Page 292 - Materials Science and Engineering An Introduction

264 • Chapter 8 / Failure

are desired. On the basis of this s
criterion, rank the metal alloys
listed in Table B.5, Appendix
B, as to critical crack size, from
longest to shortest. 2a
(b) An alternative design that is p
also often utilized with pressure t p p
vessels is termed leak-before- r
break. On the basis of principles p p
of fracture mechanics, allow-
ance is made for the growth of p p s
a crack through the thickness p
of the vessel wall prior to the
occurrence of rapid crack prop-
agation (Figure 8.11). Thus, the Figure 8.11 Schematic diagram showing the cross section
crack will completely penetrate of a spherical tank that is subjected to an internal pressure p
the wall without catastrophic and that has a radial crack of length 2a in its wall.
failure, allowing for its detection by the leaking of pressurized fluid. With this criterion the
critical crack length a c (i.e., one-half the total internal crack length) is taken to be equal to the
pressure vessel thickness t. Allowance for a c t instead of a c t/2 ensures that fluid leakage
will occur prior to the buildup of dangerously high pressures. Using this criterion, rank the
metal alloys in Table B.5, Appendix B, as to the maximum allowable pressure.
For this spherical pressure vessel, the circumferential wall stress s is a function of the pressure
p in the vessel and the radius r and wall thickness t according to
pr
s = (8.8)
2t
For both parts (a) and (b), assume a condition of plane strain.

Solution
(a) For the first design criterion, it is desired that the circumferential wall stress be less than
the yield strength of the material. Substitution of s y for s in Equation 8.5 and incorporation of
a factor of safety N leads to

s y
K Ic = Y a b 1pa c (8.9)
N
where a c is the critical crack length. Solving for a c yields the following expression:
N 2 K Ic 2
a c = a b (8.10)
2
Y p s y
Therefore, the critical crack length is proportional to the square of K Ic /s y , which is the basis for
the ranking of the metal alloys in Table B.5. The ranking is provided in Table 8.3, where it may
be seen that the medium carbon (1040) steel with the largest ratio has the longest critical crack
length and, therefore, is the most desirable material on the basis of this criterion.
(b) As stated previously, the leak-before-break criterion is just met when one-half the internal
crack length is equal to the thickness of the pressure vessel—that is, when a t. Substitution
of a t into Equation 8.5 gives
K Ic = Ys1pt (8.11)