Parallel Tangent Research Articles

Analysis of Shape Change of a Surface Crack during Stable Fatigue Growth

The multi-point procedure is developed to predict the shape change of a semi-elliptical surface crack during stable fatigue crack growth. 3-D stress intensity factors along a crack front are calculated using the simplified 3-D J-intergral. Crack growth rate coefficient in the Paris law is assumed to be constant along the crack growth. Crack growth rate is set to be the distance between the two parallel tangent lines on the two semi-elliptic crack fronts before and after crack growth.

The use of Gibbs free energy-composition diagrams in coherent solids

Thermodynamic equilibria in coherent two-phase binary alloys is significantly different from that in incoherent systems. For example, the field boundaries on a phase diagram do not coincide with the corresponding equilibrium phase compositions, making the latter dependent on the bulk alloy composition and invalidating the use of common tangent as a means of determining phase equilibrium. A new method for determination of phase boundaries and equilibrium compositions has been introduced. The method is based on the use of molar Gibbs free energy-composition diagrams and a parallel tangent construction. The effect of elastic coherency on the nucleation of a precipitate from a supersaturated matrix is also discussed.

Design and Analysis of Pulse Tests With Unequal Pulse and Shut-In Periods

Introduction Pulse testing is a relatively new well-testing procedure Pulse testing is a relatively new well-testing procedure in which a series of flow disturbances is generated at one well (the pulsing well) and the pressure is recorded in another well (the responding well), The flow disturbances at the pulsing well are generated by alternate periods of flow and shut in (or injection and shut in). At periods of flow and shut in (or injection and shut in). At the responding well, a sensitive differential pressure gauge records the pressure response and, consequently, the response amplitudes. The response amplitude (Fig. 1) is the pressure increment between the tangent to two consecutive valleys and the parallel tangent at the peak between them, or the pressure increment between the tangent to two consecutive peaks and the parallel tangent at the valley between them. The elapsed time between the end of a certain period and the point of tangency is called the time lag.The advantages of pulse tests are that the pressure response from a pulse test can be distinguished easily from other trends in the reservoir pressure and the pulse-test values are sensitive to the formation pulse-test values are sensitive to the formation properties between the two wells used in the test. properties between the two wells used in the test. The effects of different factors on the response of pulse tests have been reported. Studies have shown pulse tests have been reported. Studies have shown the response of pulse testing for a wide range of reservoir properties, when areal heterogeneities exist and for a two-zone reservoir. However, none of these papers supply enough information for an engineer to papers supply enough information for an engineer to design and analyze a pulse test. Only one paper shows how to design and analyze a pulse test; and even this paper is limited because it can be used only when the paper is limited because it can be used only when the pulse and the shut-in periods are equal, and it assumes pulse and the shut-in periods are equal, and it assumes that all the time lags are equal.In this paper, a graphical method and an analytical method that can be used to design and analyze pulse tests with unequal pulse and shut-in periods are presented. These methods allow the engineer to design presented. These methods allow the engineer to design and analyze pulse tests simply and accurately. These methods will enable the use of pulse tests without limitations on the length of the different periods. Thus, the optimum ratio between the pulse and shut-in periods that gives the maximum pulse-test response can be used. Theoretical Background To design and analyze any well test is simply to relate the test parameters to the reservoir and well properties. In the case of pulse testing, the test parameters are the pulse period, the shut-in period, the time lags, and pulse period, the shut-in period, the time lags, and the response amplitude. The reservoir properties are the formation permeability, porosity and thickness, the fluid viscosity, the total compressibility, and the distance between the pulsing and the responding wells.The reservoir properties and the test parameters can be used to define the following dimensionless groups. Pulse Ratio: Pulse Ratio:pulse periodR' =pulse period + shut-in period = = ..........(1) Dimensionless Cycle Period: = ....................(2) JPT P. 205

Optimum control of epidemics

This paper presents an application of a dynamic optimization procedure to the problem of finding the most economical use of active and passive immunization to control an infectious disease in a closed population. The mathematical model used is suitable and accurate for representing most such diseases, taking into consideration the latent and infectious periods and their statistical variations. Variables representing active and passive immunization controls are incorporated in the model and the resulting model includes functions of present and past values of both state and control variables. Conditions for cost optimization are obtained from the Minimum Principle and numerical solutions of the problem, obtained by the use of parallel tangent gradient techniques, are presented.

A generalization of parallelism in Riemannian geometry, the $C\sp{\omega }$ case

The concept of parallelism along a curve in a Riemannian manifold is generalized to parallelism along higher dimensional immersed submanifolds in such a way that the minimal immersions are self parallel and hence correspond to geodesics. Let g: N M be a (not necessarily isometric) immersion of Riemannian manifolds. Let G: T(N) T(M) be a tangent bundle isometry along g, that is, G covers g and maps fibers isometrically. By mimicing the construction used for isometric immersions, it is possible to define the mean curvature vector field of G. G is said to be parallel along g if this vector field vanishes identically. In particular, minimal immersions have parallel tangent maps. For curves, it is shown that the present definition reduces to the definition of Levi-Civita. The major effort is directed toward generalizations, in the real analytic case, of the two basic theorems for parallelism. On the one hand, the existence and uniqueness theorem for a geodesic in terms of data at a point extends to the well-known existence and uniqueness of a minimal immersion in terms of data along a codimension one submanifold. On the other hand, the existence and uniqueness theorem for a parallel unit vector field along a curve in terms of data at a point extends to a local existence and uniqueness theorem for a -parallel tangent bundle isometry in terms of mixed initial and partial data. Since both extensions depend on the Cartan-Kahler Theorem, a procedure is developed to handle both proofs in a uniform manner using fiber bundle techniques. 0. Introduction. The geodesics in a Riemannian manifold M, of dimension m, can be described in two independent ways: they are the critical points for the variational problem for minimal 1-dimensional area with fixed endpoints and they are also the auto-parallels, the curves whose tangent vector fields are parallel. More generally, the critical points for the variational problem for minimal p dimensional area, 1 _p <im, with fixed boundary are the minimal immersions. Received by the editors December 30, 1969. AMS 1969 subject classifications. Primary 5372, 5304, 5350, 5370, 5374; Secondary 3503, 3504, 3596, 5730.

Fermi Surface of Lead from Kohn Anomalies

The dispersion relations for phonons in lead determined by neutron spectrometry exhibit a large number of Kohn anomalies, which may all be related to the Fermi surface in a consistent manner by considering both electron transitions diametrically across the Fermi surface and nondiametral transitions between points with parallel tangent planes. Factors affecting the size and shape of anomalies are reviewed. The detailed interpretation of anomalies leads to a mapping of the Fermi surface, and the result may be compared with that of Anderson and Gold, who used the de Haas-van Alphen method. There is fair agreement, with significant particular differences. The sizes of anomalies have been interpreted in terms of a screened ion-electron interaction.

Über die Bestimmung der Anzahl dreiachsiger Ellipsoide aus der Anzahl ihrer Schnittellipsen in zufälligen Schnittebenen

A diagram is given for reading off the shape-factor\(\bar D\) oftri-axial ellipsoids (\(\bar D\) = average distance between parallel tangent planes over all orientations).

Parallel tangent deformation, concircular transformation and concurrent vector field

Parallel tangent deformation, concircular transformation and concurrent vector field