Inclusion invisibility, stress annihilation and stress reduction in antiplane elasticity

An infinite class of nonuniform antiplane shear fields is considered for a linear elastic isotropic space in which a void or a rigid inclusion is present. This void or inclusion has the shape of an isotoxal star-shaped polygon or a hypocycloid. This problem is solved through complex potential techniques and multinomial theorem. Usually solutions evidence stress singularities at the inclusion tips, but it is shown that there are certain circumstancies in which the singularities are absent and the stress falls to zero or to values smaller than those that would be present at the same point when the inclusion is absent. In the former case a stress annihilation is reached, while stress reduction occurs in the latter. It is also shown that in special situations a star-shaped crack or a star-shaped stiffener can leave the ambient field completely unchanged, so that the inclusion becomes "invisible". These results may find application in the design of ultra-strong composites.