.. _models-j2_plasticity:

****************************
J\ :sub:`2` Plasticity Model
****************************

Theory
======

The :math:`J_2` plasticity model is a generic implementation of a von Mises yield surface with kinematic and isotropic hardening features. Unlike other models (*e.g.* Elastic-Plastic, Elastic-Plastic Power Law) more flexible, general hardening forms are implemented enabling different isotropic hardening descriptions and some rate and/or temperature dependence. 

As is common to other plasticity models in LAMÉ, the :math:`J_2` plasticity model uses a hypoelastic formulation. As such, the total rate of deformation is additively decomposed into an elastic and plastic part such that

.. math::

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

The objective stress rate, depending only on the elastic deformation, may then be written as,

.. math::

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

where :math:`\mathbb{C}_{ijkl}` is the fourth-order elastic, isotropic stiffness tensor.

The yield surface for the :math:`J_2` plasticity model, :math:`f`, may be written,

.. math::
   :label: eqn:j2:theory:yield

   f\left(\sigma_{ij},\alpha_{ij},\bar{\varepsilon}^p,\dot{\bar{\varepsilon}}^p,\theta\right)=\phi\left(\sigma_{ij},\alpha_{ij}\right)-\bar{\sigma}\left(\bar{\varepsilon}^p,\dot{\bar{\varepsilon}}^p,\theta\right),

in which :math:`\alpha_{ij},\bar{\varepsilon}^p,\dot{\bar{\varepsilon}}^p,` and :math:`\theta` are the kinematic backstress, equivalent plastic strain, equivalent plastic strain rate, and absolute temperature, respectively, while :math:`\phi` and :math:`\bar{\sigma}` are the effective stress and a generic form of the flow stress. Broadly speaking, the effective stress describes the shape of the yield surface and kinematic effects while the flow stress gives the size of the current yield surface. It should also be noted that in writing the yield surface in this way, the dependence on the state variables is split between the effective stress and flow stress functions.

For :math:`J_2` plasticity, the effective stress is given as,

.. math::

   \phi^2\left(\sigma_{ij},\alpha_{ij}\right)=\frac{3}{2}\left(s_{ij}-\alpha_{ij}\right)\left(s_{ij}-\alpha_{ij}\right),

with :math:`s_{ij}` being the deviatoric stress defined as :math:`s_{ij}=\sigma_{ij}-(1/3)\sigma_{kk}\delta_{ij}`.  For the flow stress, a general representation of the form,

.. math::

   \bar{\sigma}\left(\bar{\varepsilon}^p,\dot{\bar{\varepsilon}}^p,\theta\right)=\sigma_y\hat{\sigma}_{\text{y}}\left(\dot{\bar{\varepsilon}}^p\right)\breve{\sigma}_{\text{y}}\left(\theta\right)+K\left(\bar{\varepsilon}^p\right)\hat{\sigma}_{\text{h}}\left(\dot{\bar{\varepsilon}}^p\right)\breve{\sigma}_{\text{h}}\left(\theta\right),

is allowed. In this fashion, the effects of rate (:math:`\hat{\sigma}_{\text{y,h}}`) and temperature (:math:`\breve{\sigma}_{\text{y,h}}`) dependence on yield (:math:`\sigma_y`) and isotropic hardening (:math:`K\left(\bar{\varepsilon}^p\right)`) are decomposed. Separate temperature and rate dependencies may be be specified for yield (subscript y) and hardening (h). This assumption is an extension of the multiplicative decomposition of the Johnson-Cook model [:footcite:`mat:ref:johncook`, :footcite:`mat:ref:johncook2`]. It should be noted that not all effects need to be included and the default parameterization of the hardening classes is such that the response is rate and temperature independent. The following section on plastic hardening will go into more detail on possible choices for functional representations.

An associated flow rule is utilized such that the plastic rate of deformation is normal to the yield surface and is given by,

.. math::

   D_{ij}^{\text{p}}=\dot{\gamma}\frac{\partial\phi}{\partial\sigma_{ij}}=\dot{\gamma}\frac{3}{2\phi}s_{ij},

where :math:`\dot{\gamma}` is the consistency multiplier enforcing :math:`f=0` during plastic deformation. Given the form of :math:`f`, it can also be shown that :math:`\dot{\gamma}=\dot{\bar{\varepsilon}}^p`.

Additional discussion on options for failure models and adiabatic heating may be found in [:footcite:`mat:ref:lester:2019`, :footcite:`mat:ref:lester:2020`] and [:footcite:`mat:ref:lester:2019:2`], respectively.

.. include:: hardening/hardening.rst

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

.. include:: hardening/flow_stress.rst

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

.. include:: hardening/decoupled_flow_stress.rst

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

The :math:`J_2` plasticity model is implemented using a radial return predictor-corrector algorithm. First, an elastic trial stress state is calculated. This is done by assuming that the rate of deformation is completely elastic,

.. math::

   T^{tr}_{ij} = T^{n}_{ij}
   + \Delta t \left( \lambda\delta_{ij}d_{kk}
   + 2\mu d_{ij} \right).

The trial stress state is decomposed into a pressure and a deviatoric stress

.. math::

   p^{tr} = \frac{1}{3} T^{tr}_{kk}
   \;\;\; ; \;\;\;
   s^{tr}_{ij} = T^{tr}_{ij} - p^{tr}\delta_{ij}.

A trial yield function value, :math:`f^{tr}`, is calculated by assuming purely thermoelastic deformations (:math:`\dot{\bar{\varepsilon}}^p=0, \bar{\varepsilon}^p_{tr}=\bar{\varepsilon}^p_n`) such that,

.. math::

   f^{tr} \left( s^{tr}_{ij},\alpha_{ij}^n,\bar{\varepsilon}^{p}_n,\dot{\bar{\varepsilon}}^p_{tr}=0,\theta_{n+1} \right) = \phi^{tr}\left(s^{tr}_{ij},\alpha_{ij}^n \right) - \bar{\sigma}\left( \bar{\varepsilon}^{p}_n,\dot{\bar{\varepsilon}}^p_{tr}=0,\theta_{n+1} \right).

If :math:`f^{tr} \leq 0` then the strain rate is elastic and the stress update is finished. If :math:`f^{tr} > 0` then plastic deformation has occurred and a radial return algorithm determines the extent of plastic deformation.

The normal to the yield surface is assumed to lie in the direction of the trial stress state. This gives the following expression for :math:`N_{ij}`,

.. math::

   N_{ij} = \frac{\left(s_{ij}^{tr}-\alpha^n_{ij}\right)}{\|\left(s_{ij}^{tr}-\alpha_{ij}^n\right)\|}.

Using a backward Euler algorithm, the final deviatoric stress state is

.. math::

   s^{n+1}_{ij} = s_{ij}^{tr} - \Delta \, t 2\mu d_{ij}^{\text{p}},

where the plastic strain increment is

.. math::

   \Delta d_{ij}^{\text{p}} = \sqrt{\frac{3}{2}} \Delta\bar{\varepsilon}^{p} N_{ij}.

The equation for the change in the equivalent plastic strain over the load step is found as the solution to

.. math::

   3 \mu \Delta\bar{\varepsilon}^{p} + \bar{\sigma}\left( \bar{\varepsilon}_{n} + \Delta\bar{\varepsilon}^{p}, \Delta t, \theta_{n+1} \right) - \phi^{tr} + f_{n} = 0,

in which the plastic strain rate is approximated as, :math:`\dot{\bar{\varepsilon}}^p=\Delta\bar{\varepsilon}^p/\Delta t`.

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

The :math:`J_2` plasticity model is verified through a series of uniaxial stress and pure shear tests considering a variety of hardening models.  Specifically, the boundary value problems of :ref:`Appendix A <appendix-common-constant-eqps-rate>` are used. Throughout these tests, the elastic properties are maintained as :math:`E=70` GPa and :math:`\nu = 0.25`. 

Additional verification exercises for the various failure models and adiabatic heating capabilities may be found in [:footcite:`mat:ref:lester:2019`], [:footcite:`mat:ref:lester:2020`] and [:footcite:`mat:ref:lester:2019:2`], respectively.

Plastic Hardening
-----------------

For the verification of the :math:`J_2` model, a series of tests using different rate *independent*, rate *dependent*, and combinations of these hardening models are investigated for both uniaxial stress and pure shear.  For these cases, by imposing a constant plastic strain rate as described in :ref:`Appendix A <appendix-common-constant-eqps-rate>` the model response may be analytically determined as a function of time.  For the rate *independent* cases, a constant rate of :math:`\dot{\bar{\varepsilon}}^p=1\times10^{-4} \text{s}^{-1}` is used to replicate quasi-static conditions.

The various rate *dependent* and rate *independent* hardening coefficients are found in :numref:`tabj2rateDepVerProps` while the remaining model parameters are unchanged from the previous verification exercises.  For the current verification exercises, the rate *independent* hardening models (linear, Voce, and power-law) and rate *dependent* forms (Johnson-Cook, power-law breakdown) are examined. 

.. _tabj2rateDepVerProps:

.. csv-table:: The model parameters for the hardening verification tests used with the :math:`J_2` plasticity model during verification tests.  Parameters for the rate *independent* hardening functions, :math:`\tilde{\sigma}_y`, are also given and denoted with a :math:`\tilde{\cdot}` while the subscript refers to the functional form.
   :align: center
   :delim: &

   :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
   :math:`\tilde{H}_{\text{Linear}}` & 200 MPa &  & 
   :math:`\tilde{A}_{\text{PL}}` & 400 MPa & :math:`\tilde{n}_{\text{PL}}` & 0.25
   :math:`\tilde{A}_{\text{Voce}}` & 200 MPa & :math:`\tilde{n}_{\text{Voce}}` & 20
   :math:`\sigma_y` & 200 MPa & & 

Rate-Independent
^^^^^^^^^^^^^^^^

First, the ability of the built-in rate *independent* hardening models is assessed in both uniaxial stress and pure shear. Specifically, the linear, power-law, and Voce hardening models are considered and the results determined analytically and numerically via Sierra are presented in :numref:`fig-j2-verRIHard`. As expected, excellent agreement is noted between the two sets of results. Importantly, as the responses of all three rate *independent* isotropic hardening models are presented in the same figures, the corresponding behaviors can be seen. Note, the given parameterizations are not selected for any form of equivalency. Nonetheless, the linear post-yielding behavior of the linear model can be seen and compared to the non-linear responses of the Voce and power-law implementations. The critical difference between the latter two being that the Voce response saturates at a stress level while the power-law continues to grow.

.. _fig-j2-verRIHard:

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

   .. image:: ../../_static/figures/j2/j2_normal_rate_independent.png
      :alt: Uniaxial Stress
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_shear_rate_independent.png
      :alt: Pure Shear
      :scale: 35 %

   Analytical and numerical (Sierra) (a) uniaxial stress-strain and (b) pure shear responses of the :math:`J_2` plasticity model with linear, power-law, and Voce rate *independent* isotropic hardening.  Solid lines are analytical while open symbols are numerical.

Rate-Dependent
^^^^^^^^^^^^^^

With the performance of the model under rate *independent* conditions established, next the capabilities of the rate *dependent* (Johnson-Cook and power-law breakdown) formulations are considered.  Note, the flow-stress and decoupled flow-stress models that incorporate more flexible descriptions of isotropic hardening and rate and temperature dependence are left to later sections. With the current Johnson-Cook and power-law breakdown models, user-defined analytic functions are used for each of the specified rate *independent* hardening functions.

The uniaxial stress-strain responses are interrogated for the Johnson-Cook and power-law breakdown rate *dependent* hardening models considering linear, power-law, and Voce isotropic hardening in :numref:`fig-j2-verRDHardNormal`.  Five decades of plastic strain rates :math:`\dot{\bar{\varepsilon}}^p=1\times10^{-3}\rightarrow1\times10^{1}\text{s}^{-1}` are considered.  In comparing the analytical and numerical results between all of the cases exceptional agreement is noted between every case.

.. _fig-j2-verRDHardNormal:

.. subfigure:: AB|CD|EF
   :subcaptions: below
   :align: center

   .. image:: ../../_static/figures/j2/j2_normal_lin_jc.png
      :alt: Linear, Johnson-Cook
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_lin_plb.png
      :alt: Linear, Power-Law Breakdown
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_pl_jc.png
      :alt: Power-Law, Johnson-Cook
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_pl_plb.png
      :alt: Power-Law, Power-Law Breakdown
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_voce_jc.png
      :alt: Voce, Johnson-Cook
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_voce_plb.png
      :alt: Voce, Power-Law Breakdown
      :scale: 35 %

   Uniaxial stress-strain responses of the :math:`J_2` plasticity model with (a,b) linear, (c,d) power-law, and (e,f) Voce isotropic hardening with the (a,c,e) Johnson-Cook and (b,d,f) Power-law breakdown rate *dependent* hardening models.  Solid lines are analytical while open symbols are numerical (Sierra).

Similarly, the pure shear responses of the six hardening combinations over the five plastic strain rates are given in :numref:`fig-j2-verRDHardShear` for both analytical and numerical approaches. As with the normal cases, outstanding agreement is noted between the various results. Thus, between the plethora of problems presented in :numref:`fig-j2-verRDHardNormal` and :numref:`fig-j2-verRDHardShear` the performance of the rate-dependent models may be considered verified.

.. _fig-j2-verRDHardShear:

.. subfigure:: AB|CD|EF
   :subcaptions: below
   :align: center

   .. image:: ../../_static/figures/j2/j2_shear_lin_jc.png
      :alt: Linear, Johnson-Cook
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_shear_lin_plb.png
      :alt: Linear, Power-Law Breakdown
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_shear_pl_jc.png
      :alt: Power-Law, Johnson-Cook
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_shear_pl_plb.png
      :alt: Power-Law, Power-Law Breakdown
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_shear_voce_jc.png
      :alt: Voce, Johnson-Cook
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_shear_voce_plb.png
      :alt: Voce, Power-Law Breakdown
      :scale: 35 %

   Pure shear responses of the :math:`J_2` plasticity model with (a,b) linear, (c,d) power-law, and (e,f) Voce isotropic hardening with the (a,c,e) Johnson-Cook and (b,d,f) Power-law breakdown rate *dependent* hardening models.  Solid lines are analytical while open symbols are numerical (Sierra).

Flow Stress
^^^^^^^^^^^

As a next step in verification, the capabilities of the *flow-stress* hardening model incorporating rate- and temperature-dependence is assessed.  To this end, :numref:`fig-j2-verFSHardJCNormal` presents uniaxial stress-strain responses considering linear, power-law, and Voce isotropic hardening models with both Johnson-Cook and power-law breakdown rate dependent multipliers and Johnson-Cook type temperature dependence.  Five decades of strain rates along with temperatures spanning 180 K are considered in the various figures.  In all of the results, agreement is noted between analytical and numerical results.

.. _fig-j2-verFSHardJCNormal:

.. subfigure:: AB|CD|EF
   :subcaptions: below
   :align: center

   .. image:: ../../_static/figures/j2/j2_normal_fs_lin_jc_jc.png
      :alt: Linear, Johnson-Cook
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_fs_lin_plb_jc.png
      :alt: Linear, Power-Law Breakdown
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_fs_pl_jc_jc.png
      :alt: Power-Law, Johnson-Cook
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_fs_pl_plb_jc.png
      :alt: Power-Law, Power-Law Breakdown
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_fs_voce_jc_jc.png
      :alt: Voce, Johnson-Cook
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_fs_voce_plb_jc.png
      :alt: Voce, Power-Law Breakdown
      :scale: 35 %

   Uniaxial stress-strain responses of the :math:`J_2` plasticity model using the flow-stress hardening model comprised of (a,b) linear, (c,d) power-law, and (e,f) Voce isotropic hardening, (a,c,e) Johnson-Cook and (b,d,f) power-law breakdown rate multipliers, and (a-f) Johnson-Cook temperature multipliers. Solid lines are analytical while open symbols are numerical (Sierra).

To complement the uniaxial results, pure shear results are given in :numref:`fig-j2-verFSHardJCShear`. These results consider the same combinations of linear, power-law, and Voce isotropica hardening multiplier, Johnson-Cook and power-law breakdown rate multipliers, and Johnson-Cook temperature dependence. The same ranges of rates and temperatures are considered. As with the uniaxial cases, good agreement is noted between the analytical and numerical results.

.. _fig-j2-verFSHardJCShear:

.. subfigure:: AB|CD|EF
   :subcaptions: below
   :align: center

   .. image:: ../../_static/figures/j2/j2_shear_fs_lin_jc_jc.png
      :alt: Linear, Johnson-Cook
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_shear_fs_lin_plb_jc.png
      :alt: Linear, Power-Law Breakdown
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_shear_fs_pl_jc_jc.png
      :alt: Power-Law, Johnson-Cook
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_shear_fs_pl_plb_jc.png
      :alt: Power-Law, Power-Law Breakdown
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_shear_fs_voce_jc_jc.png
      :alt: Voce, Johnson-Cook
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_shear_fs_voce_plb_jc.png
      :alt: Voce, Power-Law Breakdown
      :scale: 35 %

   Pure shear responses of the :math:`J_2` plasticity model using the flow-stress hardening model comprised of (a,b) linear, (c,d) power-law, and (e,f) Voce isotropic hardening, (a,c,e) Johnson-Cook and (b,d,f) Power-law breakdown rate multipliers and (a-f) Johnson-Cook temperature multipliers. Solid lines are analytical while open symbols are numerical (Sierra).

Decoupled Flow Stress
^^^^^^^^^^^^^^^^^^^^^

As a further extension, the verification of the *decoupled flow-stress* model is explored.  To this end, :numref:`fig-j2-verDFSHardJCTINormal0` and :numref:`fig-j2-verDFSHardJCTINormal` present uniaxial stress-strain results of various combinations of linear, power-law, and Voce isotropic hardening functions with rate-independent, Johnson-Cook, and power-law breakdown rate multipliers applied in different combinations to yield and hardening.  Hardening is taken to be temperature-independent while yield has a Johnson-Cook temperature multiplier.  The considered cases span five decades of applied strain rates and a range of temperatures. In these cases, the various analytical and numerical results are in agreement.

.. _fig-j2-verDFSHardJCTINormal0:

.. subfigure:: AB|CD|EF
   :subcaptions: below
   :align: center

   .. image:: ../../_static/figures/j2/j2_normal_dfs_lin_jc_ti_jc_plb.png
      :alt: (L), Yield (JC) Hardening (PLB)
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_dfs_lin_jc_ti_ri_jc.png
      :alt: (L), Yield (-) Hardening (JC)
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_dfs_lin_jc_ti_plb_ri.png
      :alt: (L), Yield (PLB) Hardening (-)
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_dfs_pl_jc_ti_jc_plb.png
      :alt: (PL), Yield (JC) Hardening (PLB)
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_dfs_pl_jc_ti_ri_jc.png
      :alt: (PL), Yield (-) Hardening (JC)
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_dfs_pl_jc_ti_plb_ri.png
      :alt: (PL), Yield (PLB) Hardening (-)
      :scale: 35

   Uniaxial stress-strain responses of the :math:`J_2` plasticity model using the decoupled flow-stress hardening model comprised of (a-c) linear ("L") and (d-f) power-law ("PL"), (a-f) temperature independent hardening, (a-f) Johnson-Cook type temperature multiplier for yield, (a,d) Johnson-Cook ("JC") and power-law breakdown ("PLB") type yield and hardening rate multipliers, respectively, (b,e) rate-independent (-) yield with Johnson-Cook type hardening rate dependence, and (c,f) power-law breakdown yield rate dependence with rate-independent hardening. Solid lines are analytical while open symbols are numerical (Sierra).

.. _fig-j2-verDFSHardJCTINormal:

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

   .. image:: ../../_static/figures/j2/j2_normal_dfs_voce_jc_ti_jc_plb.png
      :alt: (V), Yield (JC) Hardening (PLB)
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_dfs_voce_jc_ti_ri_jc.png
      :alt: (V), Yield (-) Hardening (JC)
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_dfs_voce_jc_ti_plb_ri.png
      :alt: (V), Yield (PLB) Hardening (i)
      :scale: 35 %

   Uniaxial stress-strain responses of the :math:`J_2` plasticity model using the decoupled flow-stress hardening model comprised of (a-c) Voce isotropic hardening ("V"), (a-c) temperature independent hardening, (a-c) Johnson-Cook type temperature multiplier for yield, (a) Johnson-Cook ("JC") and power-law breakdown ("PLB") type yield and hardening rate multipliers, respectively, (b) rate-independent (-) yield with Johnson-Cook type hardening rate dependence, and (c) power-law breakdown yield rate dependence with rate-independent hardening. Solid lines are analytical while open symbols are numerical (Sierra).

While the previous results considered temperature-dependence on yield only, the temperature dependence on hardening is examined in :numref:`fig-j2-verDFSHardTIJCNormal0` and :numref:`fig-j2-verDFSHardTIJCNormal`.  As with the previous case, linear, power-law, and Voce isotropic hardening laws are considered in conjunction with different combinations of Johnson-Cook, power-law breakdown, and rate-independent rate multipliers spanning large ranges of strain rates and temperatures.  Once again, excellent agreement is noted between analytical and numerical results.

.. _fig-j2-verDFSHardTIJCNormal0:

.. subfigure:: AB|CD|EF
   :subcaptions: below
   :align: center

   .. image:: ../../_static/figures/j2/j2_normal_dfs_lin_ti_jc_plb_jc.png
      :alt: (L), Yield (JC) Hardening (PLB)
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_dfs_lin_ti_jc_jc_ri.png
      :alt: (L), Yield (-) Hardening (JC)
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_dfs_lin_ti_jc_ri_plb.png
      :alt: (L), Yield (PLB) Hardening (-)
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_dfs_pl_ti_jc_plb_jc.png
      :alt: (PL), Yield (JC) Hardening (PLB)
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_dfs_pl_ti_jc_jc_ri.png
      :alt: (PL), Yield (-) Hardening (JC)
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_dfs_pl_ti_jc_ri_plb.png
      :alt: (PL), Yield (PLB) Hardening (-)
      :scale: 35 %

   Uniaxial stress-strain responses of the :math:`J_2` plasticity model using the decoupled flow-stress hardening model comprised of (a-c) linear ("L") and (d-f) power-law ("PL") hardening, (a-f) temperature independent yield, (a-f) Johnson-Cook type temperature multiplier for hardening, (a,d) power-law breakdown ("PLB") and Johnson-Cook ("JC") rate multipliers for yield and hardening, respectively (b,e) rate-independent (-)hardening with Johnson-Cook type yield rate dependence, and (c,f) power-law breakdown hardening rate dependence with rate-independent yield. Solid lines are analytical while open symbols are numerical (Sierra).

.. _fig-j2-verDFSHardTIJCNormal:

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

   .. image:: ../../_static/figures/j2/j2_normal_dfs_voce_ti_jc_plb_jc.png
      :alt: (V), Yield (JC) Hardening (PLB)
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_dfs_voce_ti_jc_jc_ri.png
      :alt: (V), Yield (-) Hardening (JC)
      :scale: 35 %

   .. image:: ../../_static/figures/j2/j2_normal_dfs_voce_ti_jc_ri_plb.png
      :alt: (V), Yield (PLB) Hardening (i)
      :scale: 35 %

   Uniaxial stress-strain responses of the :math:`J_2` plasticity model using the decoupled flow-stress hardening model comprised of (a-c) Voce ("V") isotropic hardening, (a-c) temperature independent yield, (a-c) Johnson-Cook type temperature multiplier for hardening, (a) power-law breakdown ("PLB") and Johnson-Cook ("JC") rate multipliers for yield and hardening, respectively (b) rate-independent (-)hardening with Johnson-Cook type yield rate dependence, and (c) power-law breakdown hardening rate dependence with rate-independent yield. Solid lines are analytical while open symbols are numerical (Sierra).

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

.. code-block:: sierrainput

   BEGIN PARAMETERS FOR MODEL J2_PLASTICITY
     #
     # Elastic constants
     #
     YOUNGS MODULUS = <real>
     POISSONS RATIO = <real>
     SHEAR MODULUS  = <real>
     BULK MODULUS   = <real>
     LAMBDA         = <real>
     TWO MU         = <real>
     #
     # Yield surface parameters
     #
     YIELD STRESS = <real>
     BETA         = <real> (1.0)

.. include:: hardening/flow_user_input_1.rst

.. include:: hardening/flow_user_input_2.rst

.. include:: failure_user_input.rst

.. include:: hardening/adiabatic_heat_user_input.rst

.. code-block:: sierrainput

   END [PARAMETERS FOR MODEL J2_PLASTICITY]

In the command blocks that define the :math:`J_2` plasticity model:

.. \item See \ifdevelopment the \Theusersguide\ Section:numref:`user:mat:elasticConstants` \else Section: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.

- The beta parameter defines if hardening is isotropic.

.. include:: hardening/user_output.rst

.. include:: hardening/flow_stress_user_output.rst

.. include:: hardening/decoupled_flow_stress_user_output.rst

Output variables available for this model are listed in :numref:`out-tab-j2pstvar`.

.. _out-tab-j2pstvar:

.. csv-table:: State Variables for J2 PLASTICITY Model
   :align: center
   :delim: &
   :header: Name, Description

   ``EQPS``  & equivalent plastic strain, :math:`\bar{\varepsilon}^{p}`
   ``EQDOT`` & equivalent plastic strain rate, :math:`\dot{\bar{\varepsilon}}^{p}`
   ``SEFF``  & effective stress, :math:`\phi`
   ``TENSILE_EQPS`` & tensile equivalent plastic strain, :math:`\bar{\varepsilon}^{p}_{t}`
   ``DAMAGE`` & damage, :math:`\phi`
   ``VOID_COUNT`` & void count, :math:`\eta`
   ``VOID_SIZE`` & void size, :math:`\upsilon`
   ``DAMAGE_DOT`` & damage rate, :math:`\dot{\phi}`
   ``VOID_COUNT_DOT`` & void count rate, :math:`\dot{\eta}`
   ``PLASTIC_WORK_HEAT_RATE`` & plastic work heat rate, :math:`\dot{Q}^p`

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