Linear Elastic Solutions for Slotted Plates

Two-dimensional problems of finite-length blunted cracks cut into infinite plates subject to remote tractions are solved using complex variable theory. The slot geometry is composed of two flat surfaces connected by rounded ends. This special geometrical shape was derived by Riabouchinsky in the study of two-dimensional ideal fluid flow around parallel plates. The simpler antiplane slotted plate problem is addressed initially for this geometry. From this exact solution, the equivalent of a Westergaard stress potential is found and applied to the two other principal modes of fracture, which are plane elasticity problems. For a plate subject to uniform radial tension at infinity, an analytical solution is obtained that will reduce to the familiar mode I singular crack solution as the separation between the parallel faces of the slot becomes zero. For finite-width mode I slots, the rounded ends have tensile tractions which terminate at the adjoining flat surfaces of the slot, which remain traction-free. In this respect, the finite-width mode I slot problem resembles a Barenblatt cohesive zone model of a plane crack or a Dugdale plastic strip model of a plane crack, although the tractions will vary in magnitude along the slot ends rather than remaining uniform as in the former type of crack problems. Similarly, in the case of the finite-width mode II slot problem, the rounded ends of the slot have shear tractions, while the flat surfaces remain load-free. A distinguishing feature of the mode II slot solution over the mode I slot problem is that the maximum in-plane shear stress is constant along the rounded ends of the slot. Because of this, those particular regions of the boundary can represent incipient plastic yield based on either the Mises or Tresca yield condition under plane strain loading conditions. In this way, the problem resembles the plastic strip models of Dugdale, Cherepanov, Bilby-Cottrell-Swinden, and others. Notably, the mode III slot problem also has a constant maximum shear stress along the curved portions of the slot, while the entire slot boundary remains traction-free, unlike the mode II slot problem. Consequently, the mode III slot problem represents both a generalization of the standard mode III crack problem geometry, while simultaneously satisfying the boundary conditions of a plastic strip model.