Page 150 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

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130 Fracture Mechanics: Fundamentals and Applications

With large-scale yielding (Figure 3.23(c)), the size of the finite strain zone becomes significant
relative to L, and there is no longer a region uniquely characterized by J. Single-parameter fracture
mechanics is invalid in large-scale yielding, and critical J values exhibit a size and geometry dependence.
In certain configurations, the K and J zones are vanishingly small, and a single-parameter descrip-
tion is not possible except at very low loads. For example, a plate loaded in tension with a through-
thickness crack is not amenable to a single-parameter description, either by K or J. Example 2.7 and
Figure 2.39 indicate that the stress in the x direction in this geometry deviates significantly from the
elastic singularity solution at small distances from the crack tip because of a compressive transverse
(T) stress. Consequently the K-dominated zone is virtually nonexistent. The T stress influences stresses
inside the plastic zone, so a highly negative T stress also invalidates a single-parameter description
in terms of J. See Section 3.61 for further details about the T stress.
Recall Section 2.10.1, where a free-body diagram was constructed from a disk-shaped region
removed from the crack tip of a structure loaded in small-scale yielding. Since the stresses on the
boundary of this disk exhibit a 1 r singularity, K uniquely defines the stresses and strains within
I
5
the disk. For a given material, dimensional analysis leads to the following functional relationship
for the stress distribution within this region:
σ ij  K I 2 
θ
σ o = F ij  σ 2 o r θ ,   for 0 ≤≤r r s () (3.59)

where r is the radius of the elastic singularity dominated zone, which may depend on q. Note that the
s
1 r singularity is a special case of F, which exhibits a different dependence on r within the plastic
zone. Invoking the relationship between J and K for small-scale yielding (Equation 3.18) gives
I
σ ij  EJ 
′
θ
σ o = F ij  σ o 2 r θ ,   for 0 ≤≤r r J () (3.60)
where r is the radius of the J-dominated zone. The HRR singularity (Equation (3.24a)) is a special
J
case of Equation (3.60), but stress exhibits a r − n+1/( 1) dependence only over a limited range of r.
For small-scale yielding, r = r , but r vanishes when the plastic zone engulfs the elastic
s
s
J
singularity dominated zone. The J-dominated zone usually persists longer than the elastic singularity
zone, as Figure 3.23 illustrates.
It is important to emphasize that the J dominance at the crack tip does not require the existence
of an HRR singularity. In fact, J dominance requires only that Equation (3.60) is valid in the process
zone near the crack tip, where the microscopic events that lead to fracture occur. The HRR
singularity is merely one possible solution to the more general requirement that J uniquely define
crack-tip stresses and strains. The flow properties of most materials do not conform to the ideali-
zation of a Ramberg-Osgood power law, upon which the HRR analysis is based. Even in a Ramberg-
Osgood material, the HRR singularity is valid over a limited range; large strain effects invalidate
the HRR singularity close to the crack tip, and the computed stress lies below the HRR solution
at greater distances. The latter effect can be understood by considering the analytical technique
employed by Hutchinson [7], who represented the stress solution as an infinite series and showed
that the leading term in the series was proportional to r − n+1/( 1) (see Appendix 3.4). This singular
term dominates as r → 0; higher-order terms are significant for moderate values of r. When the
computed stress field deviates from HRR, it still scales with J/(s r), as required by Equation (3.60).
o
Thus J dominance does not necessarily imply agreement with the HRR fields.

5 A complete statement of the functional relationship of σ ij should include all material flow properties (e.g., α and n for a
Ramberg-Osgood material). These quantities were omitted from Equation (3.59) and Equation (3.60) for the sake of clarity,
since material properties are assumed to be fixed in this problem.