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 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]
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 Gc 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 Gc and is not a law property to be provided.
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). 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 the value of [math]\displaystyle{ f_{peak} }[/math] above. 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 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 |