Page 422 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C009.fm Page 402 Monday, May 23, 2005 3:58 PM

402 Fracture Mechanics: Fundamentals and Applications

Since Equation (9.31) gives an expression for the P-∆ curve for a stationary crack, it is
p
possible to compare J estimates from Equation (9.29) and Equation (9.32) with Equation (9.35).
pl
According to Equation (9.31), the P-∆ curve follows a power law, where the exponent is the
p
same as in the material’s true stress–true strain curve. The plastic energy absorbed by the specimen
is as follows:
∫ ∆ p Pd ∆ p n P∆ = p

0 n +1
n  P  n +1
= P aαε h   (9.36)
n +1 o o 3  P o 

Thus the plastic J is given by

n a  P  n+1
J = η P ε α h   (9.37)
pl
n +1 po o b 3  P 
o
Comparing Equation (9.29) and Equation (9.37) and solving for h gives
p
2
η = n +1 σ bh 1 3 (9.38a)
o
PWh
n
p
o
Alternatively, if J is given by Equation (9.34)
pl
2
o
η = n +1 σ bh 1 3 (9.38b)
p
n
PWh
o
Consider an edge-cracked bend specimen in plane strain. The EPRI fully plastic J solution for
this configuration is tabulated in Table A9.8. The reference load, assuming unit thickness and the
standard span of 4W, is given by
0 364σ b 2
.
P = W o (9.39)
o
Substituting Equation (9.39) into Equation (9.38a) gives

η = 1 n +1 W h 1 (9.40)
p
0 364 n a h
.
3
Equation (9.40) is plotted in Figure 9.10 for n = 5 and n = 10. According to the equation that was
derived in Section 3.2.5, h = 2. This derivation, however, is valid only for deep cracks, since it
p
assumes that the ligament length b is the only relevant length dimension. Figure 9.10 indicates that
Equation (9.40) approaches the deep crack limit with increasing a/W. For n = 10, the deep crack
formula appears to be reasonably accurate beyond a/W~0.3. Note that the h values computed from
p
Equation (9.40) for deep cracks fluctuate about an average of ~1.9, rather than the theoretical value
of 2.0. The deviations from the theoretical value for deep cracks may be indicative of numerical
errors in the h and h values.
3
1