Page 76 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 56 Thursday, April 14, 2005 6:28 PM

56 Fracture Mechanics: Fundamentals and Applications

FIGURE 2.25 Uncracked body subject to an
arbitrary boundary traction P(x), which results
in a normal stress distribution p(x) acting on
Plane A-B.

2.6.5 WEIGHT FUNCTIONS

When one performs an analysis to infer a stress intensity factor for a cracked body, the K value
that is computed applies only to one particular set of boundary conditions; different loading
conditions result in a different stress intensity factors for that geometry. It turns out, however, that
the solution to one set of boundary conditions contains sufficient information to infer K for any
other boundary conditions on that same geometry.
Consider two arbitrary loading conditions on an isotropic elastic cracked body in plane stress
or plane strain. For now, we assume that both loadings are symmetric with respect to the crack
plane, such that pure Mode I loading is achieved in each case. Suppose that we know the stress
(2)
intensity factor for loading (1) and we wish to solve for K , the stress intensity factor for the
I
(1)
second set of boundary conditions. Rice [15] showed that K and K are related as follows:
(2)
I
I
E′  u ∂ () u ∂ () 
1
1
K () = ()  ∫ T i d Γ ∫ F + i dA  (2.49)
2
I
1
2 K I  Γ i a ∂ A i a ∂ 

FIGURE 2.26 Application of superposition to replace a boundary traction P(x) with a crack face traction
p(x) that results in the same K I .