Cubic Step Function Softening
The Softening Law
A cubic step softening law has the following values:
[math]\displaystyle{ f(\delta,s) = 1 + 2\left( {\delta\over \delta_{max} }\right)^3 - 3\left( {\delta\over \delta_{max} }\right)^2 }[/math] with [math]\displaystyle{ \delta_{max} = 2sG_c }[/math]
where s is the softening scaling term and Gc is toughness of the law (and the law's only property). The critical cracking strain, which depends on mesh size and crack orientation, is calculated above and is not a law property to be provided.
The area (or energy dissipation term) is
[math]\displaystyle{ A(\delta,s) = {\delta\over 2}\left(1. +\left( {\delta\over \delta_{max} }\right)^2\left((1. - \left( {\delta\over \delta_{max} }\right)\right)\right) }[/math]
The stability condition is:
[math]\displaystyle{ \max\bigl(-f'(\delta,s)\bigr) = {1.5\over \delta_{max}} = {3\over 4sG_c} }[/math]
Compared to
Softening Law Properties
Only one property is needed to define a cubic step function softening law:
| Property | Description | Units | Default |
|---|---|---|---|
| Gc | The toughness associated with the this softening law | energy release units | none |