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

Abstract

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: σ ij = r f( z)−2∼ σ ij ( θ, Tz) ( i, j = x, y), where f( z) is a function of triaxial stress constraint Tz. The transverse shear stresses σ yz and σ xz are of the order of unity. The corresponding in-plane strains ε ij ( i, j = x, y) have singularity of order n( f( z) − 2), while ε yz and ε xz are of the order of unity, ε zz 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.