Isotropic Phase Field Softening Material - Revision history

Nairnj: /* Constitutive Law */

2026-03-29T17:51:11Z

Constitutive Law

← Older revision		Revision as of 17:51, 29 March 2026
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	== Constitutive Law ==		== Constitutive Law ==

	This [[Material Models#Softening Materials\|MPM softening material]] models fracture using a phase field. In phase field fracture modeling, sharp crack surfaces are replaced by diffuse phase fields. The growth of these diffuse cracks is controlled by energy flow through the material and the material's fracture toughness. This material implements phase field fracture methods using methods described in Miehe <ref name="Miehe"/> and extends it in a few areas. MPM methods ~~used~~ are covered in Nairn.<ref name="transport"/>		This [[Material Models#Softening Materials\|MPM softening material]] models fracture using a phase field. In phase field fracture modeling, sharp crack surfaces are replaced by diffuse phase fields. The growth of these diffuse cracks is controlled by energy flow through the material and the material's fracture toughness. This material implements phase field fracture methods using methods described in Miehe <ref name="Miehe"/> and extends it in a few areas. MPM methods are covered in Nairn.<ref name="transport"/>

	== Phase Field Methods ==		== Phase Field Methods ==

Nairnj at 17:50, 29 March 2026

2026-03-29T17:50:47Z

← Older revision		Revision as of 17:50, 29 March 2026
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	where <math>G_c</math> is toughness, <math>\ell</math> is a length scale (describing width of diffuse cracks), and <math>\mathcal{H}</math> is history variable given by <math>\max(\Psi^{(+)})</math>. This equation is a diffusion equation that describes evolution of damage driven by damaging energy. The first two terms lead to diffusive cracks that decay exponentially from complete damage state with exponential decay distance determined by <math>\ell</math>. The last term is a source term to causes crack propagation. In other words, crack propagation is affected by the methods used to partition energy into <math>\Psi^{(+)}</math> and <math>\Psi^{(-)}</math>.		where <math>G_c</math> is toughness, <math>\ell</math> is a length scale (describing width of diffuse cracks), and <math>\mathcal{H}</math> is history variable given by <math>\max(\Psi^{(+)})</math>. This equation is a diffusion equation that describes evolution of damage driven by damaging energy. The first two terms lead to diffusive cracks that decay exponentially from complete damage state with exponential decay distance determined by <math>\ell</math>. The last term is a source term to causes crack propagation. In other words, crack propagation is affected by the methods used to partition energy into <math>\Psi^{(+)}</math> and <math>\Psi^{(-)}</math>.

	Simulations using viscous regularization must couple solution of phase field diffusion equation to the material's mechanics calculations. For that to work, all simulations using this material must include a [[Additional Transport Calculations\|Diffusion]] command and choose <tt>(style)</tt> of "fracture". If needed, you can set [[Setting Velocity and Transport Values#Other Transport Conditions\|grid based phase value boundary conditions]] using the "fracture" option to define damage conditions. Although you can set [[Setting Forces and Fluxes#Transport Flux Conditions\|damage flux on particle surfaces]], such conditions may have few applications in real-world modeling. Note that MPM solutions of transport equations can develop oscillations that prevent output of viable phase field descriptions of cracks. This problem can be solved by using FMPM(k) for transport tasks with the [[PeriodicXPIC Custom Task]]<ref name="~~MPMTransport~~"/>		Simulations using viscous regularization must couple solution of phase field diffusion equation to the material's mechanics calculations. For that to work, all simulations using this material must include a [[Additional Transport Calculations\|Diffusion]] command and choose <tt>(style)</tt> of "fracture". If needed, you can set [[Setting Velocity and Transport Values#Other Transport Conditions\|grid based phase value boundary conditions]] using the "fracture" option to define damage conditions. Although you can set [[Setting Forces and Fluxes#Transport Flux Conditions\|damage flux on particle surfaces]], such conditions may have few applications in real-world modeling. Note that MPM solutions of transport equations can develop oscillations that prevent output of viable phase field descriptions of cracks. This problem can be solved by using FMPM(k) for transport tasks with the [[PeriodicXPIC Custom Task]]<ref name="transport"/>

	=== Energy Partitioning ===		=== Energy Partitioning ===
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	<math>0 = G_c \ell \nabla^2\phi - \frac{G_c}{\ell}\phi - g'(\phi)\mathcal{H}</math>		<math>0 = G_c \ell \nabla^2\phi - \frac{G_c}{\ell}\phi - g'(\phi)\mathcal{H}</math>

	The diffusion equation approaches this solution in steady state (<i>i.e.</i> when <math>d\phi/dt\to 0</math>) or in the low viscosity limit (<i>i.e.</i> when <math>\eta\to 0</math>). The problem with the time-independent equation is that it does not match well with dynamic MPM modeling. It is possible (and some MPM papers do it<ref name="Kako"/>) to solve this equation on the background grid using standard linear finite element methods. This approach has two problems. First, one can only use linear finite element analysis when <math>g'(\phi)</math> is linear, which implies it can only be used for quadratic softening laws (this limitation might be another reason that most of the literature uses quadratic softening laws). Second, this approach in MPM is much slower (more than 30X slower<ref name="~~MPMTransport~~"/>).		The diffusion equation approaches this solution in steady state (<i>i.e.</i> when <math>d\phi/dt\to 0</math>) or in the low viscosity limit (<i>i.e.</i> when <math>\eta\to 0</math>). The problem with the time-independent equation is that it does not match well with dynamic MPM modeling. It is possible (and some MPM papers do it<ref name="Kako"/>) to solve this equation on the background grid using standard linear finite element methods. This approach has two problems. First, one can only use linear finite element analysis when <math>g'(\phi)</math> is linear, which implies it can only be used for quadratic softening laws (this limitation might be another reason that most of the literature uses quadratic softening laws). Second, this approach in MPM is much slower (more than 30X slower<ref name="transport"/>).

	Although [[OSParticulas]] has a option to solve the time-independent equation, that option is not provided here. Instead, one must use viscous regularization and that approach adds a burden of confirming choice of viscosity has not affected the results. In well behaved problems, results will be independent of viscosity provided viscosity is sufficiently small. If viscosity gets too high, it inhibits crack diffusion, slows down crack propagation, and gives non-physical results. A good rule of thumb is to pick viscosity such that it does not cause the time step needed for diffusion modeling to be smaller than time step needed to solve mechanics equations. A good starting point is to pick:		Although [[OSParticulas]] has a option to solve the time-independent equation, that option is not provided here. Instead, one must use viscous regularization and that approach adds a burden of confirming choice of viscosity has not affected the results. In well behaved problems, results will be independent of viscosity provided viscosity is sufficiently small. If viscosity gets too high, it inhibits crack diffusion, slows down crack propagation, and gives non-physical results. A good rule of thumb is to pick viscosity such that it does not cause the time step needed for diffusion modeling to be smaller than time step needed to solve mechanics equations. A good starting point is to pick:
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	<math>\eta \sim \frac{2G_c\ell}{v_{max}\Delta x}</math>		<math>\eta \sim \frac{2G_c\ell}{v_{max}\Delta x}</math>

	where <math>v_{max}</math> is wave speed of the elastic material and <math>\Delta x</math> cell size in the background grid. Simulations with different viscosites should also be run to determine if this value is small enough or if it needs to be smaller. Reference <ref name="~~MPMTransport~~"/> gives more details on choosing viscosity.		where <math>v_{max}</math> is wave speed of the elastic material and <math>\Delta x</math> cell size in the background grid. Simulations with different viscosites should also be run to determine if this value is small enough or if it needs to be smaller. Reference <ref name="transport"/> gives more details on choosing viscosity.

	== Material Properties ==		== Material Properties ==

Nairnj: /* References */

2026-03-29T17:50:06Z

References

← Older revision		Revision as of 17:50, 29 March 2026
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	== References ==		== References ==

	<references>		<references>

	<ref name="Miehe">C. Miehe, M. Hofacker, and F. Welschinger, "A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits," <i>Computer Methods in Applied Mechanics and Engineering</i>, <b>199</b>, 2765–2778 (2010).</ref>		<ref name="Miehe">C. Miehe, M. Hofacker, and F. Welschinger, "A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits," <i>Computer Methods in Applied Mechanics and Engineering</i>, <b>199</b>, 2765–2778 (2010).</ref>

	<ref name="PDSplit">H. Amor, J.-J. Marigo, and C. Maurini, "Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments," <i>Journal of the Mechanics and Physics of Solids</i>, <b>57(8)</b>, 1209-1229 (2009).</ref>		<ref name="PDSplit">H. Amor, J.-J. Marigo, and C. Maurini, "Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments," <i>Journal of the Mechanics and Physics of Solids</i>, <b>57(8)</b>, 1209-1229 (2009).</ref>
	<ref name="~~MPMTransport~~">J. A. Nairn. "Coupling Transport Equations to Mechanics in the Material Point Method Using an Approximate Full Capacity Matrix Inverse," ~~<i>~~Computer Methods in Applied Mechanics and Engineering~~</i>~~, ~~submitted~~ (~~2023~~)</ref>.
			<ref name="transport">J. A. Nairn, "Coupling Transport Equations to Mechanics in the Material Point Method Using an Approximate Full Capacity Matrix Inverse," ''Computer Methods in Applied Mechanics and Engineering'', '''420''', 116757 (2024). ([https://www.cof.orst.edu/cof/wse/faculty/Nairn/papers/TransportPaper.pdf See PDF])</ref>.

	<ref name="Kako">E. G. Kakouris and S. P. Triantafyllou, "Phase-field material point method for brittle fracture," <i>Int J Numer Meth Engng.</i>, <b>112</b>, 1750-1776(2017).</ref>		<ref name="Kako">E. G. Kakouris and S. P. Triantafyllou, "Phase-field material point method for brittle fracture," <i>Int J Numer Meth Engng.</i>, <b>112</b>, 1750-1776(2017).</ref>

	</references>		</references>

Nairnj: /* Constitutive Law */

2026-03-29T17:48:17Z

Constitutive Law

← Older revision		Revision as of 17:48, 29 March 2026
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	== Constitutive Law ==		== Constitutive Law ==

	This [[Material Models#Softening Materials\|MPM softening material]] models fracture using a phase field. In phase field fracture modeling, sharp crack surfaces are replaced by diffuse phase fields. The growth of these diffuse cracks is controlled by energy flow through the material and the material's fracture toughness. This material implements phase field fracture methods using methods described in Miehe <ref name="Miehe"/> and extends it in a few areas.		This [[Material Models#Softening Materials\|MPM softening material]] models fracture using a phase field. In phase field fracture modeling, sharp crack surfaces are replaced by diffuse phase fields. The growth of these diffuse cracks is controlled by energy flow through the material and the material's fracture toughness. This material implements phase field fracture methods using methods described in Miehe <ref name="Miehe"/> and extends it in a few areas. MPM methods used are covered in Nairn.<ref name="transport"/>

	== Phase Field Methods ==		== Phase Field Methods ==

Nairnj: /* Constitutive Law */

2023-11-18T17:49:52Z

Constitutive Law

← Older revision		Revision as of 17:49, 18 November 2023
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	== Constitutive Law ==		== Constitutive Law ==

	In phase field fracture modeling, sharp crack surfaces are replaced by diffuse phase fields. The growth of these diffuse cracks is controlled by energy flow through the material and the material's fracture toughness. This material implements phase field fracture methods using methods described in Miehe <ref name="Miehe"/> and extends it in a few areas.		This [[Material Models#Softening Materials\|MPM softening material]] models fracture using a phase field. In phase field fracture modeling, sharp crack surfaces are replaced by diffuse phase fields. The growth of these diffuse cracks is controlled by energy flow through the material and the material's fracture toughness. This material implements phase field fracture methods using methods described in Miehe <ref name="Miehe"/> and extends it in a few areas.

	== Phase Field Methods ==		== Phase Field Methods ==

Nairnj: /* Material Properties */

2023-11-17T17:34:32Z

Material Properties

← Older revision		Revision as of 17:34, 17 November 2023
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	The results in Miehe <ref name="Miehe"/> correspond to partition = 0, gd = 0, and garg = 1. These choices give poor results in some problems, especial when shear is important. Other choices are encouraged when these basic ones look questionable.		The results in Miehe <ref name="Miehe"/> correspond to partition = 0, gd = 0, and garg = 1. These choices give poor results in some problems, especial when shear is important. Other choices are encouraged when these basic ones look questionable.

	If needed, you can set [[Setting Forces and Fluxes#Initial Particle Phase Field\|set initial phase field value]] for particles using [[Particle Based Boundary Conditions]]		If needed, you can set [[Setting Forces and Fluxes#Initial Particle Phase Field\|set initial phase field value]] for particles using [[Particle-Based Boundary Conditions\|particle boundary conditions]].

	== History Variables ==		== History Variables ==

Nairnj: /* Material Properties */

2023-11-17T17:33:18Z

Material Properties

← Older revision		Revision as of 17:33, 17 November 2023
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	The results in Miehe <ref name="Miehe"/> correspond to partition = 0, gd = 0, and garg = 1. These choices give poor results in some problems, especial when shear is important. Other choices are encouraged when these basic ones look questionable.		The results in Miehe <ref name="Miehe"/> correspond to partition = 0, gd = 0, and garg = 1. These choices give poor results in some problems, especial when shear is important. Other choices are encouraged when these basic ones look questionable.

	If needed, you can set [[Setting Forces and Fluxes#Initial Particle Phase Field\|set initial phase field value]] for particles.		If needed, you can set [[Setting Forces and Fluxes#Initial Particle Phase Field\|set initial phase field value]] for particles using [[Particle Based Boundary Conditions]]

	== History Variables ==		== History Variables ==

Nairnj: /* Material Properties */

2023-11-17T17:32:32Z

Material Properties

← Older revision		Revision as of 17:32, 17 November 2023
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	The results in Miehe <ref name="Miehe"/> correspond to partition = 0, gd = 0, and garg = 1. These choices give poor results in some problems, especial when shear is important. Other choices are encouraged when these basic ones look questionable.		The results in Miehe <ref name="Miehe"/> correspond to partition = 0, gd = 0, and garg = 1. These choices give poor results in some problems, especial when shear is important. Other choices are encouraged when these basic ones look questionable.

			If needed, you can set [[Setting Forces and Fluxes#Initial Particle Phase Field\|set initial phase field value]] for particles.

	== History Variables ==		== History Variables ==

Nairnj: /* Phase Field Methods */

2023-11-17T17:31:43Z

Phase Field Methods

← Older revision		Revision as of 17:31, 17 November 2023
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	where <math>G_c</math> is toughness, <math>\ell</math> is a length scale (describing width of diffuse cracks), and <math>\mathcal{H}</math> is history variable given by <math>\max(\Psi^{(+)})</math>. This equation is a diffusion equation that describes evolution of damage driven by damaging energy. The first two terms lead to diffusive cracks that decay exponentially from complete damage state with exponential decay distance determined by <math>\ell</math>. The last term is a source term to causes crack propagation. In other words, crack propagation is affected by the methods used to partition energy into <math>\Psi^{(+)}</math> and <math>\Psi^{(-)}</math>.		where <math>G_c</math> is toughness, <math>\ell</math> is a length scale (describing width of diffuse cracks), and <math>\mathcal{H}</math> is history variable given by <math>\max(\Psi^{(+)})</math>. This equation is a diffusion equation that describes evolution of damage driven by damaging energy. The first two terms lead to diffusive cracks that decay exponentially from complete damage state with exponential decay distance determined by <math>\ell</math>. The last term is a source term to causes crack propagation. In other words, crack propagation is affected by the methods used to partition energy into <math>\Psi^{(+)}</math> and <math>\Psi^{(-)}</math>.

	Simulations using viscous regularization must couple solution of phase field diffusion equation to the material's mechanics calculations. For that to work, all simulations using this material must include a [[Additional Transport Calculations\|Diffusion]] command and choose <tt>(style)</tt> of "fracture". If needed, you can set[[Setting Velocity and Transport Values#Other Transport Conditions\|grid based phase value boundary conditions]] using the "fracture" option to define damage conditions. Although you can set [[Setting Forces and Fluxes#Transport Flux Conditions\|damage flux on particle surfaces]], such conditions may have few applications in real-world modeling. Note that MPM solutions of transport equations can develop oscillations that prevent output of viable phase field descriptions of cracks. This problem can be solved by using FMPM(k) for transport tasks with the [[PeriodicXPIC Custom Task]]<ref name="MPMTransport"/>		Simulations using viscous regularization must couple solution of phase field diffusion equation to the material's mechanics calculations. For that to work, all simulations using this material must include a [[Additional Transport Calculations\|Diffusion]] command and choose <tt>(style)</tt> of "fracture". If needed, you can set [[Setting Velocity and Transport Values#Other Transport Conditions\|grid based phase value boundary conditions]] using the "fracture" option to define damage conditions. Although you can set [[Setting Forces and Fluxes#Transport Flux Conditions\|damage flux on particle surfaces]], such conditions may have few applications in real-world modeling. Note that MPM solutions of transport equations can develop oscillations that prevent output of viable phase field descriptions of cracks. This problem can be solved by using FMPM(k) for transport tasks with the [[PeriodicXPIC Custom Task]]<ref name="MPMTransport"/>

	=== Energy Partitioning ===		=== Energy Partitioning ===

Nairnj: /* Phase Field Methods */

2023-11-17T17:30:52Z

Phase Field Methods

← Older revision		Revision as of 17:30, 17 November 2023
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	where <math>G_c</math> is toughness, <math>\ell</math> is a length scale (describing width of diffuse cracks), and <math>\mathcal{H}</math> is history variable given by <math>\max(\Psi^{(+)})</math>. This equation is a diffusion equation that describes evolution of damage driven by damaging energy. The first two terms lead to diffusive cracks that decay exponentially from complete damage state with exponential decay distance determined by <math>\ell</math>. The last term is a source term to causes crack propagation. In other words, crack propagation is affected by the methods used to partition energy into <math>\Psi^{(+)}</math> and <math>\Psi^{(-)}</math>.		where <math>G_c</math> is toughness, <math>\ell</math> is a length scale (describing width of diffuse cracks), and <math>\mathcal{H}</math> is history variable given by <math>\max(\Psi^{(+)})</math>. This equation is a diffusion equation that describes evolution of damage driven by damaging energy. The first two terms lead to diffusive cracks that decay exponentially from complete damage state with exponential decay distance determined by <math>\ell</math>. The last term is a source term to causes crack propagation. In other words, crack propagation is affected by the methods used to partition energy into <math>\Psi^{(+)}</math> and <math>\Psi^{(-)}</math>.

	Simulations using viscous regularization must couple solution of phase field diffusion equation to the material's mechanics calculations. For that to work, all simulations using this material must include a [[Additional Transport Calculations\|Diffusion]] command and choose <tt>(style)</tt> of "fracture". If needed, you can set [[Setting Velocity and Transport Values#Other Transport Conditions\|phase value ~~on the grid~~]] using the "fracture" option to define damage conditions. Although you can set [[Setting Forces and Fluxes#Transport Flux Conditions\|damage flux on particle surfaces]], such conditions may have few applications in real-world modeling. Note that MPM solutions of transport equations can develop oscillations that prevent output of viable phase field descriptions of cracks. This problem can be solved by using FMPM(k) for transport tasks with the [[PeriodicXPIC Custom Task]]<ref name="MPMTransport"/>		Simulations using viscous regularization must couple solution of phase field diffusion equation to the material's mechanics calculations. For that to work, all simulations using this material must include a [[Additional Transport Calculations\|Diffusion]] command and choose <tt>(style)</tt> of "fracture". If needed, you can set[[Setting Velocity and Transport Values#Other Transport Conditions\|grid based phase value boundary conditions]] using the "fracture" option to define damage conditions. Although you can set [[Setting Forces and Fluxes#Transport Flux Conditions\|damage flux on particle surfaces]], such conditions may have few applications in real-world modeling. Note that MPM solutions of transport equations can develop oscillations that prevent output of viable phase field descriptions of cracks. This problem can be solved by using FMPM(k) for transport tasks with the [[PeriodicXPIC Custom Task]]<ref name="MPMTransport"/>

	=== Energy Partitioning ===		=== Energy Partitioning ===