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 G_c 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]

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