M.P. Stallybrass

A crack perpendicular to an elastic half-plane

Abstract An exact solution is obtained for an elastic half-plane containing a crack perpendicular to the free surface, when the faces of the crack are subjected to a particular, but rather general, distribution of pressure. Expressions are given for the stress intensity factor, the strain energy of the system and the normal displacement of the open end of the crack. A relation is obtained between this displacement and the stress factor. The solution is based on an application of the Wiener-Hopf technique to a governing integral equation. This integral equation can also be used as a basis for computing the stress and displacement fields for an arbitrary distribution of crack pressure.

On the concentrated loading of certain elastic half-space problems and related external crack problems. A new approach

Abstract Using a new integral equation connecting different types of fundamental solutions, a variety of external crack problems are solved when the crack is situated in an infinite space and occupies the region exterior to a circle, with the loading on the crack faces being of a concentrated nature. As a consequence of a particular integral representation we are also able to discuss certain “punch” problems for an elastic half-space when the punch is rigid and frictionless, with a circular cross section, but with an arbitrary base profile

A semi-infinite crack perpendicular to the surface of an elastic half-plane

Abstract A solution is obtained for an elastic half-plane containing a semi-infinite crack perpendicular to its free surface. The crack is opened symmetrically either by a pressure distribution applied to its faces or by means of remote loading. Expressions are given for the stress intensity factor. A comparison is made between the exact solution for a particular remote loading and the result obtained from Neubers approximate method. The solution is based on an application of the Wiener-Hopf technique.

Some comments on external crack problems in 2-dimensional elastostatics

Abstract The problem of two collinear external line cracks in an infinite 2-dimensional elastic medium is discussed when a normal distribution of pressure is applied to the faces of the cracks and when there is no loading at infinity. Problems of this type have been the source of some confusion. Reference is made to the formulation and solution of similar crack problems subjected to different types of surface loading.

A class of exact solutions of the navier-stokes equations. Plane unsteady flow

Abstract We indicate a class of solutions of the Navier-Stokes equations, representing plane unsteady motions having non-uniform vorticity.

Drop induced impact response of a printed wiring board

Abstract Analytical solution is given to the Duffing equation in its free response induced by the initial velocity. Maximum deflection and acceleration are analyzed for its behavior in both hard and soft springs. The Duffing equation is derived for the transient response of a laminated printed wiring board (PWB) as a hard spring system. Effect of the system parameters on the nonlinear response is analyzed for the PWB. Results are generated to characterize the response behavior with respect to the modal parameters and structure design of PWB. It is found that the maximum deflection is almost linearly proportional to the initial velocity induced by the impact momentum.

Forced vertical vibration of a rigid elliptical disc on an elastic half-space

Abstract The forced vertical vibration of a rigid frictionless elliptical disc on the surface of an elastic half-space is considered. This mixed boundary value problem is reduced to a (two-dimensional) integral equation. An approximation is obtained for the displacement of the disc by using a variational procedure.

Elliptic integral solutions for fluid discharge rates from punctured horizontal cylindrical vessels

Abstract The problem of determining fluid discharge rates from the side (as opposed to the bottom) of punctured cylindrical and spherical vessels was recently addressed by Crowl1. Both vertical and horizontal cylindrical configurations were examined. Analytical solutions to the spherical and vertical cylindrical problems were obtained in rather straightforward fashion. However, the geometry is considerably more complex in the case of the horizontal cylindrical configuration. Crowl does not present an analytical solution to this more complicated problem, but employs numerical methods in order to integrate the governing differential material balance equation. The purpose of this communication is to present a direct analytical solution to the problem of side drainage from a punctured horizontal cylindrical vessel with the use of elliptic integrals, values of which are tabulated in numerous mathematical handbooks2–4. Because of their inherently greater accuracy, analytical solutions can serve as standards or benchmarks against which to check the reliability of numerical solutions and, in addition, are generally readily programmable on computers and/or spreadsheets.

Flow equations for parabolic and elliptical weirs

Abstract Steady‐state flow across parabolic and elliptical weirs is addressed. This situation may arise in the cases of actual flow through such cross‐sections, as in an overflow pipe in the side of a vessel, through saturator troughs as employed in the textile finishing industry, or across weirs of these shapes. An exact analytical solution is found in the parabolic case. Specifically, the volumetric flowrate of fluid across a parabolic weir is shown to be directly proportional to the square of the liquid crest height; this result may be useful in certain process or environmental control applications. The solution for the elliptical configuration is a generalization of that presented earlier by Stevens (1957) in his classical paper on circular weirs. Whether the cross‐section is elliptical or circular, the resulting solutions contain the same types of elliptic integrals.

A pointwise variational principle for mixed boundary value problems in time harmonic elastodynamics

Abstract Stationary functionals are constructed for the displacement field at a specified but arbitrary point , for certain classes of mixed boundary value problems associated with time harmonic linear elastodynamics. The utility of such variational principles is indicated for certain half-space problems with an elliptical line of separation in the boundary conditions.