Double Exponential Softening - Revision history

Nairnj at 03:57, 30 January 2023

2023-01-30T03:57:15Z

← Older revision		Revision as of 03:57, 30 January 2023
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	__TOC__		__TOC__
	== The Softening Law ==		== The Softening Law ==
	The main goal of his [[Softening Laws\|softening law]] is to provide another softening law that can initiate at low stress, <math>\sigma_{0}</math> (from the [[Damage Initiation Laws\|damage initiation law]]), rise to a peak, <math>\sigma_0*f_{peak}</math>, and then decay to zero ~~(currently [[OSParticulas]] only)~~. With that goal in mind, the function is		The main goal of his [[Softening Laws\|softening law]] is to provide another softening law that can initiate at low stress, <math>\sigma_{0}</math> (from the [[Damage Initiation Laws\|damage initiation law]]), rise to a peak, <math>\sigma_0*f_{peak}</math>, and then decay to zero. With that goal in mind, the function is

Nairnj: /* The Softening Law */

2021-09-08T18:23:37Z

The Softening Law

← Older revision		Revision as of 18:23, 8 September 2021
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	</math>		</math>

	For <math>\beta=0</math>, the stability factor simplifies to <math>\eta = 1</math> which is identical to [[Exponential Softening]] (in fact the entire law is identical to [[Exponential Softening]] and that law should therefore be used instead). Although this stability factor goes to zero as <math>\beta\to1</math>, 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 figure on the right plots this ''effective'' stability factor as a function of <math>\beta</math> for various values of <math>\alpha</math>. For <math>\beta>0</math>, this law is slightly more stable than [[Exponential Softening]]. For <math>\beta<0</math>, this law decreases in stability as <math>\alpha</math> increases, because the second exponential term ~~cause~~ magnitude of the initial slope to increase. Using this law with <math>\beta<0</math> and a large <math>\alpha</math> may cause stability problems. Typically, this law should set <math>0<\beta<1</math> for which '' effective'' stability will range from 1 to 1.49.		For <math>\beta=0</math>, the stability factor simplifies to <math>\eta = 1</math> which is identical to [[Exponential Softening]] (in fact the entire law is identical to [[Exponential Softening]] and that law should therefore be used instead). Although this stability factor goes to zero as <math>\beta\to1</math>, 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 figure on the right plots this ''effective'' stability factor as a function of <math>\beta</math> for various values of <math>\alpha</math>. For <math>\beta>0</math>, this law is slightly more stable than [[Exponential Softening]]. For <math>\beta<0</math>, this law decreases in stability as <math>\alpha</math> increases, because the second exponential term causes magnitude of the initial slope to increase. Using this law with <math>\beta<0</math> and a large <math>\alpha</math> may cause stability problems. Typically, this law should set <math>0<\beta<1</math> for which ''effective'' stability will range from 1 to 1.49.

	=== Minimum <math>f(\delta,s)</math> ===		=== Minimum <math>f(\delta,s)</math> ===

Nairnj: /* Softening Law Properties */

2021-07-20T20:15:08Z

Softening Law Properties

← Older revision		Revision as of 20:15, 20 July 2021
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	== Softening Law Properties ==		== Softening Law Properties ==

	A double exponential softening law ~~requires~~ three ~~parameters~~:		A double exponential softening law has four parameters (three of much must always be provided):

	{\| class="wikitable"		{\| class="wikitable"
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	\|-		\|-
	\| beta \|\| relative magnitude of the the second exponential (must be less than 1) \|\| none \|\| none		\| beta \|\| relative magnitude of the the second exponential (must be less than 1) \|\| none \|\| none
			\|-
			\| min \|\| Minimum <math>f(\delta,s)</math> or law is failed if it gets below this value \|\| none \|\| 0.01
	\|}		\|}

Nairnj: /* The Softening Law */

2021-07-20T20:13:20Z

The Softening Law

← Older revision		Revision as of 20:13, 20 July 2021
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	For <math>\beta=0</math>, the stability factor simplifies to <math>\eta = 1</math> which is identical to [[Exponential Softening]] (in fact the entire law is identical to [[Exponential Softening]] and that law should therefore be used instead). Although this stability factor goes to zero as <math>\beta\to1</math>, 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 figure on the right plots this ''effective'' stability factor as a function of <math>\beta</math> for various values of <math>\alpha</math>. For <math>\beta>0</math>, this law is slightly more stable than [[Exponential Softening]]. For <math>\beta<0</math>, this law decreases in stability as <math>\alpha</math> increases, because the second exponential term cause magnitude of the initial slope to increase. Using this law with <math>\beta<0</math> and a large <math>\alpha</math> may cause stability problems. Typically, this law should set <math>0<\beta<1</math> for which '' effective'' stability will range from 1 to 1.49.		For <math>\beta=0</math>, the stability factor simplifies to <math>\eta = 1</math> which is identical to [[Exponential Softening]] (in fact the entire law is identical to [[Exponential Softening]] and that law should therefore be used instead). Although this stability factor goes to zero as <math>\beta\to1</math>, 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 figure on the right plots this ''effective'' stability factor as a function of <math>\beta</math> for various values of <math>\alpha</math>. For <math>\beta>0</math>, this law is slightly more stable than [[Exponential Softening]]. For <math>\beta<0</math>, this law decreases in stability as <math>\alpha</math> increases, because the second exponential term cause magnitude of the initial slope to increase. Using this law with <math>\beta<0</math> and a large <math>\alpha</math> may cause stability problems. Typically, this law should set <math>0<\beta<1</math> for which '' effective'' stability will range from 1 to 1.49.


	=== Minimum <math>f(\delta,s)</math> ===		=== Minimum <math>f(\delta,s)</math> ===

Nairnj: /* The Softening Law */

2021-07-20T20:13:01Z

The Softening Law

← Older revision		Revision as of 20:13, 20 July 2021
Line 45:		Line 45:

	For <math>\beta=0</math>, the stability factor simplifies to <math>\eta = 1</math> which is identical to [[Exponential Softening]] (in fact the entire law is identical to [[Exponential Softening]] and that law should therefore be used instead). Although this stability factor goes to zero as <math>\beta\to1</math>, 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 figure on the right plots this ''effective'' stability factor as a function of <math>\beta</math> for various values of <math>\alpha</math>. For <math>\beta>0</math>, this law is slightly more stable than [[Exponential Softening]]. For <math>\beta<0</math>, this law decreases in stability as <math>\alpha</math> increases, because the second exponential term cause magnitude of the initial slope to increase. Using this law with <math>\beta<0</math> and a large <math>\alpha</math> may cause stability problems. Typically, this law should set <math>0<\beta<1</math> for which '' effective'' stability will range from 1 to 1.49.		For <math>\beta=0</math>, the stability factor simplifies to <math>\eta = 1</math> which is identical to [[Exponential Softening]] (in fact the entire law is identical to [[Exponential Softening]] and that law should therefore be used instead). Although this stability factor goes to zero as <math>\beta\to1</math>, 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 figure on the right plots this ''effective'' stability factor as a function of <math>\beta</math> for various values of <math>\alpha</math>. For <math>\beta>0</math>, this law is slightly more stable than [[Exponential Softening]]. For <math>\beta<0</math>, this law decreases in stability as <math>\alpha</math> increases, because the second exponential term cause magnitude of the initial slope to increase. Using this law with <math>\beta<0</math> and a large <math>\alpha</math> may cause stability problems. Typically, this law should set <math>0<\beta<1</math> for which '' effective'' stability will range from 1 to 1.49.


			=== Minimum <math>f(\delta,s)</math> ===

			This law requires selection of minimum value for <math>f(\delta,s)</math> below which the material point is marked as failed. If we define the minimum value as ''c'', this choice defines an effective maximum cracking strain by numerically solving for


			<math>f(\delta_{max},c) = c </math>

			Without a minimum value, double exponential softening would never fail. If you want to prevent failures, set the minimum value to a very small number (but it cannot be zero).

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

Nairnj: /* The Softening Law */

2021-07-20T20:07:10Z

The Softening Law

← Older revision		Revision as of 20:07, 20 July 2021
Line 44:		Line 44:
	</math>		</math>

	For <math>\beta=0</math>, the stability factor simplifies to <math>\eta = 1</math> which is identical to [[Exponential Softening]] (in fact the entire law is identical to [[Exponential Softening]] and that law should therefore be used instead). Although this stability factor goes to zero as <math>\beta\to1</math>, 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 figure on the right plots this ''effective'' stability factor as a function of <math>\beta</math> for various values of <math>\alpha</math>. For <math>\beta>0</math>, this law is slightly more stable than [[Exponential Softening]]. For <math>\beta<0</math>, this law decreases in stability as <math>\alpha</math> increases, because the second exponential term cause magnitude of the initial slope to increase. Using this law with <math>\beta<0</math> and a large <math>\alpha</math> may cause stability problems. Typically, this law should set <math>0<\beta<1</math> for which '' ~~effecrtive~~'' stability will range from 1 to 1.49.		For <math>\beta=0</math>, the stability factor simplifies to <math>\eta = 1</math> which is identical to [[Exponential Softening]] (in fact the entire law is identical to [[Exponential Softening]] and that law should therefore be used instead). Although this stability factor goes to zero as <math>\beta\to1</math>, 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 figure on the right plots this ''effective'' stability factor as a function of <math>\beta</math> for various values of <math>\alpha</math>. For <math>\beta>0</math>, this law is slightly more stable than [[Exponential Softening]]. For <math>\beta<0</math>, this law decreases in stability as <math>\alpha</math> increases, because the second exponential term cause magnitude of the initial slope to increase. Using this law with <math>\beta<0</math> and a large <math>\alpha</math> may cause stability problems. Typically, this law should set <math>0<\beta<1</math> for which '' effective'' stability will range from 1 to 1.49.

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

Nairnj: /* The Softening Law */

2021-07-20T20:06:52Z

The Softening Law

← Older revision		Revision as of 20:06, 20 July 2021
Line 44:		Line 44:
	</math>		</math>

	For <math>\beta=0</math>, the stability factor simplifies to <math>\eta = 1</math> which is identical to [[Exponential Softening]] (in fact the entire law is identical to [[Exponential Softening]] and that law should therefore be used instead). Although this stability factor goes to zero as <math>\beta\to1</math>, 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 figure on the right plots this ''effective'' stability factor as a function of <math>\beta</math> for various values of <math>\alpha</math>. For <math>\beta>0</math>, this law is slightly more stable than [[Exponential Softening]]. For <math>\beta<0</math>, this law decreases in stability as <math>\alpha</math> increases, because the second exponential term cause magnitude of the initial slope to increase. Using this law with <math>\beta<0</math> and a large <math>\alpha</math> may cause stability problems. Typically, this law should set <math>0<\beta<1</math>.		For <math>\beta=0</math>, the stability factor simplifies to <math>\eta = 1</math> which is identical to [[Exponential Softening]] (in fact the entire law is identical to [[Exponential Softening]] and that law should therefore be used instead). Although this stability factor goes to zero as <math>\beta\to1</math>, 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 figure on the right plots this ''effective'' stability factor as a function of <math>\beta</math> for various values of <math>\alpha</math>. For <math>\beta>0</math>, this law is slightly more stable than [[Exponential Softening]]. For <math>\beta<0</math>, this law decreases in stability as <math>\alpha</math> increases, because the second exponential term cause magnitude of the initial slope to increase. Using this law with <math>\beta<0</math> and a large <math>\alpha</math> may cause stability problems. Typically, this law should set <math>0<\beta<1</math> for which '' effecrtive'' stability will range from 1 to 1.49.

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

Nairnj: /* The Softening Law */

2021-07-20T19:52:30Z

The Softening Law

← Older revision		Revision as of 19:52, 20 July 2021
Line 44:		Line 44:
	</math>		</math>

	For <math>\beta=0</math>, the stability factor simplifies to <math>\eta = 1</math> which is identical to [[Exponential Softening]] (in fact the entire law is identical to [[Exponential Softening]] and that law should therefore be used instead). Although this stability factor goes to zero as <math>\beta\to1</math>, 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 figure on the right plots this ''effective'' stability factor as a function of <math>\beta</math> for various values of <math>\alpha</math>. ~~Overall~~, ~~the stability of~~ this law is ~~comparable to the stability of a single~~ [[Exponential Softening]] law.		For <math>\beta=0</math>, the stability factor simplifies to <math>\eta = 1</math> which is identical to [[Exponential Softening]] (in fact the entire law is identical to [[Exponential Softening]] and that law should therefore be used instead). Although this stability factor goes to zero as <math>\beta\to1</math>, 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 figure on the right plots this ''effective'' stability factor as a function of <math>\beta</math> for various values of <math>\alpha</math>. For <math>\beta>0</math>, this law is slightly more stable than [[Exponential Softening]]. For <math>\beta<0</math>, this law decreases in stability as <math>\alpha</math> increases, because the second exponential term cause magnitude of the initial slope to increase. Using this law with <math>\beta<0</math> and a large <math>\alpha</math> may cause stability problems. Typically, this law should set <math>0<\beta<1</math>.

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

Nairnj: /* The Softening Law */

2021-07-20T19:38:47Z

The Softening Law

← Older revision		Revision as of 19:38, 20 July 2021
Line 44:		Line 44:
	</math>		</math>

	For <math>\beta=0</math>, the stability factor simplifies to <math>\eta = 1</math> which is identical to [[Exponential Softening]] (in fact the entire law is identical to [[Exponential Softening]] and that law should therefore be used instead). Although this stability factor goes to zero as <math>\beta\to1</math>, the ''effective'' stability is better compared to monotonically decreasing laws by multiplying <math>\eta</math> by its peak ~~values~~ 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 figure on the right plots this ''effective'' stability factor as a function of <math>\beta</math> for various values of <math>\alpha</math>. Overall, the stability of this law is comparable to the stability of a single [[Exponential Softening]] law.		For <math>\beta=0</math>, the stability factor simplifies to <math>\eta = 1</math> which is identical to [[Exponential Softening]] (in fact the entire law is identical to [[Exponential Softening]] and that law should therefore be used instead). Although this stability factor goes to zero as <math>\beta\to1</math>, 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 figure on the right plots this ''effective'' stability factor as a function of <math>\beta</math> for various values of <math>\alpha</math>. Overall, the stability of this law is comparable to the stability of a single [[Exponential Softening]] law.

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

Nairnj: /* The Softening Law */

2021-07-20T19:37:40Z

The Softening Law

← Older revision		Revision as of 19:37, 20 July 2021
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	<math>sG_c = \int_0^{\infty} f(\delta,s) = \frac{\alpha-\beta}{k\alpha(1-\beta)s}		<math>sG_c = \int_0^{\infty} f(\delta,s) = \frac{\alpha-\beta}{k\alpha(1-\beta)}
	\quad{\rm or}\quad k = \frac{\alpha-\beta}{\alpha(1-\beta)sG_c}</math>		\quad{\rm or}\quad k = \frac{\alpha-\beta}{\alpha(1-\beta)sG_c}</math>