Bodies Of Finite Size Research Articles

An expansion in power series of mutual potential for gravitating bodies with finite sizes

This investigation offers an expansion of mutual potential between two gravitating bodies of finite size in a power series of the relative coordinates of their centres of mass; the coefficients of this power series are dependent upon the generalised products of inertia of the individual bodies which are easily computable in comparison with the similar coefficients of the conventional harmonic expansion. Our coefficients require partial derivatives of different orders of the reciprocal of the distance between the two centres of mass which also can be computed easily from the recurrence relations derived in this work. Computed results pertaining to the derivatives up to seven orders are included in a table, while our Fortran program has the capability to carry out such computations to any order. The method has been derived without assuming any approximation and is valid universally without loss of any generality.

CORRECTIONS FOR CONDUCTIVITY ESTIMATES IN INDUCTION PROSPECTING OF SULPHIDE DYKES IN A LAYERED ENVIRONMENT*

ABSTRACTIn a weathered environment estimates of depth and conductance of metallic sulphide dykes from conventional anomaly index diagrams for a vertical half‐plane in air have to be corrected, besides the usual corrections, for:1. moderate conductivity of the host rocks, and2. finiteness of strike length S and depth extent D.Model experiments have been carried out to evaluate the response variation of a vertical planar conductor with varying depth extent and strike length for both insulating and conductive surroundings. The results indicate:1. A conductor with finite depth extent (D/L < 2.5) or strike length (S/L < 5.0) in an insulating medium yields a lower estimate of conductance (mineralization) and a greater depth.2. A moderately‐conductive host rock enhances the anomaly and rotates the phase so that the conductor appears to be more resistive (less mineralized) and shallower.The results have practical significance since in weathered surroundings a highly‐mineralized body of finite size could be missed, or misjudged, because of low estimates of conductivity and depth.

Equations of motion in fourth approximation

The equations of motion in fourth approximation for a system of massive bodies of finite size moving in the gravitational field of the system are obtained.

Stable growth of fracture in brittle aggregate materials

Fracture of concrete is analyzed by combining the resistance curve ( R-curve) approach with linearly elastic solutions for the energy release rate resulting from the quasi-static crack model of Wnuk, analogous to the D-BCS model of a stationary crack used in describing quasi-brittle fracture in metals. The R-curve, representing the crack length dependence of the energy consumed per unit fracture extension, is calculated using the concept of the energy separation rate associated with a finite crack growth steps. To simplify calculations, the tensile stress transmitted across the nonlinear zone ahead of the fracture front is assumed to be uniformly distributed over the entire nonlinear zone, even though in reality it must be a gradually declining stress resulting in strain-softening; and an infinite elastic medium loaded at infinity is assumed. These assumptions permit an easy solution with the help of Green's function for an infinite elastic medium. Application to bodies of finite size then requires assuming the nonlinear zone (fracture process zone) to be negligible with regard to specimen dimensions, crack length and ligament length. Even though this assumption is not always realistic, the end results, which are of practical importance, appear reasonable. The analysis leads to a nonlinear first-order ordinary differential equation for the R-curve, which is integrated numerically. The R-curves calculated in this manner can be closely fitted to data from previous fracture tests. Only two parameters, characterizing the initial and the final lengths of the nonlinear zone, need to be adjusted to test data.

On the influence of heat transfer and blowing on the structure of laminar hypersonic flow behind a body

The asymptotic solution of the Navier-Stokes equations in the problem of hypersonic flow at large distances behind a body of finite size is studied. The principal term of the expansion for the perturbation of the longitudinal velocity component in the region of the trace, decaying exponentially in the direction of the outer boundary of the trace, is derived. The stream function is assumed constant along the axis of the trace and, generally speaking, non-zero. The solution constructed, enables not only the effect of the resistance, but also of the heat and mass flux from the streamlined body to be taken into account. The second terms of the asymptotic expansions of the flow parameters are obtained for the whole region of perturbed motion.

Asymptotic theory of separation flows

A review is given of the results of theoretical investigations of the separation of laminar and turbulent boundary layers in an incompressible fluid obtained on the basis of matched asymptotic expansions valid at large Reynolds numbers (Re). The global picture of the separation flow behind a body of finite size as Re → ∞ is investigated.

Variational formulation of a contact problem for linearly elastic and physically nonlinear shallow shells

A variational formulation is proposed for the problem of thin shell interaction with a smooth absolutely rigid stamp without taking friction into account in the contact domain. The shell material can be linearly or nonlinearly elastic. (Elastic-plastic problem /1/ can be reduced to the latter case under certain assumptions). Application of the Lions—Stampachia method of variational inequalities reduces the task to the problem of minimizing a Lagrange functional in a set of allowable displacements. The existence and uniqueness of the solution are proved under definite assumptions about the properties of the strain diagram. Investigation of contact problems for finite size bodies by the Lions— Stampachia method of variational inequalities was executed in /2,3/. Problems on the bending of thin plates with unilaterial constraints were examined in /4/.

Numerical investigation of an incompressible viscous fluid flow about a body of finite size

THE PROBLEM of the flow, uniform at infinity, of a homogeneous incompressible viscous fluid about the exterior of a circular cylinder is considered. The method of splitting the physical factors is used. The results of systematic calculations are presented and a comparison is made with the calculations of other authors.

Using paired integral equations in certain problems involving the diffraction of waveguide modes by bodies of finite size

Using paired integral equations in certain problems involving the diffraction of waveguide modes by bodies of finite size

A constructive method, based on factorization, for solving boundary value problems of diffraction at bodies of finite size

A METHOD is outlined for reducing Wiener-Hopf integral equations to an infmite system of linear algebraic equations, which can be solved by a reduction method in which the approximations converge well. The method is applied to the solution of some boundary value problems of diffraction at obstacles of finite size.

The conservative “flow” method and the calculation of the flow of a viscous heat-conducting gas past a body of finite size

A METHOD for the numerical solution of gas dynamical problems, based on a difference approximation of the conservation laws, is presented. Examples of calculations of the flow past a body of finite size are given. This paper presents a numerical method, based on a difference approximation of the conservation laws, recorded for each cell of the difference grid. In the field variables the method generates an explicit difference scheme which, as is shown by calculations and studies of a model example, is conditionally stable and monotonic. From the method of construction, the scheme is conservative with respect to mass, the momentum components and the total energy. The method is used to calculate the flow past a sphere of a viscous, heat conducting gas at a given surface temperature. Various formulations of this problem have been considered in [1–3]. In [1] the “shortened” Navier—Stokes equations were solved by the method of integral ratios; in [2] a numerical solution of the complete Navier—Stokes equations was found by a finite-difference method. In [3] the method of integral ratios was also used for the complete system of Navier-Stokes equations.

Determination of penetration depth of a field into bodies of finite size

Determination of penetration depth of a field into bodies of finite size

Exact fields of charge and mass distributions in general relativity

Presented in this paper are exact solutions to Einstein’s field equations generated via static spherically symmetric mass and charge distributions. In the limit in which the mass density vanishes but the charge density does not, the mass seen by an observer at infinity does not vanish and the metric exterior to the source can be the Ressner-Nordstrom or Schwarzschild solution depending upon the charge distribution. This mass, generated by the energy density of the electromagnetic field, cannot be set equal to zero for a body of finite size. If the charge is concentrated in a thin shell and the charge and mass as seen by an observer at infinity are set equal to those of an electron, the radius of the shell is half the classical electron radius. Finally, we exhibit a class of solutions in which the red-shift (from a point in the body to infinity) is maximum at the surface rather than at the center.

Low stress fracture initiation at a stress concentration: Application of dislocation type models

In recent years the problem of fracture initiation at stress concentrations has been studied using the theory of continuous distributions of dislocations, thereby allowing the elastic-plastic characteristics of the material to be taken into account. Particular attention has been given to the behavior of an isolated defect in an infinite body, the condition for fracture initiation being the achievement of a critical displacement at the defect tip, and for low stress fractures it has been demonstrated that the fracture criterion reduces to that derived from energetic considerations by Griffith, Orowan and Irwin; it was also shown that the length of plastic zone at fracture is independent of the length of the defect and the applied stress. The general analysis of the present paper indicates that these two conclusions are also valid for bodies that contain distributions of defects, even though these may be very close to each other and thus also for the case of defects in bodies of finite size; two particular problems are discussed as illustrative examples of the general situation.

Multiple scattering

A theory of the multiple scattering of particles passing through a body of finite size is proposed. The quantal multiple-scattering equations are used. It is assumed that the centres of force are distributed randomly and N 0 a 2 λ ⪡ 1 where λ is the wave length of the incident particle, a the amplitude and N 0 the concentration of the centres of force. An explicit expression for the intensity of scattered particles outside and inside the body is obtained.

Ferromagnetic resonance line width in insulating materials

It has recently been recognized that the energy spectrum of spin waves in a ferromagnetic body of finite size is very different from the spectrum in an infinitely extended body. In a finite sample a degeneracy exists between the spin wave of wave vector k = 0 and a manifold of short wavelength spin waves. In the presence of irregularities in the magnetic lattice, such as are necessarily present in magnetic insulators having an inverted spinel structure, the degeneracy leads to a resonance line width comparable to those experimentally observed. This line width depends upon the magnetization of the material, the shape of the sample, the Curie temperature and the degree of ordering of the magnetic lattice.

The wave pattern of a doublet in a stream

The following paper is a study of the surface waves caused by a doublet in a uniform stream, and in particular the variation in the pattern with the velocity of the stream or the depth of the doublet. In most recent work on this subject attention has been directed more to the wave resistance, which can be evaluated with less difficulty than is involved in a detailed study of the waves; in fact, it would seem that it is not necessary for that purpose to know the surface elevation completely, but only certain significant terms at large distances from the disturbance. Recent experimental work has shown con­siderable agreement between theoretical expressions for wave resistance and results for ship models of simple form, and attempts have been made at a similar comparison for the surface elevation in the neighbourhood of the ship. In the latter respect it may be necessary to examine expressions for the surface elevation with more care, as they are not quite determinate; any suitable free disturbance may be superposed upon the forced waves. For instance, it is well known that in a frictionless liquid a possible solution is one which gives waves in advance as well as in the rear of the ship, and the practical solution is obtained by superposing free waves which annul those in advance, or by some equivalent artifice. This process is simple and definite for an ideal point disturbance, but for a body of finite size or a distributed disturbance the complete surface elevation in the neighbourhood of the body requires more careful specification as regards the local part due to each element. It had been intended to consider some expressions specially from this point of view, but as the matter stands at present it would entail a very great amount of numerical calculation, and the present paper is limited to a much simpler problem although also involving considerable computation. A horizontal doublet of given moment is at a depth f below the surface of a stream of velocity c ; the surface effect may be described as a local disturbance symmetrical fore and aft of the doublet together with waves to the rear. Two points are made in the following work.