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Book Review MANAGEMENT AND MEN. By Meyer Bloomfield, Boston. The Century Co., New York, 1919, pp. 584, H.X. 5 1/2. $3.50. This hook is an exposition of English efforts to solve the problem of labor unrest in England. The first six chapters are reprinted from articles published in the Satur-day Evening Post, for whom the author went to England to study the problem at first hand. These articles deal with housing, increased production, eliminating the internal friction in the industrial machine and a chapter each devoted to the view-point of employer and employee. These chapters are written in the easily readable magazine style and give a complete general survey of the particular solution being tried. Perhaps their greatest., certainly not their least, worth lies in the fact that they form a vehicle for unusually voluminous appendices containing in full the famous Whitley Reports which are the basis of the new movement, together with various reports, which, except for this publication, would be largely inaccessible to those inter-ested in this country. The Whitley scheme considers each industry a unit and proposes to establish therein a Joint Standing Industrial Council composed of representatives of employers and employees to consider appropriate matters affecting the industry at large. There is to be a District Council for each industry which will, as its name implies, deal with the clatters pertaining to that interest in. the particular district. The third and last body in this scheme of Industrial Parliament is the Works Committee, which is com-posed of representatives of the management and of the workers employed in particu-lar plants or mines to act in close cooperation With the district quid national machinery. The Whitley or similar schemes have been tried in many communities in England and the book contains interesting comments on the result obtained, it being notable that most of the objections conic from plants which have not tried the scheme. Meyer Bloomfield, the author; is the specialist in industrial personnel problems who organized the industrial service activities of the United States Shipping Board, Emergency Fleet Corporation. P. E. B.

Jan 10, 1919

By A. Chakravarti, C. C. Harris

First-order kinetics adequately describe the recovery of a single floatable species. The conditions under which cell behazior can be simulated by a single-phase model are discussed; deviations from the multiphase model are accountable. The rate coefficient depends upon operating conditions besides particle size and mineralogy and the physicochemical properties of the pulp constituents. The model for a real flotation system combines the first -order, single-phase model with a distribution of noninteracting species. Methods for determining the distributions are discussed. Applications of the procedure to flotation studies are outlined. The mathematical description of flotation as a rate process can be traced to Garcia1 who reported cases in which the fraction of solid remaining in a cell in semi-batch operation, F (for Notation used throughout, see Table l), decays exponentially with experimental time T. F = e-k>r. (1 The differential equation leading to Eq. 1 is dF ----= -k,F (2) dT which is recognized as the algebraic analog of the chemical kinetics first-order reaction equation.' This has led to suggested mechanisms for flotation and mathematical relationships other than first order: with very little foundation other than analogy. The flotation rate coefficient kl can be accounted for as the product of several probabilities concerning bub-ble-particle attachment and detachment, aggregate entry from pulp into froth and return from froth to pulp, and removal from froth as concentrate. These factors clearly contribute to the magnitude of the overall flotation rate, and there are others also, but describing flotation merely as the overall probability of consecutive gross events hardly appears a significant advance in understanding. More recently, it has been described in terms of a stochastic process, which promises to provide a more satisfactory approach? The present paper will be more concerned with a phenomeno-logical description of the flotation process than with an account of the internal mechanics of the process. This is not a statement of priorities, or a judgement of the relative importance of these options but rather an assessment of what is presently possible. Previous investigators'" have developed methods for interpreting semi-batch flotation data in terms of species distribution. These methods have been used in an attempt to explain flotation behavior in response to changes in operating conditions." Inspection of examples given in the literature" reveals discrepancies between theory and experiment, and data obtained in the Colum-

Jan 1, 1971

By L. A. Rapport, J. R. McCord, W. P. Drews, K. H. Coats

Water movement about an oil or gas reservoir can be predicted provided an aquifer influence function is known. This function is generally determined from an electric analyzer study or by fitting an idealized mathematical model to field data. A prohlem arises in determining that influence function which best reproduces field data and satisfies certain smoothness requirements, without making any idealizations concerning aquifer geometry and heterogeneity. This study shows the problem to be solvable through the linear programming technique and presents applications to one oil reservoir and two gas storage reservoirs. In addition, a theorem is stated and proven which confirms influence junction smoothness requirements heretofore accepted on physical intuition and establishes additiotzal smoothness constraints formerly not recognized. INTRODUCTION The pressure decline accompanying production results in water encroachment into oil and gas reservoirs situated adjacent to aquifers. The importance of this water movement derives from the significant dependence of production rate upon reservoir pressure and of pressure, in turn, upon water encroachment. Pressure and rate of water movement are uniquely related through an "influence" function which may be in the form of reservoir pressure response to a unit water encroachment rate or of water movement caused by a unit reservoir pressure drop. This influence function reflects the heterogeneity and geometry of the aquifer and is therefore particular to each reservoir. If the influence function is known, then reservoir pressure may be predicted from a production rate schedule, or vice-versa, with the aid of the material balance equation which relates water influx to reservoir production and pressure. The problem of determining this influence function for a given reservoir has been approached in three different ways. Van Everdingen and Hurst calculated and tabulated certain influence functions pertinent to idealized mathematical models which satisfy simplifying assumptions such as aquifer homogeneity and elementary reservoir-aquifer geometries.' Other studies developed additional tables for a variety of elementary reservoir-aquifer geometrical con- figurations and demonstrated their validity and utility.:' Hicks, et al. described the use of an electrical network to determine an influence function giving "best" match of field history and allowing prediction of future performance.' Finally, Hutchinson and Sikora5 and Katz, Tek and Jones attempted to develop a calculation method for deriving the influence function directly from field data with no idealizing assumptions concerning aquifer geometry and homogeneity. They encountered difficulties with inaccuracies in field data and with satisfaction of well-known smoothness requirements of the influence function. This study is concerned with the determination of influence functions from field data and, more specifically, treats the following problem: given field pressure-production data of arbitrary accuracy, how does one determine the influence function which, subject to satisfaction of known smoothness requirements, gives best agreement with field history? A rigorous solution to this problem is ohtained through linear programming as described below. Example applications to one oil and two gas storage reservoirs are presented. In addition, a theorem is proved which confirms influence function smoothness requirements heretofore accepted on physical intuition and establishes additional smoothness requirements formerly not recognized.

Jan 1, 1965

By David, Cole

THE years 1914-15-16 were a pioneering period in mining, milling, and copper metallurgy generally. It was uncertain just what path the crushing, grinding, and concentrating processes would take. This situation was quite embarrassing to the directors of some of our large porphyry properties, then being equipped for production. Inspiration had already made several designs for its proposed 15,000-ton concentrator, incorporating in them the latest devices developed in the art of gravity concentration, only to have the work vitiated by the rapid march of events. In a little improvised plant, built on ground that is now in the crater of subsidence over the Inspiration- Miami orebodies, flotation was being experimentally ' applied to Inspiration ore by Minerals Separation engineers in 1914, with results indicating that this process would be of major importance in the concentration of the sulfides. How best to apply it, and especially how to crush and grind the ore to the very much finer sizes required for the new process, was a serious question, properly to be answered only by doing it, in full- sized machines, under regular 'operating conditions.

Jan 1, 1931

By N. J. Themelis, R. V. Flint

In a four-year period between 1973-76, the Metal Mining Division of Kennecott Copper Corp. conducted a company-wide program of operations improvement which involved line and staff personnel from all company functions. Specialist teams performed detailed studies of all operational aspects at Kennecott's four copper mines and concentrators, four smelters, and two electrore-fineries. The specific recommendations that resulted may transform the Metal Mining Division into the country's lowest cost producer of copper by the mid-1980s. In this article, N. J. Themelis and R. V. Flint describe the mechanics of MMD's integrated approach to operations improvement.

Jan 6, 1979

The third special war meeting of the New York Section was held on Thursday evening, Sept. 27, in joint session with the New York Section of the American Electrochemical Society, in conjunction with the Third National Exposition -of-Chemical Industries, at the Grand Central Palace. The topic for discussion was ferro-alloys, and the supply' of raw materials for their manufacture. After a few introductory remarks by President Philip N. Moore, of the Institute, and President Colin G. Fink of the American Electrochemical Society, Mr. C. S. Newcomb presented his paper on "Chrome Mining and Concentration."

Jan 11, 1917

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 R. G. Aspden

Crystals with an (001) [loo] initial orientation of an iron-base alloy containing 3 pct Si were cold rolled with and without the use of constraints. A major difference in the rolling and annealing textures was observed between crystals rolled with and without constraints. These data show that the contribution of constraints at grain boundaries in a poly crystalline sheet should be considered in applying textural data on single crystals to grains in an aggregate. SILICON-iron alloys with a cube texture have been recently developed and their magnetic characteristics reported.1-4 Of interest in the development of this texture were the textural changes of single crystals accompanying rolling and annealing and the influence of constraints at grain boundaries in an aggregate on the behavior of individual grains. The present study was primarily concerned with the effect of constraints during rolling on the textures of 3 pct Si-Fe crystals initially (001)[100]. Barrett and Levenson5 were among the first to observe an influence of constraints at grain boundaries on the textural changes of individual grains during deformation. They tested Taylor's6 theory of plastic deformation of face-centered-cubic metals in which deformation textures were predicted. About one-third of the grains in poly crystalline aluminum did not rotate as predicted. Grains of the same initial orientation were observed to rotate in different directions under the influence of applied stress and anisotropic flow of neighboring grains. Recently, the various inhomogeneities of flow of crystals in an aggregate have been studied7'8 and reviewed.9-11 Barrett and Levenson" rolled (001) [loo] iron single crystals inserted in close-fitting holes in copper to limit lateral flow and to simulate rolling of grains in an aggregate. Deformation bands were formed after a 90 pct reduction in thickness, and the cold-rolling texture contained two components described by rotating the (001)[100] about 35 deg in both directions around the normal of the rolling plane. No annealing textures were reported. Chen and Maddin13 rolled molybdenum single crystals initially (001) [loo]. The crystals were mounted between two hardened silicon-iron plates and 96 pct reduced in thickness by rolling at a low rate of reduction, about 0.0001 in. per pass. The deformation texture had the mean orientation of (001) [loo], and the azimuthal spread included orientations described by rotating (001) [loo] about 35 deg in both directions about the pole of the rolling plane. The presence of deformation bands were not reported by Chen and Maddin or detected in subsequent work of Ujiiye and Maddin.14 The ideal orientation of the annealing texture was (001) [loo]. Recently, Walter and Hibbard 15 reported on the textures of 3 pct Si-Fe alloy crystals initially near (001) [loo]. Each crystal was in an aggregate cut from a columnar ingot. After 66 pct reduction by rolling, the texture consisted of two symmetrical components which had the orientations described by rotating (001) [loo] about 30 deg in both directions about the pole of the rolling plane. Annealing texture was near (001) [loo]. In the above work, the textures of body-centered-cubic crystals were studied after rolling under the influence of constraints. The deformation textures varied from (001) [loo] to near the (001) [110] type and appeared sensitive to the manner in which the crystals were rolled. No textural data were available on the effect of rolling (001) [loo] crystals with and without constraints. The purpose of the present work was to evaluate the influence of constraints during rolling on the textures of 3 pct Si-Fe crystals initially (001) [loo]. Rolling and annealing textures were studied for a) crystals rolled with no constraints at different rates of reduction, and b) crystals rolled with constraints imposed by neighboring grains and by plates between which a crystal was "sandwiched". PROCEDURES AND EXPERIMENTAL TECHNIQUES Data are presented on four crystals which are representative of several crystals studied. The orientation of each crystal prior to rolling was (001) [loo] as determined by the Laue X-ray back-reflection method," i.e., each crystal had an (001) within 3 deg of the rolling plane and [100] within 3 deg of the rolling direction. These crystals were obtained from two iron-base alloys containing 3 pct Si by weight which were prepared by vacuum melting electrolytic iron and a commercial grade of silicon. Crystals 1, 2, and S-1 were cut from a large single crystal grown from the melt of one alloy by the Bridgman technique17 in an apparatus described by

Jan 1, 1960

By S. H. Ward, R. A. Barker

THE Juniper Prospect is in Carleton County, N. B., at approximately 46º 31' north latitude, 67º 20' west longitude. During the summer of 1955 an area in west-central New Brunswick was selected for aerial electromagnetic prospecting with American Metal Co.'s helicopter-mounted equipment. Generalized, small-scale government geologic maps had suggested that this area was located along the western border of the northeast-trending granitic backbone of northwestern New Brunswick and might be underlain by the Ordovician-Silurian sediments and volcanics that are the host rocks for the recently discovered sulfide deposits of the Bathurst-Newcastle district.

Jan 1, 1958

E. DEGOLYER,-New York, N. Y. (written discussion*).-Since the eastern part of the Gulf Coastal Plain is receiving considerable attention from various operators at the present time, it occurs to me that it brief review of most recent operations may be of interest in this connection, as well as a discussion of some of the points suggested by Mr. Hager. During the latter part of January, 1918, the Alahoma Oil Co. completed and abandoned a 3400-ft. (1036-m.) dry hole in Section 25, Township-8 North, Range 1 West. This well was drilled near the axis of the

Jan 4, 1918

By L. O. Howard

THE CHAIRMAN (WALTER DOUGLAS, New York, N. Y.).--I presume, gentlemen, that Mr. Howard's experience with the Great Falls basic-lined converter has been as regards the tonnage produced from a single lining, perhaps, a world's record. About 6 months ago Mr. Howard shut down and tore out what was left of the first basic-lining which was installed at the Old Dominion, the lining having produced something over 70,000,000 lb. of copper. This was due probably not only to the very excellent grade of the matte which was produced by the smelter there,

Jan 12, 1916

Improvement of drilling operations through the development of new, unconventional drill bits is a prime research target at Sandia Laboratories, Albuquerque, N.M. Sandia is currently focusing efforts on four new bit designs which its engineers believe may eventually improve penetration and reduce operating costs. The Energy Research and Development Administration (ERDA) has budgeted $500,000 in support of this program to date. Two of the drill bits-one that chips rock with high voltage spark discharges, and another which fires small projectiles into the rock ahead of a regular roller bit-are designed to increase both penetration rates and downhole bit life.

Jan 8, 1975

By Niels Hansen

The substructure formed by drawing at room temperature in aluminum (99.5 and 99.998 pct purity) and in recrystallized aluminum aluminum-oxide products containing from 0.2 to 4.7 wt pct of aluminum -oxide was examined by transmission electron microscopy, and the flow stress of the drawn materials was measured by tensile testing at room temperature. A sub-grain structure was present after a reduction in area by drawing of 10 to 20 pct, and the subgrain size was observed to decrease with increasing deformation. The tensile data show that the increase in flow stress (0.2 pct offset) by drawing from 10 to 95 pct depends on the reduction in area, not on the composition of the materials. Dispersion strengthening and subgrain bowzdary strengtlzening contribute to the flow stress, and these strengthening processes have been found to be linearly additive. The flow stress (0) can be related to the subgrain size &) by the Petclz relation = uo + k . t, where go is dependent on the composition of the products and k is approximately the same far all materials. THE microstructure of dispersion-strengthened aluminum aluminum-oxide products consists of small oxide plates distributed in an aluminum matrix. The matrix structure depends on the manufacturing history, and in hot-worked as well as cold-worked products the matrix is divided by subgrain (or dislocation) boundaries. For hot-extruded products it has been shown1 that dispersion strengthening and subgrain boundary strengthening are linearly additive, and the flow stress (0.2 pct offset.) at room temperature has been related to the subgrain size (ts) by a Petch equation,2,3 s = so + k . ts-1/2, where s0, increases with increasing oxide content. For cold-worked products containing subgrains no systematic work has been reported, and it was the aim of the present study to examine the microstructure and the relationship between the flow stress and the subgrain size for such products. The behavior of' aluminum aluminum-oxide products depends on the purity of the aluminum matrix, and aluminum of the matrix purity (99.5 pct) was included in the investigation. The literature contains few data about the behavior of this impure aluminum, and aluminum of a higher purity (99.998 pct) was therefore also examined. As regards the relationship between the flow stress and the subgrain size in cold-worked dispersion-strengthened products, no systematic work has been reported. For aged cold-worked structures containing fine precipitates (Fe-Mo carbide) a Petch relation has been found,4 and it has been shown that the k value NlELS HANSEN is Head, Metallurgy Department, Danish Atomic Energy Commission, Research Establishment Riso, Denmark. Manuscriot submitted January 9, 1969. IMD is approximately the same as in iron, whereas the s0 value is higher owing to the presence of the precipi-tates. Investigations of metals such as tungsten,' ferrous metals,4,6 and molybdenum7 cold drawn or swaged at room temperature have shown that the flow stress can be related to the subgrain size by a Petch relation when ts is taken as the subgrain size perpendicular to the direction of deformation. For aluminum no work has been reported on the relationship between the flow stress and the subgrain size after deformation at room temperature, whereas for aluminum tensile strained at different temperatures in the range -183" to 375°C a Petch relation has been found by taking ts equal to the subgrain size.' In the present study two aluminum materials (99.998 and 99.5 pct) and three aluminum aluminum-oxide products (containing 0.2, 1.0, and 4.7 wt pct oxide) were drawn at room temperature to reductions in area from about 10 to about 95 pct. The structures were studied by transmission electron microscopy, and the flow stress (0.2 pct offset) was measured at room temperature. EXPERIMENTAL Materials. The materials are given in Table I together with the chemical analysis. The three aluminum aluminum-oxide products were manufactured from aluminum powder that had been compacted and Table I. Chemical Analysis of Materials Al203 Fe SI Material wt pct wt pct wt pct 99.998 pct* - 0.0004 0.0012 Aluminum 99.5 pctt - 0.36 0.16 Aluminum AlMD13† 0.2 0.16 0.12 Aluminum- -Oxide AlMD105† 1.0 0.26 0.18 Products SAPISML960s† 4.7 0.22 0.19 *Other impurities: O.0004 pct max each of Cu and Zn (supplier's analysis). †Other impurities: 0.03 pct max Cu. 0.02 pct max each of Mn, Mg, Zn, Ti. Table 11. Mean Diameter of Aluminum-Oxide Particles in Extruded and in Cold--Drawn Aluminum Aluminum-Oxide Products Mean Diam. of A1203-Plates* Material State A AlMD105 Extruded 540 Cold Drawn 97 pct 510 SAP ISML 960 Extruded 770 Cold Drawn 95 pct 820 *The standard deviation of the mean is approx. 5 ±pct.

Jan 1, 1970

By AIME AIME

THIS journal is fully indexed-every issue, as soon as published-in a set of card?, which thus brings ready to your hand any article on any subject in this journal or in any one of 1700 other journals in all languages. This Engineering Index Service, instituted in January, 1928, is to be found in the Public Libraries of Newark, Cleveland, Bridgeport, Baltimore and the John Crerar Library, Chicago. Engineers in any field who live or work in New Jersey will find this mammoth index to the latest engineering literature in the Public Library of Newark, N. J.

Jan 1, 1929

New York Meets first Wednesday after first Tuesday of each month. W. FRANCIS, Chairman, WILLARD S. MORSE, Vice-Chairman. THOMAS T. READ, Secretary, Woolworth Bldg., New York, N. Y. P. A. MOSMAN, Treasurer. LOUIS D. HUNTOON. WILLIAM A. POMEROY. Boston Meets first Monday of each winter month. HENRY L. SMYTH, Chairman. ALFRED C. LANE, Vice-Chairman AUGUSTUS H. EUSTIS, Secretary-Treasurer, 131 State St., Boston, Mass. ROBERT H. RICHARDS, ALBERT SAUVEUR. Columbia FRANK A. ROSS Chairman. RUSH J. WHITE, Vice-Chairman. LYNDON K. ARMSTRONG, Secretary-Treasurer, P. O. Drawer 2154, Spokane, Wash. FREDERIC KEFFER, FRANCIS A.,THOMSON.

Jan 4, 1915

One of the first all-hydraulic crawler drills in service at a US open-pit mine has significantly exceeded the penetration rates of comparable pneumatic crawler rigs, according to Atlas Copco Inc. Overall production with the hydraulic unit has been double that of pneumatic crawlers in use. The installation is at the Foote Mineral Co. surface mine at Kings Mountain, N.C., where the company is engaged in mining spodumene, a combination of lithium, aluminum, and silicates. The ore is found in pegmatite dikes, a tough and abrasive host rock with compressive strength of tip to 5800 kPa (40,000 psi).

Jan 9, 1978

Following the suggestion of Engineering Council and at the direction of, the President, representatives from each of the Federal Government bureaus interested in map-making were called into a conference for the purpose of coordinating all of the Government's map-making activities. This conference has now reported back to the President the results of its findings, which indicate that Engineering Council's recommendation in this matter was well founded. While the conference report states that its members were of the opinion that little duplication of effort existed, it recommends the formation of a permanent Board of Surveys and Maps. An extended review of the map-making activities, which is given as an appendix to the report, indicates that this recommendation is a logical step. The conference recommendations concerning this Board of Surveys and Maps are as follows: (1) the Board is to he com-posed of representatives of mapping bureaus to be assisted by outside engineering organizations, which have been named above; (2) the estab-lishment of an information office in one of the Government bureaus, preferably the Geological Survey, which is to be under the supervision of the Board and is to be a mapping information clearing house for the Government and outside engineering organizations; (3) copyright laws are to be so changed that copies of all. maps made in the United States and with Governmental authority, will go to the central information office; (4) the Government departments are to receive orders to cooperate, with the Board; (5) future topographic maps of all kinds are to conform with specifications for the standard topographic map; (6) the U. S. Coast and Geodetic Survey to have supervision of final adjustment of all important control data under specifications approved by Board of Surveys and Maps; (7) surveys in the form of maps and publications are to be issued as soon as possible after field work has been completed; (8) program of the Interdepartmental Committee on Aerial Surveying was approved. Outside technical and scientific organizations that are called in for consultation are to operate in an advisory capacity only, responsibility for action on Governmental work is. to rest solely with the Governmental agencies.

Jan 12, 1919

B. F. TILLSON,* Franklin, N. J.-A preliminary study of this paper did not, in my case, succeed in checking with the shovel dimensions of one of the large shovel manufacturers. I therefore raise a question as to there being some difference in standards used by different manufacturers in regard to the dimensions of the shovels for similar sizes and styles, so it may be rather confusing to some of us when we attempt to use the table in this paper and classify the capacity of the shovels as given here with some that are on the market.

Jan 4, 1919

(Members are urged to send in for this column any notes of interest concerning themselves or their fellow-members.) Members and guests who registered at Institute headquarters during December: William M. Peschel, Bradford, Mass. Samuel T. Wellman, Cleveland, Ohio. Herbert B. Cox, Boston, Mass. Samuel S. Wyer, Columbus, Ohio. C. Q. Schlereth, Mexico, D. F. H. H. Stoek, Urbana, Ill. H. W. Caldwell, Oxford, N. C. E. Harms, Torreon, Mexico. Charles E. Schwarz, St. Louis, Mo. G. A. Roush, So. Bethlehem, Pa. Dwight Furniss, Guanajuato, Mex. M. K. Rodgers, Los Angeles, Cal. F. W. DeWolf, Urbana, Ill.

Jan 1, 1914

By E. P. Dahlberg, R. E. Reed-Hill

On the assumption that stress or strain relaxation occurs as the result of a thermally activated process, equations are derived relating to tensile experiments that give the strain as a function of the time under the condition of constant stress, and the stress as a function of the time for constant strain. It is demonstrated that if the strain-rate equation i = previously proPosed by Kuhlmann., is used as a starting point, then the relaxation of strain at constant stress may be expressed by the equation c = (-RT/(Y) 1tz tanh (t + is the strain capable of being relaxed at any given instant. Similarly, it is shown that the relaxation of stress at constant strain may be given by a = (-RT/B) In tanh (t + t0)/27, where a is the instantaneois value of the relaxable stress. The fact that these relationships reduce to well-known empirical equations at both large and small values of the stress Or strain is also shozcn. The present theory is shown to agree well with experimental data obtained from tensile elastic aftereffect experiments on a zirconium specimen prestrained at 77 k as to make it strongly anelastic. It is also demonstrated that elastic aftereffect data obtained using torsional specimens ?,Lay agree reasonably well with the equation derived for the case of tension. RELAXATION experiments are often employed as a means of studying metallic deformation mechanisms.' The simplest and most commonly employed techniques involve stress relaxation at constant strain and strain relaxation at constant stress. In general, however, investigations of this nature have been seriously handicapped in the past by a lack of suitable equations giving the time dependence of the relaxing variable over an interval that extends from small strains up into the region where internal-friction experiments become strain-amplitude dependent. This paper presents a derivation of such a set of equations for the case where the time-dependent part of the strain is anelastic or recoverable and the specimens are loaded in simple tension. The relaxation of strain under the condition of constant stress will be considered first. Let us assume that strain relaxation occurs as the result of a reversible thermally activated process that occurs at a number of relaxation centers lying in an elastic matrix. Then, following Kuhlmann,2 we may express the rate of strain relaxation as follows: where C is the strain rate, AFx the free energy of activation of the process controlling strain relaxation, a, the effective or average resolved stress at the relaxation centers, u an activation volume, R the universal gas constant, T the absolute temperature, > a factor with dimensions of a volume that accounts for the strain contribution of a successful operation of a unit process, N the number of relaxation centers per unit volume, and v the Debye frequency. The first term on the right of Eq. [I] represents a strain rate in the direction favored by the stress, while the second term represents the rate in the opposite direction. It is implied in Eq. [I.] that both F and v are symmetrical with respect to the two basic directions of operation of a relaxation process. Eq. [I] may also be written where and S and Q are the activation entropy and activation energy, respectively, of the relaxation process. In the following, A will be considered a constant. This is compatible with a set of experimental conditions where the relaxation rate is controlled by a single basic reversible process in which it may be assumed that the temperature dependence of the product ?Nv is negligible in comparison with the temperature variation of the exponential term. It is also implied that v, 7, and N do not depend strongly on a, . In deriving a relationship for the strain as a function of the time from her equation, equivalent to Eq. [2], Kuhlmann2 chose to consider only the limiting cases where the time was either very small or very large. It will now be shown that it is possible to integrate Eq. [2] to obtain a single equation valid over a wide range of strains if the concept of relaxable strain is introduced. The use of this quantity, which is the difference between the instantaneous value of the strain and the value of the strain at complete relaxation, represents the primary point of departure of the present theory from that of earlier workers. Let us express the effective stress at the relaxation centers in terms of strain. For this purpose we may use the following equation derived by zener3 for the case of strain relaxation at slip bands: where M? and M are the relaxed and unrelaxed moduli respectively, go the applied constant tensile stress, and m an average orientation factor that takes account of the fact that a,, the effective stress at the relaxation center, may not be a tensile stress (i.e., a

Jan 1, 1967