Study has been carried out to clarify the applicable range of COD and J_c -criteria which have been proposed as useful criteria in the fracture of non-linear charactristic materials. At first, evaluation methods of COD and J - integral and the applicable range of these methods were made clear in particular, simplified formulas for estimating J -integral of a shallow notched specimen were newly established. Both COD- and J - values estimated by a proper procedure dominate the distribution of the equivalent plastic strain in the vicinity of the blunting tip of a notch ; however, they can no longer control the nominal stress σ_⊥ which is related to the initiation of cleavage crack. Therefore, the critical δ _c - and J_c - values depend essentially upon the plastic constraint in the vicinity of notch tip. J_c -δ _c criterion has been newly proposed as a useful criterion for unstable brittle fracture. Using J_c -δ _c criterion, it could be expected that fracture toughness testing for estimating plane strain fracture toughness becomes more easy.