Harmonic Holes—An Inverse Problem in Elasticity

A fundamental problem in elasticity is investigated: to determine the geometry of a boundary from prescribed conditions that must be satisfied by the final field stresses. For the planar problem the hole shape termed “harmonic” is the one which satisfies a specific design requirement that the first invariant of the original field remains everywhere unchanged. Using the complex variable technique a general functional equation for the hole geometry is obtained in integral form simply in terms of the original free field and hole boundary loading. The biaxial field is considered as an example with a uniformly distributed normal stress applied to the unknown hole boundary. It is shown that for this field, a properly proportioned and oriented elliptic hole satisfies the design requirement and simultaneously produces the minimum stress concentration for any possible unreinforced hole shape. For an unloaded boundary, this optimum hole exists only when the principal stresses in the original field have the same sign.