (5-1)C3cos(2 1)6 — (g—1)Cusin(%-1)6 at e = 0 n _ 1’1 §+l)—"(_ n-2 2 - l)C3 or Cl — 5:? C3 cl( Therefore, 108 ¢ = r(“/2+1)[[c3(sin(g-1)e - gig-s1n(§+1)e] + Cu[cos(%—1)6 - cos(g+1)e]] Writing the equation in terms of the bisector angle w = 6 - fl ¢ = r(n/2+l)[c3[sin(§-1)(w+n) - gig-sin(§+1)(w+w)3 + Cufcos(%-l)(w+w) - cos(3+1)(¢+n)]] (l) (2) ,¢ = r(“/2+1)[c3[sin<%1)wcos(-—1)n + cos(% l)wsin(% 1)n (3) (A) _ 2+2 [sin(—+1)wcos(§+l)n + cos(—+1)Wsin(~+l)fl]] (5) (6) + C “[cos(? l)wcos(——l)n + sin(——1)wsin(% 1)n (7) (8) — cos(%+l)¢cos(%+l)n - sin(%+1)wsin(%+l)n]] 1,3,5,7,°°° terms (l).(3),(5),(7) will dr0p out for n 2,A,6,8,lO,-'- terms (2),(U),(6),(8) will drop out for n 2N Then, for terms (l),(3),(5),(7) let n for terms (2),(A),(6),(8) let n 2N—l Then, f(2N) + f(2N-1) = f(n) N=l,2,3 N=l,2,3 n=1,2,3 109 in f(2N) all Odd terms will appear in f(2N-1; all even terms will appear or + _ _ — ¢ = rN 1[C3[sin(N-l)w(-1)N 1 - §:%-[s1n(w+1)w(-1)N 1J] + cu[c0s(N-1)w(-1)N’l - cos(N+l)w(-1)N-l]] rw<-1>N- 33;; [cosN r‘N*1/2)[c toosw - [cos<-1> cu (2N [A(r)A"(e)<-1)N'1c A(r)C 3(2N-1) 114 [ANB (N+%) r(N-l/2) 1 5) r(N-3/2) (N+%)(N— (2N—3)(N+1/2) (2N+1) (N-%)sin(N-%)w — sin(N+%)¢ 2 2 (N-%) cos(N-%)w — (2N-%g§§§%/2) cos(N+%)w (N+l) rN N(N+l) r(N'1) (N—l)sin(N—l)w - (N+l)sin(N+l)w (N-1)2cos(N-1)w - (N+1)2cos(N+1)w + B"(r)B(9)(-1)NCu ] cos26 3(2N-1) (2N) 3(2N-1) ] sinecose < N) r sin26 N + B'( B(e)(-1) c l r) u<2N> r 3(2N-1) 3(2N-1) ] sinacose 2 ) r l\) 4» *oo h) 7 N + B(r)B"(e)(-1)NCu ‘ J 5 3(2N-1) (2- 115 For areas very close to the crack tip observe that for n = 1 the B terms will have no r dependency. The A term will all have a —é dependency. For n > 1 all terms will depend on a positivgrpower of r. Thus for areas very close to the crack tip the n = l A terms will dominate. _ .. 2 _ , , sinecose Oyyr+o - A (r)A(6)C3(l)cos 6 2A (r)A (6)C3(l) ———;———— 2 + A'(r)A(e)c3 Si“ 6 + 2A(r)A'(6)C3 ”Sine?” (l) (l) r + A(r)A"(8)C Sin29 3(1) r2 C3 ny = %"7%ll [- cos(:%) - 008(3/2£)] cosZw 3 — l - 3/2 3 sin cos - 2[§/r c3(l)[- @- sin<—§g> + —3—- sin(§\b)3] ——$.———59 2 3 :2 cos(3/2w) sin 6 + 2/r C [- cos( ) - ] .i 3(1) 2 3 1“ + 2Er3/203( )E- 2— lsin<=ib+ +3—3— sin(—ID)]] W l r2 + [r3/203(1)[E cos<fl>+ +9—3—— cos<2w>JJ 3% 116 C 3 =- 4%)- [[(cos - 52"- (<- % 003% +i—s1n2w) -S—s1n2w)3 yy /P + cos %W[- % cos2w - sinZw + % sinzw] + sin(:%)[% sinwcosw sinwcosw] + sin(%w)[sinwcosw - sinwcosw] After considerable algebra C = _ 3(1) r cos % (l + sin % sinw %) o yy from the Westerguard analysis previously discussed (at the tip of the crack) = I cos 3 [l — sin % sin g3] yy /2nr Therefore, _ KI ’ C3 ‘ ‘2: (l)' /2w K = - 0 f2? 2 3(1) Thus, the first term of the series in the solution to the problem by a boundary collocation of the Williams function is proportional to the stress intensity factor. From the first approximation to the strain energy release rate regarding the specimens as a pair of built in cantilever beams 117 2/§ Pa KI ‘ gg§72‘ This approach, however, has neglected some effects occurring in the Specimen. Brown and Srawley have carried out calcula- tions for various H5 and aS height and crack length. The 2/§ in the above equation has been found to be a limiting coefficient as H/a + 0. They determined that 3/2 KI BH P H a 3.u6 + 2.38 — a A 01” P ... a LI KI EE§7§ [3.“6 + 2.38 a] By squaring the quantity in brackets and taking the square root ,P 2 1/2 - a g 2 g KI - §§§7§ [12 + (6.92)(2.38) L + (2.38) (L) J from Gillis and Gillman 4 I (F2L2/2EIB)[1 + (kEI/AGL2) + (n+l)Can-2] kEBH3 N-2 F2L2/2EB2H3 [12 + 2 BGL J 4 II + l2(N+l)ClL 118 F2L2 2kH2(l+v) 23[12+ 2 28 H L E7 = + 12(n+1)clL“’2] At conditions of crack initiation 88 = 8U 2y = G 2 2 2 2GB = F2L3 [12 + BKH (3+“) + (n+1)12clL(“‘2)] 2B H L 2 ‘ 1/2 a x—— = FL 3, (n-2) KI GE Egg§7§ [12 + 2k(l+V)(L) + (n+1)1201L 1 Let C depend on H or C1 = C2H and let n = 1; then 2 1/2 - FL 1L 2 KI ——§7§ [12 + 2k(l+v) 2 + 2th L] BH L Comparing this with Brown and Srawley's result 1/2 2 _ FL fl 2 H KI - EE§7§ [12 + (6.92)(2.38) L + (2.38) E5] 119 2.183 2k(l+Y) = (2.38)2 ; k 2u02 = (6.92)(2.38) ; 02 = .69 2 l2 - bending term ; 2k(l+v) flE-shear term 3 L 20 a>energy past crack tip term 2 L The shear term will be unimportant when %.< éfiw The tip energy terms, however, will be important until %-< §%5' Thus, for accurate values of KI both these factors must be taken into account. The second factor which enters into the system is the force parallel to the axis of the sample. Gillis and Gillman have explored the effect of a compressive force applied to this type of sample. They solved the case of bending only with no end rotation and even though the error using this approach could be 30% to 40% these results indicate that for our sample the error due to end force will be very small F tan(w+¢) as P where w = angle of friction and ¢ = wedge angle P 2 F tan 30° z F(.5) Therefore, 120 22 z a ( 5) e z .125 FL L Lz2 __ FL”1€”32“' Looking at the graph of the solution the error due to the end force is vanishingly small. It appears, then, that the major effects that will cause error from the standard strength of materials approach is the shear and energy past the crack tip. Both of these and the effect of end force can be accounted for by using compliance techniques. During an increment of crack extension which creates_ new crack surface dA the work done by the loading force is PdA where A is the displacement of the force in its own direction. The stored energy V is always positive and contributes —dV during crack extension. Therefore, .3 G = PdA _ dV dA 5K Load is related to displacement by A = AP where A is the inverse spring constant. The strain energy can be written as the work done during loading at constant crack length. 121 3(192 G = P 3(1P) _ 2 8A 31 G PEP 31 + 1 33 12 33_— 33 £1 8A 3A a 2 A _ P2 31 _ A2 31 _ 12 a1 _ A2 31 Q ' 2"3K ’ 2 5K ‘ 2 3(157" 2 8(L) 21 21 21 b From this G for the particular sample used in this investiga- tion can be obtained and compared with that calculated from the other methods. The effect of the slot can then be determined. APPENDIX II MATERIALS USED 122 : canoaom : m-n coauouucoucou m>mom a< mm.ma o.mm m-h ma . n «In 360m . : m-~ coaumuuauucou s>mmm ~¢ mo.qH o.~m m-m.~ ma 1 a ~-H cowuuuucmucoo s>umm H< Ho.¢H o.mm m.H ma a mud cowumuucoocoo usmqa n oH-o cofluuuucmucou s>mom Hm H< so.¢H n.5w m-“ m a «an meow 1 mum sawumuucoocoo m>mom Hm H< No.¢a m.ww mum.~ e : ~1N\H ouwm camuw enemas: H< qw.qa m.om m.H m 1 OHuo coaumuuaoocoo >>mom : mud coauauucooaoo uswaa Hm H< mm.¢a o.ww mus o a mum cagumuucoocoo aamam 1 mum coaumuuamocoo >>mom H< «m.mH m.Hm mum.~ m moao> . ouoma .: ~-H cowumuucmocoo s>mmm Hm H< m~.mH m.~m m.H m ousuosuum Anamouom oo\aw < Haozxoom n pouwwoa uamnoo N .u3 mufimcon mmmcvumm oNHm cflmuu Hmwuoumz mnm<fi 311 CS 5- 2 9% Cobalt 1.5u GS 9% Cobalt 2.5-3p CS 9% Cobalt 8:1 GS Cobalt 3% 5n GS 1 2.5-3u GS APPENDIX III APPARATUS AND SETUP 126 H muswam Figure 2 FIGURE 3 APPARATUS FOR CALIBRATION 0F INSTRUMENTED WEDGE m MMDlo anmoomm - «35.35 1 755 v 5.258 saga l .2 L SE. . 38 355% . a. \\\\ 2 1.. duo 3.3 zofimzH mums. 52m / Ad