Annular crack in an elastic half-space

Abstract

In this paper, an axisymmetric problem of an annular crack is analytically investigated. We address the case where the crack is subjected to a uniform pressure and located below the free surface of a semi-infinite elastic medium. Such a problem is formulated as a three-part mixed boundary value problem which cannot be simplified by geometrical symmetry. With the help of Boussinesq’s stress functions and Hankel transforms technique, the problem is reduced to a system of coupled triple integral equations depending on both zeroth and first-order Bessel functions of the first kind. We seek the solution as a product of Bessel Function series. By means of some integral relations and the Gegenbauer addition formula, the system is further reduced to a set of simultaneous linear algebraic equations. We derive closed-form formulas expressed in terms of the solution of the obtained infinite algebraic system, for displacement and stress components, mixed mode I−II stress intensity factors, as well as the crack compliances. Sets of dimensionless plots are presented to show the influence of the radii and depth of the crack on the elastic fields, on the stress intensity factors and on the crack compliance contributions. The results of an annular crack problem in an infinite medium are also obtained as a limiting case of this study.