Interface crack onset at a circular cylindrical inclusion under a remote transverse tension. Application of a coupled stress and energy criterion

Abstract

The plane strain problem of a single circular cylindrical inclusion embedded in an unbounded matrix subjected to a remote uniform uniaxial transverse tension is studied. A theoretical model for the simultaneous prediction of \emph{the initial size of a crack originated at the inclusion/matrix interface} (or equivalently the initial polar angle of this crack) and of \emph{the critical remote tension required to originate this crack} is developed. Isotropic and linear elastic behaviour of both materials, with the inclusion being stiffer than the matrix, is assumed. The interface is considered to be strong (providing continuity of displacements and tractions across the interface surface) and brittle. The model developed is based on the classical analytic solutions for the above-mentioned inclusion problem without and with a crack situated at the inclusion/matrix interface and a recently introduced coupled stress and energy criterion of failure by Leguillon (\emph{Eur. J. Mech. A/Solids, 21, pp. 61--72, 2002}). A new dimensionless structural parameter $\gamma$, depending on bimaterial and interface properties together with the inclusion radius $a$, which plays a key role in characterizing the interface crack onset, is introduced. Asymptotic behaviour of the predicted critical remote tension and the interface crack length/polar angle at the onset are characterized for small and large values of $\gamma$ and $a$. A size effect inherent to this problem is predicted and analysed. The following asymptotic characteristics of this size effect are noteworthy: \emph{i)} for small inclusion radii $a$, the polar angle of the crack at onset is constant (independent of $a$), whereas the critical remote tension increases with decreasing $a$, being inversely proportional to the square root of $a$; \emph{ii)} for large inclusion radii $a$, the length of the crack at onset and the critical remote tension are approximately constant.