Bounds on effective fracture toughness of multiphase materials

Effective fracture toughness of composite or multiphase materials in which the constituent phases are randomly distributed, is considered. Upper and lower bounds on the effective fracture toughness of a two-phase material in terms of the values of the fracture toughness of the constituent phases and their volume fractions are derived. The results are generalized for a material consisting of an arbitrary number of phases. Several limiting cases of practical interest are considered. The bounds derived are examined in the light of extensive experimental data. The assumptions made in the derivations are discussed.