Difference between revisions of "Cubic Step Function Softening"
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<math>f(\delta,s) = 1 + 2\left( {\delta\over \delta_{max} }\right)^3 - 3\left( {\delta\over \delta_{max} }\right)^2</math> | <math>f(\delta,s) = 1 + 2\left( {\delta\over \delta_{max} }\right)^3 - 3\left( {\delta\over \delta_{max} }\right)^2</math> | ||
with <math>\delta_{max} = | with <math>\delta_{max} = 2sG_c</math> | ||
where ''s'' is the [[Softening Laws#Normalized Softening Law|softening scaling term]] and ''G<sub>c</sub>'' is toughness of the law (and the law's only property). | where ''s'' is the [[Softening Laws#Normalized Softening Law|softening scaling term]] and ''G<sub>c</sub>'' is toughness of the law (and the law's only property). | ||
Revision as of 11:03, 4 September 2017
The Softening Law
A cubic step softening law has the following values:
[math]\displaystyle{ f(\delta,s) = 1 + 2\left( {\delta\over \delta_{max} }\right)^3 - 3\left( {\delta\over \delta_{max} }\right)^2 }[/math] with [math]\displaystyle{ \delta_{max} = 2sG_c }[/math]
where s is the softening scaling term and Gc is toughness of the law (and the law's only property). The critical cracking strain, which depends on mesh size and crack orientation, is calculated above and is not a law property to be provided.
The area (or energy dissipation term) is
[math]\displaystyle{ A(\delta,s) = {\delta\over 2} }[/math]
The stability condition is:
[math]\displaystyle{ \max\bigl(-f'(\delta,s)\bigr) = {1\over \delta_{max}} = {1\over 2sG_c} }[/math]
Softening Law Properties
Only one property is needed to define a linear softening law:
| Property | Description | Units | Default |
|---|---|---|---|
| Gc | The toughness associated with the this softening law | energy release units | none |