Imperfect Interfaces - Revision history

Nairnj: /* Limitations */

2026-03-29T18:21:20Z

Limitations

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	=== Limitations ===		=== Limitations ===

	Imperfect interface define traction as a function of interfacial displacement discontinuity. The tractions are applied on nodes where two materials or two crack surfaces are in contact. But, in MPM, if the surface are separated by one to two cells (depending on contact location with the cells), the materials (or crack surfaces) will not be detected as in contact and thus treated as if not connected by an interface. In other words, all imperfect interface traction laws will have an effective failure point for normal separation between one and two cells. This numerical separation criterion is mesh dependent. As long an normal component remains in contact, the tangential component can see separations of any magnitude, but even tangential traction will disappear when normal separation is too large (''i.e.'', when not in contact, no tractions are added for either direction).		Imperfect interface define traction as a function of interfacial displacement discontinuity. The tractions are applied on nodes where two materials or two crack surfaces are in contact. But, in MPM, if the surface are separated by one to two cells (depending on contact location with the cells), the materials (or crack surfaces) will not be detected as in contact and thus treated as if not connected by an interface. In other words, all imperfect interface traction laws will have an effective failure point for normal separation between one and two cells. This numerical separation criterion is mesh dependent. As long as normal component remains in contact, the tangential component can see separations of any magnitude, but even tangential traction will disappear when normal separation is too large (''i.e.'', when not in contact, no tractions are added for either direction).

	All imperfect interface laws are treated as elastic. In other words, the interfacestraction laws are reversible and interfacial energy is area under the interface traction law.		All imperfect interface laws are treated as elastic. In other words, the interfacestraction laws are reversible and interfacial energy is area under the interface traction law.

Nairnj: /* Limitations */

2026-03-29T18:20:24Z

Limitations

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	All imperfect interface laws are treated as elastic. In other words, the interfacestraction laws are reversible and interfacial energy is area under the interface traction law.		All imperfect interface laws are treated as elastic. In other words, the interfacestraction laws are reversible and interfacial energy is area under the interface traction law.

	An alternative for modeling without numerical separation is to model interfaces using [[Defining Cracks\|cracks with traction laws]]. This approach limits imperfect interface modeling ~~wot~~ simulations with [[MPM Input Files#Using Explicit Cracks\|xxplicit cracks]]. Unlike imperfect interfaces, however, they can ~~can~~ inelastic response and irreversible failure.		An alternative for modeling without numerical separation is to model interfaces using [[Defining Cracks\|cracks with traction laws]]. This approach limits imperfect interface modeling to simulations with [[MPM Input Files#Using Explicit Cracks\|xxplicit cracks]]. Unlike imperfect interfaces, however, they can have inelastic response and irreversible failure.

	== Imperfect Interfaces in FEA ==		== Imperfect Interfaces in FEA ==

Nairnj: /* References */

2026-03-29T17:11:13Z

References

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	<ref name="Nair97">J. A. Nairn and Y. C. Liu, "Stress transfer into a fragmented, anisotropic fiber through an imperfect interface," ''Int. J. Solids Structures'', '''34''', 1255–1281 (1997).</ref>		<ref name="Nair97">J. A. Nairn and Y. C. Liu, "Stress transfer into a fragmented, anisotropic fiber through an imperfect interface," ''Int. J. Solids Structures'', '''34''', 1255–1281 (1997).</ref>

	<ref name="Nairn04">J. A. Nairn, "Numerical implementation of imperfect interfaces," ''Computational Materials Science'', '''40''', 525–536 (2007).</ref>		<ref name="Nairn04">J. A. Nairn, "Numerical implementation of imperfect interfaces," ''Computational Materials Science'', '''40''', 525–536 (2007). ([https://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/Interface.pdf See PDF])</ref>

	<ref name="Nair13">J.A. Nairn, "Modeling Imperfect Interfaces in the Material Point Method using Multimaterial Methods," ''Computer Modeling in Eng. & Sci.'', '''92''', 271-299 (2013). ([https://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/MMInterfaces.pdf See PDF])</ref>		<ref name="Nair13">J.A. Nairn, "Modeling Imperfect Interfaces in the Material Point Method using Multimaterial Methods," ''Computer Modeling in Eng. & Sci.'', '''92''', 271-299 (2013). ([https://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/MMInterfaces.pdf See PDF])</ref>

Nairnj at 17:10, 29 March 2026

2026-03-29T17:10:35Z

← Older revision		Revision as of 17:10, 29 March 2026
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	Modeling interfaces, which are often finite-thickness interphases, in composite materials is difficult. In numerical modeling, it would seemingly be straight-forward to discretize interphase zones and thereby explicitly model all effects. This approach has two problems. First, interphase zones may be much smaller than the bulk materials. Resolving both bulk materials and a thin interphase would require a highly refined model, which may exceed computational capacity. Second, interphase properties may be unknown and/or may vary within a transition zone from one material to another. Alternative methods for interphase modeling are needed to overcome these challenges. This need is especially important in nanocomposites because the amount of interphase per unit volume of reinforcement greatly exceeds the amount of interphase in composites with micron or larger reinforcement phases. As a consequence, interphases are expected to play a larger role, good or bad, in nanocomposite properties.		Modeling interfaces, which are often finite-thickness interphases, in composite materials is difficult. In numerical modeling, it would seemingly be straight-forward to discretize interphase zones and thereby explicitly model all effects. This approach has two problems. First, interphase zones may be much smaller than the bulk materials. Resolving both bulk materials and a thin interphase would require a highly refined model, which may exceed computational capacity. Second, interphase properties may be unknown and/or may vary within a transition zone from one material to another. Alternative methods for interphase modeling are needed to overcome these challenges. This need is especially important in nanocomposites because the amount of interphase per unit volume of reinforcement greatly exceeds the amount of interphase in composites with micron or larger reinforcement phases. As a consequence, interphases are expected to play a larger role, good or bad, in nanocomposite properties.

	One way to model interphases is to abandon attempts for explicit modeling and instead replace 3D interphases with 2D interfaces<ref name="Hash90"/> The interphase effects are reduced to modeling the response of 2D interfaces due to tractions normal and tangential to the interfacial surface, which can be modeled by interface traction laws. Elimination of 3D interphases removes the resolution problem. The use of interface traction laws also replaces numerous unknown and potentially unmeasurable interphase properties with a much smaller number of interface parameters. If interface traction laws can be determined, one can potentially model interphase effects well. Examples of using this for analytical modeling of interface effects in composite materials are in Hashin (1991)<ref name="Hash91a"/><ref name="Hash91b"/ and Nairn and Liu (1997).<ref name="Nair97"~~>J. A. Nairn and Y. C. Liu, "Stress transfer into a fragmented, anisotropic fiber through an imperfect interface," ''Int. J. Solids Structures'', '''34''', 1255–1281 (1997).<~~/~~ref~~> Implementation of imperfect interfaces in FEA and in MPM is described in Nairn (2007).<ref name="Nairn04"~~>J. A. Nairn, "Numerical implementation of imperfect interfaces," ''Computational Materials Science'', '''40''', 525–536 (2007).<~~/~~ref~~> An alternative method for modeling imperfect interfaces in MPM using [[Multimaterial MPM\|multimaterial MPM]] is described in Nairn (2013).<ref name="Nair13">		One way to model interphases is to abandon attempts for explicit modeling and instead replace 3D interphases with 2D interfaces<ref name="Hash90"/> The interphase effects are reduced to modeling the response of 2D interfaces due to tractions normal and tangential to the interfacial surface, which can be modeled by interface traction laws. Elimination of 3D interphases removes the resolution problem. The use of interface traction laws also replaces numerous unknown and potentially unmeasurable interphase properties with a much smaller number of interface parameters. If interface traction laws can be determined, one can potentially model interphase effects well. Examples of using this for analytical modeling of interface effects in composite materials are in Hashin (1991)<ref name="Hash91a"/><ref name="Hash91b"/> and Nairn and Liu (1997).<ref name="Nair97"/> Implementation of imperfect interfaces in FEA and in MPM is described in Nairn (2007).<ref name="Nairn04"/> An alternative method for modeling imperfect interfaces in MPM using [[Multimaterial MPM\|multimaterial MPM]] is described in Nairn (2013).<ref name="Nair13"/></ref>
	~~J.A. Nairn, "Modeling Imperfect Interfaces in the Material Point Method using Multimaterial Methods," ''Computer Modeling in Eng. & Sci.'', '''92''', 271-299 (2013).~~</ref>

	=== Imperfect Interface Traction Laws ===		=== Imperfect Interface Traction Laws ===
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	<math>T_n = f_n([u_n],[u_t]) \qquad {\rm and } \qquad T_t = f_t([u_n],[u_t])</math>		<math>T_n = f_n([u_n],[u_t]) \qquad {\rm and } \qquad T_t = f_t([u_n],[u_t])</math>

	Interfaces in composite materials may develop potential energy that is needed for effective property analysis.<ref name="Hash92">Z. Hashin, "Extremum principles for elastic heterogenous media with imperfect interfaces and their application to bounding of effective moduli." ''Journal of the Mechanics and Physics of Solids'', '''40''', 767–781 (1992).</ref> For an elastic interface, interfacial potential energy is:		Interfaces in composite materials may develop potential energy that is needed for effective property analysis.<ref name="Hash92"/>Z For an elastic interface, interfacial potential energy is:


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	== References ==		== References ==

	<references>		<references>

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	<ref name="Hash91a">Z. Hashin, "Composite materials with viscoelastic interphase: Creep and relaxation," ''Mech. of Materials'', '''11''', 135–148 (1991).</ref><ref name="Hash91b">Z. Hashin, "Thermoelastic properties of particulate composites with imperfect interface," ''Journalof the Mechanics and Physics of Solids'', '''39''', 745–762 (1991).</ref>		<ref name="Hash91a">Z. Hashin, "Composite materials with viscoelastic interphase: Creep and relaxation," ''Mech. of Materials'', '''11''', 135–148 (1991).</ref><ref name="Hash91b">Z. Hashin, "Thermoelastic properties of particulate composites with imperfect interface," ''Journalof the Mechanics and Physics of Solids'', '''39''', 745–762 (1991).</ref>

	<ref name="Nairn">J.A. Nairn, "Modeling Imperfect Interfaces in the Material Point Method using Multimaterial Methods," ''Computer Modeling in Eng. & Sci.'', '''92''', 271-299 (2013). ([https://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/MMInterfaces.pdf See PDF])</ref>		<ref name="Nair97">J. A. Nairn and Y. C. Liu, "Stress transfer into a fragmented, anisotropic fiber through an imperfect interface," ''Int. J. Solids Structures'', '''34''', 1255–1281 (1997).</ref>

			<ref name="Nairn04">J. A. Nairn, "Numerical implementation of imperfect interfaces," ''Computational Materials Science'', '''40''', 525–536 (2007).</ref>

			<ref name="Nair13">J.A. Nairn, "Modeling Imperfect Interfaces in the Material Point Method using Multimaterial Methods," ''Computer Modeling in Eng. & Sci.'', '''92''', 271-299 (2013). ([https://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/MMInterfaces.pdf See PDF])</ref>

			<ref name="Hash92">Z. Hashin, "Extremum principles for elastic heterogenous media with imperfect interfaces and their application to bounding of effective moduli." ''Journal of the Mechanics and Physics of Solids'', '''40''', 767–781 (1992).</ref>

	</references>		</references>

Nairnj: /* References */

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References

← Older revision		Revision as of 17:07, 29 March 2026
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	<ref name="Hash90">Z. Hashin, "Thermoelastic properties of fiber composites with imperfect interface," ''Mech. of Materials'', '''8''', 333–348 (1990).</ref>		<ref name="Hash90">Z. Hashin, "Thermoelastic properties of fiber composites with imperfect interface," ''Mech. of Materials'', '''8''', 333–348 (1990).</ref>

			<ref name="Hash91a">Z. Hashin, "Composite materials with viscoelastic interphase: Creep and relaxation," ''Mech. of Materials'', '''11''', 135–148 (1991).</ref><ref name="Hash91b">Z. Hashin, "Thermoelastic properties of particulate composites with imperfect interface," ''Journalof the Mechanics and Physics of Solids'', '''39''', 745–762 (1991).</ref>

	<ref name="Nairn">J.A. Nairn, "Modeling Imperfect Interfaces in the Material Point Method using Multimaterial Methods," ''Computer Modeling in Eng. & Sci.'', '''92''', 271-299 (2013). ([https://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/MMInterfaces.pdf See PDF])</ref>		<ref name="Nairn">J.A. Nairn, "Modeling Imperfect Interfaces in the Material Point Method using Multimaterial Methods," ''Computer Modeling in Eng. & Sci.'', '''92''', 271-299 (2013). ([https://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/MMInterfaces.pdf See PDF])</ref>

	</references>		</references>

Nairnj: /* Imperfect Interface Theory */

2026-03-29T17:07:00Z

Imperfect Interface Theory

← Older revision		Revision as of 17:07, 29 March 2026
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	Modeling interfaces, which are often finite-thickness interphases, in composite materials is difficult. In numerical modeling, it would seemingly be straight-forward to discretize interphase zones and thereby explicitly model all effects. This approach has two problems. First, interphase zones may be much smaller than the bulk materials. Resolving both bulk materials and a thin interphase would require a highly refined model, which may exceed computational capacity. Second, interphase properties may be unknown and/or may vary within a transition zone from one material to another. Alternative methods for interphase modeling are needed to overcome these challenges. This need is especially important in nanocomposites because the amount of interphase per unit volume of reinforcement greatly exceeds the amount of interphase in composites with micron or larger reinforcement phases. As a consequence, interphases are expected to play a larger role, good or bad, in nanocomposite properties.		Modeling interfaces, which are often finite-thickness interphases, in composite materials is difficult. In numerical modeling, it would seemingly be straight-forward to discretize interphase zones and thereby explicitly model all effects. This approach has two problems. First, interphase zones may be much smaller than the bulk materials. Resolving both bulk materials and a thin interphase would require a highly refined model, which may exceed computational capacity. Second, interphase properties may be unknown and/or may vary within a transition zone from one material to another. Alternative methods for interphase modeling are needed to overcome these challenges. This need is especially important in nanocomposites because the amount of interphase per unit volume of reinforcement greatly exceeds the amount of interphase in composites with micron or larger reinforcement phases. As a consequence, interphases are expected to play a larger role, good or bad, in nanocomposite properties.

	One way to model interphases is to abandon attempts for explicit modeling and instead replace 3D interphases with 2D interfaces<ref name="Hash90"/> The interphase effects are reduced to modeling the response of 2D interfaces due to tractions normal and tangential to the interfacial surface, which can be modeled by interface traction laws. Elimination of 3D interphases removes the resolution problem. The use of interface traction laws also replaces numerous unknown and potentially unmeasurable interphase properties with a much smaller number of interface parameters. If interface traction laws can be determined, one can potentially model interphase effects well. Examples of using this for analytical modeling of interface effects in composite materials are in Hashin (1991)<ref name="Hash91a"~~>Z. Hashin, "Composite materials with viscoelastic interphase: Creep and relaxation," ''Mech. of Materials'', '''11''', 135–148 (1991).<~~/~~ref~~><ref name="Hash91b"~~>Z. Hashin, "Thermoelastic properties of particulate composites with imperfect interface," ''Journalof the Mechanics and Physics of Solids'', '''39''', 745–762 (1991).<~~/~~ref>~~ and Nairn and Liu (1997).<ref name="Nair97">J. A. Nairn and Y. C. Liu, "Stress transfer into a fragmented, anisotropic fiber through an imperfect interface," ''Int. J. Solids Structures'', '''34''', 1255–1281 (1997).</ref> Implementation of imperfect interfaces in FEA and in MPM is described in Nairn (2007).<ref name="Nairn04">J. A. Nairn, "Numerical implementation of imperfect interfaces," ''Computational Materials Science'', '''40''', 525–536 (2007).</ref> An alternative method for modeling imperfect interfaces in MPM using [[Multimaterial MPM\|multimaterial MPM]] is described in Nairn (2013).<ref name="Nair13">		One way to model interphases is to abandon attempts for explicit modeling and instead replace 3D interphases with 2D interfaces<ref name="Hash90"/> The interphase effects are reduced to modeling the response of 2D interfaces due to tractions normal and tangential to the interfacial surface, which can be modeled by interface traction laws. Elimination of 3D interphases removes the resolution problem. The use of interface traction laws also replaces numerous unknown and potentially unmeasurable interphase properties with a much smaller number of interface parameters. If interface traction laws can be determined, one can potentially model interphase effects well. Examples of using this for analytical modeling of interface effects in composite materials are in Hashin (1991)<ref name="Hash91a"/><ref name="Hash91b"/ and Nairn and Liu (1997).<ref name="Nair97">J. A. Nairn and Y. C. Liu, "Stress transfer into a fragmented, anisotropic fiber through an imperfect interface," ''Int. J. Solids Structures'', '''34''', 1255–1281 (1997).</ref> Implementation of imperfect interfaces in FEA and in MPM is described in Nairn (2007).<ref name="Nairn04">J. A. Nairn, "Numerical implementation of imperfect interfaces," ''Computational Materials Science'', '''40''', 525–536 (2007).</ref> An alternative method for modeling imperfect interfaces in MPM using [[Multimaterial MPM\|multimaterial MPM]] is described in Nairn (2013).<ref name="Nair13">
	J.A. Nairn, "Modeling Imperfect Interfaces in the Material Point Method using Multimaterial Methods," ''Computer Modeling in Eng. & Sci.'', '''92''', 271-299 (2013).</ref>		J.A. Nairn, "Modeling Imperfect Interfaces in the Material Point Method using Multimaterial Methods," ''Computer Modeling in Eng. & Sci.'', '''92''', 271-299 (2013).</ref>

Nairnj: /* References */

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References

← Older revision		Revision as of 17:06, 29 March 2026
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	== References ==		== References ==
	<references>		<references>

			<ref name="Hash90">Z. Hashin, "Thermoelastic properties of fiber composites with imperfect interface," ''Mech. of Materials'', '''8''', 333–348 (1990).</ref>

	<ref name="Nairn">J.A. Nairn, "Modeling Imperfect Interfaces in the Material Point Method using Multimaterial Methods," ''Computer Modeling in Eng. & Sci.'', '''92''', 271-299 (2013). ([https://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/MMInterfaces.pdf See PDF])</ref>		<ref name="Nairn">J.A. Nairn, "Modeling Imperfect Interfaces in the Material Point Method using Multimaterial Methods," ''Computer Modeling in Eng. & Sci.'', '''92''', 271-299 (2013). ([https://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/MMInterfaces.pdf See PDF])</ref>

	</references>		</references>

Nairnj: /* Imperfect Interface Theory */

2026-03-29T17:06:03Z

Imperfect Interface Theory

← Older revision		Revision as of 17:06, 29 March 2026
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	Modeling interfaces, which are often finite-thickness interphases, in composite materials is difficult. In numerical modeling, it would seemingly be straight-forward to discretize interphase zones and thereby explicitly model all effects. This approach has two problems. First, interphase zones may be much smaller than the bulk materials. Resolving both bulk materials and a thin interphase would require a highly refined model, which may exceed computational capacity. Second, interphase properties may be unknown and/or may vary within a transition zone from one material to another. Alternative methods for interphase modeling are needed to overcome these challenges. This need is especially important in nanocomposites because the amount of interphase per unit volume of reinforcement greatly exceeds the amount of interphase in composites with micron or larger reinforcement phases. As a consequence, interphases are expected to play a larger role, good or bad, in nanocomposite properties.		Modeling interfaces, which are often finite-thickness interphases, in composite materials is difficult. In numerical modeling, it would seemingly be straight-forward to discretize interphase zones and thereby explicitly model all effects. This approach has two problems. First, interphase zones may be much smaller than the bulk materials. Resolving both bulk materials and a thin interphase would require a highly refined model, which may exceed computational capacity. Second, interphase properties may be unknown and/or may vary within a transition zone from one material to another. Alternative methods for interphase modeling are needed to overcome these challenges. This need is especially important in nanocomposites because the amount of interphase per unit volume of reinforcement greatly exceeds the amount of interphase in composites with micron or larger reinforcement phases. As a consequence, interphases are expected to play a larger role, good or bad, in nanocomposite properties.

	One way to model interphases is to abandon attempts for explicit modeling and instead replace 3D interphases with 2D interfaces<ref name="Hash90"~~>Z. Hashin, "Thermoelastic properties of fiber composites with imperfect interface," ''Mech. of Materials'', '''8''', 333–348 (1990).<~~/~~ref~~> The interphase effects are reduced to modeling the response of 2D interfaces due to tractions normal and tangential to the interfacial surface, which can be modeled by interface traction laws. Elimination of 3D interphases removes the resolution problem. The use of interface traction laws also replaces numerous unknown and potentially unmeasurable interphase properties with a much smaller number of interface parameters. If interface traction laws can be determined, one can potentially model interphase effects well. Examples of using this for analytical modeling of interface effects in composite materials are in Hashin (1991)<ref name="Hash91a">Z. Hashin, "Composite materials with viscoelastic interphase: Creep and relaxation," ''Mech. of Materials'', '''11''', 135–148 (1991).</ref><ref name="Hash91b">Z. Hashin, "Thermoelastic properties of particulate composites with imperfect interface," ''Journalof the Mechanics and Physics of Solids'', '''39''', 745–762 (1991).</ref> and Nairn and Liu (1997).<ref name="Nair97">J. A. Nairn and Y. C. Liu, "Stress transfer into a fragmented, anisotropic fiber through an imperfect interface," ''Int. J. Solids Structures'', '''34''', 1255–1281 (1997).</ref> Implementation of imperfect interfaces in FEA and in MPM is described in Nairn (2007).<ref name="Nairn04">J. A. Nairn, "Numerical implementation of imperfect interfaces," ''Computational Materials Science'', '''40''', 525–536 (2007).</ref> An alternative method for modeling imperfect interfaces in MPM using [[Multimaterial MPM\|multimaterial MPM]] is described in Nairn (2013).<ref name="Nair13">		One way to model interphases is to abandon attempts for explicit modeling and instead replace 3D interphases with 2D interfaces<ref name="Hash90"/> The interphase effects are reduced to modeling the response of 2D interfaces due to tractions normal and tangential to the interfacial surface, which can be modeled by interface traction laws. Elimination of 3D interphases removes the resolution problem. The use of interface traction laws also replaces numerous unknown and potentially unmeasurable interphase properties with a much smaller number of interface parameters. If interface traction laws can be determined, one can potentially model interphase effects well. Examples of using this for analytical modeling of interface effects in composite materials are in Hashin (1991)<ref name="Hash91a">Z. Hashin, "Composite materials with viscoelastic interphase: Creep and relaxation," ''Mech. of Materials'', '''11''', 135–148 (1991).</ref><ref name="Hash91b">Z. Hashin, "Thermoelastic properties of particulate composites with imperfect interface," ''Journalof the Mechanics and Physics of Solids'', '''39''', 745–762 (1991).</ref> and Nairn and Liu (1997).<ref name="Nair97">J. A. Nairn and Y. C. Liu, "Stress transfer into a fragmented, anisotropic fiber through an imperfect interface," ''Int. J. Solids Structures'', '''34''', 1255–1281 (1997).</ref> Implementation of imperfect interfaces in FEA and in MPM is described in Nairn (2007).<ref name="Nairn04">J. A. Nairn, "Numerical implementation of imperfect interfaces," ''Computational Materials Science'', '''40''', 525–536 (2007).</ref> An alternative method for modeling imperfect interfaces in MPM using [[Multimaterial MPM\|multimaterial MPM]] is described in Nairn (2013).<ref name="Nair13">
	J.A. Nairn, "Modeling Imperfect Interfaces in the Material Point Method using Multimaterial Methods," ''Computer Modeling in Eng. & Sci.'', '''92''', 271-299 (2013).</ref>		J.A. Nairn, "Modeling Imperfect Interfaces in the Material Point Method using Multimaterial Methods," ''Computer Modeling in Eng. & Sci.'', '''92''', 271-299 (2013).</ref>

Nairnj: /* References */

2026-03-29T17:05:12Z

References

← Older revision		Revision as of 17:05, 29 March 2026
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	<references>		<references>

	<ref name="">J.A. Nairn, "Modeling Imperfect Interfaces in the Material Point Method using Multimaterial Methods," ''Computer Modeling in Eng. & Sci.'', '''92''', 271-299 (2013). ([https://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/MMInterfaces.pdf See PDF])</ref>		<ref name="Nairn">J.A. Nairn, "Modeling Imperfect Interfaces in the Material Point Method using Multimaterial Methods," ''Computer Modeling in Eng. & Sci.'', '''92''', 271-299 (2013). ([https://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/MMInterfaces.pdf See PDF])</ref>

	<references/>		</references>

Nairnj: /* References */

2026-03-29T17:04:04Z

References

← Older revision		Revision as of 17:04, 29 March 2026
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	== References ==		== References ==
			<references>

			<ref name="">J.A. Nairn, "Modeling Imperfect Interfaces in the Material Point Method using Multimaterial Methods," ''Computer Modeling in Eng. & Sci.'', '''92''', 271-299 (2013). ([https://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/MMInterfaces.pdf See PDF])</ref>

	<references/>		<references/>