Page 186 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

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166 Fracture Mechanics: Fundamentals and Applications

2
in small-scale yielding) unless a = a and b = 4l EJ /s . Equation (A3.51) does not predict a
Ic
fy
ssy
o
steady-state limit where T = 0; rather this relationship predicts that T actually increases as the
ligament becomes smaller.
The foregoing analysis implies that crack-growth-resistance curves obtained from specimens
with fully yielded ligaments are suspect. One should exercise extreme caution when applying
experimental results from small specimens to predict the behavior of large structures.
A3.5.2 Steady State Crack Growth

The RDS analysis, which assumed a local failure criterion based on crack-opening angle, indicated
that crack growth in small-scale yielding reaches a steady state, where dJ/da → 0. The derivation
that follows shows that the steady-state limit is a general result for small-scale yielding; the R curve
must eventually reach a plateau in an infinite body, regardless of the failure mechanism.

Generalized Damage Integral

Consider a material element a small distance from a crack tip, as illustrated in Figure A3.7. This
material element will fail when it is deformed beyond its capacity. The crack will grow as consecutive
material elements at the tip fail. Let us define a generalized damage integral Θ, which characterizes
the severity of loading at the crack tip:

Θ ∫ ε eq Ω = (σε ij ) ε eq (A3.52)
d
,
ij
0
where e is the equivalent (von Mises) plastic strain and Ω is a function of the stress and strain
eq
tensors (s and e , respectively). The above integral is sufficiently general that it can depend on the
ij
ij
current values of all stress and strain components, as well as the entire deformation history. Referring
to Figure A3.7, the material element will fail at a critical value of Θ. At the moment of crack initiation
or during crack extension, the material near the crack tip will be close to the point of failure. At a
distance r* from the crack tip, where r* is arbitrarily small, we can assume that Θ = Θ c.
The precise form of the damage integral depends on the micromechanism of fracture. For
example, a modified Rice and Tracey [47] model for ductile hole growth (see Chapter 5) can be
used to characterize ductile fracture in metals:

 R  ε eq  .15σ 
0
Θ= ln   R o   = .283 ∫ 0 exp   σ e m   dε eq (A3.53)

FIGURE A3.7 Material point a distance r* from the
crack tip.