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By M. I. Brittan

Discussion Dr R. E. Robinson (Fellow): The author must be congratulated on a very meticulous and self-contained piece of work. It is indeed a pleasure to read a paper that is so clearly and systematically laid out, and where the conclusions and the testwork conducted have been so clearly described. The paper is complete in itself, which makes it very difficult for someone who is not intimately involved in the whole Torco project to make any comments on its content. However, two points that, strictly speaking, fall outside the scope of the paper are of considerable interest. The first relates to the particle size of the material treated. In the paper, the testwork is confined to one standard particle size (minus 60 plus 100 mesh). The essential feature of the paper is to indicate that the rate-controlling reaction in the whole segregation process is the rate of reaction of the ore particles with the reducing agent and the hydrogen chloride. One wonders, therefore, to what extent this relatively slow rate of reaction is affected by the particle size of the ore itself. One imagines that the reaction must take place by contact of the hydrogen chloride with the surface of the mineral particles, and it is reasonable to suspect that the rate of diffusion of the copper ions to the surface is a relatively slow process and is thus the limiting factor in this particular rate of reaction. It is possible, for example, that the improvement obtained, when the ore is subjected to reducing conditions before the chlorination, is due to a breakdown in the crystal structure of the original particle. This breakdown is brought about by the reduction and by the consequent increase in surface area available for reaction with hydrogen chloride. Can the author indicate whether any work has been done along these lines, and whether it has been established that the reaction depends on the surface area available? The second point relates to the application of this kinetic study to the actual operation of a Torco reactor. It was once planned to feed the sodium chloride, together with the reducing agent, into the top of the segregation chamber. In the paper, the author mentions that it has now been established that the segregation chamber behaves, to all intents and purposes, as a fluidized bed, and that there is, therefore, a rapid evolution of gas in the lower regions of the chamber, which, it is imagined, displaces the gas phase rapidly. Since the reaction between sodium chloride, water vapour, and the aluminium silicates in the ore is extremely rapid, one wonders how much of the hydrogen chloride produced is removed from the reaction zone before it has had time to react with the copper minerals. The extremely low consumption of sodium chloride (which is a vital feature of the Torco process) must depend on an extremely rapid circulation of the hydrogen chloride gas to all the ore particles in the segregation chamber. One wonders, therefore, if a system for the introduction of the sodium chloride into the bottom regions of the chamber might not result in even greater efficiency in the utilization of sodium chloride. ProC D. D. Howat (Fellow): All of us who have been concerned with the study of chemical reactions at high temperatures are keenly interested in kinetics and are well aware that this is not an easy study experimentally. Dr. Brittan is to be congratulated on the development of neat experimental methods and for his full discussion of the results obtained. Although the segregation process for the extraction of copper from oxide and silicate ores has been known for almost fifty years, the fundamental chemical and physical changes involved have been little understood and the fundamental data are very scanty. The work now in progress at A.A.R.L., together with that sponsored by the Anglo American Corporation in other research institutions throughout the world, is bound to produce new fundamental data and a much more complete understanding of this rather fascinating process. It is already apparent that some of the old and well-worn chemical reactions that were postulated to occur, just cannot take place in the way which was formerly accepted. Dr Brittan's work, carefully conducted and thoroughly analyzed as it has been, still leaves us with one great outstanding problem. The thermodynamic data and the possible reactions set out in Table I (page 281) of his paper leave us asking, in complete despair, how can copper be converted into a volatile chloride in the presence of HCl, CO and carbon at temperatures about 800°C? The thermodynamics all combine to show that copper should be reduced to metal as the first step in the process. This brings us right up against the second problem. If copper were reduced to the metal how would HCI convert it to the volatile chloride? On top of these problems is the unknown reason for the very high speed of reaction between CO, HCI and the ground copper ore. Still further into the region of the unknown is the reaction by which gaseous hydrochloric acid is produced in the actual process. Perhaps Dr Brittan is feeling grateful that he doesn't have to try to explain this reaction-at this stage of the research programme at least. The results very clearly show that both CO and HCl gas are essential for rapid production of the volatile copper chloride. Dr Brittan states that 18 minutes were required to attain 83 per cent extraction with HCl gas alone and this was reduced to 4 minutes when CO was

Jan 2, 1970

By A. Bruynesteyn

I would like to talk to you today about microbiological leaching or, the solubilization of metal values due to the action of Thiobacillus ferrooxidans. Bacterial attack of sulfide minerals has been going on for centuries but only recently was it discovered that the acid and iron often found in coal mine drainage is produced by bacterial action on the iron sulfides in the coals. Not until 1950 was it proven that bacteria were playing a role in the dump leaching of sulfide ores. B. C. Research became involved in microbiological leaching in 1955 while examining mine drainage from the copper-zinc mine at Britannia Beach, north of Vancouver. Since 1955, B.C. Research has maintained a continuous re¬search program on microbiological leaching which, due to its applied nature, was put under industrial sponsorship in 1962. Most of the information pro¬vided in my talk today has been gathered under this sponsored research program and I would like to express our appreciation to the many sponsors who have made our work possible.

Jan 1, 1970

By William T. Ingram

Early in the 60's the author served as Chairman of the Engineering Foundation's Coordinating Committee on Air Pollution Research. One of our objectives was the development of a study on the measurement of stack emissions. Three publications of the Foundation resulted from these efforts to focus attention on the stack sampling problem.1,2,3 Two of these were annotated bibliographies covering 197 references. The third reported the results of a workshop meeting at which some 21 individuals from research institutes, industry and government discussed the possible and probable areas of research that would add to or increase the ease, precision and utility of sampling gaseous and particulate matter emitted from stacks. The American Society of Mechanical Engineers has had in its Power Test Codes a Standard Code for Determining the Dust Concentration in a Gas stream4 since 1941, The American Industrial Hygiene Association has published, in its Air Pollution Manual,5 a Section on Stack Sampling. Industry and regulatory agency requirements for stack sampling have utilized some modification of the alternatives in methodology. There has been ho single guide to stack sampling that met the needs of those who have the need to know.

Jan 1, 1970

By Eugene E. Brucker

In recent years, there has been increased interest in construction problems presented by karst topography. Twenty-five years ago, karst areas were by-passed for development. Today, the high cost of land and population pressures are requiring utilization of karst areas. Areas exhibiting karst features can be satisfactorily developed so that construction can proceed in a normal manner; or the areas may be left undeveloped and the underground drainage system abused by improper use causing or contributing to structural collapse, excessive erosion, and a grave threat to the stability of land development and the environment. The term "karst" is a comprehensive term applied to areas underlain by soluble rock that possess a topography formed chiefly by the dissolving of rock and the diversion of surface waters to underground routes. There are four features which contribute to the maximum development of karst. 1. A soluble rock, preferably limestone, must be present at or near the surface of the ground. Occasionally, dolomite will suffice.

Jan 1, 1970

By R. A. Wattenbarger, H. J. Ramey

The trend in gas well testing has been to rely more on the early-time flow data of drawdown and buildup tests than on stabilized flow tests. The stabilized testing methods often are not adequate for complicated modern gas reservoirs because many of these reservoirs have extremely low permeabilities and the transient flow period of a well test can last for days or weeks. It has become common to augment the flow capacity of gas wells by hydraulically fracturing their producing formations. In deep reservoirs the induced fractures are generally vertical and tend to follow a single plane of weakness. The presence of a vertical fracture at the wellbore complicates the transient flow behavior of a low permeability gas well. The flow is further complicated when turbulence occurs near the wellbore. Russell and Truittl published transient drawdown solutions for vertically-fractured liquid wells. They developed methods of drawdown and buildup testing utilizing these solutions, which were based on numerical simulation. Clark2 applied the basic Russell-Truitt solutions to analyze fractured water injection wells by falloff tests. Field examples were given to substantiate the method of analysis. The Russell-Truitt solutions and well testing methods can be extended to gas well flow. Millheim and Cichowicz3 presented the drawdown equations for *Now with Scientific Software Corp., Denver. ideal gas flow, including the effects of formation damage and turbulence. Two actual field cases were presented, both of which exhibited extensive fractures and turbulent flow. For the pressure range of these well tests, the use of ideal gas equations was acceptable. These field cases were significant in being the first published data showing the occurrence of turbulence in fractured gas wells. We, also, observed such cases in practice. Our purpose here is to extend the theory of fractured gas well testing to the flow of real gases. To determine the effects of wellbore storage and turbulence on well test interpretation, we developed a finite-difference model to simulate well test conditions. Since gas wells usually are widely spaced and have high compressibility, the emphasis has been put on the early transient behavior before the effects of the outer boundary are noticeable at the wellbore. The Mathematical Model The geometry of our mathematical model is similar to that used by Prats.4 The well is centered in a circular uniform reservoir. A vertical fracture of infinite flow capacity penetrates the formation and passes through the wellbore. The wellbore itself is not important. Fig. 1 shows a sketch of the problem. Fig. 2 shows the idealized model. The equation for the flow of real gas in a porous medium with uniform porosity, permeability and thickness is5-7

Jan 1, 1970

By S. A. Levy, J. D. Wood, J. F. Libsch

A study of the preparation and characteristics of a sevies of Fe-C-N alloys has been conducted. X-ray, microhardness, and metallographic data from a series of single-phase alloys produced by controlled nitriding of carburized foils are reported. The existence of a, y', and E carbonitride has been confirmed and it has been found that y' is the hardest of these three phases. These results have been used to interpret the structures observed in nitrided steels. ThIS paper represents the second phase of a three-part study concerning combined nitriding and induction hardening. The first part' reports the engineering properties (depth hardness, fatigue, corrosion, and tempering resistance) obtained when seven different steels were subjected to the combined surface hardening treatment. The present paper concerns the nature of a series of single-phase alloys prepared by controlled nitriding of carburized foils. The objective was to simulate the various layers present in the as-nitrided steels. A third paper will report data concerning induction hardening of the foils and will attempt to set forth a mechanism for the high hardness obtained with the combined treatment. Studies of nitriding generally fall into three categories. The first were performed with bars using conventional nitriding procedures, thus yielding a nitrogen gradient and a series of phase layers near the surface.2-6 Such studies were complicated by the presence of the nitrogen gradient, for it was difficult to correlate microhardness and X-ray data with chemical composition. There were always complications due to penetration of the X-ray beam which yielded data influenced by underlying layers. Similar difficulty was experienced in taking hardness readings on a tapered specimen or in isolating various layers by machining. A second group of investigations was primarily concerned with establishment of regions in the Fe-N7,9,10 or the Fe-C-N phase diagram.8 These were performed with powder specimens which were nitrided in am-monia-hydrogen mixtures to produce a series of single-phase specimens. Unfortunately, the powder specimens were not amenable to either microstruc-ture or microhardness study. The third was essentially a compromise of the first two. Bose and Hawkes," using 0.005-in.-thick iron foils, were able to obtain microhardness, microstruc-ture, composition, and X-ray data on Fe-N alloys of eutectoid composition by equilibration in ammonia-hydrogen mixtures at 1290°F. To obtain a better understanding of the nitriding process and the characteristics of the phases in ni-trided plain carbon steels, the present investigation was undertaken. The investigation was patterned after that of Bose and Hawkes with the exception that nitriding was performed at 930°F (below the eutectoid temperature), as in the case of conventional nitriding. In addition to gaining some understanding of the source of high hardness of nitrided and induction-hardened steels, found in a previous study,' this investigation was concerned with the possibility of controlling the phases present in a nitrided case through the use of ammonia-hydrogen mixtures. Knowledge of the properties of each phase and the conditions necessary for its production could make it possible to avoid undesirable surface layers in nitriding. EXPERIMENTAL PROCEDURE Specimen Preparation. A specimen thickness of 0.005 in. was selected to insure that a uniform nitrogen concentration could be obtained in a reasonable length of time by equilibration in an ammonia-hydro-gen atmosphere. The starting material was a vacuum-cast ingot of pure iron, the major impurities of which were: C, 0.003 pct; 0, 0.014 pct; S, 0.004 pct; N, 0.003 pct.* 'Compositions are quoted as weight percent except as noted. The ingot was heated to 1500°F in a nitrogen atmosphere and forged to a slab 0.25 in. thick. The material was then reduced to 0.005 in. by alternately cold rolling and annealing in a nitrogen atmosphere at 1600°F. This material was divided into five groups of specimens which, after carburizing, yielded the following carbon contents: 0.003, 0.28, 0.36, 0.46, and 0.74 pct C. The carburizing was performed by the Leeds and Northrup Research Laboratories, using a controlled "carburizing potential" which yielded a uniform carbon concentration throughout the cross section.

Jan 1, 1970

By Richard H. Johnson

White Pine Copper Company, a subsidiary of Copper Range Company located in Michigan's Upper Peninsula, mines a Precambrian sedimentary deposit containing finely disseminated chalcocite, native cop- per and native silver. Silver occurs predominately in certain strata and presently provides a mill head averaging 0.25 ounces per ton. This technical publication traces the development of silver recovery at White Fine beginning in early 1961. White Pine refined copper contains an excess of silver over and above that required to impart Lake Copper properties. The copper does not require electrolytic refining to meet industry purity requirements, nor is there sufficient silver to justify electrolysis.

Jan 1, 1970

Aspects of the formation, decomposition and acid solubility of zinc ferrite were examined to find potential ways of increasing the recovery of zinc from a calcination process of mineral concentrates. Ferrite can begin to form within 2 hours at 450¦C. while complete reaction can occur after 35 minutes at 700¦C. The nature of the starting materials has a pronounced effect on the rate of formation and below 700¦C. ZnSO4 and Fe2O3 proved to be the least reactive of the materials investigated. It is possible that iron in a mineral admixed h sphalerite can be converted to a relatively inactive form, however, it does not seem possible to prevent zince f formation when iron is in the sphalerite lattice. At the two temperatures examined, 650¦C. and 700¦C, the presence of SO2 in the atmosphere accelerate .e formation of ferrite. Prolonged exposure to such atmospheres, however, caused the ferrite to decompose to ZnSO4 and Fe203. The optimum temperature for decomposition was 650¦C. and it was virtually complete within 2 hours in an atmosphere of 10% SO2 in air. Zinc ferrite was also decomposed by CO at low concentrations, and by heating with limestone at 1000¦C. for at least 2 hours. The solubility of zinc ferrite in acid was influenced by the starting materials used. As both the reaction temperature and the time at temperature increased the acid solubility decreased. The acid solubility was a minimum when the ___i_.. n - I -r~...................c -..... F, fe..i+e --A frn... A01- +^ Z(lo/

Jan 1, 1970

By E. R. DeMen

In late 1965 General Mills announced the availability of a new liquid ion exchange reagent, LIX®-64, for the recovery of copper from various solutions, such as dump leach liquors. Acceptance of this material by the copper industry has been gratifying. A number of pilot plants have been operated, and at this writing there are four commercial plants either in operation or under construction. The properties and operating characteristics of this reagent have been described in previous papers.1,2 LIX-64 is quite effective and selective for copper extractions although partially limited in pH ranges, losing extraction efficiency as the acidity of the feed solution increases to below pH 1.0. Also, the LIX-64 content in the kerosine carrier was limited to about 10 volume percent (except for special cases) because of rapidly decreasing phase disengagement rates above this content. This in turn led to large plant capital costs, as the organic flow rates had to be in- creased (rather than LIX-64 content) to extract the copper from high content liquors. This was true in all circuits where the feed copper content ran above 2.0 g/l copper. A reagent was needed which would extract copper from more acidic liquors (or higher copper content liquors) and which exhibited improved phase disengagement characteristics with reagent contents up to 30 volume percent. General Mills is pleased to announce the availability in commercial quantities of a new and improved reagent, L+-64~. This reagent broadens the pH range in which copper can be extracted, and exhibits superior phase disengagement characteristics. The improvement in the reagent has been noted in laboratory circuits, as well as pilot plant runs and commercial operations.

Jan 1, 1970

By R. E. Cerullo

Many computer programs have been written which are capable of performing a least squares fit, both linear and nonlinear, to data gathered by the petroleum researcher. However, when the petroleum engineer tries to program methods of analysis he has used manually in the past, such as forecasting production by decline-curve analysis, more than a knowledge of the use of existing least squares programs becomes necessary. It is the intent of the authors to report the techniques used in the development of such a decline-curve program.

Jan 1, 1970

By Lee A. Goetz

Cost control is a lot like motherhood - everyone proclaims its need and its worthiness, but is it really appreciated and used to successfully complete a project on time and within budget? Good Cost Control is a yardstick - requiring fixed definition but this does not mean rigidity for the project. It provides the means for telling: 1. Where you are going 2. When you will complete and 3. What it will finally cost. Cost Control does not fix the price of dreams nor the cost of projects loosely defined. It does not control costs in itself but provides the tools whereby engineering decisions can be made which will ultimately affect the project cost It does not even guarantee that a project will be completed within the original funds - but only, if this reflects management desires.

Jan 1, 1970

By A. C. Bluemel, J. S. Harper, J. H. Jett

Can the mining industry in the United States accept the change and challenge of the "Now Generation" of an entirely new high-performance automated transportation system? Advanced transportation systems for the mining industry can be defined as those which lend themselves to automation and meet definite performance and cost objectives. The term automation shall mean that control of transportation system is accomplished by a closed loop electronic circuit which, when started on command, will continue to cycle as programmed until such time as an intelligent feed-back causes the system to shut down.

Jan 1, 1970

By Harry K. James

Experimental studies of the physical and explosion characteristics of pure hydrazine nitrate and multicomponent systems containing hydrazine nitrate, supplemented with an extensive Literature survey, are presented. Such proper- ties as melting point, heat of fusion, density, viscosity, surface tension, thermal stability, decomposition process, detonation velocity, impact sensitivity, and TNT equivalence are included.

Jan 1, 1970

By Dawson CF

About 17% of the world's present day oil supplies comes from wells drilled in the continental shelf of some 28 countries. The demand for oil is increasing each year and is expected to quintuple in the next decade. In the same period, the demand for gas will climb to at least 20 times its present rate. In the next decade, the percentage of oil and gas obtained from offshore is expected to rise to nearly half the world's consumption. The offshore oil industry is currently striving to achieve new technological advances in the field of deep ocean operations, especially in face of adverse seas. While the emphasis was put on drilling vessels for exploration, the industry was slow in developing effective offshore construction vessels. Until a year ago, construction vessels were but large barges with construction equipment mounted on their decks. These vessels all shared the same severe limitation: When the sea exceeds a five-foot wave height (and in some cases as little as three feet) the conventional construction barge is forced to suspend operations. In face of an oncoming storm they must run for shelter. Their utilization, therefore, is limited and in rough weather areas, such as the North Sea or the Bass Strait is lower than 50%.

Jan 1, 1970

By John P. Capp

Information is summarized on the major uses of fly ash and prospects for utilization in agriculture and in brick and mineral-wool manufacture. Technological aspects of utilization are discussed and references are given to the work of many of the leaders in the fields of fly ash production, processing, and utilization.

Jan 1, 1970

The phrase "Total-Energy" has, during the past few years, become widely used to describe a wide range of very different operations. However, if accurately described they all have one basic factor in common - energy usage is optimised with the prime mover being powered by gas or oil to generate shaft horsepower and the products of combustion being put to some use. The item which distinguishes a Total Energy system from any other engine or turbine driven project is that orate pains are taken to utilize the waste heat, whether this be in the form of exhaust gases and/or engine jacket ling water. n areas where grid supplies of electricity are readily available, the economics of most projects undoubtedly hinge upon the successful application of waste heat. However, there has been a growing awareness of other benefits, such as reliability of supply, which can be incorporated into the system. In particular these factors sway managements of hospitals and companies with complex computer facilities to examine closely the feasibility of Total Energy, Consequently, in America for example, where many systems have proved themselves the list of applications is wide - shopping centres, apartment blocks, office buildings, airline terminals, universities, schools, motels. And these do not cover the industrial sectors where the range is wider. The total number in North America is now approximately 500 and this to some extent-speaks for itself. In Australia, the real work involved in carrying out feasibility studies for Total Energy is only just beginning. 'in some quarters, the concept is not well received in that it is too new. Therefore, a typical approach is to play safe and wait until some other company has proved its operation "under Australian conditions". This is certainly a realistic approach when considering something revolutionary but as far as Total Energy.is"concerned all the individual components and sometimes combinations, (engines, generators, boilers, air-conditioning units) have been in operation for many years. The only innovation is that they are now all being brought together under one roof. This should not be forgotten when the question "where can I see something like this operating?" is asked.

Jan 1, 1970

By Andrew J. Freyman, George S. Barry

"Mineral endowment in northern British Columbia and the Yukon Territory was assessed on the basis of geological opinion which was gathered in a quantitative framework of subjective probabilities. The survey area encompassed 364,000 square miles and was divided into 310 cells having dimensions of one degree of longitude by one-half degree of latitude. The response solicited related to tonnage, grade and number of deposits of a particular mineral in each cell. The questionnaire sought an expression of mineral endowment down to mine-able depth as distinct from economic expectations. Details on the questionnaire and responses are presented in a series of contour maps on individual commooities, which illustrate the collective occurrence picture of the respondents. The approach is largely dependent on the cooperation of an adequate number of geologists familiar with the region being examined. This approach to an assessment of the mineral endowment of a region has a wide range of applicability, from the focusing of exploration effort to aiding government and development bodies in the formulation of regional strategies and the planning, for example, of road, rail and power facilities."

Jan 1, 1970

By R. L. Blake

The crystal structure of hematite, a-Fe203, has been refined as part of a continuing study of the behavior of phases in the Fe-O system during reduction and oxidation. Three-dimensional diffraction intensities were collected on two spherical single crystals of hematite with a semiautomated Buerger single-crystal diffractometer. The first crystal refined was a natural Elba hematite with minor impurities. The other, a synthetic crystal, was refined at the request of Mossbauer spectroscopy investigators concerned with impurities in the natural crystal. The structure has been refined with a least-squares program and the final structure for the natural crystal irradiated with CuKa-radiation gave a discrepancy factor (R) of 3.65 percent using 49 unrejected reflections of the 51 measured. For the synthetic crystal, using MoKa-radiation, the discrepancy factor converged to 2.8 percent using 95 unrejected reflections and was 3.3 percent for all 109 reflections. The structure model of Pauling and Hendricks has been confirmed with essentially no change in the iron coordinates and approximately a 5-percent change in the oxygen coordinates. Selected interatomic distances and angles were calculated as were anisotropic temperature factors.

Jan 1, 1970

By J. B. Knaebel

I have been asked to discuss the "Development of the Twin Buttes Mine for Production." This property, which is a large low-grade open pit copper mine about 25 miles south of Tucson, Arizona, was recently brought into production by The Anaconda Company under a long-term lease agreement with the Banner Mining Company. Since the scope of this paper is limited to the subject broadly indicated by its title I will confine discussion of historical, environmental and other general matters to a brief summary sufficient, I hope, to establish a background of those conditions which influenced our approach to the technical and economic problems that faced us at Twin Buttes. The paper deals primarily with our studies and activities related to the solution of these problems and with the resulting engineering and business decisions that were taken - decisions which culminated last September in the initial commercial operation of another major open pit copper producer unique and interesting in several respects.

Jan 1, 1970

By W. L. Mamme, G. Y. Chin

The problem of selection of the active slip systems for a crystal undergoing an arbitrary strain was analyzed by Taylor and by Bishop and Hill in terms of a minimum (internal) and a maximum (external) work criterion, respectively. These two criteria have now been generalized to include crystallographic slip on several sets of slip systems, twinning mixed with slip, and slip by (noncrystallographic) pencil glide. The generalized treatment also takes into account the possibility of a Bauschinger effect and of unequal hardening among the shear systems, which were considered in the Bishop and Hill work. Optimization techniques of linear and nonlinear programming are shown to be applicable for the numerical calculation of the minimum or maximum work. In the case of crystallographic shear, the constraint functions are linear and hence the optimal work is obtained as the saddle value of the lagrangian function Wi(y) e minimum and W,(u) + (a) for the maximum, where Wi is the (internal) work, We is the (external) work, Y is the crystallographic shear strain, u is the applied stress, and and are constraints. It is shown that the Lagrangians are functionally the same and the saddle value of one problem is identical to the saddle value of the other, proving that the two analyses are completely equivalent. In the case of pencil glide, although the constraint functions are nonlinear and neither convex nor concave, the equivalence of the optimal values to the saddle value of the Lagrangian (which is again identical for both problems) is still valid. WHEN a crystal deforms plastically by crystallographic shear, five independent shears are generally required to accommodate five independent strain components specifying the deformation. Assuming slip as the only shear mechanism, Taylor1 in 1938 analyzed the deformation in terms of a minimum work criterion. He hypothesized that of all combinations of five slip systems which are capable of accommodating the deformation, the active combination is that one for which the internal work C is a minimum, where 1 TI is the critical resolved shear stress for slip on the 1-th slip system and is the corresponding simple shear. By further assuming equal 72 for all equivalent slip systems and no Bauschinger effect, Taylor re- duced the minimum work problem to one of minimum and applied the analysis to the case of axisym- Metric flow by {111}(110) slip in fcc crystals. However, he did not consider the question of whether the resolved shear stress has in fact attained the critical value for slip on the newly found active systems without exceeding it on the inactive systems. In 1951 Bishop and ill' put forth the maximum work analysis in which slip is again assumed as the only deformation mechanism. In this analysis, the work o1 done in a given strain ij by a stress ujj not violating the yield condition is maximized. In addition, the analysis takes into account the possibility that the critical resolved shear stress for slip may not be equal among the slip systems and that the slip behavior may exhibit the Bauschinger effect. As with Taylor, a single set of slip systems—{111)(110) — was analyzed numerically. It thus appears that the Bishop and Hill treatment is on a more sound physical basis than the Taylor treatment. However, Bishop and Hill showed that where there is equal hardening among all slip systems and when there is no Bauschinger effect, Eq. [11 ] of Ref. 2, as assumed by Taylor, the results of their maximum work analysis are the same as those of Taylor's minimum work analysis. Hence at least under those conditions there is an implication that the Taylor analysis does lead to a critical resolved shear stress for slip on the predicted active systems without violating the yield condition on the inactive systems. Recently, the Taylor analysis was applied for numerical solutions of the axisymmetric flow problem, for slip on {110}(111), {112}(111). {123)(111) systems as well as a mixture of all three sets of svstems."1 Computational techniques based on the optimization theories of linear and nonlinear programming4 were employed in these solutions. The same techniques were employed in the solutions of an axisymmetric flow problem of deformation by slip on (111) (110) systems and twinning on (111)(112) systems5 which had been considered theoretically from a modified Taylor approach. The utilization of these techniques has led to the realization that the solutions of Taylor's minimum work problem imply the solutions of Bishop and Hill's maximum work problem. The two problems turn out to be dual problems in the well known sense of mathematical programming. It is thus the purpose of this paper to first generalize the minimum and maximum work analyses to include crystallographic slip on several sets of slip systems, twinning mixed with slip, and slip by (non-crystallographic) pencil glide, as well as the possibility of a Bauschinger effect and of unequal hardening

Jan 1, 1970