Finite Slope Research Articles

Indentation of a Membrane on an Elastic Half Space

An analysis is presented of the deformation of a homogeneous isotropic elastic half space whose surface behavior departs from the classical due to the presence of a pretensed membrane or a material surface which exhibits surface tension. Two related problems are considered: the axisymmetric displacements caused by a uniform circular load and by a concentrated normal force. In contrast to the classical solution, the uniform load problem with surface tension is found to have a finite slope and zero shear stress over the entire boundary. The analysis is used to interpret data from indentation tests on inflated dog lobes and the mechanical properties of both the lung parenchyma and the surface membrane are obtained. The observed shape of the indented surface agrees well with the prediction of the analysis.

Current transfer in multifilamentary superconductors. I. Theory

In short-specimen critical current measurements, inadequate separation between current and voltage contacts results in a finite linear slope in the measured voltage-current characteristic at low currents. A simple approximate analytic expression is developed for estimating the magnitude of this slope from wire parameters. Current-transfer lengths are evaluated for multifilamentary NbTi and Nb3Sn composite superconductors.

Effective stress finite element slope analysis : Byrne, P M Proc 29th Canadian Geotechnical Conference, Vancouver, 13–15 Oct 1976, P8.35–8.55

Effective stress finite element slope analysis : Byrne, P M Proc 29th Canadian Geotechnical Conference, Vancouver, 13–15 Oct 1976, P8.35–8.55

Convex Cubic Splines

Control curves used in industry are usually planar cubic splines, continuous and single-valued for a specific independent variable. If large slopes are specified at certain points, the spline coefficients, which are computed in order to preserve second derivative continuity of the spline, invariably lead to wildly oscillatory curves. Normally the large slopes occur at the ends of the curve, and this paper deals with such cases, developing methods which have been used successfully to combat the problems encountered. The end point where the difficulty occurs is ignored for the moment and a curve (spline or not) is designed for the remaining points. Then, with the point and slope adjacent to the difficult point known, the design is completed with one or more spline segments so that the resulting curve, which connects the difficult point to the next point, is convex (or concave), and has at least first derivative continuity. For the large finite slope case, two methods are described. The first method constructs the desired curve as a sequence of spline segments with finite slope everywhere, whereas the second expands the interval so that a single spline segment with infinite end slope passes through the difficult point with the required slope. The case of infinite end slope is also treated. In this and the preceding cases, second derivative continuity can be preserved at the juncture under certain conditions.

Electron-phonon interaction in EuSe from optical absorption

The optical absorption of EuSe near the edge from K = 1 to 10 5 cm -1 has been investigated as function of temperature in the paramagnetic region (27–300 K). The absorption edge can be described with Urbach's rule, using a typical energy of acoustic phonons, but shows a finite slope even without zero point zero point vibrations.

Shape of a streamline water table near inflow and outflow boundaries

A streamline water table is usually pictured to leave an inflow boundary perpendicular to the surface and to approach an outflow boundary tangential to the surface for vertical surfaces or for inclined surfaces with the porous medium ‘below’ the surface. The hodograph solution for the problem of flow through a vertical walled dam proves that theoretically, the water table streamline leaves the inflow boundary horizontally and approaches the outflow boundary vertically but shows that at a practical scale the water table meets the boundaries at finite nonzero slopes. The actual water table shape deviates considerably from the usual way that the water table is sketched. The cause of the difference lies in an intuitive but incorrect assumption that the slope of the water table does not change dramatically near the endpoints of the water table.Supplement is available with entire article on microfiche. Order from American Geophysical Union, 1909 K Street, N.W., Washington, D. C. 20006. Document W77‐002; $1.00. Payment must accompany order.

An inequality on the slope of the diffraction peak in elastic collisions of particles of arbitrary spins

We obtain inequalities linking the slope of the diffraction peak, the total cross-section and the elastic cross-section of the form d dt log dσ A dω t=0>R 1 18π (σ tot) 2 σ el , for elastic collisions of particles of arbitrary spins ( dσ A/d μ is the “absorptive” differential cross-section). In the limit of infinite energy we get R > 0.9965, to be compared to R = 1 for the scalar case. Results for finite energies and finite slopes are also obtained, such as R > 0.989 as soon as the logarithmic derivative with respect to cos θ of the absorptive differential cross-section is greater than 33.

Wave reflections from slotted tunnel portals

Various modifications to tunnel portals may be employed to reduce the sudden pressure jump which occurs as a train passes through the entrance or exit. Tunnel entrance gives rise to pressure waves which may travel down the length of the tunnel and reflect back to cause ear discomfort to the passengers. Various distortions are introduced to the shape of these waves if they reflect off a modified tunnel portal. A linearized analysis of a pressure impulse and a pressure step reflecting off a portal consisting of a long slotted section is presented. A closed-form solution is obtained. It is shown in both cases that the infinitely steep rise of the incident wave is reflected as a pressure rise of finite slope, and the pressure amplitude may exceed that of the input for some points in time. The apparent mass of flow through a narrow slot is calculated. It is shown that the magnitude of pressure waves produced by tunnel entry may be sufficiently large that significant non-linear acoustic effects may be present.

Exchange of Stabilities in Autonomous Systems—II. Vertical Bifurcation

In the theory of singularly perturbed initial‐value problems, the principal assumption concerns a certain Jacobian matrix: all its eigenvalues should have negative real parts at each point of the reduced (or degenerate) path. If the reduced path contains a point of bifurcation, this assumption is violated. The simplest kind of bifurcation with exchange of stabilities involves just two smooth curves intersecting at a single point. The analysis of the singular perturbation theory in the case when bifurcation is present depends on whether or not both curves have finite slopes at the point of bifurcation. The case when both slopes are finite was treated in [1]; the case when the bifurcating curve has a vertical tangent is treated in the present paper.

Spin glasses: beyond the molecular field approximation

The static properties of spin glasses are discussed within the Bethe-Peierls-Weiss approximation. The effect of a finite number of nearest neighbours is reflected in the expressions for the spin-glass transition temperature, the order parameter and the susceptibility. In contrast to the results of the classical molecular field approximation, the susceptibility approaches the transition temperature from below with a finite positive slope.

Finite element slope analysis by limiting equilibrium : Thesis. Figs, Tabls, Refs. M.S. THESIS, UNIV. ARIZONA, 1973

Finite element slope analysis by limiting equilibrium : Thesis. Figs, Tabls, Refs. M.S. THESIS, UNIV. ARIZONA, 1973

Thermal conductivity of niobium in the mixed state

The magnetic field dependence of the thermal conductivity of four niobium samples in the mixed state has been investigated at several temperatures and three different field orientations. The samples are of moderate purity, that is, l e ≈ ξ0. At low temperatures, just above H c1, the phonon conductivity can be easily separated. Good agreement is obtained with qualitative predictions of Canel and Vinen for the scattering of phonons by flux lines. At high temperatures both the phonons and the electrons are scattered by the flux lines. Though it is evident that the scattering mechanism is more effective in the case that H ⊥ ▽T than for H ∥ ▽T, only a qualitative discussion can be given. Near H c2 the finite slopes of the λ(H) curves are in qualitative agreement with the gapless superconductivity theory of Caroli and Cyrot, but the dependence on the electron mean free path can be better understood from the theory of pure type II superconductors of Houghton and Maki. Some aspects of pure type II superconductivity are also recognized in the anisotropic behavior near H c2.

Effects of pressure on the dielectric properties of and the vanishing of the ferroelectricity in SbSI

The effects of pressure and temperature on the dielectric properties and ferroelectric (FE) transition in SbSI were investigated. Among the important features of the results are (i) the observation of the vanishing of the FE state at 0 K and high pressure (> ∼ 9.5 kbar), and (ii) the observation of a new, pressure-induced effect in some samples of this material. The FE transition temperature Tc → 0 K with a finite slope dTc/dP, as we have observed for other displacive soft mode ferroelectrics. In these ferroelectrics the strictly harmonic frequency ω0 of the soft FE mode in the paraelectric phase is imaginary at 1 bar, and all of the stabilization of the mode frequency is provided by high-order anharmonicities. At sufficiently high pressure ω0 becomes real and the crystal becomes stable against the FE mode at all temperatures, i.e. the FE state vanishes. It is postulated that the new effect, which is manifested by a peak in the static susceptibility which grows rapidly with pressure, is associated with crystalline defects. Two other interesting aspects of the behavior of SbSI under high pressure are the possible existence of (i) a Curie critical (or tricritical) point, and (ii) a triple point. The available evidence for each is examined, and it is concluded that the question of the Curie critical point remains unsettled.

The Flow Stress of the Flux Line Lattice and the Magnetization Curve of Low‐x Type II Superconductors

AbstractIn low‐x type II superconductors there exists a long range attractive interaction between flux lines. This leads to a step B0 in the magnetization curve at Hc1. Experimental results frequently deviate from the ideal behaviour. The deviations are caused by the necessity to perform a plastic deformation of the intruding flux line lattice in order to accommodate the specimen geometry, and by flux line pinning. It is shown that both mechanisms lead to a finite slope of the magnetization curve near Hc1. The flow stress of the FLL and the volume pinning force may be obtained from magnetization measurements.

Low lying excitations in a pair-potential well

Recently a new method has been developed for solving the Bogoliubov equations in the absence of magnetic field. In this work we apply the method to calculate low-Lying bound states in a paramagnetic-superconducting contact. It is assumed that the pairpotential drops from its bulk value inside the superconductor to zero at the interface, with finite slope. Impurity scattering is neglected. No sharp energy gap is found in the limit of vanishing thickness of the paramagnetic normal layer contrary to former work. The influence of neglecting a small term in the kinetic energy investigated. It is shown that for bound states the correction to the energy due to this term in the first order perturbation theory is zero.

Pressure fields over hypersonic wing-bodies at moderate incidence

Delta wings with conically subsonic cones-bodies mounted on their compressive side are analysed in the hypersonic small disturbance regime. The weakly three-dimensional conditions associated with slender parabolic Mach cones are used to validate a linearized rotational approximation of the flow field. A combined integral–series representation is obtained for the pressure distribution between the wing-body and shock wave for arbitrary body cross-sections, and is specialized to give analytical formulae for arbitrary-order polynomial transversal contours. Numerical results are presented for wedge, parabolic and higher order cross-sections illustrating the dominant character of the cross-flow stagnation singularity associated with sharp wing-body intersections having a finite slope discontinuity. It is shown that the pressure has a logarithmic infinity at this secondary leading edge, as in corresponding Prandtl–Glauert irrotational flows. The relation of this finding to Lighthill's theorem on cross-stream vorticity is discussed. Other features of the pressure field are considered with particular emphasis on their relationship to a recently derived area rule for such configurations, and possibilities for favourable interference.

Dynamic Susceptibility of a Model Polymer in Several Limits-Theory and Calculations

We study the susceptibility of a model polymer having rotating dipolar sidechains with nearest neighbor interactions and diagonal visocosity. The susceptibility can be found exactly (1) at zero frequency for arbitrary interaction strength, and (2) at zero interaction strength for arbitrary frequency and viscosity. In addition, a high-temperature expansion exists for arbitrary frequency both in the viscous limit and for finite viscosity. We also study the time-dependent correlation function 〈 cosφi(t) cosφi+1(0)〉 for arbitrary viscosity and weak interaction, and the possibility of a more complex diagonal friction law. In no case are we able to produce a Cole-Cole plot which approaches the origin with finite slope at high frequencies, as does much experimental data.

Piecewise Linear Programming Solutions of Transportation Costs as Obtained from Rate Tariffs

Abstract Cost functions for shipping a product by common carrier (rail and truck) are found to differ from the classical transportation problem. These cost functions are piecewise linear rather than proportional to the number of miles and/or quantity shipped. Cost functions for shipping by rail are continuous, but the cost functions for shipping by truck have a discontinuity that is similar to the fixed-charge problem. By approximating the fixed-charge step with a finite slope, both types of problems can be solved with separable-convex programming. An example is given.

Tides in shallow water: Spectroscopy

Non-linear distortions of long waves (tides) in shallow water are analysod. The effects of finite wave amplitude, bottom slope, friction, and multi-chromatic interactions are investigated separately and in combination, using both perturbation analysis and numerical solutions of the full non-linear equations. The distortions caused by each source of non-linear behavior are presented in the time and frequency domains, and some implications of the results for local and ocean-wide tide predictions are discussed. DOI: 10.1111/j.2153-3490.1971.tb00581.x

Regge Pole Model with the Diffractive Correction. I

)/(drJ/dt)(rp-fn +n» and the cross section for the n± production with polarized photon are calculated. In the neighbourhood of forward direction, our results fit experiment quite well. pion photoproductionl )~3) are accumulated. The differential cross sections for the n+ photoproduction are extensively measured over a wide energy range and in particular at small four-momentum transfer. The n-/n+ ratio from deuterium as well as the cross section for the charged pion photoproduction with a polarized photon are also measured at intermediate energies. 4 ),5) Many attempts have been made to explain the experimental results on the n± photoproduction in terms of the conspiracy model,6),7) the background model}) the cut mechanism 9 ) and the vector-meson dominance model. lo ) However, we try to explain the above-mentioned experimental results using the evasive Regge pole model with the diffractive corrections, for the following reasons: Although the Regge pole model has succeeded in explaining the two­ body scattering phenomena in a very wide range at high energy, it is difficult to explain high energy phenomena consistently by a simple Regge pole model. ll ),12) Some of the evidences of the discrepancy between the Regge pole model and experimental results are summarized below. i) The first of these dfficulties is related to the Pomeranchuk trajectory. The one is related to the violation of the Froissart bound l3 ) in the multi-Regge model which uses the repeated exchange of the Pomeranchukon. The other is related to the finite slope of the Pomeranchuk trajectory.14) ii) The second is the necessity of the additional Regge poles including a conspirator to explain the polarization and the forward peak.15) iii) The third is the inconsistency between the cross-over phenomenon and the dip in other reactions. We demand the zero for residues of ()) trajectory at t = - 0.15 (Ge V / C)2 to explain the pp and pp cross-over phenomena. Factorization also predicts the dip at the same t value for the reactions dominated by ()) trajec­ tory such as np~pN and rp~nop, but no dip is seen experimentally.