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By R. I. Jaffee, J. W. Roberts, D. N. Williams

DISPERSION hardening as an alloying process has aroused increasing interest in the past few years. This alloying procedure, in which an insoluble phase is dispersed randomly through a metal or alloy, is believed to be especially suited for the attainment of creep strength at elevated temperatures since the dispersed phase is metallurgically stable. Another advantage is that the physical properties of the matrix phase, such as electrical conductivity, remain relatively unaffected by alloying. These two characteristics resulted in interest in the applicability of dispersion hardening to copper and copper alloys. A study of copper alloys prepared by powder metallurgy methods has shown some strengthening from oxide dispersions.' However, powder metallurgy processing would not be economical for large scale production of dispersion hardened copper-base alloys. Therefore, the research described in this article was initiated to examine methods of forming dispersion alloys which would not require powder metallurgy processing. If powder metallurgy processing is to be avoided, a method must be found by which the dispersion material can be introduced into and dispersed through the melt before casting. When the dispersed phase is an oxide or similar nonmetallic, it is extremely difficult to form a satisfactory liquid: solid dispersion since successful mixing requires that the melt wet the dispersed phase. It is almost impossible to force a nonwetted particle beneath the melt surface. This difficulty can be avoided, however, if the oxide particles are produced in molten copper by reaction within the melt. The problem would then reduce to maintaining randomness until solidification. Four procedures were considered for producing an oxide particle within molten copper by reaction. In all cases, the intent was to react oxygen (0) with a reactive metal (X) to form an oxide (XO) within the copper melt. These reactions were as follows: 1) (Cu-X), + (O)g (Cu)1 + (XO)S [1] 2) (Cu-X)1 + (Cu2O)s (Cu)x + (XO)S [2] 3) (Cu-OK)1 + (Cu-X)1 (Cu)1+(XO)9 [3] 4) (Cu-OK)1 +(X)s (Cu)1 +(XO)s [4] The reactions were carried out in an 8-lb melt of 99.98 pct Cu maintained in the vicinity of 2200oF by induction heating. Where possible the amount of the reactants was controlled to give 1 to 5 vol pct oxide. Mechanical stirring during the reaction period was used to minimize agglomeration of the reaction product. The presence of a reaction product was determined metallographically by study of an ingot cast from the melt at the end of the reaction period. Reaction [I] was studied by dissolving aluminum in the melt followed by forcing oxygen through the melt. The melts became quite sluggish during the reaction period suggesting that A12O3 was formed. However, the reaction product rose out of the melt rapidly, possibly due to the flushing action of the oxygen stream, and little dispersed phase was formed. Reaction [2] was examined by dissolving aluminum, silicon, or nickel i; the melt and then adding Cu20 (melting point 2250F). This procedure was unsuccessful. Although Cu2O was easily stirred into cop-

Jan 1, 1961

By S. L. Couling, D. J-P. Adenis

i2) twinning in poly crystalline magnesium does not usually occur homogeneously but is concentrated in localized regions or bands inclined at about 45 deg to the stress axis. These bands of twinning are in many ways similar to the Luders bands seen in mild steel alzd Al-Mg alloys. Within a hand the percentage of the volume twinned may approach 100 with no twinning outside the band. Bands are not produced in specimens of grain size a3 x 10-= in. or in samples deformed at temperatures 300°F. Annealing of heavily twinned regions leads to re crystallization textures markedly different from those found in the starting material. The exact preferred orientation appears to depend on the starting grain size, the amount of compres-sive strain and the manner in which it is introduced. MAGNESIUM, with its hexagonal crystal structure, deforms plastically at room temperature by a variety of deformation mechanisms, one of the most important of which is (1012) twinning. In single crystals such twinning occurs on compression parallel to the basal plane or tension perpendicular to it. In wrought polycrystalline magnesium, such as rolled sheet, the preferred orientation is such that almost all the grains have their basal planes nearly parallel to the sheet surface. Therefore, if sheet material is deformed by cold rolling or by tension in the plane of the sheet, deformation of the grains by (1012) twinning is not favored. On the other hand, twins form readily on compression parallel to the plane of the sheet as in a compression test, on the compression side of a bent sample, or at the outer rim of a drawn cup. The first part of this paper deals with the morphology of (1012) twinning in polycrystalline magnesium. Some aspects of this subject were covered by Barrett and ~aller.' Of direct bearing on the work to be reported here are the comments of Jetter' in discussion to Ref. 1. He shows that Mg + 1.5 pct Mn alloy sheet when bent about a transverse axis develops a pattern of transverse bands on the surface of compression. A longitudinal section through the sheet shows that the bands are wedge-shaped and inclined to the rolling direction. The bands were identified as "outcrops of regions in which deformation is localized as the result of the reorientation brought about by (1012) twinning". The second part of the paper treats the recrys-tallization texture of the specimens twinned by compression. Magnesium deformed and then annealed generally recrystallizes with no change in orientation.1'3'4 This statement is true provided that the amount of (1012) twinning occurring during deformation is small. In an isolated observation, Anse15 has shown that, when deformation produces much (1072) twinning, annealing causes new maxima to appear in the pole figure. The second part of the paper elaborates on this point. I) MATERIALS AND PROCEDURES All of the experimental work to be reported was done using the commercial sheet and plate alloy AZ31B (nominal composition: 3 pct Al, 1 pct Zn, 0.4 pct Mn, balance magnesium). This alloy is available commercially in two tempers, -0 (fully annealed) and -H24 (partially annealed), and specimens of both tempers were used in various experiments. The average grain. diameter (AGD) for -0 temper 0.125-in. sheet is about 0.5 x 10- in.; where material of larger grain size was desired, anneals at elevated temperature were used to grow the grains. compression tests were made on a screw-driven "hard" machine. The specimens were clamped in a

Jan 1, 1964

By Hsun Hu, R. S. Cline

The formation of annealing textures during the course of recrystallization in 2 pct Al-Fe crystals rolled in the (111) [112], (112) [111], and (112) [Till orientations has hem studied in detail. When the rolling texture is composed of both (111) [121] and (001)[110] components, the annealing texture consists of mainly (110) [001] and (113) [332] components. As the (001) [110] component diminishes from the surface to the interior of the rolled crystal, the relative concentration of the (113) [332] component in the annealing texture decreases accordingly, whereas that of the (110)[001] component increases with the (111)[112] component in the rolling texture. Such dependence of the annealing texture on the composition of the rolling texture is in accmdance with the oriented growth mechanism. Grain-growth characteristics during recrystallization, hence the annealing texture, can he considerablv different in (112) [111]-type crystals depending sensitively on the initial orientation of the crystal. In a previous publication,' the formation of rolling textures in 2 pct A1-Fe single crystals with initial orientations of approximately (1ll)[112], (112)[111], and (112)[111] was studied in detail. The deformation texture of these crystals consisted of either a single (111)[112] or a combination of (111)[112] and (001)[110] components in various concentrations. For the (lll)[112] crystal, the deformation texture was a single (lll)[112] up to 70 pct rolling reduction, but it became (lll)[112] plus (001)[110] after -90 pct reduction. For the (112)[113.]-type crystals, the relative concentration of the (111)[ 112] and (001)[110] components varied with the depth below the surface of the crystal, as well as with the amount of deformation. These series of specimens, having deformation textures with a range of concentration of the (111)[112] and (001)[110] components, could therefore be used for a thorough investigation of the effect of deformation-texture components on the formation of annealing textures. In a study of rolling and annealing textures in Si-Fe crystals, Dunn and Koh3 noted that the addition of a (001)[110] component to the (111)[112]-type deformation texture had practically no effect on the recrystallization texture.* According to the ori- ented growth mechanism for the formation of annealing textures, nuclei related to the deformation texture by approximately 30-deg rotation around a common [110] axis have the highest rate of growth and the resulting annealing textures generally have such an orientation relationship with respect to the deformation textures.314 It was reasoned by Dunn and Koh3 that if the oriented growth mechanism operated the recrystallization texture developed from a deformation texture containing both (111)[112] and (001)[110] components should be strongly centered around (113)[332], because (113)[332] is approximately midway between the (111)[112] and (001)[110], and is related to both of these two orientations by [110] rotations of 25 to 30 deg. Hence, nuclei of (113)[332] orientation should have a high rate of growth in the deformed matrix. However, their results were not in accord with this prediction. It was felt by the writers that a detailed study was needed to clarify the effect of deformation-texture components on annealing-texture formation. For this reason, the present investigation was conducted. EXPERIMENTAL PROCEDURE Specimens used for the present investigation were taken from the crystals rolled previously for deformation-texture studies.' In order to follow the progress of annealing-texture formation during the course of recrystallization, a single specimen was taken from each rolled crystal and its textural changes examined after successive anneals until recrystallization was complete. The specimen was carefully cut from the rolled strip with a jeweler's saw. Prior to annealing, the sawed edges were etched to remove distorted metal, while both faces of the specimen were protected from the etching solution by acid-proof plastic tape. After annealing, the specimen was etched from the "bottom" face only (the reference or "top" face of the specimen was protected by plastic tape) to one half of its original thickness, so that the texture at the surface and at the central section of the strip could be determined by the reflection technique. The

Jan 1, 1965

By D. J. Schmatz

DISCUSSION OF RESULTS The qualitative correlation between low-temperature ductility and prior high-temperature creep strain in nickel obtained in this investigation confirms the result obtained on high-purity brass by Chen and Machlin. However, the quantitative effect is much smaller in nickel. Comparison can be made on the basis of a prior high-temperature strain that is a certain percentage of the total high-temperature elongation to rupture. In the research of Chen and machlin, the prior high-temperature strain of 1/2 pct equaled 8.5 pct of the total high-temperature elongation at rupture. For nickel, in our experiments, 8.5 pct of 21.2 pct equals 1.8 pct. For eh = 1.8 pct, the loss in ductility in percent equals x 100 = b.5 pct. That is, for equivalent bo Formation of Beta Manganese-Type Structure in Iron-Aluminum-Manganese Alloys D. J. Schmatz IRON-aluminum alloys have gained much interest in the past few years for use as a base material for elevated-temperature alloys because of their excel- high-temperature strain the loss in low-temperature ductility is 8.5 pct for nickel as compared to 60 pct for pure a, brass. It is also interesting to note the observation that in nickel the low-temperature fractures, even at the most extreme high-temperature pre-strain, were partly trans crystalline and partly intercrystalline. In the case of a brass, these fractures were completely intercrystalline. According to the data on the percent of the grain boundary area that is cracked, at 22 pct high-temperature elongation, about 12.5 pct of the grain boundary area does not carry load. Thus, the inter-crack areas are carrying a stress that is about 11 pct higher than the nominal stress. A simple model predicts that the overall elongation, which is primarily the elongation in the bulk of the specimen, will correspond to the nominal stress when the stress in the intercrack area equals the tensile strength. Quantitatively this model predicts the equation The form of this equation will agree with that found if the strain-hardening exponent, n, equals unity. Actually this model is too gross an approximation to yield good quantitative results. The question naturally arises as to whether in ordinary hot-working, such as hot-rolling, intercrys-talline cracks are introduced that affect the low-temperature properties. It appears that the answer is that such a possibility cannot be overlooked, if the hot-working induces tensile stresses at temperatures above the equicohesive temperature and if the material tends to be notch sensitive at low temperature. ACKNOWLEDGMENT This research was supported by the Aeronautical Research Laboratory, Directorate of Research, Wright Air Development Center, United States Air Force. REFERENCES lC. Zener: Fracturing of Metals, 1948, Cleveland, ASM. 'H. C. Chang and N. J. Grant: Mechanism of Intercrystalline Fracture. AIME Trans., 1956, voL 8, p. 544; lownal of Metals, May, 1956. 'E. S. Machlin. CreepRupt ure by Vacancy Condensation, AIME Trans., 1956, vol. 8, p. 106; lownd of Metals, February, 1956 'C. Chen and E. S. Machlin: On a Mechanism of High-Temper at ure Inter-crystalline Cracking, lownd of Metals, July. 1957. p. 829. Alloys lent oxidation resistance. Binary alloys of aluminum in iron are ferritic and consequently have poor hot strength. Alloys of iron-aluminum-manganese were made in an attempt to produce an austenitic alloy containing sufficient aluminum to retain the oxidation resistance of binary alloys of iron-alumi-

Jan 1, 1960

By R. Webeler

In order to study the role of work hardening in the fatigue process, use was made of the great sensitivty of the resistivity of AuCu to cold work. A change of the resistivity of AuCu of the order of 1 to 2 pct at the temperature of liquid nitrogen was found to occur as a consequence of severe fatigue. ACCORDING to Orowan's theory,' the process of fatigue In metals 1s associated with the production of a number of small regions which have undergone strain hardening. This phenomenon is supposed to occilr even if the stress applied during fatiguing is always smaller than the yield stress. In an attempt to verity the existence of such regions, Welber and Webeler' undertook to detect the stored energy associated with severe fatigue in copper. Previous experiments" had shown that the energy stored in a sample of copper which has been cold worked by torsion is released in the temperature range between 150" and 250°C when the sample is heated from room temperature and that no more energy is released (or absorbed) between 250" and 450°C. In particular the stored energy amounted to 0.41 cal per g for a case in which the mechanical energy expended in twisting the sample was 11.9 cal per g. In the case of fatigued copper, however, no release of stored energy could be detected between 150" and 250°C, so that the experimental error of &0.02 cal per g represents an upper limit for the amount of energy stored in strain hardening., It seemed desirable to attack the problem in a new fashion. For this purpose, it was decided to make use of the fact that, if an alloy capable of undergoing the order-disorder transition is ordered and then cold worked, the resistivity, p, increases very greatly above the value for the ordered state even if the deformation is very small. Some insight into the nature of the fatigue process may be obtained then by measuring the resistivity of an ordered sample before and after subjecting it to fatigue. For reasons which will become apparent from the following remarks, considerably more can be learned by carrying out the resistivity measurements at two different temperatures. In the case of a material containing impurities, vacancies, dislocations, or other imperfections of essentially atomic dimensions, the resistivity, p, according to Matthiessen's rule, can be represented as a sum of two terms p = p, + p, where p, is the (temperature dependent) resistivity of the pure metal, and p, is the temperature independent contribution of the imperfections. Briefly, the physical basis for this rule is the following: The main contribution of the impurities in question to the resistivity results from the fact that they interrupt the periodicity of the lattice and thus scatter the conduction electrons with a probability which is almost independent of temperature. In order that this be the case, it is necessary that the' extension of the impurities be small enough—roughly less than one electron mean free path—so that their main effect on the resistivity occurs for the foregoing reason. If an alloy like AuCu is partly or completely disordered by quenching from an appropriate temperature, Matthiessen's rule also applies to a very good approximation* with p, representing in this case the resistivity po of the ordered sample and p, the additional (temperature independent) resistivity due to the disorder. In general, the disorder can be represented in terms of atoms which are displaced from their "proper" positions in the superlattice and which thus qualitatively represent the imperfections in the superlattice responsible for the term p,. Since the misplaced atoms are distributed at random throughout the super-lattice, their contribution to the resistivity still can be considered in terms of the scattering of conduction electrons by lattice defects. The situation is somewhat more complex in the case of an alloy disordered by cold work because the process of disordering here does not involve a random redistribution of the atoms; however, Matthiessen's rule also holds in this case. Whenever Matthiessen's rule does apply, the values of the quantity /3 = (p? — /(T, — T,), where p, and p, are the values of the resistivity at two fixed temperatures, T, and T,, respectively, is constant (independent of p,) for a given alloy or metal. In particular, if a sample of AuCu is subjected to ordinary cold work, the value of /3 remains equal to Po, the value for the ordered material. According to Orowan's theory,' as remarked before, a fatigued sample contains a large number of isolated severely cold-worked regions, which make up only a small proportion of the metal. Thus, if a sample of AuCu initially in the ordered state is fatigued, more or less disordered regions will be produced within the ordered material. If these regions are small enough so that Matthiessen's rule applies, then it follows from the previous discussion that /3 again will remain equal to Po. If the effect of fatigue is to produce cold-worked regions which are macroscopic—of the order of at least several electron mean free paths—the effective resistivity, p, has to be computed by use of the ordinary laws of large-scale electrodynamics. For the sake of simplicity, it will be assumed here that the cold-worked regions are completely disordered and have a resistivity, p,. For a given proportion A of disordered regions the effective resistivity, p, for the current in a given direction depends on the geometrical configuration of these regions. In any case, the value of p for such

Jan 1, 1956

By P. Lelong, P. Lacombe, J. Herenguel

IN previous studies on the effect of orientation on the rate of anodic oxidation of A1-3 pct Mg mono-crystals of high purity, certain anomalies were observed on the cold-worked metal. These anomalies were shown to correspond to regions of changed orientation within the grains resulting from plastic deformation.' In fact, they are deformation bands revealed by differences in rate of oxidation and produced by more or less continuous bending of the lattice about a [Ill] axis common to the original crystal and the band.2,3 Preliminary observations made after relatively large degrees of cold working revealed an approximate twin relationship between the bands and the surrounding crystal. Fig. 1 shows the etch figures formed on such a band and on the original monocrystal after a 67 pct rolling reduction. It is evidently not a true twinning, since there is a broad transition zone between the two systems, however the two extreme orientations are nearly in a twin relationship. It was decided to investigate these preliminary observations by progressive rolling-reductions on monocrystals of varied initial orientation. From these studies, which form the subject of the present article, it became apparent that the initial observation of an approximate twin relation between the bands and the monocrystal was simply a special case. Our experiments show that deformation in rolling proceeds by a mechanism analogous to those which have already been observed in the simpler cases of tension and compression. Micrography—in particular, the formation of etch figures—has been applied to the quantitative study of these orientations. In addition, observations have been made of slip lines, of thin oxide films yielding interference colors, of thick films in polarized light, and of etchants giving "engraving." These micrographic methods for studying orientation have established the orientation relationship without any ambiguity, whereas X-ray methods would have given only a statistical result because we were often dealing with thin bands (2 to 100 microns) and even narrower connecting zones covering a maximum of 20 microns. Specimen Preparation and Methods The Al-3 pct Mg alloy was prepared from 99.95 pct A1 and is of the type used industrially in France for brilliant anodized finishes under the name of Brillalumag. This material combines the normal commercial properties with an enduring brilliance due to the protection of a thick, but perfectly transparent, film of alumina.' The monocrystals were prepared by the method of critical deformation followed by prolonged annealing. Great care was taken to insure perfect homogeneity of composition, to eliminate included crystals,6,7 and to avoid the formation of undulations during oxidation. The critical deformation for this alloy is about 1.2 pct (1-l0/10x100) and the subsequent annealing was for one week at 500 °C after gradual increase of temperature (35°C per hr). The crystals thus prepared had a section of 10x5 mm and a length of 20 to 40 mm. Each of these crystals was polished electrolytically and etched for the formation of etch-figures with the classical reagent.' The orientation was then calculated from their angles, as measured with a goniometer microscope.

Jan 1, 1954

By C. Feng, P. Levesque

HE addition of rhenium to molybdenum is known to produce alloys with good workability.&apos; Lawley and Maddin found that the critical stress for twinning in this system was lowered by the addition of 35 pct Re, which suggests that twinning may re- place the usual slip operation as the major element of deformation. The twinning elements in bcc Mo-Re alloys have been reported as the (112) planes and the <111> directions.3 This note represents the continuation of this investigation, in which the authors found that the occurrence of twinning promotes the formation of a bcc tetragonal phase, previously identified as u by Greenfield and Beck.4 Specimens of Mo-35 pct Re alloy, prepared as described previusly, were solution annealed at 2400o, cooled to room temperature, and plastically deformed by compression to induce twinning. After subsequent aging for 8 hr at 1300°C, traces of a were observed along the grain boundaries and at the interfaces between the twins and the matrix, Fig. 1. With increased aging time the cr phase grew into larger clusters and finally occupied the entire twinned area, Fig. 2. Experiments also indicated that if twinning did not precede aging, there was no visible evidence of a in the microstructure.

Jan 1, 1962

By E. E. Gardner, P. A. Schumann

A description of a new four-point probe configuration which permits measurement of epitaxial layer resistivity is given. An analytic solution to the potential distribution due to a point current source on a multilayered structure is briefly described. Typical solutions for this probe configuration are presented. Probe spacing, layer thickness, substrate resistivity, and substrate thickness enter into the calculation. Correction factors for typical cases are given. A brief description of the construction of a probe which can achieve a small spacing of 15 is presented. The prohe is constructed so that all points are adjustable by micrometers and the probe spacing and configuration can be easily altered. Measurements of layer resistivity with the four-point probe are compared with those taken with the three-Point probe and diode-capacitance techniques. TWO techniques of obtaining profiles of N/N+ epitaxial structures are given. The reproducihility of the probe spacing and resistivity measurement is discussed. The effects upon the resistivity measurement of errors in the measurement of the lager thickness, substrate resistil)itv, and substrate thickness are presented. SINCE its introduction by valdes&apos; in 1954, the four-point probe has been the most widely used tool for determining semiconductor resistivity. Its usefulness ranges from the evaluation of bulk semiconductor ingots to the characterization of impurity profiles in diffused structures. In general the four points, two current probes and two potential probes, are placed on one surface of the material being tested either in an in-line array or in a square array. Numerous correction factors1 8 have been worked out for these configurations which are dictated by the finite geometry of the sample being tested. One method of evaluating thin wafers has been proposed which has two probesg on each side of the wafer. Further modifications have added a fifth&apos;&apos; and a sixth" probe. One paper1&apos; has discussed the accuracy of the four-point probe and the precautions necessary in its use. With the exception of the method where two points are placed on opposite sides of the wafer, the four-point probe is restricted to evaluation of wafers with a homogeneous resistivity or to the case where an isolating junction exists between the substrate and the layer (epitaxial or diffused) being evaluated. Due to the relative resistivities and thickness, the resistance of the epitaxial layer is much greater than that of the substrate, and consequently a majority of the current flows through the substrate. The potential distribution at relatively large distances from the current sources is determined by the substrate resistance, and hence the AV/I value is a measure of the substrate resistivity and not that of the epitaxial-layer resistivity. In the immediate vicinity of a current source the potential distribution is determined by the layer resistivity among other factors. Consequently, if the potential probe is placed very close to the current source, then with appropriate correction factors it is possible to determine the layer resistivity. This was the theoretical basis for using two closely spaced probes on opposite sides of an epitaxial wafer. However, due to mechanical problems, it is desirable to place the four probes on the same side of the wafer. This paper discusses the potential distribution in

Jan 1, 1965

By H. Conrad, E. Stofel

The fracture behavior of 60-deg-oriented sapphire crystals was investigated for both tension and compression. Plastic flow on the basal plane was found to be a factor in reducing the tensile stress required to produce fracture in the temperature RECENT experiments1-&apos; have shown that single crystals of a alumina (sapphire) can deform plastically, particularly at temperatures greater than 1000°C. The effect of this plastic flow on the fracture behavior of sapphire was not established, but range 1100º to 1500ºC. Mechanical twins, both (0001) and (0111) types, were produced by compression in the temperature range 1100" to 1300°C. Cracks propagated more readily along the twin boundaries than in the nontwinned crystal. experiments with other ceramic crystals4&apos;5 have shown that plastic flow can initiate cracks that lead to catastrophic failure. This suggests by analogy that plastic flow might also be an important factor in the fracture of sapphire crystals. A recent experimental investigationS undertaken to determine the mechanism of plastic flow in sapphire provided an opportunity to study the type of fracture produced after various amounts of plastic shear strain. The results of this study of fracture are reported in the present paper. During the course of

Jan 1, 1963

By R. W. Hertzberg, R. W. Kraft

A study of the fracture behavior of unidirectwn-ally solidified Cu-Cr eutectic alloy was performed. Fractured whiskers, grain boundaries, and the interface between proeutectic Cr dendrites and the Cu matrix were the predominant failure initiation sites where grain size was larger. Strain induced void formation was restricted at lower solidification rates. This was attributed to a lack of adequate plastic constraint and a probable preferred orientation. A modification of the Crussard et al. model is presented to explain the appearance of "elongated dimples" on the shear fracture surface. A Cu-Cr alloy containing about 1.8 at. pct Cr solidifies by a eutectic reaction forming filamentary crystals of chromium within a continuous copper solid solution matrix.&apos; Webb and Forgeng2 have shown by bend tests that the extracted chromium filaments, which are ordinarily curved and branched (due to random solidification), have the high strengths associated with whiskers. If an ingot of the eutectic alloy is unidirectionally solidified and appropriate control exercised over the thermal gradient, growth rate, and impurity level, the chromium fibers can be forced to solidify as fibers which are substantially parallel to the growth direction, Fig. 1. Uniaxial tensile tests on these straight chromium fibers extracted from controlled ingots3 have confirmed the earlier bend test results on curved fibers. Combining the facts that the strength and modulus of the chromium filaments are higher than the more ductile copper matrix and assuming that a "good bond" exists between the two phases, the Cu-Cr controlled eutectic system fulfills the requirements for effective fiber reinforcement of ductile matrices as established by studies on fiber glass reinforced plastics and other two-phase systems.4 Therefore, it was decided to study the fracture characteristics of morphologically simple specimens of chromium fiber reinforced copper (i.e., specimens of the controlled eutectic with substantially parallel reinforcing fibers) with the objective of establishing the dominant aspects of fracture. PREVIOUS WORK Comparatively little work has been done to understand the manner in which ductile alloys fail be- cause of the limited demand for this kind of information. Ordinarily ductility is specified for safety or similar reasons and when a part deforms plastically its useful life is terminated. In recent years, however, a number of investigators have begun to

Jan 1, 1963

By Frank E. Hauser, Philip R. Landon, John E. Dorn

The flow and fracture strengths of polycrystalline aggregates of high purity magnesium and a solid solution of aluminum in magnesium were determined as functions of temperature and grain size. Magnesium was found to obey two distinct fracture laws. Over the high temperature range, the fracture stress decreased with increasing test temperatures in a manner that closely paralleled the flow stress-temperature relationship. However, over the low temperature range, the fracture stress was independent of the test temperature though highly dependent on grain size, following the trend that is typical for the low temperature brittle fracturing of polycrystalline aggregates of body-centered-cubic and close-packed-hexagonal metals. Even at 78°K, however, plastic strains of 1 to 7 pet were obtained preceding onset of brittle fracturing. Over the brittle fracture range of temperatures, the fracture stress increased linearly with the reciprocal of the square root of the mean grain diameter, while over the entire range of temperatures investigated, the flow strength was observed to increase linearly with the reciprocal of the square root of the mean grain diameter. PREVIOUS investigations&apos;, &apos;have shown that deformation of coarse grained aggregates of high purity magnesium is interrupted by fracturing at very small strains. Such tendency toward brittle fracturing increased with increasing grain size and decreasing test temperature. In several respects, however, low temperature fracturing of magnesium appeared to differ from the truly brittle type exhibited by zinc, molybdenum, tungsten, and iron, so often characterized as a cleavage mode of fracturing. The absence of a well-defined cleavage mechanism for fracturing and the fact that some plastic deformation precedes fracturing might suggest that magnesium does not exhibit the same kind of brittle fracturing that is common to body-centered-cubic metals and to other close-packed-hexagonal metals. But, as will be shown, the low temperature fracture laws for magnesium are analogous to those that apply to iron, molybdenum, tungsten, and zinc in spite of differences in mechanisms and in plastic strains to fracture. In view of these observations. the low temperature fracturing of magnesium will also be called brittle fracturing. This, of course, does not imply that the brittle fracturing of metals is synonymous with the brittle fracturing of glass or other nonmetallics. Whereas the brittle fracturing of glass probably obeys Griffith&apos;s law, the low temperature brittle fracturing of metals arises from stresses induced by a series of dislocations piled up at an opaque grain boundary. Experimental Results All tests were made on tensile bars having gage sections that were 2 in. long and 0.375 or 0.25 in. wide. In order to develop a series of grain sizes, the

Jan 1, 1957

By Robert T. Ault

The nature of fracture in unnotched tensile and notched tensile sheet and round specimens and V -notched and precracked Charpy-type sheet specimens of both wrought stress -relieved and re-crystallized molybdenum was investigated over the temperature range of -78° to 300°C. The sharp rise in fracture stress, for unnotched tensile samples, as the temperature is increased above the brittleness transition temperature (Tb), is found to be a result of an increase in flow stress due to plastic constraint and an increased strain rate which results from the onset of necking at Tb . Quasi-brittle fracture in notched tensile samples is found to occur when the tensile stress at the elastic-plastic interface, beneath the root of the notch, reaches a maximum critical value, which is independent of test temperature over the range from —78° to 25°C, but dependent on microstructure. In the temperature range between 150° and 300°C, unnotched tensile and notched tensile samples alike are found to fracture by a ductile fibrous tearing process which is discontinuous in nature, as a result of the competition between the processes of tearing, through continued plastic flow, and local work hardening. Results from the V-notched and fatigue-cracked Charpy-type impact tests demonstrate that crack initiation is the governing factor in the low-temperature (-78° to 25°C) fracture process for molybdenum. In recent years, there has been considerable interest, and a commensurate number of investigations, concerning the duc tile-to-brittle transition in refractory metals. There has not developed, however, a satisfactory explanation for the uniaxial tensile fracture-stress transition which accompanies the well-known ductility transition. This matter therefore warranted investigation. In a similar manner, the increased importance of notched tensile strength values in evaluating materials and in design criteria requires a better understanding of the factors which control the fracture behavior of notched tensile samples. In a previous investigation1 of the nature of initial yielding and fracture in notched sheet molybdenum at room temperature, it was suggested that plastic constraint was the controlling factor in governing the fracture behavior of notched samples. The final portion of this investigation was concerned with the fracture toughness of molybdenum. Of particular interest was the comparison of effective surface energies for fracture in V-notched samples with fatigue-cracked, Charpy-type samples in order to ascertain the relative importance of the initiation and propagation phases of the fracture process. MATERIALS AND TEST PROCEDURE Sheet tensile, notched tensile, and V-notched Charpy specimens were prepared from 50-mil, stress-relieved molybdenum sheet.* Half of these specimens were vacuum-annealed for 1 hr at 1200°C and 7.5 x l0-5 Torr and furnace-cooled to produce a relatively uniform grain size of 0.30 mm diameter. Round notched and unnotched tensile specimens were prepared from warm-rolled and swaged 5/8-in.-diam bar.* These specimens were vacuum-annealed for 1 hr at 1350°C and 5 x lom5 Torr and furnace-cooled to produce a grain diameter of 0.11 mm. Material analyses in ppm were: C N O H Sheet 290 10 4 1 Bar 50 30 30 2 The unnotched tensile sheet specimens had a 1/4-in. gage width and a 1-in. gage length. The unnotched tensile round specimens had a 9/32-in. gage diameter and a 1-1/4-in. gage length. The sheet and round notch tensile specimens are shown in Fig. 1. All of the notched tensile specimens tested were tandem-notched in order to study the location and mode of fracture initiation. The tandem-notched specimens were carefully machined so that the largest variation in notch depth on a single specimen was 0.001 in. Thus, when fracture occurs, the extent of plastic deformation that exists in the un-fractured notch section is that which exists just prior to fracture. All of the tandem-notched sheet specimens were electrolytic ally polished and chemically etched prior to testing. The V-notched Charpy-type specimens machined from the 50-mil sheet material had the standard dimensions of 2.125 in. long, 0.394 in. deep, and a 0.010-in. root radius. The unnotched tensile and notched tensile tests were conducted over the temperature range of -78° to 300°C, in a 10,000-lb Instron Universal Testing Machine, at a constant crosshead speed of 0.020 in. per min. The V-notched Charpy-type

Jan 1, 1964

By C. J. Beevers

Tlze influence of zirconium hydride precipitate mprphology on the fructure of Zr-H alloys tested at strain rates of 10- sec at 20° and - 196°C and at strain rates of -500 sec.-1 at 20°C has been inves-tigated. The critical stage in the embrittlement of these alloys was the fracture of the zirconium hyhide precipitates to form large interval cracks at a stress of 25,000 10 137,000 psi. In specimens con-temping up to 100 ppm H the preciptates existed as isolafed platelets in the zirconium matrix and their fracture as enhanced by low temperature (—196°C) and high strain rate (-500 set-f) loading. These resl~lts lead to the conclusion that the rate of strain hardening of the zirconium metal controls the frac-tzrm of the hydride platelets. Increased hydrogen contents of 500 to 2000 ppm resulted in an increase in the 1)volume fraction and size of precipitates and crlso led to crack propogation in the hydride at both 20 and —196C at the low strain rates (10&apos;* sec-&apos;). Quenching led to refinerment of hydride precipitate size and reduced the degree of embrittlement at — 196°C; this effect is attributed to the absence of fracture of the fine precipitates. The formation of zirconium hydride precipitates has been shown to be the cause of hydrogen embrittlement of zirconium.&apos; Several investigators&apos;-" have studied the problem in broad outline, demonstrating that impact-loading conditions, testing temperatures below room temperature, and increasing hydrogen content produce increased brittleness. For example, Muehlenkamp and schwope2 showed that the transition temperature for notched impact tests was raised from 150" to 350°C on increasing the hydrogen content from 45 to 490 ppm. The degree of embrittlement is also controlled by the morphology of the zirconium hydride precipitates. The solubility of hydrogen in zirconium decreases from 600 ppm at 550°C to 10 ppm at 20"C,~ so that the form of the zirconium hydride precipitates (ZrH2) can be modified by varying the cooling rate through the Zr-aZr + ZrH2 phase boundary.= Forscher showed that quenching a specimen containing 35 ppm H from above 315°C resulted in a reduction in area of 70 pct at -196"C, whereas slowly cooling from above 315°C resulted in a ductility of half this value. The role of the hydride precipitate in the fracture process has been examined by Young and schwartz6 and more recently by westlake.&apos; From metallo-graphic observations on slowly cooled Zr-H alloys they have suggested that cracks are nucleated in the hydride precipitates by the interactions of twins in the zirconium metal with the hydride. The experiments reported in this paper show that crack propagation in the zirconium hydride precipitates is the critical stage in the embrittlement of Zr-H alloys. The factors influencing fracture of the hydride precipitates and crack propagation in the zirconium metal will be related to the problem of embrittlement. EXPERIMENTAL TECHNIQUES The zirconium used in these experiments was produced from an ingot of vacuum arc-melted iodide zirconium forged down to 1 in. diameter at 800&apos; to 850°C. The 1-in.-diameter bar was surface-machined and cold-swaged to 0.375 in. diameter. After a vacuum anneal at 800°C for 1 hr the hardness was 60 to 70 Vdh. The total impurity content was approxi-

Jan 1, 1965

By G. C. Gould

The nature of freckle segregation is determined by chemical analyses, microradiograplzy, and electron microprobe. In addition, the influence of chemistry variation on freckle formation is studied in laboratory experiments. It is demonstrated that controlled laboratory solidification can produce freckle segregation. "FRECKLE" segregation is a discontinuous, rod-like segregation observed in vacuum consumable -electrode ingots. Figs. 1 and 2 show this segregation as pencil-like, dark etching areas of segregation found in longitudinally and transversely sectioned consumable-electrode ingots. In most cases the segregation is perpendicular with respect to gravity and parallel to the longitudinal axis of the ingot. The segregation is most often encountered at or around midradius in large vacuum consumable-electrode ingots. The austenitic alloys, A-286* and 718, † have shown particular susceptibility to this mode of segregation. The origin of these rodlike pockets of segregation is not well-understood. It has been suggested that the segregation is a result of a portion of the electrode dropping off and passing through the arc un-melted. On the other hand, smeltzer1 has demonstrated the freckle segregation in A-286 to be comprised of low-melting constituents relative to the matrix melting point. The discontinuous nature of the segregation is the most perplexing facet of the freckle-segregation problem. It was to gain some insight into this part of the problem that this work was undertaken. I) PROCEDURE At the outset of the work identification of the phases present and the elements concentrated in the freckle areas was undertaken. The electron micro-probe was used to determine general enrichment as well as -specific phase chemistry. Microradiography using cobalt and chromium radiation was also used on the freckle areas again to determine the elements involved in the segregation. X-ray diffraction patterns were made from portions of these same foil specimens. Optical microscopy and Knoop hardness measurements were also used

Jan 1, 1965

By J. H. Smith, H. W. Paxton

ALTHOUGH many structural and kinetic investigations have been made for alloys of iron and nickel, only meager data exist from thermodynamic investigations. The purpose of this note is to estimate the free energy of the Fe-Ni solid solutions from thermodynamic data available for ternary alloys containing iron and nickel. The lack of a description of the thermodynamic behavior for the Fe-Ni system has required the structural and kinetic analyses of the system to assume some form of solution behavior, usually regular, e.g., the mar-tensite transformation analysis of Kaufman and Cohen, 1 consistent with the limited thermodynamic data. Indications that this system could exhibit ideal solution departures might have been suggested by the large deviations from additivity rules observed in the density and lattice-parameter measurements of Jette and Foote2 who interpreted this in terms of possible solute-solvent interactions. Later Turkdogan et al. 3 and more recently Ward and wright 4 found a minimum in the graphite solubility in liquid Fe-Ni solutions near the concentration of FeNi3 with deviations from linearity persisting at all concentrations. This was interpreted by Ward and wright4 to indicate negative heats of solution and was suggestive of short-range ordering. The infinitely dilute activity coefficient of carbon (?oC) was also observed by smith5 to pass through a maximum near the FeNi3 concentration in Fe-Ni solid solutions. Recent investigations of the liquid solutions over the entire concentration range by Zellars et a1.&apos; and Speiser et a1.," using vapor-pressure determinations, revealed both negative departures from Raoult&apos;s law and a nonregular solution behavior. The solid alloys have been studied by Oriani 8 and Kubaschew-ski and von Goldbeck6 employing gaseous H2/H2O equilibria with Fe and FeO. Both investigations were in substantial agreement, indicating very small departures from ideality. However, Oriani observed the formation of Fe3O4 (instead of FeO) at XFe= 0.441 which severed the equilibrium and precluded experimental investigation of the nickel-rich alloys. Heat-capacity and heat-content measurements have been evaluated by Montgomery10 and found generally unsatisfactory for gleaning thermodynamic information. This is due to the unknown final states of the alloys caused by the interference of the martensite transformation on cooling iron-rich concentrations and superlattice formation, magnetic transformation, and the varied heat treatments used by investigators of nickel-rich concentrations. As an alternative to the experimental difficulties, indirect estimates of the solid-solution behavior have been made by Fleischer and Elliott11 and Smith et a1.12 From their measurements of the solubility of Fe-Ni alloys in liquid lead, Fleischer and Elliott calculated the nickel activity (aNi) (and integrated for aFe) by assuming: 1) liquid lead has no solubility in solid Fe-Ni alloys, 2) Henry&apos;s law is obeyed throughout the liquid lead-rich phase (0— 12 at. pct Nil, and 3) Henry&apos;s law constant is independent of relative iron and nickel concentrations. The estimate of Smith et a1.12 was obtained

Jan 1, 1964

By R. W. Gurry, L. S. Darken

The solubility of cementite in austenite is computed by thermodynamic methods from the observed solubility of graphite. It is found that the solubility of cementite is greater than that of graphite in the entire austenite temperature range. Thus the prior discrepancy between this portion of the phase diagram and the observed metastability of cementite is partially resolved. FOR several years it has been apparent that a serious discrepancy exists in certain observations pertaining to the Fe-C system. Direct experimental data indicate: 1-That the curve representing the solubility of graphite in austenite&apos;" crosses that for cementite4 at about 945°C as shown in Fig. 1. 2— That graphite and austenite saturated therewith are stable1 relative to cementite at all temperatures between the eutectoid, about 723°C, and the eutectic, about 1153°C. These observations are mutually incompatible in view of a well-known consequence of the second law of thermodynamics, namely, that the stable phase (presumably graphite at all temperatures) has a lower solubility than the metastable phase (presumably cementite). Hence either 1 or 2 above must be in error; either the measured solubility of graphite or of cementite, or the observation as to the stability of graphite, is in error. Our present purposes are: 1—To collate the available data on cementite and prepare a table of HO — HO° and of (Fa— HO°)/T. This tabular method seems the best and most convenient way to represent the heat and free energy of formation. 2—To determine whether the measured solubilities of cementite and graphite in austenite are in accord with the above table and the measured thermodynamic properties of austenite, in order to resolve, if possible, the existing paradox of these two solubilities. Enthalpy of Cementite The enthalpy of cementite at 273 °K, H279 — HOcem, was computed by integration of the low temperature heat capacity as measured by Seltz, McDonald, and Wells" 68" to 298 °K) and as given by the Debye functions recommended by them (0" to 68°K); the result is included in Table I. At higher temperature HCem — H,,"" is given by Kelley who used the data of Naeser7 (298" to 1037°K) and of Umino8,9 (273° to 1923°K). We have recalculated Hcem — H273 cem from the data of UminoO in the following way. In the equilibrated alloy at temperature the weight fraction of cementite Pct c -Pct cr is PctC -PctCr designated f(C), and the

Jan 1, 1952

By C. Law McCabe, R. G. Hudson

The Knudsen cell has been employed to determine the free energy of formation of Mn7Cs in the temperature range 800" to 950°C. A value of 66,440 cal was found for hH°o for a-manganese. Measurements of the pressure of manganese over a mixed carbide, (Fe,Mn),C, points to a power relationship between aun7cs and N.4,. RECENTLY Kuo and Perssonl have reported that the carbide of manganese which is in equilibrium with graphite at temperatures up to 1100° C is Mn7Ca. There are no published data on the thermo-dynamic properties of this compound. In order to determine the stability of Mn7Ca, it appeared that, by obtaining the pressure of manganese above 8-manganese and also above Mn,C, in equilibrium with graphite, the free energy of formation of Mn7Ca from 8-manganese and graphite could be obtained. In addition, the vapor pressure of manganese, reported by Kelley From data of Bauer and Brunner,&apos; is subject to some uncertainty and further determinations of the vapor pressure of manganese seemed warranted. In this investigation of the pressure of manganese vapor above pure manganese and also above the carbide of manganese in equilibrium with graphite the apparatus used is the Knudsen orifice cell. The same apparatus, experimental procedure, and method of calculating the pressure was used in this investigation as in one previously reported.~ Care was taken to insure that the cells were at constant weight before using them in a run. The manganese charged in the cell was CP grade powder, carbon free, obtained from the Fisher Scientific Co. A spectroscopic analysis of the manganese after appreciable amounts of it had vaporized from the Knudsen cell showed that no element was present in sufficient quantities to contribute to a weighable weight loss or to decrease the vapor pressure of manganese to any appreciable extent. The spectro-graphic analysis was 0.002 pct Cu, 0.05 pct Fe, 0.002 pct Pb, and 0.002 pct Ni. 8-manganesea is the allo-tropic form of manganese which was present in the cell at temperatures used in this investigation. The manganese carbide, Mn,Ca, was made in the following way: In a closed graphite cell manganese powder was added to graphite powder, which was made from graphite rods for spectrographic use. The manganese powder was the same as that described previously; 5 pct excess graphite was added over that required for the formation of Mn7C,. The mixture was heated in a closed graphite cell for approximately 20 hr at 1350°K under vacuum. X-ray analysis revealed that there was no manganese present after this treatment, but that the lines due to Mn,C, were present. In order to prove that there was no volatile carbide of manganese which was effusing out of the cell, the following experiment was performed: A graphite effusion cell containing graphite power, in excess of that to form Mn,C, of a desired amount, was brought to constant weight on heating at 1228°K. The required amount of manganese was accurately weighed and then added to the graphite effusion cell. The cell was placed in a vacuum at 1228°K for one week, which was the time calculated for the manganese to have effused completely, assuming instantaneous formation of Mn,C8. The cell was then weighed again. This experiment was carried out on two different occasions and both times the weight loss of the cell came within 1 pct of the weight of manganese originally charged minus the weight of manganese left in the cell, as determined by chemical analysis. These data are summarized in Table I. This agreement is considered to be within experimental error and is taken as proof that no carbide of manganese is volatile in this temperature range. It was established, by X-ray analysis, that Mn,C, formed before appreciable amounts of manganese vaporized from the metal powder which was charged. The identification of the carbide of manganese which was present in the Knudsen cell in equilibrium with graphite and manganese vapor was carried out by Kehsin Kuo at the University of Uppsala. He established that the authors&apos; sample, which was submitted to him for analysis, contained the phase

Jan 1, 1958

By Robert A. Rapp

The standard molar free energy of formation of MOO,was determined between 750o to 1050o C in galvanic cell measurements involving the solid electrolyte Zr0.85 Ca0.15 O1.85. Use of the reference electrodes Fe + FexO and Ni + NiO Provided two independent determinations of FoMoo2. The data may be represented by the equation FMoO2 = -137,500 +40.0T(°K) which is in good agreement with previous CO/CO, gas equilibrium measurements of Gleiser and Chipman. The standard molar free energy of formation of molybdenum dioxide has recently been reported over a large range of high temperatures from the calorimetric experiments of King, Weller, and christensen.&apos; Gleiser and chipman2 have also recently determined ?FmoO2 from 926o to 1068oC in CO/CO2 gas equilibrium measurements. Earlier gas equilibrium studies have been made and reviewed by Gokcen.&apos; Over the temperature range studied by Gleiser and Chipman their values for ?FoMoO2 differed from the calorimetric data by about 700 cal. This investigation was undertaken to resolve this difference and thereby establish the two phase mixture of Mo + MOO, as an electrode of well-known oxygen potential for galvanic cell applications at very high temperatures. The galvanic cell technique involving the Zr,.,, Ca,0.15 01.85 solid electrolyte was introduced for the

Jan 1, 1963

By G. Horvay, J. G. Henzel

Nomograms and charts are provided which permit rapid determination of the mold-casting interFace temperature and the speed of solidification when a semiinfinite ingot is cast into a semiinfinite mold. The physical properties of the mold, the frozen metal, and the liquid metal are not assumed to be equal, but they are assumed to be temperature independent. The results are correlated with charts prepared by Paschkis and Hlinka—using Columbia University&apos;s Analog Computer—for the freezing of finite ingot slabs when a constant surface temperature is impressed, and with experimental cooling curves obtained by Pellini in sand and iron molds for 7 in.-sq steel castings. x = coordinate directed to right of zero (freezing direction) y = coordinate directed to left of zero (melting direction) e = position of freezing front (in x direction) ? = position of melting front (in y direction) $ = temperature (function of position and time) ? = fixed temperature 2L = width of finite slab Y = a fixed position in the semiinfinite mold, or the width of a finite mold (see Section 7) Properties y = density, lb per cu ft c = specific heat, Btu per lb deg F k = conductivity, Btu per ft hr deg F X = heat of fusion, Btu per lb k = k/Yc = temperature diffusivity, sq ft per hr* b - = = - heat diffusivity, Btu per sq ft deg F Vhr Subscripts 2 = pertaining to the still-liquid metal (case of freezing), or the medium which furnishes the heat (case of melting) 1 = pertaining to solidified metal (case of freezing), or to molten metal (case of melting)* 0 = pertaining to mold (case of freezing), or to the still-solid metal (case of melting) f = pertaining to fusion condition (eg.,?f - fusion temperature) i = pertaining to the location x = 0 (the mold-ingot "interface")

Jan 1, 1960

By R. H. Tien

Temperature distribution as well as position of the solidified front is solved by means of "heat balance integral", for the case of freezing a slab with time-dependent surface temperature. Numerical solutions for the cases of linear variation and exponential decay for surface temperature are given. Application also includes solidification of casting. An increased interest in the heat-transfer problem involving a change of phase has been developed in recent years. Practical problems include solidification of a casting, freezing and thawing of soil, ice formation, and others. Despite the importance of the topic, a literature search reveals relatively few solutions which may be extended to practical problems. Due to mathematical complexity, which comes from the nonlinearity of governing differential equations, exact analytical solutions which have been published are limited to a few cases with simple boundary conditions. Although numerical solutions of these problems have become available after highspeed digital computers were introduced, lack of simple and general methods of programming for computation makes this approach impractical. The case of the semi-infinite slab with constant surface temperature was solved in the 19th century. In honor of Neumann, who first found an analytical solution, this kind of problem is generally referred to as the Neumann solution. As has been shown in Fig. 1, the problem to be analyzed is the freezing of a semi-infinite region (x > 0) which is initially liquid at the melting temperature with the surface (x = 0) maintained at time-dependent temperature. This analysis can be applied to the study of many casting processes where most of the heat removal is essentially unidirectional. "Heat balance integral" which has been used by Goodman1 is employed in this analysis. The concept of "heat balance integral" is to define a quantity, d(t), called the thermal layer. Beyond this thermal layer, for all practical purposes, there is no heat transfer, and the region is at the equilibrium temperature. "Heat balance integral" is obtained by integrating the conduction equation with respect to the space variable over this thermal layer d(t). Choosing some plausible temperature distribution, usually a polynomial or a finite trigonometric series in the space variable, this integral equation becomes an ordinary differential equation, with time as the independent variable, which can be solved by classical methods. The thermal layer and "heat balance integral" are equivalent, respectively, to the boundary-layer thickness and the momentum integral in hydrodynamics. A modern account of Prandtl&apos;s concept of the boundary layer and von Karman&apos;s momentum integral may be found in Ref. 2. Although the problem considered here has been limited initially to uniform melting temperature, the present method is equally applicable to the case with arbitrary temperature distribution in the liquid. STATEMENT OF THE PROBLEM The physical situation and coordinate system used in the problem to be analyzed are shown in Fig. 1. Notations used are listed in the table of nomenclature. Let us assume a semi-infinite region extending over positive x, initially at the uniform melting temperature. At the surface x = 0, a time-dependent temperature, which has a numerical value

Jan 1, 1965