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Elastic-Plastic Fracture Mechanics 165

be proportional to the ligament length. They made a rough estimate of R ~ b/4 for the fully plastic
case.
The constant a depends on crack-tip triaxiality and thus may differ for small-scale yielding
and fully yielded conditions. For highly constrained configurations, such as bend specimens, a for
the two cases should be similar.
In small-scale yielding, the definition of J is unambiguous, since it is related to the elastic stress
intensity factor. The J integral for a growing crack under fully plastic conditions can be computed
in a number of ways, however, and not all definitions of J are appropriate in the large-scale yielding
version of Equation (A3.38).
Assume that the crack-growth resistance behavior is to be characterized by a J-like parameter
J . Assuming J depends on the crack length and displacement, the rate of change in J should be
x
x
x
linearly related to the displacement rate and : ˙ a
J ˙ x ˙ + ξ χ∆ ˙ a = (A3.47)

where x and c are functions of displacement and crack length. Substituting Equation (A3.47) into
Equation (A3.38) gives

R
δ ˙ α ξ= ∆   β + σ o ln   + α χ   ˙ a (A3.48)
˙
r
σ o  E   σ o 
In the limit of a rigid ideally plastic material, s /E = 0. Also, the local crack-opening rate must be
o
proportional to the global displacement rate for a rigid ideally plastic material:
δ ˙ ψ = ∆ ˙ (A3.49)

Therefore, the term in square brackets in Equation (A3.48) must vanish, which implies that c = 0,
at least in the limit of a rigid ideally plastic material. Thus, in order for the RDS model to apply
to large-scale yielding, the rate of change in the J-like parameter must not depend on the crack-
growth rate:
. .
a
J x J ≠ x ( ˙) (A3.50)

Rice et al. showed that neither the deformation theory J nor the far-field J satisfy Equation (A3.50)
for all configurations.
Satisfying Equation (A3.50) does not necessarily imply that a J -R curve is geometry indepen-
x
dent. The RDS model suggests that a resistance curve obtained from a fully yielded specimen will
not, in general, agree with the small-scale yielding R curve for the same material. Assuming R = b/4
for the fully plastic case, the RDS model predicts the following tearing modulus:

 β    α
T T = o −  ln  b/4   ssy (A3.51)
2
  α ssy  λ EJ σ IC / o    α fy

where the subscripts ssy and fy denote small-scale yielding and fully yielded conditions, respec-
tively. According to Equation (A3.51), the crack-growth-resistance curve under fully yielded
conditions has a constant initial slope, but this slope is not equal to T (the initial tearing modulus
o