Crack edge stress fields

Cases for which the crack faces move apart symmetrically with respect to the crack plane without relative sliding are termed mode I or opening mode deformations. The application of a tensile load in a direction normal to the plane of a crack in an isotropic material normally results in mode I deformation. Cases for which the crack faces slide with respect to each other in the direction normal to the crack edge without relative opening are termed mode II or in-plane shearing mode deformations. Finally, cases for which the crack faces slide with respect to each other in the direction parallel to the crack edge without a relative opening are termed mode III or out- 

Similarly, the crack edge singular fields for mode II loading are given by 

An important relationship between stress intensity factors and the energy release rate for planar crack growth under equilibrium conditions was established by Irwin (1960) as 

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Mathematically, the stress field surrounding a geometric comer, such as a crack tip, a re-entrant comer or the edge of a bi-material interface can be given as a series expansion of the form ... 

A number of additional trends can be noted. For example, if the stress concentration factor is larger than 2 - it is approximately 3 for a circular groove in a tension field - then the site is more effective as a source of dislocations. If the configuration of the stress concentrator is a notch with a very high curvature of the notch surface at its root, then the stress concentration factor can be very large compared to 2, but the spatial extent of the localized stress field is significantly reduced from that of the circular stress concentrator. For the case of a planar crack, which is the ultimate sharp notch, the issue of dislocation nucleation has been modeled by Rice and Thomson (1973) and Rice and Beltz (1994). Similar techniques have been adapted for the study of dislocation nucleation at the edge of an epitaxial island (Johnson and Freund 1997). 

The factor 2 in front of the second integral accounts for the fact that two surfaces, each with surface energy Us, are created from the surface S in the reference configuration. Continuum stress and deformation fields near a geometrical discontinuity such as a crack edge are potentially singular. In such a case, the differentiability requirements on these fields for interpretation of the continuum equilibrium equation or the divergence theorem may not be satisfied consequently, a special interpretation is required (Freund 1990). 

The stress within the contact area as given by (8.103) is square-root singular as r a , so that the edge of the contact zone has the character of an elastic crack (Lawn 1993) see Section 4.2.3. The elastic stress intensity factor K associated with the singular stress field along this crack edge is... 

It was noted in Section 2.3.2 that most of the current interfacial fracture mechanics methodologies describe steady-state crack propagation, but not the initiation of interfacial cracks. A recent approach to the prediction of initiation is based on the calculation of the singular stress field at the free edge of a bimaterial system loaded on the top layer [59,60]. Because the crack is assumed not to exist initially in this analysis, a very different singular field is predicted, and the results can be used to predict initiation of cracks in residually stressed coatings. Because the predictions of this theory sometimes contradict the predictions of the Suo and Hutchinson approach, we shall briefly review it as a final note. 

An elastic field, although complex, remains well defined up to critical loading, at which point a cone-shaped crack suddenly develops in the sample. Cracking always starts just beyond the contact edge where surface defects occur and where the stress is highest. 

For numerical investigations of stress localizations in laminates, the discretizational effort can be reduced significantly if only the boundary needs to be discretized, as it is for e -ample the case in the classical boundary element method (BEM). But in this method a fundamental solution is needed which is in many cases difficult to achieve or even unknown. The Boundary Finite Element Method (BFEM) to be presented here does not require such a fundamental solution, because the element formulation is based on the finite element method (FEM), Thus the BFEM can be characterized to be a finite element based boundary discretization method. This method was originally developed from Wolf and Song [10] under the name Consistent Finite Element Cell Method for time-dependent problems in soil-mechanics. The basic assumption of this method is that a stiffness matrix describing the force-displacement relation at discrete degrees of freedom at the boundary of the continuum is scalable with respect to one point in three-dimensional space, the so-called similarity center, if similar contours within the continuum are considered. In contrast to this, the current work deals with the case of equivalent cross-sectional properties, i.e., that cross-sections parallel to the boundary can be described by the same stiffness matrix, which is the appropriate formulation for the case of the free-edge effect and the matrix crack problem. The boundary stiffness matrix results from a Matrix-Riccati equation. The field quantities inside of the continuum can be calculated from an ordinary differential equation. 

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