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] (see plots on the right). This behavior is valid for softening laws provided the modulus still monotonically softens. When using [math]\displaystyle{ k\gt 0 }[/math], you enter the initial stress into the initiation law. If you want to set 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(\frac{4k_2-1}{3}-k_2{\delta\over \delta_{max}}\right)\right) }[/math]

      [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 factor is:

      [math]\displaystyle{ \eta = \frac{12k_2}{(1+2k_2)^2k_6} }[/math]

If [math]\displaystyle{ k=0 }[/math], the stability factor simplifies to [math]\displaystyle{ \eta = 4/3 }[/math], or slightly less stable than a linear softening law. Potentially, the zero derivative at [math]\displaystyle{ \delta_{max} }[/math] could improve modeling of failure. The stability factor decreases as [math]\displaystyle{ k }[/math] increases.

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
k	Initial slope of the law (must be nonnegative) (OSParticulas only)	none	0