The cohesive zone model and the local failure criterion in the form of the average stress limitation in the cohesive zone ahead of the crack/notch tip have been employed to describe the failure assessment diagram for solids with a crack or notch under uniaxial and biaxial loading. The local stress on the crack extension line has been described by the exact elastic Westergaard’s solution taking into account the stress parallel to the crack plane. The cohesive stress ahead of the crack or notch tip is treated according to von Mises yield criterion as a property of the material, the applied stress and the stress biaxial ratio. Effect of the biaxial ratio is discussed for plates with a crack and notch. The failure assessment curve for a plate with a crack shows no significant differentiation among the three loading cases for plane stress, whereas the biaxial assessment curves move outward from the uniaxial case for plane strain.