Fracture

Consider the stress concentration factor (SCF) for an elliptical hole in a large, linear-elastic plate subjected to a remote, uniaxial stress

It can be shown that SCF can be expressed as:

K_t = \frac{\sigma_\text{max}^\text{el}}{\sigma_\text{nom}} = 1 + 2\frac{a}{b}

Therefore as b \rightarrow 0, the hole degenerates to a crack and $\frac ab \rightarrow \infty$ \therefore K_t \rightarrow \infty, provided the material behaviour remains linear elastic

Basis of the Energy Approach to Fracture Mechanics

There are three modes of loading cases: K_\text{I}, K_\text{II}, K_\text{III}.

Generally, for geometries with finite boundaries, K_\text{I} is used:
```
K_\text{I} = Y\sigma\sqrt{a\pi}
```
where Y is a function of the crack and a is never mentioned in the slides (slide 6)

TODO: find out what Y and a are
K_\text{I} is the Mode-1 stress-intensity factor which defined the magnitude of the elastic stress field in the vicinity of the crack tip
K_\text{II} and K_\text{III} are similar
The energy release release rate under mixed loading is given by
```
K_\text{total} = K_\text{I} + K_\text{II} + K_\text{III}
```

for a wide range of conditions, there is a logarithmic linear between crack growth rate and intensity factor range during cyclic loading of cracked components
it allows crack growth to be modelled and estimated based on
- knowledge of crack and component geometry
- loading conditions
- empirical crack growth data

Considering a load cycle as shown in figure \ref{fig:p-vs-t} which gives rise to a load acting on a cracked body

The load range and crack geometry gives rise to a cyclic variation in stress intensity factor:
```
\Delta K = K_\text{max} - K_\text{min}
```
Paris showed that subsequent crack growth can be modelled by following equation
```
\frac{\mathrm{d}a}{\mathrm{d}N} = C\Delta K^m
```
where C and m are empirically determined material constants.
Fatigue crack growth data is often as \log \frac{\mathrm{d}a}{\mathrm{d}N} against \log{\Delta K}
Below K_\text{th}, no observable crack growth occurs
Region II shows a near linear relationship---this is the region which fail by fatigue failure spend most of their life
In region III rapid crack grown occurs and little life is involved
the fatigue crack growth life of the component can be obtained by integrating the Paris equation between the limits of the initial crack size and final crack size, given that you know the stress intensity factor

Material	`\Delta K_\text{th}`	`m`	`\Delta K` for `\frac{\mathrm{d}a}{\mathrm{d}N} = 10^{-6}`
Mild Steel	4 to 7	3.3	6.2
Stainless Steel	4 to 6	3.1	6.3
Aluminium	1 to 2	2.9	2.9
Copper	1 to 3	3.9	4.3
Brass	2 to 4	4.0	4.3 to 66.3
Nickel	4 to 8	4.0	8.8