.. _models-modular_plane_stress_plasticity:

*************************************
Modular Plane Stress Plasticity Model
*************************************

Theory
======

Like the plane stress plasticity model of :numref:`models-plane_stress_rate_plasticity`, the modular plane stress plasticity (MPSP) model is a plane stress implementation of a :math:`J_2` plasticity formulation largely following and motivated by the works of Simo and Taylor [:footcite:`mat:ref:simo:86`] and Simo and Hughes [:footcite:`SimoHughes:1998`].  However, the modular plane stress plasticity model differs from those prior works and the aforementioned plane stress plasticity formulation via its specification of the hardening.  Specifically, in the current case kinematic hardening is neglected and expanded isotropic hardening and rate-dependence are considered by leveraging various modular hardening capabilities used with a variety of solid plasticity models (*i.e.* the :math:`J_2` plasticity model in :numref:`models-j2_plasticity`).

Like other plasticity models, the components of the objective stress rate, :math:`\stackrel{\circ}{\sigma}_{ij}`, are written as,

.. math::

   \stackrel{\circ}{\sigma}_{ij} = \mathbb{C}_{ijkl}D_{kl}^{\text{e}}

where :math:`\mathbb{C}_{ijkl}` are the components of the fourth-order, isotropic elasticity tensor and :math:`D_{ij}^{\text{e}}` are the components of the elastic part of the total rate of deformation tensor.  An additive split of the total rate of deformation tensor into elastic and plastic contributions is assumed such that,

.. math::

   D_{ij}=D_{ij}^{\text{e}}+D_{ij}^{\text{p}}.

With a :math:`J_2` plasticity model, the effective stress measure, :math:`\phi`, may be written,

.. math::

   \phi^2=\frac{3}{2}s_{ij}s_{ij}~\qquad~;~\qquad~s_{ij}=\sigma_{ij}-\frac{1}{3}\sigma_{kk}\delta_{ij},

with :math:`s_{ij}` begin the deviatoric stress tensor.  After enforcing the plane-stress conditions (:math:`\sigma_{13}=\sigma_{23}=\sigma_{33}=0`), there are only three non-zero stress components.  As such, the problem may be simplified by introducing the projection matrix, :math:`\underline{\underline{\bar{P}}}`, of Simo and Taylor [:footcite:`mat:ref:simo:86`],

.. math::

   \underline{\underline{\bar{P}}}=\frac{1}{3}\left[\begin{array}{ccc}2 & -1 & 0 \\ -1 & 2 & 0 \\ 0 & 0 & 3\end{array}\right],

so that,

.. math::

   \underline{s}=\underline{\underline{\bar{P}}}\,\underline{\sigma}

where,

.. math::

   \underline{\sigma}=\left[\begin{array}{c} \sigma_{11} \\ \sigma_{22} \\ \sigma_{12} \end{array}\right] ~\qquad~;~\qquad~
   \underline{s} = \left[\begin{array}{c} s_{11} \\ s_{22} \\ s_{12}\end{array}\right].

Note, in the previous and following relations an explicit matrix notation is used to denote variables in the projected stress space to reinforce that these terms are not tensors.  To this end, a single underline (:math:`\underline{x}`) is used for a vector while a twice underlined variable (:math:`\underline{\underline{X}}`) is a matrix.

The projected effective stress measure, :math:`\bar{\phi}`, may then be taken to be,

.. math::

   \bar{\phi}^2=\underline{\sigma}^T\underline{\underline{P}}\,\underline{\sigma},

in which a superscript T denotes transpose and,

.. math::

   \underline{\underline{P}}=\frac{1}{3}\left[\begin{array}{ccc} 2 & -1 & 0 \\ -1 & 2 & 0 \\ 0 & 0 & 6\end{array}\right].
 
Written in this fashion, there is a small difference between the projected effective stress (:math:`\bar{\phi}`) and the traditional 3D form (:math:`\phi`) associated with a constant premultiplier.  This is due to subtle differences in notation used by the plane-stress references [:footcite:`mat:ref:simo:86`, :footcite:`SimoHughes:1998`] and is accounted for in the definition of the yield surface radius, :math:`R`, ensuring equivalence in forms.

A corresponding yield function, :math:`f`, is introduced such that,

.. math::

   f=\bar{\phi}\left(\underline{\sigma}\right)-R^2\left(\bar{\varepsilon}^p,\dot{\bar{\varepsilon}}^p\right),

where :math:`R` is the yield surface radius in the deviatoric :math:`\pi`-plane that isotropically hardens via dependencies on the equivalent plastic strain (isotropic hardening variable) and its rate that are denoted :math:`\bar{\varepsilon}^p` and :math:`\dot{\bar{\varepsilon}}^p`, respectively.  The radius may be related to the current yield stress, :math:`\bar{\sigma}`, via,

.. math::

   R=\sqrt{\frac{2}{3}}\bar{\sigma}\left(\bar{\varepsilon}^p,\dot{\bar{\varepsilon}}^p\right).

The distinguishing feature of the modular plane stress plasticity model is a flexible definition of the isotropic hardening in 
which the current yield stress is generically written,

.. math::

   \bar{\sigma}=\sigma_y\hat{\sigma}_y\left(\dot{\bar{\varepsilon}}^p\right)+K\left(\bar{\varepsilon}^p\right)\hat{\sigma}_h\left(\dot{\bar{\varepsilon}}^p\right),

with :math:`\sigma_y`, :math:`K`, and :math:`\hat{\sigma}_{y,h}` being the constant initial yield stress, isotropic hardening, and separate rate multipliers for yield and hardening, respectively.  A variety of different forms may be assumed as described below.

To complete the theoretical formulation, the flow rules are specified as,

.. math::

   d\underline{\varepsilon}^p & = \lambda\underline{\underline{P}} \; \underline{\sigma}, \\
   d\bar{\varepsilon}^p & = \lambda\sqrt{\frac{2}{3}}\bar{\phi},

where :math:`\lambda` is the consistency multiplier enforcing :math:`f=0` during plastic deformation and :math:`d\underline{\varepsilon}^p` is the plastic strain increment in the constrained subspace.  It is emphasized here that the current yield function described in is *not* homogeneous of degree one like in other three-dimensional formulations presented in this manual. As such, the consistency multiplier and equivalent plastic strain increment are not equivalent. As an example of this, by consideration of the preceding relations, it is apparent that :math:`\lambda` has units of one over stress.

For more information about the modular plane stress plasticity model, consult [:footcite:`mat:ref:simo:86`, :footcite:`SimoHughes:1998`].

.. include:: hardening/hardening.rst

Flow Stress
-----------

.. include:: hardening/flow_stress_no_temp.rst

Decoupled Flow Stress
---------------------

.. include:: hardening/decoupled_flow_stress_no_temp.rst

Implementation
==============

The integration approach for the modular plane stress plasticity model follows largely from the elastic-predictor/inelastic-corrector radial return approaches of Simo and Taylor [:footcite:`mat:ref:simo:86`] (and Simo and Hughes [:footcite:`SimoHughes:1998`]) with the exception of an extra line-search step and slightly modified treatment for the hardening. To this end, the total strain increment :math:`\underline{d\varepsilon}=\dot{\underline{\varepsilon}}\Delta t` is given as,

.. math::

   \underline{d\varepsilon}=\Delta t\left[\begin{array}{c} d_{11} \\ d_{22} \\ 2d_{12} \end{array}\right]

where :math:`\Delta t= t_{n+1}-t_{n}` in which :math:`t=t_{n}` and :math:`t=t_{n+1}` are a completely known state and the state 
to be determined.  The trial stress may then be written,

.. math::
   :label: mpsp:imp:trial

   \underline{\sigma}^{tr}=\underline{\underline{\bar{C}}}\left[\underline{\varepsilon}_n+\underline{d\varepsilon}-\underline{\varepsilon}^{\text{p}}_n\right]

with

.. math::

   \underline{\underline{\bar{C}}}=\frac{E}{1-\nu^2}\left[\begin{array}{ccc} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac{1-\nu}{2} \end{array}\right]

and :math:`E` and :math:`\nu` being the elastic modulus and Poisson's ratio, respectively.  The trial yield function is then simply,

.. math::

   f^{tr}=\bar{\phi}^2\left(\underline{\sigma}^{tr}\right)-R^2\left(\bar{\varepsilon}_n^{\text{p}},0\right).

For the case of plastic loading, if a fully implicit backward Euler scheme is adopted the plastic strain flow rules are,

.. math::
   :label: mpsp:imp:flow

   \underline{\varepsilon}^{\text{p}}_{n+1} = \underline{\varepsilon}^{\text{p}}_{n}+\lambda\underline{\underline{P}}\,\underline{\sigma}_{n+1},

.. math::
   :label: mpsp:imp:flow2

   \bar{\varepsilon}^{\text{p}}_{n+1} = \bar{\varepsilon}^{\text{p}}_n+\lambda\sqrt{\frac{2}{3}}\bar{\phi}_{n+1}.

By introducing,

.. math::

   \underline{\sigma}_{n+1}=\underline{\underline{\bar{C}}}\left(\underline{\varepsilon}_{n+1}-\underline{\varepsilon}_{n+1}^{\text{p}}\right),

and using relations :eq:`mpsp:imp:trial` and :eq:`mpsp:imp:flow`, the updated stress may be shown to be,

.. math::
   :label: mpsp:imp:sigNew

   \left[\underline{\underline{I}}+\lambda\underline{\underline{\bar{C}}}\,\underline{\underline{P}}\right]\underline{\sigma}_{n+1}=\underline{\sigma}^{tr},

with :math:`\underline{\underline{I}}` being the identity matrix. As noted by Simo and Taylor [:footcite:`mat:ref:simo:86`], :math:`\underline{\underline{\bar{C}}}` and :math:`\underline{\underline{P}}` share characteristic subspaces :math:`\underline{\underline{Q}}` enabling a principle decomposition such that,

.. math::

   \underline{\underline{P}} = \underline{\underline{Q}}\,\underline{\underline{\Lambda}}^{\text{P}}\,\underline{\underline{Q}}^T,~\qquad~;~\qquad~
      \underline{\underline{\bar{C}}} = \underline{\underline{Q}}\,\underline{\underline{\Lambda}}^{\text{C}}\,\underline{\underline{Q}}^T,

in which,

.. math::

   \underline{\underline{Q}}^T=\frac{1}{\sqrt{2}}\left[\begin{array}{ccc} 1 & 1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & \sqrt{2}\end{array}\right]~~;~~
   \underline{\underline{\Lambda}}^{\text{P}} = \left[\begin{array}{ccc}\frac{1}{3} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2\end{array}\right]~~;~~
   \underline{\underline{\Lambda}}^{\text{C}} = \left[\begin{array}{ccc} \frac{E}{1-\nu} & 0 & 0 \\ 0 & 2\mu & 0 \\ 0 & 0 & \mu\end{array}\right].


In this space, a transformed stress, :math:`\underline{\eta}`, may be given as,

.. math::

   \underline{\eta}=\underline{\underline{Q}}^T\,\underline{\sigma},

which, when substituted into :eq:`mpsp:imp:sigNew` yields,

.. math::
   :label: mpsp:imp:etaNew

   \left[\underline{\underline{I}}+\lambda\underline{\underline{\Lambda}}^{\text{C}}\,\underline{\underline{\Lambda}}^{\text{P}}\right]\underline{\eta}_{n+1} = 
     \underline{\eta}^{tr}.

Importantly, in :eq:`mpsp:imp:etaNew` the matrix on the left-hand side is diagonal and easily inverted. The updated transformed stress is thus a function of the consistency multiplier alone. Substituting the corresponding evaluation of the stress into the definition of the effective stress produces a scalar function of :math:`\lambda` such that,

.. math::

   \bar{\phi}^2=\frac{\frac{1}{3}\left(\eta^{tr}_{11}\right)^2}{\left[1+\lambda\frac{E}{3\left(1-\nu\right)}\right]^2}+
     \frac{\left(\eta_{22}^{tr}\right)^2+2\left(\eta_{12}^{tr}\right)^2}{\left[1+\lambda2\mu\right]^2}.

With the effective stress written as a function of :math:`\lambda` alone and the flow rules in :eq:`mpsp:imp:flow` and :eq:`mpsp:imp:flow2` only an appropriate approximation for the effective plastic strain rate is needed to arrive at the single scalar consistency equation to be solve.  To that end, using :eq:`mpsp:imp:flow2`, the effective plastic strain rate is taken to be,

.. math::

   \dot{\bar{\varepsilon}}^{\text{p}}\left(\lambda\right)\approx\frac{\bar{\varepsilon}^{\text{p}}_{n+1}-\bar{\varepsilon}^{\text{p}}_{n}}{\Delta t}
          = \frac{\lambda}{\Delta t}\sqrt{\frac{2}{3}}\bar{\phi}_{n+1}\left(\lambda\right).

The updated yield function is now written as,

.. math::

   f_{n+1}\left(\lambda\right)=\bar{\phi}^2_{n+1}\left(\lambda\right)-R^2\left(\lambda\right)=0.

This non-linear equation may be readily solved via a line-search augmented Newton-Raphson approach (see [:footcite:`mat:ref:wmsch4`]) by recasting the consistency condition as a residual,

.. math::

   r^f\left(\lambda\right)=f\left(\lambda\right)=0.

Which, when linearized as,

.. math::

   r^f_{k+1}=r_k^f+\frac{df}{d\lambda}\Delta\lambda,

with k being the non-linear correction iteration and :math:`\Delta\lambda` is the consistency increment yields the solution (with :math:`r_{k+1}^f=0`),

.. math::

   \Delta\lambda=\frac{-r_k^f}{\frac{df}{d\lambda}}.

The derivative is simply given as,

.. math::

   \frac{df}{d\lambda}=\frac{d}{d\lambda}\left(\bar{\phi}^2\right)-\frac{d}{d\lambda}\left(R^2\right),

where

.. math::

   \frac{d}{d\lambda}\left(\bar{\phi}^2\right) = -2\left[\frac{E}{3\left(1-\nu\right)}\frac{\frac{1}{3}\left(\eta_{11}^{tr}\right)^2}
      {\left[1+\lambda\frac{E}{3\left(1-\nu\right)}\right]^3}+2\mu\frac{\left(\eta_{22}^{tr}\right)^2+2\left(\eta_{12}^{tr}\right)^2}{\left[1+\lambda2\mu\right]^3}\right],

and

.. math::

   \frac{d}{d\lambda}\left(R^2\right)=\frac{4}{3}\left(\sigma_y\hat{\sigma}_y+K\hat{\sigma}_h\right)\left[\hat{\sigma}_h\frac{dK}{d\bar{\varepsilon}^{\text{p}}}
        \frac{d\bar{\varepsilon}^{\text{p}}}{d\lambda}+\frac{d\dot{\bar{\varepsilon}}^{\text{p}}}{d\lambda}\left(\sigma_y\frac{d\hat{\sigma}_y}
        {d\dot{\bar{\varepsilon}}^{\text{p}}}+K\frac{d\hat{\sigma}_h}{d\dot{\bar{\varepsilon}}^{\text{p}}}\right)\right],

in which

.. math::

   \frac{d\bar{\varepsilon}^{\text{p}}}{d\lambda} & = \sqrt{\frac{2}{3}}\left(\phi+\lambda\frac{d\phi}{d\lambda}\right), \\
   \frac{d\dot{\bar{\varepsilon}}^{\text{p}}}{d\lambda} & = \frac{1}{\Delta t}\sqrt{\frac{2}{3}}\left(\phi+\lambda\frac{d\phi}{d\lambda}\right).

As only a single equation needs to be solved, a merit function, :math:`\psi`, is simply given as,

.. math::

   \psi\left(\lambda\right)=\frac{1}{2}\left(\frac{r^f}{\sigma_y^2}\right)^2

which may be solved via the quadratic approximation line-search scheme of [:footcite:`mat:ref:wmsch4`].

Verification
============

Given the modular nature of the modular plane stress plasticity (MPSP) model, a variety of tests are constructed to ascertain performance under different loadings and and combinations of hardening models. The model parameters needed for such tests are given below in :numref:`tab-mpsp-daProps`. While a large number of combinations of the hardening and/or rate multipliers have been tested under different conditions (>100 tests), here, for brevity only a sampling of these tests are presented.

.. _tab-mpsp-daProps:

.. csv-table:: Model parameters for verification tests used with the modular plane stress plasticity (MPSP) model.
   :align: center
   :delim: &

   :math:`E` & 70 GPa & :math:`\nu` & 0.33 (-)
   :math:`\sigma_y` & 200 MPa & :math:`H^{\prime}` & 500 MPa
   :math:`A_{\text{PL}}` & 400 MPa & :math:`n_{\text{PL}}` & 0.25 (-)
   :math:`A_{\text{Voce}}` & 200 MPa & :math:`n_{\text{Voce}}` & 20 (-)
   :math:`C` & 0.1 (-) & :math:`\dot{\varepsilon}_0` & :math:`1\times 10^{-4}` s\ :math:`^{-1}`
   :math:`g` & 0.21 s\ :math:`^{-1}` & :math:`m` & 16.4 (-)

Uniaxial stress
---------------

For the uniaxial stress tests, the constant equivalent plastic strain boundary value problem of :ref:`Appendix A <appendix-common-constant-eqps-rate>` is used.  Although that discussion is for 3D formulations, the plane stress assumptions agree with the assumed boundary conditions (*e.g.* traction free out-of-plane stress) enabling the same results to be used here.  Results for such tests and their corresponding analytical solutions are shown in :numref:`fig-mpsp-verUniStress` for constant strain rates of :math:`\dot{\bar{\varepsilon}}^p=1\times 10^{-3}`\ s\ :math:`^{-1}` (:numref:`fig-mpsp-verUniStress`\ (a)) and :math:`\dot{\bar{\varepsilon}}^p=1`\ s\ :math:`^{-1}` (:numref:`fig-mpsp-verUniStress`\ (b)).

.. _fig-mpsp-verUniStress:

.. subfigure:: AB
   :subcaptions: below
   :align: center

   .. image:: ../../_static/figures/mpsp_uni_stress_EM3.png
      :scale: 20 %

   .. image:: ../../_static/figures/mpsp_uni_stress_EM0.png
      :scale: 20 %

   Analytical and numerical constant equivalent plastic strain rate verification tests of the modular plane stress plasticity models with a uniaxial stress state and strain rates of (a) :math:`\dot{\varepsilon}^p=1\times 10^{-3}`\ s\ :math:`^{-1}` and (b) :math:`\dot{\varepsilon}^p=1`\ s\ :math:`^{-1}` with linear, power-law, and voce isotropic hardening and power-law breakdown rate-dependence. Solid lines are analytical and open symbols are from finite element calculations.

.. _models-mpsp-balancedBiaxial:

Balanced Biaxial
----------------

To assess performance of the model with multiple stress components, a constant equivalent plastic strain rate balanced biaxial test is considered. For this test, a stress-state (in the projected plane stress space) of

.. math::

   \underline{\sigma}\left(t\right)=\left[\begin{array}{c} \sigma\left(t\right) \\ -\sigma\left(t\right) \\ 0 \end{array}\right]

is assumed. Note, such a loading is equivalent to a pure shear loading in a rotated frame of reference. As such, many of the pure shear results of :ref:`Appendix A <appendix-common-constant-eqps-rate>` may be leveraged.  To that end, if elasticity effects are included the total strain, :math:`\varepsilon\left(t\right)`, may be found to be

.. math::

   \varepsilon\left(t\right)=\frac{1}{\sqrt{3}}\frac{\sigma_y+\hat{\sigma}_hK\left(\dot{\bar{\varepsilon}}^p\left(t-t^{\text{el}}\right)\right)}{2\mu}
      + \frac{\sqrt{3}}{2}\dot{\bar{\varepsilon}}^p\left(t-t^{\text{el}}\right),

with :math:`t^{\text{el}}` being the time at yield (elastic limit).  To produce the desired stress state, the corrresponding displacements are :math:`u_1\left(t\right)=\exp\left(\varepsilon\left(t\right)\right)-1` and :math:`u_2\left(t\right)=\exp\left(-\varepsilon\left(t\right)\right)-1`.  Results of such tests and their corresponding analytical solutions are presented below in :numref:`fig-mpsp-verBalBiax` with constant strain rates of :math:`\dot{\bar{\varepsilon}}^p=1\times 10^{-3}`\ s\ :math:`^{-1}` (:numref:`fig-mpsp-verBalBiax`\ (a)) and :math:`\dot{\bar{\varepsilon}}^p=1`\ s\ :math:`^{-1}` (:numref:`fig-mpsp-verBalBiax`\ (b)).

.. _fig-mpsp-verBalBiax:

.. subfigure:: AB
   :subcaptions: below
   :align: center

   .. image:: ../../_static/figures/mpsp_bal_biax_EM3.png
      :scale: 20 %

   .. image:: ../../_static/figures/mpsp_bal_biax_EM0.png
      :scale: 20 %

   Analytical and numerical constant equivalent plastic strain rate verification tests of the modular plane stress plasticity models with a balanced biaxial stress state and strain rates of (a) :math:`\dot{\varepsilon}^p=1\times 10^{-3}`\ s\ :math:`^{-1}` and (b) :math:`\dot{\varepsilon}^p=1`\ s\ :math:`^{-1}` with linear, power-law, and voce isotropic hardening and power-law breakdown rate-dependence.  Solid lines are analytical and open symbols are from finite element calculations. Positive valued stresses correspond to :math:`\sigma_{11}` while negative values are :math:`\sigma_{22}`.

Biaxial Shear
-------------

As a final set of tests, the pure shear response is probed.  To accomplish this loading, the previous balanced biaxial test is reconsidered 
with the geometry rotated 45\ :math:`^{\circ}` about the out of plane direction producing a stress state of,

.. math::

   \underline{\sigma}\left(t\right)=\left[\begin{array}{c} 0 \\ 0 \\ \sigma_{xy}\left(t\right)\end{array}\right].

The previous results from :numref:`models-mpsp-balancedBiaxial` regarding the solution for the balanced biaxial problem may again be used with :math:`\sigma_{xy}\left(t\right)=\sigma\left(t\right)`. Result for this case, both analytical and finite element, are given in :numref:`fig-mpsp-verBalBiax` with constant applied strain rates of :math:`\dot{\bar{\varepsilon}}^p=1\times 10^{-3}`\ s\ :math:`^{-1}` and :math:`\dot{\bar{\varepsilon}}^p=1`\ s\ :math:`^{-1}` in :numref:`fig-mpsp-verBiaxShear`\ (a) and :numref:`fig-mpsp-verBiaxShear`\ (b), respectively.

.. _fig-mpsp-verBiaxShear:

.. subfigure:: AB
   :subcaptions: below
   :align: center

   .. image:: ../../_static/figures/mpsp_biax_shear_EM3.png
      :scale: 20 %

   .. image:: ../../_static/figures/mpsp_biax_shear_EM0.png
      :scale: 20 %

   Analytical and numerical constant equivalent plastic strain rate verification tests of the modular plane stress plasticity models with a pure shear stress state and strain rates of (a) :math:`\dot{\varepsilon}^p=1\times 10^{-3}`\ s\ :math:`^{-1}` and (b) :math:`\dot{\varepsilon}^p=1`\ s\ :math:`^{-1}` with linear, power-law, and voce isotropic hardening and power-law breakdown rate-dependence.  Solid lines are analytical and open symbols are from finite element calculations.


User Guide
==========

.. code-block:: sierrainput

   BEGIN PARAMETERS FOR MODEL MODULAR_PLANE_STRESS_PLASTICITY
      #
      # Elastic constants
      #
      YOUNGS MODULUS = <real>
      POISSONS RATIO = <real>
      SHEAR MODULUS  = <real>
      BULK MODULUS   = <real>
      LAMBDA         = <real>
      TWO MU         = <real>
      #
      YIELD STRESS   = <real>
      #

.. include:: hardening/flow_user_input_1.rst

.. include:: hardening/flow_no_temp_user_input.rst

.. code-block:: sierrainput

   END [PARAMETERS FOR MODEL MODULAR_PLANE_STRESS_PLASTICITY]

In the command blocks that define the Modular Plane Stress Plasticity model:

.. - See \ifdevelopment the \Theusersguide\ :numref:`user:mat:elasticConstants` \else :numref:`mat:elasticConstants` \fi for more information on elastic constants input.

- The reference nominal yield stress, :math:`\bar{\sigma}`, is defined with the ``YIELD STRESS`` command line.

.. include:: hardening/user_output.rst

.. include:: hardening/flow_stress_no_temp_user_output.rst

.. include:: hardening/decoupled_flow_stress_no_temp_user_output.rst

Output variables available for this model are listed in :numref:`out-tab-mpspstvar`.

.. _out-tab-mpspstvar:

.. csv-table:: State Variables for MODULAR PLANE STRESS PLASTICITY Model
   :align: center
   :delim: &
   :header: Name, Description

   ``RADIUS``       & yield surface radius in deviatoric :math:`\pi`-plane, :math:`R`
   ``EQPS``         & equivalent plastic strain, :math:`\bar{\varepsilon}^{p}`
   ``EQDOT``        & equivalent plastic strain rate, :math:`\dot{\bar{\varepsilon}}^{p}`
   ``TENSILE_EQPS`` & tensile equivalent plastic strain, :math:`\bar{\varepsilon}^{p}_{t}`


.. raw::
   html

   <hr>

.. footbibliography::
