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Handbook for Damage Tolerant Design

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Section Developing Stress Intensity Factor Solutions

There are a number of methods that are available for developing stress-intensity factor solutions for crack body problems.  Review articles and textbook chapters that summarize these methods are provided in Sneddon & Lowengrub (1969), Rice (1968), Paris & Sih (1965), Sih (1973a, 1973b), Tada, et al. (1973), Rooke & Cartwright (1976), Wilhelm (1970), Parker (1981), Broek (1974) and Goodier (1969).  The basic solutions for simple geometries can be derived by means of classical methods of elasticity which employ complex stress functions Sneddon & Lowengrub (1969), Rice (1968), Westergaard (1939) and Mushkelishvili (1953).

For finite size bodies containing cracks, the boundary conditions usually prohibit a closed form solution.  In such cases, numerical solutions can be obtained using methods such as the finite element method, the boundary collocation technique [Gross, et al., 1964; Newman, 1971], or the boundary integral method [Cruse, 1972; Cruse & Besuner, 1975].  Solutions for multiple load path geometries can sometimes be obtained from basic stress field solutions combined with displacement compatibility requirements for all the structural members involved [Swift & Wang, 1969].  Section 4 describes this method and provides an example based on the displacement compatibility method.

There are also several experimental methods that have been used to obtain (or verify) the stress-intensity factor for cracked structural members.  These experimental methods include:  The compliance method, the photoelastic method [Smith, 1975; Kobayashi, 1973], the fatigue crack growth (inverse) method [James & Anderson, 1969; Grandt & Hinnericks, 1974; Gallagher, et al., 1974], and the interferometric method [Packman, 1975; Pitoniak, et al., 1974].

While a general knowledge of each stress-intensity factor solution method might be useful for attacking specific problems, detailed knowledge is required before any method can be applied to solve a given problem.  Beyond what is described elsewhere in these guidelines, an engineer can also utilize two separate solution techniques to solve any two-dimensional structural geometry or loading situation without access to a damage tolerant specialist.  One solution technique involves the generation of the stress for an uncracked body along the expected path of crack propagation.  (The finite element method provides a powerful tool for generating stress at any point in an uncracked body).  The second solution technique involves the generation of the stress-intensity factor solution via an integral calculation that employs the stresses obtained for the case of the uncracked body along the expected path of the crack.  Two integral calculation technique types are available:  the Green’s function technique [Cartwright & Rooke, 1979, 1978; Cartwright, 1979; Hsu & Rudd, 1978; Hsu, et al., 1978] and the weight function technique [Cartwright & Rooke, 1978; Cartwright, 1979; Bueckner, 1971; Rice, 1972; Grandt, 1975].  These two crack-line loading techniques are reviewed in the following subsections.