Page 78 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 58 Thursday, April 14, 2005 6:28 PM

58 Fracture Mechanics: Fundamentals and Applications

FIGURE 2.27 Through crack configuration analyzed in Example 2.6: (a) definition of coordinate axes and
(b) arbitrary traction applied to crack faces.

If we apply a surface traction of ±p(x) on the crack faces, the Mode I stress intensity factor for the two
crack tips is as follows:
K = 1 ∫ 2 a px() x dx
Ix=2 a) π a 0 a 2 x −
(
−
K = 1 ∫ 2 a px() 2 ax dx
Ix=0 ) π a 0 x
(

The weight function concept is not restricted to two-dimensional bodies, Mode I loading, or
isotropic elastic materials. In their early work on weight functions, Rice [15] extended the theory
to three dimensions, Bueckner [16] considered combined Mode I/II loading, and both allowed for
anisotropy in the elastic properties. Subsequent researchers [17–22] have shown that the theory
applies to all linear elastic bodies that contain an arbitrary number of cracks.
For mixed-mode problems, separate weight functions are required for each mode: h , h , and
I
II
h . Since the stress intensity factors can vary along a three-dimensional crack front, the weight
III
functions also vary along the crack front. That is
η
h a h = a x (, ) (2.52)
i
where α( = 1, 2, 3) indicates the mode of loading and η is the crack front position.
Given that any loading configuration in a cracked body can be represented by equivalent crack-
face tractions, the general mixed-mode, three-dimensional formulation of the weight function
approach can be expressed in the following form:

K a ∫ T () i hη a x= i d S ( , ) η (2.53)
S c
where T are the tractions assumed to act on the crack surface S .
i
c
See Chapter 9 for examples of practical applications of weight functions.
2.7 RELATIONSHIP BETWEEN K AND G

Two parameters that describe the behavior of cracks have been introduced so far: the energy release
rate and the stress intensity factor. The former parameter quantifies the net change in potential