Page 148 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 128 Monday, May 23, 2005 5:42 PM

128 Fracture Mechanics: Fundamentals and Applications

where the variation in b during the loading history is taken into account. The deformation theory
leads to the following relationship for J in this specimen:
J D . o b = 073σ Ω (3.58)

The two expressions are obviously identical when the crack is stationary.
Finite element calculations of Dodds et al. [16, 17] for a three-point bend specimen made from
a strain hardening material indicate that J and J are approximately equal for moderate amounts
f
D
of crack growth. The J integral obtained from a contour integration is path-dependent when a crack
is growing in an elastic-plastic material, however, and tends to zero as the contour shrinks to the
crack tip. See Appendix 4.2 for a theoretical explanation of the path dependence of J for a growing
crack in an inelastic material.
There is no guarantee that either the deformation J or J will uniquely characterize crack-tip conditions
D
f
for a growing crack. Without this single-parameter characterization, the J-R curve becomes geometry
dependent. The issue of J validity and geometry dependence is explored in detail in Section 3.5 and
Section 3.6.

3.5 J-CONTROLLED FRACTURE

The term J-controlled fracture corresponds to situations where J completely characterizes crack-tip
conditions. In such cases, there is a unique relationship between J and CTOD (Section 3.3); thus
J-controlled fracture implies CTOD-controlled fracture, and vice versa. Just as there are limits to
LEFM, fracture mechanics analyses based on J and CTOD become suspect when there is excessive
plasticity or significant crack growth. In such cases, fracture toughness and the J-CTOD relationship
depend on the size and geometry of the structure or test specimen.
The required conditions for J-controlled fracture are discussed below. Fracture initiation from
a stationary crack and stable crack growth are considered.

3.5.1 STATIONARY CRACKS

Figure 3.23 schematically illustrates the effect of plasticity on the crack tip stresses; log (s ) is
yy
plotted against the normalized distance from the crack tip. The characteristic length scale L
corresponds to the size of the structure; for example, L could represent the uncracked ligament
length. Figure 3.23(a) shows the small-scale yielding case, where both K and J characterize crack-
tip conditions. At a short distance from the crack tip, relative to L, the stress is proportional to
1 r ; this area is called the K-dominated region. Assuming monotonic, quasistatic loading, a
J-dominated region occurs in the plastic zone, where the elastic singularity no longer applies. Well
inside of the plastic zone, the HRR solution is approximately valid and the stresses vary as r − n+1/( 1) .
The finite strain region occurs within approximately 2d from the crack tip, where large deformation
invalidates the HRR theory. In small-scale yielding, K uniquely characterizes crack-tip conditions,
despite the fact that the 1 r singularity does not exist all the way to the crack tip. Similarly, J
uniquely characterizes crack-tip conditions even though the deformation plasticity and small strain
assumptions are invalid within the finite strain region.
Figure 3.23(b) illustrates elastic-plastic conditions, where J is still approximately valid, but there is
no longer a K field. As the plastic zone increases in size (relative to L), the K-dominated zone disappears,
but the J-dominated zone persists in some geometries. Thus although K has no meaning in this case,
the J integral is still an appropriate fracture criterion. Since J dominance implies CTOD dominance, the
latter parameter can also be applied in the elastic-plastic regime.