Exponential Softening

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The Softening Law

A exponential softening law has the following values:

      [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) \lt k = {1\over sG_c} }[/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