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By J. P. Zannaras

Referring to the article by R. J. Charles and P. L. de Bruyn, let us assume that W = weight of glass bar; P = weight of hammer; e = total deformation; E = unit of deformation; K = potential stress energy; E = modulus of elasticity; L = length of the bar; 7, = coefficient of inertia; h = height, ft; V = velocity, fps; and a = cross section area. The portion of kinetic energy which is effective in producing stress energy in a fixed bar struck horizontally is given by the formula [ ]

Nov 1, 1956

By P. Eadington

I feel sure that the apparent anomolies between his findings and ours arise as a result of the different experimental methods used. The modem oxygen electrode method he used is sensitive to nano moles of oxygen uptake whereas the method we used was only sensitive to micro moles at best. Furthermore, our experimental method did not permit measurements to be made in the first ten to fifteen minutes of reaction, the very time he is observing. Direct comparison of his data to ours is made difficult because of not knowing the specific area of the samples he used. However, we found that precipitation of PbS using sodium sulphide gave rise to samples of area approximately 30 m2 g - 1. Noticing the rapid decline in oxidation rate over ten minutes, as reported in Table 1, one can conclude that we could not measure the rate after ten minutes using our method. Taking the above area I estimate that at these rates only a small fraction of surface would be oxidised in a period of hours. I can only conclude that at some point a change in oxidation mechanisms occurs giving rise to a much faster rate such as we observed. Finally, I would like to comment on the apparent inertness of the homogeneously precipitated lead sulphide sample 11. This method of preparation we found to give specific areas of 20 to 30 times less than other methods and this would account for the very low oxidation rate.

Jan 1, 1982

By John S. Rinehart, Norman Ahlquist

The model which Birkimer has developed to explain the increase in fracture strength of brittle, rock-like materials with increasing rate of loading may have some applicability. It should be noted that since his analysis presupposes elastic behavior he could equally well express his result as an increase of stress to fracture with increasing strain rate, a fact which is well known. Certainly his experimental results do not contradict his strain rate to the one third power law. On the other hand, we do feel that the data are insufficient and inadequate to establish the relation- ship definitely. 'In metals, the whole fracture picture is greatly complicated by concomitant plastic flow and, in rocks, by the presence of macroscopic flaws and inhomogeneties. Both lead to uncertainties as to the state of stress during the fracture process. He quotes in support of his theory Rinehartfs results on rock in which the latter found that under impulsive loading by explosives, rocks appear to be many times stronger than under static loading. We suspect that the observed increase has little specific relevance to the theory which Birkimer puts forward. Rinehart's measurements were made by spalling rocks using a sawtooth shaped stress wave of known intensity and duration. The rise time of such a pulse is essentially zero, certainly less than 0.1 microseconds, so that it is not possible to apply Birkimerfs theory quantitatively. We favor attributing the increase in fracture strength to two factors. First, the action takes place very quickly in an exceedingly thin region so that statistically the probability is small of encountering one or more of the large flaws that contribute so significantly to the reduction of strength of most rocks under tension. It is also conceivable that the compressive wave that has just passed by the region of fracture has strengthened the rock by collapsing some of the larger flaws. Second, the state of stress obtaining at the time of fracture is not all that existing when the

Jan 1, 1971

By E. J. Garrison

A.C. Noble presents a clear and concise summary of the factors pertinent to a competent reserve evaluation of a mineral occurrence. However, I believe that he confuses the difference between a resource and a reserve. By definition, a reserve estimate includes all known economic, legal, mining, metallurgical and environmental factors impacting the recovery of the product(s) found in the deposit. A resource estimate, however, is an estimate of what is there - what is available for exploitation and should not include economic and other considerations. Only after defining the resource, its extent, location, physical and chemical properties, can an intelligent selection process begin of the extraction and processing technologies most appropriate to the circumstances encountered. The early and almost certainly inappropriate application of these often cookbook choices has probably lead many companies to walk away from good properties. (How many mines have been found in others rejects?) An example of the process is the early selection of metallurgical samples, after encouraging values are found in a few holes or surface samples, in an attempt to determine the economics of the prospect. Unfortunately, the sampler does not know how representative of the occurrence the sample is because he does not know what the deposit looks like. As a consequence, the property becomes saddled with a bad metallurgical sample that is often inappropriately used in evaluating the deposit which it does not represent.

Jan 1, 1994

By D. G. Krige

I read the paper “The jackknife approach” with interest. There appears to be a widespread misunderstanding of the kriging variance as a measure of the error variance of the kriged grade estimate of an ore block. Virtually all mineral deposits display skew frequency distributions and are subject to the well-known proportional effect. Thus, it is usual to find in an ore body, when subdivided a really into subpopulations, that the variances of the grades (or the total sills of the variograms) for the subpopulations will be correlated with the corresponding mean grades (usually linearly with the squares of the mean grades). Thus, the variogram will be stationary.

Jan 1, 1996

By D. Nilsson, B. Aaro

To find out if there are any potential savings in "in-the-wall haulage for open pits," the Swedish companies ASEA and Kiruna-Truck in 1984 gave us financial support to study this solution in more detail. In the study, the use of in-the-wall haulage was studied for three hypothetical open pits with different sizes and shapes. Annual ore production rates also varied. A part of this thor¬ough study was published by the Swedish Mining Research Foundation (see Report F 8444, "The Use of Electric Trucks and In-the-Wall Haulage in Open-Pit Mining," in Swedish with an English Summary). Although in-the-wall haulage is of interest in some open pits, the local terrain is important for profitability. The authors do not think in-the-wall haulage is of any major interest for the mining industry. The following is a summary of some of the author's findings. To place the haul road in the wall is not of interest in open pits with declining metal contents in the bedrock. In such mines, the volumes from the haul roads will not yield reve¬nues. The cost per m3 for an underground haul road is much higher than for a haul road in the pit. This means that it is only of interest to place the haul road in the wall when mining gets deeper than about 50 m (165 ft). In-the-wall haul roads will reduce flexibility in the pit, and it will make necessary the use of smaller equipment with lower productivity and higher costs per ton. As Hustrulid, et al. show (Table 4), excavation savings of material hauled are very low, $0.10 to $0.25/t (0.09 to 0.27 per st). The extra operating cost, due to lower productive equip¬ment, will normally be much higher and thus destroy the whole idea with in-the-wall haulage. If electric trucks are profitable for in-the-wall haulage, it is normally also profitable to use electric trolley assist for trucks on the haul road in the pit. But the profitability of using electric power is different in different countries, and depends on the relation between the cost for electric power and diesel fuel. In the US, diesel fuel is inexpensive compared with electricity, but in a country like Sweden, diesel fuel is much more expensive. In most open pits, the trucks have to move a considerable distange from the loader until they reach a final haul road in the wall. A trolley line along such temporary haul roads will be exposed to flyrock. It is normally less expensive to perform rock support from the open pit than from underground ramps in the wall. In Fig. 8, Hustrulid et al. gives the impression that the underground haul roads will be very close to each other. This is seldom the case. Figure 1 shows a haul road in the wall. The haul road passes through each cross-section only once. Ar¬ranging a reliable dewatering and rock support system from only one underground ramp is probably impossible. Many more drifts are therefore necessary. In our report, we also studied what would happen if the final pit slope is increased by 5°, using underground drifts in the wall. Our conclusion was that the extra cost for drifts, rock support, pumps, etc. destroys the whole idea, and that it was better to accept a higher stripping ratio with the haul road in the pit and to use conventional low-cost open pit equipment. Finally, we think it is would be difficult to try to reduce the safety factor when determining the slope angle by moving the haul road in the wall. Minimizing the risk for pit wall collapse is also important with in-the-wall haulage, primarily because men and equipment will be working in the pit, but also to guarantee many accesses between the pit and the haul road in the wall.

Jan 1, 1989

By G. Sanders

The studies reported on in this paper were initiated to draw attention to the severe contamination problem in the Section 30 drilling program at Chimney Creek. The lithologic-subset sampling study reached a different conclusion from that presented in your paper, and I wish to comment on your subsequent analysis of the data and your conclusions. Request for more complete data In the section on subsampling, you mention that the subordinate lithologies were separated and sampled, yet only the dominant-lithology gold value is plotted in Fig. 4. In a contamination study, the reader is interested in the assay values for the individual subsets. Please include a table of the subsample assay data in your reply. Also, please indicate which analytical methods were used to arrive at the gold values in the subsampling study. Turning barren rock into low-grade ore Figure 5 is very revealing and typical of all of the cross sections in Section 30. Note the long strings of low-grade mineralization spread out for hundreds of feet below the ore zones. There were some very high gold values found in certain contaminated fractions during the subset sampling. The conclusion, here, was that the distinctive, strongly-mineralized dolomite layer was probably loose and crumbly and continued to disintegrate during drilling. This caused salting of the unmineralized rock samples below.

Jan 1, 1993

By R. C. Kirkman

Your interesting and well-written paper brings up the issue of the purpose or use of a mineral property appraisal, specifically the value of these evaluations when the quality and quantity of the mineral resource to be exploited have not been thoroughly investigated. I am not a practicing professional in regards to valuation of limestone investment opportunities. I simply wish to make a few generic observations about the appraisal process in order to stimulate meaningful thinking by appraisal professionals whose reports may be reviewed by professional investors and "laypersons" alike. It is my contention that the analysis given in your paper is very thorough, considering the data that was made available to you. But I would not like to be the person responsible for investment decision making without having a feel for the risk involved. Therefore, the reviewer must classify this type of analysis as relatively "subjective" because they have not received an indication of the inherent risk involved in exploiting a mineral resource. This risk can best be quantified by using probabilistic methods in conjunction with the quality and quantity of the mineral to be produced. Most appraisals passively ignore this problem. Specifically, you have accurately identified the various costs and values entering the economic appraisal process. These can usually be quantified with a great deal more certainty than the capability of the resource to provide 25 years of quality material. Also, in Cases A (West Virginia) and B (Alabama), you are fortunate enough to have previous operations data. What if there had never been production from these properties? How does a potential mineral property investor evaluate the resource capability to meet various projected financial outcomes without spending a good deal of up-front capital to drill and delineate enough quality resources for a 20-plus year operation.

Jan 1, 1990

By Eldred D. Wilson

Apropos of metals low in the series illustrated by Dr. Butler's Table II, it is interesting to note that Cress and Feldman reported traces of platinum metals in several samples of alunite. They found, chemically and spectrographically, several traces of platinum, palladium, and rhodium in alunite from Sugar Loaf Peak, near Quartzite, Ariz.; silver, platinum, and rhodium in alunite from Rio Grande County, Colorado; and traces of rubidium and gallium in various samples of alunite. These authors stated that the platinum metals appear to be present as oxides and suggested deposition by hydrothermal solutions under near-surface conditions.

Dec 1, 1956

By G. E. Agar

In the April issue of Mining Engineering there is a technical note "New size laboratory ball mill for Bond Work Index determination" written by M. Nematollahi which is very confusing. My recollection is that a description of an experimental procedure should be such that someone skilled in the art should be able to reproduce the results. Applying this test to this particular article would according to me cause a failure to be registered. There are apparently three batch laboratory test procedures referred to in the article. One of these is the traditional Bond test with all its limitations such as dry grinding followed by exclusively dry screening, two attributes which fail to simulate conventional industrial grinding. There is a second procedure (NSBM) which the author is advocating as an improvement of the Bond procedure but from the description I doubt that anyone would be able to repeat the procedure. For instance, is the grinding to be done wet or dry, if wet at what pulp density, and how is the screening to be done, wet or dry or a combination?

Jan 1, 1995

By L. Adler

The paper under discussion provides an excellent and concise study of compartment ergonomics for thin seam equipment. However, in accomplishing this, some very important operational aspects were ignored. Since these are likely to be determinant, the authors' response is solicited. The novel shuttle car with its cross-car, end-mounted compartment is operationally very different from the current one. The latter can be considered a highly mobile extension of the continuous miner or loader, with a pass-through conveyor, dump boom, and extra storage. As such, the vehicle's kinematics are the simple "I," or shuttle pattern, bent for going around pillars, with an intermediate stop at the "change-out point" (Adler, 1986). The action of the cable reel adjusts well to this pattern. The proposed shuttle car's motion either superimposes a "Y," time is lost (5 seconds) for each cycle, space is needed, line of sight is compromised, and a cable anchor is probably required at the dump. A chute is almost prohibitive in construction time and cost. The lack of a dump boom further curtails flexibility in the proposal. The value of a standard hand and opposite standard hand configuration also needs consideration. Many front-end loaders have a rocker (overshot) capability that is designed to achieve an "I" pattern. And crawler loaders sometimes use a side dumping bucket variant for the same purpose. These examples indicate the pattern's importance. For underground, thin seam application, the ram car has become popular. This is an articulated truck, battery-powered, and with an ejector dump. Its operational patterns would be similar to the novel shuttle car, except without the loading flexibility provided by a conveyor. Since the cab location of the proposed shuttle car largely obviates some of the major advantages of the present shuttle car, the question arises whether this configuration could be adapted to the current ram car and scoop.

Jan 1, 1990

By J. A. Richardson

J. A. RICHARDSON[t]-Mr. Stuckey's account of the geology of the kaolins of North Carolina suggests that they exhibit some features similar to those of the Malay Peninsula. About one half of British Malaya (some 26,000 square miles) is occupied at the surface by igneous rocks, of which granite and its associated derivatives constitute fully 90 pct. The average granite is a biotic variety. It is predominantly potassic but carries, in addition, notable amounts of soda and lime so that albite and albiteoligoclase are generally common constituents of the rock. The minor derivatives from granite "magma" likewise include potassic and sodi-potassic members. Decomposition of the granitic rocks has led to the formation of kaolin in many localities, but, in general, the deposits are rather small and, at present, none is exploited commercially. True primary or residual kaolins (as defined by Stuckey; p. 48 Vol. 173, p. 2 TP 2219) have been developed from feldspar veins, pegmatites and from other granitic differentiates which cut through the larger masses of granite; they are usually small and narrow. Kaolin has been derived from aplite, quartz porphyry, quartz micro-porphyry, felsites and microgranites belonging to a complex known as the Chegar Perah-Pulai intrusion in northwest Pahang and southwest Kelantan. The bulk of the component rocks has been more or less strongly sheared. Kaolin-like end products have been formed from both sheared and nonsheared varieties. Kaolin has been formed also by the complete breakdown of trondjemite flanking the northwestern and western margins of the complex Bukit Ranjut syenite intrusion in northwest Pahang. These trondjemites range in composition from essentially quartz-free albiteplagioclasites to quartz-free microcline rocks; they are, in effect, the syenitic homologues of white sodic and potassic granite-aplites. The granites in Malaya have generally undergone more or less severe surficial weathering leading to the production of a spongy and rather cellular or a soft "mealy" material (sandy clay) down to depths of about 100 ft; in places the granite is strongly decomposed down to 150 ft. There has occurred, therefore, widespread partial kaolinization, but, in general, no extensive deposits of good quality kaolin seem to have been formed by these processes of decay. Frequently, the katamorphic processes of desilicification have advanced beyond the production of kaolin towards the formation of lateritic end products. The more important Malayan kaoiins seem to be confined to the western alluvial plains, particularly in areas either where granite crops out at surface, or where it forms the bedrock of the alluvium. There are numerous excellent examples around Gopeng and elsewhere in the Kinta Valley of Perak. Such kaolins belong to Stuckey's groups of secondary deposits, and, like them, these also may be bedded. Field observations have established the existence of what for want of a better name is referred to by the Malayan Geological Survey as "granite wash." It is believed that granite, undergoing extensive deeply penetrative and probably rather rapid destruction under humid tropical weathering (temperature around 90°F; rainfall about 100 in. per annum; high humidity) "flowed," was moved by "creep" and, in addition, probably slumped in large waterlogged masses of sandy clay from topographic salients onto the adjacent flats. Either during this period of transmigration or later when the sludged granite detritus, after having more or less come to rest, was attacked by flowing water, differential sorting and erosion took place with the result that the resorted debris was moved and later deposited as bedded alluvium. In this way, banded "granite wash" was produced comprising alternating bands of clayey sand (resorted granite sludge) and white

Jan 1, 1947

By E. S. Cheney

Minter (1990) has provided an excellent summary of the sedimentology and theories of origin of the Witwatersrand paleoplacer Au/U deposits. However, recent stratigraphic, metamorphic and structural-tectonic research is radically changing the concepts about the Witwatersrand that now occur in most textbooks, review articles and North American minds. Presently, these concepts are biased by the past sedimentological research so well described by Minter. Specifically, the Witwatersrand Supergroup is Archean, does not occupy a depositional basin and is significantly metamorphosed and deformed. Thus, many of its sedimentary features, including the mineralogy of its ores, have been modified. The importance of the Witwatersrand ores cannot be overestimated. Since their discovery in 1886, they have produced 52% of the gold mined in all of human history (Pretorius, 1989).

Jan 1, 1991

By D. Leppert

Mr. Garrett brings up several interesting points with respect to the proposed API specification for "nontreated bentonite." However, although not explicitly stated, these specifications are being written based strictly on bentonites from the Wyoming/South Dakota area. Other sodium bentonites may be suitable for use as drill mud but will be unacceptable since the specifications are designed for Wyoming bentonites. Regardless of the API's intent that its specifications are intended only as guidelines, they are generally adhered to on an international basis and, for practical purposes, might as well be chiseled in stone. Estes (1986) notes that the quality of bentonite available from Wyoming producers has steadily declined since the early 1960's. The plastic viscosity: yield point (PV:YP) ratio was included in the current API 13A specification primarily to prevent excessive polymer treatment of lower quality bentonites to make drill mud. However, with development of polymers designed to control YP, this specification cannot ensure that only high-quality bentonites are used for drill mud. Nor will the new specifications for untreated bentonites. Other sodium bentonites are available. Bentonite produced in southeast Oregon demonstrates superior performance characteristics without addition of polymer or other chemicals. However, the exceptionally high purity of these deposits leads to a relatively high YP:PV ratio since the naturally high YP probably results from the higher purity and smaller particle size of this bentonite (Rath, 1986 and personal communication). These bentonites typically have a smectite component exceeding 90% (Rath, 1985), compared with 70% to 75% smectite for typical Wyoming bentonites, as determined by X¬ray diffraction and tests for colloid content. As evident from the accompanying table, the untreated Oregon bentonite has excellent rheological properties, though different from Wyoming bentonites due to the high purity. Additionally, this bentonite reacts well with polymers typically used in the oil industry, making it eminently suitable for use in a low-solids mud system. The extremely low grit content, less that 0.5% on a 75 mm (200-mesh) wet sieve, greatly exceeds the standards set for Wyoming bentonite. Optimum rheological properties are obtained at a higher moisture content than Wyoming bentonites. At 10% moisture, API 13A maximum, the clay is overdried, again reflecting the high smectite content. In summary, the specifications proposed by the API Task Group will ensure that only Wyoming producers can supply "nontreated" bentonite to the oilfield since specifications are

Jan 1, 1989

By J. Parker

Gentlemen; Rainy day in Upper Michigan, a good time to catch up on reading. Am doing that, and have a couple of questions concerning the K and P paper on roof problems and supports. I wonder if the authors can enlighten me. The compressive strength of the coal was reported as 4.4 MPa (633 psi) and compressive stresses measured in the coal seam were reported at around 27.6 and 48.2 MPa (4000 and 7000 psi ). Can such a soft coal actually contain such a high level of stress? Can it be measured as if the material were elastic, or does the coal behave plastically? Can that stress level be confirmed with hydraulic load cells? And if the coal is so soft, and if it is plastic, why is the stress field not hydrostatic? The intact roofrock appears to be unusually strong, with compressive strength reported at around 145 MPa (21,000 psi), and some failures were noted as soon as the rock was exposed. Were tests run to find out how that rock responded to the addition

Jan 1, 1991

By G. T. Lineberry, R. S. Mateer

Often only after years of "technical dormancy" do errors in published literature become apparent and meaningful. To prevent duplication of effort for future generations, the fol¬lowing corrections are offered. These errors were discovered during a recent study in which a computer program for mine hoist rope selection was developed (Lineberry and Mateer, 1988). In order of increasing gravity: Captions for Figs. 3 and 4 are transposed. • Purported to be a more accurate method of rope selection than the process of successive approximation, the direct method assumes linearity between a given rope's minimum breaking strength, T, and unit rope weight, Wv. Because of the sensitive location of the constant, K, in equation (8) for breaking strength, the significance of this nonlinearity is magnified. For a typical 6 x 37 class fiber core wire rope (Wire Rope Corp. of America, 1985), K actually varies from 43,650 to 52,190, depending upon rope diameter. Although the direct solution permits a rapid selection of hoist rope, an indirect (iterative) solution procedure is favored because of its im¬proved accuracy and ease of computerization. • Equation (8) for minimum breaking strength is in error. It should read:

Jan 1, 1989

By A. Unal

Barbaro et al. (1990) implemented a tedious study aimed at delineating the short-scale spatial variability of sulfur in a coal seam. It is not possible, however, to extend their conclusions to any other coal seam or use theirs in any other fashion, because the background geology was not presented. Nor were the conclusions accompanied by geological interpretations in the paper. In addition, an unfortunate printing error occurred where the total sulfur (determined by High Temperature Combustion) variogram in Fig.3 was duplicated in Fig. 2 instead of the seam height variogram. A more serious error, however, has been committed in the definition of the variable, seam height. What is defined as seam height by Barbaro et al. (1990) is, in essence, mining height, and is not a regionalized variable that should be studied by geostatistical methods directly. Despite the fact that the variable, seam height, is not defined in the paper explicitly, the following quotation discloses the fallacy: "All roof rock that exceeded 1.8 m (6 ft) from the floor was not taken because the longwall was operated to not mine more than 1.8 m (6 ft) unless the coal seam height exceeded 1.8 m (6 ft)." From this definition, the seam height, and therefore the sampling height, is equal to the actual coal thickness, if the coal thickness is greater than 1.8 m (6 ft). Otherwise, it is equal to the sum of the coal thickness plus the thickness of the roof rock that complements the thickness to 1.8 m (6 ft). In the latter case, the seam height is constant and equal to 1.8 m (6 ft). It is possible to conduct a variogram study on a pool of samples that are realizations of two different variables. But the conclusions derived would not belong to any one of the two variables uniquely and, therefore, do not possess any significance. Geostatistical analysis is irrelevant for the sum of multiple regionalized variables formed by arbitrary selections. In a two¬seam setting, for example, the mining height, as defined by the thickness of one seam at one location and the thickness of both seams at another location (due to quality and/or minimum thickness considerations perhaps), should not be used in the calculation of one common variogram. The two seams should be modeled separately. They can then be combined according to the specific purposes of the study. On the other hand, if a constant is added to a regionalized variable (to incorporate dilution perhaps), the variogram of the new variable will not change. Barbaro et al.( 1990), surprisingly, does not give any geologi¬cal interpretation of their results despite the fact that most of them can be explained by the origins of sulfur in a coal seam. The presentation of the results of a geostatistical study with no reference to the geology of the deposit is uninformative. It may also be misleading for the potential users of geostatistics. It is not unusual to find nuggets of pyrite in coal seams. In such cases, pyritic sulfur will probably display a spatial structure for only a very small distance that will appear in the experimen-

Jan 1, 1991

By E. Arioglu

Y. Luo and S.S. Peng are to be congratulated for their outstanding paper providing new results from surface subsidence monitoring over longwall panels under different conditions. This writer found the authors' paper very interesting. In reference to this paper, the following contributions are offered: • In the paper, the maximum possible subsidence (?Smax) induced by chain pillar convergence has been related to a parameter (h/I -p) by linear regression. (Notations are expressed in this discussion's Fig. I (b).) It is a well-known fact that in a goaf the roof lowers onto broken material that gradually compacts. Then the physical behavior of goaf may closely follow the compaction characteristics of broken materials. In light of this assumption, a mathematical means (Fig. 1(a)) of defining the expected compaction (i.e. surface subsidence) represents an exponential expression between ?Smax/m vs. h/1-p. This is based on Luo and Peng's field observations from the Appalachian, Illinois and Alabama coal fields. (The coefficient of regression "r" under consideration was computed to be 0.883). • According to Wardell's (1969) findings for board and pillar mining, the maximum possible subsidence can be expressed in terms of maximum pillar loading (depth, ratio of extraction) and the height-to-width ratio of the pillars left. Using Luo and Peng's 1991 results, this writer discovered a regression equation between the above-mentioned factors. As displayed in Fig. 1(b), the functional expression approximates as closely as possible the observed subsidence data.

Jan 1, 1992

By Robert T. Walker

Robert T. Walker and Woodville J. Walker (Walker Engineering Corp., Salt Lake City)-Mr. Kinkel's article embodies, in condensed form, the results of the first detailed and complete geological survey that has ever been made of the long neglected but potentially very important West Shasta Copper-Zinc district, in Shasta County, California. In view of the difficulties of the terrain and the complexity of the geological structure, this represents a notable achievement, and it is with regret, therefore, that the writers find it necessary to differ from Mr. Kinkel's conclusions on two important points. The first has to do with the interpretation of two members of the stratigraphic column of formations in the district; for purposes of comparison, the description of these two members according to Mr. Kinkel is shown parallel to the description of these members according to the writers, see Fig. 14, Mr. Kinkel's nomenclature being used

Mar 1, 1956

P. W. BRIDGMAN.-Owing to a misunderstanding, I did not see Dr. McAdam's and Dr. MacGregor's remarks on my paper on Flow and Fracture (Metals Technology, December 1944, Pp. 32-38), until after their publication. Both Dr. McAdam and Dr. MacGregor speak of my analysis for the stress distribution at the neck of a tension specimen as containing several assumptions; in particular, that the strain is uniform across the section of the neck, that there is no hoop tension at the periphery, and that the plasticity conditions of von Mises hold. As far as the mathematics goes, the situation is no different in this problem of plastic flow than in a conventional problem of elasticity. That solution is unique which satisfies the boundary conditions on the stresses, the stress equations of equilibrium, and in addition the constitutive equations connecting stress and strain in the elastic case, or a corresponding set of equations, of which the von Mises relations are a special form, in the plastic case. In the particular case of the necked tension specimen I have found that there is a solution satisfying the boundary conditions, the stress equations of equilibrium, and the von Mises plasticity conditions in which the strain is uniform across the section, and there is no hoop tension at the periphery. This is therefore the solution, and the uniformity of strain and vanishing of hoop tension at the periphery are to be described as discoveries, over which the operator has no control, rather than as assumptions. The only part of the solution that can properly be described as an assumption is the von Mises plasticity condition. With regard to this it is to be noted that the condition is needed only in a much weakened form, for the analysis uses the condition only at the neck, where the strain is approximately constant. Thus, in the extreme case referred to in a footnote of my paper (p. 37, Met. Tech. Dec. 1944) the actually measured strain across the section varies from 2.4 to 2.6. Obviously it is a much weaker assumption to postulate that the von Mises condition holds in the strain range 2.4 to 2.6 than to assume it to hold for the entire range of strain from o up to 2.6, which would be necessary if the solution applied to the entire tension specimen rather than to the immediate region of the neck only. Furthermore, it is to be commented that under the special conditions of this problem the von Mises condition becomes identical with the maximum shearing-stress criterion of plastic flow; this amounts also to a weakening of the restrictions imposed by the von Mises condition. Dr. MacGregor refers to actual experiments of his own in which the strain was found not to be uniform across the neck. It is to be emphasized, however, that these were experiments on bars of rectangular section, and that the nonuniformity established was a nonuniformity around the perimeter, not a nonuniformity from the surface toward the axis. My method of solution applies only to a circular cross section. It can be said that the solution that satisfies the boundary conditions, etc., would definitely not be expected to demand a uniform strain for noncircular sections. In discussing my results, Dr. McAdam uses a definition of "flow stress" different from mine, so that the immediate applica-

Jan 1, 1945