# Chapter 5 Schwalbe

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5.1

PART B: FATIGUE5. MATERIAL RESISTANCE 5.1 Fatigue Life under Constant Amplitude Loading 5.1.1 The S N curve The strength of a material under repeated loading is lower than that under a single loading. One observes that repeated loading (usually called cyclic loading) with a stress below the strength of the material under consideration the material breaks after a certain number of load cycles. This effect is called fatigue since repeated loading lowers the stress the material is able to sustain. Therefore, the strength under repeated loading is termed fatigue strength. The simplest form of cyclic loading is characterised by constant values of the stress amplitude and the mean stress, Fig. 5.1.

m :

Average stress a : Stress amplitude

max :

Maximum stress : Cyclic stress min : Minimum stress

Figure 5.1: Definition of characteristic stress values during a load cycle.

The ratio R=

min max

(5.1)

is called stress ratio and can vary as follows: Tension Range: Tension/Compression:

min 0, min < 0,

0 R 0,Compression Range:

max < 0,

5.2 Failure occurs after a certain number of load cycles - number of load cycles to failure, N f which increases with decreasing loading, Fig. 5.2. It is common belief that in the case of ferritic steels one of the most common structural materials at stress amplitudes below the endurance limit E life is infinite. It will be shown later, however, that according to more recent research this is not the case. The data points ( a , N f ) form the S N curve (in German literature: Whlerkurve).

Figure 5.2: S-N curve (Whlerkurve).

Failure under cyclic loading conditions consists of three different events: Formation of cracks Crack propagation Final failure

which are shown in Fig. 5.3. Fatigue crack propagation and final fracture can be described using the methods of fracture mechanics.

Figure 5.3: The three phases of fatigue fracture.

The transition from crack formation to crack propagation is very difficult and is subject to arbitraryness; it is a matter of the resolution of the tools for observing and measuring a crack, Fig. 5.4. In more application oriented work, the early stages of crack propagation up to several hundredths of a millimetre are regarded as part of the crack formation process since cracks

5.3 must be detectable by the available techniques of non-destructive inspection (NDI). Therefore, crack propagation covers only a relatively small percentage of the total life. On the other hand, fundamental studies using high resolution techniques such as electron microscopy demonstrate that cracks appear in a very early stage of life.

Figure 5.4: Fatigue crack propagation during the life of a specimen or component for various Crack initiation sites [5.1].

As a rule, fatigue cracks are formed at the surface of a specimen or component; their formation depends strongly on the local surface conditions and is therefore subject to large scatter. In the following, the individual processes of fatigue will be neglected, only the final result: the number offload cycles to failure, N f , will be of interest. The S N curve can be partitioned into three ranges, Fig. 5.5:

5.4 Low cycle fatigue, with N f 10.000 High cycle fatigue Endurance range.

The endurance range begins at a load cycle limit, N lim , which is N lim = 10 7 N lim = 108

for structural steels for non-ferrous metals.

Figure 5.5: The three ranges of an S-N curve.

The endurance limit is that stress amplitude which can be applied to a material with an infinite number of load cycles. Non-ferrous metals and alloys usually do not exhibit a true endurance limit. In that case, the endurance limit is defined at 10 8 load cycles. The S-N curve is determined by keeping either the mean stress or the stress range constant. The stress amplitude, a , is plotted either linearly, Fig. 5.5, or logarithmically, Fig. 5.6, as a function of log(N f ). In the high cycle fatigue range, the equations log(N f ) = a - b a or log(N f ) = a blog( a ) (5.3) (5.4)

are frequently being used.

5.5

Figure 5.6: High cycle fatigue and endurance region in log-log scales.

5.1.2 Parameters affecting the S N curve A large number of factors affect the behaviour under cyclic loading. Fig. 5.7 provides an overview on some important effects on the S-N curve. Under the same stress amplitude, a higher strength of the material to tested leads to smaller micro-plastic deformations. Consequently, fatigue life and endurance limit are increased, since fatigue is a result of cyclic plastic deformations. The loading frequency affects the fatigue strength insofar as air is a mild corrosive agent which may affect the fatigue life. Corrosion processes are time dependent so that at higher frequencies there is less time per load cycle available than at lower frequencies. Therefore, fatigue life and endurance limit increase. Similarly, if fatigue loading occurs in media more aggressive than air, then the fatigue life decreases and the endurance limit can disappear. Hardening of the surface layer and introduction of residual stresses are the mechanisms by which shot peening improves the fatigue properties. However, this very positive effect can be partially or entirely removed if shot peening is carried out beyond the optimum peening parameters. Stress concentrations, such as notches, reduce fatigue life and endurance limit. Under low cycle fatigue conditions, however, the notch effect may be reversed. A further important parameter is represented by the mean stress. The fatigue life is reduced if the mean stress is raised.

5.6

Figure 5.7: Some parameters affecting fatigue strength.

Figure 5.8: Effects of material strength and stress concentrations on endurance limit.

5.7 The combined effect of strength and stress concentrations is illustrated in Fig. 5.8. As already mentioned above, increasing strength leads to higher endurance limit. This, however, is only valid for smooth and polished surfaces. In this case, the following relationships describe the endurance limit under rotating bending conditions [5.4]: Normalised carbon steels (R m = 340700MPa):E

= (0.454R m + 8.4) MPa

(5.5)

Heat treated carbon steels (R m = 4001300MPa): E = (0.515R m - 24)MPa Heat treated alloyed steels (R m = 8001300)MPa: E = (0.383R m + 94)MPa Stainless steels (R m = 5001300)MPa: E = 0.484MPa

(5.6)

(5.7)

(5.8)

Wrought aluminium alloys (R m < 300MPa): E = 0.4R m

(5.9)

The endurance limit of notched parts, En , is determined by the notch acuity, given by the elastic stress concentration factorn

=

max n

,

(5.10)

where - max designates the maximum stress in the notch root, and - n is the nominal stress in the cross section. The notch effect on the endurance limit is given by the notch effect factorn

=

E En

,

(5.11)

where - E is the endurance limit of unnotched specimens, and - En is the endurance limit of notched specimens, expressed in terms of nominal stress, n .

n from which it follows that notched components can be loaded higher than expected from n , Fig. 5.9. Close examination of theThe notch effect factor is usually smaller than notch root shows that the material fails indeed by the formation of fatigue cracks which, however, do not propagate as they run into a region of lower local stress, Fig. 5.10. This in turn means that it is not only the maximum stress at the notch root that matters, the stress gradient in the interior of the material is also important.

5.8

2

n

1 2

n

3

Figure 5.9: Correlation between the notch effect on the endurance limit, n, and the elastic stress concentration factor, n.

Figure 5.10: Crack in the stress field of a notch.

The quotient q= k 1 k 1 (5.11)

is known as notch sensitivity. The values of k and q depend on material and notch geometry. For q = 0, a material is completely insensitive to the presence of notches; this is the case for grey cast iron where due to the high acuity of the internal notches caused by the vermicular graphite, the macroscopic external notch has no effect any more. A value of q = 1 means complete notch sensitivity.

5.9 The effect of the mean stress on the endurance limit can be approximated by D,m = E 1 ( m ) n Rm

(5.12)

where 1