Page 51 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 31 Thursday, April 14, 2005 6:28 PM

Linear Elastic Fracture Mechanics 31

FIGURE 2.4 A penny-shaped (circular) crack embed-
ded in a solid subjected to a remote tensile stress.

The Griffith approach can be applied to other crack shapes. For example, the fracture stress
for a penny-shaped flaw embedded in the material (Figure 2.4) is given by

E
 πγ  12 /
σ =  s  (2.20)
f
2
)
 21 ( − va
where a is the crack radius and ν is Poisson’s ratio.

2.3.1 COMPARISON WITH THE CRITICAL STRESS CRITERION
The Griffith model is based on a global energy balance: for fracture to occur, the energy stored
in the structure must be sufficient to overcome the surface energy of the material. Since fracture
involves the breaking of bonds, the stress on the atomic level must be equal to the cohesive
stress. This local stress intensification can be provided by flaws in the material, as discussed in
Section 2.2.
The similarity between Equation (2.13), Equation (2.14), and Equation (2.19) is obvious.
Predictions of the global fracture stress from the Griffith approach and the local stress criterion
differ by less than 40%. Thus, these two approaches are consistent with one another, at least in the
case of a sharp crack in an ideally brittle solid.
An apparent contradiction emerges when the crack-tip radius is significantly greater than the
atomic spacing. The change in the stored energy with crack formation (Equation (2.16)) is insen-
sitive to the notch radius as long as a >> b; thus, the Griffith model implies that the fracture stress
is insensitive to ρ. According to the Inglis stress analysis, however, in order for σ to be attained
c
at the tip of the notch, σ must vary with 1 . ρ
f