Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1969"
TL;DR: Van Dyke as mentioned in this paper measured the potentials at which viscous jets or drops first appear in a parallel electric field and compared with calculations of A. B. Basset and found that their stability is due to mechanical rather than electrical causes, like a stretched string, which is straight when pulled but bent when pushed.
Abstract: Fine jets of slightly conducting viscous fluids and thicker jets or drops of less viscous ones can be drawn from conducting tubes by electric forces. As the potential of the tube relative to a neighbouring plate rises, viscous fluids become nearly conical and fine jets come from the vertices. The potentials at which these jets or drops first appear was measured and compared with calculations. The stability of viscous jets depends on the geometry of the electrodes. Jets as small as 20 μm in diameter and 5 cm long were produced which were quite steady up to a millimetre from their ends. Attempts to describe them mathematically failed. Their stability seems to be due to mechanical rather than electrical causes, like that of a stretched string, which is straight when pulled but bent when pushed. Experiments on the stability of water jets in a parallel electric field reveal two critical fields, one at which jets that are breaking into drops become steady and another at which these steady jets become unsteady again, without breaking into drops. Experiments are described in which a cylindrical soap film becomes unstable under a radial electric field. The results are compared with calculations by A. B. Basset and after a mistake in his analysis is corrected, agreement is found over the range where experiments are possible. Basset’s calculations for axisymmetrical disturbances are extended to those in which the jet moves laterally. Though this is the form in which the instability appears, calculations about uniform jets do not seem to be relevant. In an appendix M. D. Van Dyke calculates the attraction between a long cylinder and a perpendicular plate at a different potential.
1,364 citations
TL;DR: In this article, a direct measurement of the forces in air between two cylindrical sheets of mica arranged with their axes mutually at right angles is made using multiple beam interferometry using fringes of equal chromatic order.
Abstract: A direct measurement has been made of the forces in air between two cylindrical sheets of mica arranged with their axes mutually at right angles. The contact resembles that between a sphere and a flat. The mica sheets are glued to glass formers, their concave face being first slightly silvered. The contact region and the distance of approach are measured by multiple beam interferometry using fringes of equal chromatic order (f. e. c. o.). This gives an accuracy for the distances between the surfaces of ± 0.3 nm. Since the surfaces are molecularly smooth it is possible to bring them very close to one another. One surface is held on a rigid support, the other on a light cantilever beam. The surfaces are slowly brought together and at a critical separation ‘flick’ together. The ‘flick’ distance depends on the stiffness of the cantilever and this in turn provides a direct measure of the surface forces. By using cantilevers of different stiffnesses the method has proved effective for separations ranging from 5 to 30 nm. The results show that for separations less than about 10 nm the forces operating are ‘normal’ van der Waals forces whilst for distances greater than 20 nm they are ‘retarded’ van der Waals forces.
668 citations
TL;DR: In this paper, a rational theory is developed to explain the initial pressure rise and consequent separation of a laminar boundary layer when it interacts with a moderately strong shock, where the region of interest is divided into three parts: the major part of the boundary layer, which is shown to change under largely inviscid forces, the supersonic main stream just adjacent to the boundary surface, and a region close to the wall, on boundary-layer scale, in which the relative variation of the velocity is controlled by the incompressible boundary layer equations.
Abstract: A rational theory is developed to explain the initial pressure rise and consequent separation of a laminar boundary layer when it interacts with a moderately strong shock. In this theory, which is firmly based on the linearized theory of Lighthill (1953), the region of interest is divided into three parts: the major part of the boundary layer, which is shown to change under largely inviscid forces, the supersonic main stream just adjacent to the boundary layer in which the pressure variation is small; and a region close to the wall, on boundary-layer scale, in which the relative variation of the velocity is large but is controlled by the incompressible boundary-layer equations, together with novel boundary conditions. We find that the first two parts can be handled in a straightforward way and the problem of self-induced separation reduces, in its essentials, to the solution of a single problem in the theory of incompressible boundary layers. It is found that this problem has three solutions, one of which corresponds to undisturbed flow and another describes a boundary layer which, spontaneously, generates an adverse pressure gradient and a decreasing skin friction which eventually vanishes and then downstream a reversed flow is set up. The third solution generates a favourable pressure gradient and is not relevant to the present study. Although there has hitherto been no valid numerical method of integrating a boundary layer with reversed flow, we find that an ad hoc method seems to lead to a stable solution which has a number of the properties to be expected of a separated boundary layer. Comparison with experiment gives qualitatively good agreement, but quantitatively errors of the order of 20% are found. It is believed that these errors arise because the Reynolds numbers at which the experiments were carried out are too small.
504 citations
TL;DR: In this article, the authors made measurements of the forces required to peel a thin layer of a model visco-elastic adhesive off a rigid substrate and derived a single master relation in terms of peel rate when reduced to a reference temperature by means of the Williams, Landel & Ferry rate equivalence for viscous materials.
Abstract: Measurements have been made of the forces required to peel a thin layer of a model visco-elastic adhesive off a rigid substrate. Over a wide range of temperature and rate of peel the results were found to yield a single master relation in terms of peel rate when reduced to a reference temperature by means of the Williams, Landel & Ferry rate–temperature equivalence for viscous materials. The relations obtained were complex, however, with two main features: a cohesive–adhesive failure transition at low rates and a sharp decrease in peel strength at high rates. The first effect is shown to be due to a change in the deformation process in the adhesive, from a liquid-like to a rubber-like response. An approximate relation between peel strength and the tensile stress–strain behaviour of the adhesive is developed in terms of a single empirically-determined parameter, the interfacial bond strength. Values of this parameter are deduced for several substrates. Measurements of peeling under hydrostatic pressure showed that cavitation of the adhesive did not occur in these experiments. The second effect is shown to be due to the transition from a rubber-like to a glass-like response of the adhesive. It is also shown to depend on the way in which separation is effected. Some general principles governing the mode of failure and the development of maximum peel strength are outlined.
311 citations
TL;DR: In this paper, the authors describe the effects of this instability on the formation of a depression in the surface when the load is concentrated into a few small parts of the surface causing high local temperatures.
Abstract: If two sliding solids are nominally in contact over a large area, the inevitable irregularities in the surfaces will cause the pressure distribution to be non-uniform. The generation of heat due to friction at the interface will also be non-uniform and the solids will be distorted by thermal expansion. The highest parts of the surface will carry the greatest pressure, reach the highest temperature and consequently expand more than the surroundings. Thus the thermal distortion tends to exaggerate the initial irregularity of the surface. The wear at the interface has the opposite effect, but under suitable conditions the process can be unstable. Experiments are described in which the effects of this instability are observed: the load is concentrated into a few small parts of the surface causing high local temperatures. After a few seconds, the load is transferred to a new part of the surface and, when the original contact area has returned to the temperature of the surroundings, it contracts leaving an observable depression in the surface. The scale of this process is large in comparison with the size of the surface asperities. An experimental model has been produced which demonstrates the characteristics of the instability in a simplified form. The thermal expansion drives the surfaces apart at the beginning of the cycle and this movement has been observed experimentally. An analysis of the instability is produced and a good correlation is obtained with experiment.
291 citations
TL;DR: The float process for flat glass has been studied in detail in this paper, where the authors put the float process into perspective by describing briefly, and in simple terms, the methods used for making flat glass before and at the time of the invented float process and then to describe the development of the process itself and the position it occupies in the flat glass industry today.
Abstract: My subject is the float process for making flat glass. I would like, first of all, to put the float process into perspective by describing briefly, and in simple terms, the methods used for making flat glass before and at the time of the invention of the float process and then to describe the development of the process itself and the position it occupies in the flat glass industry today. Finally, I would like to describe in as much depth as time allows, three of the main problems which had to be tackled in developing this process. The Egyptians seem to have been the first people to realize what could be done with glass when it is hot and plastic, and they made vessels for cosmetics and perfumes by, it is assumed, trailing molten glass around a shaped core. By Roman times glass was being blown and moulded, cut and engraved, painted and gilded, and the Romans had mastered the plastic character of heat softened glass so fundamental to today’s processes.
248 citations
TL;DR: In this paper, the authors used a model in which the flow properties of the ice are simplified as Newtonian viscous, rather than obeying a more realistic nonlinear flow law.
Abstract: A glacier slides over its irregular rock bed by a combination of regelation and plastic deformation (Weertman 1957). An exact calculation for this combined process is made possible by using a model in which the flow properties of the ice are simplified as Newtonian viscous, rather than obeying a more realistic nonlinear flow law. The bed is represented as a smooth plane on which there are perturbations of general three-dimensional form but small slope, and the ice is assumed to maintain contact with the bed everywhere. The first-order solution for the velocity field leads to an expression for the drag, which is a second-order effect. It is found that the velocities due to regelation and to viscous flow are additive only when the bed consists of a single sine wave. In the general case the total drag is a summation of the drags due to each of the Fourier components of the bed relief taken separately. The total drag is expressible in terms of a single average property of the bed relief, namely, the product of its mean square amplitude and its autocorrelation function, or, alternatively, its power spectrum. Numerical illustrations are given for a Gaussian autocorrelation function.
241 citations
TL;DR: In this article, audio and radio-frequency dielectric constants, of far infrared transmission spectra and of refractive indices through the visible spectrum were used to assess critically the current theoretical treatments and interpretations of the electronic structures and interionic forces in ionic crystals.
Abstract: Dielectric and spectroscopic measurements have been made on the alkali halides and several thallium and silver halides, at 2 and 290 $^\circ$K, in order to determine, for each, values for the parameters which characterize the dispersion of the dielectric constant due to lattice vibrations. These have then been used to assess critically the current theoretical treatments and interpretations of the electronic structures and interionic forces in ionic crystals. The measurements were of audio and radio-frequency dielectric constants, of far infrared transmission spectra and of refractive indices through the visible spectrum.
214 citations
TL;DR: In this article, it was shown that a set of equations which determine the orbitals and correlation function can be derived so that they only involve integrals of up to nine dimensions.
Abstract: It has been widely thought that the use of wavefunctions with full electronic correlation would involve integrals of 3 N dimensions, where N is the number of electrons. Here it is shown that by a method similar to that of variation theory a set of equations which determine the orbitals and correlation function can be derived so that they only involve integrals of up to nine dimensions. Even these nine-dimensional integrals have some special characteristics which make them equivalent to six-dimensional integrals in some methods of integration. The method is formulated for the particular canonical choice of correlation function that has been previously investigated by the authors and is based on a particular trans-correlated kind of wave equation and on some particular convergence properties recently shown for bi-variational problems. This appears to provide a solution to the problem of including all r ij - quantities in wavefunctions: a problem which has been variously discussed for the last thirty years.
214 citations
TL;DR: In this article, the de Gennes theory of the proximity effect has been used to account quantitatively for the behaviour of thin-film lead-copper-lead junctions with the lead in the superconducting state.
Abstract: The resistance of thin-film lead-copper-lead junctions has been studied with the lead in the superconducting state. The junctions will sustain a supercurrent up to a certain critical value above which a voltage appears, rising smoothly from zero as the current is increased. The effect of a magnetic field upon the critical current has demonstrated that the sandwiches behave phenomenologically as Josephson junctions. The critical current rises rapidly as the temperature is lowered, decreases exponentially with increasing thickness of copper and mcreases with increase of the mean free path of the copper. A simplified version of the de Gennes theory of the proximity effect has been used to account quantitatively for this behaviour. The experiments show that the coherence length of the paired electrons in the copper increases as the temperature decreases, implying that thermal fluctuations govern the decay of the pairs. From the value of the decay length, the interaction parameter in copper is estimated to lie between +0$\cdot$06 and +0$\cdot$14. The properties of these junctions are compared with those of junctions with insulating barriers.
162 citations
TL;DR: In this paper, a theoretical analysis of the loss of energy by wave breaking in a random sea state in terms of the spectral density function was made, and it was shown that α ≑ -1/8/ln[1600C 3/2 ( ρ air/ ρ water)].
Abstract: A theoretical calculation is made of the loss of energy by wave breaking in a random sea state in terms of the spectral density function. In the special case of the equilibrium spectrum F(σ) = αg 2 σ -5 the proportion ɷ of energy lost per mean wave cycle is found to be given by ω ≑ e -1/8α irrespective of the low-frequency cut-off in the spectrum. Assuming that in the equilibrium state the loss of energy by breaking is comparable to that supplied by the wind, one can estimate the constant α in terms of the drag coefficient of the wind on the sea surface. It is found that α≑ -1/8/ln[1600C 3/2 ( ρ air/ ρ water)]. Taking a representative value of C one finds α ≑ 1.3 x 10 -2 , which falls within the range of observed values of α. The above equation for α is rather insensitive to the various assumptions made in the analysis. There is some evidence, derived from observation, that α may not in fact be quite constant, but may decrease slightly as the wave age ( gt/U ) or the non-dimensional fetch ( gx/U 2 ) is increased. It is suggested that the drag coefficient may behave similarly.
TL;DR: In this paper, a new wave generation mechanism based on the idea of a maser-like action of the short waves on the longer waves was proposed, which is capable of imparting energy to sea waves at the rate observed.
Abstract: Recent observations of the growth of sea waves under the action of wind have established that the rate of growth is several times greater than has yet been accounted for. In this paper a new mechanism of wave generation is proposed, based on the idea of a maser-like action of the short waves on the longer waves. It is shown that when surface waves decay they impart their momentum to the surrounding fluid. Short waves are readily regenerated by shear instability. But a longer wave passing through shorter waves causes the short waves to steepen on the long-wave crests. Hence the short waves impart more of their momentum to the crests of the long waves, where the orbital motion of the long waves is in the direction of wave propagation. If the short waves are decaying only weakly (under the action of viscosity), the effect on the long waves is slight. But when the short waves are forced to decay strongly by breaking on the forward slopes of the long waves the gain of energy by the latter is greatly increased. Calculations suggest that the mechanism is capable of imparting energy to sea waves at the rate observed.
TL;DR: In this paper, it was shown that stable causality is the necessary and sufficient condition that there should exist a cosmic time function which increases along every future directed timelike or null curve.
Abstract: It is shown that stable causality is the necessary and sufficient condition that there should exist a cosmic time function which increases along every future directed timelike or null curve. Stable causality means that there are no closed timelike or null curves in any Lorentz metric that is sufficiently close to the space-time metric.
TL;DR: In this article, the authors used optical anisotropy arising from resin shrinkage on to the fibres during cure to study debonding in polyester films of polyester resins containing glass fibres.
Abstract: Thin films of polyester resin containing glass fibres of several different compositions have been exposed to water at three different temperatures (20, 60, 100 $^\circ$C) and examined by means of optical and scanning electron microscopy. The incidence and development of fibre debonding has been studied by using the optical anisotropy arising from resin shrinkage on to the fibres during cure. To aid interpretation of these experiments, measurements have been made of resin dimensional changes produced by water immersions. At each temperature, the first response to diffused water is resin swelling and, in hot water, this is superseded by shrinkage, the magnitude of which becomes considerable after prolonged immersion, e.g. 8% linear shrinkage after 2000 h in boiling water. Evidence is reported suggesting that this shrinkage is mainly due to leaching of low molecular weight material from the resin. The interfacial bond between resin and clean glass fibres is rapidly destroyed by diffused water at all three temperatures. However, the use of a coupling agent produces vast improvements in bond life. In fact, debonding in the presence of a coupling agent has been observed only for hot water immersions; even then, the bond withstands the interfacial tension present during early resin swelling and is destroyed very much later when the interfacial stress is compressive owing to resin shrinkage. The mechanism by which this debonding is eventually initiated depends on glass composition. With 'E' and 'C' glass fibres, bond fracture is due to osmotic pressure generated at the interface by water soluble constituents leached from the fibre, and is often accompanied by the growth of cracks into the resin from the fibre surface. With fused silica fibres, which contain negligible amounts of impurities, small regions of debonded interface appear at fibre ends after much longer immersion times and are attributed to high interfacial shear stresses caused by resin shrinkage. Debonding facilitates relative longitudinal movement between fibre and resin, enabling each fibre end to act as a rigid indentor pushing into the adjacent resin. In hot water, resin indentation cracks result and subsequent resin shrinkage and gross plastic deformation lead to their displacement along the fibres, followed by the successive nucleation and displacement of further indentation cracks.
TL;DR: In this article, the variational method is applied to weak second-order resonant interactions among waves whose amplitudes and phase angles vary slowly with position and time, leading to the proof of local conservation of total mean wave energy and momentum laws.
Abstract: Whitham’s variational method is formulated so as to apply to weak second-order resonant interactions among waves whose amplitudes and phase angles vary slowly with position and time. The method is applied in detail to capillary-gravity wave interactions. An internal gravity waves problem is also discussed briefly. The method leads to new and substantial simplifications of the interaction equations. This makes possible the proof of local conservation of total mean wave energy and momentum laws. These, together with another integral of the motion, are found to be of central importance in classifying and characterizing the slow modulations of planewave-like form. Such a classification is given in detail for all initial values of phase angles and relative amplitudes. All progressive uniform waves in the capillary range are found to be unstable with perturbation growth rates which can be of first order in the wave slopes. In this formulation amplitude dependent first-order corrections of classical frequency and/or wave-number arise for all waves participating in a resonance. A few predictions which could be verified by simple experiments are made.
TL;DR: In this article, a numerical solution has been obtained for the development of the flow from the initial unsteady state described by Rayleigh to the ultimate steady states described by Blasius.
Abstract: A numerical solution has been obtained for the development of the flow from the initial unsteady state described by Rayleigh to the ultimate steady state described by Blasius. The usual formulation of the problem in two independent variables is dropped, and three independent variables, in space and time, are reverted to. The boundary-layer problem is unconventional in that the boundary conditions are not completely known. Instead, it is known that the solution should satisfy a similarity condition, and use is made of this to obtain a solution by iteration. A finite-difference technique of a mixed, explicit-implicit, type is employed. The iteration converges rapidly. It is terminated where the maximum errors are estimated to be about 0.04%. A selection of the results for the velocity profiles and the surface shear stress is presented. One striking feature is the rapidity of the transition from the Rayleigh to the Blasius state. The change is practically complete, at a given station on the plate, by the time the plate has moved a distance equal to four times the distance from the station to the leading edge of the plate.
TL;DR: In this paper, a formal theory of radiative transfer in one-dimensional scattering media of arbitrary physical constitution is presented, based on an extension of the treatment of Redheffer, in which the response of a layer of arbitrary thickness to fluxes incident on its boundaries is described by a certain linear operator.
Abstract: This paper sets out a formal theory of radiative transfer in one-dimensional scattering media of arbitrary physical constitution. The theory is based on an extension of the treatment of Redheffer, in which the response of a layer of arbitrary thickness to fluxes incident on its boundaries is described by a certain linear operator. Juxtaposition of two such layers gives a third layer, whose operator can be related to those of its constituents by an operation designated as the star product. It is shown that this set of operators constitutes a semigroup under the star product, and that the infinitesimal generators of the semigroup can be computed in term s of the physical properties of the medium , point by point. This makes it possible to write equivalent discrete and differential equations from both of which transmission and reflexion operators, the emission due to internal sources, and the internal fluxes at prescribed levels in the medium can be obtained.
TL;DR: In this paper, it is shown that it is possible to introduce an extra condition which does not affect the wave function but which causes the correlation factors to have the least effect on the Slater determinant.
Abstract: It can easily be seen that there is an infinity of different ways of expressing a correlated wave-function which consists of a Slater determinant multiplied by two-electron correlation functions. It is shown that it is possible to introduce an extra condition which does not affect the wavefunction but which causes the correlation factors to have the least effect on the Slater determinant. Here such a contraction condition is proposed in a form which only involves six-dimensional integrals and which has been found to give a very systematic procedure in a thorough calculation of a correlated wavefunction for neon. Certain correlation expansion functions can be defined so that any combination of these gives a function which satisfies this contraction condition. This contracted form provides canonical expressions for the correlation function and the orbitals so that if two different calculations are made in this form it is possible to compare the two correlation functions and the two sets of orbitals because in the completely accurate wavefunction they should be the same. Such a property has not occurred in previous calculations or theories.
TL;DR: In this paper, it was shown that the mechanism and extent of deformation of materials by wedges depends upon the angle of the indenter, and on the elasticity of the material as well as on its yield point.
Abstract: It is shown that the mechanism and extent of the deformation of materials by wedges depends upon the angle of the indenter, and on the elasticity of the indented material as well as on its yield point. The theory for a plastic rigid solid applies only when the angle of the wedge is acute and the indented materials are of low elasticity. At more acute angles, the mechanism of deformation has been shown by Mulhearn to become one of radial compression. It is shown here that this mode of deformation is also favoured when more elastic materials are used. It is also shown that the indentation pressure approximates to that for the expansion of a semicylindrical cavity in an infinite elastic medium with definite yield point and takes the form $P/Y = m \ln (E/Y)+C$ where m and C are constants, E is the Young modulus, Y the yield point in simple extension and P the flow pressure. The value of C depends upon the angle of the wedge. The result is analogous to that obtained by Marsh using a 136$^\circ$ pyramidal indenter, but elastic effects are more important in wedge indentation than when indenting with a pyramid. For blunt wedges and elastic materials, elastic effects predominate in importance; the process can then be regarded as that of the indentation of an ideally elastic solid by a rigid wedge. The pressure distribution on the indentation has been determined. It is shown that elastic theory satisfactorily predicts this distribution of pressure, as well as its mean value. For the indentation of highly elastic materials by acute angled wedges, the elastic and plastic deformations become comparable and the mode of deformation is very complex.
TL;DR: In this paper, it was shown that a bifurcate Killing horizon N in a four-dimensional Lorentz manifold is defined as the union of intersecting (smooth) Killing horizons (of the same group G ).
Abstract: Geodesic orbits of a one-dimensional group G of isometries of a semi-Riemannian manifold are classified into complete and incomplete orbits. It is shown that the latter (which are null), if extendable, define fixed points of G . A bifurcate Killing horizon N in a four-dimensional Lorentz manifold is defined as the union of intersecting (smooth) Killing horizons (of the same group G ). By means of an analysis of the action of G near a fixed point, the theorem is established that a Killing horizon N is contained, as a ‘branch’, in a bifurcate Killing horizon N if and only if it contains an incomplete, extendable, null geodesic orbit. Examples in familiar relativistic space times are pointed out.
TL;DR: In this article, a theory was developed to account for the transport of charge in ultra thin, aggregated metal films, using quantum mechanical electron tunnelling, and thermionic emission theories.
Abstract: A theory is developed to account for the transport of charge in ultra thin, aggregated metal films, using quantum mechanical electron tunnelling, and thermionic emission theories. The current density in the film is expressed as a direct function of the structure of the film, the electrical properties of the substrate, the applied voltage and the temperature. It is shown that for a typical aggregated metal film conduction is by electron tunnelling, and that the tunnelling path is through the substrate. The trap hopping model of conduction on an active substrate is re-examined and limits are established for each of the mechanisms. It is shown that the activation energy associated with conduction in very thin films is due to the interaction of Coulombic fields surrounding small, discrete, metal particles. Four types of film structures are analysed, depending on whether the particles and the gaps between them are large or small. For small particles and small gaps the basic mechanism of conduction is an activated quantum mechanical tunnelling; for large gaps conduction in the substrate dominates, whereas for large particles with small gaps a simple, unactivated, tunnelling process is dominant.
TL;DR: In this paper, the first calculation of a molecular wavefunction and energy by the solution of the appropriate transcorrelated wave equation (C -1 HC-W )ϕ = 0 has been made for LiH.
Abstract: The first calculation of a molecular wavefunction and energy by the solution of the appropriate transcorrelated wave equation ( C -1 HC – W )ϕ = 0 has been made for LiH. The results are in accord with the high accuracy found for the Ne atom by Handy & Boys (1968 c ). The spherical symmetry of the Ne problem gave such advantages that the integrals could virtually be evaluated exactly. The new problem of evaluating these integrals for functions about the many nuclei has now been overcome by a particular numerical integration procedure which gives a much higher accuracy in the energy than corresponds to the accuracy of integration for an ordinary integral. Hence a set of points which is much more restricted in number than otherwise can be used. The error per electron is nearly as small as 1 % of a bond energy, and it is reasonable to expect that later applications of this method will easily surpass this accuracy for a wide range of molecules, and for reacting systems. The method depends on the direct introduction of functions of r ij into the correlation factor in a way applicable to any molecule and the successful implementation of this for the first molecule may provide a turning point in methods of theoretical chemical prediction. The preparation of the whole set of programs requires less than one man year and the general characteristics are those of a method which is much simpler than the previous methods which do not appear capable of this level of accuracy.
TL;DR: In this article, the problem of accommodation of constrained deformation by slip and twinning has been analyzed based on Taylor's least-work hypothesis, and the operative combination of slip-and twinning systems is found by minimizing the orientation factor M = ($Sigma\_i$s$\_I$ + $\alpha \Sigma ε + ε ϵ + ϵπ ϵσon), where ϵ is the ratio of the critical resolved shear stress for twinning against slip, and ϵ = the external strain.
Abstract: The problem of accommodation of constrained deformation by slip and twinning has been analysed. The analysis is based on Taylor's least work hypothesis. In this analysis, the operative combination of slip and twinning systems is found by minimizing the orientation factor M = ($\Sigma\_i$s$\_i$ + $\alpha \Sigma\_i$t$\_i$)/$\epsilon$, where s$\_i$ and t$\_i$ are the simple shears resulting from slip and twinning respectively, $\alpha$ is the ratio of the critical resolved shear stress for twinning against slip, and $\epsilon$ is the external strain. Detailed calculations have been made for face-centred cubic crystals deformed by plane strain compression. Experimental observations on deformed single crystals of a Co-8% Fe alloy indicate good agreement with the analysis. Implications of the present study to the twinning observations of Heye & Wassermann on rolled Ag crystals are discussed.
TL;DR: In this article, it was shown that the error in the eigenvalue is proportional to the amount by which the pre-and post-expansion functions are not able to fit the adjoint and direct eigenfunctions of the operator.
Abstract: If an approximate eigenfunction is obtained by the solution of a dissymmetric set of secular equations formed by the use of different expansion functions to precede and follow the operator, it is shown here that the error in the eigenvalue is proportional to $\mu^\dagger\mu$ where $\mu^\dagger$ and $\mu$ are measures of the amounts by which the pre- and post-expansion functions are not able to fit the adjoint and direct eigenfunctions of the operator This replaces the $\mu^2$ error in Rayleigh-Ritz variation theory This result is of considerable value for the circumstances where the use of different sets makes the integrals evaluable for specially desirable post-expansion functions The introduction of direct electronic correlation into wavefunctions is a case where the integrals can be evaluated with different sets of functions but not with the same set Further, these results show how a particular use of numerical integration gives eigenvalues with errors of lower order than those associated with the same integration procedure in normal integrals
TL;DR: In this paper, the vacuum ultraviolet flash photolysis of C 3 O 2 in the 159.0 nm absorption band has been investigated and a number of pressure independent reactions involving C( 3 P), C( 1 D), and C(1 S) with CO, CH 4, N 2, NO, O 2, and H 2 have been qualitatively measured and discussed in detail.
Abstract: The vacuum ultraviolet flash photolysis of C 3 O 2 in the 159.0 nm absorption band has been investigated. The major primary products are C( 1 S), C( 1 D), C( 3 P), and CO. The species C 2 and C 3 have also been observed but are of minor importance in the overall reaction scheme. A number of pressure independent reactions involving C( 3 P), C( 1 D), and C( 1 S) with CO, CH 4 , N 2 , NO, O 2 , and H 2 have been observed by means of the kinetic-spectroscopic method. The rate constants measured at room temperature are summarized here (cm 3 s -1 molecule -1 ): C( 3 P) + CH 4 → C 2 H 4 (?) k -15 (7) C( 1 D) + CH 4 → C 2 H 2 + H 2 k = 3.2 x 10 -11 (8) C( 1 D) + N 2 → C( 3 P) + N 2 k ≈ 2.5 x 10 -12 (10) C( 3 P) + NO → CN + O k = 1.1 x 10 -10 (12) C( 1 D) + NO → CN + O k = 9.2 x 10 -11 (13) C( 1 D) + H 2 → CH + H k = 4.15 x 10 -11 (18) C( 1 S) + H 2 → CH + H(?) k -12 (19) C( 3 P ) + O 2 → CO + O k = 3.3 x 10 -11 (20) C( 1 D) + O 2 → CO + O(?) k -12 (22) The pressure dependent reaction rates of C( 3 P) with N 2 , CO, and H 2 have been qualitatively measured and are discussed in detail.
TL;DR: In this paper, the two beam theory of electron diffraction including absorption is used to interpret these defects as planes of discontinuity where neighbouring slabs of MoO$\_3$ type structure are displaced relative to each other by the displacement vectors.
Abstract: Thin specimens of molybdenum trioxide, after heating in the electron microscope, develop planar defects on {120} planes. The two beam theory of electron diffraction including absorption is used to interpret these defects as planes of discontinuity where neighbouring slabs of MoO$\_3$ type structure are displaced relative to each other by the displacement vectors $\mathbf{R} = \pm\frac{1}{2}\mathbf{a}+\frac{1}{7}\mathbf{b}$. These 'shear' planes are chemical defects since oxygen atoms must be removed before the shear can occur. The density of shear planes is such that the non-stoichiometric range MoO$\_{3-x}$ (0 < x < 0.05) may be described as a disordered shear structure whereby the MoO$\_3$ type matrix is interrupted by irregularly spaced shear planes whose density, area and orientation vary according to the exact composition of the specimen. The shear planes may be observed to nucleate and grow within domains of an intermediate phase. The crystal structure of the domains is studied using the displacement of bend extinction contours at the domain boundaries and dark field observations in addition to selected area diffraction. The composition MoO$\_{2.9975}$ is estimated for the domain structure from the electron micrographs. A 7 x 7 MoO$\_3$ superlattice of oxygen vacancies is deduced for the domain structure. These observations provide direct evidence of the nature, distribution and mechanism of production of the defects responsible for the non-stoichiometric range MoO$\_{3-x}$ (0 < x < 0.05). On the basis of our observations models for the thermal decomposition of molybdenum trioxide and for the production of the disordered shear structure are discussed. The latter is compared with three conflicting models recently proposed in the literature, without direct experimental evidence, for the mechanism of production of ordered and disordered shear structures in non-stoichiometric transition metal oxides.
TL;DR: In this paper, a perturbation theorem is proved that a class of real, bounded perturbations of norm ϵ to real self-adjoint operators preserve the reality of the simple eigenvalues for ϵ sufficiently small.
Abstract: A perturbation theorem is proved: a class of real, bounded (non-self-adjoint) perturbations of norm ϵ to real self-adjoint operators preserve the reality of the simple eigenvalues for ϵ sufficiently small. A bound is obtained on ϵ. Application is made to Benard convection with constant heat sources, radiation, particular time-dependent profiles and nonlinear equations of state and to instability of circular Couette flow for a range of gap widths. In each case the growth rate is the eigenvalue and hence if ϵ c , travelling waves (either growing or decaying) are forbidden.
TL;DR: In this article, an appropriate extension of the virial method developed by Chandrasekhar is used to systematically re-examine the equilibrium and stability of an incompressible dielectric fluid drop situated in a uniform electric field.
Abstract: In this paper an appropriate extension of the virial method developed by Chandrasekhar is used to systematically re-examine the equilibrium and stability of an incompressible dielectric fluid drop situated in a uniform electric field. The equilibrium shapes are initially assumed to be ellipsoidal; and it is shown that only prolate spheroids elongated in the direction of the applied field are compatible with the moment equations of lowest order. The relation between the equilibrium elongation a/b and the dimensionless parameter x = FRIfTi, where F is the applied field, R = (ab2)*, and T is the surface tension, is obtained for every dielectric permeability e. This relation is monotonic if e 20.801 exhibit instability while those having e < 20.801 do not. In the former case, where there are three different equilibrium configurations for the same value of x, only the middle one is unstable.
TL;DR: In this paper, the authors show that if the data admit infinite conductivity, there is an infinite-dimensional affine space of surface velocity fields which will produce the observed secular variation from the observed geomagnetic field, but not with the velocity field recently proposed by Kahle, Ball & Vestine.
Abstract: Backus has observed that infinite core conductivity implies the vanishing of the time derivative of the magnetic flux through any patch on the core surface bounded by a 'null-flux curve', on which the radial magnetic field vanished. Field model GSFC (12/66) is consistent with this criterion only if features with scales smaller than angular degree 7 are important or if the dipole time derivatives are deleted. Deleting the dipole is reasonable if the dipole decays in its 3rd radial mode or core conductivity is 4 x 4$^4$ mho/m. If the data admit infinite conductivity, Backus has also shown that there is an infinite-dimensional affine space of 'eligible' surface velocity fields which will produce the observed secular variation from the observed geomagnetic field, but that at any point on any null-flux curve all eligible flows have the same component normal to the curve. Using only secular variation harmonic coefficients with angular degrees 2 to 6, we obtain velocity components normal to the null-flux curves which are compatible with primarily latitude-dependent westward drift, but not with the velocity field recently proposed by Kahle, Ball & Vestine (1967b).
TL;DR: In this paper, the presence of a charge compensating interstitial F - ion (which reduces the symmetry of the rare-earth site to tetragonal) has been confirmed by the observation of endor lines from its nucleus.
Abstract: This paper describes endor of 19 F nuclei in crystals of calcium fluoride containing the impurities Ce 3+ and Yb 3+ at low concentrations. The presence of a charge compensating interstitial F - ion (which reduces the symmetry of the rare-earth site to tetragonal) has been confirmed by the observation of endor lines from its nucleus. Each rare-earth ion has eight nearest neighbour (n. n.) F - ions which are separated into two inequivalent sets of four by the presence of the interstitial F - ion. The hyperfine parameters for each of these sets and for the interstitial have been measured. Formulae have been deduced for the contributions to these parameters from covalent bonding with the 4 f electrons, but for n. n. the formulae are too complicated for precise fitting with the experimental parameters, and for the interstitial the simple covalent bonding theory gives the wrong predictions. The most likely cause of the discrepancy is core polarization of the rare-earth ion which allows core electrons to take part in the covalent bonding. The hyperfine interactions with next nearest neighbours (n. n. n.) have also been measured. These have almost pure dipolar form so that it is possible to use them to map out the distortion of the lattice due to the interstitial F - ion.