.. _models-barlat:

***********************
Barlat Plasticity Model
***********************

.. _models-barlat-theory:

Theory
======

The Barlat plasticity model is a hypoelastic, rate-independent plasticity model.  The underlying yield surface is both 
anisotropic and non-quadratic [:footcite:`mat:ref:barlat:05`].  With respect to the former, linear transformations of the deviatoric 
stress are used to capture texture and anisotropy effects.  The rate form of this model assumes an additive split of the 
rate of deformation into an elastic and plastic part

.. math::

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

The stress rate only depends on the elastic rate of deformation

.. 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.

To describe anisotropy in the yield-behavior, two linear transformation tensors, :math:`C^{\prime}_{ijkl}` and 
:math:`C^{\prime\prime}_{ijkl}`, are introduced such that,

.. math::

    s_{ij}^{\prime}=C^{\prime}_{ijkl}s_{kl} \qquad ; \qquad s_{ij}^{\prime\prime}=C^{\prime\prime}_{ijkl}s_{kl},

with :math:`s_{ij}` being the deviatoric stress tensor (:math:`s_{ij}=\sigma_{ij}-1/3\sigma_{kk}\delta_{ij}`) and :math:`s_{ij}^{\prime}` and 
:math:`s_{ij}^{\prime\prime}` being transformed stresses.  Two transformations are used to 
capture both the anisotropy of the yield surface and flow rule.  In Voigt notation the two transformation tensors are given as,

.. math::

    \left[C^{\prime}\right] = \left[\begin{array}{cccccc}
   0 & -c_{12}^{\prime} & -c_{13}^{\prime} & 0 & 0 & 0 \\
   -c_{21}^{\prime} & 0 & -c_{23}^{\prime} & 0 & 0 & 0 \\
   -c_{31}^{\prime} & -c_{32}^{\prime} & 0 & 0 & 0 & 0 \\
   0 & 0 & 0 & c_{44}^{\prime} & 0 & 0 \\
   0 & 0 & 0 & 0 & c_{55}^{\prime} & 0 \\
   0 & 0 & 0 & 0 & 0 & c_{66}^{\prime} \end{array}\right]

.. math::

    \left[C^{\prime\prime}\right] = \left[\begin{array}{cccccc}
   0 & -c_{12}^{\prime\prime} & -c_{13}^{\prime\prime} & 0 & 0 & 0 \\
   -c_{21}^{\prime\prime} & 0 & -c_{23}^{\prime\prime} & 0 & 0 & 0 \\
   -c_{31}^{\prime\prime} & -c_{32}^{\prime\prime} & 0 & 0 & 0 & 0 \\
   0 & 0 & 0 & c_{44}^{\prime\prime} & 0 & 0 \\
   0 & 0 & 0 & 0 & c_{55}^{\prime\prime} & 0 \\
   0 & 0 & 0 & 0 & 0 & c_{66}^{\prime\prime} \end{array}\right] .

Alternatively, the transformed stresses may be written in terms of the Cauchy stress tensor as,

.. math::

    s_{ij}^{\prime}=L^{\prime}_{ijkl}\sigma_{kl} \qquad ; \qquad s^{\prime\prime}_{ij}=L^{\prime\prime}_{ijkl}\sigma_{kl},

where :math:`L^{\prime}_{ijkl}=C^{\prime}_{ijmn}I_{mnkl}` and :math:`L^{\prime\prime}_{ijkl}=C^{\prime\prime}_{ijmn}I_{mnkl}`.  In this 
case, :math:`I_{ijkl}` is the symmetric deviatoric projection tensor and takes the form of, 

.. math::

   I_{ijkl}=\frac{1}{2}\left(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right)-\frac{1}{3}\delta_{ij}\delta_{kl}.

In reduced form,

.. math::

    \left[L^{\prime}\right] =\frac{1}{3} \left[\begin{array}{cccccc}
    c_{12}^{\prime}+c_{13}^{\prime} & -2c_{12}^{\prime}+c_{13}^{\prime} &c_{12}^{\prime}-2c_{13}^{\prime} & 0 & 0 & 0 \\
    -2c_{21}^{\prime}+c_{23}^{\prime} & c_{21}^{\prime}+c_{23}^{\prime} & c_{21}^{\prime}-2c_{23}^{\prime} & 0 & 0 & 0 \\
    -2c_{31}^{\prime}+c_{32}^{\prime}& c_{31}^{\prime}-2c_{32}^{\prime} & c^{\prime}_{31}+c_{32}^{\prime} & 0 & 0 & 0 \\
    0 & 0 & 0 & 3c_{44}^{\prime} & 0 & 0 \\
    0 & 0 & 0 & 0 & 3c_{55}^{\prime} & 0 \\
    0 & 0 & 0 & 0 & 0 & 3c_{66}^{\prime} \end{array}\right],

and an analogous expression may be written for :math:`L^{\prime\prime}_{ijkl}`.  

The yield surface, :math:`f`, is given as,

.. math::

   f\left(\sigma_{ij},\bar{\varepsilon}^p\right)=\phi\left(\sigma_{ij}\right)-\bar{\sigma}\left(\bar{\varepsilon}^p\right)=0,

in which :math:`\phi\left(\sigma_{ij}\right)` is the effective stress and :math:`\bar{\sigma}\left(\bar{\varepsilon}^p\right)` is the current yield stress 
that may depend on rate and/or temperature.  The effective stress is written in terms of the principal transformed stresses (:math:`s^{\prime}_i` and :math:`s^{\prime\prime}_i`, respectively) 
and the yield surface exponent, :math:`a`, as,

.. math::

   \phi\left(\sigma_{ij}\right) = & \bigg\{\frac{1}{4}\Big[
     |s^{\prime}_1-s^{\prime\prime}_1|^a+
     |s^{\prime}_1-s^{\prime\prime}_2|^a+
     |s^{\prime}_1-s^{\prime\prime}_3|^a
   \nonumber \\
   & + 
     |s^{\prime}_2-s^{\prime\prime}_1|^a+
     |s^{\prime}_2-s^{\prime\prime}_2|^a+
     |s^{\prime}_2-s^{\prime\prime}_3|^a
   \\
   & +
     |s^{\prime}_3-s^{\prime\prime}_1|^a+
     |s^{\prime}_3-s^{\prime\prime}_2|^a+
     |s^{\prime}_3-s^{\prime\prime}_3|^a
   \Big]\bigg\}^{1/a} . \nonumber

An example of such a yield surface is given in :numref:`models-barlat-theory-yield` along with examples of previously presented (von Mises, Hosford, Hill) 
surfaces.  The presented Barlat surface corresponds to that of 2090-T3 aluminum first characterized by Barlat *et al.* [:footcite:`mat:ref:barlat:05`].  
In :numref:`models-barlat-theory-yield`, both the anisotropy and non-quadratic nature of the yield surface is evident leading to differing 
strengths and flow directions at various stresses from any of the other models.

.. _models-barlat-theory-yield:

.. figure:: ../../_static/figures/barlatSurface.png
   :align: center
   :scale: 25 %

   Example Barlat yield surface, :math:`f_{Barlat}\left(\sigma_{ij},\bar{\varepsilon}^p=0\right)`, of 2090-T3 aluminum presented in the deviatoric :math:`\pi`-plane. Comparison von Mises (:math:`J_2`), Hosford (with :math:`a=8`), and Hill surfaces are also presented for comparison.

The orientation of the principal material axes with respect to the global Cartesian axes may be specified by the user.
First, a rectangular or cylindrical reference coordinate system is defined.
*Spherical coordinate systems are not currently implemented for the Barlat model.*
The material coordinate system can then be defined through 
two successive rotations about axes in the reference rectangular or cylindrical coordinate system.
In the case of the cylindrical coordinate system this allows the principal material axes to vary point-wise in a given element block.

The plastic rate of deformation, as with the isotropic models, assumes associated flow

.. math::

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

in which :math:`\dot{\gamma}` is the consistency multiplier.  Given the form for :math:`\phi`, :math:`\dot\gamma` is equal to the rate of the equivalent plastic strain, :math:`\dot{\bar{\varepsilon}}^{p}`.  As the yield surface is cast in transformed stress space, determining the flow direction in Cartesian space may be done via the chain rule (details may be found in [:footcite:`mat:ref:wmsch4`]) leading to an expression of the form,

.. math::
   :label: eqn:bp-normal

   \frac{\partial\phi}{\partial\sigma_{ij}}=\sum_{k=1}^3\left(\frac{\partial\phi}{\partial s^{\prime}_k}
               \frac{\partial s^{\prime}_k}{\partial s^{\prime}_{mn}}L^{\prime}_{mnij}+\frac{\partial\phi}
               {\partial{s^{\prime\prime}_k}}\frac{\partial s^{\prime\prime}_k}{\partial s^{\prime\prime}_{mn}}
               L^{\prime\prime}_{mnij}\right).

For more information about the Barlat plasticity model, consult [:footcite:`mat:ref:barlat:05`, :footcite:`mat:ref:wmsch4`].  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

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

Like the Hill and Hosford models, the Barlat plasticity model uses a elastic predictor-inelastic corrector 
closest point projection (CPP) return mapping algorithm (RMA) for integration.  Details of the numerical scheme and forms of the 
necessary derivatives may be found in the work of Scherzinger [:footcite:`mat:ref:wmsch4`].  For this approach, 
given a rate of deformation, :math:`d_{ij}`, and a time step, :math:`\Delta t`, a trial 
stress state is calculated based on an elastic response

.. math::

   T_{ij}^{tr} = T_{ij}^{n} + \Delta t \, \mathbb{C}_{ijkl}d_{kl}.

If the trial stress state lies outside the yield surface, i.e. if :math:`\phi(T_{ij}^{tr}) > \bar{\sigma}`, then the model uses an implicit, backward 
Euler algorithm to return the stress to the yield surface.  To perform this task, two nonlinear equations need to be solved.
The first is associated with the satisfaction of the flow-rule and ensures that the plastic strain increment is in the correct direction.  
Such a relation leads to a residual of the form,

.. math::
   :label: eq:bp-imp:resid

   R_{ij} = \Delta d_{ij}^{\text{p}} - \Delta\gamma \frac{\partial\phi}{\partial T_{ij}} = 0.

while the second equation to be addressed enforces that the converged stress state is on the yield surface and is written as,

.. math::
   :label: eq:bp-imp:con

   f = \phi\left( T_{ij} \right) - \bar{\sigma}\left( \bar{\varepsilon}^{p} \right) = 0.

 
The primary method for solving these equations is a Newton-Raphson algorithm.
With :math:`\Delta \gamma` (which is equal to :math:`\Delta\bar{\varepsilon}^{p}`) and :math:`T_{ij}` being the solution variables, an iterative algorithm is utilized 
such that

.. math::

   \Delta \gamma^{(k+1)} & = \Delta \gamma^{(k)} + \Delta\left(\Delta\gamma\right) \nonumber \\
   \\
   T_{ij}^{(k+1)} & = T_{ij}^{(k)} + \Delta T_{ij}, \nonumber 

with :math:`\Delta\gamma^{(0)} = 0` and :math:`T_{ij}^{(0)} = T_{ij}^{tr}`.  The plastic rate of deformation correction is then simply

.. math::

   \Delta d_{ij}^{\text{p}} = \mathbb{C}^{-1}_{ijkl} \left( T_{kl}^{tr} - T_{kl} \right).

After linearizing the residual and consistency equations (Equations :eq:`eq:bp-imp:resid` and :eq:`eq:bp-imp:con`), the set of 
nonlinear equations may be solved for the correction increments leading to expressions of the form,

.. math::

   \Delta\left( \Delta \gamma \right) & = \frac{\displaystyle f^{(k)} - R_{ij}^{(k)} \, \mathcal{L}^{(k)}_{ijkl} \, \frac{\partial\phi^{(k)}}{\partial T_{kl}}}
   {\displaystyle \frac{\partial\phi^{(k)}}{\partial T_{ij}} \, \mathcal{L}^{(k)}_{ijkl} \, \frac{\partial\phi^{(k)}}{\partial T_{kl}} + H^{\prime \, (k)}} \nonumber \\
   \\
   \Delta T_{ij} & = - \mathcal{L}^{(k)}_{ijkl} \, \left( R_{kl}^{(k)} + \Delta\left(\Delta\gamma\right)\frac{\partial\phi^{(k)}}{\partial T_{kl}}\right),

and :math:`\mathcal{L}^{\left(k\right)}_{ijkl}` is the Hessian of the RMA problem (*not* the yield surface) and is given as,

.. math::

   \mathcal{L}^{\left(k\right)}_{ijkl}=\left(\mathbb{S}_{ijkl}+\Delta\gamma^{\left(k\right)}\frac{\partial^2\phi^{\left(k\right)}}{\partial\sigma_{ij}\partial\sigma_{kl}}\right)^{-1},

and :math:`\mathbb{S}_{ijkl}=\mathbb{C}_{ijkl}^{-1}`.

Unfortunately, a straightforward Newton-Raphson algorithm does not always converge, so the RMA is augmented with a line search algorithm 
producing modified incrementation relations with

.. math::

   \Delta \gamma^{(k+1)} & = \Delta \gamma^{(k)} + \alpha \Delta\left(\Delta\gamma\right), \nonumber \\
   \\
   T_{ij}^{(k+1)} & = T_{ij}^{(k)} + \alpha \Delta T_{ij}, \nonumber 

where :math:`\alpha \in (0,1]` is the line search parameter which is determined from certain convergence considerations.
If :math:`\alpha = 1` then the Newton-Raphson algorithm is recovered.
The line search algorithm greatly increases the reliability of the return mapping algorithm.

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

To verify the Barlat plasticity model a similar approach to that used for the Hill plasticity model (:numref:`models-hill-verification`) 
is utilized.  

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.

Specifically, both uniaxial stress and pure shear loadings are considered.  To this end, 
the response of a 2090-T3 aluminum [:footcite:`mat:ref:barlat:05`] with Voce hardening of the form,

.. math::

   \bar{\sigma}\left(\bar{\varepsilon}^p\right)=\sigma_y+A\left(1-\exp\left(-b\bar{\varepsilon}^p\right)\right),

is used. The corresponding elastic, plastic, and anisotropy model parameters are given in :numref:`tab-bp-verProps`.

.. _tab-bp-verProps:

.. csv-table:: The material and model parameters for the Barlat plasticity model used for verification testing. The anisotropy coefficients correspond to 2090-T3 aluminum.
   :align: center
   :delim: &

   :math:`E` & 70 GPa & :math:`\nu` & 0.25
   :math:`a` & 8 & :math:`\sigma_{y}` & 200 MPa
   :math:`A` & 200 MPa & :math:`b` & 20
   :math:`c^{\prime}_{12}` &  -0.069888 & :math:`c^{\prime\prime}_{12}` & 0.981171
   :math:`c^{\prime}_{13}` & 0.936408 & :math:`c^{\prime\prime}_{13}` & 0.476741
   :math:`c^{\prime}_{21}` & 0.079143 & :math:`c^{\prime\prime}_{21}` & 0.575316
   :math:`c^{\prime}_{23}` & 1.003060 & :math:`c^{\prime\prime}_{23}` & 0.866827
   :math:`c^{\prime}_{31}` & 0.524741 & :math:`c^{\prime\prime}_{31}` & 1.145010
   :math:`c^{\prime}_{32}` & 1.363180 & :math:`c^{\prime\prime}_{32}` & -0.079294
   :math:`c^{\prime}_{44}` & 1.023770 & :math:`c^{\prime\prime}_{44}` & 1.051660
   :math:`c^{\prime}_{55}` & 1.069060 & :math:`c^{\prime\prime}_{55}` & 1.147100
   :math:`c^{\prime}_{66}` & 0.954322 & :math:`c^{\prime\prime}_{66}` & 1.404620

Finally, the coordinate system used in these calculations is a rectangular coordinate system with 
the :math:`e^{1}_i,e^{2}_i,e^{3}_i` axes aligned with the :math:`x,y,z` axes.

Uniaxial Stress
---------------

First, the response of the material subject to a uniaxial stress is considered.  As such, 
the Cauchy stress tensor takes the form :math:`\sigma_{ij}=\sigma\delta_{i1}\delta_{j1}`.  In the transformed stress space, 
this uniaxial tensor becomes,

.. math::
   :label: eqn:bp-transStress

   s_{ij}^{\prime} = \frac{1}{3}\sigma\left[\begin{array}{ccc} c^{\prime}_{12}+c^{\prime}_{13} & 0 & 0 \\
                            0 & -2c^{\prime}_{21}+c^{\prime}_{23} & 0 \\
                            0 & 0 & -2c^{\prime}_{31}+c^{\prime}_{32} \end{array}\right] \nonumber \\
   \\
   s_{ij}^{\prime\prime} = \frac{1}{3}\sigma\left[\begin{array}{ccc} c^{\prime\prime}_{12}+c^{\prime\prime}_{13} & 0 & 0 \\
                            0 & -2c^{\prime\prime}_{21}+c^{\prime\prime}_{23} & 0 \\
                            0 & 0 & -2c^{\prime\prime}_{31}+c^{\prime\prime}_{32} \end{array}\right] . \nonumber

It is noted from :eq:`eqn:bp-transStress` the that two transformed stress tensors are purely diagonal and therefore in a principal 
state.  The actual ordering of the components into the corresponding principal stresses depends on the anisotropy coefficients.  
By inspection of :numref:`tab-bp-verProps` it is clear in this instance that tensors are already ordered 
(:math:`s^{\prime}_1=s^{\prime}_{11},~s^{\prime\prime}_1=s^{\prime\prime}_{11}` *etc*.).  With this observation, 
the effective stress may be reduced to,

.. math::

   \phi\left(\sigma_{ij}\right)=\omega|\sigma|,

where :math:`\omega` is a constant dependent on model parameters and is written as,

.. math::

   \omega = & \frac{1}{3}\bigg\{\frac{1}{4}\Big[
       |c_{12}^{\prime}+c_{13}^{\prime}-c_{12}^{\prime\prime}-c_{13}^{\prime\prime}|^a +
       |c_{12}^{\prime}+c_{13}^{\prime}+2c_{21}^{\prime\prime}-c_{23}^{\prime\prime}|^a +
       |c_{12}^{\prime}+c_{13}^{\prime}+2c_{31}^{\prime\prime}-c_{32}^{\prime\prime}|^a 
   \nonumber \\
   & +
       |c_{23}^{\prime}-2c_{21}^{\prime}-c_{12}^{\prime\prime}-c_{13}^{\prime\prime}|^a +
       |c_{23}^{\prime}-2c_{21}^{\prime}+2c_{21}^{\prime\prime}-c_{23}^{\prime\prime}|^a +
       |c_{23}^{\prime}-2c_{21}^{\prime}+2c_{31}^{\prime\prime}-c_{32}^{\prime\prime}|^a  
   \\
   & +
       |c_{32}^{\prime}-2c_{31}^{\prime}-c_{12}^{\prime\prime}-c_{13}^{\prime\prime}|^a +
       |c_{32}^{\prime}-2c_{31}^{\prime}+2c_{21}^{\prime\prime}-c_{23}^{\prime\prime}|^a +
       |c_{32}^{\prime}-2c_{31}^{\prime}+2c_{31}^{\prime\prime}-c_{32}^{\prime\prime}|^a 
   \Big]\bigg\}^{1/a}. \nonumber

Axial Stresses
^^^^^^^^^^^^^^

To determine the axial stress, it is first noted that during plastic deformation,

.. math::

    \phi\left(\sigma_{ij}\right)=\omega\sigma=\bar{\sigma}\left(\bar{\varepsilon}^p\right),

where the fact that a tensile loading will be investigated (:math:`\sigma>0`) is leveraged.  The stress is then simply,

.. math::
   :label: eqn:bp-verStress

   \sigma=\frac{\bar{\sigma}\left(\bar{\varepsilon}^p\right)}{\omega}.

This shows that during plastic deformation the stress state can be calculated from the hardening law and anisotropy parameters.

To evaluate the axial stress, a relationship between the equivalent plastic strain and axial strain is needed.  
By noting the uniaxial stress state and equating the rate of plastic work, it is evident that,

.. math::

   \bar{\sigma} \dot{\bar{\varepsilon}}^{p} = \sigma \left( \dot\varepsilon - \dot\varepsilon^{\text{e}} \right)
   \;\;\; \rightarrow \;\;\;
   \dot{\bar{\varepsilon}}^{p} = \frac{1}{\omega}\left( \dot\varepsilon - \dot\varepsilon^{\text{e}} \right)

which, when integrated, gives an implicit equation for the equivalent plastic strain that is written as

.. math::
   :label: eqn:bp-eqps

   \bar{\varepsilon}^{p} = \frac{1}{\omega} \left( \varepsilon - \frac{\bar{\sigma}(\bar{\varepsilon}^{p})}{\omega E} \right).

The equivalent plastic strain can then be used in :eq:`eqn:bp-verStress` to find the axial stress, :math:`\sigma`.  Corresponding 
stress-strain results determined analytically in this fashion and numerically via Adagio are presented below in 
:numref:`fig-bp-ver-uniStressStrain`.

.. _fig-bp-ver-uniStressStrain:

.. figure:: ../../_static/figures/uniStressStrainBarlat.png
   :align: center
   :scale: 25 %

   Axial stress-strain response determined analytically and numerically for 2090-T3 aluminum using the Barlat plasticity model with Voce hardening.

Lateral Strains
^^^^^^^^^^^^^^^

To determine the plastic strain, the derivatives of the yield surface with respect to the Cauchy stress (:math:`\partial\phi/\partial\sigma_{ij}`) 
are needed.  From :eq:`eqn:bp-normal` it can be seen that these relations are quite complex and the reader is referred to 
[:footcite:`mat:ref:wmsch4`] for a detailed discussion of how to rigorously evaluate these derivatives under arbitrary conditions.  
In this effort, the fact that the principal directions of the transformed stresses (:math:`\hat{e}^{k\prime}_i` and :math:`\hat{e}^{k\prime\prime}_i`) 
are aligned with the global coordinate system (:math:`\hat{e}^{1\prime}_i=e^1_i` *etc*.) simplifies the problem 
sufficiently to allow for an analytical treatments.  In this case,

.. math::

   \frac{\partial s_k^{\prime}}{\partial s_{ij}^{\prime}}=e^k_ie^k_j.

With this observation, the lateral flow directions may be written as,

.. math::
   :label: eqn:bp-ver-22dir

   \frac{\partial\phi}{\partial\sigma_{22}} = & \frac{1}{3}\Big[
                                   \frac{\partial\phi}{\partial s^{\prime}_1}\left(c^{\prime}_{13}-2c^{\prime}_{12}\right)
                                  + \frac{\partial\phi}{\partial s^{\prime}_2}\left(c^{\prime}_{21}+c^{\prime}_{23}\right)
                                  + \frac{\partial\phi}{\partial s^{\prime}_3}\left(c^{\prime}_{31}-2c^{\prime}_{32}\right)
                                \nonumber \\
                                 &  + \frac{\partial\phi}{\partial s^{\prime\prime}_1}\left(c^{\prime\prime}_{13}-2c^{\prime\prime}_{12}\right)
                                  + \frac{\partial\phi}{\partial s^{\prime\prime}_2}\left(c^{\prime\prime}_{21}+c^{\prime\prime}_{23}\right)
                                  + \frac{\partial\phi}{\partial s^{\prime\prime}_3}\left(c^{\prime\prime}_{31}-2c^{\prime\prime}_{32}\right)
                            \Big] 

.. math::
   :label: eqn:bp-ver-33dir

   \frac{\partial\phi}{\partial\sigma_{33}} = & \frac{1}{3}\Big[
                                   \frac{\partial\phi}{\partial s^{\prime}_1}\left(c^{\prime}_{12}-2c^{\prime}_{13}\right)
                                  + \frac{\partial\phi}{\partial s^{\prime}_2}\left(c^{\prime}_{21}-2c^{\prime}_{23}\right)
                                  + \frac{\partial\phi}{\partial s^{\prime}_3}\left(c^{\prime}_{31}+c^{\prime}_{32}\right)
                                \nonumber \\
                                 &  + \frac{\partial\phi}{\partial s^{\prime\prime}_1}\left(c^{\prime\prime}_{12}-2c^{\prime\prime}_{13}\right)
                                  + \frac{\partial\phi}{\partial s^{\prime\prime}_2}\left(c^{\prime\prime}_{21}-2c^{\prime\prime}_{23}\right)
                                  + \frac{\partial\phi}{\partial s^{\prime\prime}_3}\left(c^{\prime\prime}_{31}+c^{\prime\prime}_{32}\right)
                            \Big]  ,

in which the various :math:`\partial\phi/\partial s^{\prime}_i` derivatives are functions of the anisotropy coefficients and explicit forms 
may be found in [:footcite:`mat:ref:wmsch4`].

The total strain is written simply as,

.. math::

   \varepsilon_{ij}=\varepsilon_{ij}^{\text{e}}+\varepsilon_{ij}^{\text{p}},

with the elastic strain being

.. math::

   \varepsilon_{22}^{\text{e}}=\varepsilon_{33}^{\text{e}}=-\nu\frac{\sigma}{E},

and the plastic strains found via the flow rules as,

.. math::

   \varepsilon_{22}^{\text{p}}=\bar{\varepsilon}^p\frac{\partial\phi}{\partial\sigma_{22}} \
   \qquad ; \qquad
   \varepsilon_{33}^{\text{p}}=\bar{\varepsilon}^p\frac{\partial\phi}{\partial\sigma_{33}}.

The flow directions were given previously in :eq:`eqn:bp-ver-22dir` and :eq:`eqn:bp-ver-33dir` while the equivalent 
plastic strain may be found via :eq:`eqn:bp-eqps`.  :numref:`fig-bp-ver-uniLateralStrain` presents the lateral strains 
as a function of the axial.  Clear agreement may be observed both in :numref:`fig-bp-ver-uniStressStrain` 
and :numref:`fig-bp-ver-uniLateralStrain` verifying the model.  Additionally, the effect of the anisotropy is plainly evident 
in :numref:`fig-bp-ver-uniLateralStrain` in which the two lateral strains differ by approximately a factor of four. 

.. _fig-bp-ver-uniLateralStrain:

.. figure:: ../../_static/figures/uniLateralStrain.png
   :align: center
   :scale: 25 %

   Lateral strain as a function of axial strain of 2090-T3 aluminum with Voce hardening as determined by the Barlat plasticity model both analytically and numerically.

To test the other directions and further examine the anisotropic character of the model, the coordinate system rotation input options are 
used to align the 2 and 3 directions of the material with the applied load.  Analytical expressions may be determined 
by similarly rotating the coefficients in the previous expressions, although these are not repeated here for brevity.  The 
corresponding results for the loading aligned with the 2 and 3 directions are presented in :numref:`fig-bp-verUniYDir` 
and :numref:`fig-bp-verUniZDir`, respectively.  All of the results are given with respect to the original coordinate system to avoid 
confusion.  Clear agreement between analytical and simulation results is noted in both cases further verifying the capabilities of the model.  
Importantly, by comparing the various stress-strain and lateral strain curves, the influence of the material and model anisotropy 
on the responses may readily be observed. 

.. _fig-bp-verUniYDir:

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

   .. image:: ../../_static/figures/uniStressStrain_YY.png
      :alt: Stress-strain
      :scale: 35 %

   .. image:: ../../_static/figures/uniLatStrain_YY.png
      :alt: Lateral strains
      :scale: 35 %

   Stress-strain (a) and lateral strain (b) responses of 2090-T3 aluminum with Voce hardening and the Barlat plasticity model. The material is rotated such that the loading is aligned with the 2 direction.

.. _fig-bp-verUniZDir:

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

   .. image:: ../../_static/figures/uniStressStrain_ZZ.png
      :alt: Stress-strain
      :scale: 35 %

   .. image:: ../../_static/figures/uniLatStrain_ZZ.png
      :alt: Lateral strains
      :scale: 35 %

   Stress-strain (a) and lateral strain (b) responses of 2090-T3 aluminum with Voce hardening and the Barlat plasticity model. The material is rotated such that the loading is aligned with the 3 direction.

Pure Shear
----------

In this section, the pure shear response of the Barlat model is interrogated to assess its performance under such conditions.
Before proceeding, it is important to recall the ordering of the shear stresses in Sierra/SM.  Specifically, the :math:`\sigma_{12},\ \sigma_{23},` and :math:`\sigma_{31}` stresses are associated with the 44, 55, and 66, respectively, anisotropy coefficients.

To explore the shear performance of the Barlat plasticity model, a stress tensor of the form 
:math:`\sigma_{ij}=\tau\left(\delta_{i1}\delta_{j2}+\delta_{i2}\delta_{j1}\right)` is considered.  The ordered principal stresses 
of the transformed stress tensors are,

.. math::
   :label: eqn:bp-ver-1shear

   s^{\prime}_i=\left[\begin{array}{c} c_{44}^{\prime}\tau \\ 0 \\ -c_{44}^{\prime}\tau\end{array}\right]
   \qquad ; \qquad
   s^{\prime\prime}_i=\left[\begin{array}{c} c_{44}^{\prime\prime}\tau \\ 0 \\ -c_{44}^{\prime\prime}\tau\end{array}\right] ,

thereby simplifying the effective stress to,

.. math::

   \phi\left(\sigma_{ij}\right)=\tau\zeta,

with

.. math::

   \zeta=\bigg\{\frac{1}{2}\Big[|c^{\prime}_{44}-c^{\prime\prime}_{44}|^a+|c^{\prime}_{44}+c^{\prime\prime}_{44}|^a+|c^{\prime}_{44}|^a+|c^{\prime\prime}_{44}|^a \Big]\bigg\}^{1/a}.

During plastic flow, 

.. math::

   \phi=\tau\zeta=\bar{\sigma}\left(\bar{\varepsilon}^p\right),

producing an expression for the stress in terms of equivalent plastic strain as,

.. math::

   \tau = \frac{1}{\zeta}\bar{\sigma}\left(\bar{\varepsilon}^p\right).

A relationship between the equivalent plastic and axial strains may be determined by first considering the equivalency 
of plastic work,

.. math::

   \bar{\sigma}\dot{\bar{\varepsilon}}^p=2\tau\left(\dot{\varepsilon}_{12}-\dot{\varepsilon}_{12}^{\text{e}}\right) \qquad\rightarrow\qquad
   \dot{\bar{\varepsilon}}^p=\frac{2}{\zeta}\left(\dot{\varepsilon}_{12}-\dot{\varepsilon}^{\text{e}}\right).

Integrating leads to an implicit expression of the form,

.. math::
   :label: eqn:bp-ver-FinalShear

   \bar{\varepsilon}^p=\frac{2}{\zeta}\left(\varepsilon_{12}-\frac{\bar{\sigma}\left(\bar{\varepsilon}^p\right)}{\zeta G}\right).

The preceding relations may be used to analytically determine the shear stress-strain response.  Corresponding results, along with 
those produced by Adagio, are presented in :numref:`fig-bp-ver-pureShear`.  Shear responses are also presented for stress tensors 
of the form :math:`\sigma_{ij}=\tau\left(\delta_{2i}\delta_{3j}+\delta_{3i}\delta_{2j}\right)` (23) and 
:math:`\sigma_{ij}=\tau\left(\delta_{1i}\delta_{3j}+\delta_{3i}\delta_{1j}\right)` (31).  Analytically, these 
results were determined by substituting the relevant anisotropy coefficients in :eq:`eqn:bp-ver-1shear`-:eq:`eqn:bp-ver-FinalShear`.  
For the results from Adagio, the coordinate system input commands were used to rotate the material coordinate system accordingly.  

In all the cases presented in :numref:`fig-bp-ver-pureShear` excellent agreement is noted.  This not only verifies the 
performance of the current model under pure shear loadings but also demonstrates the impact of the anisotropy and 
exercises the coordinate system rotation capabilities.

.. _fig-bp-ver-pureShear:

.. figure:: ../../_static/figures/pureShearFig.png
   :align: center
   :scale: 25 %

   Shear stress-strain results for 2090-T3 aluminum determined analytically and numerically by the Barlat plasticity model with Voce Hardening

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

To verify the capabilities of the hardening models, rate *independent* and rate *dependent* alike, the constant equivalent plastic strain rate, 
:math:`\dot{\bar{\varepsilon}}^p`, uniaxial stress and pure shear verification tests described in :ref:`Appendix A <appendix-common-constant-eqps-rate>` are utilized.  In these 
simplified loading cases, the material state may be found explicitly as a function of time knowing the prescribed equivalent strain rate.  
For the rate *independent* cases, a strain rate of :math:`\dot{\bar{\varepsilon}}^p=1\times10^{-4} \text{s}^{-1}` is used for ease in simulations 
although the selected rate does note affect the results.  Through this testing protocol, the hardening models are not only tested at different rates but also 
in different principal material directions to consider the anisotropy of the Barlat yield surface.  Additionally, the rate *dependent* models are tested 
for a wide range of strain rates (over five decades) with all three rate *independent* hardening functions (:math:`\tilde{\sigma}_y` in the previous 
theory section).  Although linear, Voce, and power-law rate *independent* representations are utilized in the rate *dependent* tests, in those 
cases the hardening models are prescribed via user-defined analytic functions.  The rate *independent* verification exercises, on the other hand, 
examine the built-in hardening models.  This distinction necessitates the different considerations and treatments.

The rate *dependent* and rate *independent* hardening coefficients are found in :numref:`tab-bp-rateDepVerProps` while the remaining 
model parameters are unchanged from the previous verification exercises.  For the current verification exercise, the rate *independent* hardening 
models (linear, Voce, and power-law) will first be considered and then the rate *dependent* forms (Johnson-Cook, power-law breakdown).

.. _tab-bp-rateDepVerProps:

.. csv-table:: The model parameters for the hardening verification tests used with the Barlat 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

Linear
^^^^^^

For the rate *independent* linear hardening model, verification is considered via the uniaxial stress and pure shear exercises of :ref:`Appendix A <appendix-common-constant-eqps-rate>`.  
As the anisotropic Barlat yield surface is being used for this examination, the uniaxial stress response is determined for loading in three different 
principal material planes while the pure shear response is found along three shear planes.  Results determined analytically and numerically are 
presented in :numref:`fig-barlat-verLinHard`.  Clear agreement is evident between the dual solution approaches.  Additionally, the linear response and constant 
tangent modulus during plastic deformation highlights the characteristic feature of the current model. 

.. _fig-barlat-verLinHard:

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

   .. image:: ../../_static/figures/barlat/barlat_normal_linear.png
      :alt: Stress-strain
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/barlat_shear_linear.png
      :alt: Lateral strains
      :scale: 35 %

   Uniaxial stress-strain (a) and pure shear (b) responses of the Barlat plasticity model with rate *independent*, linear hardening. Solid lines are analytical while open symbols are numerical

Power-Law
^^^^^^^^^

To probe the power-law rate *independent* hardening model, analytical and numerical results to the uniaxial stress and pure shear problems of 
:ref:`Appendix A <appendix-common-constant-eqps-rate>` are determined.  Given the anisotropic nature of the current model, responses are determined along the three principal 
and three shearing planes for the uniaxial stress and pure shear cases and all six cases are shown in :numref:`fig-barlat-verPLHard`.  In considering 
:numref:`fig-barlat-verPLHard`, it is apparent that the numerical and analytical responses agree quite well verifying this specific response.  
These cases also highlight the initially stiff plastic response that eventually evolves into a more compliant linear like response that 
is associated with a power-law hardening model.

.. _fig-barlat-verPLHard:

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

   .. image:: ../../_static/figures/barlat/barlat_normal_power_law.png
      :alt: Stress-strain
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/barlat_shear_power_law.png
      :alt: Lateral strains
      :scale: 35 %

   Uniaxial stress-strain (a) and pure shear (b) responses of the Barlat plasticity model with rate *independent*, power-law hardening. Solid lines are analytical while open symbols are numerical.

Voce
^^^^

Verifying the Voce model is addressed through the methods of :ref:`Appendix A <appendix-common-constant-eqps-rate>`.  To this end, analytical and numerical 
uniaxial stress and pure shear responses are determined along three different principal directions and shear planes, respectively.  
The results for these various cases are presented in :numref:`fig-barlat-verVoceHard` and unambiguous agreement is readily seen 
between the analytical and numerical results providing further credence to hardening model capabilities.  Responses in :numref:`fig-barlat-verVoceHard` 
also exhibit the clear saturation of hardening with sufficient plastic strain that is usually associated with the Voce model. 

.. _fig-barlat-verVoceHard:

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

   .. image:: ../../_static/figures/barlat/barlat_normal_voce.png
      :alt: Stress-strain
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/barlat_shear_voce.png
      :alt: Lateral strains
      :scale: 35 %

   Uniaxial stress-strain (a) and pure shear (b) responses of the Barlat plasticity model with rate *independent*, Voce hardening. Solid lines are analytical while open symbols are numerical.

Johnson-Cook
^^^^^^^^^^^^

To investigate the uniaxial response of the Johnson-Cook rate *dependent* hardening model, the problem discussed in :ref:`Appendix A <appendix-common-uniaxial-stress>` is 
considered.  In this analysis, the response depends only on time and the various :math:`c^{\prime}_i` and :math:`c^{\prime\prime}_i` Barlat yield surface coefficients.  
For a full-spectrum verification, forty-five different cases are evaluated using three different material principal directions (:math:`\hat{e}_1,~\hat{e}_2,` and :math:`\hat{e}_3`), 
five different rates (:math:`\dot{\bar{\varepsilon}}^p=~1\times10^{-3},~1\times10^{-2},~1\times10^{-1},~1\times10^{0}` and :math:`1\times10^{1}~\text{s}^{-1}`), and 
three different rate *independent* hardening models (linear, Voce, and power-law).  All forty-five analytical and numerical results are presented in 
:numref:`fig-barlat-verJCUni0` and :numref:`fig-barlat-verJCUni` and quite notable agreement is observed in each instance.

.. _fig-barlat-verJCUni0:

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

   .. image:: ../../_static/figures/barlat/normal_jc_lin_xx.png
      :alt: Linear Hardening -- 11
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/normal_jc_lin_yy.png
      :alt: Linear Hardening -- 22
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/normal_jc_lin_zz.png
      :alt: Linear Hardening -- 33
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/normal_jc_pl_xx.png
      :alt: Power-Law Hardening -- 11
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/normal_jc_pl_yy.png
      :alt: Power-Law Hardening -- 22
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/normal_jc_pl_zz.png
      :alt: Power-Law Hardening -- 33
      :scale: 35 %

   Uniaxial stress-strain response of the Barlat plasticity model (:math:`a=8`) with rate *dependent*, Johnson-Cook type hardening with (a-c) linear and (d-f) power-law rate *independent* hardening. Solid lines are analytical results while open symbols are numerical.

.. _fig-barlat-verJCUni:

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

   .. image:: ../../_static/figures/barlat/normal_jc_voce_xx.png
      :alt: Voce Hardening -- 11
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/normal_jc_voce_yy.png
      :alt: Voce Hardening -- 22
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/normal_jc_voce_zz.png
      :alt: Voce Hardening -- 33
      :scale: 35 %

   Uniaxial stress-strain response of the Barlat plasticity model (:math:`a=8`) with rate *dependent*, Johnson-Cook type hardening with (a-c) Voce rate *independent* hardening. Solid lines are analytical results while open symbols are numerical.

For the pure shear case, the forty-five different permutations are again explored.  The same five rates and three hardening models are used 
although three different shearing planes are used instead of the three principal directions.  The solution of the pure shear problem is 
described in :ref:`Appendix A <appendix-common-pure-shear>` and the analytical and numerical results are presented in :numref:`fig-barlat-verJCShear0` and :numref:`fig-barlat-verJCShear`. As with the uniaxial stress response excellent correspondence is noted between the two sets of results.

.. _fig-barlat-verJCShear0:

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

   .. image:: ../../_static/figures/barlat/shear_jc_lin_xy.png
      :alt: Linear Hardening -- 11
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/shear_jc_lin_yz.png
      :alt: Linear Hardening -- 22
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/shear_jc_lin_zx.png
      :alt: Linear Hardening -- 33
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/shear_jc_pl_xy.png
      :alt: Power-Law Hardening -- 11
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/shear_jc_pl_yz.png
      :alt: Power-Law Hardening -- 22
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/shear_jc_pl_zx.png
      :alt: Power-Law Hardening -- 33
      :scale: 35 %

   Stress-strain response of the Barlat plasticity model (:math:`a=8`) with rate *dependent*, Johnson-Cook type hardening in pure shear with (a-c) linear and (d-f) power-law rate *independent* hardening.  Solid lines are analytical results while open symbols are numerical.

.. _fig-barlat-verJCShear:

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

   .. image:: ../../_static/figures/barlat/shear_jc_voce_xy.png
      :alt: Voce Hardening -- 11
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/shear_jc_voce_yz.png
      :alt: Voce Hardening -- 22
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/shear_jc_voce_zx.png
      :alt: Voce Hardening -- 33
      :scale: 35 %

   Stress-strain response of the Barlat plasticity model (:math:`a=8`) with rate *dependent*, Johnson-Cook type hardening in pure shear with (a-c) Voce rate *independent* hardening.  Solid lines are analytical results while open symbols are numerical.

Power-Law Breakdown
^^^^^^^^^^^^^^^^^^^

In the case of the power-law Breakdown model, verification is again pursued through the problem of :ref:`Appendix A <appendix-common-uniaxial-stress>`
and using the same forty-five cases discussed with the Johnson-Cook model.  Corresponding results are given in :numref:`fig-barlat-verPLBUni0` and :numref:`fig-barlat-verPLBUni` and as with the preceding results substantial convergence is noted between the analytical and numerical results giving further credence to 
the hardening models.

.. _fig-barlat-verPLBUni0:

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

   .. image:: ../../_static/figures/barlat/normal_plb_lin_xx.png
      :alt: Linear Hardening -- 11
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/normal_plb_lin_yy.png
      :alt: Linear Hardening -- 22
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/normal_plb_lin_zz.png
      :alt: Linear Hardening -- 33
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/normal_plb_pl_xx.png
      :alt: Power-Law Hardening -- 11
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/normal_plb_pl_yy.png
      :alt: Power-Law Hardening -- 22
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/normal_plb_pl_zz.png
      :alt: Power-Law Hardening -- 33
      :scale: 35 %

   Uniaxial stress-strain response of the Barlat plasticity model (:math:`a=8`)  with rate *dependent*, power-law breakdown type hardening with (a-c) linear and (d-f) power-law rate *independent* hardening.  Solid lines are analytical while open symbols are numerical.

.. _fig-barlat-verPLBUni:

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

   .. image:: ../../_static/figures/barlat/normal_plb_voce_xx.png
      :alt: Voce Hardening -- 11
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/normal_plb_voce_yy.png
      :alt: Voce Hardening -- 22
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/normal_plb_voce_zz.png
      :alt: Voce Hardening -- 33
      :scale: 35 %

   Uniaxial stress-strain response of the Barlat plasticity model (:math:`a=8`)  with rate *dependent*, power-law breakdown type hardening with (a-c) Voce rate *independent* hardening.  Solid lines are analytical while open symbols are numerical.

As with the uniaxial stress case, the pure shear capabilities are interrogated through the procedure of :ref:`Appendix A <appendix-common-pure-shear>` using the 
same forty-five cases outlined in the Johnson-Cook discussion.  The analytical and numerical results are presented in 
:numref:`fig-barlat-verPLBShear0` and :numref:`fig-barlat-verPLBShear`.  Again, the two result sets align beautifully enabling further capability credibility.

.. _fig-barlat-verPLBShear0:

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

   .. image:: ../../_static/figures/barlat/shear_plb_lin_xy.png
      :alt: Linear Hardening -- 11
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/shear_plb_lin_yz.png
      :alt: Linear Hardening -- 22
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/shear_plb_lin_zx.png
      :alt: Linear Hardening -- 33
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/shear_plb_pl_xy.png
      :alt: Power-Law Hardening -- 11
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/shear_plb_pl_yz.png
      :alt: Power-Law Hardening -- 22
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/shear_plb_pl_zx.png
      :alt: Power-Law Hardening -- 33
      :scale: 35 %

   Stress-strain response of the Barlat plasticity model (:math:`a=8`) with rate *dependent*, power-law breakdown type hardening in pure shear with (a-c) linear and (d-f) power-law rate *independent* hardening.  Solid lines are analytical results while open symbols are numerical.

.. _fig-barlat-verPLBShear:

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

   .. image:: ../../_static/figures/barlat/shear_plb_voce_xy.png
      :alt: Voce Hardening -- 11
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/shear_plb_voce_yz.png
      :alt: Voce Hardening -- 22
      :scale: 35 %

   .. image:: ../../_static/figures/barlat/shear_plb_voce_zx.png
      :alt: Voce Hardening -- 33
      :scale: 35 %

   Stress-strain response of the Barlat plasticity model (:math:`a=8`) with rate *dependent*, power-law breakdown type hardening in pure shear with (a-c) Voce rate *independent* hardening.  Solid lines are analytical results while open symbols are numerical.

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

.. code-block:: sierrainput

   BEGIN PARAMETERS FOR MODEL BARLAT_PLASTICITY
     #
     # Elastic constants
     #
     YOUNGS MODULUS = <real>
     POISSONS RATIO = <real>
     SHEAR MODULUS  = <real>
     BULK MODULUS   = <real>
     LAMBDA         = <real>
     TWO MU         = <real>
     #
     # Material coordinates system definition
     #
     COORDINATE SYSTEM             = <string> coordinate_system_name
     DIRECTION FOR ROTATION        = <real> 1|2|3
     ALPHA                         = <real> (degrees)
     SECOND DIRECTION FOR ROTATION = <real> 1|2|3
     SECOND ALPHA                  = <real> (degrees)
     #
     # Yield surface parameters
     #
     YIELD STRESS = <real>
     A            = <real> (4.0)
     CP12         = <real> (1.0)
     CP13         = <real> (1.0)
     CP21         = <real> (1.0)
     CP23         = <real> (1.0)
     CP31         = <real> (1.0)
     CP32         = <real> (1.0)
     CP44         = <real> (1.0)
     CP55         = <real> (1.0)
     CP66         = <real> (1.0)
     CPP12        = <real> (1.0)
     CPP13        = <real> (1.0)
     CPP21        = <real> (1.0)
     CPP23        = <real> (1.0)
     CPP31        = <real> (1.0)
     CPP32        = <real> (1.0)
     CPP44        = <real> (1.0)
     CPP55        = <real> (1.0)
     CPP66        = <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 BARLAT_PLASTICITY]

In the command blocks that define the Barlat plasticity model:

.. - See :numref:`mat:elasticConstants` for more information on elastic constants input.

.. - See :numref:`mat:coordSys` for more information on material coordinates system definition commands.

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

- The exponent for the yield surface description, :math:`a`, is defined with the ``A`` command line.

- The transformation coefficient, :math:`c^{'}_{12}`, is defined with the ``CP12`` command line.  It is defaulted to 1.0.

- The transformation coefficient, :math:`c^{'}_{13}`, is defined with the ``CP13`` command line.  It is defaulted to 1.0.

- The transformation coefficient, :math:`c^{'}_{21}`, is defined with the ``CP21`` command line.  It is defaulted to 1.0.

- The transformation coefficient, :math:`c^{'}_{23}`, is defined with the ``CP23`` command line.  It is defaulted to 1.0.

- The transformation coefficient, :math:`c^{'}_{31}`, is defined with the ``CP31`` command line.  It is defaulted to 1.0.

- The transformation coefficient, :math:`c^{'}_{32}`, is defined with the ``CP32`` command line.  It is defaulted to 1.0.

- The transformation coefficient, :math:`c^{'}_{44}`, is defined with the ``CP44`` command line.  It is defaulted to 1.0.

- The transformation coefficient, :math:`c^{'}_{55}`, is defined with the ``CP55`` command line.  It is defaulted to 1.0.

- The transformation coefficient, :math:`c^{'}_{66}`, is defined with the ``CP66`` command line.  It is defaulted to 1.0.

- The transformation coefficient, :math:`c^{''}_{12}`, is defined with the ``CPP12`` command line.  It is defaulted to 1.0.

- The transformation coefficient, :math:`c^{''}_{13}`, is defined with the ``CPP13`` command line.  It is defaulted to 1.0.

- The transformation coefficient, :math:`c^{''}_{21}`, is defined with the ``CPP21`` command line.  It is defaulted to 1.0.

- The transformation coefficient, :math:`c^{''}_{23}`, is defined with the ``CPP23`` command line.  It is defaulted to 1.0.

- The transformation coefficient, :math:`c^{''}_{31}`, is defined with the ``CPP31`` command line.  It is defaulted to 1.0.

- The transformation coefficient, :math:`c^{''}_{32}`, is defined with the ``CPP32`` command line.  It is defaulted to 1.0.

- The transformation coefficient, :math:`c^{''}_{44}`, is defined with the ``CPP44`` command line.  It is defaulted to 1.0.

- The transformation coefficient, :math:`c^{''}_{55}`, is defined with the ``CPP55`` command line.  It is defaulted to 1.0.

- The transformation coefficient, :math:`c^{''}_{66}`, is defined with the ``CPP66`` command line.  It is defaulted to 1.0.

.. include:: hardening/user_output.rst

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

.. _out-tab-barlatstvar:

.. csv-table:: State Variables for BARLAT 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` 

.. raw::
   html

   <hr>

.. footbibliography::
