CTOD - Crack Tip Opening Displacement
============================

In the early 60's, A. Wells was measuring the mode I fracture toughness ($K_{IC}$) of structural steels when he noticed that the toughness they exhibit is too high to be characterized by linear elastic fracture mechanics. 

Wells than proposed a failure criterion based on the observation that in the presence of crack-tip plasticity, the crack tip appears to blunt with the application of load before a crack emerges. 

**Moreover** Wells noted that the blunting process appeared to increase proportionally to the tested materials toughness. 

Using *Irwin's* crack-tip plasticity model, Wells was able to show that the CTOD ($\delta$) follows:

$$
\delta = \frac{4}{\pi}\frac{G}{\sigma_y}
$$

A similar result is obtained following the strip model:

$$
\delta = \frac{G}{m \sigma_y} 
$$ 
and $m$ assumes a value of $1$ for plane stress and $2$ for plane strain.

```{image} ../EPFM/CTOD.png
:alt: Kirsch's plate
:width: 800px
:align: center
```

The CTOD is defined in one of the two ways shown above. However the two expressions are only identical for a semicircle blunting crack.


```{tabbed} Fatigue crack in SLM $AlSi_{10}Mg$

![initial]( ../EPFM/a0.png)
```

```{tabbed} Blunting
![initial]( ../EPFM/ctodA.gif)
```


During the process of crack tip blunting, the crack faces ae gradually moving apart, and using the assumption of rigidity they can be assumed to rotate around a hinge point. 

While not entirely accurate, this assumption allows us to measure the **Crack Mouth Opening Displacement (CMOD)** and using similarity of triangles obtain the **CTOD**. 

$$
\delta_{CTOD} = \frac{r(W-a)}{r(W-a)+a} \delta_{CMOD}
$$

The experimentaly measured **CMOD** is decomposed into an elastic and plastic part (similar to the way you would for a stress strain curve) and thus:

$$
\delta_{CTOD} = \delta_{e} + \delta_{p} = \frac{K_I^2}{m\sigma_y E'} + \frac{r(W-a)}{r(W-a)+a} \delta_{CMOD}
$$


### J-CTOD 

From the above relation it is easy to see that when LEFM holds we obtain 

$$
J=m\sigma_y \delta_{CTOD}
$$

Using the strip model for crack tip plasticity it is simple to demonstrate that for a non-hardening elasto-plastic material undergoing crack tip blunting:

$$
J=\sigma_y \delta_{CTOD}
$$

In 1981, Shih demonstrated using the HRR solution that :

$$
\delta_{CTOD} = d_n \frac{J}{\sigma_y}
$$

Where $d_n$ is a dimensionless constant which depends on the strain hardening exponent $n$ and on $\alpha \frac{\sigma_y}{E}$.