Page 413 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C009.fm Page 393 Monday, May 23, 2005 3:58 PM

Application to Structures 393

equal to Equation (9.12) for the two reference cases and assuming a unit value for p results in
n
simultaneous integral equations:
π a a
G = ∫ hx dx (9.16a)
()
0 Q 0
π a a  x 
G = ∫ hx () dx (9.16b)
1 Q 0  a 

Substituting Equation (9.13) and Equation (9.14) into the above expressions and applying the
aforementioned boundary conditions leads to expressions for M and N . At the deepest point of
i
i
the crack, the weight function coefficients are given by
2π 24
M = G 3 ( G − ) − (9.17a)
1 Q 1 0
2 5
M = 3 (9.17b)
2
6π 8
M = G 2 G − ) + ( (9.17c)
3 Q 0 1
2 5
where the influence coefficients G and G are evaluated at f = 90°. The weight function coefficients
1
0
at the free surface are as follows:
3π
N = 2 G ( 5 ) − 8 (9.18a)
G −
1 Q 0 1
15π
N = G 3 ( G − ) + 15 (9.18b)
2 Q 1 0

3π
N = 3 ( G 10 G − ) − 8 (9.18c)
3 Q 0 1

where G and G are evaluated at f = 0°.
1
0
Once M and N have been determined for a given geometry and crack size, the stress
i
i
distribution for the problem of interest must be substituted into Equation (9.12). Numerical
integration is normally required, especially if s (x) is characterized by nodal stress results from
a finite element analysis. For some stress distributions, a closed-form integration of Equation
(9.12) is possible. For example, closed-form solutions exist for power-law crack-face pressure
(Equation (9.4)). Consequently, it is possible to solve for higher-order influence coefficients
using the weight function method.
At the deepest point of the crack (f = 90°), the influence coefficients for n = 2 to 4 are given by


G = 2 Q 16 + 1 M 16 M + 1 M +  (9.19a)
2 π  15 3 1 105 2 12 3 

G = 2 Q 32 + 1 M 32 M + 1 M +  (9.19b)
3 π  35 4 1 315 2 20 3 


G = 2 Q 256 + 1 M 256 M + + 1 M  (9.19c)
4 π  315 5 1 3465 2 30 3 