Showing papers in "Philosophical Transactions of the Royal Society A in 1921"

The Phenomena of Rupture and Flow in Solids

Abstract: In the course of an investigation of the effect of surface scratches on the mechanical strength of solids, some general conclusions were reached which appear to have a direct bearing on the problem of rupture, from an engineering standpoint, and also on the larger question of the nature of intermolecular cohesion. The original object of the work, which was carried out at the Royal Aircraft Estab­lishment, was the discovery of the effect of surface treatment—such as, for instance, filing, grinding or polishing—on the strength of metallic machine parts subjected to alternating or repeated loads. In the case of steel, and some other metals in common use, the results of fatigue tests indicated that the range of alternating stress which could be permanently sustained by the material was smaller than the range within which it was sensibly elastic, after being subjected to a great number of reversals. Hence it was inferred that the safe range of loading of a part, having a scratched or grooved surface of a given type, should be capable of estimation with the help of one of the two hypotheses of rupture commonly used for solids which are elastic to fracture. According to these hypotheses rupture may be expected if (a) the maximum tensile stress, ( b ) the maximum extension, exceeds a certain critical value. Moreover, as the behaviour of the materials under consideration, within the safe range of alternating stress, shows very little departure from Hooke’s law, it was thought that the necessary stress and strain calculations could be performed by means of the mathematical theory of elasticity.

Investigations on Lightning Discharges and on the Electric Field of Thunderstorms

Abstract: The method and apparatus used in the measurements are substantially those described in a paper "On Some Determinations of the Sign and Magnitude of Electric Discharges in Lightning Flashes." The induced charge on an exposed earthed conductor (test-plate or sphere) is used as a measure of the electric field. The testplate virtually forms part of a flat portion of the earth’s surface, and the vertical electric force or potential gradient at ground level is equal (in electrostatic measure) to 4 π Q/A, where Q is the charge on its exposed surface and A is its area. The charge Q on the earth-connected sphere of radius R, when exposed at a height h , great compared with R, is a measure of the potential at that height; the zero potential of the sphere being the resultant of the undisturbed atmospheric potential V at the height h and of the potential Q/R due to the charge on the sphere, so that Q/R = - V. The earthed conductors can be shielded from the earth’s field: the test-plate by means of an earth-connected cover, the sphere by lowering it into a conducting case resting on the ground. The quantity of electricity which flows to earth through the connecting wire on exposing or shielding the test-plate or sphere, is measured by a special type of capillary electrometer in which the readings indicate the total quantity of electricity which has traversed the instrument ; the sign and magnitude of the charge on the exposed conductor, and thus of the potential gradient, at the beginning and end of an exposure are thus determined. The sign and magnitude of sudden changes of potential gradient which occur while the conductor is exposed are indicated by the direction and magnitude of the resulting displacements of the electrometer meniscus. The total flow of electricity between the atmosphere and the test-plate or sphere during an exposure is also measured —being given by the difference between the electrometer readings before and after the exposure. The principal improvement introduced has been the provision of apparatus for giving a photographic trace of the electrometer readings; rapid changes in the field occupying less than one-tenth of a second are in this way recorded. In the observations described in the previous paper the sphere was supported in a manner which did not admit of absolute measurements being made, as the charge measured included that on the upper part of the support as well as that on the sphere itself; in these earlier measurements therefore the sphere was standardised by comparison with the test-plate. The method of supporting the sphere is now such that the charge on the sphere alone is measured, while the disturbing effect of the earthed supporting rod is small, and thus the potential at the level of the earthconnected sphere can be calculated from the charge upon it. The new method of mounting the sphere is shown in fig. 1.

Plane Stress and Plane Strain in Bipolar Co-Ordinates

Abstract: The problem of the equilibrium of an elastic solid under given applied forces is one of great difficulty and one which has attracted the attention of most of the great applied Mathematicians since the time of Euler. Unlike the kindred problems of hydrodynamics and electrostatics, it seems to be a branch of mathematical physics in which knowledge comes by the patient accumulation of special solutions rather than by the establishment of great general propositions. Nevertheless, the many and varied applications of this subject to practical affairs make it very desirable that these special solutions should be investigated, not only because of their intrinsic importance but also for the light which they often throw on the general problem. One of the most powerful methods of the mathematical physicist in the face of recalcitrant differential equations is to simplify his problem by reducing it to two dimensions. This simplification can only imperfectly be reproduced in the Nature of our three-dimensional world, but, in default of more general methods, it provides an invaluable weapon. It was shown by Airy that in the two-dimensional case the, stresses may be derived by partial differentiations from a single stress function, and it was shown later that, in the absence of body forces, this stress function satisfies the linear partial differential equation of the fourth order ∇4 X = 0, where ∇4 = ∇2. ∇2, and ∇2 is the two-dimensional Laplacian ∂2/∂ x 2 + ∂2/∂ y 2.

Tidal Friction in Shallow Seas

Abstract: The astronomical importance of the dissipation of energy that goes on in shallow seas has been shown by G. I. Taylor’s recent estimate of the amount in the Irish Sea, which is enough to account for about one-fiftieth of the secular acceleration of the moon. It also produces a considerable effect on the tides themselves, and there are probably many places where it must be taken into account before any satisfactory theory of the local tides, or even their empirical prediction, can be achieved. It is indeed very well known that there are bays and straits where the height of the tides, or the speed of the currents, or both, are greater than in the Irish Sea, and a careful examination of such places, with a view to finding the dissipation in them, is needed. There are other places where the dissipation for an equal area is less than in the Irish Sea, but which may actually contribute much more altogether on account of their greater size. The object of this paper is to discuss what regions are capable of producing notable parts of the secular acceleration; to estimate as accurately as possible from the data available the dissipation in these; and to compare this with that calculated from the secular acceleration, so as to find out whether it is necessary to assume the existence of any other important cause to account for the latter. The horizontal force of the skin friction of water over the sea bottom is 0·002 ρ V2 dynes per square centimetre, where ρ is measured in grammes per cubic centimetre and V in centimetres per second. The difficulty of the problem is in the estimation of V. The available observations of the velocities of tidal currents are given in the Admiralty Sailing Directions; hut they are never uniformly distributed, and are usually confined to the neighbourhood of the coasts, and they must be supplemented by theory before the velocities remote from the coast can be found. A few theoretical considerations that have been found useful in this process will now be mentioned.

The Aerodynamics of a Spinning Shell

Abstract: This paper contains the results, theoretical and experimental, of work undertaken, at the request of the Ordnance Committee, by the authors as Technical Officers of the Munitions Inventions Department. Permission to publish such parts as appear to be of general scientific interest has now been granted by the Ordnance Committee and the Director of Artillery. The publication of this paper has received their sanction. The experiments in question were carried out at the firing ground of H. M. S. “Excellent,” Portsmouth; the Experimental Department, H. M. S. “Excellent,” also provided the 3-inch guns used and the material for the construction of the range. The authors’ best thanks are due to the officers of this department, especially Lieut.-Commander R. F. P. Maton, O. B. E., R. N., without whose cordial co-operation these experiments could never have been carried out; also to the other officers of the Munitions Inventions Department who assisted in the heavy work of making and analysing the observations. The aeronautical measurements at low velocities, required for comparison, were made in the wind channels of the National Physical Laboratory, by arrangement with the Director and the Superintendent of the Aero­nautical Department, to whom also we wish to express our thanks.

Some Measurements of Atmospheric Turbulence

Abstract: The following notation is used throughout. The co-ordinate axes are a right-handed rectangular system x, y, h , in which 0 h is directed vertically upwards, and 0 x lies in any azimuth which happens to be convenient. Elements of distance to east and to north are denoted by de, dn , so that they are special cases of, dx, dy . The atmospheric density is ρ, the pressure is p , acceleration of gravity is g , latitude ϕ is reckoned negative in the southern hemisphere, and ω is the angular velocity of the earth. Velocities are denoted by v with a suffix to indicate the direction towards which they blow. Momenta per unit volume are denoted by m x, m Y, m H. The eddy-diffusivity is denoted by a capital K as in G. I. Taylor’s recent papers. Another, and in the author’s opinion a better, measure of turbulence is ξ discussed in a previous paper. The relation K to ξ is given by ∂/∂p(ξ ∂χ/∂p) ═ ∂χ/∂t ═ ∂2x/∂h2........(1) where χ is either potential temperature, or else mass of water or smoke per mass of atmosphere. If ρ and ξ were independent of height, then from (l) we should have ξ ═ g2 ρ 3K (2) It is suggested that ξ might be named “the turbulivity.” Its dimensions are: (mass)2 x (length)-2 x (time)-5.

A Selective Hot-Wire Microphone

Abstract: The instrument described in the following paper provides:— (i) A convenient means of detecting a note of given pitch when other sounds are present; and (ii) A method of estimating the relative intensities of sounds of the same pitch. The idea which formed the starting-point for the construction of the instrument—viz., the placing of an electrically heated grid of fine platinum wire in the orifice of an otherwise closed vessel—was originally employed by one of us (W. S. T.) in the construction of a sound-detector for the use of Sound Ranging Sections in the British Army. In its original form, the detector was intended to respond to heavily damped aerial vibrations, such as those produced by the firing of guns. Further experiments, however, showed that the detector could be tuned to respond to any continuous sound of definite frequency by suitably choosing the dimensions of the vessel and its orifice.

Researches on the Elastic Properties and the Plastic Extension of Metals

Abstract: On March 7, 1912, I described an instrument which gives photographically a load-extension diagram of a metal test piece during the process of stretching it to fracture. On February 13, 1913, I described further experiments with the instrument. A diagram was shown which was taken from a test piece broken in ten seconds. It is safe to say that up to that time no apparatus existed which would give a complete record of the load-extension relation during such a quick break.

The Stress-Strain Properties of Nitro-Cellulose and the Law of Its Optical Behaviour

Abstract: The physical characteristics of transparent bodies capable of resisting stress have been the subject of much investigation, and in particular the properties of various glasses have been studied with much thoroughness since these latter have an extensive use both for commercial and scientific purposes. In recent years many new forms of optical materials have found an industrial use, and especially nitro-cellulose compounds, which are valuable in cases where glass is not suitable. In recent years many new forms of optical materials have found an industrial use and especially nitro-cellulose compounds, which are valuable in cases where glass is not suitable.

Reduction of Error by Linear Compounding

Abstract: 1. Introductory .—This paper is a development of two earlier papers, which for brevity I call “Reduction” and “Fitting” respectively. The paper immediately preceding “Fitting” is referred to as “Factorial Moments.” These earlier papers deal with two problems, which are closely connected and have the same solution. For both of them, the data are a set of quantities , u , u 1, u 2, ... of the same kind, which we regard as representing certain true values U , U 1, U 2, ..., with errors e , e 1, e 2, ..., so that u r = U r + e r . These errors may be independent or may be correlated in any way. The first problem is based on the assumption (which defines the class of cases we are dealing with) that the sequence of U 's is fairly regular, so that differences after those of a certain order, which we will call j , are negligible. This being so, we may alter any u , or any linear compound of the u 's, such as an interpolation-formula, by adding to it any linear compound of the negligible differences. (I use the term “linear compound” in preference to “linear function,” since there is no consideration of functionality.) The problem is to find the value of the resulting sum when, by suitable choice of the coefficients in the added portion, the mean square of error of the sum is a minimum. This is the problem of “reduction of error.” For the second problem it is assumed that U r is a polynomial in r of degree j , and the problem is to find the coefficients in this polynomial by the method of least squares. This is the problem of “fitting.”