State of stress at the vertex of a quarter-infinite crack in a half-space

Spherical coordinates are r, θ, φ. The half-space extends in θ < π/2. The crack occurs along φ = 0. The region to be investigated is the solid space-triangle (or cone) between the three planes θ = π/2, φ = +0 and φ = 2π ? 0, which planes are to be taken stress-free.In this space-angle a state of stress is considered in terms of the cartesian stress components σ_xx = r^λ?_xx(λ, θ, φ); σ_xy = r^λ(λ, θ, φ); etc. Possible values λ are determined from a characteristic (or eigenvalue) equation, expressing the condition that a determinant of infinite order is equal to zero. The root of λ which gives the most serious state of stress in the vertex region (the region r → 0) is the root closest to the limiting value Re λ > ?3/2. Knowledge of this state of stress, or at least of this value of λ is essential in the determination of the three-dimensional state of stress around a crack in a plate for distances of order of the plate thickenss.Along the front of the quarter-infinite crack (z-axis) the so called stress-intensity factor behaves like z^λ+½ (z → 0) and thus tends to zero, respectively to infinity, accordingly to Re λ being >?½ or <?½. But in the region z → 0 the notion stress-intensity factor loses its meaning. The required state of stress passes into the well-known state of plane strain around a crack tip if Poisson's ratio (v) tends to zero. The computed state of stress for the incompressible medium (v = ½) is confirmed by experiment.