Page 153 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

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Elastic-Plastic Fracture Mechanics 133

the material, leaving a plastic wake, as illustrated in Figure 3.25. The R curve is flat; J does not increase
with crack extension, provided the material properties do not vary with position. Appendix 3.5.2
presents a formal mathematical argument for a flat R curve during steady-state growth; a heuristic
explanation is given below.
If Equation (3.63) applies, J uniquely describes crack-tip conditions, independent of crack exten-
sion. If the material fails at some critical combination of stresses and strains, it follows that local
failure at the crack tip must occur at a critical J value, as in the stationary crack case. This critical J
value must remain constant with crack growth. A rising or falling R curve would imply that the local
material properties varied with position.
The second stage in Figure 3.25 corresponds to the transition between the blunting of a stationary
crack and crack growth under steady state conditions. A rising R curve is possible in Stage 2. For
small-scale yielding conditions the R curve depends only on crack extension:
J R J = R a (∆ ) (3.64)

That is, the J-R curve is a material property.
The steady-state limit is usually not observed in laboratory tests on ductile materials. In typical
test specimens, the ligament is fully plastic during crack growth, thereby violating the small-scale
yielding assumption. Moreover, the crack approaches a finite boundary while still in Stage 2 growth.
Enormous specimens would be required to observe steady-state crack growth in tough materials.

3.6 CRACK-TIP CONSTRAINT UNDER LARGE-SCALE YIELDING

Under small-scale yielding conditions, a single parameter (e.g., K, J, or CTOD) characterizes crack-
tip conditions and can be used as a geometry-independent fracture criterion. Single-parameter
fracture mechanics breaks down in the presence of excessive plasticity, and the fracture toughness
depends on the size and geometry of the test specimen.
McClintock [18] applied the slip line theory to estimate the stresses in a variety of configurations
under plane strain, fully plastic conditions. Figure 3.26 summarizes some of these results. For
small-scale yielding (Figure 3.26(a)), the maximum stress at the crack tip is approximately 3s in
o
a nonhardening material. According to the slip line analysis, a deeply notched double-edged notched
tension (DENT) panel, illustrated in Figure 3.26(b), maintains a high level of triaxiality under fully
plastic conditions, such that the crack-tip conditions are similar to the small-scale yielding case.
An edge-cracked plate in bending (Figure 3.26(c)) exhibits slightly less stress elevation, with the
maximum principal stress approximately 2.5s . A center-cracked panel in pure tension (Figure 3.26(d))
o
is incapable of maintaining significant triaxiality under fully plastic conditions.
The results in Figure 3.26 indicate that for a nonhardening material under fully yielded condi-
tions, the stresses near the crack tip are not unique, but depend on geometry. Traditional fracture
mechanics approaches recognize that the stress and strain fields remote from the crack tip may
depend on geometry, but it is assumed that the near-tip fields have a similar form in all configurations
that can be scaled by a single parameter. The single-parameter assumption is obviously not valid
for nonhardening materials under fully plastic conditions, because the near-tip fields depend on the
configuration. Fracture toughness, whether quantified by J, K, or CTOD, must also depend on the
configuration.
The prospects for applying fracture mechanics in the presence of large-scale yielding are not
quite as bleak as the McClintock analysis indicates. The configurational effects on the near-tip
fields are much less severe when the material exhibits strain hardening. Moreover, single-parameter
fracture mechanics may be approximately valid in the presence of significant plasticity, provided
the specimen maintains a relatively high level of triaxiality. Both the DENT specimen and the edge-
cracked plate in bending (SE(B)) apparently satisfy this requirement. Most laboratory measurements