Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1976"

Bounds and Self-Consistent Estimates for Creep of Polycrystalline Materials

Abstract: A study of steady creep of face centred cubic (f.c.c.) and ionic polycrystals as it relates to single crystal creep behaviour is made by using an upper bound technique and a self-consistent method. Creep on a crystallographic slip system is assumed to occur in proportion to the resolved shear stress to a power. For the identical systems of an f.c.c. crystal the slip-rate on any system is taken as $\gamma =\alpha (\tau /\tau \_{0})^{n}$ where $\alpha $ is a reference strain-rate $\tau $ is the resolved shear stress and $\tau \_{0}$ is the reference shear stress. The tensile behaviour of a polycrystal of randomly orientated single crystals can be expressed as $\overline{\epsilon}=\alpha (\overline{\sigma}/\overline{\sigma}\_{0})^{n}$ where $\overline{\epsilon}$ and $\overline{\sigma}$ are the overall uniaxial strain-rate and stress and $\overline{\sigma}\_{0}$ is the uniaxial reference stress. The central result for an f.c.c. polycrystal in tension can be expressed as $\overline{\sigma}\_{0}=h(n)\tau \_{0}$. Calculated bounds to $h(n)$ coincide at one extreme $(n=\infty)$ with the Taylor result for rigid/perfectly plastic behaviour and at the other $(n=1)$ with the Voigt bound for linear viscoelastic behaviour. The self-consistent results, which are shown to be highly accurate for $n=1$, agree closely with the upper bound for $n\geq 3$. Two types of glide systems are considered for ionic crystals: A-systems, {110} $\langle 110\rangle $, with $\gamma =\alpha (\tau /\tau \_{\text{A}})^{n}$; and B-systems, {100} $\langle 110\rangle $, with $\gamma =\alpha (\tau /\tau \_{\text{B}})^{n}$. The upper bound to the tensile reference stress $\overline{\sigma}\_{0}$ is shown to have the simple form $\overline{\sigma}\_{0}\leq A(n)\tau \_{\text{A}}+B(n)\tau \_{\text{B}};A(n)$ and $B(n)$ are computed for the entire range of $n$, including the limit $n=\infty $. Self-consistent predictions are again in good agreement with the bounds for high $n$. Upper bounds in pure shear are also calculated for both f.c.c. and ionic polycrystals. These results, together with those for tension, provide a basis for assessing the most commonly used stress creep potentials. The simplest potential based on the single effective stress invariant is found to give a reasonably accurate characterization of multiaxial stress dependence.

The Deformation of Steep Surface Waves on Water. I. A Numerical Method of Computation

Abstract: Plunging breakers are beyond the reach of all known analytical approximations Previous numerical computations have succeeded only in integrating the equations of motion up to the instant when the surface becomes vertical In this paper we present a new method for following the time-history of space-periodic irrotational surface waves The only independent variables are the coordinates and velocity potential of marked particles at the free surface At each time-step an integral equation is solved for the new normal component of velocity The method is faster and more accurate than previous methods based on a two dimensional grid It has also the advantage that the marked particles become concentrated near regions of sharp curvature Viscosity and surface tension are both neglected The method is tested on a free, steady wave of finite amplitude, and is found to give excellent agreement with independent calculations based on Stokes’s series It is then applied to unsteady waves, produced by initially applying an asymmetric distribution of pressure to a symmetric, progres­sive wave The freely running wave then steepens and overturns It is demonstrated that the surface remains rounded till well after the over­turning takes place

Optical Studies of the 1.945 eV Vibronic Band in Diamond

Abstract: The 1945eV band is the dominant end product of annealing radiation damage in diamonds containing isolated substitutional nitrogen atoms In this paper uniaxial stress experiments are described which show the band to be a transition between A 1 (ground) and E (excited) electronic states of a trigonal centre, the E state being coupled predominantly to A 1 modes of vibration Annealing experiments suggest that the centre is either a substitutional nitrogen atom plus an interstitial nitrogen atom, or a substitutional nitrogen atom plus a vacancy Small differences between the luminescence and absorption bands are shown to be consistent with the double minima vibrational potential of the nitrogenvacancy model

Radiation from a moving mirror in two-dimensional space-time conformal anomaly

Abstract: The energy-momentum tensor is calculated in the two dimensional quantum theory of a massless scalar field influenced by the motion of a perfectly reflecting boundary (mirror). This simple model system evidently can provide insight into more sophisticated processes, such as particle production in cosmological models and exploding black holes. In spite of the conformally static nature of the problem, the vacuum expectation value of the tensor for an arbitrary mirror trajectory exhibits a non-vanishing radiation flux (which may be readily computed). The expectation value of the instantaneous energy flux is negative when the proper acceleration of the mirror is increasing, but the total energy radiated during a bounded mirror motion is positive. A uniformly accelerating mirror does not radiate; however, our quantization does not coincide with the treatment of that system as a 'static universe'. The calculation of the expectation value requires a regularization procedure of covariant separation of points (in products of field operators) along time-like geodesics; more naive methods do not yield the same answers. A striking example involving two mirrors clarifies the significance of the conformal anomaly.

Statistical Thermodynamics of Random Networks

Abstract: Polymeric networks exhibiting high elasticity consist, typically, of linear chains several hundred bonds in length joined at their ends to $\phi $-functional junctions $(\phi >2)$. For random unconstrained chains of this length, the density distribution of chain vectors $r$ is Gaussian in satisfactory approximation. The chains interpenetrate copiously in the network; the domain described by the set of $\phi $ junctions that are topological neighbours of a given junction encompasses many (20-100) spatial neighbours. A phantom network is expressly defined as a hypothetical one whose chains may move freely through one another; the chains act exclusively by introducing a force that is proportional to the distance between each pair of junctions so connected. The following results of James & Guth are rederived for a Gaussian phantom network using a simplified version of their procedure: (1) the mean values $\overline{r}$ of the individual chain vectors are linear functions of the tensor $\lambda $ of the principal extension ratios specifying the macroscopic strain, (2) fluctuations $\Delta r=r-\overline{r}$ about these mean values are Gaussian, and (3) the mean-square fluctuations depend only on the structure of the network and not on the strain. Additionally, we show (4) that the distribution of the average vectors $\overline{r}$ is Gaussian, and (5) that $\langle (\Delta r)^{2}\rangle =(2/\phi)\langle r^{2}\rangle \_{0}$, a result obtained previously by Graessley. It follows from (1) and (3) that the transformation of chain vectors $r$ of the phantom network is not affine in $\lambda $, and hence that junctions exchange neighbours with strain. In real networks, the mutual interpenetration of chains pendent at a given junction must obstruct this process of local rearrangement of junctions; the transformation of chain vectors may therefore be more nearly affine in $\lambda $, especially when the network is undiluted. The elastic free energy derived for a phantom network of any functionality and degree of imperfection reduces to $\Delta A\_{\text{e1}}=\frac{1}{2}\xi kT(I\_{1}-3)$, where $I\_{1}$ = trace $(\lambda ^{T}\lambda)$ is the first invariant of the strain and $\xi $ is the cycle rank of the network. If the fluctuations of junctions in a real network are suppressed for the reasons stated, the elastic free energy is $\Delta A\_{\text{e1}}^{\ast}=\xi (1-2/\phi)^{-1}\frac{1}{2}kT(I\_{1}-3)-(2\xi /\phi)^{-1}kT$ln $(V/V^{0})$, where $V$ and $V^{0}$ are the actual and reference volumes, respectively. The expected trend from $\Delta A\_{\text{e1}}^{\ast}$ to $\Delta A\_{\text{e1}}$ with dilution may account, qualitatively at least, for the effect of dilution on the stress-strain relation. A similar trend with extension may explain the familiar departure of the observed tension-elongation relation from theory.

Closed Orbits and the Regular Bound Spectrum

Abstract: The energy levels of systems whose classical motion is multiply periodic are accurately given by the quantum conditions of Einstein, Brillouin & Keller (E. B. K.). We transform the E. B. K. conditions into a representation of the spectrum in terms of a ‘topological sum’ involving only the closed classical orbits; the theory applies equally to separable and non-separable systems; stability parameters are not involved. Significant contributions come from complex closed orbits which however have real constants of the motion. Clustering of levels on different scales is demonstrated by smoothing the spectrum using the formal device, due to Balian & Bloch, of adding a variable imaginary part to the energy. The topological sum is shown to agree very well with exactly-computed spectra for circular and spherical potential wells with repulsive core.

The solution of Dirac’s equation in Kerr geometry

Abstract: Dirac’s equation for the electron in Kerr geometry is separated; and the general solution is expressed as a superposition of solutions derived from a purely radial and a purely angular equation.

A New Hierarchy of Korteweg-De Vries Equations

Abstract: We have found new hierarchies of Korteweg–de Vries and Boussinesq equations which have multiple soliton solutions. In contrast to the stan­dard hierarchy of K. de V. equations found by Lax, these equations do not appear to fit the present inverse formalism or possess the various pro­perties associated with it such as Backlund transformations. The most interesting of the new K. de V. equations is ( u nx ≡ ∂ n u /∂ x n ) ( u 4 x + 30 uu 2 x + 60 u 3 ) x + u t = 0. We have proved that this equation has N -soliton solutions but we have been able to find only two soliton solutions for the rest of this hierarchy. The above equation has higher conservation laws of rank 3, 4, 6 and 7 but none of rank 2, 5 and 8 and hence it would seem that an unusual series of conservation laws exists with every third one missing. Apart from the Boussinesq equation itself, which has N -soliton solutions, ( u xx + 6 u 2 ) xx + u xx – u tt = 0 we have found only two-soliton solutions to the rest of this second class. The new equations have bounded oscillating solutions which do not occur for the K. de V. equation itself.

Solid particle erosion of metals: the removal of surface material by spherical projectiles

Abstract: Experiments are described in which steel spheres were projected obliquely onto mild steel targets. It is shown that this successfully simulates a class of impact occurring during the erosion of metals by dust and sand particles. The dependence of the crater dimensions on impact angle and velocity is determined and, using high speed photography, the energy balance in the impact process is studied. A model of crater formation is proposed which accurately predicts the volume of material displaced and the energy lost by an impacting sphere. It is found that metal becomes detached along a band of intense subsurface shear; calculation shows that this is associated with the production of local high temperatures. The data and analysis presented provide a basis for assessing the role of the ploughing component of deformation in erosion.

The phase problem

Abstract: The paper discusses the use of the theory of entire functions for solving the phase problem. In all practical cases only three forms of logarithmic Hilbert transform could possibly be required. The paper defines them and analyses their applicability. A generating form is also put forward for cases of possible theoretical interest. The uniqueness of the phase obtained from a logarithmic Hilbert transform is investigated and the difficulties due to the presence of zeros in the complex plane are discussed. Methods are put forward for both the removal of the zeros and, when this is not possible, for locating them in order to include their effect. The paper analyses known experimental methods for phase determination from the point of view of the theory presented and highlights their unique character.

The Kinetic Friction of Ice

Abstract: An apparatus based on a pendulum hanging around a revolving drum of ice was developed to measure the kinetic friction between a slider and an ice surface under conditions commonly experienced in ice skating (temperatures from -15 to -1°C and velocities from 0.2 to 10m s -1 ). The results are explained by a quantitative development of the frictional heating theory of Bowden & Hughes (1939): heat produced by friction raises the surface to its melting point and a small amount of water is produced which lubricates the contact area. The frictional heat used in melting is usually small; most of the heat flows from the contact area at the melting point into the slider and into the ice. This makes it possible to calculate the dependence of the coefficient of friction μ on the thermal conductivity of the slider, the ambient temperature and the velocity of sliding v , without considering the detailed mechanism that produces the frictional force. For sliders of mild steel and Perspex the main heat loss is through the ice and μ is hence proportional to the temperature below the melting point and to v -½ . For these two materials the magnitude of the coefficient of friction is correctly calculated from measured and known parameters to within a factor of 2. The remaining discrepancy is probably mainly due to the difference between the real and apparent contact areas. For a copper slider the heat loss through the metal is about the same as that through the ice. There is no pressure melting in these experiments; the only effect of the lowering of the melting point by pressure is to reduce slightly the frictional heat needed to keep the contact area at the melting point. On the other hand, at temperatures above about -2°C pressure melting would be expected.

13.—The Functional Differential Equation y′(x) = ay(λx) + by(x)

Abstract: The paper discusses the asymptotic behaviour of solutions of the functional differential equation where a is a complex constant, 0 b is a constant such that Re b = 0, but b ≠ 0.

The mechanics of rubber elasticity

Abstract: The mechanical properties of a rubber may conveniently be represented in terms of the strain energy $W$, which in Rivlin's notation is expressed as a function of the strain invariants $I\_{1}$ and $I\_{2}$. The experiments of Rivlin & Saunders indicated that $\partial W/\partial I\_{1}$ was approximately constant while $\partial W/\partial I\_{2}$ varied with $I\_{2}$, in contrast to the Mooney equation, according to which both $\partial W/\partial I\_{1}$ and $\partial W/\partial I\_{2}$ are constant. More recently it has been proposed by Valanis & Landel that $W$ may be expressible in terms of separate functions $w(\lambda \_{i})$ of the principal extension ratios $\lambda \_{i}$. This hypothesis appears to be borne out experimentally, and gives promise of a more accurate analysis of experimental measurements. Application to recent data enables $w(\lambda \_{i})$ to be expressed as an explicit algebraic function, from which it appears that the earlier conclusions of Rivlin & Saunders require some modification in detail.

Computer simulations of polyatomic molecules - I. Monte Carlo studies of hard diatomics

Abstract: The Monte Carlo method has been used to study a model system of 256 hard diatomic molecules, each consisting of two fused hard spheres of diameter σ with centres separated by reduced distance L = L/σ of 0.2, 0.4 and 0.6, at densities typical of the liquid state. The orientational structure of dense, hard diatomic fluids has been studied by calculating up to sixteen terms in the expansion of the total pair correlation function, g ( r 12 , ω 1 , ω 2 ), in spherical harmonics. The coefficients g u9m ( r 12) the series have been calculated as ensemble averages in the simulation. At short distances, the system exhibits a high degree of angular correlation, which increases with increasing density and elongation; however, this correlation is relatively short ranged at all densities and elongations, and in no case is there significant angular structure at distances greater than twice the major diameter of the molecule. In the nearest neighbour shell there is a strong preference for 9T-shaped’ pair orientations. At low elongations and densities the spherical harmonic coefficients are in close agreement with those predicted both by the ‘blip function’ theory and the solution of the Percus-Yevick equation for hard diatomics. The harmonic series for the total pair correlation function, is rapidly convergent at distances greater than L + σ , but slowly convergent at smaller distances. The results are suitable for use as a non-spherical reference system for perturbation calculations.

Directed Fluid Sheets

Abstract: This paper is concerned mainly with incompressible inviscid fluid sheets but the incompressible linearly viscous fluid sheet is also considered. Our development is based on a direct formulation using the two dimensional theory of directed media called Cosserat surfaces. The first part of the paper deals with the formulation of appropriate nonlinear equations (which may include the effects of gravity and surface tension) governing the two dimensional motion of incompressible inviscid media for two categories, namely those (a) for two dimensional flows confined to a plane perpendicular to a specified direction and (b) for propagation of fairly long waves in a stream of variable initial depth. The latter development is a generalization of an earlier direct formulation of a theory of water waves when the fixed bottom of the stream is level (Green, Laws & Naghdi 1974). In the second part of the paper, special attention is given to a demonstration of the relevance and applicability of the present direct formulation to a variety of two dimensional problems of inviscid fluid sheets. These include, among others, the steady motion of a class of two-dimensional flows in a stream of finite depth in which the bed of the stream may change from one constant level to another, the related problem of hydraulic jumps, and a class of exact solutions which characterize the main features of the time-dependent free surface flows in the three dimensional theory of incompressible inviscid fluids.

The double sheath associated with a hot cathode

Abstract: A theoretical treatment of the double sheath in a hot cathode low pressure discharge is presented. The current density of thermionically produced electrons is found to have an upper limit. The energy of ions entering the sheath from the plasma region is calculated, and the spatial structure of the double sheath is computed.

Grain boundary cavitation under various states of applied stress

Abstract: Submicrometre grain boundary cavities are produced in Nimonic 80A when plastic deformation in any of three different stress states is followed by a short anneal. Tension, torsion and compression specimens were plastically strained in a systematic manner and then annealed for 2 h at 750 °C. Detailed quantitative observations with a 1 MV microscope showed that the number of cavities per unit volume was a function of the shear strain and independent of the stress state. Furthermore the measurements revealed the surprising result that most cavities were on those grain boundaries which were parallel to the maximum principal stress axis. However, Preferential cavity growth occurred during subsequent tensile creep and cavities on these parallel boundaries either remained constant in size or diminished while those on boundaries which were orthogonal to the applied stress axis grew relatively quickly, thus producing the usual appearance of cavitated tensile samples. Plastic strain was more detrimental to torsional creep ductility when the direction of torque between plastic deformation and creep was reversed which is in accordance with the anisotropic cavitated boundary distribution. The results are compatible with the hypothesis that cavities are produced by grain interior slip and stabilized by plastic deformation induced internal tensile stresses.

Backlund Transformations for the Sine-Gordon Equations

Abstract: The generalized sine-Gordon equations $z\_{,xt}=F(z)$ in two independent variables $x$, mi include the sine-Gordon $z\_{,xt}=$sin z and the multiple sine-Gordon's like $z\_{,xt}$ = sin $z+\frac{1}{2}$ sin $\frac{1}{2}z$. Among other physical applications all these sine-Gordon's are significant to the theory of intense ultra-short optical pulse propagation. The sine-Gordon itself has analytical multisoliton solutions. It also has an infinity of polynomial conserved densities and has auto-Backlund transformations which generate a second solution of the sine-Gordon from a first solution - particularly from the solution $z\equiv 0$. We prove first that the generalized multi-dimensional sine-Gordon in two or more space variables $x^{1},x^{2}$... has no auto-Backlund transformations. Next we prove that the generalized sine-Gordon's $z\_{,xt}=F(z)$ and $z\_{,xt}^{\prime}=G(z^{\prime})$ have an invertible Backlund transformation between solutions $z$ and $z^{\prime}$ if and only if $F$ and $G$ are solutions of $\ddot{F}=\alpha ^{2}F,\ddot{G}=\beta ^{2}G$ where, in general, $\beta =\alpha h^{-1},\alpha $ is a complex number and $h^{2}( eq 0)$ is real. In case $h$ = 1 and $F$ and $G$ are the same function $z\_{,xt}=F(z)$ has an auto-Backlund transformation if and only if $\ddot{F}=\alpha ^{2}F$. We exhibit the B.ts and a.B.ts in these cases as well as the other B.ts for the generalized sine-Gordon. We conclude that the multiple sine-Gordon's do not have a.B.ts and infer that, despite the soliton character of the numerical solutions, the multiple sine-Gordon's are not soluble by present simplest formulations of the two by two inverse scattering method.

Some Quantum Field Theory Aspects of the Superspace Quantization of General Relativity

Abstract: An investigation is started into a possible mathematical structure of the Wheeler-DeWitt superspace quantization of general relativity. The emphasis is placed throughout on quantum field theory aspects of the problem and topics discussed include canonical commutation relations in a triad basis, the status of the constraint equation and the role played by perturbation theory.

On the Nonlinear Transfer of Energy in the Peak of a Gravity-Wave Spectrum: A Simplified Model

Abstract: An equation given by Davey & Stewartson (1974) for the evolution of wave packets in three dimensions is employed to discuss the resonant transfer of energy within the peak of a narrow spectrum of gravity waves. It is shown that the coupling coefficient G ( k 1 , k 2 , k 3 , k 4 ) between four nearly equal wavenumbers k 1 ,..., k 4 is not zero (as had been speculated) but is equal to 4π. This implies that the exchange of energy within the peak itself is of dominant importance, and leads to a simplified discussion of the energy transfer.

The absolute photoionization cross sections of helium, neon, argon and krypton in the extreme vacuum ultraviolet region of the spectrum

Abstract: An experiment has been set up at the Daresbury Synchrotron Radiation Facility to make absolute absorption cross section measurements over a wide range of photon energies. New data are reported for helium, neon, argon and krypton over the range 340-40 A which are believed to be reliable to ± 5%. A critical evaluation of published cross section data has been carried out to produce best value data from the ionization thresholds throughout the vacuum ultraviolet and X-ray region. Agreement with theoretical calculations on helium is demonstrated to be within ± 2-3% from threshold down to the double ionization threshold at 79 eV. Comparison with recent calculations of photoionization cross sections has shown that the effect of electron correlations is significant for the heavier inert gases. Contrary to previous claims, the position of the M shell maximum in krypton is located at 184 + 10 eV in good agreement with r. p. a. e. calculations. Oscillator strength sum rules have been examined and their moments calculated. Discrepancies developing towards the heavier inert gases suggests a decrease in polarizabilities and other atomic factors from those predicted by Hartree-Fock calculations.

The Momentum of Light in a Refracting Medium

Abstract: It is shown that neither Minkowski's result, according to which the ratio of momentum to energy for a light wave in a medium of refractive index n is n/c, nor that of Abraham, who found 1/nc, is correct. For a broad wave in a uniform medium, the correct answer is given by (2.12) with $\sigma =\frac{1}{5}$. For weak refraction it is approximately equal to the average of the Abraham and Minkowski results. Abraham's formula gives correctly the part of the momentum which resides in the electromagnetic field, but not the mechanical momentum of the medium which travels with the light pulse. Minkowski's formula gives the pseudo-momentum, a quantity of physical interest. The momentum change upon reflexion or transmission usually involves also acoustic transients, these are discussed for some simple cases.

Spin-Orbit Coupling and the Intersection of Potential Energy Surfaces in Polyatomic Molecules

Abstract: Longuet-Higgins’ theorem, which shows that the existence of intersections between potential energy surfaces may be deduced from the behaviour of the wavefunction at points remote from the intersection, is generalized to cover cases where the Hamiltonian is complex. It is concluded that an intersection due to symmetry in one region of nuclear configuration space may imply that the same surfaces intersect over a region of higher dimen­sion and lower symmetry where their wavefunctions belong to the same symmetry species. It is shown that this behaviour occurs in d 1 octahedral complexes in the presence of spin-orbit coupling.

3.—A Criterion for the Existence of Solutions of Non-linear Elliptic Boundary Value Problems

Abstract: By a new method it is proved that a non-linear elliptic boundary value problem of rather general type admits a weak solution lying between a given weak lower solution ϕ and a given weak upper solution ψ≧ϕ

Rotary ‘aether drag’

Abstract: Following observations of the transverse linear Fresnel ‘aether drag’ (Jones 1972, 1975) it was conjectured that there should be a corresponding angular drag on the plane of polarization of a beam of light traversing a medium rotating about an axis parallel to the direction of the beam. This conjecture was confirmed by Pryce, who found that the angular drag per unit path length should be, to the first order, w ( n - n -1 )/ c , where w is the angular speed of rotation and n the refractive index of the medium. This expression had previously been obtained by Fermi. It has been modified by Player to w ( n g — n ϕ -1 )/ c by considering the effects of dispersion, where n g is the ratio of the group velocities in vacuo and in the medium and n ϕ is the usual phase refractive index. To detect the effect a laser polarimeter has been developed with a noise level of about 3 x 10 -8 rad for 1 s response time. The paper describes the polarimeter and the precautions that have to be taken in order to detect the rotational aether drag in a specimen rod of Schott SF 57 glass, 100 mm long and 20 mm diameter, rotating at speeds of the order 6000 rev/min. The observations appear to support the Player formula rather than others based on phase refractive index or group refractive index alone.

Outline of the Geochronology and Tectonic Units of the Basement Complex of Northeast Africa

Abstract: Results of new geological mapping with the help of air and satellite photography in Sudan together with information from adjacent territories has enabled a map to be drawn showing the dominant basement tectonic trends in a previously geologically unknown area. Over 100 age determinations, including 25 unpublished analyses, allow the recognition of Eburnian age events in Central Africa Republic and southeast Libya similar to the 1950 million year (Ma) old Ruwenzori Belt in Uganda and similar events in Zaire. A northeast trending fold belt is recognized in Central Africa, western Sudan and southeast Egypt in which 1000 Ma ages are found. The Pan African age Mozambique belt truncates older structures in eastern Uganda and southern Sudan but is covered by a greenschist volcanic assemblage along the Red Sea coast in which 550 $\pm $ 150 Ma old granites and regional metamorphism occur.

Some unsolved problems of unknown depth about operators on Hilbert space

Abstract: The paper presents a list of unsolved problems about operators on Hilbert space, accompanied by just enough definitions and general discussion to set the problems in a reasonable context. The subjects are: quasitriangular matrices, the resemblances between normal and Toeplitz operators, dilation theory, the algebra of shifts, some special invariant subspaces, the category (in the sense of Baire) of the set of non-cyclicoperators, non-commutative(i.e. operator) approximation theory, infinitary operators, and the possibility of attacking invariance problems by compactness or convexity arguments.

Computer simulation of the screw dislocation motion in b.c.c. metals under the effect of the external shear and uniaxial stresses

Abstract: An atomistic study of the motion of the 1/2 [111] screw dislocation was carried out for a shear stress applied on {112} planes and for uniaxial stresses along [012], [001] and [111]. Central force interactions described by the three different empirical potentials used in the previous work (Duesbery, Vitek & Bowen 1973) were assumed. The distortions of the core and the subsequent dislocation motion always reflected the twinning-antitwinning asymmetry of shear on {112} planes. The non-shear components of the stress tensor introduced further asymmetries which vary with interatomic forces. The application of the results of this study to the theory of slip and twinning in b. c. c. metals, is discussed.

Centrifugal Instability of a Stokes Layer: Linear Theory

Abstract: The instability of the flow induced by a circular cylinder oscillating in an infinite viscous fluid is investigated. The flow is shown to be unstable to a Taylor vortex mode of instability. A series solution of the partial differential system governing the stability of the flow is obtained. The method used has several advantages over the numerical methods used by different authors for related problems. The instability predicted by the theory leads to a flow with no mean velocity component tangential to the cylinders. The disturbance velocity field decays exponentially at the edge of the Stokes layer. The theoretical results are qualitatively confirmed by an experimental investigation of the problem.