Harmonic Shapes in Finite Elasticity Under Nonuniform Loading

We investigate the classic (inverse) problem concerned with the design of so-called harmonic shapes for an elastic material undergoing finite plane deformations. In particular, we show how to identify such shapes for a particular class of compressible hyperelastic materials of harmonic type. The “harmonic condition,” in which the sum of the normal stresses in the original stress field remains unchanged everywhere after the introduction of the harmonic hole or inclusion, is imposed on the final stress field. Using complex variable techniques, we identify particular harmonic shapes arising when the material is subjected nonuniform (remote) loading and discuss conditions for the existence of such shapes.