18.1 Introduction

In Chapters 14 and 15, we examined the conditions under which a crack was stable, and would not grow, and the condition

K=Kc

18.2 Fatigue of Uncracked Components

Fatigue tests are done by subjecting specimens of the material to a cyclically varying load or displacement (Figure 18.1). For simplicity, the time variation of stress and strain is shown as a sine wave, but modern testing machines are capable of applying almost any chosen load or displacement spectrum. The specimens are subjected to increasing numbers of fatigue cycles N, until they finally crack (when N = N_f—the number of cycles to failure).

Figure 18.2 shows how the cyclic stress and cyclic strain are coupled in a linear elastic specimen. The stress and strain are linearly related through Young’s modulus, and the stress range Δσ produces (or is produced by) an elastic strain range Δɛ^el (equal to Δσ/E).

Figure 18.3 shows how the cyclic stress and cyclic strain are coupled in a specimen that is cycled outside the elastic limit. There is no longer a simple relationship between stress and strain. Instead, they are related by a stress–strain loop, which must be determined from tests on the material. In addition, the shape of the loop changes with the number of cycles, and it can take several hundred (or even thousand) cycles before the shape of the loop stabilizes. The stress range Δσ produces (or is produced by) a total strain range Δɛ^tot—which is the sum of the elastic strain range Δɛ^el and the plastic strain range Δɛ^pl.

The best way to correlate fatigue test data is on a log-log plot of the total strain amplitude Δɛ^tot/2 versus the number of reversals to failure 2N_f (there are two reversals of load or displacement in each complete cycle). Figure 18.4 shows the shape of the curve on which the data points typically fall (although there is usually a lot of experimental scatter on either side of the curve). It is useful to know that this curve is the sum of two linear relationships on the log–log plot: (a) between the elastic strain amplitude and 2N_f, and (b) between the plastic strain amplitude and 2N_f. It can be approximated mathematically as:

Δɛtot2≈σf′(2Nf)bE+ɛf′(2Nf)c

b and c are constants determined by fitting the test data—they are generally in the range –0.05 to –0.12 for b, and –0.5 to –0.7 for c. σf′ and ɛf′ are the true fracture stress and true fracture strain (derived from a standard tensile test on the material).

The data in Figure 18.4 can be divided into two régimes:

• ▪ Low-cycle fatigue (less than about 10⁴ cycles; plastic strain > elastic strain)
  
• ▪ High-cycle fatigue (more than about 10⁴ cycles; elastic strain > plastic strain)

Until the 1950s, most fatigue studies were concerned with high-cycle fatigue (HCF), since engineering components subjected to cyclic loadings (e.g., railway axles, engine crankshafts, bicycle frames) were designed to keep the maximum stress below the elastic limit. Because of this, it is still common practice to plot HCF data on a log–log plot of the stress amplitude Δσ/2 versus 2N_f—see Figure 18.5. The test data can then be approximated as:

Δσ2≈σf′(2Nf)b

So far, we have only considered test data obtained with zero mean stress (σ_m = 0). However, in many design situations, there will be a tensile mean stress (σ_m > 0). Intuitively, we would expect that the component would be more prone to fatigue if it were subjected to a large mean stress in addition to having to cope with repeated cycles of stress. The test data confirm this—to keep the fatigue life the same when the mean stress is increased from 0 to some large tensile value, the stress (or strain) cycles must be reduced in amplitude to compensate.

In terms of the strain approach to fatigue, the test data can be approximated as follows:

Δɛtot2≈(σf′–σm)(2Nf)bE+ɛf′(2Nf)c

18.3 Fatigue of Cracked Components

Data on fatigue crack propagation are gathered by cyclically loading specimens containing a sharp crack like that shown in Figure 18.6. We define

ΔK=Kmax–Kmin=YΔσπa

The cyclic stress intensity ΔK increases with time (at constant load) because the crack grows in tension. It is found that the crack growth per cycle, da/dN, increases with ΔK in the way shown in Figure 18.7.

In the steady-state régime, the crack-growth rate is described by

dadN=A(ΔK)m