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By Morris E. Fine, Richard A. Kloske

The stress-strain beha1,ior in tension of Au-Ag alloy single crystals containing nominally 1,3, 5, 95, 97, and 9.9 at. pct Ag was studied uS strain role and lektlperalure down lo 4.2K. A slrain aging yield hob lc,rrs observed on aging under stress in the temperature range 30° lo 75°C. The species diffusing to the dislocation is thought to be a divacancy-solute complex wilh /he. solute then pinning the dislocation by short-Range ovdering and possibly Suzuki locking. At room tempevullcve the critical reso1ved shear stress .follo~c.s an empirical equation of the form tc = A +Bc' where A and B are 90 and 420 ,g per sy mm tor gold base alloys and 60 and 510 g p~r sq )11ttz .tor si1ver base alloys. This strengthening was attributed to a long-range size ejtect. The alloy slrenglliening at i.2'K is greafer than at room temperature. The additional atrloutzf was altribuled lo a uwak skovl-range i?~levacIion between the solute and dislocallon cove. The activation energy .tor deformation a/ 4.Z0h7 decreases on alloying. 11 increases irr other fee systetns. hi the 5 at. pcl Air-Ag- alloy in pmvlicular there is very extensive easy glide and lavge overshooting- of the synmetry line at 4.Z°K. There is also u sharp decrease in the vale of. strain hardenitlg near the widdle of stage II with the new rate bezng abold the sarne as tlzal In easy glide. This uws attributed to a sudden reduction of activity on the pvitrrarg sgstertr. In Au-Ag alloys the critical resolved shear stress Tc. at room temperature is essentially a parabolic function of atomic concentration:' the 50 at. pct Ag-50 at. pct Au alloy is about 10 times stronger than the pure metals. This strengthening occurs in spite of the similarity in valence and atomic size. Solid-solution strengthening in fcc metals is characteristically greater at low temperatures than at room temperature. In Ag-10 at. pct Au and Au-10 at. pet Ag; r, (20.4K)/Tc (300°K) is 2.3 and 2.5. respectively.- compared to 1.2 in pure silver.3 Suzuki' and Flinn5 explained the alloy strengthening at room temperature as a combination of Suzuki locking and short-range order hardening. The agreement between the computed and measured strengthening was very good: however, in Ag base-A1 alloys Hendrickson and Fine6 concluded that Suzuki locking resulted mainly in a yield drop effect. Recent reviews of the solid-solution strengthening in fcc metals by Fleischer2 and Haasen8, 9attributed the strengthening at room tenl-perature to the combined effect of atom nlisfit and change in shear modulus 011 the long-range stress field surrounding a dislocation. The parameter used by Fleischer was The shear tnoduli of Ag-0 to 6 at. pct Au alloys were measured by Pur-wins. Labusch. and Haasen.In While addition of 4 at. pct Au caused an appreciable increase in the C,, elastic constant. there was a decrease on increasing the solute to 6 at. pct Au: C,, and C :: were not measured in the 6 at. pct alloy. The greater solution hardening at low temperatures implies the presence of short-range interactions between the solute atoms and dislocations These include size and electronegativity effects. Whether the extra alloy strengthening at low temperatures is due to locking or friction hardening is still a controversy."".' In the present research the stress-strain behavior of Au-Ag single crystals containing 1 to 5 at. pct Ag and 1 to 5 at. pct Au was measured vs strain rate and temperature down to 4.2' K. Particular attention was paid to yield drop effects. These have not been previously reported. PROCEDURES The alloys were supplied by the Engelhard Industries. Inc.. as strips 0.050 by 0.250 by 12 in. The compositions are given in Table I. Single crystals were grown under static vacuum using a traveling molten zone technique. Crucibles shaped to the size of the strips were made of high-purity graphite prebaked for 24 hr at 1100°C in vacuum. The crucible and its charge were placed in

Jan 1, 1970

By K. F. Veith, R. L. Marovelli, T. S. Chen

An analytical study is made of thermal stress distribution in a thin circular disc subjected to a peripheral thermal shock at various rates of heat transfer. The problem is of importance in predicting the thermal shock response of a rock body of finite size. The theoretical analysis is based on radial heat flow by conduction in the disc and heat exchange by convection between the disc and the surroundings. The case of constant properties and plane stress is treated. Solutions of the stress distribution are presented for both cooling and heating shocks and an average stress theory is formulated. Preliminary experimental verification was obtained from the results of shock tests on thin rock discs insulated on both flat end faces so that heat exchange took place through the exposed peripheral surface. Physical properties vital to the analysis are Young's modulus, tensile strength, coefficient of linear thermal expansion, thermal diffusivity and thermal conductivity. Plots of these properties are presented. Historically, thermal energy has been used for primary rock removal throughout the ages. Although it is not widely known, the fire-setting method of thermal rock fragmentation was still in use 100 years ago in systematic underground mining operations in Scandinavia.' There, the availability of cheap labor and abundant wood fuel made the method competitive with the early blasting powders. By 1880, as high explosives replaced powder, use of the method declined and it was finally abandoned about 1885. Only 40 years ago, some experimental fire-setting methods were used for rock removal underground at Pribram and Zinnwald in Central Europe.2 Compressed air-oil burners replaced the earlier wood fuels. About 15 years ago, gaseous oxygen-oil burners were introduced in the United States and gained acceptance for difficult blasthole drilling and some quarry opera- tions. At present, there is renewed interest in the compressed air-oil burner designs. The literature included information on thermal secondary processes for rock weakening or particle liberation3,4,5,6,7,8 on thermal spallation of rock 9,10,11,12,13 and on the thermal fracture of materials other than rock. The 1955 Symposium on Thermal Fracture14 and the later work of Hassel-man, cover recent developments in thermal shock investigations on brittle ceramics. Although the number of published analytical and experimental investigations conducted on ceramic and refractory materials is large, the literature reveals that there is a lack of information on both the theoretical and actual response of rock to thermal shock. Many of the thermal shock studies on ceramic and refractory bodies are based on the case of heat transfer at the solid's surface by convection in either liquid or air. A similar heat transfer situation was adopted for a theoretical analysis of the thermal stresses in a thin circular disc subjected to a peripheral thermal shock. Experimental thermal shock tests were then conducted on rock specimens to check the analytical results. Specially insulated quartzite, basalt and taconite discs were shocked in either liquid or still air bath at various rates of heat transfer. The work was done in order to resolve the question as to whether theoretically determined tensile stresses are actually realized in a finite rock body undergoing thermal shock. Specifically, if theory predicts that the tensile strength of a rock should be exceeded in a particular type of controlled thermal shock, will fracture occur? This information is needed by Bureau researchers working on thermal and electrical rock fragmentation methods. The object of this paper is to present the main features of the theory and procedure being used in the authors' approach to quantitative prediction of rock response to a thermal shock. THEORETICAL ANALYSIS In the analysis to follow, simple geometry of a thin circular disc is considered. The disc is assumed to be isotropic, homogeneous and perfectly elastic, and to have constant thermal and mechanical properties. The effect of radiation is neglected so that heat is transferred by conduction inside the disc and by convection with its surrounding. In addition, the end

Jan 1, 1967

By G. S. Ansell, J. Weertman

The creep behavior of an aluminum alloy hardened with a finely dispersed phase of aluminum oxide was investigated. The as-extruded alloy shows an approximate steady-state creep in which the creed rate depends exponentially on the applied stress. The activation energy of creeb is abbroximately 150,000 cal per mole. The recrystallized alloy shows no steady-state creep. ONE method of improving the creep resistance of a metal is to introduce a finely dispersed second phase into the metal matrix. The improvement of the creep resistance has been qualitatively explained by assuming that the dispersed second-phase particles act as obstacles to dislocation motion. If the main effect of second-phase particles is simply to hinder dislocation motion then it is possible to derive a high-temperature creep equation for a dispersion-hardened alloy in a straightforward manner. In the Appendix of this paper such an equation is derived from a creep model which works very well for pure metals. Recently, F. V. Lenel has fabricated, for the first time, powder extrusions of aluminum-aluminum oxide in a large-grained recrystallized form. This alloy, designated as MD 2100, consists of a fine dispersion of aluminum oxide plates in a matrix of commercial purity aluminum. A considerable amount of investigation has been carried out concerning the microstructure and physical properties of this alloy.&apos; ) The aluminum oxide is present in the form of flakes 130A units thick and 0.3 u on edge. They are dispersed in the aluminum matrix with an average spacing of approximately 0.5 jx. The spacing varies in the range of 0.05 to 1.5 µ. The alloy structure is extremely stable at high temperatures. For this reason the alloy offers a unique opportunity for a fundamental study of creep of a very finely dispersed two-phase alloy. Lenel supplied specimens in both the unrecrystallized and recrystallized condition. This paper reports high-temperature creep experiments carried out on these specimens. The results obtained were rather unexpected. No steady-state creep was observed in the recrys-tallized material. In fact, after some transient creep which takes place upon loading, the creep rate is essentially zero ( < 10-8per min). If the second-phase particles acted solely as obstacles to the motion of dislocations, measurable steady-state creep would be expected. Since none is observed it appears that the main effect of the fine dispersion in the recrystallized material is to inactivate the dislocation sources themselves, rather than hinder the motion of dislocation loops created at these sources. EXPERIMENTAL DETAILS Specimens were tested in wire form, 0.087 in. in diam for the as-extruded alloy, and 0.035 in. in diam for the recrystallized alloy. These samples were held in friction-type wire grips; a gage length of 2.5 cm was used for all the creep tests. The specimens were held at 600°C in the test apparatus for at least 15 hr prior to each test. The tests were run under the condition of constant loading and, since the creep strains were small, can be considered as constant stress tests. The temperature of testing was held constant within 3°C. Elongations were measured with an optical cathetometer which was capable of measuring strains as smaIl as 0.00012. This allowed the measurement of strain rates as low as l0-8&apos;per rnin. In addition to the creep tests optical micrographs were made in order to determine both the grain size and microstructure of these materials. RESULTS Fig. 1 shows a few typical creep curves obtained from the as-extruded material. The elongations were somewhat erratic, but each curve shows a region of quasi-steady-state creep from which an approximate steady-state creep rate can be obtained. In general the higher the stress at a given temperature, the greater the total elongation before fracture. The lower the applied stress, the longer is the region where the creep rate is almost constant. Summary data from the creep tests of the as-extruded material are listed in Table I. The steady-state creep data of the as-extruded material for a range of stresses at a constant temperature follow a creep equation of the type creep rate = K&apos; = A exp (&) [11 where A and B are constants, k is Boltzmann&apos;s constant, T is the absolute temperature, and a the stress. The standard error of estimate of the data received is less than one order of magnitude. As shown in Fig. 2, if one compensates the creep-rate data for the effect of temperature over the range of test temperature, the steady-state creep data roughly follow a creep equation of the type Temperature compensated creep rate = K* = A exp

Jan 1, 1960

By B. C. Wonsiewicz, W. W. Wilkening

Following a procedure proposed by Wheeler and Ireland, Plane stress yield loci were constructed from Knoob hardness numbers. Basically, six differently oriented hardness measurements were made on three orthogml surfaces through pure poly crystalline magnesium sheet, a magnesium single crystal, and sheet of the magnesium alloys: Mg + 0.5 pct Th, Mg + 4 pct Li, AZ31B, and EKOO. Hardness loci were found to be in poor agreement at small strains (E < 0.05) with loci established by a more rigorous technique. At larger strains (E - 0.10) the agreement is fair, but at this stage in deformation the conventional locus has lost much of the asymmetry that characterizes these anisotropic materials. Two effects which will lead to distortions in the Khn locus are discussed with reference to the geometry of plastic flow during a hardness test. DETERMINING a material&apos;s resistance to multiaxial loading is of interest not only from a structural design viewpoint but also from that of deformation processing. Unfortunately, the determination of the yield locus, although simple in principle, involves tedious procedures if the results are to be at all rigorous.&apos; The idea, first proposed by Wheeler and 1reland2 of determining the yield locus by means of six Knoop hardness impressions along the principal directions in a material has obvious appeal. It is simple, quick, and should be applicable to very thin sheets. If such a technique could be demonstrated to produce consistently reliable results, it would be of interest to both researcher and designer. Lee, Jabara, and ackofen have compared the yield locus determined by Knoop hardness measurements (the Khn locus) to a locus determined by more rigorous techniques. They found good agreement for two titanium alloys at a plastic strain of about 1 pct. The purpose of this paper is to investigate if the Khn locus construction is a reasonable approximation to the locus of a highly anisotropic material. Examples of such materials are magnesium and magnesium alloys which have severely distorted yield loci which in turn reflect markedly dfferent yield strength in different directions.&apos; In pure magnesium, for example, the yield stress in tension along the transverse direction may be four times the yield stress in compression in the same direction and twice the tensile yield stress in the rolling direction. Predicting such large differences ought to serve as a severe test of the Khn locus construction. EXPERIMENTAL PROCEDURES Samples of rolled sheet, 0.250 in. (6.35 mm) thick, of pure magnesium and four magnesium alloys (Mg experimental materials. The pure magnesium together with the lithium and thorium alloys were those used in the study of Kelley and Hosford. The grain size was ASTM number 4 for the pure magnesium and number 6 for the alloys. HARDNESS TESTING The materials were sectioned along the rolling and transverse planes, mounted in a quick setting resin, and mechanically polished. Most of the hardness tests were performed on a surface prepared by electro-polishing (30 pct nitric acid in methanol at 0°C and 20 v) with the exception of the AZ31B and EK00 alloys which were made directly on a metallographically polished surface. However, subsequent hardness tests on the same sample after heavily electropolishing, revealed essentially the same hardness as before. At least twenty Knoop hardness impressions under a 100-g load were made in each of the six orientations shown in Fig. 1. The average hardness number and standard deviation were then calculated for each orientation. CONVENTIONAL LOCUS CONSTRUCTION Yield loci were constructed using a technique described in detail by Lee and ackofen,&apos; in which the flow stress (stress at a given plastic strain) fixes the coordinates of a point on the locus and measurements of the strain ratio serve to establish the slope of the locus at that point. The loading paths which correspond to uniaxial tension or compression tests establish the four intercepts of the locus with the coordinate axes plus one point on the balanced biaxial tension line Tensile testing was performed along the rolling and transverse (r, t) directions. Samples had a uniform rectangular gage length 1 by 4 by 4 in. (25.4 by 6.35 by 6.35 mm) and were deformed at a strain rate of 3.33 x 104 sec-&apos;. The tests were interrupted periodically to unload the sample and measure the plastic strains by means of X-Y post yield strain gages. Compression tests in the rolling, transverse, and through-thickness (r, f, z) directions were performed on 1/4 in. (6.35 mm) cubes at an initial strain rate of 8.33 x sec-&apos;. Lubrication was provided by 0.002 in. (51 pm) Teflon sheet which was renewed after unloading for micrometer measurements used to calculate the strains.

Jan 1, 1970

By R. D. Nelson, J. C. Shyne

The ß and a ß transformations in plutonium were studied with particular emphasis on the transformation kinetics and microstructure. Significant observations are: 1) The kinetic data show conclusively that the ß — a transformation in high-purity plutonium can proceed isothermally with no athermal component. 2) Plastic deformation of the stable (3 phase retards the subsequent (3 — a transformation. 3) Plastic deformation of the stable a phase accelerates the a — ß transformation; the acceleration is attributed only to residual stresses. 4) The a and a?a volume changes occur anisotroPically in textured plutonium. 5) An apparent crystallogvaphic relationship exists between the parent and the product phases of the and (3 — a transformations. 6) Both applied uniaxial compressive stresses and uniaxial tensile stresses raise the starting temperature for the ß — a transformation. 7) A given uniaxial tensile stress favors the a — ß transformation more than an equivalent applied uniaxial compressive stress opposes the transformation. These observations of the (ß —a and a — ß phase changes in plutonium are consistent with known mar-tensitic transformations. ThIS paper elucidates some of the characteristics of the a— ß and ß —a transformations in plutonium. Because considerable conjecture exists about the mechanisms by which the phase transformations occur in plutonium, experiments have been performed to provide indirect information concerning the mechanisms responsible for the a —ß and ß -a transformations. Indirect information is of particular value in the study of plutonium because of the experimental difficulties presented by the metal. Single crystals have not been produced in any of the allotropes. The large density results in high X-ray and electron-absorption factors and consequently complicating X-ray and electron diffraction. The kinetics of ß — a and a — ß transformations of plutonium and the behavior of the transformations under a variety of conditions have been investigated in detail. Information about the mechanisms of the allo-tropic transformations of plutonium was obtained indirectly from three sources: 1) the effect of plastic deformation of the stable parent phase upon the transformation kinetics; 2) the behavior of the metal transforming under applied stresses; and 3) the microstruc-tural and crystallographic features between parent and product phases. PHASE-TRANSFORMATION CHARACTERISTICS In characterizing solid-state phase transformations in metals and alloys, it is useful to define several types of transformations. An aim of the present work was to identify the low-temperature transformations in plutonium by type, i.e., as martensitic or nonmar-tensitic. Practical definitions for these terms follow. The terms commonly used to categorize phase transformations lack universally accepted definitions. This confusion arises doubtlessly because some terms specify crystallographic or morphological character while other words have a kinetic or a thermodynamic connotation. For example, martensitic specifies certain definite crystallographic restrictions. Unfortunately, martensitic is sometimes used in an ill-defined way to imply kinetic characteristics. Further confusion attends the use of such expressions as nucleation and growth, diffusional, and massive. From time to time new systems of phase-transformation nomenclature are suggested; unfortunately none of these has gained general acceptance.1,2 The authors of the present paper have no intention of entering the controversy. We recognize that some readers may object to the nomencliture used here. For exampie, the terms military and civilian have recently been used in much the same way as martensitic and non-martensitic are used in this paper. This paper is intended to describe several specific details of the low-temperature phase transformations in plutonium. The authors have found it useful to identify these transformations as martensitic; the term was chosen as the best available to describe the experimentally observed features of the transformations studied. A martensitic transformation is one that occurs by the cooperative movement of many atoms; the rearrangement of atoms from parent to product crystal structure occurs by the passage of a mobile semico-herent growth interface. The geometric features characteristic of a martensitic transformation are a specific orientation relationship between the product and parent phase lattices, a specific habit-plane orientation for the growth interface, and a shape change with a specifically oriented shear component. There is no alloy partition between the parent and product phases in a martensitic transformation. Martensitic transformations may display either athermal kinetic behavior or thermally activated isothermal kinetic behavior. Some martensitic transformations occur isothermally, although more commonly martensitic transformations are athermal. Isothermal martensitic transformations are suppressible by rapid cooling. In athermal martensitic transformations, nucleation and growth are not thermally activated and the transformations are essentially time-independent. Nucleation, growth, or both can be thermally activated in isothermal martensitic reactions. Transformation of the parent phase into a marten-

Jan 1, 1967

By J. E. Flinn, S. A. Duran

The steady-state creep behavior of poly crystalline cad mi inn was studied over a temperature range of (1.56 to 0.94 Tm. Two distinct mechanisms were found to occur over this temperature range. They were described by: where and represerqt the minimum strain rates corresponding to the low- and high-temperature regions, respectirely. The two regions of constant acti11ation energy were connected by a transition region where the strain rate was controlled by both mechanisms acting in parallel. At temperatures below a transition temperature of about 0.7 Tm the agreement between the activation energy value for creep and that for self-diffiision suggests a rate-controlling mechanism of dislocation climb. For cadwzium, steady-state creep at temperatures above 0. 7 Tm appears to be controlled by another mechanism, perhaps involving the behavior of dislocation jogs. FRENKEL et al.1 studied the high-temperature creep of polycrystalline cadmium and reported an activation energy of 21 kcal per mole for the 0.5 Tm < T < 0.8 Tm range. Based on observations of creep rate at only two temperatures, a value of 22.1 kcal per mole was determined by Medbury. These two investigations were for the purpose of showing agreement between the activation energy for creep and that for self-diffusion, reported3 as 18.2 and 19.1 kcal per mole, respectively, for diffusion parallel and perpendicular to the hexagonal axis. Gilman4 investigated prismatic glide in single crystals of cadmium over a higher-temperature range of 0.72 to 0.93 Tm, and found an activation energy of 29 kcal per mole. He also reported5 an activation energy higher than that of self-diffusion for prismatic glide in zinc single crystals deformed at temperatures near the melting point. This value was in good agreement with those found for an equivalent temperature range by Flinn and Munson6 and by Tegart and sherby7 for polycrystalline zinc. These two independent studies also disclosed at lower temperatures another value of activation energy near that for self-diffusion. It would be expected from the creep results on zinc and single-crystal cadmium that creep studies on polycrystalline cadmium, extended to temperatures near the melting point, might yield an activation-energy value higher than the 22 kcal per mole value found in earlier studies. The purpose of this paper is to report the steady-creep behavior of polycrystalline cadmium over a temperature range of approximately 0.5 to 0.9 Tm EXPERIMENTAL METHOD The cadmium used in this study was obtained in the form of as-cast rods, 0.5 in. diam, through the courtesy of the Bunker Hill Mining Co. The material was of 99.995 pct purity, as determined by spectro-chemical analysis. The creep specimens, which were 0.250 in. diam by 0.400 in. long and annealed at 300°C for 45 min to produce a stable average grain diameter 0.25 mm, were tested in compression using an apparatus similar to that described by Sherby.8 The specimen temperature was controlled to within ±0.5°C with the help of appropriate constant-temperature baths. The applied stress was maintained within 1.0 pct of the desired value by the additions of lead shot at fixed strain increments. No barreling was observed over the strains encountered during testing. Isothermal creep tests9 were used in the study with only a few differential temperature tests10 run for comparison purposes. Steady-state creep data were obtained over a temperature range of 60 to 287°C (0.56 to 0.94 Tm) at five stress levels ranging from 28.1 to 140.6 kg per sq cm. RESULTS The minimum or steady-state creep rate may be described by an equation of the following form:" where i is the minimum strain rate, S is the structure factor, F is a stress function, Qc is the energy of activation, T is the absolute temperature, and R is the gas constant. The minimum strain rates obtained in this study for cadmium were recorded on a semilogarithm plot as a function of the reciprocal absolute temperatures for the various stress levels, as shown in Fig. 1. This figure shows a characteristic transitional behavior" with a parallel interaction of two mechanisms. It is obvious that the activation energies corresponding to the individual processes are insensitive to stress because the curves are parallel. The discrete activation-energies values for the low- and high-temperature regions for the various stress levels are reported in Table I, and were determined by the least-mean-square method. For the low-temperature region, an activation energy of 20.7 ± 0.6 kcal per mole was obtained, and for the

Jan 1, 1967

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

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

Jan 1, 1970

By Arthur Phillips, R. M. Brick, A. J. Smith

A previous publication1§ has described the effect of quenching stresses on the lattice parameter values of high-purity aluminum-copper alloys particularly with reference to the solution and precipitation of CuAl². A similar study has been made of the aluminum-rich aluminum-magnesium alloys. Attention has been directed in the present investigation to the magnitude of the quenching stresses as affected by the specimen size and shape, the number of solute atoms and the quenching rate. The effect of cooling stresses in causing plastic defomation or cracking has been considered in relation to the physical properties of the alloy On reheating alloys in the two-phase field, the time required for precipitation to occur has been observed in connection with the degrec of supersaturation, the rate of diffusion of the solute atoms as a function of the comparative atomic radii, degree of strain, temperature of reheating and grain size. A few qualitative comparisons on the relative effect of these factors have been drawn between the aluminum-copper alloys and the aluminum-magnesium alloys. Strain in Solid-solution Alloys A 10 per cent magnesium alloy ill the form of chill-cast bars was furnished by the Aluminum Company of America through the courtesy of L. W. Kempf. Additional alloys were prepared using alllminum (99.95 per cent Al) and high-grade magnesium. The alloys listed in Table 1 were melted in an Acheson graphite crucible under a flux of fused MgCl², CaC12, NaCI. Chill-cast rods were sealed under vacuum in Pyrex glass, homogenized several days at 450&apos; C., air-cooled and machined to rods of from 1/8 to ¾-in. dia. Metal powders taken from these rods with a fine-toothed saw were again sealed under vacuum in

Jan 1, 1935

By R. M. Brick, Arthur Phillips, A. J. Smith

A previous publication1§ has described the effect of quenching stresses on the lattice parameter values of high-purity aluminum-copper alloys particularly with reference to the solution and precipitation of CuAl². A similar study has been made of the aluminum-rich aluminum-magnesium alloys. Attention has been directed in the present investigation to the magnitude of the quenching stresses as affected by the specimen size and shape, the number of solute atoms and the quenching rate. The effect of cooling stresses in causing plastic defomation or cracking has been considered in relation to the physical properties of the alloy On reheating alloys in the two-phase field, the time required for precipitation to occur has been observed in connection with the degrec of supersaturation, the rate of diffusion of the solute atoms as a function of the comparative atomic radii, degree of strain, temperature of reheating and grain size. A few qualitative comparisons on the relative effect of these factors have been drawn between the aluminum-copper alloys and the aluminum-magnesium alloys. Strain in Solid-solution Alloys A 10 per cent magnesium alloy ill the form of chill-cast bars was furnished by the Aluminum Company of America through the courtesy of L. W. Kempf. Additional alloys were prepared using alllminum (99.95 per cent Al) and high-grade magnesium. The alloys listed in Table 1 were melted in an Acheson graphite crucible under a flux of fused MgCl², CaC12, NaCI. Chill-cast rods were sealed under vacuum in Pyrex glass, homogenized several days at 450&apos; C., air-cooled and machined to rods of from 1/8 to ¾-in. dia. Metal powders taken from these rods with a fine-toothed saw were again sealed under vacuum in

Jan 1, 1935

By J. W. Pugh, Sam Leber

Single crystals of tungsten were made and deformed in tension at 3000°C. The slip traces so formed on these crystals were analysed to determine the apparent slip system. Results indicate that deformation occurs in the<III> direction on planes of maximum shear stress. This determination agrees with the early work of Taylor and Elam on other body-centered cubic crystals at lout temperatures. SLIP in face-centered cubic and hexagonal close-packed metals has consistently manifested itself in straight traces which indicate that glide takes place on planes of highest atomic density and in directions of closest packing. The slip traces observed on body-centered cubic metals are, on the other hand, frequently branched and wavy, and indicate that while glide takes place in the direction of closest packing, it does not necessarily choose the plane of highest density. The early work of Taylor and Elam1 4 indicated that glide took place in the close-packed direction on planes close to those defined by the maximum shear stress. Subsequent investigations have tried to rationalize these observations in terms of an elementary process on the close-packed planes (110). For example, Elam5 and Grenin~er&apos; have suggested alternate glide on nonparallel (110) planes with the observed result that the summation of these glides results in a trace which defines some plane other than{110). Madden and chen7 have recently reviewed the phenomenological and theoretical aspects of noncrystallographic slip. More recently Koehler&apos; has presented a model of dislocation propagation which permits another rationalization. He indicated that dislocation loops are transferred from one glide plane to another by double cross-slip of screw segments. It is reasonable to assume that, on the average, such transfers may result in a trace which approaches the plane of maximum shear stress. The work described in this paper will show that

Jan 1, 1961

By A. N. Mitinsky

(San Francisco Meeting, September, 1915) THE theory of elasticity, the science of the strength of materials, and all our calculations regarding engineering structures are based on Hooke&apos;s law, that in loaded bodies the deformations are, proportional to the load producing them. Without this law it would be impossible to design intelligently engineering structures such as bridges, frames, axles, etc. This being the case, the proportional limit,1 that is, the point at which deformation ceases to be proportional to load, is the most important factor in engineering design. Further, the working fiber stress should be fixed in reference to the proportional limit instead of, as is usual, in reference to the ultimate strength. The incorrect use of the ultimate strength instead of the proportional limit in setting the working fiber stress is probably clue to two reasons: (1) it is a more delicate operation to measure the proportional limit; (2) it is usually -assumed, though incorrectly, that the proportional limit is determined by the ultimate -strength. In Germany the point at which permanent set first appears is often taken instead of the proportional limit. The two points will coincide if Gerstner&apos;s law, that elastic deformations are proportional to the loads producing them both before and after permanent set, is true; and the limit of proportionality and the elastic limit or point of permanent set will both &apos;be represented on a stress-strain diagram, by the point at which the line ceases to be a straight line and becomes curved-or changes its inclination. It is to be further noted that the difference in definition between proportional limit and elastic limit introduces a difference in methods of determination in the testing laboratory. The German method of determining. the elastic limit by alternately loading and unloading

Jan 8, 1915

By S. R. Maloof

In 1945, Hollomon&apos; showed that after plastic yielding and prior to necking under simple tension, both ferrous and nonferrous materials are approximated by an equation of the following form: where, a = true stress K = strength coefficient 5 = true strain n = strain hardening exponent If a material is stress free, a plot of log o vs log 6 would be expected to be a straight line, the slope of which is the strain-hardening exponent (n) and the intercept on the ordinate axis for 6 = 1. is the logarithm of the strength coefficient K. It can be easily shown that the strain-hardening exponent in the above equation is equal to the total uniform elongation prior to necking. Thus, one can determine the total uniform elongation prior to necking from such a plot and, as pointed out by C. W. MacGregor, 2 it is the amount of strain prior to necking which determines the extent to which a material may be deep drawn. In the view of the current interest in the fabrication of beryllium shapes, it was thought that a knowledge of the strain-hardening exponent of cross-rolled beryllium sheet would be of general interest. Load-strain data after yielding and prior to necking was taken from the work of Greenspan,3 converted to true stress and true-plastic strain, and plotted on log-log paper as shown in Fig. 1. The value of Young&apos;s modulus E for beryllium was taken as 43 . l06 psi. The slope of this plot gave a value of 0.212 for the strain-hardening exponent (n) and the intercept on the ordinate axis a value of 105,000 for K. The total uniform elongation is thus 21.2 pct, in excellent agreement with the value estimated by Greenspan (about 20 pct). Furthermore,

Jan 1, 1960

By K. Heindlhofer

ESTIMATES of the "plasticity" of a metal are commonly deduced from three types of test-tensile, torsion and impact. The several results have been more or less at variance, though this disparity has attracted little notice, largely perhaps because they have been expressed in different terms. The discrepancy between the results of the three tests becomes obvious when each is made at a series of low temperatures, for the temperature range within which the "plasticity" of iron largely disappears proves to be very different in the three types of test. The rather abrupt disappearance of "plasticity" of iron within a narrow temperature range cannot be due to any physicochemical change (such as a precipitation) taking place in the material at that time, because the original plasticity is immediately regained when the temperature of the material is brought above the sensitive range. Accordingly this differ-ence in behavior shows that the "plasticity" of a metal depends upon the mode of action of the stresses which produce the deformation, and there-fore upon the method of test. Analysis of the stresses leads to the conclusion that the observed plasticity of a metal is determined by the relation between its resistance to rupture by separation and its resistance to shear (slip), on the one hand, and between the effective normal and shearing stresses on the other; and that the relation between the two resistances does not in general remain constant with change of temperature, and may somehow be altered to some extent by the previous treatment of the specimen. This suggests the view that the results of none of the usual mechanical tests are directly comparable even for a single metal tested at a series of tem-peratures, or for a series of metals at one temperature, because the propor-tion of the total resistance of the metal to the test contributed by the

Jan 1, 1934

By Fred Leighton, Wilson Blake

The microseismic method of detecting instability and high-stress zones in underground mines was developed by the U.S. Bureau of Mines (USBM) in the early 1940&apos;s.l,2 For about 25 years this method has been used throughout the world by mining and construction companies in assessing the stability of underground and surface excavations. Rock noises are the release of small amounts of stored strain energy propagating outward from their source as a seismic wave that can be detected with special geophysical equipment. This release of strain energy is caused by a rock mass undergoing load changes resulting in small-scale displacement adjustments such as microcracking, shearing, sliding, and crushing of crystal grains, some even as small as the elastic displacements themselves. No detailed field studies of any of these mechanisms have been made, and knowledge about the origin of rock noises is relatively incomplete. Laboratory tests have shown that when rocks are stressed in a testing machine, the rock noise rate-the number of rock noises generated per unit of time increases with increasing applied stress.3 This increase is most pronounced as the ultimate strength of the rock is approached. The standard field procedure has been to monitor a rock structure at intervals and plot rock noise rates vs. time, thus gaining a semiquantitative estimate of the behavior and stability of the structure. The success of such monitoring, however, has depended almost entirely on the experience and knowledge of the operator, his interpretation of the data, and his equipment. Very little study has been devoted to the basic properties of the vibration record of a rock noise. USBM decided that the microseismic method was being used sufficiently by various mining and construction companies

Jan 1, 1970

By Norbert R. Morgenstern

INTRODUCTION An understanding of the role of water in controlling the stability of rock masses is central for a rational approach to the subject. The presence of water can hinder mining opera- tions in two ways: 1. excessive discharge 2. excessive pressure It should be noted that if the rock mass is very tight, excessive pressures can exist without large discharges. Alternatively, if abundant water is available and the rock mass is open, then substantial discharges can occur under quite small pressures. The problems associated with excessive discharge are usually well understood by the mining engineer. Vehicle mobility is inhibited, some areas are flooded and hence become inaccessible; at best, a costly nuisance exists. The methods of solving these problems are also well-known to mining engineers. They generally involve tapping the source of the excessive discharge and channeling it away in such a manner that any hindrance to the mining operation is minimized. The effects of water pressure on the stability of rock masses are less obvious and the following will be devoted solely to this aspect of the problem. The manner in which water pressures enter into the consideration of stability is made clear by an understanding of the concept of effective stress. THE CONCEPT OF EFFECTIVE STRESS The simplest mechanical model for the problem of rock slope stability consists of a rough, rigid block sliding down an inclined plane under the influence of gravity. This is illustrated in [Fig. 1]. The down-slope component of the weight of the block tends to produce movement which is resisted by the shearing force generated by friction along the base of the block. The shear stress per unit area inducing movement is

Jan 1, 1971

By K. Thirumalai

Although the term "spalling" has long been known, Norton l first referred to its usage for the fracture or disintegration of materials subjected to rapid temperature changes. Spalling of ceramic materials at high temperatures has received attention in the past.2-4 Removal of the char formed during ablation in reentry of space vehicles has been referred to as mechanical spalling.5 The first experiments to evaluate the process of rock fragmentation by thermal spalling were conducted by Stoces6 using a series of gasoline burners on the quartz veins of the Zinnwald deep mines in Germany. After the development of rocket-jet burners, the principles of thermal spalling combined with high fluid momentum found wide practical application in rocks. The well-known jet piercing process for drilling blastholes in taconite and for jet channeling in the quarrying of granite are among the more notable of these application.7 For the purposes of this study, thermal spalling or spallation of rocks is defined as progressive failure caused by thermal stresses induced by a sudden application of heat from a high temperature heat source. Gray observes that the size of the spall is dependent on the pattern of induced thermal stresses before failure. The spallability or the relative susceptibility of rocks to failure by thermal spalling is a complex but potentially useful parameter for qualitative comparisons. Fundamental differences exist in the degree of spallability among different types of rocks. Accurate prediction of spallability of any particular rock has so far not been possible. Measurement of the state of stress and temperature in the rock during the rapid thermal spalling process poses difficult problems that must be solved. Identifying the parameters that influence the thermal spalling proc-

Jan 1, 1970

By Frank W. Osterwald

Violent, spontaneous destruction of coal faces and ribs during, what are commonly called, bumps endangers and at times destroys life and property in mines of the Book Cliffs coalfield, Carbon County, Utah. Because these mines supply much of the coking coal for the West Coast steel industries, mining is necessary, although costs are high and operating conditions are very difficult at times. As operations continue, mining will go to greater depths, and the difficulties will increase. Whether or not mining can be continued successfully depends in part on whether or not incidence of bumps can be controlled successfully. For this reason, the U.S. Geological Survey, at the request of and in cooperation with the U.S. Bureau of Mines, undertook a study of the Sunnyside No. 1 mine to determine what geologic factors were related to the incidence of bumps. This report presents the preliminary results of that study and indicates that some observable geologic features in the mine may be related directly or indirectly to bumps, and that other features may be used to infer stress distribution around workings that will aid in design of roof support measures. Coal is mined in the Sunnyside No. 1 mine from depths as great as 2500 ft by room-and-pillar methods. The surface topography is rugged with many steep cliffs and deep canyons, hence the load on the coal varies greatly within short distances. Much of the Sunnyside coal is under high stress, and while being mined, it shatters continuously&apos; and falls from the face. Continuous shattering (bumping) makes mining easy, but if the shattering is interrupted, greater stress is built up and violent bumps may result when the coal is mined.

Jan 4, 1962

By William G. Pariseau

There are important phenomena in rock and soil mechanics that cannot be explained in terms of theories of homogeneous, isotropic materials. Subsidence of strata about mine openings is an example. In-situ stress measurements in nonisotropic sedimentary and metamorphic rocks is another. Patterns of joints, faults, fractures, and similar geologic features impart directional mechanical properties to rock masses that are locally isotropic or anisotropic. On the geologic scale, the nonhomogeneity of composite rock masses may, in certain instances, be dealt with as homogeneous anisotropy. The mechanics of large rock masses can, therefore, be expected to involve considerations of anisotropy. The source of the anisotropy is not relevant. Once the scale of observation is selected and the material is defined accordingly, the analysis of stress and deformation proceeds without regard to the numerical values of the material constants. There are also important phenomena in rock and soil mechanics that cannot be explained in terms of linear elasticity, theory. For the mechanical description of anisotropic geologic materials capable of deforming beyond the limit of purely elastic strains, a plasticity theory is required. The purpose of this chapter is to present a plasticity theory for anisotropic rocks and soils. The proposed theory is based on the work of Hill in anisotropic metal plasticity and represents an extension of his work so as to include anisotropic geologic materials. The point of departure from metal plasticity theory begins with the inclusion of the normal stresses as linear terms in the yield condition, which plays a central role in plasticity. Once the yield condition is established, the plastic strain increments are obtained immediately through the principle of normality. For practical applications, some simplification of the three-dimensional theory is re-

Jan 1, 1972

By W. F. Carew, F. A. Crossley

The stress for rupture in 500 hr at 1000° F has been reported to be about 13,000 psi higher for Widmanstitten than for equiaxed microstructures for the Ti-7A1-3Mo alloy.1,2 Also, limited data indicated Widmanstatten and equiaxed micro-structures to have approximately the same creep resistance at 1000°F.&apos; Recently, data have been obtained which show Widmanstatten structures to be significantly more creep resistant than equiaxed structures at 1000°F, see Fig. 1. These data represent several ingots and several heat treatments of the solution treat and age type. The thermal history and tensile properties corresponding to the creep data are given in Table I. The creep data were taken from creep-rupture tests. The associ- ated rupture data are given in Table 11. It may be noted that total creep deformations are considerably greater for equiaxed structures. Previous evaluations of stability indicated that all of the structures tested are stable with respect to exposure to stress at elevated temperature.1,2 For a minimum creep rate of 10-4 in. per in. per hr the difference in stress levels is indicated to be about 17,000 psi. Holden,et found an irreversible contraction upon heating three a-ß titanium alloys in the equiaxed condition above their ß-transus temperatures. The alloys were: Ti-6Al-4V, Ti-155 A (5Al-1.3Cr-1.5Fe-1.2Mo), and C-130 AM (44-4Mn). The net contractions in length between the original equiaxed and the final Widmanstatten structures measured at room temperature were: 0.4, 0.4, and 0.7 pct, respectively. The greater creep and rupture resistance of the WidmanstStten structure of the Ti-7Al-3Mo alloy may be associated with the phenomenon

Jan 1, 1959

By P. W. Ramsey, D. P. Kedzie

INCREASING demand for improved strength-weight ratios made on aircraft structures has resulted in a gradual increase in the tensile strength requirements for steels used in such applications. As the cyclic loads often are more critical than static loads, and the fatigue properties are generally the design criteria, the question arises as to whether an improvement in fatigue properties accompanies the increased tensile strength. The present investigation was undertaken to determine the relationship between fatigue and tensile properties for an aircraft quality steel in the 155 to 280 ksi (1000 psi) tensile strength range, using the Prot method of fatigue testing. Procedure and Results After preliminary work to establish proper heat treating procedures for this heat of aircraft quality electric furnace steel (a Ni-Cr-Mo-V steel," AMS- Prot Fatigue Testing Method—The determination of the endurance limit by the conventional Wohler method can be time-consuming and easily require 20 specimens when statistical variance is to be established. This is also true when endurance limit loads are determined on actual parts or subassem-blies, and the desirability of a reliable accelerated method has been evident for some time. The primary advantage of the Prot method of fatigue testing is in reducing the number of specimens and the time required for each test. Unlike the Wohler method, every Prot specimen tested con- tributes to the determination of the endurance limit, and the authors have been able to duplicate Wohler results obtained with 20 specimens with only half the specimens and a fourth of the total cycles. This new technique for defining the endurance limit was proposed in 1948&apos; by Marcel Prot and consists of progressively increasing cyclic stress until fatigue failure occurs. By varying the load rate, a, a useful relationship is found to exist with failure stress, so that for steel the failure stress appears to

Jan 1, 1958