Page 188 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

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168 Fracture Mechanics: Fundamentals and Applications

Thus the J integral remains constant during crack extension (dJ/da = 0) when Equation (3.69)
is satisfied. Steady-state crack growth is usually not observed experimentally because large-scale
yielding in finite-sized specimens precludes characterizing a growing crack with J. Also, a signif-
icant amount of crack growth may be required before a steady state is reached (Figure 3.25); the
crack tip in a typical laboratory specimen approaches a free boundary well before the crack growth
is sufficient to be unaffected by the initial blunted tip.

A3.6 NOTES ON THE APPLICABILITY OF DEFORMATION PLASTICITY
TO CRACK PROBLEMS
Since elastic-plastic fracture mechanics is based on the deformation plasticity theory, it may be
instructive to take a closer look at this theory and assess its validity for crack problems.
Let us begin with the plastic portion of the Ramberg-Osgood equation for uniaxial deformation,
which can be expressed in the following form:

 σ  n−1 σ
ε α =   (A3.59)
p
 σ  E
o
Differentiating Equation (A3.59) gives

 σ  n−2 σσ
d
d α = nε   (A3.60)
p
 σ o  E σ o
for an increment of plastic strain. For the remainder of this section, the subscript on strain is
suppressed for brevity; only plastic strains are considered, unless stated otherwise.
Equation (A3.59) and Equation (A3.60) represent the deformation and incremental flow theo-
ries, respectively, for uniaxial deformation in a Ramberg-Osgood material. In this simple case, there
is no difference between the incremental and deformation theories, provided no unloading occurs.
Equation (A3.60) can obviously be integrated to obtain Equation (A3.59). Stress is uniquely related
to strain when both increase monotonically. It does not necessarily follow that deformation and
incremental theories are equivalent in the case of three-dimensional monotonic loading, but there
are many cases where this is a good assumption.
Equation (A3.59) can be generalized to three dimensions by assuming deformation plasticity
and isotropic hardening:

e
ε 3 α =   σ   n−1 S ij (A3.61)
ij
2  σ  E
o
where s is the effective (von Mises) stress and S is the deviatoric component of the stress tensor,
e
ij
defined by
S = − σ 1 σ δ (A3.62)
ij
ij
3 kk ij
where d is the Kronecker delta. Equation (A3.61) is the deformation theory flow rule for a Ramberg-
ij
Osgood material. The corresponding flow rule for incremental plasticity theory is given by

ij
d 3 α = nε   σ   n−2 S dσ e (A3.63)
e
ij
2  σ o  E σ o