Page 187 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 167 Monday, May 23, 2005 5:42 PM

Elastic-Plastic Fracture Mechanics 167

where
R = void radius
R = initial radius
o
s = mean (hydrostatic) stress
m
s = effective (von Mises) stress
e
Failure, in this case, is assumed when the void radius reaches a critical value.

Stable Crack Growth

Consider an infinite body that contains a crack that is growing in a stable, self-similar, and
10
quasistatic manner. If the crack has grown well beyond the initial blunted tip, dimensional analysis
indicates that the local stresses and strains are uniquely characterized by the far-field J integral, as
stated in Equation (3.69). In light of this single-parameter condition, the integrand of Equation
(A3.52) becomes

 J 
Ω Ω =   σ o r θ ,   (A3.54)

We can restrict this analysis to q = 0 by assuming that the material on the crack plane fails during
Mode I crack growth. For a given material point on the crack plane, r decreases as the crack grows,
and the plastic strain increases. If strain increases monotonically as this material point approaches
the crack tip, Equation (A3.54) permits writing Ω as a function of the von Mises strain:

Ω Ω = (ε eq ) (A3.55)

Therefore the local failure criterion is given by

Θ c ∫ ε * Ω = ( ε e q d ) ε e q (A3.56)
0
where e* is the critical strain (i.e., the von Mises strain at r = r*). Since the integrand is a function
only of e , the integration path is the same for all material points ahead of the crack tip, and e*
eq
is constant during crack growth. That is, the equivalent plastic strain at r* will always equal e*
when the crack is growing. Based on Equation (3.69) and Equation (A3.54), e* is a function only
*
of r and the applied J:
Jr
ε ε* = *( , *) (A3.57)
*
Solving for the differential of e gives

∂ ε ∂ ε * *
dε* = dJ + dr * (A3.58)
J ∂ ∂ r *

Since e* and r* are both fixed, de* = dr* = dJ = 0.

10 In practical terms ‘‘infinite’’ means that external boundaries are sufficiently far from the crack tip so that the plastic zone
is embedded within an elastic singularity zone.