In the other subsections of Section 2, the emphasis has been on developments of linear elastic fracture mechanics (LEFM) specifically based on the crack tip characterizing parameter K, the stress-intensity factor.  This parameter has provided the major damage tolerance design tool for aerospace engineers since the early sixties.  It was discovered and justified for its universal capability for describing the magnitude of the crack tip stress field by Irwin [1957; 1960] and Williams [1957].  Irwin discovered this relationship through his studies of the energy balance equation associated with fracture.  Prior to 1957, fracture research concentrated on extending the original energy balance equation given by Griffith [1921]  In 1957, Irwin [1960] linked the “driving force”, G, in the energy balance equation to the stress-intensity factor, K, and suggested how the stress-intensity factor could be used as the driving force for crack tip behavior.  Subsequent to Irwin’s initial stress-intensity factor analysis, and as a result of the success of the LEFM approach for solving major fracture problems, interest in the energy approach to fracture waned. 

In the late sixties, Rice [1968b] published a paper that again heightened the interest in the energy approach.  Rice’s specific contribution was to develop an integral, the J-integral, which could be used to account for observed non-linear behavior during the fracture process.  This integral also has the useful property that it reduces to the elastic “driving force”, G, when the localized plastic deformation is well contained by the elastic crack tip stress field.  Because many of the materials utilized in aerospace structures have exhibited typical LEFM behavior, aerospace engineers have not assumed a leadership role in the development of the J-integral technology.

Engineers interested in the damage tolerance analysis of more ductile pressure vessels and welded steel structures have provided the major developments here.  Aerospace applications are being recognized each day, however, for this technology, e.g., residual strength analysis of tough materials and sub-critical crack growth behavior of aircraft gas turbine engine structures.

Another analysis approach for characterizing the level of the local stress-strain behavior at the tip of a crack was initiated in Britain in the early sixties.  Wells [1961] suggested that the localized behavior at the tip of the crack was controlled by the amount of crack opening, which was referred to as the crack opening displacement, COD.  The value of the technology built on the COD concept, like that of the J-Integral technology, is that it allows for the coupling of the LEFM analysis and its results to the solution of problems in which the behavior approaches unconstrained yielding.

The subsections below further describe the analysis methodologies based on the three fracture mechanics parameters: the strain energy release rate (driving force - G), the J-Integral (J), and the crack opening displacement (COD), respectively.  Each subsection outlines the analytical basis for the parameter and provides the principal equations that tie the parameter to the LEFM parameter K.  Further information on these parameters can be obtained by the references cited in the text.