Cubic Step Function Softening

The Softening Law

This softening law was originally a step function (hence the name), but was latter generalized to allow it to be a cubic function that rises to a peak and then decays to zero at [math]\displaystyle{ \delta_{max} }[/math]. The function is

      [math]\displaystyle{ f(\delta,s) = \left(1+2k_2{\delta\over \delta_{max} }\right)\left(1-{\delta\over \delta_{max} }\right)^2 \quad{\rm where}\quad k_2 = 1+\frac{k}{2} }[/math]

This cubic function has [math]\displaystyle{ f(0)=1 }[/math], [math]\displaystyle{ f'(0)= k/\delta_{max} }[/math], and [math]\displaystyle{ f(\delta_{max})=f'(\delta_{max})=0 }[/math]. The value for [math]\displaystyle{ \delta_{max} }[/math] is found 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]

Here 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.

If [math]\displaystyle{ k=0 }[/math], this law is a cubic step function with [math]\displaystyle{ f'(0)=0 }[/math], but if [math]\displaystyle{ k\gt 0 }[/math], this function rises to a peak and then decays to failure at [math]\displaystyle{ \delta_{max} }[/math]. This behavior is valid for softening laws provided the modulus still monotonically softens. When using [math]\displaystyle{ k\gt 0 }[/math], you enter initiation stress for initation of failure when [math]\displaystyle{ \delta=0 }[/math]. If you prefer to control the peak stress, the initiation stress should be calculated from the desired peak using

      [math]\displaystyle{ \sigma_{0} = \frac{\left(1+\frac{k}{2}\right)^2}{\left(1+\frac{k}{23}\right)^3}\sigma_{peak} }[/math]

The area (or energy dissipation term) is

      [math]\displaystyle{ A(\delta,s) = {\delta\over 2}\left(1 +\left( {\delta\over \delta_{max} }\right)^2\left(1 - {\delta\over \delta_{max}} \right)\right) }[/math]

The stability condition is:

      [math]\displaystyle{ \max\bigl(-f'(\delta,s)\bigr) = {1.5\over \delta_{max}} = {3\over 4sG_c} }[/math]

Compared to a linear softening law, a cubic step function softening has zero derivative at both δ=0 and at δ=δ_max. This change might smooth the damage process. The law, however, needs numerical methods to evolve damage (which are not needed by linear softening law) and will need slightly smaller elements (because of 50% higher maximum slope).

Softening Law Properties

Only one property is needed to define a cubic step function softening law:

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