# Fatigue and Fracture

Fatigue is the progressive and localized structural damage that occurs when a material is subjected to cyclic loading. The maximum stress values are less than the yield stress limit of the material. One or many cracks initiate and propagate into a solid, leading to fracture.

Fatigue and Fracture

## Contents

### The Wöhler curves

Fatigue behavior of a material can be characterised by its Wöhler (or S-N) curve. Such curves are obtained by imposing to a specimen an cyclic load. The applied stress is plot in terms of the number of cycles the specimen can undergo before failure. The following figure shows the aspect of Wöhler curves.

Fatigue failure is characterized by a three stages history. First a microcrack initiates in the specimen and propagates. Its growth is driven by metallurgical parameters (the crack growth direction is determined by crystal orientation, etc). The second stage starts when the microcrack reaches a critical length. Its growth is driven by mechanical parameters (the crack growth at each cycle is determined by the stress intensity factor, the crack plane is perpendicular to the direction of maximal stress, etc). Eventually, fracture occurs (ruled by Linear elastic fracture mechanics).

If the applied alternating stress is below a given limit, failure will not happen. The fatigue strength is usually called the endurance limit, it is often set to the failure stress at 108 cycles.

The results of fatigue test show substantial scattering, as it is ruled by random process. As a convention, the Wöhler curves show the stress at which 50% of the specimens fail. As a first approximation, it can be assumed that, at a given number of cycles, the stress failure fits a Normal distribution. Thus, at a given alternating stress, the number of cycle before failure can be assumed to fit a log Normal distribution.

### Variable stress amplitude

Palmgren-Miner rule can be used to handle variable amplitude load. If a specimen undergoes different stress amplitudes, it will fail when

$D = \sum \frac{n_{i}}{N_{i}}$

where the ni are the number of cycles stress amplitude σi that the structure undergoes and Ni is the number of cycle before failure given by the Wöhler curve. The previous equation does not give an accurate description when the applied stress vary over a wide range.

As illustrated in the following figure, failure happens when

$\frac{n_1}{N_1} + \frac{n_2}{N_2} = 1$

where n1 and n2 are the number of cycles at stress σ1 and σ2 (respectively).

### Effects of mean stress

The fatigue behavior of metals is affected by the mean stress. As the mean stress varies, the number of cycles to failure is taken as a reference and a corrective factor is added to the alternating stress (to keep the number of cyles to failure constant).

Several equations describing the relationship among fatigue limit and tensile strength exist. Experiments have to be performed in order to find the law that best describes each material behavior. The failure stress σa after N cycles is obtained by multiplying the stress read on the Wöhler curve (after N cycles) by a correcting factor. Gerber proposed[1]:

$\sigma_{a}=\sigma_{f}\left( 1-\left( \frac{\sigma_{m}}{\sigma_{t}}\right) ^{2}\right)$

where σa is the stress variation amplitude, σf is the fatigue strength at zero mean stress, σt is the tensile strength, σm is the mean stress. Goodman proposed to use a straight line:

$\sigma_{a}=\sigma_{f}\left( 1-\left( \frac{\sigma_{m}}{\sigma_{t}}\right) \right)$

See for instance[2] for more details.

### Crack Propagation

 Please improve this article by expanding it. HMP: the part on crack propagation needs to be extended

A cracked solid might undergo three modes of loading: tensile stress perpendicular to the crack plane, shear in the crack plane or shear perpendicular to the crack plane (see the figure below). A cracked solid can undergo any combination of the three modes. The first mode (opening) is far more damaging, and only this mode will be studied here.

## References

1. W. Gerber. Bestimmung der zulässigen Spannungen in Eisenkonstruktionen. Zeitschrift des Bayerischen Architekten- und Ingenieur-Vereins, 6:101–110, 1874.
2. L Pook. Metal fatigue. Springer, 2007. ISBN 140205596X