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# Section 11.2.2.2. The Weight Function Technique

The weight function technique can be derived using the definition of the strain energy release rate [Parker, 1981; Cartwright, 1979; Bueckner, 1971; Rice, 1972].  The stress-intensity factor is obtained from the difference between the strain energy of a cracked structure and of the identical structure without a crack, and is given by: (11.2.15)

where the function m(x,a) is the Bueckner weight function, a function which is unique for the given geometry and is independent of the loading from which it was derived.  The weight function is defined as a function of

1)      material properties,

2)       a known stress-intensity factor (K*) for the given geometry under a defined loading, and

3)      the crack opening n*(x,a) corresponding to K*: (11.2.16)

H is a material constant that is given by: =E for plane stress (11.2.17) for plane strain

with m = shear modulus and k is defined as a function of the stress state and Poisson’s ratio (n)

 k = for plane stress (11.2.18) for plane strain

For the infinite plate center crack problem K*, n*, , and m are given by the following equations: (11.2.19) for -a < x

and (11.2.22)

The stress-intensity factor associated with a symmetrical pressure loading of s(x) on the central crack faces is then given by (11.2.23)

The reader is cautioned to note that Equations 11.2.23 and 11.2.12 differ.  However, both equations yield exactly the same stress-intensity factor solution when the pressure stress s is a symmetrical function, i.e., the stress at x = xo is equal to the stress at x = -xo (0 < xo < a).  The reason that Equations 11.2.23 and 11.2.12 differ is that the Bueckner function in Equation 11.2.12 was derived for a symmetrical loading whereas the Green’s function was derived for the more general case of unsymmetrical loading.  Thus, when deriving the weight function one should seek to locate stress-intensity factor (K*) and crack displacement (v*) solutions which are representative of the loading symmetry associated with the problems that are to be solved.

A weight function for radially and diametrically cracked holes was developed by Grandt  for through-thickness type cracks.  His solution is given by (11.2.24)

where KB represents the appropriate (radial or diametrical) Bowie stress-intensity factor (Sectoin 11.3), and the crack opening displacement h was obtained from finite-element solutions.  The displacements h were described by the conic section equation: (11.2.25)

Here ho is the displacement at the crack mouth (x=0) and m is the conic section coefficient from (11.2.26)

In this instance, Y is the Bowie geometric factor (11.2.27)

The finite-element results for the crack mouth displacement ho were closely represented by the least squares expression (11.2.28)

where the coefficients Di are given in Table 11.2.2.

Table 11.2.2.  Least Squares Fit Of Finite Element Data For Crack Mouth Displacement

[Grandt, 1975] Coefficient Single Crack Double Crack D0 -1.567 x10-6 1.548 x10-5 D1 6.269 x10-4 5.888 x10-4 D2 -6.500 x10-4 -4.497 x10-4 D3 4.466 x10-4 3.101 x10-4 D4 -1.725 x10-4 -1.162 x10-4 D5 3.485 x10-5 2.228 x10-5 D6 -2.900 x10-6 -1.694 x10-6

Grandt has applied the weight function technique to a number of fastener-type cracked hole problems.  Using finite-element descriptions of the stress along the expected crack path for a hole that has been cold-worked (loaded) to a 0.006 inch radial expansion and then unloaded, Grandt was able to derive the stress-intensity factor shown in Figure 11.2.11 for a remote stress loading of 40 ksi.  Figure 11.2.11 also provides the stress-intensity factor solution for a remote stress loading of 40 ksi applied to a radially cracked hole without cold-working.  The dramatic difference in stress-intensity factors from the two cases has been shown to translate itself into orders of magnitude difference in crack growth rate behavior. Figure 11.2.11.  Stress-Intensity Factor Calibration for a Cold Worked Hole [Grandt, 1975]