Page 391 - Mechanical Behavior of Materials

392 Chapter 8 Fracture of Cracked Members

However, before proceeding, it is useful to take note of the concept of a fully plastic yielding
load. For a given cracked member, the fully plastic force P o , or moment M o , is the load or
moment necessary to cause yielding to spread across the entire remaining uncracked portion of
the cross-sectional area. Large and unstably increasing strains and deflections occur once P o or
M o is exceeded. If the stress–strain curve of the material is idealized as being elastic, perfectly
plastic, then lower bound estimates of fully plastic loads may be made, as described in Appendix A,
Section A.7. Some particular results are given in Fig. A.16.
Three approaches to extending fracture mechanics beyond linear elasticity will now be
introduced: (1) the plastic zone adjustment, (2) the J-integral, and (3) the crack-tip opening
displacement (CTOD).

8.9.1 Plastic Zone Adjustment
Consider the redistributed stress near the plastic zone, as in Fig. 8.43. The stresses outside of the
plastic zone are similar to those for the elastic stress field equations for a hypothetical crack of
length a e = a + r oσ , that is, a hypothetical crack with its tip near the center of the plastic zone. This
in turn leads to modifying K, increasing it, to account for this yielding by using a e in place of the
actual crack length a.
√
Where the form K = FS πa is used, the modified value is

2

√ 1 K e
K e = F e S πa e = F e S π(a + r oσ ), where r oσ = (8.42)
2π σ o
The F used is the value corresponding to a e /b, and r oσ is calculated by using K e in Eq. 8.36.
An iterative calculation is generally involved in using this equation, as F e = F(a e /b) cannot be
determined in advance, since r oσ and hence a e depend on K e .If F is not significantly changed for
the new crack length a e , then no iteration is required, and the modified value K e is related to the
√
unmodified value K = FS πa by

K
(8.43)
K e =
2
1 FS

1 −
2 σ o
In some situations with a high degree of constraint, such as embedded elliptical cracks or half-
elliptical surface cracks, it may be appropriate to use the plane strain plastic zone size to make the
adjustment. Replacing r oσ with r oε in the previous equations gives a relationship similar to Eq. 8.43,
1
differing in that the 1 in the denominator is replaced by .
2 6
Such modified K values allow LEFM to be extended to somewhat higher stress levels than
permitted by the limitation of Eq. 8.39. However, large amounts of yielding still cannot be analyzed.
The use of even adjusted crack lengths becomes increasingly questionable if the stress approaches
a value that would cause yielding fully across the uncracked section of the member. It is suggested
that the use of plastic zone adjustments be limited to loads below 80% of the fully plastic force or
moment—that is, below 0.8P o or 0.8M o .