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 G_c 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