The problem of fatigue

Low-amplitude vibration causes no permanent damage in materials. Increase the amplitude, however, and the material starts to suffer fatigue. You will, at some time or other, have used fatigue to flex the lid off a sardine can or to break an out-of-date credit card in two. The cyclic stress hardens the material and causes damage in the form of dislocation tangles to accumulate, from which a crack nucleates and grows until it reaches the critical size for fracture. Anything that is repeatedly loaded (like an oil rig loaded by the waves or a pressure vessel that undergoes pressure cycles), or rotates under load (like an axle), or reciprocates under load (like an automobile connecting rod), or vibrates (like the rotor of a helicopter) is a potential candidate for fatigue failure.

Figure 9.2 shows in a schematic way how the stress on the underside of an aircraft wing might vary during a typical flight, introducing some important practicalities of fatigue. First, the amplitude of the stress varies as the aircraft takes off, cruises at altitude, bumps its way through turbulence and finally lands. Second, on the ground the weight of the engines (and the wings themselves) bends them downward, putting the underside in compression. Once in flight, aerodynamic lift bends them upward, putting the underside in tension. The amplitude of the stress cycles and their mean values change during flight. Fatigue failure depends on both. And of course it depends on the total number of cycles. Aircraft wings bend to and fro at a frequency of a few hertz. In a trans-Atlantic flight, tens of thousands of loading cycles take place; in the lifetime of the aircraft, it is millions. For this reason, fatigue testing needs to apply tens of millions of fatigue cycles to provide meaningful design data.

Figure 9.2 Schematic of stress cycling on the underside of a wing during flight.

The food can and credit card are examples of low-cycle fatigue, meaning that the component survives for only a small number of cycles. It is typical of cycling at stresses above the yield stress, σ_y, like that shown in Figure 9.1(c). More significant in engineering terms is high-cycle fatigue: here the stresses remain generally elastic and may be well below σ_y, as in cycle (b) of Figure 9.1; cracks nonetheless develop and cause failure, albeit taking many more cycles to do so, as in Wöhler’s railway axles.

Figure 9.1 Cyclic loading. (a) Very low amplitude acoustic vibration. (b) High-cycle fatigue: cycling well below general yield, σ_y. (c) Low-cycle fatigue: cycling above general yield (but below the tensile strength σ_ts).

In both cases we are dealing with components that are initially undamaged, containing no cracks. In these cases, most of the fatigue life is spent generating the crack. Its growth to failure occurs only at the end. We call this initiation-controlled fatigue. Some structures contain cracks right from the word go, or are so safety-critical (like aircraft) that they are assumed to have small cracks. There is then no initiation stage—the crack is already there—and the fatigue life is propagation controlled, that is, it depends on the rate at which the crack grows. A different approach to design is then called for—we return to this later.

High-cycle fatigue and the S–N curve

Figure 9.3 shows how fatigue characteristics are measured and plotted. A sample is cyclically stressed with amplitude Δσ/2 about a mean value σ_m, and the number of cycles to cause fracture is recorded. The data are presented as Δσ − N_f (’S–N’) curves, where Δσ is the peak-to-peak range over which the stress varies and N_f is the number of cycles to failure. Most tests use a sinusoidally varying stress with an amplitude σ_a of

Figure 9.3 Fatigue strength at 10⁷ cycles, the endurance limit, σ_e.

(9.1)

and a mean stress σ_m of

(9.2)

all defined in the figure. Fatigue data are usually reported for a specified R-value:

(9.3)

An R-value of −1 means that the mean stress is zero; an R-value of 0 means the stress cycles from 0 to σ_max. For many materials there exists a fatigue or endurance limit, σ_e (units: MPa). It is the stress amplitude σ_a, about zero mean stress, below which fracture does not occur at all, or occurs only after a very large number (N_f > 10⁷) of cycles. Design against high-cycle fatigue is therefore very similar to strength-limited design, but with the maximum stresses limited by the endurance limit σ_e rather than the yield stress σ_y.

Experiments show that the high-cycle fatigue life is approximately related to the stress range by what is called Basquin’s law:

(9.4)

where b and C₁ are constants; the value of b is small, typically 0.07 and 0.13. Dividing Δσ by the modulus E gives the strain range Δε (since the sample is elastic):

(9.5)

or, taking logs,

This is plotted in Figure 9.4, giving the right-hand, high-cycle fatigue part of the curve with a slope of 2b.

Figure 9.4 The low- and high-cycle regimes of fatigue and their empirical description of fatigue.

Low-cycle fatigue

In low-cycle fatigue the peak stress exceeds yield, so at least initially (before work hardening raises the strength), the entire sample is plastic. Basquin is no help to us here; we need another empirical law, this time that of Dr Lou Coffin:

(9.6)

where Δε^pl means the plastic strain range—the total strain minus the (usually small) elastic part. For our purposes we can neglect that distinction and plot it in Figure 9.4 as well, giving the left-hand branch. Coffin’s exponent, c, is much larger than Basquin’s; typically it is 0.5.

High-Cycle fatigue: mean stress and variable amplitude

These laws adequately describe the fatigue failure of uncracked components cycled at constant amplitude about a mean stress of zero. But as we saw, real loading histories are often much more complicated (Figure 9.2). How do we make some allowance for variations in mean stress and stress range? Here we need yet more empirical laws, courtesy this time of Goodman and Miner. Goodman’s rule relates the stress range for failure under a mean stress σ_m to that for failure at zero mean stress :

(9.7)

where σ_ts is the tensile stress, giving a correction to the stress range. Increasing σ_m causes a small stress range to be as damaging as a larger applied with zero mean (Figure 9.5(a)). The corrected stress range may then be plugged into Basquin’s law.

Figure 9.5 (a) The endurance limit refers to a zero mean stress. Goodman’s law scales of the stress range to a mean stress σ_m. (b) When the cyclic stress amplitude changes, the life is calculated using Miner’s cumulative damage rule.

The variable amplitude problem can be addressed approximately with Miner’s rule of cumulative damage. Figure 9.5(b) shows an idealised loading history with three stress amplitudes (all about zero mean). Basquin’s law gives the number of cycles to failure if each amplitude was maintained throughout the life of the component. So if N₁ cycles are spent at stress amplitude Δσ₁ a fraction N₁/N_f1 of the available life is used up, where N_f1 is the number of cycles to failure at that stress amplitude. Miner’s rule assumes that damage accumulates in this way at each level of stress. Then failure will occur when the sum of the damage fractions reaches 1—that is, when

(9.8)

Goodman’s law and Miner’s rule are adequate for preliminary design, but they are approximate; in safety-critical applications, tests replicating service conditions are essential. It is for this reason that new models of cars and trucks are driven over rough ‘durability tracks’ until they fail—it is a test of fatigue performance.

The discussion so far has focused on initiation-controlled fatigue failure of uncracked components. Now it is time to look at those containing cracks.

Fatigue loading of cracked components

In fabricating large structures like bridges, ships, oilrigs, pressure vessels and steam turbines, cracks and other flaws cannot be avoided. Cracks in castings appear because of differential shrinkage during solidification and entrapment of oxide and other inclusions. Welding, a cheap, widely used joining process, can introduce both cracks and internal stresses caused by the intense local heating. If the cracks are sufficiently large that they can be found it may be possible to repair them, but finding them is the problem. All nondestructive testing (NDT) methods for detecting cracks have a resolution limit; they cannot tell us that there are no cracks, only that there are none longer than the resolution limit, c_lim. Thus, it is necessary to assume an initial crack exists and design the structure to survive a given number of loadings. So how is the propagation of a fatigue crack characterised?

Fatigue crack growth is studied by cyclically loading specimens containing a sharp crack of length c like that shown in Figure 9.6. We define the cyclic stress intensity range, ΔK, using equation (8.4), as

Figure 9.6 Cyclic loading of a cracked component. A constant stress amplitude Ds gives an increasing amplitude of stress intensity, as the crack grows in length.

(9.9)

The range ΔK increases with time under constant cyclic stress because the crack grows in length: the growth per cycle, dc/dN, increases with ΔK in the way shown in Figure 9.7. The rate is zero below a threshold cyclic stress intensity ΔK_th, useful if you want to make sure it does not grow at all. Above it, there is a steady-state regime described by the Paris law:

Figure 9.7 Crack growth during cyclic loading.

(9.10)

where K and m are constants. At high ΔK the growth rate accelerates as the maximum applied K approaches the fracture toughness K_1c. When it reaches K_1c the sample fails in a single load cycle.

Safe design against fatigue failure in potentially cracked components means calculating the number of loading cycles that can safely be applied without the crack growing to a dangerous length. We return to this in Chapter 10.

Material damping: the mechanical loss coefficient

There are many mechanisms of material damping. Some are associated with a process that has a specific time constant; then the energy loss is centred about a characteristic frequency. Others are frequency independent; they absorb energy at all frequencies. In metals a large part of the loss is caused by small-scale dislocation movement: it is high in soft metals like lead and pure aluminum. Heavily alloyed metals like bronze and high-carbon steels have low loss because the solute pins the dislocations; these are the materials for bells. Exceptionally high loss is found in some cast irons, in manganese–copper alloys and in magnesium, making them useful as materials to dampen vibration in machine tools and test rigs. Engineering ceramics have low damping because the dislocations in them are immobilised by the high lattice resistance (which is why they are hard). Porous ceramics, on the other hand, are filled with cracks, the surfaces of which rub, dissipating energy, when the material is loaded. In polymers, chain segments slide against each other when loaded; the relative motion dissipates energy. The ease with which they slide depends on the ratio of the temperature T to the glass temperature, T_g, of the polymer. When T/T_g < 1, the secondary bonds are ‘frozen’, the modulus is high and the damping is relatively low. When T/T_g > 1, the secondary bonds have melted, allowing easy chain slippage; the modulus is low and the damping is high.

Fatigue damage and cracking

A perfectly smooth sample with no changes of section, and containing no inclusions, holes or cracks, would be immune to fatigue provided neither σ_max nor σ_min exceeds its yield strength. But that is a vision of perfection that is unachievable. Blemishes, small as they are, can be deadly. Rivet holes, sharp changes in section, threads, notches and even surface roughness concentrate stress in the way described in Chapter 7, Figure 7.7. Even though the general stress levels are below yield, the locally magnified stresses can lead to reversing plastic deformation. Dislocation motion is limited to a small volume near the stress concentration, but that is enough to cause damage that finally develops into a tiny crack.

In high-cycle fatigue, once a crack is present it propagates in the way shown in Figure 9.9(a). During the tensile part of a cycle a tiny plastic zone forms at the crack tip, stretching it open and thereby creating a new surface. On the compressive part of the cycle the crack closes again and the newly formed surface folds forward, advancing the crack. Repeated cycles make it inch forward, leaving tiny ripples on the crack face marking its position on each cycle. These ‘striations’ are characteristic of a fatigue failure and are useful, in a forensic sense, for revealing where the crack started and how fast it propagated.

Figure 9.9 (a) In high-cycle fatigue a tiny zone of plasticity forms at the crack tip on each tension cycle; on compression the newly formed surface folds forward. (b) In low-cycle fatigue the plastic zone is large enough for voids to nucleate and grow within it. Their coalescence further advances the crack.

In low-cycle fatigue the stresses are higher and the plastic zone larger, as in Figure 9.9(b). It may be so large that the entire sample is plastic, as it is when you flex the lid of a tin to make it break off. The largest strains are at the crack tip, where plasticity now causes voids to nucleate, grow and link, just as in ductile fracture (Chapter 8).

Choosing materials that are strong

The chart of Figure 9.8 established the close connection between endurance limit and tensile strength. Most design is based not on tensile strength but on yield strength, and here the correlation is less strong. We define the fatigue ratio, F_r, as

Figure 9.10 shows data for F_r for an age-hardening aluminum alloy and a low alloy steel. Both can be heat-treated to increase their yield strength, and higher strength invites higher design stresses. The figure, however, shows that the fatigue ratio F_r falls as strength σ_y increases, meaning that the gain in endurance limit is considerably less than that in yield strength. In propagation-controlled fatigue, too, the combination of higher stress and lower fracture toughness implies that smaller crack sizes will propagate, causing failure (and the resolution of inspection techniques may not be up to the job).

Figure 9.10 The drop in fatigue ratio with increase in yield strength for an aluminum alloy and a steel, both of which can be treated to give a range of strengths.

Making sure they contain as few defects as possible

Mention has already been made of clean alloys—alloys with carefully controlled compositions that are filtered when liquid to remove unwanted particles that, in the solid, can nucleate fatigue cracks. Nondestructive testing using X-ray imaging or ultrasonic sensing detects dangerous defects, allowing the part to be rejected. And certain secondary processes like hot isostatic pressing (HIPing), which heats the component under a large hydrostatic pressure of inert argon gas, can seal cracks and collapse porosity.

Providing a compressive surface stress

Cracks only propagate during the tensile part of a stress cycle; a compressive stress forces the crack faces together, clamping it shut. Fatigue cracks frequently start from the surface, so if a thin surface layer can be given an internal stress that is compressive, any crack starting there will remain closed even when the average stress across the entire section is tensile. This is achieved by treatments that plastically compress the surface by shot peening (Figure 9.11): leaf-springs for cars and trucks are ‘stress peened’, meaning that they are bent to a large deflection and then treated in the way shown in the figure; this increases their fatigue life by a factor of 5. A similar outcome is achieved by sandblasting or burnishing (local deformation with a smooth, polished tool), or diffusing atoms into the surface, expanding a thin layer, which, because it is bonded to the more massive interior, becomes compressed.

Figure 9.11 Shot peening, one of several ways of creating compressive surface stresses.