# Showing papers in "Philosophical Magazine Series 1 in 1951"

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TL;DR: In this paper, a general relationship between stress and plastic strain in polycrystalline aggregate is derived for any metal in which individual crystals deform by slipping over preferred planes under a critical shear stress.

Abstract: Summary A general relationship between stress and plastic strain in a polycrystalline aggregate is derived for any metal in which individual crystals deform by slipping over preferred planes under a critical shear stress. Full account is taken of the non-uniform distortion due to mutual constraints between the grains of an aggregate. It is shown that a plastic potential exists which is identical with the yield function. Upper and lower bounds are obtained for an approximate calculation of this function for any applied system of combined stresses.

1,251 citations

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TL;DR: In this article, a method for finding the equilibrium positions of a set of like dislocations in a common slip-plane under the influence of a given applied stress is given by the roots of a certain set of orthogonal polynomials.

Abstract: A method is given for finding the equilibrium positions of a set of like dislocations in a common slip-plane under the influence of a given applied stress. Their positions are given by the roots of a certain set of orthogonal polynomials. The case of a set of free dislocations piled up against a fixed dislocation by a constant applied stress is discussed in detail and the resulting stress-distribution is compared with that produced by a crack with freely slipping surfaces.

787 citations

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TL;DR: In this paper, it was shown that the work-hardness of an isotropic aggregate of face-centred cubic crystals is a function only of the total plastic work if the grains hardened equally.

Abstract: Summary In continuation of a previous paper (Bishop and Hill 1951) it is conjectured that the work done in plastically deforming a polycrystal is approximately equal to that which would be done if the grains were free to deform equally. In conjunction with the principle of maximum plastic work, this enables the yield function of an aggregate to be calculated. This is done for an isotropic aggregate of face-centred cubic crystals, following a determination of the stresses needed to produce multi-slip. The theoretical yield criterion lies between those of Tresca and von Mises, in good agreement with observaton for copper and aluminum. It is shown further that the work-hardening of an aggregate would be a function only of the total plastic work if the grains hardened equally ; the departure from this functional relation is expressed explicitly in terms of the non-uniform hardening.

702 citations

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TL;DR: In this article, a straight crack moving through an elastic medium is calculated and the stresses depend on the velocity and reduce to Inglis' solution when the velocity is zero, and the results may be applied to the spicular fracture of glass.

Abstract: Summary The stress field is calculated about a straight crack moving through an elastic medium. The stresses depend on the velocity and reduce to Inglis' solution when the velocity is zero. The results may be applied to the spicular fracture of glass.

663 citations

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TL;DR: The phenomenological theory of barium titanate given in an earlier paper has been extended, and expressions obtained for the piezoelectric constants, elastic coefficients for constant field and dielectric constant for constant strain in terms of other physical constants of the material are drawn showing these quantities as functions of temperature as discussed by the authors.

Abstract: Summary The phenomenological theory of barium titanate given in an earlier paper has been extended, and expressions obtained for the piezoelectric constants, elastic coefficients for constant field and dielectric constants for constant strain in terms of other physical constants of the material. Curves are drawn showing these quantities as functions of temperature, and some comparison made with experiment. The relations of the constants of the ceramic to those of the single crystal are then briefly discussed.

477 citations

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TL;DR: In this paper, it was shown that a dislocation can move steadily from plane to plane in a crystal, and a theory of mechanical twinning, which is formally analogous to Frank's theory of crystal growth, can be developed.

Abstract: Summary The principal difficulty in developing a dislocation theory of mechanical twinning has been to explain how a twin grows through a finite thickness of crystal. By extending the recent theory of slip bands, proposed by Frank and Read, it is possible to show that a dislocation can move steadily from plane to plane in a crystal. A theory of mechanical twinning, which is formally analogous to Frank's theory of crystal growth, can then be developed. The theory is applied to the body- centred and face-centred cubic lattices.

344 citations

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TL;DR: In this article, the basic concepts of crystal dislocation theory are defined in a manner independent of particular crystal lattices or models or special orientations of the dislocation, and it is shown how their essential properties arise, in so far as they are independent of the particular crystal in which they occur.

Abstract: Summary The elementary concepts of crystal dislocation theory—in particular the Burgers vector, perfect and imperfect dislocations, sessile dislocations, extended dislocations, twin and twinning dislocations—are defined in a manner independent of particular crystal lattices or models or special orientations of the dislocation : and it is shown how their essential properties arise, in so far as they are independent of the particular crystal in which they occur.

286 citations

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TL;DR: A dislocation reaction in the face-centred cubic lattice was studied in this paper, where it was shown that the dislocation can be caused by a dislocation in the lattice's face.

Abstract: (1951). A dislocation reaction in the face-centred cubic lattice. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science: Vol. 42, No. 334, pp. 1327-1331.

216 citations

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TL;DR: In this paper, it is shown that in certain cases the quantum description requires an extra parameter, which is the potential of a new velocity field superimposed on the classical motion, and the condition for the existence of this new parameter is that the probability, in the equilibrium state, of a particle having a very large de Broglie wavelength (that is, a negligible momentum) is finite.

Abstract: The quantum mechanics of a system of identical interacting particles must lead to the classical hydrodynamic equations of motion at high temperatures, because of the correspondence principle. On the other hand, the behaviour of helium II shows that this is not always the case at low temperatures. In this paperit is shown that' in certain cases the quantum description requires an extra parameter, which is the potential of a new velocity field superimposed on the classical motion. Expressed in semi-classical terms, the condition for the existence of this new parameter is that the probability, in the equilibrium state, of a particle having a very large de Broglie wavelength (that is, a negligible momentum) is finite. This condition is satisfied in one model of a superfluid system, a condensed Bose-Einstein gas, but not in a crystal. A tentative theoretical interpretation of two basic equations of the empirical two-fluid theory of helium II is given, in which this new parameter determines the velocit...

190 citations

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TL;DR: In this article, it is shown that polytypism may occur as a result of growth through the rotation of a spiral growth step, which is a feature of carborundum growth.

Abstract: A discussion is given of certain features of the growth of carborundum, as illustrated in Verma's paper. It is shown that polytypism may occur as a result of growth through the rotation of a spiral growth step.

190 citations

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TL;DR: In this paper, the authors show that the stress due to a screw dislocation passing normally through an infinite plate or a disc is largely confined to the neighbourhood of the dislocation line, in contrast to the case of a dislocation in an infinite medium.

Abstract: The stress due to a screw dislocation passing normally through an infinite plate or a disc is largely confined to the neighbourhood of the dislocation line, in contrast to the case of a dislocation in an infinite medium. Two screw dislocations in a plate attract or repel one another with a short-range force in place of the inverse first power law for infinite parallel dislocations. The stress due to an edge dislocation is not essentially different in the plate and infinite body so long as the plate remains flat, but in some circumstances the stress may be largely relieved by buckling of the plate.

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TL;DR: In this paper, it is shown that part of the plastic zone at the yield point, where deformation occurs, depends only on the current surface tractions and not on the previous loading programme.

Abstract: Summary The yield point of a plastic-rigid body is defined as the moment when deformation first becomes possible as the load is increased. The practical significance of the yield-point load for an actual plastic-elastic body is discussed. It is then shown that part of the plastic zone at the yield point, namely the part where deformation occurs, depends only on the current surface tractions and not on the previous loading programme. By means of the principle of maximum plastic work and its complimentary minimum principle (Hill 1950) approximations from above and below to the yield-point load can be obtained; some examples are given. Recent American work on the plastic limit design of structures is critically reviewed and shown, when properly regarded, to be a particular application of the principle of maximum plastic work.

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TL;DR: In this article, the modifications produced in the Rayleigh-Jeffreys theory of slow thermal convection by magneto-hydrodynamic effects in a conducting fluid placed in a magnetic field are examined.

Abstract: Summary The modifications produced in the Rayleigh-Jeffreys theory of slow thermal convection by magneto-hydrodynamic effects in a conducting fluid placed in a magnetic field are examined. Even for a non-viscous fluid, a critical temperature gradient β0 must be exceeded in order that convection occur. In this, the place of viscosity η occurring in the Rayleigh-Jeffreys formula is taken by a quantity η H depending on the conductivity of the fluid, the magnetic field strength H and depth of fluid d, thus η H= (4/27)(d 2/π2)(σμ 2H2/c 2). If the fluid is viscous a multiple of the normal viscosity depending on H must be added to η H. An estimate of the critical gradient is made tor somewhat artificial boundary conditions and it is found large enough to be experimentally detectable. The applicability of Jeffreys' method of marginal stability is discussed and the nature of possible oscillations investigated.

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TL;DR: In this article, the authors introduced the concept of content and showed that the total content of any system in motion is an invariant property in non-linear or non-reactive systems.

Abstract: Summary This paper introduces some ideas and general theorems pertaining to non-linear or mechanical systems. Thus in the case of a resistive network, the dissipation is divided into two parts—the “content” and “co-content”—which are duals of each other. The dissipation itself has stationary properties in linear but not (in general) in non-linear networks, but it can be shown that the “content” and “co-content” have stationary and additive properties in the non-linear case. The idea of “content” is extended to reactive systems, and it is shown that the total content of any system in motion is an invariant. This paper is intended as a companion to the paper by E. C. Cherry.

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TL;DR: In this paper, the static field of an infinitesimal loop of dislocation lying in its glide plane is calculated as a function of time and the field produced by the sudden creation of an infinite loop is calculated.

Abstract: Summary The static field of an infinitesimal loop of dislocation lying in its glide plane is calculated. The field of any plane loop lying in its glide plane may be obtained by integration of the result. The field produced by the sudden creation of an infinitesimal loop is calculated as a function of time.The field of an arbitrarily moving plane loop may be obtained by integration. The method is illustrated by simple examples.

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TL;DR: In this paper, a theory is suggested to describe the distribution of stress and velocity in a block of ductile material symmetrically indented on opposite sides by two smooth flat dies.

Abstract: Summary A theory is suggested to describe the distribution of stress and velocity in a block of ductile material symmetrically indented on opposite sides by two smooth flat dies. The theory is two- dimensional and an ideal plastic-rigid material is assumed. Slip-line fields are proposed for all ratios of width of dies to height of block greater than one. It is found that the velocity field transforms into its corresponding slip-line field. Other plane plastic problems where this correspondence occurs are cited. Since the velocities are known along only one slip-line boundary, the usual numerical procedure for calculating the field has to be modified and the use of influence coefficients is introduced. Solutions are computed for width/height ratios between one and two, and it is observed that simple analytical relations hold between these ratios and the corresponding velocity discontinuities in the field. The calculated mean pressures on the dies are compared with upper and lower bound curves derived by Hi...

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TL;DR: In this paper, a number of growth spirals with shapes ranging from circular to regular hexagonal have been observed on carborundum crystals of types I. and II by coating the crystal faces with a thin film of silver of reflectivity nearly 90 per cent and then working in reflection, using phase contrast illumination.

Abstract: Numerous growth spirals with shapes ranging from circular to regular hexagonal have been observed on carborundum crystals of types I. and II., by coating the crystal faces with a thin film of silver of reflectivity nearly 90 per cent and then working in reflection, using phase contrast illumination. The shapes of these spirals, in relation to the crystal structure, are shown to be in accordance with the predictions of Frank's theory. On a type II. crystal, hexagonal interlaced spirals have been observed. The annihilation of growth fronts where they meet, and other properties of growth fronts are illustrated and hence the observed patterns for two or more screw dislocations are explained. The observed density of dislocations ranges from small values up to a maximum of 104 per sq. cm. For a typical circular spiral the calculated values of the radius of the critical nucleus is 2μ and supersaturation is 0·2 per cent. By the application of multiple-beam interference, using both Fizeau fringes and frin...

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TL;DR: In this article, a pulse height analyser is described suited particularly to the requirements of nuclear physics, where information is stored in a supersonic delay line and displayed on a cathode ray tube.

Abstract: Summary A pulse height analyser is described suited particularly to the requirements of nuclear physics. Information is stored in a supersonic delay line and displayed on a cathode ray tube. The accuracy of channel width and the linearity of response are both one part in a thousand. The number of channels and storage capacity per channel may be varied at will to give 60 channels of up to 106 pulses, 80 channels of up to 3 × 104 pulses, or 120 channels of up to 1023 pulses. The maximum counting rate is 1600 per second.

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TL;DR: In this article, the elastic constants of aluminium have been calculated and an explanation for the fact that aluminium crystals are almost elastically isotropic is given, and it is shown that the density of states near the Fermi surface may depend on electron concentration in aluminium solid solutions.

Abstract: Summary A calculation is made of the elastic constants of aluminium which correspond to pure shears without change of volume, and an explanation is given for the fact that aluminium crystals are almost elastically isotropic. Conclusions are drawn about the nature of the density of states near the Fermi surface, and it is shown how the elastic constants may be expected to depend on electron concentration in aluminium solid solutions.

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TL;DR: The velocity of second sound in liquid helium near the absolute zero was investigated in this paper, where it was shown that it is a function of the velocity of the second sound of liquid helium.

Abstract: (1951). The velocity of second sound in liquid helium near the absolute zero. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science: Vol. 42, No. 326, pp. 314-316.

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TL;DR: In this paper, an analysis of the secondary particles ejected from nuclear disintegrations observed in electron sensitive emulsions exposed to the cosmic radiation at 68,000 ft. was made.

Abstract: Summary An analysis has been made of the secondary particles ejected from nuclear disintegrations observed in electron sensitive emulsions exposed to the cosmic radiation at 68,000 ft. Scattering and grain density measure-ments have been carried out on the tracks of 2000 particles associated with these stars. In addition, the grain density and angular distribution of 3070 shower particles and 1508 “grey” tracks have been measured. For 200 stars, the energy of the primary particle which produced the disintegration was measured. A detailed analysis was made of such events. Single fast τ-mesons of kinetic energy less than 1 BeV. are found to interact strongly with nuclear matter. An estimate of the frequency of occurrence of neutral mesons was made from a consideration of the energy balance in stars of low multiplicity, n s .

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TL;DR: In this article, the concept of co-energy is introduced for circuits of electrical or mechanical elements with non-linear properties. But the concept is not applicable to the case of reactive elements.

Abstract: Summary The theorems and concepts presented here are generally applicable to circuits of electrical or mechanical elements with non-linear characteristics. The quantity called the co-energy (the dual of energy) is shown to possess stationary properties (maximum or minimum) and superposition properties ; this is sufficient to establish the concept of an “ equivalent element ” for any 2-terminal system of like elements (all-inductor, all-capacitor, all-springs, etc.). The unfamiliar “ rectangle representation ” of a circuit of linear resistors is explained and extended to the non-linear case, including reactive elements. It is shown that the equations of motion of a non-linear system, possessing reactance, may be expressed in Lagrangian form, thus emphasizing the importance of co-energy and also showing that the Principle of Duality is applicable. Finally, systems are considered possessing mutual inductance and moving magnetic circuits (as in rotating machines). This paper is intended as a companion paper t...

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TL;DR: In this paper, the authors have discussed the propagation of radio-atmospherics by assuming that the space between the earth's surface and the ionosphere behaves like a wave-guide.

Abstract: Summary Previous authors have discussed the propagation of radio-atmospherics by assuming that the space between the earth's surface and the ionosphere behaves like a wave-guide. If the surfaces of such a guide are treated as perfect conductors, then there is one mode of propagation, the “zero-order” mode, which is unattenuated at all frequencies. In such a guide the wave-form of the received e.m.f. due to one mode at a great distance from a lightning flash can be calculated, and for modes of non-zero order it would be of an oscillatory type similar to that observed in many atmospherics from distant sources. But it would also include a large undistorted impulse due to the energy propagated in the zero-order mode, and this is not observed in practice. Moreover, recent measurements of the amplitudes of the component frequencies of atmospherics have shown that frequencies below about7·5 kc./s. are heavily attenuated during propagation from the source to the receiver, and this is inconsistent with the presenc...

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TL;DR: In this paper, it has been shown that at high angles of elevation the radio fluctuations are correlated with the occurrence of “spread” ionospheric echoes from the F region.

Abstract: Summary It has been shown previously that the fluctuations in the intensity of radio energy from the galactic sources have a local terrestrial origin. This paper describes further observations of the two most intense sources, in the constellations of Cygnus and Cassiopeia, taken over a wide range of angles of elevation. At high angles of elevation the radio fluctuations are shown to be correlated with the occurrence of “spread” ionospheric echoes from the F region. When the sources are low on the northern horizon fluctuations are always observed; these are probably introduced by the passage of radio waves through the continuously disturbed ionospheric regions at high magnetic latitudes. Spaced receiver observations taken over base lines of 0·1, 4 and 11 km. enable the scale of the radio energy diffraction pattern across the ground to be determined.

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TL;DR: The results of five independent determinations of the scattering constant for G.5 nuclear research emulsions, details of which are to be presented in the following papers, are compared and discussed as discussed by the authors.

Abstract: Summary The theories of multiple scattering of Williams, Moliere, Snyder and Scott are briefly surveyed. The results of five independent determinations of the scattering constant for G.5 nuclear research emulsions, details of which are to be presented in the following papers, are compared and discussed. The constant has been found to vary with cell-size and β in a manner which is in agreement with theory. The effect of rejecting large angles due to plural scattering is considered, and also those errors which arise from the presence of spurious scattering.

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TL;DR: In this paper, the density of states curve for the 3D electrons in nickel was calculated and the density was shown to be a function of the number of electrons in the atom.

Abstract: (1951). Calculation of the density of states curve for the 3d electrons in nickel. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science: Vol. 42, No. 324, pp. 106-109.

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TL;DR: In this paper, the binding energies of the nuclei H3, He3 and He4 were calculated using the standard variation method with a simple two-parameter wave function of a new form.

Abstract: The binding energies of the nuclei H3, He3 and He4 are calculated, using the standard variation method with a simple two-parameter wave-function of a new form. Purely central interactions of the Yukawa and exponential types are considered. It is found that, when the nuclear parameters are chosen to fit the low-energy scattering and deuteron data, reasonable values are obtained for the three-particle nuclei, but the a-particle binding energy is too high. This result is in agreement with the calculations of Rarita and Present for the exponential potential, and those of Svartholm for the Yukawa potential. In contrast with Svartholm's method of calculation, the potential and kinetic energies of the nuclei are obtained explicitly. In the case of the a-particle, the Yukawa potential yields exceptionally high values for these quantities, and results also in a very concentrated nucleus. The exponential potential, on the other hand, does not give rise to such high values for the potential and kinetic ener...