Peter Schiavone

On the Elliptic Inclusion in Anti-Plane Shear

In this paper, we consider the anti-plane shear problem of an elliptic inclusion embedded in an infinite, isotropic, elastic medium, subjected at infinity to a uniform stress field. Using complex variable methods and the theory of analytic functions, we prove that the state of deformation in the inclusion is a simple shear if and only if the curve enclosing the inclusion is an ellipse.

The Effects of Surface Elasticity on an Elastic Solid With Mode-III Crack: Complete Solution

We examined the effects of surface elasticity in a classical mode-III crack problem arising in the antiplane shear deformations of a linearly elastic solid. The surface mechanics are incorporated using the continuum based surface/interface model of Gurtin and Murdoch. Complex variable methods are used to obtain an exact solution valid everywhere in the domain of interest (including at the crack tip) by reducing the problem to a Cauchy singular integro-differential equation of the first order. Finally, we adapt classical collocation methods to obtain numerical solutions, which demonstrate several interesting phenomena in the case when the solid incorporates a traction-free crack face and is subjected to uniform remote loading. In particular, we note that, in contrast to the classical result from linear elastic fracture mechanics, the stresses at the (sharp) crack tip remain finite.

New phenomena concerning the effect of imperfect bonding on radial matrix cracking in fiber composites

Abstract In this paper we study the effects of imperfect bonding on stress intensity factors (SIFs) calculated at a radial matrix crack in a fiber (inclusion) composite subjected to various cases of mechanical loading. We use analytic continuation to adapt and extend the existing series methods to obtain series representations of deformation and stress fields in both the inclusion and the surrounding matrix in the presence of the crack. The interaction between the crack and the inclusion is demonstrated numerically for different elastic materials, geometries and varying degrees of bonding (represented by imperfect interface parameters) at the interface. Some qualitatively new phenomena are predicted for radial matrix cracking, specifically the influence of imperfect bonding at the inclusion–matrix interface on the direction of crack growth. For example, in the case of an inclusion perfectly bonded to the surrounding matrix, the SIF at the nearby crack tip is greater than that at the distant crack tip only when the inclusion is more compliant than the matrix. In contrast, the effects of imperfect bonding at the inclusion–matrix interface allow for the SIF at the nearby crack tip to be greater than that at the distant crack tip even when the inclusion is stiffer than the matrix . In fact, for any given case when the inclusion is stiffer than the matrix, we show that there is a corresponding critical value of the imperfect interface parameter below which a radial matrix crack grows towards the interface leading eventually to complete debonding. In particular, this critical value of the imperfect interface parameter tends to a non-zero finite value when the stiffness of the inclusion approaches infinity. To our knowledge, these results provide, for the first time, a clear quantitative description of the relationship between interface imperfections and the direction of propagation of radial matrix cracks.

A circular inclusion with inhomogeneously imperfect interface in plane elasticity

A general method is presented for the rigorous solution of a circular inclusion embedded within an infinite matrix in plane elastostatics. The bonding at the inclusion-matrix interface is considered to be imperfect with the assumption that the interface imperfections are circumferentially inhomogeneous. Using analytic continuation, the basic boundary value problem for four analytic functions is reduced to two coupled first order differential equations for two analytic functions. The resulting closed-form solutions include a finite number of unknown constants determined by analyticity and certain other auxiliary conditions. The method is illustrated using a particular class of inhomogeneous interface. The results from these calculations are compared to the corresponding results when the imperfections in the interface are circumferentially homogeneous. These comparisons illustrate, for the first time, how the circumferential variation of the parameter describing the imperfection has a pronounced effect on the average stresses induced within the inclusion.

Effect of surface elasticity on an interface crack in plane deformations

We consider the effect of surface elasticity on an interface crack between two dissimilar linearly elastic isotropic homogeneous materials undergoing plane deformations. The bi-material is subjected to either remote tension (mode-I) or in-plane shear (mode-II) with the faces of the (interface) crack assumed to be traction-free. We incorporate surface mechanics into the model of deformation by employing a version of the continuum-based surface/interface theory of Gurtin & Murdoch. Using complex variable methods, we obtain a semi-analytical solution valid throughout the entire domain of interest (including at the crack tips) by reducing the problem to a system of coupled Cauchy singular integro-differential equations, which is solved numerically using Chebychev polynomials and a collocation method. It is shown that, among other interesting phenomena, our model predicts finite stress at the (sharp) crack tips and the corresponding stress field to be size-dependent. In particular, we note that, in contrast to the results from linear elastic fracture mechanics, when the bi-material is subjected to uniform far-field stresses (either tension or in-plane shear), the incorporation of surface effects effectively eliminates the oscillatory behaviour of the solution so that the resulting stress fields no longer suffer from oscillatory singularities at the crack tips.

A clarification of the role of crack-tip conditions in linear elasticity with surface effects:

We examine the role of crack-tip conditions in the reduction of stress at a crack tip in a theory of linear elasticity with surface effects. The maximum number of allowable end conditions for complete removal of a stress singularity is demonstrated for both plane and anti-plane problems. In particular, we show that the necessary and sufficient conditions for bounded stresses at a crack tip cannot be satisfied with a first-order (curvature-independent) theory of surface effects, which leads, at most, to the reduction of the classical strong square-root singularity to a weaker logarithmic singularity.

Uniformity of Stresses Within a Three-Phase Elliptic Inclusion in Anti-Plane Shear

In this paper, we show that a three-phase elliptic inclusion under uniform remote stress and eigenstrain in anti-plane shear admits an internal uniform stress field provided that the interfaces are two confocal ellipses. The exact closed-form solution is used to quantify the effect of the interphase layer on the residual stresses within the inclusion and the dependency of this effect on the aspect ratio of the elliptic inclusion.

An elliptic inclusion with imperfect interface in anti-plane shear

Abstract A semi-analytic solution is developed for the problem associated with an elliptic inclusion embedded within an infinite matrix in anti-plane shear. The bonding at the inclusion-matrix interface is assumed to be homogeneously imperfect. The interface is modeled as a spring (interphase) layer with vanishing thickness. The behaviour of this interphase layer is based on the assumption that tractions are continuous but displacements are discontinuous across the interface. Complex variable techniques are used to obtain infinite series representations of the stresses induced within the inclusion. The results obtained demonstrate how the (non-uniform) stress field and the average stresses inside the inclusion vary with the aspect ratio of the inclusion and the parameter describing the imperfect interface. In addition, it is shown that, in some cases (depending on the aspect ratio of the ellipse), it is possible to identify specific values of the interface parameter which correspond to maximum peak stress along the interface.

Stress Analysis of an Elliptic Inclusion with Imperfect Interface in Plane Elasticity

In this paper, a semi-analytic solution of the problem associated with an elliptic inclusion embedded within an infinite matrix is developed for plane strain deformations. The bonding at the inclusion-matrix interface is assumed to be homogeneously imperfect. The interface is modeled as a spring (interphase) layer with vanishing thickness. The behavior of this interphase layer is based on the assumption that tractions are continuous but displacements are discontinuous across the interface.Complex variable techniques are used to obtain infinite series representations of the stresses which, when evaluated numerically, demonstrate how the peak stress along the inclusion-matrix interface and the average stress inside the inclusion vary with the aspect ratio of the inclusion and a representative parameter h (related to the two interface parameters describing the imperfect interface in two-dimensional elasticity) characterizing the imperfect interface. In addition, and perhaps most significantly, for different aspect ratios of the elliptic inclusion, we identify a specific value (h*) of the (representative) interface parameter h which corresponds to maximum peak stress along the inclusion-matrix interface. Similarly, for each aspect ratio, we identify a specific value of h (also referred to as h* in the paper) which corresponds to maximum peak strain energy density along the interface, as defined by Achenbach and Zhu (1990). In each case, we plot the relationship between the new parameter h*and the aspect ratio of the ellipse. This gives significant and valuable information regarding the failure of the interface using two established failure criteria.

Integral equation methods in plane asymmetric elasticity

The boundary integral equation method is used to solve the interior and exterior Dirichlet, Neumann and mixed problems of plane micropolar elasticity. In the exterior case, a specific far-field pattern for the displacements and microrotation is introduced without which the classical scheme fails to work. Finally, we discuss the direct method and establish a connection with results obtained previously.