On an integral equation governing an internally indented penny-shaped crack

The paper examines the problem of a flat rigid circular disc that is compressed between two finitely deformed incompressible elastic halfspace regions with smooth surfaces. This problem yields a unilateral contact problem where the zone of separation needs to be determined. The analysis of the unilateral contact problem is reduced to the solution of a set of triple integral equations associated with the internal indentation of a penny-shaped crack and the internal tensile pressurization of an annular crack. The solutions to these problems can be obtained in an approximate series form in terms of the non-dimensional parameter involving the radius of the rigid disc to the radius of the separation zone. The extent of the separation zone is determined from the vanishing of the contact stress at the point of separation. Specific solutions are developed for the case where the initial finite deformation is for halfspace regions with a strain energy function of the Mooney-Rivlin form.

The paper examines the in-plane loading of a bonded rigid disc inclusion of finite thickness that is embedded at a smooth pre-compressed elastic interface composed of two halfspace regions. The in-plane loading induces a pure translation of the inclusion without rotation. The paper first develops estimates for the in-plane elastic stiffness of the disc inclusion. Upon de-bonding of the inclusion-elastic medium interfaces, the peak force is established by considering a Coulomb friction response of fully debonded interfaces. When the detached interface exhibits dilatant processes the displacement-dependent peak force at the detached surfaces can be estimated using a work-plastic energy dissipation relationship. When the bonded faces of the inclusion exhibit an elasto-plastic response, the force–displacement response of the in-plane loaded inclusion can exhibit a transition from the fully bonded response to partial detachment. The analysis of this problem is achieved using a computational approach that accounts for the development of failure and separation at the pre-compressed interface.

Interfaces possess complex mechanical responses that are governed by several factors including the type of material, the local topography of the interacting surfaces, the stress state and the mode of deformation. This paper examines the mechanics of a mated smooth interface that is subjected to a normal stress and where the contact is perturbed by a circular patch that can experience dilatancy under shear. The analysis of the static stress drop occurring during shear at the interface is examined using a contact mechanics approach that accounts for the separation at the pre-compressed geological interface induced by the development of dilatancy of the patch during relative shear. This paper presents an elementary model of the mechanics that takes into consideration the normal stress evolution during dilatant shearing of the interface. The problem is of particular interest to the modelling of local phenomena that can occur at material interfaces and geological faults that are subjected to steady movement.

In the framework of linear elastic continuum mechanics, an analytical formulation is presented for the axisymmetric axial interaction of a rigid disk in frictionless contact with the face of a penny-shaped crack in a transversely isotropic solid. The problem is reduced to an integral equation and is shown to be degenerated to the formulation of isotropic materials in the literature. As the closed-form solution is not possible, by means of a numerical procedure, the obtained integral equation is solved and the accuracy of numerical procedure and mathematical formulation is verified through comparison with the available results for the relevant analysis in isotropic solids. Through several numerical displays, the effect of material anisotropy on the stiffness of the interacting system and the stress intensity factors at the tip of penny-shaped crack is surveyed.

The paper examines the axisymmetric problem related to the indentation of the plane surface of a penny-shaped crack by a smooth rigid disc inclusion. The crack is also subjected to a far-field compressive stress field which induces closure over a part of the crack. The paper presents the Hankel integral transform development of the governing mixed boundary value problem and its reduction to a single Fredholm integral equation of the second kind and an appropriate consistency condition which considers the stress state at the boundary of the crack closure zone. A numerical solution of this integral equation is used to develop results for the axial stiffness of the inclusion and for the stress intensity factors at the tip of the penny-shaped crack.