Cubic Step Function Softening - Revision history

Nairnj: /* Softening Law Properties */

2023-01-30T04:00:43Z

Softening Law Properties

← Older revision		Revision as of 04:00, 30 January 2023
Line 49:		Line 49:
	\| 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
	\|-		\|-
	\| k \|\| Initial slope of the law (must be nonnegative~~) ([[OSParticulas]] only~~) \|\| none \|\| 0		\| k \|\| Initial slope of the law (must be nonnegative) \|\| none \|\| 0
	\|}		\|}

Nairnj: /* The Softening Law */

2023-01-30T03:55:18Z

The Softening Law

← Older revision		Revision as of 03:55, 30 January 2023
Line 22:		Line 22:


	<math>{\tt sigmac} = \sigma_{~~init~~} = \frac{\left(1+\frac{k}{2}\right)^2}{\left(1+\frac{k}{3}\right)^3}\sigma_{peak}</math>		<math>{\tt sigmac} = \sigma_{0} = \frac{\left(1+\frac{k}{2}\right)^2}{\left(1+\frac{k}{3}\right)^3}\sigma_{peak}</math>

	The area (or energy dissipation term) is		The area (or energy dissipation term) is

Nairnj: /* The Softening Law */

2023-01-30T03:54:13Z

The Softening Law

← Older revision		Revision as of 03:54, 30 January 2023
Line 22:		Line 22:


	<math>\sigma_{0} = \frac{\left(1+\frac{k}{2}\right)^2}{\left(1+\frac{k}{3}\right)^3}\sigma_{peak}</math>		<math>{\tt sigmac} = \sigma_{init} = \frac{\left(1+\frac{k}{2}\right)^2}{\left(1+\frac{k}{3}\right)^3}\sigma_{peak}</math>

	The area (or energy dissipation term) is		The area (or energy dissipation term) is

Nairnj: /* The Softening Law */

2021-07-20T19:33:41Z

The Softening Law

← Older revision		Revision as of 19:33, 20 July 2021
Line 35:		Line 35:
	<math>\eta = \frac{12k_2}{(1+2k_2)^2k_6}</math>		<math>\eta = \frac{12k_2}{(1+2k_2)^2k_6}</math>

	If <math>k=0</math>, the stability factor simplifies to <math>\eta = 4/3</math>, or slightly less stable than a [[Linear Softening\|linear softening law]]. Although this stability factor goes to zero as <math>k</math> increases, the ''effective'' stability is better compared to monotonically decreasing laws by multiplying <math>\eta</math> by its peak value squared (''i.e.'', [[Softening Laws#Law Stability\|minimum cell size]] is related to <math>\eta/\sigma_0^2</math> and <math>\sigma_0</math> decreases if set by ~~reducing peak stress by~~ <math>f_{peak}</math>). The ``effective'' stability of this law only ~~decreased~~ from 4/3 to 0.79 as <math>k</math> ~~gets large~~.		If <math>k=0</math>, the stability factor simplifies to <math>\eta = 4/3</math>, or slightly less stable than a [[Linear Softening\|linear softening law]]. Although this stability factor goes to zero as <math>k</math> increases, the ''effective'' stability is better compared to monotonically decreasing laws by multiplying <math>\eta</math> by its peak value squared (''i.e.'', [[Softening Laws#Law Stability\|minimum cell size]] is related to <math>\eta/\sigma_0^2</math> and <math>\sigma_0</math> decreases if set by scaling it from a fixed <math>\sigma_{peak}</math>). The ''effective'' stability of this law only decreases from 4/3 to 0.79 as <math>k</math> goes from zero to infinity.

	A difference between this law and [[Linear Softening\|linear softening law]] is that <math>f'(\delta_{max})=0</math>. Thus, despite a reduction in stability, it is possible this zero derivative could reduce numerical effects caused by decohesions.		A difference between this law and [[Linear Softening\|linear softening law]] is that <math>f'(\delta_{max})=0</math>. Thus, despite a reduction in stability, it is possible this zero derivative could reduce numerical effects caused by decohesions.

Nairnj: /* The Softening Law */

2021-07-20T19:31:47Z

The Softening Law

← Older revision		Revision as of 19:31, 20 July 2021
Line 35:		Line 35:
	<math>\eta = \frac{12k_2}{(1+2k_2)^2k_6}</math>		<math>\eta = \frac{12k_2}{(1+2k_2)^2k_6}</math>

	If <math>k=0</math>, the stability factor simplifies to <math>\eta = 4/3</math>, or slightly less stable than a [[Linear Softening\|linear softening law]]. As <math>k</math> increases the stability decreases. A difference between this law and [[Linear Softening\|linear softening law]] is that <math>f'(\delta_{max})=0</math>. Thus, despite a reduction in stability, it is possible this zero derivative could reduce numerical effects caused by decohesions.		If <math>k=0</math>, the stability factor simplifies to <math>\eta = 4/3</math>, or slightly less stable than a [[Linear Softening\|linear softening law]]. Although this stability factor goes to zero as <math>k</math> increases, the ''effective'' stability is better compared to monotonically decreasing laws by multiplying <math>\eta</math> by its peak value squared (''i.e.'', [[Softening Laws#Law Stability\|minimum cell size]] is related to <math>\eta/\sigma_0^2</math> and <math>\sigma_0</math> decreases if set by reducing peak stress by <math>f_{peak}</math>). The ``effective'' stability of this law only decreased from 4/3 to 0.79 as <math>k</math> gets large.

			A difference between this law and [[Linear Softening\|linear softening law]] is that <math>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 ==		== Softening Law Properties ==

Nairnj: /* Softening Law Properties */

2021-07-20T19:14:12Z

Softening Law Properties

← Older revision		Revision as of 19:14, 20 July 2021
Line 39:		Line 39:
	== Softening Law Properties ==		== Softening Law Properties ==

	~~Only one property is needed to define a~~ cubic step function softening law:		A cubic step function softening law as two parameters (only one is required):

	{\| class="wikitable"		{\| class="wikitable"

Nairnj: /* The Softening Law */

2021-07-09T18:09:07Z

The Softening Law

← Older revision		Revision as of 18:09, 9 July 2021
Line 12:		Line 12:

	<math>sG_c = \int_0^{\delta_{max}} f(\delta,s) = {\delta_{max}\over 2}\left(1+\frac{k}{6}\right)		<math>sG_c = \int_0^{\delta_{max}} f(\delta,s) = {\delta_{max}\over 2}\left(1+\frac{k}{6}\right)
	\quad{\rm or}\quad \delta_{max} = \frac{2sG_c}{1+\frac{k}{6}}</math>		\quad{\rm or}\quad \delta_{max} = \frac{2sG_c}{k_6} \quad{\rm where}\quad k_6 = 1+\frac{k}{6}</math>

	Here ''s'' is the [[Softening Laws#Normalized Softening Law\|softening scaling term]] and ''G<sub>c</sub>'' is toughness of the law.		Here ''s'' is the [[Softening Laws#Normalized Softening Law\|softening scaling term]] and ''G<sub>c</sub>'' is toughness of the law.

Nairnj: /* The Softening Law */

2021-07-09T18:06:18Z

The Softening Law

← Older revision		Revision as of 18:06, 9 July 2021
Line 14:		Line 14:
	\quad{\rm or}\quad \delta_{max} = \frac{2sG_c}{1+\frac{k}{6}}</math>		\quad{\rm or}\quad \delta_{max} = \frac{2sG_c}{1+\frac{k}{6}}</math>

	Here ''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)~~.		Here ''s'' is the [[Softening Laws#Normalized Softening Law\|softening scaling term]] and ''G<sub>c</sub>'' is toughness of the law.
	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 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.

Nairnj: /* The Softening Law */

2021-07-09T18:04:09Z

The Softening Law

← Older revision		Revision as of 18:04, 9 July 2021
Line 35:		Line 35:
	<math>\eta = \frac{12k_2}{(1+2k_2)^2k_6}</math>		<math>\eta = \frac{12k_2}{(1+2k_2)^2k_6}</math>

	If <math>k=0</math>, the stability factor simplifies to <math>\eta = 4/3</math>, or slightly less stable than a [[Linear Softening\|linear softening law]]. As <math>k</math> increases the ~~law's~~ stability decreases. A difference between this law in [[Linear Softening\|linear softening law]] is that <math>f'(\delta_{max})=0</math>. Thus, despite a reduction in stability, it is possible this zero derivative ~~oculd~~ reduce numerical effects caused by ~~decohesion~~.		If <math>k=0</math>, the stability factor simplifies to <math>\eta = 4/3</math>, or slightly less stable than a [[Linear Softening\|linear softening law]]. As <math>k</math> increases the stability decreases. A difference between this law and [[Linear Softening\|linear softening law]] is that <math>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 ==		== Softening Law Properties ==

Nairnj: /* The Softening Law */

2021-07-09T18:02:44Z

The Softening Law

← Older revision		Revision as of 18:02, 9 July 2021
Line 35:		Line 35:
	<math>\eta = \frac{12k_2}{(1+2k_2)^2k_6}</math>		<math>\eta = \frac{12k_2}{(1+2k_2)^2k_6}</math>

	If <math>k=0</math>, the stability factor simplifies to <math>\eta = 4/3</math>, or slightly less stable than a [[Linear Softening\|linear softening law]]. ~~Potentially, the zero derivative at~~ <math>~~\delta_{max}~~</math> ~~could improve modeling of failure. The~~ stability ~~factor~~ decreases as <math>k</math> ~~increases~~.		If <math>k=0</math>, the stability factor simplifies to <math>\eta = 4/3</math>, or slightly less stable than a [[Linear Softening\|linear softening law]]. As <math>k</math> increases the law's stability decreases. A difference between this law in [[Linear Softening\|linear softening law]] is that <math>f'(\delta_{max})=0</math>. Thus, despite a reduction in stability, it is possible this zero derivative oculd reduce numerical effects caused by decohesion.

	== Softening Law Properties ==		== Softening Law Properties ==