3.1.7.19.2. Plastic Flow Directions

Specifies the direction of plastic flow \(\mathbf{m}\) used to define the evolution of the plastic strain (see ASDPlasticMaterial Theory).

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

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 the location in stress space for the axis of the Von-Mises cylinder.

Parameters required

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

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

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