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 figure on the right shows double exponential softening functions for various values of [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math]. The law peaks whenever [math]\displaystyle{ 1/\alpha\lt \beta\lt 1 }[/math]. This behavior is valid for softening laws provided the modulus still monotonically softens (which is does for any valid parameters). 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 value is:

      [math]\displaystyle{ f_{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]

When using this law, you enter the initiation stress into the initiation law. If you want to set the peak stress instead, the initiation stress should be reduced from that peak stress by this value of [math]\displaystyle{ f_{peak} }[/math]. When [math]\displaystyle{ \beta\lt 1/\alpha }[/math], this law monotonically decreases from 1 to 0. For [math]\displaystyle{ \beta\lt 0 }[/math], this law is a simple double-exponential decay function.

The value for [math]\displaystyle{ k }[/math] is is found from

      [math]\displaystyle{ sG_c = \int_0^{\infty} f(\delta,s) = \frac{\alpha-\beta}{k\alpha(1-\beta)s} \quad{\rm or}\quad k = \frac{\alpha-\beta}{\alpha(1-\beta)sG_c} }[/math]

Here s is the softening scaling term and G_c is toughness of the law. The decay rate, [math]\displaystyle{ k }[/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 stability factor depends on relative values of [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math]:

      [math]\displaystyle{ \eta = \left\{ \begin{array}{ll} \frac{\alpha^2(1-\beta)^2(\alpha^2\beta)^{\frac{1}{\alpha-1}}}{(\alpha-\beta)(\alpha-1)} & {\rm for\ } \beta\gt 1/\alpha^2 \\ \frac{\alpha(1-\beta)^2}{(\alpha-\beta)(1-\alpha\beta)} & {\rm for\ } \beta \lt 1/\alpha^2 \end{array}\right. }[/math]

For [math]\displaystyle{ \beta=0 }[/math], the stability factor simplifies to [math]\displaystyle{ \eta = 1 }[/math] which is identical to Exponential Softening (in fact the entire law is identical to Exponential Softening and that law should therefore be used instead). Altough this stability factor is often much lower then other softening laws, the minimum cell size is also determined by [math]\displaystyle{ 1/\sigma_0^2 }[/math]. Because that stress may be low when using this law with a peak, this provides similar stability to other laws in practice.

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