in shape and magnitude to the real material's load response. The elastic-plastic failure parameter is designated JIc and is conventionally converted to KIc using the equation below. Also note that the J integral approach reduces to the Griffith theory for linear-elastic behavior. The mathematical definition of J-integral is as follows: where is an arbitrary path clockwise around the apex of the crack, is the density of strain energy, are the components of the vectors of traction, are the components of the displacement vectors, is an incremental length along the path , and and are the stress and strain tensors. Since engineers became accustomed to using KIc to characterise fracture toughness, a relation has been used to reduce JIc to it: where for plane stress and for plane strain. Cohesive zone model When a significant region around a crack tip has undergone plastic deformation, other approaches can be used to determine the possibility of further crack extension and the direction of crack growth and branching. A simple technique that is easily incorporated into numerical calculations is the cohesive zone model method which is based on concepts proposed independently by Barenblatt and Dugdale in the early 1960s. The relationship between the Dugdale-Barenblatt models and Griffith's theory was first discussed by Willis in 1967. The equivalence of the two approaches in the context of brittle fracture was shown by Rice in 1968. Transition flaw size Let a material have a yield strength and a fracture toughness in mode I . Based on fracture mechanics, the material will fail at stress . Based on plasticity, the material will yield when . These curves intersect when . This value of is called as transition flaw size ., and depends on the material properties of the structure. When the , the failure is governed by plastic yielding, and when the failure is governed by fracture mechanics. The value of for engineering alloys is 100 mm and for ceramics is 0.001 mm. If we assume that manufacturing processes can give rise to flaws in the order of micrometers, then, it can be seen that ceramics are more likely to fail by fracture, whereas engineering alloys would fail by plastic deformation. See also – Fracture mechanics and fatigue crack growth analysis software , a formulation of continuum mechanics that is oriented toward deformations with discontinuities, especially fractures References Further reading Buckley, C.P. "Material Failure", Lecture Notes (2005), University