Double Exponential Softening

The Softening Law

The main goal of his softening law is to provide another softening law that can initiate at low stress, [math]\displaystyle{ \sigma_{0} }[/math] (from the damage initiation law), rise to a peak, [math]\displaystyle{ \sigma_0*f_{peak} }[/math], and then decay. With that goal in mind, the function is

      [math]\displaystyle{ f(\delta,s) = \frac{1}{1-\beta}\left(e^{-k\delta} - \beta e^{-\alpha k\delta}\right) }[/math]

where [math]\displaystyle{ \alpha\gt 1 }[/math] (to have second term decay faster) and [math]\displaystyle{ \beta\lt 1 }[/math] (to keep [math]\displaystyle{ f(\delta,s)\gt 0 }[/math]. The peak value of this function is located at:

      [math]\displaystyle{ \delta_{peak} = \left\{ \begin{array}{ll} \frac{ \ln \alpha\beta }{k(\alpha-1)} & 1/\alpha\lt \beta\lt 1 \\ 0 & \beta \le 1/\alpha \end{array}\right. }[/math]

The corresponding peak values is:

      [math]\displaystyle{ \sigma_{peak} = \left\{ \begin{array}{ll} \frac{\alpha-1}{\alpha(1-\beta)(\alpha\beta)^{\frac{1}{\alpha-1}}} & 1/\alpha\lt \beta\lt 1 \\ 1 & \beta \le 1/\alpha \end{array}\right. }[/math]

      [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}\left(1+\frac{k}{6}\right) \quad{\rm or}\quad \delta_{max} = \frac{2sG_c}{k_6} \quad{\rm where}\quad k_6 = 1+\frac{k}{6} }[/math]

Here s is the softening scaling term and G_c is toughness of the law. 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}{3}\right)^3}\sigma_{peak} }[/math]

The area (or energy dissipation term) is

      [math]\displaystyle{ \Omega(\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]

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. As [math]\displaystyle{ k }[/math] increases the stability decreases. A difference between this law and linear softening law is that [math]\displaystyle{ f'(\delta_{max})=0 }[/math]. Thus, despite a reduction in stability, it is possible this zero derivative could reduce numerical effects caused by decohesions.

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