Page 341 - Mechanical Behavior of Materials

342 Chapter 8 Fracture of Cracked Members

8.2.4 Effects of Cracks on Brittle Versus Ductile Behavior

Consider the crack length where the failure stress predicted by LEFM equals the yield strength,
identified as a t in Fig. 8.5. Substituting S c = σ o into Eq. 8.3 gives its value:

1 K c 2
a t = (8.4)
π σ o

On an approximate basis, cracks longer than this transition crack length will cause the strength to
be limited by brittle fracture, rather than by yielding. Thus, if cracks of length around or greater than
the a t of a given material are likely to be present, fracture mechanics should be employed in design.
Conversely, for crack lengths below a t , yielding dominated behavior is expected, so that there will
be little or no strength reduction due to the crack.
Note that Eq. 8.4 is based on the assumed case of a wide, center-cracked plate, and that a t
will differ for other geometric cases. It is nevertheless useful to employ a t values from Eq. 8.4 as
representative quantities in comparing different materials.
Consider two materials, one with low σ o and high K c , and the other with an opposite
combination, namely, high σ o and low K c . These combinations of properties cause a relatively large
a t for the low-strength material, but a small a t for the high-strength one. Compare Figs. 8.6(a)
and (b). Thus, cracks of moderate size may not affect the low-strength material, but they may
severely limit the usefulness of the high-strength one. Such an inverse trend between yield
strength and fracture toughness is fairly common within any given class of materials. Low strength
in a tension test is usually accompanied by high ductility and also by high fracture toughness.
Conversely, high strength is usually associated with low ductility and low fracture toughness. Trends
of this nature for AISI 1045 steel are illustrated in Fig. 8.7.

yielding alone a
fracture alone t σ high
actual behavior o

a t σ low S, Stress σ (b)
S, Stress ut
o

(a) K high
c
K low
c
a
i
0 0
a, Crack Length a, Crack Length
Figure 8.6 Transition crack length a t for a low-strength, high-toughness material (a), and
for a high-strength, low-toughness material (b). If (b) contains internal flaws a i , its strength
in tension σ ut is controlled by brittle fracture.