Abstract Using Green functions in an infinite medium, the constraint stress field of an inclusion with general shape is given. The stress free strains of the inclusion may be functions of position. On this basis, all calculating formulas for plane problems are given. We consider cracks or holes as special inhomogeneities with elastic constants equal to zero. For a body stressed by the applied field, the stress-free strains of the equivalent inclusion have been calculated. For oblate inclusions, near the end of major axis of ellipse, the stress field exhibits a r-1/2 stress singularity similar to that of a crack. Some applications, including the interaction of a hole with the applied field, micro-crack nucleation due to martensite plates and deformation twins, are discussed.