The effect of out-of-plane constraint on the stress fields near the front of a crack in a thin ductile plate

Classical plane solutions based on the elastic-plastic fracture mechanics are applied widely in practical engineering. These plane solutions follow the plane stress or plane strain assumption. However, in the case of a thin ductile plate with a through-thickness crack under tension, plane stress conditions exist at a distance of about one half of the plate thickness ahead of the crack front cross the thickness of the plate. What is the stress state in the region where both plane strain and plane stress conditions cannot be met? In the current paper, a semi-analytical method is presented to investigate the problem. Three dimensional Maxwell stress functions, the minimum complementary potential energy principle and three dimensional J-integrals are employed to obtain solutions for crack front fields in a thin ductile plate. Three-dimensional finite element (FE) analyses are carried out to verify the current solutions. FE results reveal that the out-of-plane constraint level Tz increases with increasing remote loading near the crack front. FE results also show that the in-plane stress fields near the crack front can be characterized by the current J-Tz solutions. Both FE and theoretical results illustrate that opening stresses decrease gradually to the corresponding plane stress HRR-field solutions with increasing radial distance r from the crack front in the mid-plane when remote loading is large enough.