Trilinear Traction Law - Revision history

Nairnj: /* The Traction Law */

2021-01-03T05:16:07Z

The Traction Law

← Older revision		Revision as of 05:16, 3 January 2021
Line 23:		Line 23:
	<math>J_2 = {1\over2}\sigma_2\delta_c</math>		<math>J_2 = {1\over2}\sigma_2\delta_c</math>

	For example, the first peak might be associated with crack tip fracture toughness while the tail models a process zone such as fiber bridging. In fact, in the limit as σ<sub>1</sub> → ∞ and δ<sub>2</sub> → 0, a simulation with a pre-existing cohesive zone should approach a simulation propagating a crack tip using fracture mechanics along with a linear softening traction law having σ<sub>2</sub> as the cohesive stress. [[NairnMPM]] can do both these simulations and the results will be similar. The pure cohesive zone simulation is done by using a trilinear traction law on an existing crack. The combined fracture mechanics/cohesive zone simulation starts with a crack that propagates when energy release rate at the crack tip is equal to J<sub>1</sub>. Furthermore, the [[Crack Propagation Commands\|crack propagation commands]] are set to leave a cohesive zone in the wake of the crack. This cohesive zone can be a [[Triangular Traction Law\|\|triangular traction law]] with its toughness equal to J<sub>2</sub> and stress equal to σ<sub>2</sub>. Both these types of simulations are described in Nairn (2009).<ref name="czm"/>		For example, the first peak might be associated with crack tip fracture toughness while the tail models a process zone such as fiber bridging. In fact, in the limit as σ<sub>1</sub> → ∞ and δ<sub>2</sub> → 0, a simulation with a pre-existing cohesive zone should approach a simulation propagating a crack tip using fracture mechanics along with a linear softening traction law having σ<sub>2</sub> as the cohesive stress. [[NairnMPM]] can do both these simulations and the results will be similar. The pure cohesive zone simulation is done by using a trilinear traction law on an existing crack. The combined fracture mechanics/cohesive zone simulation starts with a crack that propagates when energy release rate at the crack tip is equal to J<sub>1</sub>. Furthermore, the [[Crack Propagation Commands\|crack propagation commands]] are set to leave a cohesive zone in the wake of the crack. This cohesive zone can be a [[Triangular Traction Law\|triangular traction law]] with its toughness equal to J<sub>2</sub> and stress equal to σ<sub>2</sub>. Both these types of simulations are described in Nairn (2009).<ref name="czm"/>

	== Failure ==		== Failure ==

Nairnj: /* Traction History Variables */

2021-01-03T05:15:09Z

Traction History Variables

← Older revision		Revision as of 05:15, 3 January 2021
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	# Maximum normal opening displacement (or equal to normal δ<sub>e</sub> prior to initiation).		# Maximum normal opening displacement (or equal to normal δ<sub>e</sub> prior to initiation).
	# Maximum shear opening displacement (or equal to tangential δ<sub>e</sub> prior to initiation).		# 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 */

2021-01-03T05:12:52Z

Traction Law Properties

← Older revision		Revision as of 05:12, 3 January 2021
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	\| ([[Common Traction Law Properties\|other]]) \|\| Properties common to all traction laws, where sigmaI and sigmaII apply to the first break point stress \|\| varies \|\| varies		\| ([[Common Traction Law Properties\|other]]) \|\| Properties common to all traction laws, where sigmaI and sigmaII apply to the first break point stress \|\| varies \|\| varies
	\|}		\|}

			== Traction History Variables ==

			This material tracks two history variables:

			# Maximum normal opening displacement (or equal to normal δ<sub>e</sub> prior to initiation).
			# 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.

	== References ==		== References ==

Nairnj: /* Failure */

2020-12-22T18:10:05Z

Failure

← Older revision		Revision as of 18:10, 22 December 2020
Line 27:		Line 27:
	== Failure ==		== Failure ==

	This laws uses uncoupled mode I and mode II cohesive laws. The failure is handled by the same methods used for the [[Triangular Traction Law]]. This ~~cubic~~ cohesive law can also be used for mode I or mode II laws in coupled mixed-mode modeling<ref name="mmzone"/> by using the [[Mixed Mode Traction Law]].		This laws uses uncoupled mode I and mode II cohesive laws. The failure is handled by the same methods used for the [[Triangular Traction Law]]. This trilinear cohesive law can also be used for mode I or mode II laws in coupled mixed-mode modeling<ref name="mmzone"/> by using the [[Mixed Mode Traction Law]].

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

Nairnj at 18:08, 22 December 2020

2020-12-22T18:08:09Z

← Older revision		Revision as of 18:08, 22 December 2020
Line 23:		Line 23:
	<math>J_2 = {1\over2}\sigma_2\delta_c</math>		<math>J_2 = {1\over2}\sigma_2\delta_c</math>

	For example, the first peak might be associated with crack tip fracture toughness while the tail models a process zone such as fiber bridging. In fact, in the limit as σ<sub>1</sub> → ∞ and δ<sub>2</sub> → 0, a simulation with a pre-existing cohesive zone should approach a simulation propagating a crack tip using fracture mechanics along with a linear softening traction law having σ<sub>2</sub> as the cohesive stress. [[NairnMPM]] can do both these simulations and the results will be similar. The pure cohesive zone simulation is done by using a trilinear traction law on an existing crack. The combined fracture mechanics/cohesive zone simulation starts with a crack that propagates when energy release rate at the crack tip is equal to J<sub>1</sub>. Furthermore, the [[Crack Propagation Commands\|crack propagation commands]] are set to leave a cohesive zone in the wake of the crack. This cohesive zone can be a [[Triangular Traction Law\|\|triangular traction law]] with its toughness equal to J<sub>2</sub> and stress equal to σ<sub>2</sub>. Both these types of simulations are described in Nairn (2009).<ref name="czm"~~>J. A. Nairn, "Analytical and Numerical Modeling of R Curves for Cracks with Bridging Zones" <i>Int. J. Fracture<~~/~~i>, <b>155</b>, 167-181 (2009). ([http://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/JBridging.pdf See PDF])</ref~~>		For example, the first peak might be associated with crack tip fracture toughness while the tail models a process zone such as fiber bridging. In fact, in the limit as σ<sub>1</sub> → ∞ and δ<sub>2</sub> → 0, a simulation with a pre-existing cohesive zone should approach a simulation propagating a crack tip using fracture mechanics along with a linear softening traction law having σ<sub>2</sub> as the cohesive stress. [[NairnMPM]] can do both these simulations and the results will be similar. The pure cohesive zone simulation is done by using a trilinear traction law on an existing crack. The combined fracture mechanics/cohesive zone simulation starts with a crack that propagates when energy release rate at the crack tip is equal to J<sub>1</sub>. Furthermore, the [[Crack Propagation Commands\|crack propagation commands]] are set to leave a cohesive zone in the wake of the crack. This cohesive zone can be a [[Triangular Traction Law\|\|triangular traction law]] with its toughness equal to J<sub>2</sub> and stress equal to σ<sub>2</sub>. Both these types of simulations are described in Nairn (2009).<ref name="czm"/>

	== Failure ==		== Failure ==
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	== References ==		== References ==

	<references/>		<references>
			<ref name="czm">J. A. Nairn, "Analytical and Numerical Modeling of R Curves for Cracks with Bridging Zones" <i>Int. J. Fracture</i>, <b>155</b>, 167-181 (2009). ([http://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/JBridging.pdf See PDF])</ref><ref name="mmzone">J. A. Nairn and Y. E. Aimene "A re-evaluation of mixed-mode cohesive zone modeling based on strength concepts instead of traction laws" <i>in preparation</i> (2020).</ref>
			</references>

Nairnj: /* Failure */

2020-12-22T18:06:21Z

Failure

← Older revision		Revision as of 18:06, 22 December 2020
Line 27:		Line 27:
	== Failure ==		== Failure ==

	The failure ~~criterion under pure more or possible mixed-mode conditions~~ is ~~identical to~~ the ~~one~~ used in the [[Triangular Traction Law~~#Failure\|triangular traction~~ law]].		This laws uses uncoupled mode I and mode II cohesive laws. The failure is handled by the same methods used for the [[Triangular Traction Law]]. This cubic cohesive law can also be used for mode I or mode II laws in coupled mixed-mode modeling<ref name="mmzone"/> by using the [[Mixed Mode Traction Law]].

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

Nairnj: /* The Traction Law */

2020-06-21T20:45:06Z

The Traction Law

← Older revision		Revision as of 20:45, 21 June 2020
Line 11:		Line 11:
	<math>J_c = {1\over2}\bigl(\sigma_1\delta_2 + \sigma_2(\delta_c-\delta_1)\bigr)</math>		<math>J_c = {1\over2}\bigl(\sigma_1\delta_2 + \sigma_2(\delta_c-\delta_1)\bigr)</math>

	The traction law for each mode depends up seven properties - (δ<sub>1</sub>,σ<sub>1</sub>) and (δ<sub>2</sub>,σ<sub>2</sub>) breakpoints, critical COD (δ<sub>c</sub>), total area or toughness (J<sub>c</sub>), and the initial elastic slope (k). You must enter exactly 5 of these seven properties for each mode. Furthermore, if one of the 5 is the initial slope, then δ<sub>c</sub> along with either σ<sub>1</sub> or δ<sub>1</sub> for that mode (but not both) must be among the specified properties. The two remaining unspecified parameters will be calculated from the five provided parameters. The final law must satisfy 0 < δ<sub>1</sub> ≤ δ<sub>2</sub> ≤ δ<sub>c</sub>, δ<sub>c</sub> > 0, σ<sub>1</sub> ≥ 0, σ<sub>2</sub> ≥ 0, and σ<sub>1</sub>+σ<sub>2</sub> > 0. Note that these restriction prohibit infinite initial stiffness (''i.e.'', δ=0) but allow zero initial stiffness (''i.e.'', σ<sub>1</sub>=0). It is not clear is zero initial stiffness has any realistic applications.		The traction law for each mode depends up seven properties - (δ<sub>1</sub>,σ<sub>1</sub>) and (δ<sub>2</sub>,σ<sub>2</sub>) breakpoints, critical COD (δ<sub>c</sub>), total area or toughness (J<sub>c</sub>), and the initial elastic slope (k). You must enter exactly 5 of these seven properties for each mode. Furthermore, if one of the 5 is the initial slope, then δ<sub>c</sub> along with either σ<sub>1</sub> or δ<sub>1</sub> for that mode (but not both) must be among the specified properties. The two remaining unspecified parameters will be calculated from the five provided parameters. The final law must satisfy 0 < δ<sub>1</sub> ≤ δ<sub>2</sub> ≤ δ<sub>c</sub>, δ<sub>c</sub> > 0, σ<sub>1</sub> ≥ 0, σ<sub>2</sub> ≥ 0, and σ<sub>1</sub>+σ<sub>2</sub> > 0. Note that these restriction prohibit infinite initial stiffness (''i.e.'', δ=0) but allow zero initial stiffness (''i.e.'', σ<sub>1</sub>=0). It is not clear if zero initial stiffness has any realistic applications.

	One useful interpretation of a trilinear law is that it is modeling two physically mechanisms. The total toughness is the total area under the curve and is given above. The first failure mechanism can be identified with the area under the first peak and bounded by the dotted red line in the above figure. This partial area is the toughness for the first mechanism and is equal to		One useful interpretation of a trilinear law is that it is modeling two physically mechanisms. The total toughness is the total area under the curve and is given above. The first failure mechanism can be identified with the area under the first peak and bounded by the dotted red line in the above figure. This partial area is the toughness for the first mechanism and is equal to

Nairnj: /* The Traction Law */

2020-06-21T20:44:27Z

The Traction Law

← Older revision		Revision as of 20:44, 21 June 2020
Line 11:		Line 11:
	<math>J_c = {1\over2}\bigl(\sigma_1\delta_2 + \sigma_2(\delta_c-\delta_1)\bigr)</math>		<math>J_c = {1\over2}\bigl(\sigma_1\delta_2 + \sigma_2(\delta_c-\delta_1)\bigr)</math>

	The traction law for each mode depends up seven properties - (δ<sub>1</sub>,σ<sub>1</sub>) and (δ<sub>2</sub>,σ<sub>2</sub>) breakpoints, critical COD (δ<sub>c</sub>), total area or toughness (J<sub>c</sub>), and the initial elastic slope (k). You must enter exactly 5 of these seven properties for each mode. Furthermore, if one of the 5 is the initial slope, then δ<sub>c</sub> along with either σ<sub>1</sub> or δ<sub>1</sub> for that mode (but not both) must be among the specified properties. The two remaining unspecified parameters will be calculated from the five provided parameters. The final law must satisfy 0 < δ<sub>1</sub> ≤ δ<sub>2</sub> ≤ δ<sub>c</sub>, δ<sub>c</sub> > 0, σ<sub>1</sub> ≥ 0, σ<sub>2</sub> ≥ 0, and σ<sub>1</sub>+σ<sub>2</sub> > 0. Note ~~this~~ restriction prohibit infinite initial stiffness (''i.e.'', δ=0) but allow zero initial stiffness (''i.e.'', σ<sub>1</sub>=0). It is not clear is zero initial stiffness has any realistic applications.		The traction law for each mode depends up seven properties - (δ<sub>1</sub>,σ<sub>1</sub>) and (δ<sub>2</sub>,σ<sub>2</sub>) breakpoints, critical COD (δ<sub>c</sub>), total area or toughness (J<sub>c</sub>), and the initial elastic slope (k). You must enter exactly 5 of these seven properties for each mode. Furthermore, if one of the 5 is the initial slope, then δ<sub>c</sub> along with either σ<sub>1</sub> or δ<sub>1</sub> for that mode (but not both) must be among the specified properties. The two remaining unspecified parameters will be calculated from the five provided parameters. The final law must satisfy 0 < δ<sub>1</sub> ≤ δ<sub>2</sub> ≤ δ<sub>c</sub>, δ<sub>c</sub> > 0, σ<sub>1</sub> ≥ 0, σ<sub>2</sub> ≥ 0, and σ<sub>1</sub>+σ<sub>2</sub> > 0. Note that these restriction prohibit infinite initial stiffness (''i.e.'', δ=0) but allow zero initial stiffness (''i.e.'', σ<sub>1</sub>=0). It is not clear is zero initial stiffness has any realistic applications.

	One useful interpretation of a trilinear law is that it is modeling two physically mechanisms. The total toughness is the total area under the curve and is given above. The first failure mechanism can be identified with the area under the first peak and bounded by the dotted red line in the above figure. This partial area is the toughness for the first mechanism and is equal to		One useful interpretation of a trilinear law is that it is modeling two physically mechanisms. The total toughness is the total area under the curve and is given above. The first failure mechanism can be identified with the area under the first peak and bounded by the dotted red line in the above figure. This partial area is the toughness for the first mechanism and is equal to

Nairnj: /* The Traction Law */

2020-06-21T20:44:08Z

The Traction Law

← Older revision		Revision as of 20:44, 21 June 2020
Line 11:		Line 11:
	<math>J_c = {1\over2}\bigl(\sigma_1\delta_2 + \sigma_2(\delta_c-\delta_1)\bigr)</math>		<math>J_c = {1\over2}\bigl(\sigma_1\delta_2 + \sigma_2(\delta_c-\delta_1)\bigr)</math>

	The traction law for each mode depends up seven properties - (δ<sub>1</sub>,σ<sub>1</sub>) and (δ<sub>2</sub>,σ<sub>2</sub>) breakpoints, critical COD (δ<sub>c</sub>), total area or toughness (J<sub>c</sub>), and the initial elastic slope (k). You must enter exactly 5 of these seven properties for each mode. Furthermore, if one of the 5 is the initial slope, then δ<sub>c</sub> along with either σ<sub>1</sub> or δ<sub>1</sub> for that mode (but not both) must be among the specified properties. The two remaining unspecified parameters will be calculated from the five provided parameters. The final law must satisfy 0 δ<sub>1</sub> ≤ δ<sub>2</sub> ≤ δ<sub>c</sub>, δ<sub>c</sub> > 0, σ<sub>1</sub> ≥ 0, σ<sub>2</sub> ≥ 0, and σ<sub>1</sub>+σ<sub>2</sub> > 0.		The traction law for each mode depends up seven properties - (δ<sub>1</sub>,σ<sub>1</sub>) and (δ<sub>2</sub>,σ<sub>2</sub>) breakpoints, critical COD (δ<sub>c</sub>), total area or toughness (J<sub>c</sub>), and the initial elastic slope (k). You must enter exactly 5 of these seven properties for each mode. Furthermore, if one of the 5 is the initial slope, then δ<sub>c</sub> along with either σ<sub>1</sub> or δ<sub>1</sub> for that mode (but not both) must be among the specified properties. The two remaining unspecified parameters will be calculated from the five provided parameters. The final law must satisfy 0 < δ<sub>1</sub> ≤ δ<sub>2</sub> ≤ δ<sub>c</sub>, δ<sub>c</sub> > 0, σ<sub>1</sub> ≥ 0, σ<sub>2</sub> ≥ 0, and σ<sub>1</sub>+σ<sub>2</sub> > 0. Note this restriction prohibit infinite initial stiffness (''i.e.'', δ=0) but allow zero initial stiffness (''i.e.'', σ<sub>1</sub>=0). It is not clear is zero initial stiffness has any realistic applications.

	One useful interpretation of a trilinear law is that it is modeling two physically mechanisms. The total toughness is the total area under the curve and is given above. The first failure mechanism can be identified with the area under the first peak and bounded by the dotted red line in the above figure. This partial area is the toughness for the first mechanism and is equal to		One useful interpretation of a trilinear law is that it is modeling two physically mechanisms. The total toughness is the total area under the curve and is given above. The first failure mechanism can be identified with the area under the first peak and bounded by the dotted red line in the above figure. This partial area is the toughness for the first mechanism and is equal to

Nairnj: /* The Traction Law */

2020-06-21T19:23:05Z

The Traction Law

← Older revision		Revision as of 19:23, 21 June 2020
Line 11:		Line 11:
	<math>J_c = {1\over2}\bigl(\sigma_1\delta_2 + \sigma_2(\delta_c-\delta_1)\bigr)</math>		<math>J_c = {1\over2}\bigl(\sigma_1\delta_2 + \sigma_2(\delta_c-\delta_1)\bigr)</math>

	The traction law for each mode depends up seven properties - (δ<sub>1</sub>,σ<sub>1</sub>) and (δ<sub>2</sub>,σ<sub>2</sub>) breakpoints, critical COD (δ<sub>c</sub>), total area or toughness (J<sub>c</sub>), and the initial elastic slope (k). You must enter exactly 5 of these seven properties for each mode. Furthermore, if one of the 5 is the initial slope, then δ<sub>c</sub> along with either σ<sub>1</sub> or δ<sub>1</sub> for that mode (but not both) must be among the specified properties. The two remaining unspecified parameters will be calculated from the five provided parameters. The final law must satisfy δ<sub>1</sub> ≤ δ<sub>2</sub> ≤ δ<sub>c</sub>, δ<sub>c</sub> > 0, σ<sub>1</sub> ≥ 0, σ<sub>2</sub> ≥ 0, and σ<sub>1</sub>+σ<sub>2</sub> > 0.		The traction law for each mode depends up seven properties - (δ<sub>1</sub>,σ<sub>1</sub>) and (δ<sub>2</sub>,σ<sub>2</sub>) breakpoints, critical COD (δ<sub>c</sub>), total area or toughness (J<sub>c</sub>), and the initial elastic slope (k). You must enter exactly 5 of these seven properties for each mode. Furthermore, if one of the 5 is the initial slope, then δ<sub>c</sub> along with either σ<sub>1</sub> or δ<sub>1</sub> for that mode (but not both) must be among the specified properties. The two remaining unspecified parameters will be calculated from the five provided parameters. The final law must satisfy 0 δ<sub>1</sub> ≤ δ<sub>2</sub> ≤ δ<sub>c</sub>, δ<sub>c</sub> > 0, σ<sub>1</sub> ≥ 0, σ<sub>2</sub> ≥ 0, and σ<sub>1</sub>+σ<sub>2</sub> > 0.

	One useful interpretation of a trilinear law is that it is modeling two physically mechanisms. The total toughness is the total area under the curve and is given above. The first failure mechanism can be identified with the area under the first peak and bounded by the dotted red line in the above figure. This partial area is the toughness for the first mechanism and is equal to		One useful interpretation of a trilinear law is that it is modeling two physically mechanisms. The total toughness is the total area under the curve and is given above. The first failure mechanism can be identified with the area under the first peak and bounded by the dotted red line in the above figure. This partial area is the toughness for the first mechanism and is equal to