Linear Softening

The Softening Law

A linear softening law has the following values:

      [math]\displaystyle{ f(\delta,s) = 1 - {\delta\over \delta_{max} } = 1 - {\delta\over 2sG_c} }[/math]

which follows from

      [math]\displaystyle{ sG_c = \int_0^{\delta_{max}} f(\delta,s) = {\delta_{max}\over 2} \quad{\rm or}\quad \delta_{max} = 2sG_c }[/math]

where s is the softening scaling term and G_c is toughness of the law (and the law's only property). The critical cracking strain, [math]\displaystyle{ \delta_{max} }[/math], which depends on mesh size and crack orientation, is calculated from s and G_c and is not a law property to be provided.

The area (or energy dissipation term) is

      [math]\displaystyle{ \Omega(\delta,s) = {\delta\over 2} }[/math]

The stability condition is:

      [math]\displaystyle{ \max\bigl(-f'(\delta,s)\bigr) = {1\over \delta_{max}} = {1\over 2sG_c} \qquad\implies\qquad \eta=2 }[/math]

Softening Law Properties

Only one property is needed to define a linear softening law:

Property	Description	Units	Default
Gc	The toughness associated with the this softening law	energy release units	none