Boundary-integral formulation of three-dimensional problems of steady vibrations of an infinite body with a crack located on an open Lyapunov surface

Abstract

We construct integral representations of the solutions of three-dimensional problems in the form of a combination of Helmholtz potentials [1] whose coefficients are determined in terms of the geometric parameters of the crack. By satisfying the boundary conditions we obtain two-dimensional integral equations in the displacement jumps on the opposite surfaces of the crack when it is loaded by harmonic forces.The method of boundary integral equations has been used previously to solve three-dimensional dynamic problems of the theory of elasticity for an infinite body with one crack [3] or a system of planar cracks [5], as well as the axisymmetric problem [6,7] for a hemispherical crack.Consider an infinite elastic isotropic body with a crack located along an arbitrary open Lyapunov surface [1] (see figure). On the opposite surfaces of the crack S'there are self-balancing forces varying in time t according to a harmonic law. These forces are N § =-N-= iqexp (-iot), where lq is the amplitude of the vibrations and co is the frequency of vibrations. Partial contact of opposite surfaces of the crack is not taken into account, and we require that the Sommerfeld radiation conditions hold at infinity [I]. In this formulation the exponential time factor drops out of the solution and the problem reduces to finding the corresponding amplitude characteristics, which we denote by the superscript A over the functions.Imagining an arbitrary nonplanar surface of the crack S = S § partitioned into elementary regions AS, S = U AS, and assuming that these elementary regions are flat because of their smallness, to determine the displacements %, j = 1,---3, at a point y of the body from the jumps Au s of the displacements on the surface AS we apply the formulas of [5], which we write as follows: uj(y)=0hr22 0(.~3,-

Effect of scattering of elastic harmonic waves in the far zone by a spatial crack

“…The potential theory being used, the solution of problem (1)-(3) is reduced to the solution of the following system of three BIEs with respect to the jumps of displacements of the opposite surfaces of the crack ∆u j (j = 1, 3 ) in the direction of the coordinate axes [14]:…”