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By W. S. Hyler, L. W. Berger, R. I. Jaffee

Hydrogen additions of 390 ppm to A-55 titanium and 368 ppm to Ti-8 pet Mn have no deleterious Hydrogenadditionseffect on the unnotched and notched rotating-beam fatigue properties of these materials. 'These amounts of hydrogen, however, are sufficient to cause severe notch-impact thesematerials.embrittlement in A-55 titanium and pronounced loss of tensile ductility in Ti-8 pet Mn. The lack of embrittling effect in fatigue in the latter alloy is consistent with the postulated strain-aging mechanism of hydrogen embrittlement in a-ß alloys. There is a significant strain-agingincrease in the unnotched endurance limit of A-55 titanium with the addition of hydrogen. This increase may be explained as the result of internal heating effects which would dissolve the hydride and cause solid-solution strengthening. TITANIUM and its alloys may be seriously embrittled by relatively small amounts of hydrogen. The form which this embrittlement takes has been shown to vary with alloy type. The a alloys, for example, suffer most strongly from loss of notch-bend impact toughness' when sufficient hydrogen is added, and this effect has generally been associated with the presence of hydride phase in the micro-structure. In a-ß alloys, on the other hand, hydrogen is most detrimental to tensile ductility in slow-speed tests,2-1 and the embrittlement may be detected in a most convincing manner by means of rupture tests at room temperature. This particular kind of embrittlement has not been associated with a change in microstructure, but has been classified rather generally as associated with a strain-aging type of mechanism.' In the present paper, the effect of an embrittling amount of hydrogen on the rotating-beam fatigue properties of both an a and an a-ß titanium alloy is covered. For this study, annealed commercially pure (A-55) titanium was chosen as an a alloy, and equilibrated and stabilized Ti-8 pet Mn as representative of a typical a-ß alloy. Nominal hydrogen levels of 20 and 400 ppm were evaluated, the latter amount having been shown previously to be severely detrimental to the impact toughness of commercially pure titanium and to cause pronounced strain-aging embrittlement in the Ti-8 pet Mn alloy. The only report of the effect of hydrogen on the fatigue properties of titanium is given by Anderson et al.,° in which a push-pull type of fatigue test was conducted on as-received commercial-purity titanium sheet. Much scatter was found in the results, but generally the presence of hydrides slightly decreased the fatigue strength of unnotched specimens in the longitudinal direction. The results of notched tests were masked too greatly by scatter to be significant. Experimental Procedure Preparation of Materials—Analyses of the A-55 titanium and the Ti-8 pet Mn alloy used in this investigation are given in Table I, which indicates the 8 pet Mn alloy to be more nearly a 6 pet Mn alloy. This alloy will be referred to as Ti-8 pet Mn, however, since this is the commercially designated composition. Both alloys were received in the form of5/8-in. diam rod and, after suitable surface preparation, 5-in. lengths were vacuum annealed at 820°C. Half of the rods for each material were then hydrogenated at 820°C to a nominal hydrogen content of 400 ppm. The hydrogenated and vacuum-annealed A-55 rods were hot swaged at 700°C from 5/8-in. diam to 1/4-in. diam, and then annealed 1 hr at 800°C and air cooled prior to preparation into test specimens. Fabrication of the Ti-8 pet Mn alloy was by hot swaging to 3/8-in. diam at 760" and then 1/4-in. diam at 704°C. This material was then annealed 1 hr at 704", followed by furnace cooling to 593"C, and finally air cooling to room temperature. Evaluation—In order to examine more completely the effects of hydrogen on the particular materials studied, slow-speed tensile and notch-bend impact properties were determined in addition to fatigue data. Tensile specimens were of the standard ASTM type with a reduced section of 1/8-in. diam and a gage length of 1/2 in. A subsize cylindrical Izod specimen was used for impact tests. These specimens had a 45" notch with a 0.005-in. radius and a 0.150-in. root diam, and the stress concentration factor of this notch in bending was Kr = 3. Both the ten-

Jan 1, 1959

By P. C. Claspy, P. H. Smakula

The electro-optic coefficients of LiNbO, and KTao.,5Nboe3,O3 (KTN) have been experimentally determined at the HeNe laser wavelengths of 0.6328, 1.15, and 3.39 µ. The coefficients were calculated fronz the retardation induced by an applied electric field. The retardations were measured by observing one or move transmission maxima and minima as a function of the applied field, and by measuring the retardation directly with a Babinet-Soleil compensator. The values obtained agree quite well with those previo~lsly reported by others. Extinction ratios as high as 3000:l have been observed. Resistivities and Curie temperatures have also been measured. Light-induced optical inhornogeneities in LiNbO, repovted by others have been observed. CONSIDERABLE interest has recently been shown in the electro-optic properties of LiNb03 (Refs. 1 and 2) and KTN.3, 4 These crystals have a number of advantages over such well-known materials as KDP. These advantages include: 1) the infrared transmission of KTN and LiNb03 extends to 5 p whereas that of KDP cuts off at 1.5 p ; 2) driving voltages required for modulators using these materials are considerably lower than those required for KDP. After first discussing some theoretical considerations of the electro-optic effect, we will report the results of measurements of the dc electro-optic coefficients of these materials at wavelengths of 0.6328, 1.15, and 3.39 p. THEORY The generalized equation for the electro-optic effect can be written: Although lattice polarization is the fundamental variable, rather than the applied electric field, we follow the convention of expressing the linear electro-optic effect in terms of the applied field. For LiNbO the quadratic terms are negligible. Since KTN has a center of symmetry the rijk tensor is 0. After applying the usual symmetry considerations and index contractions,* the rijk and gijkl tensors can be expressed as follows: LiNbOaT If field directions are now chosen, the change in refractive index and the angle between the induced optic axes and the crystallographic axes can be found. The induced rotations of the principal axes of the index ellipsoid for fields in principal crystallographic directions are given in Table I. Since KTN is initially isotropic, the principal axes in the zero-field condition are parallel to the crystallographic axes. The induced retardations are given in Table 11. EXPERIMENTAL We obtained the various electro-optic coefficients by measuring the induced retardation as a function of

Jan 1, 1968

By Edward H. Robie

Engineers' Centennial THIS summer, in Chicago, will be celebrated the "Centennial of Engineering" and half a hundred engineering societies will have meetings of one kind or another in the two weeks following Labor Day. It is really the 100th anniversary of the founding of the American Society of Civil Engineers. Centennial of Engineering is merely a convenient label to put on the celebration of 100 years of engineering progress in this country since the ASCE was founded. Engineering, of course, is much older than is thus indicated-thousands of years older. But Americans have a penchant for dramatizing progress by celebrating assumed anniversaries. Four years ago a special 30 stamp was issued on which a Brahma chicken was prominently displayed, with the legend, "1848-1948. Centennial of the American Poultry Industry." We feel reasonably sure that the first dozen eggs was sold considerably earlier than that. Until a hundred or so years ago engineers were generally classified as either military or civil. Apparently a group of civil engineers met for the first time in Augusta, Ga. but no records were kept and the first meeting of which we have definite knowledge was at Barnum's Hotel in Baltimore, Feb. 11, 1839. Forty engineers were present and they decided to form a society. Among the aims of these engineers was "the greatest amount of useful effect at the smallest cost," which is as good a slogan for the engineer of today as it was 113 years ago. It was mentioned that the existence and prosperity of the society depended in no small degree on "the personal conduct and exertions" of the members. "Talents and respectability are preferable to numbers ... from too easy and promiscuous ad mission, unavoidable, and not infrequently incurable, inconveniences perplex most societies."

Jan 1, 1952

By C. L. Mantell

Although so common and well known a metal, tin is really a less abundant element than many of those less familiar and usually ranked with the scarce or rare elements, such as cerium, yttrium, lithium, beryllium, titanium, zirconium and vanadium. The color of tin is white, with a slightly bluish tinge. When it is compared with nickel, the latter metal is brown. When placed alongside of tin, chromium is definitely blue. Tin is whiter than silver or zinc. Exceedingly thin films of tin show various shades of brown by transmitted light. The metal has a brilliant luster, and when mirror-polished, has high light reflectivity. The luster depends to a large extent upon the pouring temperature at which the metal is cast. If the temperature be too high, iridescent colors may show on the surface. These are the result of oxide film formation. If the pouring temperature be too low, the surface is dull. Small amounts of foreign metals, such as lead, arsenic, antimony and iron, decrease the luster of tin and impart a yellowish tinge to the metal. The structure is decidedly crystalline. The cast metal is a mass of crystals. When a bar of tin is bent, it emits a characteristic creaking sound. This is called the "cry" of tin, resulting from the grinding of the crystals one against the other during the bending of the metal. Tin metal may lose its crystalline structure by cold working, but grain growth occurs rapidly under the influence of heat. Metal which has been cast at a low temperature may become amorphous as the result of mechanical work. If such material be heated to temperatures well below the melting point, say at 110° C. for a half hour or for a few minutes at 150" C. recrystallization occurs. If the surface of tinned plate, tin foil, or tin meta1 in cast form be etched with hydrochloric acid containing a little free chlorine, attack occurs at the intererystalline faces. The surface shows patterns resembling the frost flowers on window panes in winter. The etched appearance is called moire metallique. Fine dendritic surface crystals of tin may be obtained by casting the metal on a surface of polished steel. In commercial tin plate, the intercrystalline boundaries are shown by fine grooves or channels, .which result from the method

Jan 1, 1929

By C. L. Mantell

Although so common and well known a metal, tin is really a less abundant element than many of those less familiar and usually ranked with the scarce or rare elements, such as cerium, yttrium, lithium, beryllium, titanium, zirconium and vanadium. The color of tin is white, with a slightly bluish tinge. When it is compared with nickel, the latter metal is brown. When placed alongside of tin, chromium is definitely blue. Tin is whiter than silver or zinc. Exceedingly thin films of tin show various shades of brown by transmitted light. The metal has a brilliant luster, and when mirror-polished, has high light reflectivity. The luster depends to a large extent upon the pouring temperature at which the metal is cast. If the temperature be too high, iridescent colors may show on the surface. These are the result of oxide film formation. If the pouring temperature be too low, the surface is dull. Small amounts of foreign metals, such as lead, arsenic, antimony and iron, decrease the luster of tin and impart a yellowish tinge to the metal. The structure is decidedly crystalline. The cast metal is a mass of crystals. When a bar of tin is bent, it emits a characteristic creaking sound. This is called the "cry" of tin, resulting from the grinding of the crystals one against the other during the bending of the metal. Tin metal may lose its crystalline structure by cold working, but grain growth occurs rapidly under the influence of heat. Metal which has been cast at a low temperature may become amorphous as the result of mechanical work. If such material be heated to temperatures well below the melting point, say at 110° C. for a half hour or for a few minutes at 150" C. recrystallization occurs. If the surface of tinned plate, tin foil, or tin meta1 in cast form be etched with hydrochloric acid containing a little free chlorine, attack occurs at the intererystalline faces. The surface shows patterns resembling the frost flowers on window panes in winter. The etched appearance is called moire metallique. Fine dendritic surface crystals of tin may be obtained by casting the metal on a surface of polished steel. In commercial tin plate, the intercrystalline boundaries are shown by fine grooves or channels, .which result from the method

Jan 1, 1929

By T. Prof. Egleston

DURING the last winter we have been actively engaged in the School of Mines in search for boracic acid. This has been owing to the fact that Mr. M. W. Iles, assistant in the qualitative laboratory, has discovered a test for that substance which makes its discovery exceedingly simple instead of being, as previously, very difficult. It consists in powdering the mineral, calcining it, then moistening it with sulphuric acid, and heating on a platinum wire until the sulphuric acid is expelled. It is then moistened with glycerin and made to take fire. If boracic acid is present it infallibly gives a green flame. It is not at all surprising, under such circumstances, the test being so easy, that boracic acid has been found by the students in a great many cases where its presence was not previously suspected, and in some where its detection was previously too difficult to warrant looking for it. We were not prepared, however, for its discovery in Lake Superior iron ores, and were disposed at first to think that a mistake had been made. After, however, it had been frequently reported in these ores it was considered worth while

Jan 1, 1877

By T. Egleston

During the last winter we have been actively engaged in the School of Mines in search for boracic acid. This has been owing to the fact that Mr. M. W. Iles, assistant in the qualitative laboratory, has discovered a test for that substance which makes its discovery exceedingly simple instead of being, as previously, very difficult. It consists in powdering the mineral, calcining it, then moistening it with sulphuric acid, and heating on a platinum wire until the sulphuric acid is expelled. It is then moistened with glycerin and made to take fire. If boracic acid is present it infallibly gives a green flame. It is not at all surprising, under such circumstances, the test being so easy, that boracic acid has been found by the students in a great many cases where its presence was not previously suspected, and in some where its detection was previously too difficult to warrant looking for it. We were not prepared, however, for its discovery in Lake Superior iron ores, and were disposed at first to think that a mistake had been made. After, however, it had been frequently reported in these ores it was considered worth while

By M. Perišic

Requirements concerning determination of geometric characteristics and estimate of reserves and grade may in various mineral deposits significantly differ and by no means can be completely envisaged. Therefore, program writing which would in advance meet all requirements is from a practical point of view impossible. Programming for each particular project is also unacceptable solution, due to many facts, among which is ever growing scope of work and the terms to be met. Therefore, conventional method of programming proved to be ineffective. The team of the Mining Institute experts came to the conclusion that the solution of facing problems lies in the idea of introducing specialized programs, instead of complex ones. Each of them represents a functional entirety. Procedure of complete mineral deposits computer processing is described by a problem-oriented programming language which was designed for that particular application. One or more than one specialized programs are called by one instruction so that the program in that language can have a few hundred time less instructions than an equivalent FORTRAN program. Besides description of the program system and programming language the paper also covers some aspects of its application in solving complex problems.

Jan 1, 1977

The fundamentals of blasting involve both the properties of explosives and of the rock being blasted. Four of the most important explosive properties appear to be energy density, bulk density, rate of energy release and the pressure- time history of the gases produced. Important rock properties are density and porosity, strength, and energy absorption including the effective moduli of elasticity. Rock structure also has a marked effect on its properties: for example, joints, bedding, and fracture alteration. Several mechanisms enter into rock fragmentation by blasting. These include slabbing, radial fracturing, crushing and bursting by the gas bubble. All of these factors are closely related to properties of both explosives and rock. Slabbing depends upon the first generated stress wave, whose shape and magnitude are a result of the pressure generated by the explosive, its time history and the resultant response of the rock, including its ability to sustain the stress wave. Actual slabbing, in turn, is a function of the magnitude and shape of the pressure pulse and the tensile strength of rock. Further fragmentation depends to a great extent upon the amount of gas in the bubble and its energy and pressure.

Jan 10, 1967

By R. V. Ramani, R. M. Mishra, Y. J. Wang

The validity of the concept of mine resistance in ventilation systems is examined. Mine resistance is the proportionality constant relating the head generated by a fan and the quantity flowing through it when the fan is connected to a mine circuit. It is the total effect of all branches in the mine ventilation circuit. Its importance lies in the fact that for new mine projections, and in the extensions of existing mines, ventilation calculations are usually based upon adjustments of the mine resistance. In ventilation circuits with regulation, and/or with multi-fans, simple adjustments of the mine resistance curve do not lead to correct solution. A ventilation system consists of several interconnected subsystems where one subsystem behavior affects that of the others, and the total system. Through a series of simple but realistic examples, the authors demonstrate the importance of the systems concept in mine ventilation, and its practical application to coal mine ventilation planning. Starting from an initial system with no leakage and one fan, more complex ventilation networks with several leakages, sections and fans, are analyzed to show that ventilation system design on the basis of total systems approach is the correct procedure.

Jan 1, 1979

By C. G. Black

THE purpose of this paper is to present data obtained from an investi-gation conducted on the grindability of coal by the American Society for Testing Materials Tentative Standard ball-mill method. This investiga-tion has been conducted over a period of a year, during which time several hundred tests were run on a wide selection of different coals coming from Pennsylvania, Ohio, West Virginia and Kentucky. The data herewith presented tend to show that the tentative standard ball-mill method might give results that are misleading and that a volume measure might be more indicative of the true grindability 'of materials of varying specific gravity. A means whereby the time element might be reduced has been studied and other phases of this work such as preparation of samples, mechanical screening, reporting results and duplicating tests have been investigated. It is hoped that all criticism offered in this paper will be taken as constructive, with the purpose in mind of perfecting the method now tentative to show the true conditions and of making optional the manner in which certain parts of the test might be conducted to meet diversified laboratory conditions.

Jan 1, 1936

By T. A. Rickard

In a previous paper (ante, p. 411) some description was given of the occurrence of the auriferous alluvium, extensive deposits of which lie among the highlands of the province of Otago, New Zealand. The present paper will describe some of the methods by which that alluvium is exploited. The alluvial mining of New Zealand is not yet hindered by " debris legislation." The mines are for the most part situated in a country of little agricultural value, and large rivers with rapid currents act as tailings conveyors. In time, however, steps will have to be taken to restrict the discharge into the Clutha of the tailings which are slowly raising the river-bottom. The methods of California have been followed in Otago with such alterations as peculiar local conditions require. The distinctive feature of the alluvial mining of Otago is the prominence given to the hydraulic elevator. The all-important factor in this form of mining is the control of an abundant water-supply at a suitable height above the level where it is to be utilized; and in this respect the gold-fields of the south island of New Zealand are most favora-

Jan 1, 1893

By G. R. Countryman, G. H. Thomas, I. Fatt

Miscible displacement in both thewetting and non-wetting phase has been studied in two-phase systems. Experimental data show that dispersion is a function of saturation and cannot be predicted from dispersion theory developed for single-phase miscible displacement. Some of the effect of saturation on dispersion is a result of the development of dead-end pores or a dendritic pore structure in multiphase systems. The presence of dead-end pores or dendritic structure is confirmed by pressure transient studies. INTRODUCTION Displacement by flooding with a miscible liquid is a possible means for recovering the estimated two-thirds of the oil that remains behind after primary production. The immense economic importance of a process that can recover such large quantities of oil has led to extensive laboratory studies of miscible displacement. As usual in production research, most laboratory studies of miscible displacement have not attemped to reproduce all of the conditions existing in a petroleum reservoir. In the early stages of investigation the need has been for information on the broad, basic principles of the phenomena involved in miscible displacement. The effects of overburden pressure, reservoir temperature, and wettability have been considered of secondary importance. However, one property of a petroleum reservoir which is expected to be of major importance and yet has been omitted from many laboratory studies is the presence of interstitial water. Two possible effects of interstitial water on the displacement mechanism in the hydrocarbon phase immediately come to mind. First, from the general1y accepted theory that capillarity governs the distribution of oil and water in porous rock one would expect that for water-wet rock the water will be in the small pores and oil in the large pores. A miscible displacement of oil carried out in the presence of water is operating in a pore size distribution shown in Fig. lb, whereas if the same test had been performed with only one phase present the pore size distribution is as shown in Fig. la. Although there is not yet a theory of miscible displacement which explains in detail the effect of pore size distribution, one would expect the differences between Figs. la and lb to influence the displacement efficiency. A second factor which may make a multiphase system different from a single-phase system is the presence in the multiphase system of dead-end pores or dendritic structure. Experiments of various kinds on reservoir rock have led to the belief that all pores in the network structure of a porous rock take part in conducting fluid during single phase fluid flow.1,2 There are then no dead-end pores and no fingers or dendritic structures containing stagnant fluid. In a multiphase system, however, the second phase may

By H. M. Kraner, R. C. Devries, K. H. Gee, E. F. Osborn

On the basis of liquidus measurements in the system COO-Mg0-Al2O3-Sio, and previously published data, diagrams have been constructed at 5 pct Al2O3, intervals from 5 to 35 pct Al2O3,. Liquidus temperatures and primary phase fields are shown. The optimum composition of a blast furnace slag for a given alumina content is indicated. At the optimum point, ordinary slags will be entirely liquid and will have maximum desulphurization potential and minimum viscosity. The relation of optimum composition of slags to the "plateau region" of the liquidus surface, and the application of these data on synthetic quaternary slags to actual slag compositions are discussed. Index of refraction of glasses is given as well as composition, temperature, and phase data for each mixture. THE usual blast furnace slag, as a first approximation, can be considered a mixture of the four oxides, CaO, MgO, Also,, and SiO,. Systematic, extensive data for the quaternary system, CaO-MgO-A1,0,-SiO, is therefore required for the understanding of the properties of such a slag as a function of composition and temperature. With sufficient data, it should be possible to fix unequivocably the relative amounts of lime, magnesia, and silica required for a given alumina content to result in a slag composition having optimum properties. Other constituents, such as FeO, MnO, Ti02, and S, have an effect on the properties of a slag, but as long as they are present in small and approximately constant amounts they produce only a second-order effect superimposed upon the major changes in properties of slags determined by the relative amounts of the four major components. Viscosity and desulphurization data have been reported for mixtures approaching blast furnace slags in composition, but systematic liquidus data have not been obtained. Inasmuch as a slag must be all liquid in order to be most effective as well as practicable, clearly the temperature above which a mixture is all liquid (liquidus temperature) is a type of data needed. Such data have been obtained through the study of 446 compositions and are pre- sented in this paper along with a discussion of their bearing on the optimum composition of blast furnace slags. For an orderly presentation of the data, the quaternary system is viewed as a tetrahedron with one component at each apex. Fig. 1 shows a tetrahedron representing the system Ca0-Mg0-A1,0,-Si0,. The base is the ternary system Ca0-Mg0-SiO,, A1,0, is the top apex, and the front face, the system CaO-

Jan 1, 1955

Discussion of the paper of W. G. MATTESON, presented at the New York meeting, February, 1918, and printed in Bulletin No. 134, February, 1915, pp. 429 to 468. WILLIAM KENNEDY, Fort Worth, Tex. (written discussion*). -In his discussion of this paper, Mr. Sherburne Rogers offers voluminous figures to show that the salt forming the. cores of the various domes could not have been derived from sea water, and gives as an illustration the immense quantity of water required to form the salt found in the Humble dome. He refers to the theory advanced by Van der Gracht apparently in support of this contention, and claims to have disposed of the views held by some that the domes were, in a great measure, formed by lateral secretion. Van der Gracht says the immense salt plugs found throughout North Germany and other countries in that region were apparently formed by orogenic movements acting upon what he calls the Mother Bed of Salt, which he gives a thickness of about 1000 ft., and an extent of several thousand square miles. The age of this "mother bed" is given as Permian. Harris, in addition to his growing-crystal theory, also says probably a great portion of the salt found in the Coastal dories may have been derived from the salt water of Permian time. Both of these theories require an abundance of water, and consequently oppose Mr. Rogers' idea that the immensity of water required disproves the formation of the salt from that source. I do not suppose, however, that either Van der Gracht or Harris absolutely limits his view to the Permian as being the only source from which the salt could have come, but would probably agree to consider any other age as being the beginning. Although the Permian is very strongly represented throughout North Texas, we have none anywhere throughout the region from which the salt may be considered as having been derived. Many deep wells have been drilled along the eastern outcropping of the Carboniferous and each has passed directly from Cretaceous to Carboniferous formations. Thus we may eliminate the Permian as a source of salt in this region. But, although the Permian may be absent, we have with us the Carboniferous with probably a thickness of 3000 to 4000 ft., and something like 3000 ft. of Cretaceous, to say nothing of approximately 8000 ft. of Tertiary and later beds, all of which, if we may judge by the quantity and density of the waters obtained from the numerous wells drilled, carry immense quantities

Jan 7, 1918

Jan 1, 1967

Jan 1, 1967

Jan 1, 1967

Jan 1, 1968

By R. L. Fullman

IN the investigation of metallurgical transformations and the relationships between microstructure and properties of metals, it frequently is desirable to obtain a measurement of the relative amounts of the various phases present and of the mean size of particles into which each phase is dispersed. The relative amounts of the phases can be measured by the classical methods of area, lineal, and point analysis,1-5 in accordance with the principle that the volume fraction of a phase, the fraction of a polished cross section occupied by the phase, the fraction of a random line occupied by the phase, and the fraction of randomly arrayed points occupied by the phase are all equal. The validity of this relationship depends only on the attainment of a truly random sample of area, length, or points, and not on the size, shape, or distribution of the particles constituting the phase. Smith and Guttman8 have derived a relationship between the interface area per unit volume S, and the measurable quantities L., the interface length per unit area on a cross section, and NL, the number of interfaces per unit length intersected by a random line. Their equation, Sv = — L8 = 2NL is also valid regardless of the distribution of particle sizes and shapes. In contrast to the situation concerning measurement of relative fractions of phases and of interface area, the measurement of particle sizes in opaque samples has not been subjected to a complete analysis. It has been common to measure some lineal or area dimension of particles on a polished cross section and to use the mean value as a qualitative measure of particle size. In the present paper, quantitative relationships are established among the various mean dimensions on a polished cross section and the actual dimensions of the particles present. Particles of Uniform Size Spheres: If a metal sample contains particles of a phase a dispersed in the form of spheres of uniform size, a polished cross section through the sample will reveal circular areas of phase a with radii from 0 to ?, the radius of the spheres. Consider a cube of unit dimensions to be cut from the sample. If a cross section parallel to one of the cube faces is examined, the average number of particles per unit area (N,) equals the number of particles per unit volume (Nv) times the probability p1 that the plane would intersect a single sphere positioned at random within the unit cube. Since, of the various possible positions for the cross-sectional plane over the unit length from top to bottom of the cube, only those positions existing over the length 2r would lead to the plane intersecting the sphere, the probability of intersecting a single sphere is just 2r. N8= Nvp1 = Nd-2r [1] Applying the equality of area and volume fractions, the relationship is found between sphere size and average area s of uniform spheres intersected by a random cross section, 4 - f = NV V = Nr . — pra = N s = Nd . 2rs S = —pr2 [2] A similar analysis reveals the average traverse length across spheres of uniform size when random lines are passed through the sample. If a randomly oriented unit cube is cut from the sample and a randomly positioned line is passed through the cube parallel to a cube edge, the number of spheres intersected by the line (Nl) equals the number of spheres per unit volume times the probability p1 of the line hitting a single randomly placed sphere in the cube. Since possible positions of the line occupy unit area, and possible positions for which it will pass through the sphere occupy an area of pr2, the probability of the line hitting a randomly placed single sphere is pr2. NL = Nv p1 = Nvpr2 [3] Combining this relationship with the equality of volume and lineal fraction, the desired relationship is obtained between radius and mean lineal traverse length -i, for spheres of uniform size. 4 - - 3 l=4/3r [4] Circular Plates: Consider a sample containing particles of a phase a in the form of circular plates of uniform radius r and thickness t, where r >> t. If the plates are randomly oriented, as in a sufficiently large sample of a fine grained polycrystalline material, area and lineal analysis may be carried out with parallel cross-sectional planes and lineal traverses. If the plates are not randomly oriented, it is necessary to randomize the orientation of the cross-sectional planes and traverse directions. Let a unit cube be cut from the sample, and a cross-section plane be passed through the cube parallel to one of the cube faces. The number of plates cut by the cross-sectional plane per unit area is equal to the number of plates per unit volume times the probability of a plate intersecting a single randomly positioned and randomly oriented plate in the cube. If J is the component of the plate diameter in the direction normal to the cross-sectional planes, the probability of a plane cutting a single randomly oriented plate is equal to J, the mean value of J for all possible orientations of the plate. Let 4 be the dihedral angle between a plate and the cross-sectional plane, and let p?, d? be the probability that a plate makes an angle between 4 and ? + d? with the cross-sectional plane. Then for ran-

Jan 1, 1954