Page 89 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

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Linear Elastic Fracture Mechanics 69

FIGURE 2.36 Effect of strain-hardening on the
Mode I plastic zone; n = 5 corresponds to a high strain-
hardening material, while n = 50 corresponds to very
low hardening (cf. Equation (2.86)).

effective stress results in a much larger plastic zone. Given the difficulties of defining the plastic
zone unambiguously with a detailed analysis, the estimates of the plastic zone size and shape from
the elastic analysis (Figure 2.34) appear to be reasonable.
Figure 2.36 illustrates the effect of strain hardening on the plastic zone. A high strain-hardening
rate results in a smaller plastic zone because the material inside of the plastic zone is capable of
carrying higher stresses, and less stress redistribution is necessary.

2.9 K-CONTROLLED FRACTURE

Section 2.6.1 introduced the concept of the singularity-dominated zone and alluded to single-
parameter characterization of crack-tip conditions. The stresses near the crack tip in a linear elastic
material vary as 1/ r ; the stress intensity factor defines the amplitude of the singularity. Given the
equations in Table 2.1 to Table 2.3, one can completely define the stresses, strains, and displacements
in the singularity-dominated zone if the stress intensity factor is known. If we assume a material
fails locally at some combination of stresses and strains, then crack extension must occur at a
critical K value. This K value, which is a measure of fracture toughness, is a material constant
crit
that is independent of the size and geometry of the cracked body. Since energy release rate is
uniquely related to stress intensity (Section 2.7), G also provides a single-parameter description of
crack-tip conditions, and G is an alternative measure of toughness.
c
The foregoing discussion does not consider plasticity or other types of nonlinear material
behavior at the crack tip. Recall that the 1 r singularity applies only to linear elastic materials. The
equations in Table 2.1 to Table 2.3 do not describe the stress distribution inside the plastic zone. As
discussed in Chapter 5 and Chapter 6, the microscopic events that lead to fracture in various materials
generally occur well within the plastic zone (or damage zone, to use a more generic term). Thus,
even if the plastic zone is very small, fracture may not nucleate in the singularity-dominated zone.
This fact raises an important question: Is stress intensity a useful failure criterion in materials that
exhibit inelastic deformation at the crack tip?
Under certain conditions, K still uniquely characterizes crack-tip conditions when a plastic zone
is present. In such cases, K is a geometry-independent material constant, as discussed below.
crit
Consider a test specimen and structure loaded to the same K level, as illustrated in Figure 2.37.
I
Assume that the plastic zone is small compared to all the length dimensions in the structure and
test specimen. Let us construct a free-body diagram with a small region removed from the crack tip
of each material. If this region is sufficiently small to be within the singularity-dominated zone,
the stresses and displacements at the boundary are defined by the relationships in Table 2.1 and
Table 2.2. The disc-shaped region in Figure 2.37 can be viewed as an independent problem. The
imposition of the 1 r singularity at the boundary results in a plastic zone at the crack tip. The
size of the plastic zone and the stress distribution within the disc-shaped region are a function only
of the boundary conditions and material properties. Therefore, even though we do not know the
actual stress distribution in the plastic zone, we can argue that it is uniquely characterized by the