Exponential Softening
The Softening Law
A exponential softening law has the following form:
[math]\displaystyle{ f(\delta,s) = e^{-k\delta} = e^{-\delta/(sG_c)} }[/math]
which follows from
[math]\displaystyle{ sG_c = \int_0^{\delta_{max}} f(\delta,s) = {1\over k} \quad{\rm or}\quad k = {1\over sG_c} }[/math]
where s is the softening scaling term and Gc is toughness of the law (and the law's only property). The exponential decay rate, k, 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{ \Omega(\delta,s) = sG_c - e^{-\delta/(sG_c)}\left(sG_c+{\delta\over2}\right) }[/math]
The stability condition is:
[math]\displaystyle{ \max\bigl(-f'(\delta,s)\bigr) = k = {1\over sG_c} \qquad\implies\qquad \eta=1 }[/math]
Minimum [math]\displaystyle{ f(\delta,s) }[/math]
This law requires selection of minimum value for [math]\displaystyle{ f(\delta,s) }[/math] below which the material point is marked as failed. If we define the minimum value as c, this choice defines an effective maximum cracking strain as:
[math]\displaystyle{ \delta_{max} = -{\ln c \over k} = -s G_c \ln c }[/math]
For example, picking c = 0.01 gives
[math]\displaystyle{ \delta_{max} = 4.60517 s G_c }[/math]
Note that linear softening has 2 in place of 4.60517 for finding maximum cracking strain. Without a minimum value, exponential softening would never fail. If you want to prevent failures, set the minimum value to a very small number (but it cannot be zero).
Softening Law Properties
Only two properties are needed to define an exponential softening law:
| Property | Description | Units | Default |
|---|---|---|---|
| Gc | The toughness associated with the this softening law | energy release units | none |
| min | Minimum [math]\displaystyle{ f(\delta,s) }[/math] or law is failed if it gets below this value | none | 0.01 |