Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1987"

Some Models for Rainfall Based on Stochastic Point Processes

Abstract: Stochastic models are discussed for the variation of rainfall intensity at a fixed point in space. First, models are analysed in which storm events arise in a Poisson process, each such event being associated with a period of rainfall of random duration and constant but random intensity. Total rainfall intensity is formed by adding the contributions from all storm events. Then similar but more complex models are studied in which storms arise in a Poisson process, each storm giving rise to a cluster of rain cells and each cell being associated with a random period of rain. The main properties of these models are determined analytically. Analysis of some hourly rainfall data from Denver, Colorado shows the clustered models to be much the more satisfactory.

Stability and Instability of Solitary Waves of Korteweg-de Vries Type

Abstract: Considered herein are the stability and instability properties of solitary-wave solutions of a general class of equations that arise as mathematical models for the unidirectional propagation of weakly nonlinear, dispersive long waves. Special cases for which our analysis is decisive include equations of the Korteweg-de Vries and Benjamin-Ono type. Necessary and sufficient conditions are formulated in terms of the linearized dispersion relation and the nonlinearity for the solitary waves to be stable.

Turbulent burning velocities: a general correlation in terms of straining rates

Abstract: All known experimental values of turbulent burning velocity have been scrutinized. These number 1650, a significant proportion of which at the higher turbulent Reynolds numbers we measured in a fan-stirred bomb. Dimensionless correlations which have a theoretical basis are presented. These are in terms of flame straining rates and the effective r.m.s. turbulent velocity, as well as the laminar burning velocity of the mixture. When a flame develops from an ignition source it is not initially exposed to the lower frequencies of the turbulent spectrum. As the kernel grows the flame is affected by ever-lower frequencies and the turbulent burning velocity increases towards a fully developed value. An experimental dimensionless power spectral density function is presented, and used to show how both effective r.m.s. turbulent velocity and flame straining rate develop in an explosion. The results are relevant to a variety of practical devices, including gasoline engines, as well as atmospheric explosions.

Neutrino billiards: time-reversal symmetry-breaking without magnetic fields

Abstract: A Dirac hamiltonian describing massless spin-half particles (‘neutrinos’) moving in the plane r = ( x, y ) under the action of a 4-scalar (not electric) potential V(r) is, in position representation, H ^ = − i h c σ ^ ⋅ ∇ + V ( r ) σ ^ z , , where σ = (σ x , σ y ) and σ z are the Pauli matrices; Ĥ acts on two-component column spinor wavefunctions ψ ( r ) = ( ψ 1 , ψ 2 ) and has eigen­values ћck n . Ĥ does not possess time-reversal symmetry ( T ). If V ( r ) describes a hard wall bounding a finite domain D (‘billiards’), this is equivalent to a novel boundary condition for ψ 2 / ψ 1 . T -breaking is interpreted semiclassically as a difference of π between the phases accumulated by waves travelling in opposite senses round closed geo­desics in D with odd numbers of reflections. The semiclassical (large- k ) asymptotics of the eigenvalue counting function (spectral staircase) N ( k ) are shown to have the ‘Weyl’ leading term Ak 2 /4π, where A is the area of D, but zero perimeter correction. The Dirac equation is transformed to an integral equation round the boundary of D, and forms the basis of a numerical method for computing the k n . When D is the unit disc, geodesics are integrable and the eigenvalues, which satisfy J l ( k n ) = J l +1 ( k n ), are (locally) Poisson-distributed. When D is an ‘Africa’ shape (cubic conformal map of the unit disc), the eigenvalues are (locally) distributed according to the statistics of the gaussian unitary ensemble of random-matrix theory, as predicted on the basis of T -breaking and lack of geometric symmetry.

Mechanics of matrix cracking in brittle-matrix fibre-reinforced composites

Abstract: Energy-balance calculations for a continuum model of cracking in a uniaxially fibre-reinforced composite having a brittle matrix are presen­ted. It is assumed that the fibres are strong enough to remain intact when the matrix cracks across the entire cross section of the composite. By equating the energy availability for the cracking of continuum and discrete fibre models it is shown how the crack boundary condition relating fibre stress to crack opening must be selected. It is confirmed that the Griffith fracture criterion is valid for matrix cracking in composites. By considering the energy balance of long cracks it is shown that the limiting value of the stress intensity factor is independent of crack length and that it predicts a matrix-cracking strain that is consistent with the known result. An improved numerical method is described for solving a crack problem arising from the study of the cracking of brittle-matrix composites. Numerical results of high accuracy are obtained, which show how the cracking stress is related to the size of a pre-existing defect. Of special significance is the prediction of the correct threshold stress (i.e. matrix­-cracking stress) below which matrix cracking is impossible no matter how large the pre-existing defect.

The Bakerian Lecture, 1987. Quantum chaology

Abstract: Bounded or driven classical systems often exhibit chaos (exponential instability that persists), but their quantum counterparts do not. Nevertheless, there are new regimes of quantum behaviour that emerge in the semiclassical limit and depend on whether the classical orbits are regular or chaotic, and this motivates the following definition. Definition . Quantum chaology is the study of semiclassical, but nonclassical, behaviour characteristic of systems whose classical motion exhibits chaos. This is illustrated by the statistics of energy levels. On scales comparable with the mean level spacing (of order h N for N freedoms), these fall into universality classes: for classically chaotic systems, the statistics are those of random matrices (real symmetric or complex hermitian, depending on the presence or absence of time-reversal symmetry); for classically regular ones, the statistics are Poisson. On larger scales (of order h , i. e. classically small but semiclassically large), universality breaks down. These phenomena are being explained by representing spectra in terms of classical closed orbits: universal spectral behaviour has its origin in very long orbits; non-universal behaviour depends only on short ones.

The wave structure of monochromatic electromagnetic radiation

Abstract: The paper considers a general field of electromagnetic waves of a single frequency and identifies the salient structurally stable features of the three-dimensional pattern of polarization. The approach is geometrical rather than analytical, and it differs from previous treatments of this kind by being applicable even when the constituent plane waves are travelling in all directions. Lines and surfaces exist where the electric or magnetic vibration ellipse is singular. The field is divided into right-handed and left-handed regions by \`T surfaces', the electric and magnetic T surfaces not being coincident. Lying in the T surfaces are \`L$^T$ lines' where the vibration is linear, and cutting through the T surfaces are `C$^T$ lines' where the vibration is circular. Both kinds of lines are surrounded by characteristic patterns of vibration ellipses, which provide a singularity index, $\pm$ 1 for L$^T$ and $\pm \frac{1}{2}$ for C$^T$. The analysis is applicable in a cavity, but a loss-free resonating cavity represents a highly degenerate case.

Large-time behaviour of solutions to hyperbolic–parabolic systems of conservation laws and applications

Abstract: We study the large-time behaviour of solutions to the initial value problem for hyperbolic-parabolic systems of conservation equations in one space dimension. It is proved that under suitable assumptions a unique solution exists for all time t ≧ 0, and converges to a given constant state at the rate t − ¼ as t → ∞. Moreover, it is proved that the solution approaches the superposition of the non-linear and linear diffusion waves constructed in terms of the self-similar solutions to the Burgers equation and the linear heat equation at the rate t − ½ +α, α > 0, as t →∞. The proof is essentially based on the fact that for t → ∞ the solution to the hyperbolic-parabolic system is well approximated by the solution to a semilinear uniformly parabolic system whose viscosity matrix is uniquely determined from the original system. The results obtained are applicable straightforwardly to the equations of viscous (or inviscid) heat-conductive fluids.

Quantum phase corrections from adiabatic iteration

Abstract: The phase change $\gamma$ acquired by a quantum state |$\psi$(t)$\rangle$ driven by a hamiltonian H$\_0$(t), which is taken slowly and smoothly round a cycle, is given by a sequence of approximants $\gamma^{(k)}$ obtained by a sequence of unitary transformations. The phase sequence is not a perturbation series in the adiabatic parameter $\epsilon$ because each $\gamma^{(k)}$ (except $\gamma^{(0)}$) contains $\epsilon$ to infinite order. For spin-$\frac{1}{2}$ systems the iteration can be described in terms of the geometry of parallel transport round loops C$\_k$ on the hamiltonian sphere. Non-adiabatic effects (transitions) must cause the sequence of $\gamma^{(k)}$ to diverge. For spin systems with analytic H$\_0$(t) this happens in a universal way: the loops C$\_k$ are sinusoidal spirals which shrink as $\epsilon^k$ until k $\sim \epsilon^{-1}$ and then grow as k!; the smallest loop has a size exp \{-1/$\epsilon$\}, comparable with the non-adiabaticity.

Non-autonomous equations related to polynomial two-dimensional systems

Abstract: Periodic solutions of certain one-dimensional non-autonomous differential equations are investigated (equation (1.4)); the independent variable is complex. The motivation, which is explained in the introductory section, is the connection with certain polynomial two-dimensional systems. Several classes of coefficients are considered; in each case the aim is to estimate the maximum number of periodic solutions into which a given solution can bifurcate under perturbation of the coefficients. In particular, we need to know when there is a full neighbourhood of periodic solutions. We give a number of sufficient conditions and investigate the implications for the corresponding two-dimensional systems.

Chaos and the dynamics of biological populations

Abstract: As first emphasized in the early 1970s, the nonlinearities that are inherent in simple models for the regulation of plant and animal populations can lead to chaotic dynamics. This review deals with a variety of instances where chaotic phenomena can arise, particularly in interactions between prey and predators (including hosts and pathogens, hosts and parasitic insects, and harvested populations). Some of the complications in disentangling deterministic chaos from environmental noise will be discussed. The combination of population biology with population genetics leads to an even richer assortment of nonlinear phenomena and to the suggestion that many genetic polymorphisms may vary cyclically or chaotically (rather than being steady, as usually is assumed implicitly). I argue that complex dynamics - including chaos - is likely to be pervasive in population biology and population genetics, even in seemingly simple situations. But superimposed environmental noise, in heterogeneous natural settings, will usually complicate the dynamics, making it unlikely that population data will exhibit elegant properties (such as universalities in period doubling) associated with the underlying maps. The existence of chaotic regimes of dynamical behaviour can, however, invalidate standard techniques for analysing population data to reveal density-dependent mechanisms; this, I believe, may currently be the most significant implication of dynamical chaos for population biology.

The crystal structure of deuterated benzene

Abstract: The crystal structure of deuterated benzene has been refined by single-crystal neutron diffraction analysis at 15 and 123 K. The unit-cell dimensions were also measured at 52.6 and 80 K, and the thermal-expansion coefficients at all four temperatures were calculated. The molecules have C$\_{3v}$ symmetry with a small chair-distortion from D$\_{6h}$, which is possibly significant for the carbon atoms and significant for the deuterium atoms. The mean observed bond lengths at 15 K \[123 K\] are C-C = 1.3972 (5) A \[1.3940 (9) A\] (1 A = 10.$^{-10}$ m = 10$^{-1}$ nm); C-D = 1.0864 (7) A \[1.0838 (10) A\]. When corrected for molecular libration, the corresponding rest values are 1.3980 A \[1.3985 A\]; 1.088 A \[1.088 A\]. Ab initio molecular orbital calculations at the MP-2/6-31G* level gave energy-optimized bond lengths of 1.395 and 1.087 A for the isolated molecule at rest, in agreement with the corrected values from the crystal structure within the experimental errors.

Singularities in the Transverse Fields of Electromagnetic Waves. I. Theory

Abstract: Electromagnetic waves generally contain three kinds of singularities called C lines, S surfaces and disclinations. These singularities are features of the transverse electric and transverse magnetic fields of the waves and all three kinds usually occur in any given wavefield. We show that in the case of nominally uniformly polarized waves, the simple line zeros predicted for interference fringes by scalar wave theory in fact have an underlying polarization structure consisting of two C lines and an S surface. In consequence, virtually all monochromatic electromagnetic waves contain polarization states ranging from right-hand circular, through linear to left-hand circular polarization. Singularities of the electric and magnetic fields are not generally coincident in space; in fact they can be separated by arbitrarily large distances. The separation of the electric and magnetic S surfaces means that there are regions where the transverse electric and transverse magnetic vectors counterrotate. C lines are probably the most significant of the singularities, since they are not only structural features of polarization, but also organize the time structure of electromagnetic waves. They play a crucial role in determining the topology of disclinations in paraxial wavefields. In pulsed electromagnetic waves all three singularities move through space. Their behaviour, including interactions between pairs of C lines, S surfaces or disclinations, which are likely to be frequent events in pulsed waves, is discussed.

Radiative gravitational fields in general relativity II. Asymptotic behaviour at future null infinity

Abstract: We prove that Penrose’s requirements for asymptotic simplicity are formally satisfied by the general metric, (1), which admits both post-Minkowskian and multipolar expansions, (2), which is stationary in the past and asymptotically Minkowskian in the past, (3), which admits harmonic coordinates, and (4), which is a solution of Einstein’s vacuum equations outside a spatially bounded region. The proof is based on the setting up, by using the method of a previous work (L. Blanchet & T. Damour ( Phil . Trans . R . Soc . Lond . A 320, 379-430 (1986))), of an improved algorithm that generates a metric equivalent to the general harmonic metric of that work but written in radiative coordinates, i. e. admitting an expansion in powers of r -1 for r → ∞ and t - r fixed. The arbitrary parameters of the construction are the radiative multipole moments in the sense of K. S. Thorne ( Rev . mod . Phys . 52, 299 (1980)).

R-Matrix Theory of Electron Scattering at Intermediate Energies

Abstract: A new R -matrix theory of electron-atom and electron-molecule scatter­ing at intermediate energies is described. In this theory both the outer valence electron of the target atom or molecule and the scattered electron are expanded in terms of a continuum R -matrix basis. This enables target eigenstates as well as pseudostates representing inelastic effects to be accurately represented in an internal region. In addition, a two-dimensional R -matrix propagator approach is developed that enables the internal region to be subdivided and highly excited target states that extend out to large distances to be treated. This new theory is then combined with the T -matrix energy averaging technique introduced earlier by Burke et al . (1981) to yield accurate cross sections at inter­mediate energies. The method is illustrated by applying it to the elastic s-wave scattering of electrons by atomic hydrogen from threshold to 60 eV.

Elasticity of particle assemblies as a measure of the surface energy of solids

Abstract: A theory is suggested to explain the elasticity of particle assemblies. It is shown that a regular cubic packing of spheres has an effective Young's modulus that depends on the contacts between individual particles. In particular, it is noted that the effective modulus depends on the interfacial attractive energy between the spheres, and thus provides a direct method for measuring the surface energy of solids. However, most particle assemblies are neither cubic nor regular. The problem is to describe the properties of these real systems in terms of the packing of the grains. Theoretically, it is shown that the modulus of a powder compact should vary as the fourth power of the particle packing fraction. This result has been verified experimentally and has been used to determine the surface energies of zirconia, titania, alumina, and silica powders. The measured values were sometimes much lower than expected from theoretical calculations of surface energy. Experiment has shown that such discrepancies result from contamination of the solid surfaces.

Conformal geodesics in general relativity

Abstract: Conformal geodesics, space-time curves which are related to conformal structures in a similar way as geodesics are related to metric structures, are discussed. \`Conformal normal coordinates', \`conformal Gauss systems' and their associated \`normal connections', \`normal frames' and `normal metrics' are introduced and used to study: (i) asymptotically simple solutions of Ric(g) = $\Lambda g$ near conformal infinity, (ii) asymptotically simple solutions of Ric(g) = 0 with a past null infinity, which can be represented as the future null cone of a point i$^-$, past time-like infinity. In the first case we define an $\infty$-parameter family of (physical) Gauss systems near conformal infinity, in the second case a ten-parameter family of (physical) Gauss systems covering a neighbourhood of i$^-$. The behaviour of physical geodesics can be analysed in a particularly simple way in these coordinate systems. Each of these systems allows an extremely simple transition from the conformal analysis to the physical description of space-time. For $\Lambda\eta_{00}$ < 0 (De-Sitter type solutions) all solutions are characterized in terms of the physical space-time by their data on past time-like infinity. For $\Lambda$ = 0 the conserved quantities of Newman and Penrose are characterized as the first non-trivial coefficient, given by the value of the rescaled Weyl tensor at i$^-$, in an expansion of the physical field in a Gauss system of the type considered before.

Crystal Growth from Solutions

Abstract: Gravity-related phenomena in crystal growth from solutions are reviewed in the first section of the paper. The second and third sections deal with solution growth experiments performed in space at high and low temperatures. The basis for conceiving these experiments is discussed and the experimental procedures, both for the space experiments and the reference experiments on the ground, are explained including the development of appropriate hardware. Analysis of the results obtained allows the identification of materials and techniques relevant for space conditions.

Modulation Theory Solution for Resonant Flow Over Topography

Abstract: The near-resonant flow of a stratified fluid over topography is considered in the weakly nonlinear, long-wave limit, this flow being governed by a forced Korteweg-de Vries equation. It is proved from the modulation equations for the Korteweg-de Vries equation, which apply away from the obstacle, that no steady state can form upstream of the obstacle. This has been noted from previous experimental and numerical studies. The solution upstream and downstream of the topography is constructed as a simple wave solution of the modulation equations. Based on similarities between the method by which this solution is found and the quarter plane problem for the Korteweg-de Vries equation, the solution to the quarter plane problem is found for the special case in which a positive constant is specified at x = 0.

Nonlinear Dynamics, Chaos and Complex Cardiac Arrhythmias

Abstract: Periodic stimulation of a nonlinear cardiac oscillator in vitro gives rise to complex dynamics that is well described by one-dimensional finite difference equations. As stimulation parameters are varied, a large number of different phase-locked and chaotic rhythms is observed. Similar rhythms can be observed in the intact human heart when there is interaction between two pacemaker sites. Simplified models are analyzed, which show some correspondence to clinical observations.

Singularities in the transverse fields of electromagnetic waves. II. Observations on the electric field

Abstract: Electromagnetic waves propagating in free space contain three kinds of singularities called C lines, S surfaces and disclinations. The paper describes observations of these singularities in two different monochromatic microwave fields. The observations confirm all the theoretically predicted properties of the singularities that could be tested. As expected, the singularities were found to be prominent structural features of the fields and in consequence to provide an economical means of characterizing their structure. A notable result is the observation of both right-hand and left-hand C lines in a field that is nominally uniformly left-hand circularly polarized. This is in agreement with the previous assertion that, in general, electromagnetic wavefields contain both right-hand and left-hand polarized regions.

A new set of composition variables for the representation of reactive-phase diagrams

Abstract: A set of transformed composition variables is introduced for the representation of reactive-phase diagrams. These variables are superior to mole fractions because in the new representation, the non-reactive limits are well defined, the equilibrium surfaces are tangent at azeotropic states, and the number of linearly independent transformed composition variables coincides with the number of independent variables that describe the chemical equilibrium problem. Some examples of reactive-phase diagrams, which emphasize these features, are given.

The Bakerian Lecture, 1986. Ships from Space

Abstract: Narrow V-shaped wakes extending some 20 km behind surface ships were first found on Synthetic Aperture Radar (SAR) images from SEASAT in 1978. The V-wake geometry differed strikingly from the traditional Kelvin wake geometry consisting of divergent and transverse wave components generated by a travelling pressure point. The SAR images can be accounted for in terms of Bragg scatter from relatively short waves generated by the surface vessel. An essential ingredient of this hypothesis is that the wave generation is by an intermittent rather than a steady point source. Optical images from a hand-held camera on a 1985 space shuttle mission revealed many V-like wakes behind surface ships. There is no Bragg scattering from the ocean surface at optical wavelengths, so an alternative hypothesis is called for. We can interpret the observed features in terms of sun glitter from the tilted facets of a Kelvin wake. An essential ingredient is the generation by complex sources rather than by a single point source. We regard the present study as a step towards the interpretation of many unexplained naturally occurring features at the edge of the Sun glitter.

Supersonic vortex rings

Abstract: Experiments by Phan & Stollery (In Proc . 14 th International Symposium on Shock Tubes and Waves , pp. 519-526 (1983)) indicated the presence of a supersonic vortex ring with an embedded rearward-facing shock in the unsteady flow that follows the emergence of a normal shock from a circular tube. The vortex flow is supersonic in the sense that the on-axis flow is supersonic in the frame of reference of the vortex ring. In the present work this flow phenomenon is studied by using differential interferometry and a detailed description of the flow-starting process is given. A simplified quasi-steady calculation is performed and a comparison is made with previously measured pressures in the flow field. These results are in agreement with the physical description of the flow development.

Infinitely many solutions for some nonlinear elliptic problems in symmetrical domains

Abstract: We consider the boundary value problemwhere Ω ⊂ ℝn is a bounded domain, n≧3, 2* = 2n/(n − 2) is the critical exponent for the Sobolev embedding and λ is a real positive parameter. We prove the existence of infinitely many solutions of (*) when Ω exhibits suitable symmetries.

Transversely Isotropic Elasticity Tensors

Abstract: A representation formula for the elasticity tensor of a linearly elastic, transversely isotropic material is obtained, depending on eight constants. If, besides rotations about the axis of symmetry, reflections with respect to planes through that axis are also regarded as admissible symmetry transformations for the material, it is shown that the number of constants reduces to six. It is also shown that, no matter whether reflections belong to the collection of admitted symmetry transformations or not, only five constants are needed for hyperelastic materials.

On Converging Shock Waves

Abstract: An approximate theory of shock dynamics is used to study the behaviour of converging cylindrical shocks. For cylindrical shocks with regular polygonal-shaped cross sections, exact solutions are found, showing that an original polygonal shape repeats at successive intervals with successive contractions in scale. In this sense, these shapes are stable, and the successive Mach numbers increase according to exactly the same formula as for a circular cylindrical shock. The behaviour for initial shock shapes close to these and the general tendency of perturbed circular shapes to become polygonal, not necessarily regular, is explored numerically. Further analytical results are provided for rectangular shapes. Comments are made on the interpretation of regular reflection in this theory and on converging shocks in three dimensions.

Quantum-Electrodynamic Level Shifts between Parallel Mirrors: Analysis

Abstract: Atoms inserted between parallel conducting planes (mirrors) separated by a distance L suffer level shifts that can be understood only through a careful quantum-electrodynamic calculation embracing both electrostatics and electromagnetic retardation. The basic theory is reformulated with a view to spectroscopic experiments now under way. The requisite mathematics is systematized and made more accessible; special attention is paid to the symmetry properties of the shifts; the asymptotically leading terms are given in full for small and for large L; the role of the characteristic hydrogenic degeneracies is explored, and is found to be surprisingly unimportant in almost all situations of potential interest. For small L, the shifts are dominated by essentially electrostatic effects, of order 1/L$^3$. For large enough L, all frequency shifts are dominated by energy shifts peculiar to excited states, and decreasing only as 1/L; a simple classical model helps to elucidate this effect, and a kind of resonant enhancement to which it can lead. The following paper applies the results specifically to Rydberg (high-n) states, which present some interesting problems of their own.

The measurement of stress in steels of varying microstructure by magnetoacoustic and Barkhausen emission

Abstract: Magnetoacoustic emission (MAE) and Barkhausen emission (BE) have been measured as a function of applied magnetic field and tensile stress from mild-steel samples in a wide range of heat treatments, to develop a technique to measure stress without prior knowledge of the microstructure. The results are supplemented by measurements of magnetic coercivity and mechanical hardness. MAE is found to decrease with increasing applied stress, whereas the variation of BE is more complicated. The amplitudes of both MAE and BE, as well as the coercivity and hardness are also found to depend on the microstructure to varying degrees. Thus in ferritic-pearlitic and ferritic-pearlitic-martensitic steel MAE is much more sensitive to stress than to changes in microstructure, whereas the sensitivity of BE to stress and microstructure is similar. Above 50 MPa MAE is also more sensitive to stress in ferrite containing cementite, whereas BE both lacks a monotonic dependence upon stress and is sensitive to microstructure. In martensite, however, there is no MAE, the BE increasing monotonically with stress. Tempered martensitic structures give a weak MAE signal that is more sensitive to tempering temperature than applied stress, whereas the BE increases with stress for tempers below 500°C and decreases above. The dependence of MAE and BE on magnetic field are discussed in terms of domain-wall nucleation and irreversible motion in ferrite at higher fields, and irreversible wall motion through martensite or pearlite at lower fields. The results imply that MAE can be used alone to measure stress provided the general form of the microstructure is known; otherwise BE can be used as an additional technique to resolve any ambiguity.

Stress analysis in machining with the use of sapphire tools

Abstract: Stress birefringence in sapphire tools has been used to determine the stress boundary conditions in machining. Steel and brass specimens were machined orthogonally at speeds of up to 75 m min$^{-1}$ at a maximum feed rate of 0.381 mm per revolution to study the effect of speed and feed rate on stress distributions. The shear-difference method was used to calculate the normal and shear stresses from isochromatics and isoclinics obtained experimentally. The normal stress was found to peak at the cutting edge and decrease exponentially to zero at the end of the contact length. The shear stress was either zero or very small at the edge, increasing to a maximum in the middle of the contact length and reducing to zero at the end of the contact. The peak normal stress was found to be approximately 1.5-2 times the average normal stress.