Fracture strength referred particularly to the eccentric initiation in a fractured surface has been investigated in employing a brittle plastic sheet and FCD-50 cast iron. Micro-faults like a craze and inclusion particles play a role of a nucleus in the homogeneous plastics. Micro-bubbles introduced purposely into the plastics control the fracture initiation sites in the nonhomogeneous material. Nodular graphite particles in FCD-50 are the inevitable fracture nucleation sources. A ferrous sphere of 0.7 or 2.0mm dia. or a glass sphere of 0.5 or 2.0mm dia. artificially introduced into the plastics also controls a fracture initiation site. All such nucleation points in fractured surfaces have been analyzed by a plane eccentricity and by a relative distance to the free boundary proposed herewith. Fracture strength is shown to be functions of these. Discussion is made on these including the basic and the current knowledge on the stress concentration factor on an off site second phase and the stress distributions around a spherical solid in the continuum and the interfacial bonding condition between a second spherical phase and the matrix.