Difference between revisions of "Double Exponential Softening"
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The decay rate, <math>k</math>, which depends on mesh size and crack orientation, is calculated from ''s'' and ''G<sub>c</sub>'' and is not a law property to be provided. | The decay rate, <math>k</math>, which depends on mesh size and crack orientation, is calculated from ''s'' and ''G<sub>c</sub>'' and is not a law property to be provided. | ||
[[File:DbleExp.png|right]] | [[File:DbleExp.png|500px|right]] | ||
The figure on the right shows double exponential softening for various values of <math>\alpha>1</math> and <math>\beta<1</math>. The law peaks whenever <math>1/alpha<\beta<1</math>. This behavior is valid for softening laws provided the modulus still monotonically softens. When using this law, you enter the initial stress into the [[Damage Initiation Laws|initiation law]]. If you want to set the peak stress, the initiation stress should be reduced from the peak stress by the value of <math>f_{peak}</math> above. When <math>\beta<1/\alpha</math>, this law monotonically decrease from 1 to 0. For <math>\beta<0</math>, this law is a simple double-exponential decay functino. | The figure on the right shows double exponential softening for various values of <math>\alpha>1</math> and <math>\beta<1</math>. The law peaks whenever <math>1/alpha<\beta<1</math>. This behavior is valid for softening laws provided the modulus still monotonically softens. When using this law, you enter the initial stress into the [[Damage Initiation Laws|initiation law]]. If you want to set the peak stress, the initiation stress should be reduced from the peak stress by the value of <math>f_{peak}</math> above. When <math>\beta<1/\alpha</math>, this law monotonically decrease from 1 to 0. For <math>\beta<0</math>, this law is a simple double-exponential decay functino. | ||
Revision as of 10:05, 20 July 2021
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{ \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]
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 for various values of [math]\displaystyle{ \alpha\gt 1 }[/math] and [math]\displaystyle{ \beta\lt 1 }[/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. When using this law, you enter the initial stress into the initiation law. If you want to set the peak stress, the initiation stress should be reduced from the peak stress by the value of [math]\displaystyle{ f_{peak} }[/math] above. When [math]\displaystyle{ \beta\lt 1/\alpha }[/math], this law monotonically decrease from 1 to 0. For [math]\displaystyle{ \beta\lt 0 }[/math], this law is a simple double-exponential decay functino.
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 |