Triangular Traction Law - Revision history

Nairnj: /* Elastic-Brittle Traction */

2026-03-28T00:54:56Z

Elastic-Brittle Traction

← Older revision		Revision as of 00:54, 28 March 2026
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	A special case of this traction laws can model elastic-brittle behaviour where load increases linearly until reaching σ<sub>c</sub> at δ<sub>c</sub> followed by instant drop to zero traction (<i>i.e.</i>, brittle failure. To model with elastic-brittle traction for either mode I or mode II, set δ<sub>I,peak</sub> = 1 or δ<sub>II,peak</sub> = 1, respectively.		A special case of this traction laws can model elastic-brittle behaviour where load increases linearly until reaching σ<sub>c</sub> at δ<sub>c</sub> followed by instant drop to zero traction (<i>i.e.</i>, brittle failure. To model with elastic-brittle traction for either mode I or mode II, set δ<sub>I,peak</sub> = 1 or δ<sub>II,peak</sub> = 1, respectively.

	~~This option is currently in [[OSParticulas]] and will be added to [[NairnMPM]] soon (if not already).~~

	=== Pure Linear Softening ===		=== Pure Linear Softening ===

Nairnj: /* Elastic-Brittle Traction */

2025-07-15T23:27:13Z

Elastic-Brittle Traction

← Older revision		Revision as of 23:27, 15 July 2025
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	This option is currently in [[OSParticulas]] and will be added to [[NairnMPM]] soon (if not already).		This option is currently in [[OSParticulas]] and will be added to [[NairnMPM]] soon (if not already).

			=== Pure Linear Softening ===

			Pure linear softening, which is infinite stiffness until σ<sub>c</sub> at δ = 0 followed by linear softening until zero load at δ<sub>c</sub>, is currently not supported. It can be emulated by choosing small, but non-zero, values for δ<sub>I,peak</sub> or δ<sub>II,peak</sub>.

	== Failure ==		== Failure ==

Nairnj: /* Pure Linear Softening */

2025-07-15T23:01:59Z

Pure Linear Softening

← Older revision		Revision as of 23:01, 15 July 2025
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	<math>k = {\sigma_c\over 0.225926299 \delta_c}</math>		<math>k = {\sigma_c\over 0.225926299 \delta_c}</math>

	=== ~~Pure Linear Softening~~ ===		=== Elastic-Brittle Traction ===

	A special case of this traction laws can model ~~pure linear softening, which is infinite stiffness~~ until ~~initiation of failure at~~ σ<sub>c</sub> ~~following by linear softening until failure~~ at δ<sub>c</sub>. To model with ~~pure linear softening~~ for either mode I or mode II, set δ<sub>I,peak</sub> = 1 or δ<sub>II,peak</sub> = 1, respectively.		A special case of this traction laws can model elastic-brittle behaviour where load increases linearly until reaching σ<sub>c</sub> at δ<sub>c</sub> followed by instant drop to zero traction (<i>i.e.</i>, brittle failure. To model with elastic-brittle traction for either mode I or mode II, set δ<sub>I,peak</sub> = 1 or δ<sub>II,peak</sub> = 1, respectively.

	This option is currently in [[OSParticulas]] and will be added to [[NairnMPM]] soon (if not already).		This option is currently in [[OSParticulas]] and will be added to [[NairnMPM]] soon (if not already).

Nairnj: /* The Traction Law */

2025-07-15T22:56:59Z

The Traction Law

← Older revision		Revision as of 22:56, 15 July 2025
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	=== Pure Linear Softening ===		=== Pure Linear Softening ===

	A special case of this traction laws can model pure linear softening, which is infinite stiffness until initiation of failure at σ<sub>c</sub> following by linear softening until failure at δ<sub>c</sub>. To model with pure linear softening ~~enter~~		A special case of this traction laws can model pure linear softening, which is infinite stiffness until initiation of failure at σ<sub>c</sub> following by linear softening until failure at δ<sub>c</sub>. To model with pure linear softening for either mode I or mode II, set δ<sub>I,peak</sub> = 1 or δ<sub>II,peak</sub> = 1, respectively.
	~~This implementation disallows zero of infinite initial stiffness (~~for ~~stability reasons). In other words~~, ~~the entered parameters must resolve to k~~>~~0 and~~ δ<sub>peak</sub>>0.
			This option is currently in [[OSParticulas]] and will be added to [[NairnMPM]] soon (if not already).

	== Failure ==		== Failure ==

Nairnj: /* The Traction Law */

2025-07-15T22:54:18Z

The Traction Law

← Older revision		Revision as of 22:54, 15 July 2025
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	<math>k = {\sigma_c\over 0.225926299 \delta_c}</math>		<math>k = {\sigma_c\over 0.225926299 \delta_c}</math>

			=== Pure Linear Softening ===

			A special case of this traction laws can model pure linear softening, which is infinite stiffness until initiation of failure at σ<sub>c</sub> following by linear softening until failure at δ<sub>c</sub>. To model with pure linear softening enter
	This implementation disallows zero of infinite initial stiffness (for stability reasons). In other words, the entered parameters must resolve to k>0 and δ<sub>peak</sub>>0.		This implementation disallows zero of infinite initial stiffness (for stability reasons). In other words, the entered parameters must resolve to k>0 and δ<sub>peak</sub>>0.

Nairnj: /* Traction History Variables */

2021-01-03T05:13:17Z

Traction History Variables

← Older revision		Revision as of 05:13, 3 January 2021
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	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 (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 History Variables */

2021-01-03T05:01:16Z

Traction History Variables

← Older revision		Revision as of 05:01, 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 History Variabls */

2021-01-03T04:56:55Z

Traction History Variabls

← Older revision		Revision as of 04:56, 3 January 2021
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	The resulting laws are required to have kIe, delpkI, kIIe, and delpkII as entered (or as [[#The Traction Law\|calculated]]) must all be greater than zero. In other words, for stability reasons, the initial stiffness cannot be infinite.		The resulting laws are required to have kIe, delpkI, kIIe, and delpkII as entered (or as [[#The Traction Law\|calculated]]) must all be greater than zero. In other words, for stability reasons, the initial stiffness cannot be infinite.

	== Traction History ~~Variabls~~ ==		== Traction History Variables ==

	This material tracks two history variables:		This material tracks two history variables:

Nairnj: /* Traction Law Properties */

2021-01-03T04:56:29Z

Traction Law Properties

← Older revision		Revision as of 04:56, 3 January 2021
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	The resulting laws are required to have kIe, delpkI, kIIe, and delpkII as entered (or as [[#The Traction Law\|calculated]]) must all be greater than zero. In other words, for stability reasons, the initial stiffness cannot be infinite.		The resulting laws are required to have kIe, delpkI, kIIe, and delpkII as entered (or as [[#The Traction Law\|calculated]]) must all be greater than zero. In other words, for stability reasons, the initial stiffness cannot be infinite.

			== Traction History Variabls ==

			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 23:15, 29 December 2020

2020-12-29T23:15:23Z

← Older revision		Revision as of 23:15, 29 December 2020
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	where G<sub>I</sub> and G<sub>II</sub> are areas under mode I and mode II traction laws up to current maximum COD and ''n'' is mixed-mode failure criterion parameter. This failure criterion will mean that failure may occur when normal or shear tractions are nonzero. In other words, each traction law will have COD less then their critical COD, which means their traction will be greater than zero. Because failure releases the crack surfaces, mixed-mode failure by this criterion may result in a sudden drop of finite normal and shear traction to zero traction. Such drops might cause numerical issues.		where G<sub>I</sub> and G<sub>II</sub> are areas under mode I and mode II traction laws up to current maximum COD and ''n'' is mixed-mode failure criterion parameter. This failure criterion will mean that failure may occur when normal or shear tractions are nonzero. In other words, each traction law will have COD less then their critical COD, which means their traction will be greater than zero. Because failure releases the crack surfaces, mixed-mode failure by this criterion may result in a sudden drop of finite normal and shear traction to zero traction. Such drops might cause numerical issues.

	Although use of uncoupled mode I and mode II laws with a failure criterion is common in cohesive zone models<ref name="thouless"/>, other models have coupled the mode I and II traction laws such that they always approach zero at time of failure (''e.g.'', work of Högberg (2006),<ref name="hog/> Camacho and Ortiz (1996),<ref name=ortiz"/> and Comanho (1996)<ref name="comanho"/>). The Högberg model is implemented in the [[Coupled ~~Triangular~~ Traction Law]]. These approaches are based on defining an ''effective displacement.'' A recent re-analysis of cohesive zones, however, has shown these models are invalid unless the normal and tangential laws are scaled copies of each other.<ref name="mmzone"/> A better approach based on this re-analysos<ref name="mmzone"/> that couples mode I and mode II and remains valid for arbitrary normal and tangential laws is now the recommended model for mixed mode failure and is implemented in the [[Mixed Mode Traction Law]].		Although use of uncoupled mode I and mode II laws with a failure criterion is common in cohesive zone models<ref name="thouless"/>, other models have coupled the mode I and II traction laws such that they always approach zero at time of failure (''e.g.'', work of Högberg (2006),<ref name="hog/> Camacho and Ortiz (1996),<ref name=ortiz"/> and Comanho (1996)<ref name="comanho"/>). The Högberg model is implemented in the [[Coupled Traction Law]]. These approaches are based on defining an ''effective displacement.'' A recent re-analysis of cohesive zones, however, has shown these models are invalid unless the normal and tangential laws are scaled copies of each other.<ref name="mmzone"/> A better approach based on this re-analysos<ref name="mmzone"/> that couples mode I and mode II and remains valid for arbitrary normal and tangential laws is now the recommended model for mixed mode failure and is implemented in the [[Mixed Mode Traction Law]].

	== Traction Law Properties ==		== Traction Law Properties ==