Elastic state and orientation of needle-shaped inclusions

The inclusion problem of linear isotropic elasticity is applied to analyze the elastic strain energies of coherent, semicoherent and incoherent needle-shaped inclusions. Misfit strains of a general tetragonal type, elastic constants and orientation of the needle are treated as variables. The strain-energy minimization criterion is adopted to find the optimum orientation. Several new and general conclusions are derived for the elastic state of the needle inclusions. The predicted optimum orientation is compared with that obtained by the purely geometrical invariant-line criterion. It is found that the two criteria predict the identical orientation only under special cases.