Elastoplastic three dimensional crack border field—I. Singular structure of the field

The structure of stress and strain fields at the border of three dimensional cracks in a tension field is investigated for elastoplastic materials treated by a deformation theory. The investigation is based upon the physics of the problem and is conducted with mathematical rigour. It is found that the character of singular stresses is as follows: σij = rf(z)−2∼σij(θ, Tz) (i,j = x,y), where f(z) is a function of triaxial stress constraint Tz. The transverse shear stresses σyz and σxz are of the order of unity. The corresponding in-plane strains εij (i,j = x, y) have singularity of order n(f(z) − 2), while εyz and εxz are of the order of unity, εzz has the same order as in-plane strains at corner points but may be much weaker in the interior of the crack border. Further, it is argued that the problem can be simplified to a quasi-planar problem with the triaxial stress constraint Tz being considered. When the solution is degenerated into a plane problem by enforcing the confinement, the exact solution for a plane strain crack is obtained and some interesting phenomena are discussed in detail.