Page 304 - Materials Science and Engineering An Introduction

276 • Chapter 8 / Failure

EXAMPLE PROBLEM 8.3

Computation of Minimum Specimen Diameter to Yield a Specified Fatigue
Lifetime for Tension-Compression Tests
A cylindrical 70Cu-30Zn brass bar (Figure 8.20) is subjected to axial tension–compression
stress testing with reversed-cycling. If the load amplitude is 10,000 N, compute the minimum
7
allowable bar diameter to ensure that fatigue failure will not occur at 10 cycles. Assume a fac-
tor of safety of 2.5, data in Figure 8.20 were taken for reversed axial tension–compression tests,
and that S is stress amplitude.
Solution
6
7
2
From Figure 8.20, the fatigue strength for this alloy at 10 cycles is 115 MPa (115 10 N/m ).
Tensile and compressive stresses are defined in Equation 6.1 as
F
s = (6.1)
A 0
is the cross-sectional area. For a cylindrical bar having a
Here, F is the applied load and A 0
diameter of d 0 ,
2
d 0
A 0 = pa b
2
Substitution of this expression for A 0 into Equation 6.1 leads to

F F 4F
s = = = (8.21)
2 2
A 0 d 0 pd 0
pa b
2
We now solve for d 0 , replacing stress with the fatigue strength divided by the factor of safety
(i.e., s/N). Thus,
4F
d 0 = (8.22)
s
H pa N b

Incorporating values of F, N, and s cited previously leads to

(4)(10,000 N)
d 0 =
6
115 * 10 N>m 2
H (p)a 2.5 b
-3
= 16.6 * 10 m = 16.6 mm
Hence, the brass bar diameter must be at least 16.6 mm to ensure that fatigue failure will not occur.

8.9 CRACK INITIATION AND PROPAGATION 8
The process of fatigue failure is characterized by three distinct steps: (1) crack initiation,
in which a small crack forms at some point of high stress concentration; (2) crack propa-
gation, during which this crack advances incrementally with each stress cycle; and (3)
final failure, which occurs very rapidly once the advancing crack has reached a critical
size. Cracks associated with fatigue failure almost always initiate (or nucleate) on the

8 More detailed and additional discussion on the propagation of fatigue cracks can be found in Sections M.10 and
M.11 of the Mechanical Engineering (ME) Online Module.