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 <a
|
(11.2.20)
|

|
|
(11.2.21)
|
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 [1975] 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]