Section 22.214.171.124. The Resistance Curve Approach
126.96.36.199 The Resistance (R)
If the crack tip plastic zone size
is estimated to be on the order of the structural thickness but substantially
smaller than other geometrical variables (crack length, ligament size, height,
etc.), a linear elastic fracture mechanics analysis can still be sensibly used
to predict the catastrophic cracking event.
The failure criterion for tearing type fractures under these conditions
states that fracture will occur when (1) the stress-intensity factor K reaches or exceeds the material’s
fracture resistance KR and
(2) the rate of change of K (with
respect to crack length) reaches or exceeds the rate of change of KR (with respect to crack
length). Physically, the criterion
means that at failure, the energy available to extend the crack equals or
exceeds the material resistance to crack growth. The failure criterion becomes simply,
The corresponding applied stress, sf,
at this point is defined as the fracture stress that determines the residual
strength of the cracked structure. The
criterion presented in Equation 4.2.4 is noted to be a two-parameter criterion
rather than the single parameter criteria that was presented in paragraph
188.8.131.52. To interpret the meaning of
this criterion, first consider the structural parameters that are a function of
the geometry and stress, i.e. K and ÐK/Ða.
In general, the estimation of K involves the relationship K = sbÖpa
as given in Section 2; using this equation, the variation of K with respect to crack length (a) can be obtained for various values of
stress (s) as shown in Figure 4.2.11a.
Shown in Figure 4.2.11b is the variation of KR with respect to the crack
extension (Da) that was developed
for the given material using the procedures outlined in Figure 4.2.7. Since this R-curve is assumed to be independent of the initial crack length,
it can be superimposed on the plot of K
versus a as shown in Figure 4.2.11c.
The tangency point between the applied stress intensity factor curve (K vs. a) and the R-curve (KR vs. Da)
determines the commencement of unstable crack propagation. In general, the accurate method of
determining the tangency point involves the numerical solution based on the
experimentally obtained R-curve. Using a least squares determined polynomial
expression for R-curve and knowing an
expression for K in terms of crack
length, the common tangent point can be obtained by equating the functional
values (K = KR) and also the first derivatives with respect to the
crack length dK/da = dKR /da of these two expressions.
Figure 4.2.11. Schematic Illustration of the Individual and
Collective Parts of a KR
The slow stable tear is dependent on a structural configuration
in which the plastic zone at the crack tip
is no longer negligible but not enormous.
et al. , Srawley & Brown
, and McCabe  explain the dependence of the R-curve on structural configuration as
well as with test procedures used to evaluate the R-curve. See Section 7 for
additional information on test procedures and the Damage Tolerant Design (Data)
Handbook [Skinn, et al., 1994] for a summary of available data.