Difference between revisions of "Linear Softening"
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<math>sG_c = \int_0^{\delta_{max}} f(\delta,s) = {\delta_{max}\over 2} \quad{\rm or}\quad \delta_{max} = 2sG_c</math> | <math>sG_c = \int_0^{\delta_{max}} f(\delta,s) = {\delta_{max}\over 2} \quad{\rm or}\quad \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). | |||
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 | The area (or energy dissipation term) is | ||
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<math>\max\bigl(-f'(\delta,s)\bigr) = {1\over \delta_{max}} = {1\over 2sG_c}</math> | <math>\max\bigl(-f'(\delta,s)\bigr) = {1\over \delta_{max}} = {1\over 2sG_c}</math> | ||
== Softening Law Properties == | == Softening Law Properties == | ||
Revision as of 10:51, 27 December 2016
The Softening Law
A linear softening law has the following values:
[math]\displaystyle{ f(\delta,s) = 1 - {\delta\over \delta_{max} } = 1 - {\delta\over 2sG_c} }[/math]
which follows from
[math]\displaystyle{ sG_c = \int_0^{\delta_{max}} f(\delta,s) = {\delta_{max}\over 2} \quad{\rm or}\quad \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 |
Note that softening materials