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{ A(\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} }[/math]
Minimum [math]\displaystyle{ f(\delta,s) }[/math]
In this law, it is desirable to define and minimum value for [math]\displaystyle{ f(\delta,s) }[/math]. If we define the minimum value as c, this choice effective defines a maximum cracking strain:
[math]\displaystyle{ \delta_{max} = -{\ln c \over k} = -s G_c \ln c }[/math]
For example, picking c = -0.001 gives
[math]\displaystyle{ \delta_{max} = 6.907755 s G_c }[/math]
Note that linear softening has 2 in place of 6.907755 for finding maximum cracking strain.
Softening Law Properties
Only one property is 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 gets below this value | none | 0.01 |