Near-field Solution Research Articles

Dynamic fracture toughness of Homalite-100

Dynamic fracture toughness of Homalite-100 determined by T. Kobayashi and Dally are compared with those previously obtained by the authors where similarities in the two results for single-edged-notch specimens of various configurations are noted. Dynamic fracture toughness of Araldite B obtained by Kalthoff, Beinert and Winkler and those of Homalite-100 obtained by the authors are then compared and, again, similarities in the two results and, in particular, the scatters in experimental data for wedge-loaded DCB specimens of different sizes are found. All three teams of investigators used static near-field solution to compute the dynamic stress-intensity factors from recored dynamic isochromatics or dynamic caustics. Errors generated through this use of static near-field solutions, as well as through the use of larger isochromatic lobes, are thus discussed.

Unusual characteristics of free bending waves in thick plates with fluid loading

Exact, steady‐state solutions, representing straight‐crested free bending waves, were calculated for the infinite Timoshenko‐Mindlin plate with fluid loading. In general there are three different types waves, two of which are roughly similar to plate waves in vacuum, and a third wave which is unique to fluid‐loaded plates. The unusual behavior of this third free bending wave is the subject of this paper. Calculated solutions are presented as loci of complex free bending wave numbers for steel plates and plates of a softer material for different conditions of fluid loading. The principal finding of this paper is that the loci of free bending wave number, and the character of the waves, change drastically with fluid density. Behavior under one fluid loading condition thus cannot be easily extrapolated to predict behavior in a different fluid. As a result of this work, it is concluded that nearfield solutions to plate radiation problems are more complicated than previously thought. One result of particular interest is that free bending waves in steel plates with water on one side are qualitatively different from those occurring with water on both sides. The results therefore have a very practical value.

A Far-Field Matching Method for Transonic Computations

In solving a mixed-type (elliptic-hyperbolic) differential equation in an unbounded region, which is elliptic near infinity, some way must be found to transfer the boundary conditions at infinity to a finite artificial boundary in order to keep the discretized problem finite. The common example of this is transonic flow over an airfoil or wing with subsonic freestream. Here we present an approach which is in many ways analogous to the adaptive wind-tunnel wall concept. Iterative revision of a Dirichlet condition on the common or boundary of the near and far fields results in convergence to a far-field solution that matches the discretized near-field solution in potential and normal derivative across the matching boundary. The far-field equation is either a first-order (FO) Prandtl-Glauert, or a second-order (SO) Poisson-type approximation to the transonic equation. A parameter is easily calculated which gives a good estimate of the accuracy of the far-field solution in either case. Two-dimensional results are given showing the success of the method in reproducing the circulation and Cp for a lifting airfoil. Accurate solutions are given using far-field matching boundaries which are much closer to the airfoil than is permissible with Klunker-type far fields based on multipole expansions. The results are shown to be invariant with the location of the vortex representing the far-field circulation. Thus, we significantly reduce computer time by factors of 3 (FO) and 7 (SO) for mesh density and accuracy equivalent to those of a fixed asymptotic far-field representation. Nonlifting FO calculations for a three-dimensional rectangular wing similarly yield accurate results for a much reduced near field, cutting computer time by more than a factor of 2 in an unoptimized case where the minimum boundary size has not yet been established.

Initial-value problems for nonlinear diatomic chains

An analysis is made of the response of infinite diatomic chains to an initial velocity disturbance. An integral transform solution is obtained for the case of linear interaction force between neighboring particles. Restrictions on the form of the initial condition are indicated for a long-wavelength acoustic response. For nonlinear interactions between neighboring particles, a Korteweg-de Vries equation is obtained for the farfield response of each typical particle within a unit cell, with initial conditions for the Korteweg-de Vries equations obtained by matching to a near-field solution. Numerical solutions are obtained for the standingwave response to a spatially periodic initial condition of harmonic form, with solutions of the chain equations of motion compared to solutions of the Korteweg-de Vries equation. A close correspondence is shown between the two methods of solution for both particle velocity response and the discontinuous strains that occur in a unit cell of a general diatomic chain.

An approximate study of void expansion by ductility or creep

The growth of a spherical void in an infinite plastic matrix is studied. Based on conclusions by Rice and Tracey the analysis is directed at the behavior under high stress triaxiality, strain-hardening and creep being presently included. A symmetric near-field solution is continued into the remote field in an approximate manner. Implications are carefully studied and conclusions on the effects of hardening and creep are ventured.

Near field of scattering by a hollow semi-infinite cylinder and its application to sensor booms

The Wiener-Hopf technique is employed to derive the near-field solution for scattering by a semi-infinite conducting tube due to an axially incident plane wave. Numerical results are included and emphasis is given to the interpretation of the results in terms of the error in the measurement of the magnetic field due to the presence of the sensor boom. It is found that at interior resonance frequencies the measurement error can be as large as 100 percent in the immediate neighborhood of the end of the boom. Guidelines for reducing such errors are suggested.

Diffraction by a conducting wedge in the near field

The purpose of this paper is to derive a near-field asymptotic solution for the diffraction of waves by a perfectly conducting wedge. Thus, starting from the series solution for cylindrical-wave incidence and using an integral expression for products of the Bessel functions involved, the total field is represented as a geometrical-optics term plus a diffraction integral. Using an integral form for the field of a spherical wave, the solution for the cylindrical-wave excitation is extended to the spherical-wave case. The method of steepest descent is used to obtain an asymptotic expression for the diffraction integral in terms of Fresnel integrals. The accuracy of our expressions is established by comparison with previous results as well as the exact solutions.

Nearfield Solution for Spherical Waves of Finite Amplitude

It is desired to determine theoretically the shape of an input pulse at a small spherical source that will distort into a single N pulse in the farfield. This has been accomplished by considering the effects of nonlinearities and pulse distortion in the nearfield of a sphere whose diameter is small in comparison to the characteristic wavelength of the pulse. Two approaches have been used: (1) a simple model based on the method of characteristics for an inviscid non-heat-conducting medium, and (2) a more refined weak shock approximation, which includes the effects of viscosity and heat conduction. Results are presented in terms of the waveform as it propagates for a class of input pulses with variations in shape and amplitude. [This study has been sponsored by the Dynamics Loads Division, National Aeronautics and Space Administration, Langley Research Center, Hampton, Virginia.]

Appraisal of near-field solutions for a Hertzian dipole over a conducting half-space

Various approximations to the exact formal integral representations of the near-field and the impedance are considered. A new approximate form is obtained which appears to be valid even when the dipole is near a poorly conducting earth. Numerical comparisons with the exact integral formula verify some of the conjectures. The result shows that the ground has a significant influence on the input impedance of the dipole, particularly for low heights.

Non-linear wave propagation in a relaxing gas

An outline of the classical far-and near-field solutions for small-amplitude one-dimensional unsteady flows in a general inviscid relaxing gas is given. The structure of the complete flow field, including a non-linear near-frozen (high frequency) region at the front, is obtained by matching techniques when the relaxation time is ‘large’.If the energy in the relaxing mode is small compared with the total internal energy, the solution in the far field is, in general, more complex than that predicted by classical theory. In this case the rate process is not necessarily able to diffuse all convective steepening. An equation valid in this limit is derived and discussed. In particular, a sufficient condition for the flow to be shock-free is established. For an impulsively withdrawn piston it is shown that the solution is single-valued both within and downstream of the fan. Some useful similarity rules are pointed out.The corresponding formulation for two-dimensional steady flows is also noted in the small energy limit.

Wave propagation in an elastic solid with a line of discontinuity or finite crack

With the aid of integral transforms, a method is presented for solving the problem of scattering of plane harmonic compression and shear waves by a line of discontinuity or crack of finite width embedded in an elastic medium of infinite extent. When the incoming waves are applied in an arbitrary direction, the scattered-wave field may be determined by separating the crack-surface boundary conditions into functions even and odd with respect to the variable along the line crack. The problem is reduced to the evaluation of a system of coupled Fredholm integral equations with special emphasis placed on finding the near-field solution which consists of a knowledge of the detailed structure of the displacements and stresses in a small region around the crack vertex. Dynamic stress-intensity factors, the critical values of which govern the condition of crack propagation, are defined and found to be dependent on the incident wave length and Poisson’s ratio of the medium. At certain wave lengths, they are larger than those encountered under static loading. Such information is of particular importance in perdicting the fracture strength of structures subjected to oscillating loads.

Three-dimensional stresses around elliptical cracks in transversely isotropic solids

Potential solutions for transversely isotropic bodies containing flat elliptical cracks deformed by the application of surface pressure and/or displacement are given. In particular, the general character of the linear elastic solution near the periphery of the crack is determined such that local stresses and displacements can be expressed independently of uncertainties of both the crack geometry and magnitude of the applied load. Hence, the near-field solution remains valid for a surface of discontinuity of arbitrary shape. The familiar stress singularity of the order r − 1 2 is retained in the theory of transverse isotropy, but the intensity and detailed structure of the stresses in the proximity of the crack border differ from those of the isotropic case. The energy release rate or the force tending to spread the crack, associated with each of the three modes of crack extension, is related to the stress-intensity factor. This result and consideration of some specific examples imply that current fracture mechanics theories may be applied directly to transversely isotropic solids in many cases.

Series Expansion Prediction of the Farfield from the Nearfield

A series expansion method for determining the farfield from nearfield measurements of coherent and partially coherent sound fields is presented. Current investigations have generally been based on a Helmholtz integral or Green's-function approach, although one of the earliest methods for sinusoidal oscillation [J. Pachner, J. Acoust. Soc. Am. 28, 86–89 (1956)] was actually based on a series expansion of the nearfield solution in spherical wave-functions. By confining the measurements to a spherical surface, Pachner was able to evaluate the expansion coefficients through the orthogonality of the Legendre polynomials. However, in the method presented here, the expansion is truncated to include only the number of terms equal to the number of nearfield measuring points. In this way, the coefficients may be evaluated through a matrix inversion for nearfield measuring points at any location. A comparison of exact and predicted farfields is presented for some simple source distributions. The method has been extended for crosspower nearfield spectral measurements. [Work supported by the Office of Naval Research.]

Near-field solution for non-equilibrium transonic flow around a corner

Near-field solution for non-equilibrium transonic flow around a corner

Interactions between Sound Waves

Exact solutions for the interaction between two concentric cylindrical waves and between two concentric spherical waves are presented. The scattered-pressure amplitude in the far field is shown to be constant in the cylindrical case and to be proportional to r−1 lnr in the spherical case, where r is the distance to the source of primary waves. A near-field solution is derived for the scattered waves generated when two sharply defined, plane-wave beams of square cross section intersect at right angles. A comparison is made of the theory with recent experiments in which beams of circular cross section were used. It is concluded that if the scattered waves do exist, their amplitudes are at least 40 dB below those that are predicted by this theory. When a hard object is placed in the region of intersection, scattered waves are observed. This effect can be explained by the fact that with the addition of the hard object (a cylinder) the primary waves have components of the same symmetry. These components are the waves scattered from the primary beams by the object. Evidence is presented to show that these components, having the same symmetry, interact strongly in the volume surrounding the object.