SN Curves 
============================

### 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


```{image} ../Fatigue/sn.png
:alt: approaches
:width: 600px
:align: center
```
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](https://ci.nii.ac.jp/naid/10015102552/) in 1910 to follow:

$$
\frac{\color{blue}{\Delta \sigma}}{2} = \sigma_a = \sigma'(2N_f)^b
$$

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):

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

### 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: 

$$
D=\sum_i d_i=\sum_i \frac{n_i}{N_{fi}}
$$

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?
```

1. Strain hardening/softening
2. $\mu$-cracks nucleation

### 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. )

```{image} ../Fatigue/sn.png
:alt: strain-life
:width: 600px
:align: center
```

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

$$
\frac{\Delta \epsilon^p}{2}=\epsilon_f'(2N_f)^c
$$

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 

$$
\frac{1}{2}\Delta \epsilon^e = \frac{1}{2}\frac{\Delta \sigma }{E} = \frac{\sigma'}{E}(2N_f)^b
$$

and after substitution:

$$
\frac{\Delta \epsilon}{2}=\epsilon_f'(2N_f)^c+\frac{\sigma'}{E}(2N_f)^b
$$

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

$$
2N_t = \left (\frac{E\epsilon_f'}{\sigma'} \right )^{1/(b-c)}
$$


### 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 


#### 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.