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Jan 1, 1970

By R. G. Burdick, W. R. Peters

Shallow underground voids resulting from early coal mining and other resource recovery activities over the past several decades are now being recognized as a significant cause of ground subsidence problems. Uncertain knowledge of abondoned coal mines also imposes potential hazards in current underground mining operations, since water inundation or the release of methane gas is a serious hazard when mine excavation operations break into an abandoned mine. Requirements for detecting and mapping subversive manmade tunnels in previous work has led to the development of a surface-operated automatic earth resistivity survey system and associated computer data reduction process for military applications. Field demonstration tests of this system have resulted in successful detection of tunnels at depth-to-diameter ratios greater than 10 to one. Under the sponsorship of the Bureau of Mines, a similar high-resolution earth resistivity system was designed and constructed for use as a potential means of detecting abandoned coal mine workings. The hardware and software aspects of third new system are described together with applications of the method for surveying and mapping abandoned coal mine workings. Field tests are presented to illustrate the detection of both air-filled and water-filled mine workings.

Jan 1, 1984

By M. Bevis, A. F. Acton, P. C. Rowlands

Defor~nation twinning and transformation twinning modes most likely to be operative in Fe-Ni and Fe-Ni-C martensites have been determined using a new theory of the crystallography of deformation t~inning.~ This analysis shows that potentially important conventional and nonconventional twinning modes1 have been omitted in previous analyses. Discussion is given on the relevance of the predicted twinning modes to the lattice invariant shear associated with the martensite transformation in steels and to anomalous deformation twinning in Fe-Ni-C martensites. THE two most important criteria which appear to govern operative twinning modes in metallic structures1 are that the magnitude of the twinning shear should be small and that the twinning shear should restore the lattice or a multiple lattice in a twin orientation. The latter criterion ensures that the shuffle mechanism required to restore the structure in a twin orientation is simple. These criteria have been adhered to in the prediction of twinning modes2"6 in bcc and bct single-lattice structures with axial ratios in the range y = 1 to 1.09 as, for example, encountered in martensite occurring in steels. Refs. 2 and 3 in particular consider the martensite transformation in steels and the twinning modes in these cases relate to transformation twinning, and hence the lattice invariant shear associated with the martensite transformation. The list of twinning modes which can be compiled from these sources is incomplete and the ranges of magnitude of shear considered could be unrealistically small, particularly in the case of deformation twinning. The latter consideration is supported by the fact that twinning modes with magnitudes of shear large compared with the smallest shear consistent with a simple shuffle mechanism have been established in, for example, the single-lattice structure mercury7 and the multiple-lattice structure zirconium.&apos; In addition the anomalous deformation twins reported by Ftichrnan4 to occur in a range of Fe-Ni-C martensites still remain unexplained. It is clear that a comprehensive analysis of twinning modes likely to be operative in martensite In steels is required. The results of the application of a new theory of the crystallography of deformation twinningg to these structures are presented in this paper. The theory has been used to determine all shears which restore the lattice or a multiple lattice in a new orientation with magnitude of shear up to a required maximum. The orientation relationships between parent and twinned lattices are not restricted to the classical orientation relationships of reflection in the twin plane or a rotation of 180 deg about the shear direction. PREDICTED TWINNING MODES Twinning modes which restore all or one half of lattice points to their correct twin positions will be referred to as m = 1 and m = 2 modes, respectively. These modes are the most likely to describe operative modes in single lattice structures. The bcc m = 1 and m = 2 modes which have magnitudes of shear s in the range s < 2 and s < 1, respectively, have been given10 and are reproduced here in Tables I and 11. Detailed discussion of the crystallography of these modes and cubic modes in general will be discussed elsewhere (~evis and rocker, to be published). The four twinning elements Kl, &,ql,7)2 as well as the magnitude of shear s are given for each twinning mode, and the twinning modes are given in order of increasing shear. Two twinning modes are given in each row of the tables, the twinning mode Kl, Kz, ql, q2 and the reciprocal twinning mode with elements Kl = K,, Ki = Kl, q: = q2, and 17; = ql. The m = 1 and m = 2 twinning modes which describe twinning shears with small magnitudes of shear and simple shuffle mechanisms in bct crystals with -y = 1 to 1.09 are given in Tables I11 and IV, respectively. On increasing the symmetry of the tetragonal lattice to cubic, that is making y = 1, all modes listed in Tables 111 and IV must reduce to crystallographically equivalent variants of the modes given in Tables I and 11, respectively, or become twinning modes with both shear planes as symmetry planes in the cubic lattice and hence not considered in Tables I and 11. With the exception of this last type of mode only those tetragonal twinning modes which reduce to modes 1.1, 1.2, 2.1, and 2.2 of Tables I and I1 are considered in Tables 111 and IV. For values of y in the range -y = 1 to 1.09 the tetragonal modes and the corresponding cubic twinning modes have approximately the same magnitude of shear. The twinning modes listed in Tables 111 and IV are therefore by the criteria given above the most

Jan 1, 1969

By Edwin Berry

THE consideration of the present value of a mine is of great importance when planning for the operation of a property,, and it is particularly so in the case of mines that have large ore reserves. At various times during the past 15 years, the writer has had occasion to make rather thorough studies of the subject, and has been greatly impressed by certain important points which these studies have brought out, points which, though they may be well known, are often lost track of in the rush to obtain low mining costs and high daily tonnage output. . It is the object of this paper to present briefly, and in a general way, some of these points, and especially those which have to do with the relationship which (1) ore reserves and plant capacity, (2) grade of ore mined, and (3) ore losses in mining, bear to the present value. Present value, sometimes called present worth or actuarial value, means the value today of the future earnings of an enterprise. Great industrial, manu-facturing or railroad enterprises, if properly handled, go on indefinitely and their value is directly dependent on their estimated annual earnings. Mines, or oil properties, on the other hand, have an earning capacity only until the ore or oil is exhausted, and their present value is dependent not only on their annual earnings but also on their life, which is the length of the period over which they can produce at a profit. A mine, or an oil property, is a "wasting asset," while an estab-lished manufacturing business is not, or should not be. In other words, the present value of a mining property. represents a sum of money on which there could be paid from operating profits a determined rate of interest, say 8 per cent., and in addition an amount which, if invested at compound interest as received, would at the end of the life of the mine pay back the present value. In his "Wealth and Income of the American People," Walter Renton Ingalls says: "The fundamental principle of mine valuation is the present worth of an annual dividend accruing during the period of years corresponding to the life of the mine."

Jan 7, 1922

Institute of Metals Division SAM TOUR, Chairman J. R. FREEMAN, JR., Vice-chairman ZAY JEFFRIES, Past-chairman C. H. MATHEWSON, Vice-chairman WILLIAM M. COHSE, Secretary-Treasurer 810 Eighteenth Street, N. W. Washington, D. C. Executive Committee Until February, 1931 Until February, 1932 Until February, 1933 T. S. FULLER J. L. CHRISTIE W. M. W.M. PEIRCE G. E. JOHNSON ARTHUR PHILLIPS A. J. PHILLIPS W. A. SCHEUCH H. M. WILLIAMS GUY C. RIDDELL Research PAUL D. MERICA, Chairman S. SKOWRONSKI Data Sheet R. S. ARCHER, Chairman T. S. FULLER J. STRADSS C. H. MATHEWSON F. L. WOLF

Jan 1, 1929

By W. C. Snow

The plant of the Electrolytic Zinc Company of Australasia Limited, situated at Risdon on the estuary of the River Derwent, near Hobart, was designed and erected primarily to produce high-grade zinc from zinc concentrates produced by the mines of Broken Hill, N. S. W. Consequently the metallurgy of the process has been influenced in its development by the peculiar characteristics of the Broken Hill concentrate, which on treatment gives into solution relatively large amounts of cobalt, manganese and silica. The Risdon works (Fig. 1) produces high-grade electrolytic zinc, leaded electrolytic zinc, zinc dust, cadmium, sulphuric acid and superphosphate. This paper, however, is concerned only with the zinc plant and process. Roasting The roasting of raw concentrate is not a major feature of Risdon&apos;s process, and in this respect practice is unlike that of most zinc plants. The roasting is mostly done on the Australian mainland at centers where the sulphuric acid made from the roaster gases can be used for superphosphate manufacture. A small amount, approximately 40 tons per day, is roasted at Risdon, acid being manufactured from the gases. The amount that can be roasted there is limited by the acid requirements of the zinc plant and of the superphosphate plant. Roasting is not carried to completion at the mainland plants, and the product, known as preroast calcine, largely constitutes the raw material for the Risdon zinc plant. It contains about 6 per cent of sulphur as sulphide. Roasting is completed at Risdon for reasons that are set out later in this paper. The zinc plant proper is divided into four main divisions: (1) final roasting division, (2) leaching and purification division, (3) electrolytic division, (4) melting and casting division. There are two subsidiary divisions—the cadmium division and the preroast, acid and superphosphate division.

Jan 1, 1937

By EDWARD B. DURHAH

(San Francisco Meeting, UCtober, 1911.) THE refinery at the San Francisco Mint takes the bullion purchased by the receiving department, and carrying more than 200 parts of precious metals in 1000, or, in mint parlance; over 200 fine, and separates and refines the various metals contained therein, using electrolytic processes exclusively. Bullion containing silver is treated in cells charged with a nitric electrolyte. These cells produce fine silver and leave a residue rich in gold. The residue from the silver-cells, together with crude gold-bullion, is treated in cells having a chloride electrolyte. These produce fine gold and leave a residue containing silver chloride. The latter is reduced to the metallic state with zinc and is then treated in the silver-cells. The various waste solutions and the wash-waters, after being freed from the bulk of their precious metals, still contain copper and other metals. These are removed by scrap-iron, and are then treated in the copper-cells, having a sulphate electrolyte. These cells produce pure copper, and collect a residue containing lead, some gold and silver, and all the metals of the platinum group that were in the bullion. This residue is relatively small, and is melted into bars and stored until sufficient accumulates to warrant treating it for platinum, etc. The refinery occupies three large and three small rooms. The large ones are, a melting-room, 30 by 34 ft.; a cell-room, 39 by 46 ft.; and a wash-room, 30 by 33 ft. The small rooms are used as foreman&apos;s office, laboratory, and generator-room, respectively. The methods here described are those in use in December, 1909, when notes for the present paper were taken.

Oct 1, 1911

By Howard Martens, Pol Duwez

XPERIMENTS were performed on a Cr-Fc alloy Econtaining 44.7 pet Cr in order to determine the heat evolved during the transformation of the a solid solution into the s phase, and the change in volume accompanying the transformation. Since previous measurements on the rate of formation of a from s indicated that the reaction was quite fast as soon as the equilibrium temperature of 820°C was exceeded by some 50°, standard thermal analysis experiments were performed on heating specimens of s at the rate of approximately 40°C per sec. A definite thermal arrest was observed at 870"C, Fig. 1. This result indicates that the reverse seaction of into a is exothermic. From the heating curve of Fig. 1, taking 15°C per sec as the heating rate in the neighborhood of the transformation, 2.4 sec as the time for transformation, and assuming that the specific heat of the alloy at 850°C is an average between iron and chromium, namely 9.5 cal per mol per "C, the latent heat of transformation is approximately 342 cal per mol. The order of magnitude of this latent heat was also obtained by comparing the heating curve of Fig. 1 with a similar curve obtained with pure iron under identical experimental conditions. It was found that the length of the thermal arrest for s was about 1.8 times as long as that for iron. Since the latent heat of the a to 7 iron transformation is 203 cal per mol1, 2 the latent heat of it to a would be 365 cal per mol, which is in relatively good agreement with the value determined from the heating curve. For the determination of volume change, thermal expansion curves were recorded using a specimen containing 44.7 pet Cr and completely transformed to cr by previous treatment. The curves were recorded at a rate of heating of about 4°C per min and at approximately 840°C a sharr, increase in length of about 0.08 pet took place. Fig. 2. It is therefore established that the specific volume of a near the transformation temperature is about 0.24 pet smaller than that of a, and consequently, when the s phase forms in an a matrix, it would tend to shrink away from the matrix. Since the s phase is known to be extremely brittle, these results offer an explanation for the microscopic cracks often observed in the a phase under the microscope. The results of this study are in qualitative agreement with previously published data on stainless steel 19-9DLa and on a Cr-Fe alloy containing 45 pet Cr.&apos; The increase in density due to the transformation from n to s given in ref. 4 is 0.90 pet. These measurements, however, were made at room temperature. whereas the presently reported 0.24 pet is the change at the temperature at which the transformation takes place during heating. In the graph of Fig. 2, the percentage change in length that would be observed at room temperature is given by the length X, which is about 0.27 pet. This would correspond to 0.81 pet in volume change. Acknowledgment This work was done at the Jet Propulsion Laboratory, California Institute of Technology, under Contract No. DA-04-495-ORD-18 with the Army Ordnance Corps, Washington, D. C. The authors wish to thank this agency for permission to publish the results of this investigation. References I H. Klinkhardt: Measurements of True Specific Heats at High Temperatures. Annalen der Physik (1927) 84, pp. 167-200. - H. Esser and W. Runghardt: Influence of the Method, of Dctcrmination on the Measured Heat Content of Iron. Archtvo ffur Dct Eisenhiittenwesen 11934) 8, pp. 37-38. J. J. Gilman, P. K. Koh, and 0. Zmeskal. Delta Ferrite Formation and its influence on the Formation of Sigma in a Wrought Heat Resisting Steel. Trans. ASM 1 1949) 11, pp. 1371-1388. 1 J. M. Thomas: Discussion of ref. 3. Trans. ASM (1949) 41, pp. 3 390-1392.

Jan 1, 1957

By F. C. Wagner, E. P. Klier, D. E. Nulk

The presence or absence of plastic strain at a cleavarage surface is a question of much importance, and upon its answer depends future possible theoretical work. To investigate this question back-reflectinn X ray patterns were obained on the cleavage facets of ferrite grains. The test set-up consisting of FeK colli- mated radiation, single grain material and stationary cassette and specimen will not lead to a diffraction pattern, in general, but is an ideal arrangement for the exami-nation of surface effects. A pattern taken on hydrogen purified

Jan 1, 1949

Jan 1, 1970

Jan 1, 1948

Jan 1, 1948

By Eugene R. Rice

There are many methods for determining the meridian, but all of those in common use involve at least two separate observations, one for latitude and one for azimuth. Such observations made upon a south star at any hour are open to the objection of possible inaccuracy due to ignorance of the absolute refraction, and also to slow change in the altitude of the star when near the meridian. The method of observing a circum-polar star at elongation. besides requiring a separate observation for latitude, is open to the serious objection that clouds may obscure the star at the critical moment. Moreover, a tedious wait for the star to come to elongation is sometimes necessary. Many of us can recall &apos;long, cold, weary vigils, waiting for Polaris to reach elongation. For observing a circumpolar star at any hour, not only the latitude, but also the true time, must be known. This it is practically impossible to obtain, except in regions reached by the telegraph, and usually then one has to know the longitude also, in order to reduce " standard time" to " local time." The results obtained by this method, when the star is at culmination, are not very good, unless the true local time is absolutely known. The methods of obtaining the nleridian from the sun, by means of a solar attachment, or by direct observation, while good, are not absolute, by reason of refraction, change in declination of the sun, and small change in vertical angle, when the sun is near the meridian. For these two methods, latitude, time, and approximate longitude have to be known. In the method here described, only one observation is needed, and from it are computed azimuth, latitude, aud sidereal time. In other words, one call observe a circumpolar star—Polaris

Jan 1, 1911

By A. W. Schlechten, W. J. Kroll

As this paper is written, the only method for the commercial production of lithium metal is by the fusion electrolysis of LiC1-KC1 mixtures, as first proposed by Gunkz.2 The details of the industrial process have not been made public hut Osborg8 stated that an efficiency of more than 90 pct is obtained with a lithium metal recovery above 95 pct and that the metal is 99.5 pct pure. Pletenev9 gave a power consumption of 66 kw-hr per kilogram of lithium metal produced with a salt consumption of 9 kg of LiCl and 0.4 kg of KCI. Pletenev&apos;s figures indicate a somewhat lower recovery than that given by Osborg. In a recent report on&apos; German lithium plants, Motock7 gives figures that are also less favorable than those reported by Osborg. At the German plants the lithium recovery in electrolysis was only 83.4 pct, with a power consumption of 140 kw-hr per kilogram of lithium. The metal averaged only 97 pct pure, with sodium and potassium as the chief impurities. The presence of SO2, SO3, SiO2, Ba, Ca, and appreciable amounts of Na and Fe2O3 were especially disturbing in the electrolysis, and it was necessary to run the cell for some time to eliminate most of these impurities and condition the bath. It may be noted here that most of these impurities do not interfere with a thermal reduction process. These facts show that the present method of producing lithium is fairly efficient and would suggest that other methods could not compete. However, fusion electrolysis has certain drawbacks, such as the necessity of supplying costly, low-voltage direct current and of collecting and processing the anodic chlorine if it is not to be wasted. An even greater disadvantage is that lithium chloride must be used for an electrolyte. It is one of the most expensive salts of lithium, since it cannot be produced in anhydrous form by precipitation from aqueous solution, as is done with LiF or Li2CO3. The reduction of lithium by a vacuum process similar to the ferrosilicon method for the production of magnesium would have certain advantages. Small units such as Pidgeon retorts, would give flexibility of production to meet demand, and a small plant would require moderate capital outlay. There is a possibility of using idle Government-owned magnesium plants for such purposes. The raw materials needed for vacuum thermal reduction would not have to be extremely pure, Director, with respect to the iron and calcium Content. This paper describes a series of experiments to determine the effectiveness of various agents for the reduction of lithium compounds. The investigation is part of a program of vacuum metallurgy con-

Jan 1, 1949

Jan 1, 1971

R. Dawson Hall, who has long been a member of the editorial staff of Coal Age, has been appointed managing editor. Increase the Promise of the future. Buy W. S. S.

Jan 7, 1919

By Louise Reber

THE town of Jerome is located in Yavapai County in north central Arizona. It has a population of over 6000 people and the two important mines of the district, the United Verde and the United Verde Extension, lie almost within its limits. Perched high on the southwest side of the Verde Valley, at elevations of 4800 and 5500 ft.. (1463 and 1676 m.), both mines have long tunnels from their lower levels through which ore is hauled to their respective smelters. The two smelter towns of Clarkdale and Verde, out in. the center of the valley and about 2000 ft. (609 m.) below, form part of the wide panorama that meets the eye at Jerome. This panorama and the topographic and surface features and the geologic formations are shown in a separate pamphlet. In the vicinity of Jerome, the great southern escarpment of the Arizona plateau forms the northeast side of the Verde valley. The general level of this escarpment, or "the rim" as it is called, is about 4000 ft. above the Verde River. A corresponding elevation is reached only by the, highest parts of the Black Hills, which bound the valley to the southwest. The two lava-capped mesas, known as Mingus and Woodschute Mountains, form the culmination of the Black Hills. That part of the old Black Hills mining district which lies between their crests -and the Verde River is now known as the Jerome or Verde copper district. However, if the smelters are not included, a strip about 7 mi. long and 3 mi. wide (11 by 5 km.) embraces practically all the properties belonging to the Jerome district. This strip lies parallel to and along the side of the Verde valley from about First View, 2 mi. northwest of Jerome, to the neighborhood of Allen Spring, near the south end of Mingus. Mountain. The rank of the district in copper production, as well as features of peculiar geologic interest in connection with the ore occurrences, entitle the district to much more consideration than it has received in the literature of copper mining. Arizona stands preeminent as a copper-producing state. In 1918, it produced four-tenths of the total amount produced in the United States, and nearly equaled the, combined production of its three chief competitors, Montana, Michigan, and Utah. The production of the Jerome district places it in the sixth place among the copper

Jan 8, 1920

By D. G. Swarthout, J. W. Savage, R. W. Guard

The Nb-N constitution diagvam has been examined from 1100° to 1450°C at 1 atm N2 pressure. On the basis of- our work plus that of Brauer and coworkers the stable nitvide Phases are: a) a solid solutlion, basic niobium (bcc) with nitrogen contained interstitially; NbN,, x = <0.40; b) ß NhNx,x = 0.39 to 0.45; hexagonal (W2C type) (NbzN); a =3.058A,c = 4.961 + 0.001A c) y NbNx x = 0.75 to 0.80; tetragonal (distorted NaCl type) (Nb4N3); a = 4.386A,c = 4.332 + 0.001A; d) 6 NbN,, x = 0.88 to 0.91; cubic (NaCl type); a = 4.3885 + 0.0007A; e) E NbNx x = 0.92 to 1.00; hexagonal (Tip type); a = 2.958A, c = 11.272 * 0.0015A. In addition, a metastuble transition phase 6&apos; has been observed and occurs only during the transition 6 — E. 6&apos; NbN has these distinguishing parameters: NbNx,x = 0.96; hexagonal (anti-NiAs type); a = 2.94A, c = 5 .46A. The high-temperature phase fields in the system are governed by two reactions: a) 6 + Y + E, a entectoid reaction occurring at 1270" ± 20° C; b) o + N2 = E, a 2 012 1 reaction occurring at 1370" ± 20° C. As part of a study of the electronic low-temperature. properties of transition metal nitrides, a number of specimens of niobium nitride were prepared. Preliminary observations on these specimens showed several points of disagreement with data already in the literature. It was, therefore, decided to determine the constitution diagram for the Nb-N system between 1100" and 1450°C. The objective of this research was to define the temperature and composition regions of stability of the various phases and to explain the differences among previously proposed diagrams. Review of Previous Literature. The research on the Nb-N system to 1964 has been summarized by storms.&apos; Although he does not present a phase diagram, his review covers the research on the system thoroughly. One of the earliest works on NbN is that of Ascher-mann and coworkers2 who studied the superconductivity of the cubic 6 NbN. Subsequent work by Horn and ziegler,3 Cook, Zemansky, and Boorse,1 Rögener:5 Schroder6 and sellmaier7 has qualitatively confirmed the first results on superconductivity, although their methods of preparation differed considerably. The constitution diagrams proposed for the system by Brauer and Esselborn8 and Elliott9 are shown in Fig. l(a)and (b). The diagram in l(a) has been modified by more recent results in Brauer&apos;s laboratory (private communication). Although 6&apos; is shown on this diagram, Brauer has indicated that he now believes 6&apos; to be a metastable phase in the temperature range shown. The diagram of Elliott, Fig. l(b), is based primarily on the results of Elliott and Komjathy10 on ni-trided niobium strip or rod specimens examined by X-ray diffraction and metallography. It will be noted that these proposed diagrams are in conflict with respect to the temperature and composition ranges of

Jan 1, 1968

By M. ROWE, H. E. Gilliland, L. BARNES

The authors of this paper are to be commended for a clear exposition of a timely subject. However, some important conclusions, in addition to those presented in the paper, can be drawn from the results of their elaborate and rather complete numerical computations. First of all, the major part of the calculations presented in the paper are for semi-infinite slabs of oil shale (the authors designate this as Model I) and a question must be posed — Are these elaborate computations needed to arrive at realistic estimates of the shale oil displaced. Consider the model of Lauwerier* that corresponds to the authors Model I if thermal conduction parallel to the fracture be neglected. If one also neglects the finite (but relatively small) heat capacity of the material in the fracture, it is easily shown from Lauwerier&apos;s solution that the cumulative oil production would be:

Jan 1, 1969

By D. R. de Fontaine

By computing the values of the resolved shear stress for a great many orientations of the tensile axis on all 12 (111) <110> slip systems, Taylor and Elam&apos; were able to map out a stereogram of slip systems, given here, Fig. 1, in Diehl&apos;s notation (see, for example Hauser and Jackson&apos;). The purpose of the present note is to derive such a stereogram by mathematical reasoning and to give a simple method of defining the system with highest resolved shear stress for any direction of the tensile axis, thereby eliminating tedious computation of the Schmid factor. Let the tensile axis, of direction cosines [xyz], make the angles 0 and relative to the slip plane normal [HKL] and the slip direction [hkl] respectively. Fig. 2 shows a simple cubic cell whose edges define a Cartesian coordinate system. Let A be a unit vector of origin 0 supported by the tensile axis. Its end point [xyz] then lies on a sphere of unit radius, on the positive hemisphere of which the planes z = 0, x = y, y = z limit the so called primitive triangle of a standard [loo] stereographic projection. For any point of this triangle, the following hold: 0«2 «y *sx « 1 [l] Now, for given [xpz], the most highly stressed system will be the one for which the following expression of the Schmid factor has the largest possible value: (HKL) belonging to the form (111) and [hkl] to the form <110>, the denominator on the right hand side of Eq. [2] is equal to 1.fi.a. The unknowns H, ... h, . . . can now be determined by correctly distributing +1 and -1 to H, K, L, and +1, -1,O to h,k, 1 giving the numerator of Eq. [2] its greatest positive value (the angles 0 and cpare chosen acute) subject to the condition: Hh+Kk+Ll = 0 [3] since the slip direction must lie in the slip plane. Condition [3] can be satisfied only by having one of its terms vanish and the other two equal to +1 and -1. Taking the case of the tensile axis lying in the primitive triangle, x2y, z and therefore Hh will be chosen equal to +1 with H = h = +1 in order to retain a maximum of + signs. There remain two alternatives: either 1 = 0 or k = 0. 1) take 2 = 0 Then Kk = -1 and the numerator of [2] becomes: Retaining a maximum of + signs gives L = k = +1 and becomes: 2) take k = 0 Then L1 = -1and, likewise, the numerator of [2] becomes: xz - z2 + xy + yz [5] But the difference [5] - [4] can be written: (y - z)(x+y +z) [6] positive or zero by virtue of [I]. If [6] is positive (y > z), then the largest possible value for the numerator of [2] for any xyz inside the primitive triangle is given by [5], i.e., for: That is to say, the active slip system corresponding to an orientat:~on of the tensile axis inside the primitive triangle is:

Jan 1, 1962