Harmonic Holes for Nonconstant Fields Available to Purchase

Hole shapes which do not perturb the first invariant of the stress tensor of an isotropic stress field with superimposed linear gradients are derived. These haromonic holes, called the deloid and cardeloid, result from the solution of a nonlinear singular integral equation and can be expressed very simply in parametric form. A comparison of stresses around the boundary of a deloid with a circular hole of the same size shows that the deloid significantly reduces not only the maximum stress concentration but also the total variation of stress around the hole boundary. Deloids and cardeloids without internal boundary loading exist only in locations where the first invariant of the original field does not change sign.