Elastic strain energies of sphere, plate and needle inclusions

The inclusion problem of linear isotropic elasticity is applied to discuss the orientation and shape dependencies of the elastic strain energies of sphere, plate and needle inclusions with general misfit strains. The strain-energy minimization criterion is adopted and difference in elastic constants between the inclusion and the surrounding medium is taken into account. Regardless of the types of the misfit strains, it is generally found that the plate shape is most favorable for soft inclusions. For hard inclusions, however, the minimum-energy shape depends on the signs of the three principal misfit strains, a sphere when the three misfit strains have the same sign and a needle when they have different signs. Some other new and general characteristics on the strain energies of the coherent inclusions are newly found. Stresses and strain energies of incoherent inclusions after the occurrence of plastic accommodation are also evaluated.