Difference between revisions of "Linear Softening"
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<math>f(\delta,s) = 1 - {\delta\over 2sG_c}</math> | <math>f(\delta,s) = 1 - {\delta\over \delta_{max} } = 1 - {\delta\over 2sG_c} </math> | ||
which follows from | |||
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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> | |||
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, <math>\delta_{max}</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 area (or energy dissipation term) is | |||
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<math> | <math>\Omega(\delta,s) = {\delta\over 2}</math> | ||
The stability condition is: | |||
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<math>\delta_{max} = 2sG_c</math> | <math>\max\bigl(-f'(\delta,s)\bigr) = {1\over \delta_{max}} = {1\over 2sG_c} | ||
\qquad\implies\qquad \eta=2</math> | |||
== Softening Law Properties == | == Softening Law Properties == | ||
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| Gc || The toughness associated with the this softening law || [[ConsistentUnits Command#Legacy and Consistent Units|energy release units]] || none | | Gc || The toughness associated with the this softening law || [[ConsistentUnits Command#Legacy and Consistent Units|energy release units]] || none | ||
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Latest revision as of 09:46, 9 July 2021
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, [math]\displaystyle{ \delta_{max} }[/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 area (or energy dissipation term) is
[math]\displaystyle{ \Omega(\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} \qquad\implies\qquad \eta=2 }[/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 |