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 to zero (currently OSParticulas only). 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). Although this stability factor goes to zero as [math]\displaystyle{ \beta\to1 }[/math], the effective stability is better compared to monotonically decreasing laws by multiplying [math]\displaystyle{ \eta }[/math] by its peak values squared (i.e., minimum cell size is related to [math]\displaystyle{ \eta/\sigma_0^2 }[/math] and [math]\displaystyle{ \sigma_0 }[/math] decreases if set by reducing peak stress by [math]\displaystyle{ f_{peak} }[/math]). The figure on the right plots this effective stability factor as a function of [math]\displaystyle{ \beta }[/math] for various values of [math]\displaystyle{ \alpha }[/math]. Overall, the stability of this law is comparable to the stability of a single Exponential Softening law.

Softening Law Properties

A double exponential softening law requires three parameters:

Property	Description	Units	Default
Gc	The toughness associated with the this softening law	energy release units	none
alpha	relative decay rate of the second exponential (must be greater than 1)	none	none
alpha	relative magnitude of the the second exponential (must be less than 1)	none	none