Two-dimensional Fredholm Integral Equations Research Articles

Dynamic response of elliptical footings

Dynamic response of an elliptical footing in frictionless contact with a homogeneous elastic half-space is considered. Both vertical and horizontal vibrations are treated. In the case of the vertical vibration, the mixed boundary value problem gives rise to a set of dual integral equations. For the horizontal vibration, we have a system of dual integral equations. The dual integral equations which are two dimensional in nature are reduced to two-dimensional Fredholm integral equations of the first kind. They are then recast in a suitable form after separating out the static solution. Successive low-frequency terms are then obtained by utilising the static solution. The series solutions up to ω2, ω being the frequency, are obtained, and analytical results for the dynamic compliances are obtained. In the limiting case of a circular footing, our results are in agreement with those of previous authors.

Diffraction of elastic waves by a penny-shaped crack: analytical and numerical results

The diffraction of time-harmonic stress waves by a penny-shaped crack in an infinite elastic solid is an important problem in fracture mechanics and in the theory of the ultrasonic inspection of materials. Martin ( Proc. R. Soc. Lond . A 378, 263 (1981)) has proved that the corresponding linear boundary-value problem has precisely one solution, and that this solution can be constructed by solving a two-dimensional Fredholm integral equation of the second kind. However, this integral equation has a complicated matrix kernel and the components of its vector solution are coupled. The main purpose of the present paper is to show how Martin’s integral equation can be explicitly solved in terms of a sequence of functions, each of which satisfies a very simple scalar integral equation of the second kind; this simplification may be made for any incident wave. For an incident plane wave, further simplifications are possible. We show that the solution at an arbitrary angle of incidence can be derived from the solution at a particular angle of incidence, namely grazing incidence. The resulting computational procedure is especially attractive if only the stress-intensity factors or the far-held displacements are required. Finally, we present some numerical results for the scattering of a P-wave at normal incidence and an SV-wave at oblique incidence, and compare these with those of other authors.

Scattering of steady elastic waves by surfaces of arbitrary shape

abstract The three-dimensional problem of scattering of steady elastic waves from an arbitrarily shaped body is formulated in terms of simultaneous singular integral equations. These basic equations defining the displacement potentials at the surface are two-dimensional Fredholm integral equations of the second kind over the surface of the body. In contrast to other formulations, this procedure is not restricted to a particular shape of the scattering object or to the nature of the incoming wave. However, by assuming the scatterer to be a body of revolution, the integral equations are reduced to one-dimensional Fredholm equations along the axis of the scatterer. The formulation is presented for a fixed rigid scatterer. The integral equations are solved by approximating the potential functions by a set of polynomials and by satisfying the integral equations in the sense of least squares. The solution yields the surface potentials from which the field potentials or the field displacements are obtained by surface integrals. The numerical results are in good agreement with those obtained by the eigenfunction method for an incoming wave which is a plane shear or longitudinal wave.