Page 151 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

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Elastic-Plastic Fracture Mechanics 131

Equation (3.59) and Equation (3.60) gradually become invalid as specimen boundaries interact
with the crack tip. We can apply dimensional arguments to infer when a single-parameter description
of crack-tip conditions is suspect. As discussed in Chapter 2, the LEFM solution breaks down when
the plastic-zone size is a significant fraction of in-plane dimensions. Moreover, the crack-tip
conditions evolve from plane strain to plane stress as the plastic-zone size grows to a significant
fraction of the thickness. The J integral becomes invalid as a crack-tip-characterizing parameter
when the large strain region reaches a finite size relative to in-plane dimensions. Section 3.6 provides
quantitative information on size effects.

3.5.2 J-CONTROLLED CRACK GROWTH
According to the dimensional argument in the previous section, J-controlled conditions exist at the
tip of a stationary crack (loaded monotonically and quasistatically), provided the large strain region
is small compared to the in-plane dimensions of the cracked body. Stable crack growth, however,
introduces another length dimension, i.e., the change in the crack length from its original value.
Thus J may not characterize crack-tip conditions when the crack growth is significant compared
to the in-plane dimensions. Prior crack growth should not have any adverse effects in a purely
elastic material, because the local crack-tip fields depend only on current conditions. However,
prior history does influence the stresses and strains in an elastic-plastic material. Therefore, we
might expect the J integral theory to break down when there is a combination of significant plasticity
and crack growth. This heuristic argument based on dimensional analysis agrees with experiment
and with more complex analyses.
Figure 3.24 illustrates crack growth under J-controlled conditions. The material behind the
growing crack tip has unloaded elastically. Recall Figure 3.7, which compares the unloading
behavior of nonlinear elastic and elastic-plastic materials; the material in the unloading region of
Figure 3.24 obviously violates the assumptions of deformation plasticity. The material directly in
front of the crack also violates the single-parameter assumption because the loading is highly
nonproportional, i.e., the various stress components increase at different rates and some components
actually decrease. In order for the crack growth to be J controlled, the elastic unloading and non-
proportional plastic loading regions must be embedded within a zone of J dominance. When the
crack grows out of the zone of J dominance, the measured R curve is no longer uniquely charac-
terized by J.
In small-scale yielding, there is always a zone of J dominance because the crack-tip conditions
are defined by the elastic stress intensity, which depends only on the current values of the load and
crack size. The crack never grows out of the J-dominated zone as long as all the specimen boundaries
are remote from the crack tip and the plastic zone.
Figure 3.25 illustrates three distinct stages of crack growth resistance behavior in small-scale
yielding. During the initial stage the crack is essentially stationary; the finite slope of the R curve

FIGURE 3.24 J-controlled crack growth.