# 2. SN Curves#

## 2.1. Cyclic fatigue with constant amplitude (\(\sigma\) based)#

A Wöhler curve, or S-N curve, in its basic form, represent the number of cycles \(N_f\) which will lead to failure, given that the material was loaded with a given stress amplitude

the red line represent the **fatigue limit** of the material \(\to\) the stress below which an infinite amount of cycles is allowed.

Infinity in the context of fatigue is often set as \(N=10^7\).

For many materials, a linear relation was observed between \(log(\sigma_a)\) and \(log(2N_f)\) , This relation, was proposed by Basquin in 1910 to follow:

where \(\sigma'\) is usually taken as the failure stress of the material and \(b\) is in the empirical range of \([-0.05, -0.12]\)

A missingf piece of this approach as dicussed so far is the assumption that \(\sigma_m = 0\)

Several models have been proposed to handle that, but we will only mention two of them:

Soderberg (1939,more conservative):

\(\sigma_a = \sigma_a^{(\sigma_m=0)} \left ( 1-\frac{\sigma_m}{\sigma_y} \right )\)

Goodman(1899):

\(\sigma_a = \sigma_a^{(\sigma_m=0)} \left ( 1-\frac{\sigma_m}{\sigma_{UTS}} \right )\)

## 2.2. Cyclic fatigue with varying amplitude (\(\sigma\) based)#

For many real life scenarios (e.g. estimating life of aircraft components, dental implants etc.) it makes more sense to subject the test specimen to a spectrum load which represent in a more realistic way the loads it will experience throughout its life cycle.

To account for the varying amplitude (amongst others) we can use \(Miner's rule\) which can be written as:

As long as \(D<1\) the component is persumabley safe.

Note

Miner’s rule does not take under consideration the order in which the different loading cycles were applied.

Why is that a major drawback?

Strain hardening/softening

\(\mu\)-cracks nucleation

## 2.3. Low cycle fatigue (\(\epsilon\) based)#

For scenarios where significant plasticity might occur, we will prefer to take a strain based approach (high temperature, locally high stresses etc. )

by plotting \(\frac{1}{2}\Delta \epsilon^p\) vs \(2N_f\) we will again observe(usually) a linear relation described by (Coffin Manson 1955):

where \(c\) lies in the range of \([-0.5,-0.7]\) (empirically).

If we combine the stress approach (Basquin) with the strain one we can use simple linear elasticity to obtain

and after substitution:

Moreover, we can find the transition between elastic and plastic dominant scenarios by taling the two terms to be equall yielding:

## 2.4. Lies damn lies and statistics#

The scatter observed in fatigue tests tend to be rather large, with \(N_f(\sigma_f)\) exhibiting a log-normal distribution and \(\sigma_f(N_f)\) a normal distribution.

One of the issues arising when dealing with fatigue tests on smooth specimens is theat the crack initiation stage may take up a large portion of \(N_f\).

It is sensitive to effects such as:

grain orientation near the surface

inclusions distribution as a function of \(r\)

defects population in general

machinning defects

### 2.4.1. Cyclic hardening/softening#

many structural alloys will exhibit cyclic strain hardening (both isotropic and kinematic). For some alloys this will only be present n the first few cycles before saturating, and in some cases, a maximum will be attained which will later decrease before saturating.

Similarly, some materials (e.g. low carbon steels) exhibit softening acompanying heterogenous strainning followed by hardening with homogenous deformation.