Plastic Flow Directions

Specifies the direction of plastic flow \(\mathbf{m}\) used to define evolution of the plastic strain (see ASDPlasticTheory). The plastic flow direction determines how plastic strain components develop and influences both volumetric and deviatoric plastic deformation.

Plastic Flow Directions may define internal variables they need for their specification, as well as paramters. When specifying the ASDPlasticMaterial or ASDPlasticMaterial3D instance, one must provide internal variables mentioned below, together with their hardening function, when defining internal variables.

Note

ASDPlasticMaterial3D provides enhanced monitoring of plastic flow effects through specialized response queries like VolStrain and J2Strain to track volumetric and deviatoric plastic deformation separately.

Available functions:

VonMises_PF

Defines a plastic flow direction derived from the Von Mises Yield Criterion. Its definition. It is the stress-derivative of the VonMises_YF.

\[\newcommand{\vec}[1]{\boldsymbol{#1}} \mathbf{m} = \dfrac{\partial f}{\partial \vec{\sigma}} = \dfrac{\vec{s} - \vec{\alpha} }{ \sqrt{ (\vec{s} - \vec{\alpha}) \cdot (\vec{s} - \vec{\alpha})}}\]

Internal variables defined

IV Name	Type	Symbol	Description
BackStress	Rank-6 Tensor	\(\vec{\alpha}\)	Backstress, definining location in stress space for axis of Von-Mises cylinder.

Parameters required

None required for VonMises plastic flow. Parameters are inherited from associated yield function and hardening functions.

Usage in ASDPlasticMaterial3D

When using VonMises_PF with ASDPlasticMaterial3D, the plastic flow direction is automatically computed as the stress derivative of the Von Mises yield function. This provides:

• Associated plastic flow (flow rule equals yield function gradient)

• No volumetric plastic strain (isochoric plastic deformation)

• Purely deviatoric plastic response

You can monitor plastic strain components:

recorder Node -file plastic_strain.out -time -node 1 -dof 1 2 3 4 5 6 pstrain

DruckerPrager_PF

Defines a plastic flow direction derived from the Drucker-Prager Yield Criterion. It is the stress-derivative of the DruckerPrager_YF.

\[\mathbf{m} = \dfrac{\partial f}{\partial \vec{\sigma}} = \dfrac{\vec{s} - \vec{\alpha} }{ \sqrt{ (\vec{s} - \vec{\alpha}) \cdot (\vec{s} - \vec{\alpha})}} - \dfrac{\sqrt{2/3}k}{3} \vec{I} ;\]

Internal variables defined

It uses the same variables as the Von-Mises yield function, but the VonMisesRadius is now unitless and should be defined with respect to a reference confinement.

IV Name	Type	Symbol	Description
BackStress	Rank-6 Tensor	\(\vec{\alpha}\)	Backstress, definining the location in stress space for the axis of the Drucker-Prager cone.
VonMisesRadius	Scalar	\(k\)	Shear strength at reference confinement, definining the radius of the DruckerPrager cone in stress space as \(kp\)

Parameters required

None required for DruckerPrager plastic flow. Uses parameters from yield function.

Usage in ASDPlasticMaterial3D

DruckerPrager_PF provides non-associated flow when used with pressure-sensitive yield functions. Key characteristics:

• Pressure-dependent plastic flow (includes volumetric component)

• Associated flow with Drucker-Prager yield surface

• Suitable for geomaterials exhibiting dilatancy or contractancy

Monitor volumetric plastic strain:

recorder Node -file vol_strain.out -time -node 1 -dof 1 VolStrain

ConstantDilatancy_PF

Von-Mises PF provides no volumetric change (dilatancy) during plasticity. On the other hand Drucker-Prager PF provides a constant negative volumetric change which is proportional to the current value of the \(k\) parameter. This PF defines a controllable dilatancy during plasticity by specifying a constant dilatancy coeficient \(D\).

\[\mathbf{m} = \dfrac{\partial f}{\partial \vec{\sigma}} = \dfrac{\vec{s} - \vec{\alpha} }{ \sqrt{ (\vec{s} - \vec{\alpha}) \cdot (\vec{s} - \vec{\alpha})}} - \dfrac{D}{3} \vec{I} ;\]

Internal variables defined

IV Name	Type	Symbol	Description
BackStress	Rank-6 Tensor	\(\vec{\alpha}\)	Backstress, definining the location in stress space for the axis of the Drucker-Prager cone.

Parameters required

Parameter Name	Type	Symbol	Description
Dilatancy	Scalar	\(D\)	Defines the rate of dilatancy with plastic flow. Positive values specify negative plastic volumetric change.

Usage in ASDPlasticMaterial3D

ConstantDilatancy_PF provides non-associated flow with controllable dilatancy. This is particularly useful for:

• Concrete and rock materials (positive dilatancy)

• Dense sands (positive dilatancy initially, then contractancy)

• Loose sands (continuous contractancy)

Monitor dilatancy effects:

recorder Node -file vol_strain.out -time -node 1 -dof 1 VolStrain
recorder Node -file j2_strain.out -time -node 1 -dof 1 J2Strain