On the effect of an out-of-plane constraint on the three-dimensional crack front fields in a thin elastic plate

Two-dimensional theories of fracture are still applied widely today and provide theoretical foundations for solutions to many practical problems. These two-dimensional theories are based on the plane strain or plane stress assumption. However, strictly speaking, for a thin elastic plate with a through-thickness crack under tension, plane strain conditions can be met only at the crack front (except the corn point) and plane stress conditions exist at a distance of about one half of the plate thickness from the crack front in the mid-plane. What are the stress fields in the region where both plane strain and plane stress conditions cannot be met? In this paper, further investigations into the problem are carried out. Three-dimensional Maxwell stress functions, the principle of minimum complementary potential energy and three-dimensional J-integrals are employed to obtain an analytical solution to depict the relationship among out-of-plane constraints, three-dimensional J-integrals and stress intensity factors. Three-dimensional finite element simulations with fine meshes are carried out to verify the analytical results. Compared with the corresponding plane strain solution, the solution proposed is valid in a larger region.