Stress intensity factors for curvilinear cracks loaded by bending and torsional moments

The geometry of a thin flat plate containing a curvilinear crack of finite size is investigated. The plate is loaded by uniform bending and torsional moments at infinite distance from the crack contour, while distributed normal bending moments and torques exist at the upper and lower crack surfaces. The bending stress intensity factors for the curvilinear crack are calculated on the basis of classical plate theory. The displacements and internal moments are represented by two complex analytic functions. Extra conditions are imposed to ensure the univalence of the displacements, which is not evident because the plate comprises a multiply connected domain due to the presence of the crack. A linearization with respect to the crack-curvature function has been performed and the bending stress intensity factors are calculated as the first-order solutions for slightly curved cracks. The results are illustrated with a few examples, such as uniform loading configurations and the geometry of a crack along a circular arc. The loading of thin flat plates by a combination of tensile forces and bending moments is also investigated. In analogy with the variation of the stress components over the cross section of the plate, two combined stress intensity factors are introduced having the same dependence on the perpendicular coordinate and being related to the symmetric and anti-symmetric stress intensity factors of the separate plane-stress and bending problems. The resulting energy release rate is shown to be in full agreement with known results in the literature.