Pentagonal nanorods (PNRs) are crystalline objects with unique fivefold symmetry. They are often experimentally observed for materials with FCC crystal structure. In an ideal case a PNR consists of five elastically distorted but otherwise perfect crystalline regions divided by low-energy twin boundaries. The elastic distortions in PNRs and associated stored elastic energy are effectively described in the framework of a disclination approach. As a result of mechanical stress relaxation, the stored energy can be diminished in expense of structural defect formation in PRN interior. It is demonstrated that a perfect multiple twinned PNR structure is unstable with respect to dislocation formation, i.e. prismatic dislocation loop or straight edge dislocation, for PNRs above a certain critical diameter. A new mechanism for the relaxation processes in PNRs is theoretically investigated. This mechanism assumes the formation of the shell possessing crystal lattice mismatch with respect to the PNR core region. The optimal magnitude for core/shell crystal lattice mismatch and optimal shell thickness providing maximum energy release for this mechanism of mechanical stress relaxation, are predicted.