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280 Mechanical Engineering Design
Figure 6–15 Log da
dN
When da/dN is measured
in Fig. 6–14 and plotted on Region I Region II
log-log coordinates, the data
Crack Crack
for different stress ranges initiation propagation Region III
superpose, giving rise to a Crack
Increasing
sigmoid curve as shown. unstable
stress ratio
( K I ) th is the threshold value R
of K I , below which a crack
does not grow. From threshold
to rupture an aluminum alloy
will spend 85–90 percent of
life in region I, 5–8 percent in
region II, and 1–2 percent
in region III.

K c
(ΔK) th
Log ΔK
Table 6–1
m/cycle in/cycle
Material C, m m
Conservative Values of C, √ m √
MPa m kpsi in
Factor C and Exponent
m in Eq. (6–5) for Ferritic-pearlitic steels 6.89(10 −12 ) 3.60(10 −10 ) 3.00
Various Forms of Steel Martensitic steels 1.36(10 −10 ) 6.60(10 −9 ) 2.25
. −12 −10
(R = σ max /σ min = 0) Austenitic stainless steels 5.61(10 ) 3.00(10 ) 3.25

From J. M. Barsom and S. T. Rolfe, Fatigue and Fracture Control in Structures, 2nd ed., Prentice Hall,
Upper Saddle River, NJ, 1987, pp. 288–291, Copyright ASTM International. Reprinted with permission.

8
conditions prevail. Assuming a crack is discovered early in stage II, the crack growth in
region II of Fig. 6–15 can be approximated by the Paris equation, which is of the form
da m
= C( K I ) (6–5)
dN
where C and m are empirical material constants and K I is given by Eq. (6–4).
Representative, but conservative, values of C and m for various classes of steels are
listed in Table 6–1. Substituting Eq. (6–4) and integrating gives

N f a f
1 da

dN = N f = √ m (6–6)
0 C a i (β σ πa)
Here a i is the initial crack length, a f is the final crack length corresponding to failure,
and N f is the estimated number of cycles to produce a failure after the initial crack is
formed. Note that β may vary in the integration variable (e.g., see Figs. 5–25 to 5–30).

8 Recommended references are: Dowling, op. cit.; J. A. Collins, Failure of Materials in Mechanical Design,
John Wiley & Sons, New York, 1981; H. O. Fuchs and R. I. Stephens, Metal Fatigue in Engineering, John
Wiley & Sons, New York, 1980; and Harold S. Reemsnyder, “Constant Amplitude Fatigue Life Assessment
Models,” SAE Trans. 820688, vol. 91, Nov. 1983.