Exponential Traction Law - Revision history

Nairnj: /* Traction History Variables */

2021-01-03T05:13:54Z

Traction History Variables

← Older revision		Revision as of 05:13, 3 January 2021
Line 80:		Line 80:
	This material tracks two history variables:		This material tracks two history variables:

	# Maximum normal opening displacement		# Maximum normal opening displacement (or equal to normal δ<sub>e</sub> prior to initiation).
	# Maximum shear opening displacement (~~absolute value~~)		# Maximum shear opening displacement magnitude (or equal to tangential δ<sub>e</sub> prior to initiation).

	These history variables can be [[MPM Archiving Options#ToArchive Command\|archived]] for later plotting. The [[MPM Archiving Options#ToArchive Command\|cmdisp archiving option]] can archive total mode I and mode II cumulative dissipated energies.		These history variables can be [[MPM Archiving Options#ToArchive Command\|archived]] for later plotting. The [[MPM Archiving Options#ToArchive Command\|cmdisp archiving option]] can archive total mode I and mode II cumulative dissipated energies.

Nairnj: /* Traction Law Properties */

2021-01-03T05:03:01Z

Traction Law Properties

← Older revision		Revision as of 05:03, 3 January 2021
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	\| alphaII \|\| Mode II α coefficient \|\| Dimensionless \|\| none		\| alphaII \|\| Mode II α coefficient \|\| Dimensionless \|\| none
	\|-		\|-
	\| ([[Triangular Traction Law#Traction Law Properties\|other]]) \|\| All other traction law properties \|\| varies \|\| varies		\| ([[Triangular Traction Law#Traction Law Properties\|other]]) \|\| All other triangular traction law properties \|\| varies \|\| varies
	\|}		\|}

Nairnj: /* Traction History Variables */

2021-01-03T05:00:41Z

Traction History Variables

← Older revision		Revision as of 05:00, 3 January 2021
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	# Maximum shear opening displacement (absolute value)		# Maximum shear opening displacement (absolute value)

	These history variables can be [[MPM Archiving Options#ToArchive Command\|archived]] for later plotting.		These history variables can be [[MPM Archiving Options#ToArchive Command\|archived]] for later plotting. The [[MPM Archiving Options#ToArchive Command\|cmdisp archiving option]] can archive total mode I and mode II cumulative dissipated energies.

	== References ==		== References ==

Nairnj: /* Traction Law Properties */

2021-01-03T04:57:37Z

Traction Law Properties

← Older revision		Revision as of 04:57, 3 January 2021
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	The resulting laws are required to have alphaI and alphaII greater than zero.		The resulting laws are required to have alphaI and alphaII greater than zero.

			== Traction History Variables ==

			This material tracks two history variables:

			# Maximum normal opening displacement
			# Maximum shear opening displacement (absolute value)

			These history variables can be [[MPM Archiving Options#ToArchive Command\|archived]] for later plotting.

	== References ==		== References ==

Nairnj at 04:50, 3 January 2021

2021-01-03T04:50:37Z

← Older revision		Revision as of 04:50, 3 January 2021
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	__TOC__		__TOC__
	== The Traction Law ==		== The Traction Law ==

	~~(Currently only in [[OSParticulas]] but coming soon to [[NairnMPM]]).~~

	[[File:Exptraction.jpg\|right]]		[[File:Exptraction.jpg\|right]]

Nairnj: /* Comparison to Triangular Traction */

2020-12-28T19:53:40Z

Comparison to Triangular Traction

← Older revision		Revision as of 19:53, 28 December 2020
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	== Comparison to Triangular Traction ==		== Comparison to Triangular Traction ==

	While some a swayed that the use of exponential laws is ~~some how much more~~ realistic ~~the~~ linear laws, the details of cohesive law ~~shape~~ are of second-order importance. They have some minor effects (such as changes in predicted R curve shapes<ref name="czm"/>), but in general, any two laws with the same toughness (J<sub>c</sub>) and peak (σ<sub>c</sub>), will have very similar results.		While some a swayed that the use of exponential laws is somehow realistic while linear laws are too simplistic, the details of cohesive law shapes are of second-order importance. They have some minor effects (such as changes in predicted R curve shapes<ref name="czm"/>), but in general, any two laws with the same toughness (J<sub>c</sub>) and peak (σ<sub>c</sub>), will have very similar results.

	The only ~~different~~ between this exponential law and the [[Triangular Traction Law]] is the α parameter. As shown in the figure above, this parameter changes the decay region of the cohesive law. If one does a series of simulations keeping J<sub>c</sub> and σ<sub>c</sub> constant, but varying α (as a consequence, δ<sub>c</sub> will increase as α increases), the results will have only minor differences. Because [[Triangular Traction Law]] is more efficient, it is usually the preferred cohesive law for simulations. This exponential law is provided for comparison to others or if the minor differences become important.		The only difference between this exponential law and the [[Triangular Traction Law]] is the α parameter. As shown in the figure above, this parameter changes the decay region of the cohesive law. If one does a series of simulations keeping J<sub>c</sub> and σ<sub>c</sub> constant, but varying α (as a consequence, δ<sub>c</sub> will increase as α increases), the results will have only minor differences. Because [[Triangular Traction Law]] is more efficient, it is usually the preferred cohesive law for simulations. This exponential law is provided for comparison to others or if the minor differences become important.

	== Failure ==		== Failure ==

Nairnj at 19:52, 28 December 2020

2020-12-28T19:52:22Z

← Older revision		Revision as of 19:52, 28 December 2020
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	\quad{\rm where}\quad f(\alpha) = 2\left(\frac{1}{\alpha}-\frac{e^{-\alpha}}{1-e^{-\alpha}}\right)</math>		\quad{\rm where}\quad f(\alpha) = 2\left(\frac{1}{\alpha}-\frac{e^{-\alpha}}{1-e^{-\alpha}}\right)</math>

	The function f(α) is 1 when α=0 and decays to zero as α increases. The toughness is thus bracketed by		and δ<sub>peak</sub>=δ<sub>e</sub>/δ<sub>c</sub>. The function f(α) is 1 when α=0 and decays to zero as α increases. The toughness is thus bracketed by


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	The minimum corresponds to large α with an instant drop to zero traction at δ<sub>e</sub> while the maximum corresponds to α=0 and is area under the corresponding [[Triangular Traction Law]].		The minimum corresponds to large α with an instant drop to zero traction at δ<sub>e</sub> while the maximum corresponds to α=0 and is area under the corresponding [[Triangular Traction Law]].

	When creating this traction law, you have to enter α, <i>exactly</i> two of σ<sub>c</sub>, δ<sub>p</sub>, and k and <i>either</i> J<sub>c</sub> or δ<sub>c</sub>. These four required properties must be entered for both mode I and mode II. If you enter δ<sub>c</sub>, then the missing one of σ<sub>c</sub>, δ<sub>p</sub>, and k is calculated from one of		When creating this traction law, you have to enter α, <i>exactly</i> two of σ<sub>c</sub>, δ<sub>peak</sub>, and k and <i>either</i> J<sub>c</sub> or δ<sub>c</sub>. These four required properties must be entered for both mode I and mode II. If you enter δ<sub>c</sub>, then the missing one of σ<sub>c</sub>, δ<sub>peak</sub>, and k is calculated from one of


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	</math>		</math>

	Finally J<sub>c</sub> is found from the expression above. If instead you enter J<sub>c</sub>, the calculation of δ<sub>c</sub> depends on which two of σ<sub>c</sub>, δ<sub>p</sub>, and k are provided:		Finally J<sub>c</sub> is found from the expression above. If instead you enter J<sub>c</sub>, the calculation of δ<sub>c</sub> depends on which two of σ<sub>c</sub>, δ<sub>peak</sub>, and k are provided:


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	</math>		</math>

	Once δ<sub>c</sub> is found, the remaining property among σ<sub>c</sub>, δ<sub>p</sub>, and k is found using an expression above.		Once δ<sub>c</sub> is found, the remaining property among σ<sub>c</sub>, δ<sub>peak</sub>, and k is found using an expression above.

	== Comparison to Triangular Traction ==		== Comparison to Triangular Traction ==

Nairnj at 19:46, 28 December 2020

2020-12-28T19:46:56Z

← Older revision		Revision as of 19:46, 28 December 2020
Line 2:		Line 2:
	== The Traction Law ==		== The Traction Law ==

	(Currently only in [[OSParticulas]] but ~~common~~ soon to [[NairnMPM]]).		(Currently only in [[OSParticulas]] but coming soon to [[NairnMPM]]).

	[[File:Exptraction.jpg\|right]]		[[File:Exptraction.jpg\|right]]

Nairnj: /* Comparison to Triangular Traction */

2020-12-24T00:29:52Z

Comparison to Triangular Traction

← Older revision		Revision as of 00:29, 24 December 2020
Line 54:		Line 54:
	== Comparison to Triangular Traction ==		== Comparison to Triangular Traction ==

	While some a swayed that the use of exponential laws is some how much more realistic the linear laws, the details of cohesive law shape are of second-order importance. They have some minor effects (such as changes in predicted R curve shapes<ref name="czm">), but in general, any two laws with the same toughness (J<sub>c</sub>) and peak (σ<sub>c</sub>), will have very similar results.		While some a swayed that the use of exponential laws is some how much more realistic the linear laws, the details of cohesive law shape are of second-order importance. They have some minor effects (such as changes in predicted R curve shapes<ref name="czm"/>), but in general, any two laws with the same toughness (J<sub>c</sub>) and peak (σ<sub>c</sub>), will have very similar results.

	The only different between this exponential law and the [[Triangular Traction Law]] is the α parameter. As shown in the figure above, this parameter changes the decay region of the cohesive law. If one does a series of simulations keeping J<sub>c</sub> and σ<sub>c</sub> constant, but varying α (as a consequence, δ<sub>c</sub> will increase as α increases), the results will have only minor differences. Because [[Triangular Traction Law]] is more efficient, it is usually the preferred cohesive law for simulations. This exponential law is provided for comparison to others or if the minor differences become important.		The only different between this exponential law and the [[Triangular Traction Law]] is the α parameter. As shown in the figure above, this parameter changes the decay region of the cohesive law. If one does a series of simulations keeping J<sub>c</sub> and σ<sub>c</sub> constant, but varying α (as a consequence, δ<sub>c</sub> will increase as α increases), the results will have only minor differences. Because [[Triangular Traction Law]] is more efficient, it is usually the preferred cohesive law for simulations. This exponential law is provided for comparison to others or if the minor differences become important.

Nairnj: /* Comparison to Triangular Traction */

2020-12-24T00:29:35Z

Comparison to Triangular Traction

← Older revision		Revision as of 00:29, 24 December 2020
Line 54:		Line 54:
	== Comparison to Triangular Traction ==		== Comparison to Triangular Traction ==

	While some a swayed that the use of exponential laws is some how much more realistic the linear laws, the details of cohesive law shape are of second-order importance. They have some minor effects (such as ~~change~~ in predicted R curve shapes), but in general, any two laws with the same toughness (~~area~~) and peak (~~strength~~), will have very similar results.		While some a swayed that the use of exponential laws is some how much more realistic the linear laws, the details of cohesive law shape are of second-order importance. They have some minor effects (such as changes in predicted R curve shapes<ref name="czm">), but in general, any two laws with the same toughness (J<sub>c</sub>) and peak (σ<sub>c</sub>), will have very similar results.

	The only different between this exponential law and the [[Triangular Traction Law]] is the α parameter. As shown in the figure above, this parameter changes the decay region of the cohesive law. If one does a series of simulations keeping J<sub>c</sub> and σ<sub>c</sub>, but ~~changing~~ α (as a consequence, δ<sub>c</sub> will increase as α increases), the results will have only minor differences. Because [[Triangular Traction Law]] is more efficient, it is usually the preferred cohesive law for simulations. This exponential law is provided for comparison to others or if the minor ~~difference are~~ important.		The only different between this exponential law and the [[Triangular Traction Law]] is the α parameter. As shown in the figure above, this parameter changes the decay region of the cohesive law. If one does a series of simulations keeping J<sub>c</sub> and σ<sub>c</sub> constant, but varying α (as a consequence, δ<sub>c</sub> will increase as α increases), the results will have only minor differences. Because [[Triangular Traction Law]] is more efficient, it is usually the preferred cohesive law for simulations. This exponential law is provided for comparison to others or if the minor differences become important.

	== Failure ==		== Failure ==