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By D. R. Mash

A strong yield point with attendant enhanced mechanical properties was found in commercially pure beryllium under certain conditions of heat treatment. Beryllium specimens also responded to both quench and strain aging treatments. Based on these results, a possible means of obtaining increased normal temperature mechanical properties is suggested. COMMERCIALLY pure beryllium has been found to exhibit the yield point phenomenon and several other aging manifestations under certain conditions of heat treatment. This behavior is not surprising in view of the present accumulation of experience regarding such effects in numerous metals; but in addition to extending this experience to include beryllium, the present study is of interest because aging effects are generally associated with the presence of small amounts of solute elements. The well known lack of tensile ductility in beryllium at normal temperatures is also considered to be due to impurities in some way. In the present investigation, the view was taken that beryllium of commercial purity (greater than 98 pct) should be considered as a dilute alloy and that precipitation phenomena may well exist which could lead to enhanced mechanical properties. In connection with this possibility, published data on the variation of mechanical properties with temperature are of interest. For example, Fig. 1 illustrates the variation of ductility (percentage of elongation) and ultimate strength as a function of temperature for beryllium of various histories.* Although these curves are somewhat schematic in nature, they clearly indicate the possibility that precipitation of an unstable phase and/or aging effects occur during tests in the region 400" to 700°C. The existence of such behavior should be recognized as affecting the properties of the commercial material and perhaps can be employed advantageously. Experimental Method All testing was done at room temperature using a Baldwin universal testing machine. A microformer extensometer was used in conjunction with a micro-former-type stress-strain recorder. All tests were conducted at constant loading rate, the slowest rates being used. This corresponded to 120 lb per min in the 1200 lb full scale range and 300 lb per min in the 6000 lb full scale range. After the rate of loading was set in the elastic range of the material during the early part of the test, no further adjustments were made, allowing the test to proceed at that setting. In the plastic region of the curve above the yield region, strain rates of from 0.002 to 0.006 in. per min were regularly obtained. All experimentation was carried out on specimens machined from the same lot of beryllium, lot No. Y5036BP. Table I shows the chemical analysis as furnished by the supplier, Brush Beryllium Co., as well as the standard specifications to which commercial beryllium is produced. Both 1/4 in. thick hot pressed plate and ½ in. round hot pressed and warm extruded rods were obtained. The history of the material is reported to be as follows: 1—Beryllium metal plates, produced from a standard hot pressed QMV beryllium metal block 4x15x40 in., were cut off and finished machined to 1/4x6x12 in. flat plates. The original hot pressed block was made by hot pressing —200 mesh vacuum cast attri-tioned powder at about 1050°C under vacuum. 2—Beryllium rod was hot pressed and warm extruded. Extrusion billets were machined from the regular hot pressed blocks followed by extrusion at 425°C. The extrusion was done without jacketing, and graphite was used as a lubricant. The rods were then straightened at 720 °C, after which they were annealed at 800°C for 15 min. Both flat tensile specimens and round tensile specimens were tested. The former were machined from the plate stock into standard specimens according to specifications shown in ASTM E8-52T. The nominal cross-section was 0.25x0.500 in. and the gage length was 2 in. The round specimens were

Jan 1, 1956

By L. L. Swift

.It the present time there are four base metals being used for automo-tive bearing alloys. Of course there are numerous variations in the amounts of alloying elements added to each base metal and nearly all of these alloys have the conventional, if not classical, duplex structure; that is, they are composed of a hard constituent embedded in a soft matrix. The well-known theory is that the former will wear well, while the latter will be plastic enough to allow the bearing to be broken in and conform to the contours of the shaft. The requirements of the modern bearing have been most aptly described by W. M. Corse: In order to meet the engineering demands of today a good bearing metal must be able to carry the weight of the shaft and stand up under steady or suddenly applied loads, or even blows ill sorne types of service. A certain amount of plasticity is desired to compensate for minute irregularities in the shaft or bearing surface, and to permit yielding, thereby avoiding dangerous stresses in bearing and shaft and pre-venting overheating and rapid wear of parts. Less plasticity is required in a bearing tuetal if the surfaces of bearings and journals are machined accurately, aligned, and kept in alignment. It is desirable, of course, that a bearing metal have a low coeffi-cicnt of friction. Under most operating conditions the friction depends on the unit, pressure of the bcaring, the speed, the temperature, and the properties of the lubricant. A good bearing metal must have a low rate of wear to reduce the necessity for frequent replacements, and adjustment of the bearings. It must wear more rapidly, however, than the journal or shaft running on it, because replacement of the journal or shaft is more difficult and costly than replacement of the bearing.' The hardness of the bearing alloy is a property closely related to several of the foregoing. With a duplex microstructure, the measurement of hardness with any of the usual commercial devices means little. The Bierbaum method of measuring the hardness of the individual canstituents was developed to fill this need. MICROCHARACTER The Bierbaum MicroCharacter~,~ measures the scratch-hardness of a solid by pulling a weighted diamond point over the surface. The diamond point is ground so that it is the corner of a cube, and Mounted

Jan 1, 1939

By L. L. Swift

.It the present time there are four base metals being used for automo-tive bearing alloys. Of course there are numerous variations in the amounts of alloying elements added to each base metal and nearly all of these alloys have the conventional, if not classical, duplex structure; that is, they are composed of a hard constituent embedded in a soft matrix. The well-known theory is that the former will wear well, while the latter will be plastic enough to allow the bearing to be broken in and conform to the contours of the shaft. The requirements of the modern bearing have been most aptly described by W. M. Corse: In order to meet the engineering demands of today a good bearing metal must be able to carry the weight of the shaft and stand up under steady or suddenly applied loads, or even blows ill sorne types of service. A certain amount of plasticity is desired to compensate for minute irregularities in the shaft or bearing surface, and to permit yielding, thereby avoiding dangerous stresses in bearing and shaft and pre-venting overheating and rapid wear of parts. Less plasticity is required in a bearing tuetal if the surfaces of bearings and journals are machined accurately, aligned, and kept in alignment. It is desirable, of course, that a bearing metal have a low coeffi-cicnt of friction. Under most operating conditions the friction depends on the unit, pressure of the bcaring, the speed, the temperature, and the properties of the lubricant. A good bearing metal must have a low rate of wear to reduce the necessity for frequent replacements, and adjustment of the bearings. It must wear more rapidly, however, than the journal or shaft running on it, because replacement of the journal or shaft is more difficult and costly than replacement of the bearing.' The hardness of the bearing alloy is a property closely related to several of the foregoing. With a duplex microstructure, the measurement of hardness with any of the usual commercial devices means little. The Bierbaum method of measuring the hardness of the individual canstituents was developed to fill this need. MICROCHARACTER The Bierbaum MicroCharacter~,~ measures the scratch-hardness of a solid by pulling a weighted diamond point over the surface. The diamond point is ground so that it is the corner of a cube, and Mounted

Jan 1, 1939

By David W. Brunton

FROM the old-fashioned " grab-sample " to the modern timing- . device, which takes a machine-sample with mathematical precision, there is a wide gap which was only crossed' by many years of toil and unremitting endeavor. Even to-day, notwithstanding the advancement in the art, "grab-sampling" is still practiced-sometimes to afford the unscrupulous mine-promoter a basis for fairy-tales with which to entrap the too-gullible investor, and often by milling and smelting companies to determine the amount of moisture in custom-ores. The latter practice is almost as reprehensible as the former, and it causes more trouble and ill-feeling between seller and buyer than all other factors put together. No reputable concern to-day would think of attempting to determine by grab-sampling the amount of gold, silver, lead, or copper contained in an ore, and yet many buyers expect the miner to accept the results of grab-sampling in the determination of the amount of water contained in the ore, forgetting that accurate results are just as necessary here as in the determination of the metals, because the result determines the percentage of weight of the ore which shall be excluded and considered to have no value whatever. Samples for the determination of moisture should be taken with as great care as samples for the determination of. metallic content, and in order to avoid the extra expense of a separate operation moisture-samples should be taken from the sample-safe. As the sample reaches the sample-bin in a smaller stream and by a more circuitous route than the "reject" travels in its path to the outgoing car, it loses more moisture en route, and a constant should be added to compensate for this difference. Carefully-conducted experiments have shown that the difference in loss of moisture between the two routes does not exceed 10 per cent. in summer and 7 per cent.. in winter. For instance,

Aug 1, 1909

By F. D. Rosi, C. A. Dube, B. H. Alexander

WHEN a deformed polpcrystalline metal is heated to a sufficiently high temperature, a recrystallized structure develops which consists of small, essentially stress-free grains. This transformation is referred to as primary recrystallization. Under certain conditions, on prolonged annealing of the specimens, a type of grain coarsening occurs in which a small number of grains commence to grow at various points by devouring the neighboring finegrained primary structure, which, in itself, exhibits a marked stability toward normal grain growth. This grain coarsening process has been referred to by Burgers' as "secondary recrystallization" or "exaggerated grain growth." Although much work has been done on recrystallization and growth processes, present knowledge of the factors which determine the occurrence and growth of secondary grains is inadequate for the advancement of a theory which accounts for the observed characteristics of this recrystallization phenomenon. In this respect, it would be important to make measurements of the velocity of growth of these large secondary grains. The early work of Feitnecht2 on aluminum showed that the size of secondary grains increased with temperature and time, suggesting that such measurements would be possible. More recently, Burgers1 found that the growth velocity of these grains in aluminum was constant over a large time interval, as is generally found for the growth velocity of primary grains in a uniformly deformed test piece.3-5 It is reasonable to expect that the determination of the activation energy of the secondary growth process might lead to a better understanding of the underlying atomic mechanism. This investigation, therefore, is one of a series primarily designed to measure the growth velocity of secondary grains at various temperatures for silver, copper, and aluminum. It is hoped that these studies along with those of orientation relationships, will establish a sound basis for theorizing as to the origin and nature of the secondary transformation. Experimental Procedure The metal used in the present investigation was high-purity silver (999.9 fine), which was supplied in the form of cold-rolled plates 0.250 in. thick. These plates were annealed at 500°C for 1 hr and then straight-rolled to a thickness of 0.04 in. in three reductions of approximately 45 pct, followed in each case by an anneal at 450°C for 1/2 hr. The 0.04 in. sheets were given a final reduction of approximately 50 pct, which resulted in a thickness of 0.02 in., thereby insuring two-dimensional crystal growth. Individual specimens of this material about 1x3 cm were used for all heat treatments, unless otherwise noted. The annealing was carried out under an atmosphere of argon in an Inconel tube furnace whose temperature was controlled to within -+2°C. Metallographic examination of the secondary transformation was accomplished by mechanically polishing the specimens on one side to approximately one half their thickness through 2/0 metallographic emery paper, and then electrolytically polishing in a solution of potassium ferrocyanide and sodium cyanide. With a current density of approximately 20 amp per sq dm, the time required to obtain a satisfactory surface varied from 3 to 5 min. Standard potassium bichromate was used to etch the electrolytically polished surfaces. A heavy etch with this solution produced a sharp contrast between the secondary grains and the fine-grained primary structure, darkening the latter while leaving the large secondary grains bright. All area measurements of the secondary crystals were made with a planimeter on enlarged tracings of the grains.

Jan 1, 1953

By D. M. Boulin, E. S. Greiner

Single crystals of vanadium were bent at 293" and 77°K to a maximum fiber strain of about 1 pct. Analysis of slip line and etch markings of- crystals deformed at either temperature show the slip planes are {112}. The slip directions at 298° and 77°K, deterwined from the asterism axes of rotation of crystals compressed in a {110} direclion, were (111). Etch pits, associated with dislocations, were observed in the plastically deformed vanadium. There was no evidence of mechanical twinning. THIS paper presents results on single crystals of vanadium plastically deformed by bending and compressing at 298° and 77°K. Bcc metals have been observed to slip on {110}, (1121, and (123) planes while the slip direction has invariably been found to be ( 111) 1 Lindley and smallman2 have studied the deformation intension of polycrystalline vanadium between 298° and 20°K and reported that at and above 77°K the deformation is predominantly by slip, but at 20°K the deformation is associated with twinning. Later, Edington and smallman3 found that single crystals of vanadium compressed in a (111) direction at 77°K deformed by twinning on { 112}( 111) systems. Clough and pavlovic4 have reported that mechanical twins were observed in vanadium loaded by impact at 77°, 173°, and 195°K, respectively. PREPARATION OF SINGLE CRYSTALS The vanadium single crystals were made from commercial vanadium reported to contain 0.03 pct C, 0.03 pct 0, 0.02 pct N, 0.0X* pct Ca, 0.0X pct Si, proximately 0.8 cm diam, 4 cm long and were grown in (100) and (110) directions. A crystal for studies of etch pits was gound to have a face parallel to a {l00} plane by the Bond crystal-orienting technique.7 In order to avoid straining, the crystals were placed on the steel mount with a low-melting organic compound (glycol phthalate) before grinding. Crystals for the bending experiments were ground to have the four side faces parallel to ( 100) planes. The crystals were electropolished8 in 10 pct H2So4 using a current density of about 3 amp per sq cm to remove the strained surface material. The crystals were bent slowly by three-point loading in a stainless-steel jig designed by Sylwestrowicz.9 The end rollers of the jig were 3.8 cm apart. DEFORMATION AT 298°K The bent vanadium crystal, after removal from the jig, had a radius of curvature of 20.4 cm over the 2-cm length close to the center loading roller, where the maximum bending moment occurred. This radius of curvature corresponded to a maximum fiber strain of 0.9 pet. The side faces contained slip bands as shown in Fig. 1. These bands are straight where the strain is high, top and bottom of micrograph, but are curved on either side of the neutral axis in the center. This curvature of the slip bands in the low-stress area may be due to cross slip. The plastically deformed crystal was subsequently electropolished and then electroetched in 10 pct H2SO4 solution, using a current density of 0.25 amp per sq cm. The structure is shown in Fig. 2. The directions of the traces on two adjoining faces were plotted stereo graphically, Fig. 3.* Since there were two sets of

Jan 1, 1968

By W. T. Turrall, M. J. Cook

A discussion of modern developments in beneficiation of fine sizes of anthracite, this paper includes a description of the plant flowsheet, an analysis of operating results, and a summary of fundamentals of mineral separation relative to plant operation. DURING the past several years Lehigh Naviga-tion Coal Co. has installed equipment at Tamaqua and Lansford collieries for cleaning and sizing No. 4 buckwheat —3/32 + 3/64 in. and for recovering and cleaning flotation coal —20 + 200 mesh at Tamaqua colliery. Papers describing some of these installations have been presented to AIME. At Coaldale colliery, however, methods used to clean fine coal were inadequate and inefficient. Management decided upon a thorough investigation of all existing processes before building a plant to treat the combined three sizes mentioned. Investigation, carried out on company property and at other plants, entailed a study of hydrotators, hydroclassifiers, tables, spirals, flotation machines, methods of screening, and methods of dewatering. On the basis of this investigation a flowsheet was designed which necessarily excluded many processes in use, and the writers wish to emphasize that no criticism is implied by omission of processes or machines. A determining factor in selection of many machines was the policy of standardizing equipment wherever possible. Flowsheet and Operational Data The first principle to be considered in designing of any plant is efficiency of cleaning as related to laws of classification. Beneficiation of coal may be described as separation of two materials having different specific gravities. Although there is an inherent ash content even in the purest coal which slightly affects its specific gravity, it is recognized that many particles are true middlings. The percentage of these particles allowable in the final product depends upon the ultimate ash, or specifications of the consumer. Richards in his study of free settling and hindered settling classification' determined that particles of equal size and different specific gravities had unequal settling velocities. He also determined a definite size relationship for equal settling velocities with free settling and hindered settling conditions. Considering the design of machines used to clean No. 4 and No. 5 buckwheat, it may be assumed that both free settling and hindered settling conditions exist and that ratio of size for practical operations is between the established values. The essential factor is that the more efficiently and closely sized the feed is maintained, the more nearly perfect the resulting separation. Flowsheet design was governed primarily by factors related to the 1200 to 1500 tons per hr feed to the main plant. It should be stressed that because of the many sources of supply from mine and stripping, the physical characteristics of this feed varied considerably. The most significant influence was varying percentage of fines, necessitating that all machines be capable of operating under maximum peak load during periods of surge. Since cleaning practice is a wet process, about 9000 gal per min containing 14 to 15 pct solids, or 350 tons per hr, must be treated. This slurry contained all the fines, or —3/32 in., but also percentages of oversize, up to 1 % in., dependent on spills or breaks in screen jackets, a not uncommon occurrence in preparation plant practice. The flowsheet developed is shown in Fig. 1. Circled numbers represent points of sampling. Table I gives operational data including size, ash by size, composite ash, rate of flow in gallons per minute, percent solids dry weight, and average tons of solids per hour. It should be noted that these results are subject to human error in sampling practice. Detailed information reported is a representative evaluation of plant operation.

Jan 1, 1954

By Danforth R. Hale, R. E. Blair

Minerals for electronic and optical uses divide easily into two sections: (1) quartz and (2) minerals other than quartz. Quartz Quartz, having a great usefulness discovered by the radio communication enthusiasts about 1918, has become a subject of amazingly expanded interest in the US over the past decade as a man-made mineral and a multimillion dollar crystal-growing business. An economic summary of the commercial growing of quartz crystals has a place in a handbook directed to the mineral engineering industry when it is remembered that quartz crystals have long been an important commercial mineral, and that the raw material for "cultured quartz"-that is to say, quartz crystals grown through the ingenuity of man-is still natural quartz. Nearly all the natural crystals come from Brazil. The larger pieces which meet rigorous standards of quality are used for electronic and, to a lesser extent, optical components. Smaller pieces and fragments are used for vitreous silica. The need for high quality material in quantity led to US government-sponsored research programs in the 1940s that have resulted in the factory growing of beautiful crystals of prescribed shape, size, and quality. The development of the cultured quartz crystal is typical of man's success in adapting the product of the mine to increasingly sophisticated uses. A remarkable achievement perhaps, but foreshadowed by experiments in 1908 by Giorgio Spezia (1908), an Italian geologist studying the relative effects of temperature and alkaline environment on the solubility of quartz. Modern radio equipment is most often controlled as to frequency by the presence in the circuit of a separately added "crystal"-the 1918 discovery responsible for the existence and growth of the quartz industry. The "crystal" is quartz, but this component is a carefully oriented and prepared slice from a crystal, but not a crystal as recognized by a rock hound or seen in a museum. How quartz operates to control frequencies is not a proper subject for a handbook on industrial minerals, and references should be consulted (Cady, 1964; Mason, 1964). Quartz belongs to that class of materials called dielectrics: those that do not conduct an electric current but permit electric fields to exist and act across them. Quartz shows the "piezoelectric effect," which means that when a quartz plate is mechanically deformed against its natural stiffness, one of its surfaces becomes negatively charged, the other, positively charged. When the plate is released from the stress quickly, the charges disappear as the plate regains its original shape, but because of mechanical momentum the plate deforms in the opposite direction (to a lesser amount) and the surfaces correspondingly become charged in the opposite direction. By coating the two surfaces thinly with metal and attaching flexible wires, these charges may be brought into an electronic circuit. If the surfaces are suddenly electrically charged by movement of current through the wires, the "converse piezoelectric effect" occurs -the plate deforms. Carry the thought further and it is realized that an alternating current flowing through the wires responds to the mechanical oscillation of the plate and is keyed to this oscillation. By controlling the thickness of the plate its mechanical vibration frequency can be varied through a wide range. One type of quartz plate, called the AT-cut, having a precisely defined orientation with respect to the crystallographic axes of the crystal, vibrates on a microscopic scale much as a book would deform when placed flat on a table and the top cover moved parallel back and forth with the hand. There are at least 17 other orientations which have been studied, some of

Jan 1, 1983

By J. R. Low, M. Gensamer

During the course of an investigation into the drawability of automobile-body sheet steel, it became apparent that certain advantages would be possessed by a deep-drawing steel with a very low yield strength, but almost normal tensile strength, provided a reduction in yield strength could be accomplished without sacrifice of ductility and fine grain size. It would also be desirable to have this steel free from strain-aging; that is, free from the tendency for the yield point to return after it had been eliminated by straining. It was known that, wet hydrogen treatment acted to eliminate aging, but as carried out by other investigators, at relatively high temperatures, this treatment resulted in grain coarsening. It has been established by some of the experiments reported in this paper that all of these objectives can be accomplished by treatment in wet hydrogen at a low temperature, in the neighborhood of 72o°C. The time of treatment is relatively short (about 3 hr, to completion with 20-gauge sheet) if the water content of the hydrogen atmosphere is made fairly high, say aimost 30 per cent by volume. At this temperature. the grain size remains fine in the time required. After this treatment sheet reduced the usual amount by cold-rolling has a grain size of about A.S.T.M. No. 7. The yield point is completely eliminated, and the yield strength is reduced to a very low value, in the neighborhood of 15,000 lb. per sq. in. The stress-strain curve departs gradually from the straight-line elastic part after the fashion of nonferrous deep-drawing alloys. This elimination of the yield point is accompanied by a verv low indentation hardness; for example, 32 Rockwell B or even lower. There is no aging as measured by increase in the yield strength upon aging at room temperature or for 3 hr. at 200°C., after stretching about 10 per cent in tension. Tensile strength is only slightly less than the box-annealed sheet, seldom dropping much below 40,-ooo lb. per sq. in. Elongation is practically unchanged, or increased slightly. The term yield point, as here used, refers to the phenomenon of heterogeneous deformation occurring at a substantiallv constant load, observed principally in annealed low-carbon steels. If such a steel is loaded in tension, the load increases steadily with elastic strain, suddenly falls, fluctuates slightly about some constant value for a time and then begins to rise again as deformation proceeds. The maximum initial load before the sudden drop is generally called the "upper yield point"; the value of the load where deformation proceeds at a

Jan 1, 1944

By J. R. Low, M. Gensamer

During the course of an investigation into the drawability of automobile-body sheet steel, it became apparent that certain advantages would be possessed by a deep-drawing steel with a very low yield strength, but almost normal tensile strength, provided a reduction in yield strength could be accomplished without sacrifice of ductility and fine grain size. It would also be desirable to have this steel free from strain-aging; that is, free from the tendency for the yield point to return after it had been eliminated by straining. It was known that, wet hydrogen treatment acted to eliminate aging, but as carried out by other investigators, at relatively high temperatures, this treatment resulted in grain coarsening. It has been established by some of the experiments reported in this paper that all of these objectives can be accomplished by treatment in wet hydrogen at a low temperature, in the neighborhood of 72o°C. The time of treatment is relatively short (about 3 hr, to completion with 20-gauge sheet) if the water content of the hydrogen atmosphere is made fairly high, say aimost 30 per cent by volume. At this temperature. the grain size remains fine in the time required. After this treatment sheet reduced the usual amount by cold-rolling has a grain size of about A.S.T.M. No. 7. The yield point is completely eliminated, and the yield strength is reduced to a very low value, in the neighborhood of 15,000 lb. per sq. in. The stress-strain curve departs gradually from the straight-line elastic part after the fashion of nonferrous deep-drawing alloys. This elimination of the yield point is accompanied by a verv low indentation hardness; for example, 32 Rockwell B or even lower. There is no aging as measured by increase in the yield strength upon aging at room temperature or for 3 hr. at 200°C., after stretching about 10 per cent in tension. Tensile strength is only slightly less than the box-annealed sheet, seldom dropping much below 40,-ooo lb. per sq. in. Elongation is practically unchanged, or increased slightly. The term yield point, as here used, refers to the phenomenon of heterogeneous deformation occurring at a substantiallv constant load, observed principally in annealed low-carbon steels. If such a steel is loaded in tension, the load increases steadily with elastic strain, suddenly falls, fluctuates slightly about some constant value for a time and then begins to rise again as deformation proceeds. The maximum initial load before the sudden drop is generally called the "upper yield point"; the value of the load where deformation proceeds at a

Jan 1, 1944

By Didyk M. Marko, Vásquez Ñ. Gregorio

INTRODUCTION The Salvador Division of Codelco Chile is an important copper molybdenum production center, located 1,200 km north of Santiago and 124 km east of the Pacific Ocean, in the foothills of the Andes Range. El Salvador Mine produces by block caving 8,600,000 t of ore with a content of 1 .l0% Cu and 0.02% Mo, from a copper porphyry ore body. The geology of the ore body shows a great heterogeneity as to rocks, grade distribution, ore height and undercutting levels. This has resulted in a variety of designs for the productive units (blocks) and in the mineral handling systems. Up to now, 4 systems have been employed: grizzly and intermediate haulage; slusher drifts with intermediate ore pass and haulage; slusher drifts with scraper loading directly into trains; and, LHD. On the other hand there is a marked dispersion in the final results from each block mined out. We report here the conclusions reached after analyzing the data gathered on the mining of 163 depleted blocks, representing a total production of 102,000,000 t, with an undercut area of 340,000 sq m. The final results of each block, characterized in terms of tonnage and fine cop- per recovery, were studied as a function of a number of design, geological, and operational parameters. BASIC CONCEPTS Indices of Efficiency In order to evaluate the final result of each block, or group of blocks, the following indices of efficiency were defined : Tonnage Recovery (TR) . It is the ratio between the ore tonnage extracted and the in situ tonnage estimate for each block or group of blocks. Grade efficiency (E) . It is the ratio between the Cu mill feed grade and the in situ Cu grade estimate for each block or group of blocks. Grade efficiency is numerically equal to the ratio between fine copper recovery (FR) and tonnage recovery (TR) . Figure 1 shows the final results for 163 mined out blocks, in terms of tonnage and fine copper recovery. Different grade efficiencies are also indicated. The marked dispersion in the final results of mined out blocks must be stressed since it greatly incides in mine planning. It must be kept in mind that an eventual concentration of low overall efficiency in the blocks during a given period (e.g., one year) will make it necessary to bring into production new blocks over the standard development schedule. It is worth noting that the final results in the 163 mined out blocks corresponding to a production of 102,000,000 t of ore with a mill feed grade of 1.445% were obtained from an ore estimate of 93,000,000 t and a grade of 1.671% Cu, which represents 110% tonnage recovery and 86% grade efficiency. Parameters Studied The following parameters are considered in this study: Geological Parameters. These parameters depend exclusively on the geological and geotechnical features of the ore body. IN SITU GRADE: The initial estimate of the block grade. Four copper grade categories were considered : (a) lower than 1.3% ; (b) 1.3 to 1.6%; (c) 1.6 to 2.0%; (d) higher than 2.0%. The overall initial in situ estimate of the ore body was 1.6% Cu. IN SITU TONNAGE: Initial tonnage estimate of the block. Four categories are considered: (a) below 500,000 t; (b) 500,000 to 1,000,000 t (c) 1,000,000 to 1,500,000 t; (d) over 1,500,000 t. CONTACTS SLOPE: Since up to 4 rock types can be found in a block, resulting in a corresponding number of contacts, a horizontal con- tact was defined as that which forms an angle of less than 45" with the horizontal and a vertical contact that which forms an angle of more than 45" with the horizontal. GRADE DISTRIBUTION: As in rock types, horizontal and vertical grades were defined. ORE HEIGHT: The distance measured between the undercut level and the top of ore. Four categories are considered: (a) less than 60 m; (b) 60 to 100 m; (c) 100 to 150 m; (d)over 150 m

Jan 1, 1981

By Carl Altstetter, Roy Adams

Measurements of enthalpy changes and transformation temperatures me reported for the fcc -hcp marten-sitic transformation in pure cobalt single and multi-variant crystals andpolycrystalline specimens. From these measurements the free-energy difference between the two phases was calculated for both heating and cooling and for repeated cycles through the transformation. For heating of single crystals the average value of enthalpy change was 113 cal per mole but on cooling it was 84 cal per mole. This difference is interpreted in terms of lattice imperfections which are introduced during transformation. A calorimeter suitable for measurement of enthalpy changes of athertnal transformations is described. COBALT exhibits an allotropic phase transformation at around 417°C' where an hcp phase (a = 2.507, c = 4.0686, c/a = 1.6228)~ stable at low temperatures transforms to an fcc phase stable up to 1495°C. The transformation has a hysteresis of about 30°C— the exact magnitude of the hysteresis depending on the number of transformation cycles, grain size, impurity content, and prior treatment.'-6 The athermal character of the transformation and surface relief due to transformation shear indicate that the transformation is martensitic. Pole mechanisms for the transformation have been discussed by Sebilleau and Bibrin, Bilby, Seeger,' and Basinski and Christian. These authors have proposed that the cooling transformation is effected by rotation of a Shockley partial dislocation around a pole dislocation with a screw component of 2a/3(111). Such a rotation would change the stacking of close-packed planes from the abcabc fcc stacking to the ababab hcp stacking sequence. This mechanism would predict a lattice correspondence of {111}fcc~~(0002~cp, have been observed. One would expect the heating transformation to occur similarly, involving dislocation movements on the (0002)hcp. However, a high density of pole dislocations and high dislocation ve- locity are required to account for microscopic observations. Furthermore, the high back stress developed on the Shockley partial dislocation as it rotates toward the sessile dislocation associated with it makes it doubtful that this is in fact the transformation mechanism. A mechanism for spontaneous nucleation of partial dislocations1*l2 seems more reasonable. If a single crystal of cobalt is cycled through the transformation but never heated above 600°C, it will remain single."13 However, if the specimen is heated through the transformation and held for a short time at or above 1000°C before cooling, the cooling transformation usually involves all four {lll}fcc habits, resulting in neighboring orientations of hcp differing by 70.5 deg. This condition is termed multivariance and has been observed by Bibring and sebilleau5 and Nelson and ~ltstetter.' The latter authors used single crystals of cobalt grown in an electron beam zone refiner. They proposed that the strongly directional cooling caused the initial cooling transformation to occur on the {lll)fcc most nearly perpendicular to the temperature gradient. Transformation shear on one set of {lll}fcc planes immobilized dislocations on the other {lll}fcc planes involved in the initial transformation. Since there is only one (0002)hcp plane orientation, the product of subsequent heating transformations could have no more than two variants—the original fcc orientation or its twin. If, however, the specimen was annealed at a high temperature, such as 1000°C, there would be relaxation of dislocations into low-energy configurations, and in the absence of a sharp temperature gradient transformation on all four (111 Ifcc planes could occur. If a dislocation mechanism is responsible for the transformation, the perfection of the lattice should affect the reversibility of the transformation. Mutual constraints of neighboring grains in polycrystalline material would inhibit dislocation movement resulting in increasingly sluggish transformation. Indeed, Nelson and Altstetter found the M, to decrease and the A, to increase in going from single crystals to multi-variant specimens to polycrystals. The effect is enhanced by decreasing the grain size of a solid specimen or a powder.6'14 Data on the free-energy difference necessary to initiate the transformation gives information about the driving force of the transformation. This is an important consideration in the understanding of the transformation mechanism. The free-energy change associated with the transformation of the hcp phase to the fcc phase at some temperature T different from the equi-

Jan 1, 1969

By J. E. Pavlick, A. G. Guy

The total creep of tin alloys containing antimony in solid solution was observed to decrease with increase in antimony content. However, near the solubility limit an anomalous maximum in deformation was. found, in agreement with similiar observations by Bochvar. IT has long been recognized that alloys may deform more rapidly under stress if there is a tendency for a phase change or other structural change to occur simultaneously. However, the phenomena involved are complex and in some instances the opposite effect, strengthening, may result.' This area of research has not attracted much interest in the western world, and although Russian investigators have studied it for the past 15 years their work has not received due attention here. Recently diametrically opposing views on one aspect of this problem have been espoused by two Russian investigators, Kornilov and Bochvar. The question is this: in certain binary alloys tested in creep at the temperature where the solute is at the limit of solid solubility, does the incipient formation of the second phase strengthen or weaken the alloy? Korni-lov's arguments in favor of strengthening were presented at the 1954 Symposium at the National Physical Laboratory, =Creep and Fracture of Metals at High Temperatures.The most recent statement of Boch-var's theory of weakening is in Oding's new book on creep and rupture.3 Although Bochvar agrees that strengthening occurs in some alloy systems, he argues that in certain alloys an anomalous weakening is observed. However, the extensive work of Korni-lov's group on creep properties of alloy systems has not revealed any anomaly of this type, but instead has tended to refute Bochvar's theory. This clear-cut controversy offered a good opportunity for a simple experiment to resolve the problem. The experiment and its results supporting Bochvar are the subject of this paper. However, it will be helpful first to sketch the present status of the larger question of the effect of a phase change or other structural change on plastic deformation. PREVIOUS WORK One of the first researches in this area was by Pfei1,37 who investigated the creep resistance of an 80 Ni-20 Cr alloy containing varying amounts of titanium. He found that the resistance to creep increased rapidly as the solid solubility limit in the alloy at 775'c was approached and slightly exceeded. With further addition of titanium the creep resistance decreased, presumably because of precipitation effects. Bochvar's first work in this field was a study in 1945 of "superplasticity" in zinc alloys containing 15 to 25 pct Al.4 He found that alloys quenched from about 400 had anomalously low hardness when tested below this temperature. The following year, in a paper on the dependence of mechanical properties on the composition and structure of alloys,5 he extended some observations of Kurnakov to predict that at high temperatures of testing the hardness values for solid-solution alloys might be lower than those for the component metals. He also pointed out the influence of changes in solidus temperature in affecting curves of hardness vs composition, and he

Jan 1, 1962

By B. A. Wilcox, A. H. Clauer

The tensile and creep deformation characteristics of fiber reinforced composites have been studied, primarily at room temperature. The matrix was an alloy of Mg-14 wt pct Li-1 wt pct Al (LA141A alloy) and the reinforcing fiber consisted of continuous wires of a high strength precipitation hardening stainless steel (AFC-77 alloy). The ultimate tensile strength and strength-to-density ratio of composites were much improved over that of the mmix, both at room ternperatwe and up to 200°C. Creep studies at room temperature revealed logarithmic weep behavior for wires and composites and steady state creep for the nonreinforced matrix. It was concluded that creep of the fibers controlled composite creep. The instantaneous logarithmic weep rate of the composite, 6, was related to the average tensile stress on the fibers, where n = 3.4. Mg-Li alloys are the lightest structural metals commercially available, e.g., the alloy LA141A (Mg-14 wt pct Li-1 wt pct Al) is 27 pct less dense than beryllium Although Mg-Li alloys have moderate tensile strengths at normal strain rates and ambient temperature, their creep strength is relatively poor, even at room temperature. Reinforcement by fibers offers a means of increasing the tensile strength as well as overcoming the limitations of low creep strength. With a proper choice of reinforcing filament, composites could be designed with higher strength-to-density and modulus-to-density ratios than the matrix alloy. This communication reports a study on the tensile and creep characteristics of alloy LA141A reinforced with continuous wires of a high-strength precipitation hardening stainless steel (AFC-77 alloy). The matrix has a bcc structure, with possible precipitates of AlLi and MgLi2A1,3 and is ductile at room temperature Stainless steel wire was chosen as the -reinforcing filament since previous work at Battelle4 showed that a good bond between wire and matrix could be produced in this system. For all volume fractions of reinforcing wire, the theoretical composite strength-to-density ratio is greater than that of the matrix and the modulus-to-density ratio is somewhat lower than that of the matrix. EXPERIMENTAL PROCEDURES The 0.004-in. diam AFC-77 wire was obtained from two suppliers: National-Standard Co.* and Crucible 0.13 Si, 0.18 V, 0.011 P, 0.012 S, balance Fe. In the as-received condition, wires from both sources had room-temperature ultimate tensile strengths of 585,000 to 590,000 psi with approximately 2 pct total elongation and 40 pct reduction in area. Composite preparation was accomplished by the vacuum infiltration technique. This involved placing the continuous wires in a mild steel tube (1/8-in. ID), capping one end of the tube with a sheet of the matrix alloy, and evacuating the other end of the tube. When the capped end was placed in a crucible of the molten matrix alloy and held for 31/2 min at 70O°C, the liquid metal was drawn up the tube approximately 7 to 8 in. A cross-sectional view of a typical composite is shown in Fig. 1. This preparation technique promoted excellent bonding between the matrix and wires, and porosity was rarely observed. Test specimens for both creep and tension studies were prepared by machining off the steel tube. Tensile tests on the wires and composite specimens were made in an Instron using a cross-head speed of 0.01 in. per min. The composite specimens had no reduced section and were held in self-tightening grooved serrated grips. Most of the specimens broke in the effective 1-in. gage section. Even though several tests fractured in the grip section, their ultimate strengths were essentially the same as those of specimens which fractured in the gage section. Creep specimens of the composite were gripped in a similar manner, and

Jan 1, 1970

By J. Geertsma

The relationship between porosity and the speed of propagation of acoustic waves in fluid-saturated porous rocks as measured by the Sonic log and by ultrasonic tecbniques is analyzed. Biot's continuum theory 1 is used to explain the difference in acoustic wave propagation between a dry and a liquid-saturated porous material. The porosity is a variable in this theory. However, the acoustic wave propagation in the dry rock depends too on porosity, and this dependence is not predicted by the theory. Frequently in dry sandstones, a nearly linear relationship between reciprocal acoustic wave velocity and porosity is observed in the low-porosity range. The physics behind this behavior is outlined. An empirical relationship of the form, 1/v = A + F, applies accordingly for many porous dry rocks, provided the porosity is the only variable. The presence of a liquid in the pores changes the value of B, and this change is found to be in agreement with the Biot theory. The time-average relation introduced some years ago2 results in an equation of the same type — 1/V = F/Vf + (1 - F)/Vr — but is not based on a sound physical picture. Still, this relation sometimes predicts approximately correct A and B values. Carbonate rocks with their complicated pore structures sometimes show a different relationship between wave velocity and porosity, unfavorable for log interpretation. Examples are presented. The simultaneous presence of calcite, dolomite and anhydrite, with their different grain densities and matrix compressibilities, complicates acoustic-log interpretation in carbonate rocks still further. Other complicating effects of acoustic-log interpretation are discussed. They are related to the influence of shale streaks and natural fractures on the average wave velocity observed by the logging tool and to the effect of adsorption phenomena on wave propagation in unstressed rocks particularly in sandstones. INTRODUCTION The velocity of propagation of sound waves in porous sediments as a function of depth is a quantity frequently measured. Velocity loggers of various d-sign may be used, for which velocity is defined as the. distance between a wave generator and a wave detector (one-receiver system) or the distance between two wave detectors (two-receiver system), divided by the shortest time required for a vibrational pulse to cross this distance. Velocity loggers ale used to assist in seismic prospecting, to differentiate between the various types of sedimentary rock layers, and also to determine rock porosity and pore fluid content. This paper is especially concerned with the relation between velocity and porosity and the effect of the pore fluid. Most velocity-log interpretation in terms of porosity is based on a formula advocated by Wyllie, et al,2 suggested earlier by Hughes and Jones,3 and known as the "time-average relation". This formula app1ies for a model consisting of a layered system of parallel alternating slices of a solid and a liquid, crossed by the wave path perpendicular to the interfacrbs. This must be an unsuitable model for the derivation of wave propagation properties of liquid-saturated porous media. It suggests that only rock matrix and fluid properties influence the wave velocity. The surprising fact, nevertheless, is that applications of this formula to clean sandstones under specific conditions (sufficient effective stress) results in porosity predictions which are close to reality. Other authors, such as Gassmann,4 White and Sengbush,5 Brandt,6 and Hicks and Berry,7 make use of more realistic models to arrive at a wave velocity formula for porous sediments. Their models consist of various liquid-saturated packings of frictionless spherical grains and are, therefore, also highly idealized. A number of experimental results obtained in the laboratory can be better explained with these model theories than by the time-average relation. A study of a general character, not depending a priori on a special model of the porous structure, was still lacking.

By B. A. Cheadle, K. P. Steward

Axial tensile, ring tensile, closed end burst, and free end burst tests were done at room tempe.vature and 300°C on three batches of Zircaloy-2 sheathing zuith different textures. Knoop hardness tests were also done on all batches. The mechanical properties of the tubes were largely dependent on the percentages of cold work, but were also markedly affected by the textures, particularly with respect to deformation behavio'r. The deformation behavior observed confirmed Picklesimer's predictions, zuhereas Knoop hardness results could be correlated with texture and yield strength anisotropy after the ,method of Wheeler and Ireland. From the mechanical properties, and from efjectiue strain values deterrlzined at the fractures using a photoqid technique, it zuas concluded that tubing witlz a single preferred orientation of radial basal poles best combines strength and ductility for any given deqee of cold work. THE mechanical properties required in collapsible Zircaloy-2 fuel sheathing for water-cooled power reactors are strength to resist wrinkling deformation by coolant pressure and ductility to accommodate a small amount of deformation in service without rupturing. Zircaloy-2 tubing during fabrication develops pre- ferred orientations of the a Zr grains due to aniso-tropic deformation behavior, and these textures markedly affect mechanical properties. Picklesimer' had predicted the deformation behavior of tubes with various textures, but the texture required to provide the optimum strength and ductility for a given degree of cold work has not been reported. To investigate this problem, tests at three transverse to axial stress ratios were done at room temperature and 300°C on three batches of Zircaloy-2 tubing covering a wide range of textures and percentages of residual cold work. Knoop hardness values were also measured to correlate with preferred orientation and yield strength anisotropy after the method of Wheeler and relland.' 1) MATERIAL For this investigation, three batches of tube-reduced tubing were chosen from the Zircaloy-2 sheathing produced for the Douglas Point Reactor. Individual tubes were 191/2 in. long, 0.600 in. OD, and 0.0165 in. nominal wall thickness. The identities of the batches, together with their textures, are shown in Table I. The tubes had the normal Zircaloy-2 composition3 of 1.5 wt pct Sn, 0.3 wt pct Fe, Ni, Cr combined with the following oxygen concentrations: Batch VII 1140 to 1390 ppm 0 Batch VIII 1400 to 1750 ppm 0 Batch IX 880 to 1320 ppm 0

Jan 1, 1968

By L. L. Swift

AT the present time there are four base metals being used for automotive bearing alloys. Of course there are numerous variations in the amounts of alloying elements added to each base metal and nearly all of these alloys have the conventional, if not classical, duplex structure; that is, they are composed of a hard constituent embedded in a soft matrix. The well-known theory is that the former will wear well, while the latter will be plastic enough to allow the bearing to be broken in and conform to the contours of the shaft. The requirements of the modern bearing have been most aptly described by W. M. Corse: In order to meet the engineering demands of today a good bearing metal must be able to carry the weight of the shaft and stand up under steady or suddenly applied loads, or even blows in some types of service. A certain amount of plasticity is desired to compensate for minute irregularities in the shaft or bearing surface, and to permit yielding, thereby avoiding dangerous stresses in bearing and shaft and preventing overheating and rapid wear of parts. Less plasticity is required in a bearing metal if the surfaces of bearings and journals are machined accurately, aligned, and kept in alignment. It is desirable, of course, that a bearing metal have a low coefficient of friction. Under most operating conditions the friction depends on the unit pressure of the bearing, the speed, the temperature, and the properties of the lubricant. A good bearing metal must have a low rate of wear to reduce the necessity for frequent replacements, and adjustment of the bearings. It must wear more rapidly, however, than the journal or shaft running on it, because replacement of the journal or shaft is more difficult and costly than replacement of the hearing., The hardness of, the bearing alloy is a property closely related to several of the foregoing. With a duplex microstructure, the measurement of hardness with any of the usual commercial devices means little. The Bierbaum method of measuring the hardness of the individual constituents was developed to, fill this need. MICROCHARACTER The Bierbaum microcharacter2.3 measures the scratch-hardness of a solid by pulling a weighted diamond point over the surface. The diamond point is ground. so that it is the corner of a cube, and mounted

Jan 1, 1938

By P. A. Fisher, J. B. Wilson, D. J. Whitehead, C. J. P. Ball, A. C. Jessup

The properties of a new magnesium alloy ZT1 containing 3.0 pct Th, 2.5 pct Zn, 0.7 pct Zr are described. The alloy possesses good creep and fatigue resistance up to 650°F, is free from microporosity, and is readily sand-cast. ZT1 is shown to be superior to zinc-free Mg-Th-Zr alloys. THE advent of the jet engine created a demand for a light sand-casting alloy capable of being cast into large complex shapes and capable of resisting creep at temperatures up to 500°F. This demand has been largely met by alloys of the magnesium-cerium mischmetal-zirconium types, containing about 3 pct Ce mischmetal and 0.7 pct Zr, typified by Elektron magnesium alloys MCZ and ZREI, the latter alloy also having a zinc addition of 2.5 pct. These alloys utilize the valuable properties at elevated temperatures conferred by the addition of rare earth metals to magnesium, a fact which has been known for many years and discussed by several investigators.1-1 Murphy and Payne8 showed that the addition of the powerful grain-refining element zirconium to Mg-Ce mischmetal alloys produced alloys with attractive room temperature properties combined with good creep properties and castability. With the increasing power of modern jet engines the temperatures to which some of the magnesium parts are subject are likely to exceed the safe working range of the cerium-mischmetal-containing alloys, and at the beginning of the research considerable interest had been shown by aeroplane engine designers in a magnesium sand-casting alloy suitable for service at higher temperatures, e.g., 600°F and above. At an early stage in the development of this type of alloy a check on the fatigue properties at elevated temperatures indicated that, since the fatigue limit even at 108 eversals was appreciably higher than any permissible creep stress, the main emphasis of the research could be concentrated on creep behavior. The primary aim in developing the new alloy has therefore been to attain maximum creep resistance; but due consideration has also been given to other important properties such as castability, tensile strength, and cost. It is by now well established that it is not possible to correlate creep strength with other short time mechanical properties such as are obtained in tensile

Jan 1, 1954

By David A. Thomas, David N. French

The lrnrdness of WC crystals has been measured with the Knoop indenter at loads of 100 and 500 g on the (0001) and (1070) planes. The hardness as tneasitred on the basal plane is 2400 kg per sq mm and shows little anisotropy. The hardness on the prism plane, however, shows a marked orientation dependence, with a low value of 1000 kg -per sq mm when the long axis of the Knoop indenter is oriented parallel to the c axis and a high value of 2400 kg per sq mm when the indenter is perpendicular to the c axis. Slip lines (Ire observed surrounding the microhardness indentations and they show slip on (1010) planes, probably in [0001] and (1120) directions. This slip behavior can be explained by the crystal structure of TVC, which is simple hexagonal with a c/a ralio of 0.976. The hardness anisotropy call be explained by [0001]{1010} and (1130) {10l0] slii) and the resolved shear-stress analysis of Daniels and Dunn. HARDNESS anisotropy is a well-known phenomenon and has been reported for many metals, with both cubic and hexagonal structure.1-6 For hexagonal tungsten carbide, WC, a wide range of hardness values is reported in the literature. For example, Schwarzkopf and Kieffer7 give a hardness of 2400 kg per sq mm and report a value of 2500 kg per sq mm by Hinnüber. Foster and coworkerss give the average Knoop microhardness as 1307 kg per sq mm with a maximum value of 2105 kg per sq mm. Although these values and the structure of WC suggest the likelihood of hardness anisotropy, no such measurements have been made. We first detected a large apparent hardness anisotropy in WC crystals about 75 p large, in over-sintered cemented tungsten carbide. Prominent slip lines also occurred around many indentations. This report presents further observations and interpretations of hardness anisotropy and slip in WC crystals obtained from Kennametal, Inc. Both Knoop and diamond pyramid indenters were used on a Wilson microhardness tester with loads of 100 and 500 g. EXPERIMENTAL RESULTS The carbide crystals tended to be triangular plates parallel to the (0001) basal plane of the hexagonal structure. The side faces were parallel to the ( 1010) prism planes. Specimens were mounted approximately parallel to these two types of faces and metallographically polished. Laue back-reflection X-ray patterns were used to orient the specimens, which werethen ground to within ±1 deg of the (0001) and (1010) planes. The Knoop hardness values measured on the basal plane are plotted in Fig. 1. There is only a small anisotropy, with a hardness range of 2240 to 2510 kg per sq mm. The additional points at angles from 52.5 to 67.5 deg confirm the sharp minimum hardness at 60-deg intervals, consistent with the sixfold hexagonal symmetry. The average hardness of all values obtained on the basal plane is 2400 kg per sq mm. While the basal plane shows only slight anisotropy, the (1010) plane exhibits marked hardness anisotropy, from 1000 to 2400 kg per sq mm. Fig. 2 shows the hardness as a function of the angle between the long axis of the indenter and the hexagonal c axis, the [0001] direction. The minimum and maximum occur when the indenter is oriented parallel and perpendicular to the [0001] direction, respectively. The anisotropy of the prism plane is contrary to that reported for hexagonal zinc and hard- However, the basal-plane anisotropy is similar to these two metals.1'2 To check the accuracy and reproducibility of the measurements, a series of fifteen impressions was made at 100-g load in the same orientation in the same area of the specimen surface. The average for all was 2040 kg per sq mm, with a range of 1950 to 2130 kg per sq mm, giving an accuracy of about ± 5 pct. Thus the slight anisotropy on the basal plane is almost within experimental error. Fig. 3 shows slip lines around the Knoop indentations on the basal plane. The slip traces are in directions of the type (1130). The presence of slip steps on the basal plane indicates that the slip direction lies out of the (0001) plane. Because WC has a c/a ratio of 0.976,' the shortest slip vector is [0001], which suggests slip systems of the type [0001] (1010). Fig. 4 shows slip lines around the Knoop intentations on the (1010) plane. These slip lines are inconsistent with [0001] slip but can be

Jan 1, 1965

By C. S. Smith, L. Guttman

It is shown, from a study of geometric probabilities, that the average number of intercepts per unit length of a random line drawn through a three-dimensional structure is exactly half the true ratio of surface to volume. Since the surfaces can be internal or external, the area of grain boundary or of the interface between any two constituents in a micro-structure can be measured. Other metric relations are tabulated that may be of use in studies of the microstructure of polycrystalline, cellular, or particulate matter generally. IN many fields of scientific investigation the structure of cellular aggregates or random arrays of discrete particles imbedded in some matrix is observed on a two-dimensional section and inferences are drawn therefrom as to the real structure in three dimensions. The biologist's microtome slice, the petrologist's thin section, and the metallurgist's plane polished and etched sections are common examples, although the problem is a general one. Scientists have commonly limited their thinking to the same dimensionality as their structures, and the few attempts that have been strictly three-dimensional in character have been laborious and noteworthy. From a metallurgical standpoint it is often of considerable importance to know, in addition to the volume fraction of two or more components in an alloy, the amount of two-dimensional interface between crystals. Such grain boundaries (which may separate either two identical crystals differing only in orientation or two crystals differing in structure, and possibly also in orientation) have a determining factor upon the mechanical behavior. It is at these boundaries that melting commences, that stress-induced corrosion occurs, and that various precipitates (harmful or otherwise) first appear. The boundary is doubtless of equal importance in nonmetallic crystalline aggregates such as rocks, ceramics, and concrete, and the biologist is deeply concerned with the area of cellular membranes. Many synthetic cellular foams involve similar structural problems. The very term structure usually implies the presence of interfaces and a complete understanding of structure involves nothing but an analysis of the geometrical, metrical, and topological relations between the various interfaces (zero, one and two-dimensional) that exist in a three-dimensional structure. Even systems lacking sharp physical interfaces often have interrelated gradients of composition or velocity (as cored crystals or turbulence cells) in which a neutral surface can be treated as a two-dimensional interface. In an earlier paper by one of the authors' the question of cell shape was considered in terms of simple topological principles without regard to physical dimensions. The determination of the actual size of grains in two dimensions is carried out in a routine fashion in innumerable metallurgical laboratories (see, for example, the ASTM standard methods of grain size determination2), though this is done merely to check the uniformity of a product and has no relation to the actual three-dimensional shape or size of the grains. Some authors have discussed the three-dimensional problem but only on the basis of assumptions as to idealized grain shapes.:'-' Quantitative measurements of microstruc-tures to obtain the volumetric relations of various phases have been carried out by petrographers for many years and are of increasing popularity among metallurgists." The present paper will show how, on the basis of no assumptions other than randomness of sectioning (usually realizable in experiment), it is possible to learn a great deal about the three-dimensional structure. The relations to be derived will generally be used on random arrays of cells or other particles, although they are equally applicable to ordered arrays and even to isolated objects of complex shape provided that suitable random sections can be made. Total Area of Interfaces in a Sample Consider a typical microstructure of a single-phase polyerystalline metal, such as that shown pres- in Fig. 1. The plane cross section shown contains a network of lines which subdivides the area into two-dimensional cells. The lines, of course, represent merely the intersection of the two-dimensional plane of sectioning with the two-dimensional interfaces between adjacent three-dimensional cells. The structure also contains points at which two or more lines intersect: in three dimensions the points are, of course, lines. In the more general case there may be in three dimensions isolated particles surrounded by a single interface without contact with others, and both two and one-dimensional features which do not necessarily connect with others. On the two-dimensional section these will appear as areas delineated by a closed line (as at a and b, Fig. 2), as isolated lines

Jan 1, 1954