Arabinda Roy

Diffraction of elastic waves by an elliptic crack—II

Abstract The integral equation method is used to obtain the scattered field of a normally incident plane wave from an elliptic crack embedded in an isotropic elastic medium. It is shown that the determination of the diffracted field depends on the solution of integro-differential equation. A formal power series solution, in the low frequency limit, is obtained. Expressions are derived for the scattered amplitudes and the dynamic stress intensity factor.

Interaction between coplanar elliptic cracks—II shear loading

A recently developed integral equation method is used to solve the problem of interaction between coplanar, elliptic cracks under normal loading. The pair of dual integral equations, in a Cartesian system is first transformed to four sets of infinite systems of Fredholm integral equations of the second kind in a cylindrical polar coordinate system. For cracks which are well separated, a perturbed solution of these integral equations can be obtained in terms of the separation parameter β. Analytical expressions for the stress intensity factor and the strain energy of deformation per crack, when subjected to a constant normal loading, have been given up to the order β6. Results have been illustrated graphically.

An elliptic crack in an elastic half-space

Abstract In this paper we consider the effect of the free surface on the stress distribution of an elliptic crack aligned parallel to the free boundary and at a depth h below it. The title problem is posed as a dual integral equation in Cartesian coordinate system. By suitable transformation the dual integral equation is first reduced to an infinite system of dual integral equation in cylindrical coordinates. Then they are further reduced by a recently developed technique to an infinite system of Fredholm integral equation of the second kind. When the boundary is at a large distance away from the crack, the system of integral equation is solved by perturbation technique as powers of δ = a/h , a being the length of semi major axes and h the depth of the crack below the boundary. The analytical expressions for the three stress-intensity factors at the crack edges for normal loading are given upto order δ 5 . Effect of the boundary on the stress-intensity factor is illustrated graphically.

Pulse generation in an elastic half space by normal pressure

Abstract The generation of pulse in an elastic half space by impulsive normal pressure over a circular area on the surface has been investigated by Cagniard De-Hoop technique. The approximate representations of the displacement field near the times of arrival of various wave fronts have been derived by a limiting process.

First motions from nonuniformly moving dislocations

Abstract The response of an elastic half space to a realistic model of faulting is considered. A dislocation is assumed to be developed along a line of finite length and then moves nonuniformly along an inclined plane (fault) surface. Analytical solutions to the surface displacement in the form of double integrals are derived by Cagniard De-Hoop technique. Nature of wave arrivals at the surface are discussed both in case of a decelerating and an accelerating source. First morion responses are obtained near different wave arrivals by a limiting process.

Dynamic response of elliptical footings

Abstract 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 ω 2 , 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.

Response of an elastic solid to nonuniformly expanding surface loads

Abstract Exact expressions are obtained for the displacement field in a homogeneous isotropic elastic half space whose surface is subjected to a unit normal pressure. The load emanates from a point and expands nonuniformly and radially outwards. The displacement field is obtained in the form of triple integrals over finite ranges. Both accelerating and decelerating loads have been considered. Wave front surfaces with their regions of existence have been identified. First motion approximations near different wave arrivals have been obtained by a limiting process and do not involve any integration.

Contact Problem in Elasticity

We review the classical Hertz contact theory under normal load and formulate a new unified method valid for the Hertz contact theory and a variety of frictionless elliptic contact problem with an elliptic contact connection both for a rigid punch and a conical indenter. We also give a direct way to evaluate the stress and displacement field in the medium. As a limiting case, we derive the results for circular connection as well as line contact problems in the two-dimensional case. Relations of the contact stresses of the wheel of a locomotive rolling on the rails of straight and curved railway with failure of the rail are discussed. Use of such study in hardness testing is discussed.

Response of an elastic half-space to normal pressure over an elliptic area

Abstract Transient response of an elastic half-space subjected to a uniform normal pressure acting over an elliptic area on the boundary is obtained. This displacement field is obtained by the use of Cagniard-De-Hoop technique. Different wave fronts expected are identified and nature of approximate form of displacement near wave fronts are discussed.

On an Interface Elliptic Crack

The three-dimensional problem of an elliptic crack located at the interface between two bonded dissimilar elastic half-spaces and crack faces subjected to normal pressure equal in magnitude and opposite in direction is considered here. Considering a Cartesian coordinate system with the xOy-plane coinciding with the crack plane and origin O coinciding with the crack centre, the mixed boundary conditions on the \(z=0\) plane give rise to three pairs of dual integral equations. This typical mixed boundary value problem is solved here analytically for the first time for normal pressure prescribed on the crack faces. With uniform normal pressure, the three pairs of dual integral equations are reduced to two sets of dual integral equations, which further reduce to a Cauchy singular integral equation that is solved using Plemelj formula. The present work opens up the possibility of further research work in the field of interface elliptic crack located at the interface of bonded elastic or piezoelectric solids.