1. A brief introduction to the fatigue phenomenon
(last updated )

2. Aim
The aim of this presentation is to give a short introduction to the fatigue phenomenon.
In addition we will take a quick glance at stress concentrations, cracks and cyclic crack growth.

3. What is fatigue?
A body or structure subjected to repeated loadings can fail, even though the magnitude of the applied loading is far below any yield- or static failure limit- we refer to this phenomenon as fatigue.
Generally, the higher the load amplitude is, the shorter the life will be (in terms of loading cycles).
By obvious reasons fatigue failures may be both dangerous and expensive, and must therefore be avoided. An additional complicating factor is that fatigue failures often occur without any previous "warnings", and are therefore especially awkward. As a result, fatigue design has become an increasingly important task in the modern product design process.

4. An example of a real fatigue failure case
As an example of a real fatigue case, we can note the front wheel failures of a motorbike from the 1980's. Without any previous warnings, a simultaneous failure of all spikes around the center of the wheel could happen (e.g. when using the brakes), see below.
Part of circumferential breakage illustrated.
It was later concluded that the failures were due to fatigue. It may be noted, that in order to avoid further accidents on existing motorbikes, it was recommended to continuously check the spikes for cracks (this is for instance done in the Swedish national vehicle test programme for motorbikes in use).

5. The physical nature of fatigue
The fatigue process (in metallic materials) is typically divided into an
initiation phase, where micro-cracks are nucleated, cyclically growing, and finally merging into one dominant macro-crack
and a
propagation phase, where the created macro-crack cyclically grows until it reaches a critical size, when the final failure occurs by a static fracture
Initiation phase Propagation phase Final failure
micro- scopic cyclic slip
micro-crack nucleation
cyclic micro-crack growth
cyclic macro-crack growth
static fracture
As noted above, micro-crack nucleation is in metallic materials due to cyclic slip, i.e. cyclic plastic flow on a microscopic scale, involving the back and forth movement of dislocations.

6. The physical nature of fatigue; cont.
We note that
The final fatigue failure surface often (but not always) shows two regions
* one surface with so called striations, resulting from the cycling loading
* one rough surface, representing the final static fracture surface schematically
final failure surface
micro- crack
macro- crack
striations
crack front

7. The physical nature of fatigue; cont.
micro-cracks are typically initiated
* in shear bands (with extensive localized microscopic plastic flow)
* at the surface of a component or specimen, due to environment effects (oxidation), surface roughness and less restraint to plastic flow
the distinction micro-crack/macro-crack and initiation phase/propagation phase is not sharp
It may e.g. be based on the size or detectability of the crack; 1mm (i.e. visible to the naked eye) could be a criterion
The initiation phase is typically very much longer than the propagation phase for smooth laboratory specimens, but this is generally not true for notched specimens or real structures

8. The physical nature of fatigue; cont.
different factors may have very different effect on the initiation and the propagation phase, resp.
As an example, initiation is strongly affected by the surface conditions, while the propagation phase hardly depends on it at all
there is often a large scatter in experimentally obtained lives, especially for lower loads/longer lives, since for this case the microstructure has a larger influence on the material behavior

9. Fatigue design
As noted previously, fatigue design (i.e. designing against fatigue) is a very important topic, where life-, safety- and economic aspects are to be guaranteed and/or improved.
However, fatigue design is to its nature a complex and difficult topic. Furthermore, large uncertainties often exist regarding e.g.
actual load spectrum
the real material behavior
residual stresses and surface conditions
environment
which makes the task even more difficult.

10. Stress concentrations
A stress concentration is simply a geometrical feature (hole, notch, etc), which locally increases the stresses. Since such stress raisers always will be present in components and designs, we simply need to be able to handle their effect on e.g. the life (number of cycles to failure) of cyclically loaded components. The probably most simple, and therefore the by far most discussed case, is a circular hole in a uniaxially loaded large flat plate. By using the stress function approach of 2-dimensional linear elastostatics, one finds for the tangential stresses at the hole the following formula
As can be seen, we get a 3-fold increase of the tangential stress at the hole!

11. Stress concentrations; cont.
The nominal stress at the hole is defined as the stress that would be present if no redistribution of stresses around the hole took place, i.e.
where w is the width of the plate.
The stress concentration factor is defined as
For a small hole in a large plate
which for the case above equals 3.

12. Stress concentrations; cont.
In the case that the stress concentration becomes sharper, a higher stress concentration factor will be found. As an example, one may show (not to be done here) that we for an elliptical hole in a large flat plate have the following situation.
As can be seen, we regain the previous result for the case of a circular hole (a=b). Furthermore, we see that the more elongated the hole will get, the higher the stress concentration will be (as stated above). Imagine what will happen for a crack!

13. The stress concentration at the tip of a crack
As we saw previously, the stress concentration at the tip of an elliptical hole in a very large flat plate becomes higher as the hole becomes more elongated. In detail we have
Obviously, the stress concentration factor for a sharp crack becomes (in theory) infinite! This is of course not so in reality, since the material will yield plastically at the crack tip. However, surprisingly enough, the stress solution based on linear elasticity has been shown to be extremely useful in predicting e.g. static fracture and cyclic crack propagation!

14. The stress concentration at the tip of a crack; cont.
Uni-axially loaded plate Let us now in some more detail look at the stress state at a crack located in a very large flat plate (for which we can assume that plane stress- or plane strain conditions prevail). More specifically, we will start by considering a uni-axial loading as illustrated below.

15. The stress concentration at the tip of a crack; cont.
Uni-axially loaded plate; cont. It can be shown that the stress state near the crack tip takes the following form
As can be seen, the stress has a so called singularity at the crack tip, where the intensity of the singularity is given by the KI parameter, which is referred to as the stress intensity factor.

16. The stress concentration at the tip of a crack; cont.
Uni-axially loaded plate; cont.
The stresses at the crack tip of a very large plate can thus be written where the f's are coordinate functions
If the size of the plate becomes "finite", the expression for the stress intensity factor will also depend on the relation between the crack size and the plate size and on the type of loading, which may be written where W is the width of the plate; for details please see any formula table in fracture mechanics.

17. Different Modus
A crack can in principle be loaded in three different ways, which we refer to as loading modus, see below. The stress field at the crack tip will in each of these cases have the same basic structure. The reason why we used the name KI previously, was simply that the loading corresponded to a Mode I-loading!
Most dangerous!

18. Fracture toughness
Even though, as was pointed out previously, the obtained stress field is clearly unphysical, it is still extremely useful in determining the risk of static failure, or in calculating the propagation of a crack under cyclic loading!
Considering the former topic (static fracture), it has been shown that materials have a fracture toughness KIC , such that they fail if KI exceeds this value. Thus, the failure criterion for static failure becomes (in Mode I)
The value KIC is to be obtained under plane strain conditions. In order to ensure that the actual loading condition is of this type, the following requirements are to be fulfilled (must be checked when solving problems!)

19. Cyclic crack growth
If a crack is subjected to a varying load it may grow. We find experimentally
The linear region is described by the so called Paris law

20. Cyclic crack growth; damage tolerant approach
Let us assume that we need to design against fatigue in a structure where we have cracks smaller than a certain size (found by e.g. non-destructive testing)
When designing this structure for infinite life, we require (for a certain safety factor s)
When designing this structure for finite life, we require that an integration of Paris law from the known crack size, and for a given number of load cycles/sequences (corresponding to e.g. an inspection interval), will give (for a certain safety factor s)