Difference between revisions of "Exponential Softening"
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== The Softening Law == | == The Softening Law == | ||
A exponential [[Softening Laws|softening law]] has the following values: | |||
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<math>f(\delta,s) = e^{-\delta/(sG_c)}</math> | <math>f(\delta,s) = e^{-k\delta} = e^{-\delta/(sG_c)}</math> | ||
which follows from | |||
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<math>sG_c = \int_0^{\delta_{max}} f(\delta,s) = {1\over k} \quad{\rm or}\quad k = {1\over sG_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 exponential decay rate, ''k'', 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 | |||
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<math>A(\delta,s) = sG_c - e^{-\delta/(sG_c)}\left(sG_c+{\delta\over2}\right)</math> | <math>A(\delta,s) = sG_c - e^{-\delta/(sG_c)}\left(sG_c+{\delta\over2}\right)</math> | ||
The stability condition is: | |||
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<math> | <math>\max\bigl(-f'(\delta,s)\bigr) < k = {1\over sG_c}</math> | ||
== Softening Law Properties == | == Softening Law Properties == | ||
Revision as of 11:54, 27 December 2016
The Softening Law
A exponential softening law has the following values:
[math]\displaystyle{ f(\delta,s) = e^{-k\delta} = e^{-\delta/(sG_c)} }[/math]
which follows from
[math]\displaystyle{ sG_c = \int_0^{\delta_{max}} f(\delta,s) = {1\over k} \quad{\rm or}\quad k = {1\over sG_c} }[/math]
where s is the softening scaling term and Gc is toughness of the law (and the law's only property). The exponential decay rate, k, 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) = sG_c - e^{-\delta/(sG_c)}\left(sG_c+{\delta\over2}\right) }[/math]
The stability condition is:
[math]\displaystyle{ \max\bigl(-f'(\delta,s)\bigr) \lt k = {1\over sG_c} }[/math]
Softening Law Properties
Only one property is needed to define an exponential softening law:
| Property | Description | Units | Default |
|---|---|---|---|
| Gc | The toughness associated with the this softening law | energy release units | none |
Note that softening materials