Page 370 - Mechanical Behavior of Materials

Section 8.6 Fracture Toughness Values and Trends 371

Solution From Fig. A.7(b), the maximum stress in the vessel wall is

pr 1 (6MPa)(750 mm)
σ t = = = 225 MPa
2t 2(10 mm)
√
Combining this value with K Ic = 187 MPa m from Table 8.1, the critical crack length is
√ 2

1 K Ic 1 187 MPa m

c c = = = 0.220 m = 220 mm
π σ t π 225 MPa
This far exceeds the wall thickness of t = 10 mm, so the leak-before-break condition is met.
When the vessel leaks, the crack length along the surface is 2c = 2t, so that c = t = 10 mm.
At this point, the stress intensity factor is
√ √
K = FS πa = 1(225 MPa) π(0.01 m) = 39.9MPa m

Here, the situation is treated as a center crack in a wide plate, as in Fig. 8.12(a), with substitutions
F = 1, S = σ t , and a = c. Hence, the safety factor on K is

√
K Ic 187 MPa m
X K = = √ = 4.69 Ans.
K 39.9MPa m
This is a reasonable value, so the vessel is safe from brittle fracture.
Noting that the principal stresses are σ 1 = σ 2 = 225 MPa and σ 3 ≈ 0, we conclude that the
effective stress from Eq. 7.21 is ¯σ S = 225 MPa, and the safety factor against yielding is

σ o 760 MPa
X o = = = 3.38 Ans.
¯ σ S 225 MPa
where the yield strength is also from Table 8.1. Hence, yielding is unlikely.

8.6 FRACTURE TOUGHNESS VALUES AND TRENDS

In fracture toughness testing, an increasing displacement is applied to an already cracked specimen
of the material of interest until it fractures. The arrangement used for a bend specimen is shown
in Fig. 8.27. Growth of the crack is detected by observing the force versus displacement (P-v)
behavior, as in Fig. 8.28. A deviation from linearity on the P-v plot, or a sudden drop in force due
to rapid cracking, identifies a point P Q corresponding to an early stage of cracking. The value of
K, denoted K Q , is then calculated for this point. If there is some tearing of the crack prior to final
fracture, K Q may be somewhat lower than the value K c corresponding to the final fracture of the
specimen.