Page 182 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 162 Monday, May 23, 2005 5:42 PM

162 Fracture Mechanics: Fundamentals and Applications

In order for J to be path independent, it cannot depend on r , which was defined arbitrarily. The
2
=
radius vanishes in Equation (A3.34) only when ( + n )( −1 s ) + 2 1 0 (n + 1)(s − 2) + 1 = 0, or
s = 2 n +1
n +1
which is identical to the result obtained numerically (Equation (A3.31)). Thus the amplitude of the
stress function is given by

1
 J  +1
n
κ =   (A3.36)
I
 ασε 
oo n
Substituting Equation (A3.36) into Equation (A3.30) yields the familiar form of the HRR singularity:

1
 EJ n  +1
σ σ = o  σ ˜ θ n (, ) (A3.37)
ij
 ασ Ir  ij
2
on
since e = s /E. The integration constant is plotted in Figure 3.10 for both plane stress and plane
o
o
strain, while Figure 3.11 shows the angular variation of ˜ σ ij for n = 3 and n = 13.
Rice and Rosengren [8] obtained essentially identical results to Hutchinson (for plane strain),
although they approached the problem in a somewhat different manner. Rice and Rosengren began
with a heuristic argument for the 1/r variation of strain energy density, and then introduced an Airy
stress function of the form of Equation (A3.29) with the exponent on r given by Equation (A3.31).
They computed stresses, strains, and displacements in the vicinity of the crack tip by applying the
appropriate boundary conditions.
The HRR singularity was an important result because it established J as a stress amplitude
parameter within the plastic zone, where the linear elastic solution is invalid. The analyses of
Hutchinson, Rice, and Rosengren demonstrated that the stresses in the plastic zone are much higher
in plane strain than in plane stress. This is consistent with the simplistic analysis in Section 2.10.1.
One must bear in mind the limitations of the HRR solution. Since the singularity is merely
the leading term in an asymptotic expansion, and elastic strains are assumed to be negligible, this
solution dominates only near the crack tip, well within the plastic zone. For very small r values,
however, the HRR solution is invalid because it neglects finite geometry changes at the crack tip.
When the HRR singularity dominates, the loading is proportional, which implies a single-parameter
description of crack-tip fields. When the higher-order terms in the series are significant, the
loading is often nonproportional and a single-parameter description may no longer be possible
(see Section 3.6).

A3.5 ANALYSIS OF STABLE CRACK GROWTH
S
IN MALL-SCALE YIELDING
A3.5.1 The Rice-Drugan-Sham Analysis

Rice, Drugan, and Sham (RDS) [15] performed an asymptotic analysis of a growing crack in an
elastic-plastic solid in small-scale yielding. They assumed crack extension at a constant crack-
opening angle, and predicted the shape of J resistance curves. They also speculated about the effect
of large-scale yielding on the crack growth resistance behavior.