Exponential Softening

The Softening Law

An exponential softening law has the following values:

      [math]\displaystyle{ f(\delta,s) = e^{-\delta/(sG_c)} }[/math]

      [math]\displaystyle{ A(\delta,s) = cG_c - e^{-\delta/(sG_c)}\left(sG_c+{\delta\over2}\right) }[/math]

[math]\displaystyle{ \max\bigl(f'(\delta,s)\bigr) \lt {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). This law never fails, although the traction asymptotically approaches zero. The exponential decay rate, k, is

      [math]\displaystyle{ k = sG_c \qquad {\rm or} \qquad f(\delta) = e^{-k\delta} }[/math]

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

Note that softening materials