Page 412 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C009.fm Page 392 Monday, May 23, 2005 3:58 PM

392 Fracture Mechanics: Fundamentals and Applications

Since the flaw in Figure 9.7 is near a weld, there is a possibility that weld residual stresses will
be present. These stresses must be taken into account in order to obtain an accurate estimate of K .
I
Weld residual stresses are discussed further in Section 9.1.4.

9.1.3 WEIGHT FUNCTIONS FOR ARBITRARY LOADING

While the influence coefficient approach is useful, it has limitations. It requires that the stress
distribution be fitted to a polynomial. There are many instances where a polynomial expression
(fourth order or lower) does not provide a good representation of the actual stress field. For
example, the five-term Taylor series expansion of the hoop stress in a thick-wall pressure vessel
(Equation (9.8) and Equation (9.9)) is not accurate for R /t < 4.
i
The weight function method, which was introduced in Chapter 2, provides a means to compute
stress-intensity solutions for arbitrary loading. Consider a surface crack of depth a, subject to a
normal stress that is an arbitrary function of x, where x is oriented in the crack depth direction and
2
is measured from the free surface. The Mode I stress-intensity factor is given by

K I ∫ a h = x a x (, ) ( ) d xσ (9.12)
0

where hx a) is the weight function.
(,
For the deepest point of a semielliptical crack (f = 90°), the weight function can be represented
by an equation of the following form [13,14]:

h = 2   1+ M 1−  x  12 / + M 1−  x  + M 1−  x  32 /   (9.13)
90 − 1  a  2  a  3  a 
2π( ax)  

where the coefficients M to M depend on the geometry and crack dimensions. The corresponding
1
3
expression for the free surface (f = 0°) is given by [13,14]
2   x  12 /  x   x  32 / 
h =  1+ N + N + N  (9.14)
0 π x  1  a  2  a  3  a  

Each of these expressions contains three unspecified coefficients. However, one unknown can
be eliminated by invoking boundary conditions on the weight function [13,14]. For the deepest
point of the crack, imposing the condition that the second derivative of the weight function is zero
at x = 0 implies that M = 3. Setting h = 0 at x = a results in the following relationship:
0
2
=
N N + N + + 10 (9.15)
1 2 3
Therefore, the weight function coefficients M and N can be inferred from reference K solutions
i
i
I
for two load cases on the configuration of interest. The choice of reference load cases is arbitrary,
but it is convenient to use uniform and linear crack-face pressure (n = 0 and n = 1, respectively).
The corresponding influence coefficients for these load cases are G and G . Setting Equation (9.5)
1
0
2 Note that the present discussion is restricted to a one-dimensional normal stress distribution. That is, the pressure normal
to the crack face may vary over the depth of the flaw, but it is constant along the flaw length at a given x value. Equation
(2.53) describes the general case where the traction varies arbitrarily over the crack surface.