.. _models-hill:

*********************
Hill Plasticity Model
*********************

.. _models-hill-theory:

Theory
======

The Hill plasticity model is similar to other plasticity models except that it is not isotropic.
It is a hypoelastic, rate-independent plasticity model.
The rate form of the equation 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.

The Hill plasticity model has an orthotropic yield surface that assumes orthogonal principal material directions.  
An example of this yield surface is presented below in :numref:`fig-hill-theory-yield` along with examples of two 
isotropic surfaces -- the von Mises (:math:`J_2`) and Hosford (with :math:`a=8`).  The various surface parameters 
correspond to 2090-T3 aluminum and the specific Hill strengths are found in [:footcite:`mat:ref:wmsch4`].  By comparing 
the Hill surface to the two isotropic surfaces, the impact of the anisotropy is clear.  Additionally, substantial differences 
to the normals of the yield surfaces at points of intersection highlight the impact of the yield function selection 
on the resulting flow directions.

Like other plasticity models, the Hill yield surface, :math:`f`, is written,

.. math::

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

with :math:`\phi` being the effective stress and :math:`\bar{\sigma}` is the current yield stress that may be dependent on rate and/or temperature.  The Hill effective stress is essentially an orthotropic extension of the von Mises function. After accounting for plastic incompressibility and related constraints, there are six individual yield stresses: :math:`\sigma^{y}_{11}`, :math:`\sigma^{y}_{22}`, :math:`\sigma^{y}_{33}`, :math:`\tau^{y}_{12}`, :math:`\tau^{y}_{23}`, and :math:`\tau^{y}_{31}`. These yield stresses correspond to 3 normal and 3 shear yield stresses. Written in terms of the components, the effective stress has the form,

.. math::
   :label: mat:eq:hillyield

   \phi^{2}\left( \sigma_{ij} \right) & = F \left( \hat\sigma_{22} - \hat\sigma_{33} \right)^{2}
   + G \left( \hat\sigma_{33} - \hat\sigma_{11} \right)^{2}
   + H \left( \hat\sigma_{11} - \hat\sigma_{22} \right)^{2}
   \\
   & + 2L \hat\sigma_{23}^{2} + 2M \hat\sigma_{31}^{2} + 2N \hat\sigma_{12}^{2}.
   \nonumber

.. _fig-hill-theory-yield:

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

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

The coefficients :math:`F`, :math:`G`, :math:`H`, :math:`L`, :math:`M`, and :math:`N` were introduced by Hill.
In terms of the yield stresses they are:

.. math::

   F = \frac{\left(\bar{\sigma}\right)^{2}}{2}\left[
   \frac{1}{\left( \sigma_{22}^{y} \right)^{2}}
   + \frac{1}{\left( \sigma_{33}^{y} \right)^{2}}
   - \frac{1}{\left( \sigma_{11}^{y} \right)^{2}}\right]
   \;\;\; ; \;\;\;
   L = \frac{\left(\bar{\sigma}\right)^{2}}{2}
   \left[ \frac{1}{\left(\tau_{23}^{y}\right)^{2}} \right]

.. math::
   :label: mat:eq:hillconst

   G = \frac{\left(\bar{\sigma}\right)^{2}}{2}\left[
   \frac{1}{\left( \sigma_{33}^{y} \right)^{2}}
   + \frac{1}{\left( \sigma_{11}^{y} \right)^{2}}
   - \frac{1}{\left( \sigma_{22}^{y} \right)^{2}} \right]
   \;\;\; ; \;\;\;
   M = \frac{\left(\bar{\sigma}\right)^{2}}{2}
   \left[ \frac{1}{\left(\tau_{31}^{y}\right)^{2}} \right]

.. math::

   H = \frac{\left(\bar{\sigma}\right)^{2}}{2}\left[
   \frac{1}{\left( \sigma_{11}^{y} \right)^{2}}
   + \frac{1}{\left( \sigma_{22}^{y} \right)^{2}}
   - \frac{1}{\left( \sigma_{33}^{y} \right)^{2}} \right]
   \;\;\; ; \;\;\;
   N = \frac{\left(\bar{\sigma}\right)^{2}}{2}
   \left[ \frac{1}{\left(\tau_{12}^{y}\right)^{2}} \right].

where :math:`\bar{\sigma}` is a reference yield stress.

Rather than input the six independent yield stresses, the ratios of the yield stresses to some reference yield stress are generally used as input.
These ratios are

.. math::

   R_{11} = \frac{\sigma^{y}_{11}}{\bar{\sigma}}
   \;\;\; ; \;\;\;
   R_{12} = \sqrt{3}\frac{\tau^{y}_{12}}{\bar{\sigma}}

.. math::
   :label: mat:eq:hillratio

   R_{22} = \frac{\sigma^{y}_{22}}{\bar{\sigma}}
   \;\;\; ; \;\;\;
   R_{23} = \sqrt{3}\frac{\tau^{y}_{23}}{\bar{\sigma}}

.. math::

   R_{33} = \frac{\sigma^{y}_{33}}{\bar{\sigma}}
   \;\;\; ; \;\;\;
   R_{31} = \sqrt{3}\frac{\tau^{y}_{31}}{\bar{\sigma}}.

These ratios are set up so that if :math:`R_{ij} = 1` then the yield surface is isotropic.

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 Hill 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}}.

Given the form for :math:`\phi`, the consistency parameter, :math:`\dot\gamma` is equal to the rate of the equivalent plastic strain, :math:`\dot{\bar{\varepsilon}}^{p}`.

For more information about the Hill plasticity model, consult [:footcite:`mat:ref:hill1`].  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 Hill plasticity model uses a predictor-corrector algorithm for integrating the constitutive model.
Given a rate of deformation, \ifindex :math:`d_{ij}`\else :math:`{\bf d}`\fi, 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 \ifindex :math:`\phi(T_{ij}^{tr}) > \bar{\sigma}`\else :math:`\phi({\bf T}^{tr}) > \bar{\sigma}`\fi, then the model uses a backward Euler algorithm to return the stress to the yield surface.
There are two equations that need to be solved.
To ensure that the plastic strain increment is in the correct direction we have

.. math::

   R^{p}_{ij} = \Delta t \, d_{ij}^{p} - \Delta\gamma \frac{\partial\phi}{\partial T_{ij}} = 0

while to ensure that the stress state is on the yield surface we require

.. math::

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

The primary algorithm for solving these equations is a Newton-Raphson algorithm.
Using :math:`\Delta \gamma` (which is equal to :math:`\Delta\bar{\varepsilon}^{p}`) and \ifindex :math:`T_{ij}`\else :math:`{\bf T}`\fi as the solution variables, we set up an iterative algorithm where

.. 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 

where :math:`\Delta\gamma^{(0)} = 0` and \ifindex :math:`T_{ij}^{(0)} = T_{ij}^{tr}`\else :math:`{\bf T}^{(0)} = {\bf T}^{tr}`\fi and

.. math::

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

The Newton-Raphson algorithm gives

.. 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)
   \nonumber 

A straightforward Newton-Raphson algorithm does not always converge, so the return mapping algorithm is augmented with a line search algorithm

.. 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.

.. _models-hill-verification:

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

The Hill plasticity material model is verified for a number of loading conditions.

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.

The elastic properties used in these analyses are :math:`E = 70` GPa and :math:`\nu = 0.25`.
The parameters that are used to define the yield surface are

.. math::

   R_{11} = 1.000680 \;\;\; ; \;\;\; R_{12} = 0.909194
   \nonumber \\
   \nonumber \\
   R_{22} = 0.906397 \;\;\; ; \;\;\; R_{23} = 0.851434 \\
   \nonumber \\
   R_{33} = 1.027380 \;\;\; ; \;\;\; R_{31} = 0.799066

These parameters correspond to a parameterization of the Barlat model for 2090-T3 aluminum [:footcite:`mat:ref:barlat:05`] that is fit to the Hill model.
The hardening law used for the model is a Voce law with the following form

.. math::

   \bar{\sigma}\left(\bar{\varepsilon}^{p}\right) = \sigma_{y} + A \left( 1 - \exp(-n \bar{\varepsilon}^{p})\right)

For these calculations :math:`\sigma_{y} = 200` MPa, :math:`A = 200` MPa, and :math:`n = 20`.
Finally, the coordinate system used in these calculations is a rectangular coordinate system with the :math:`e_{1},e_{2},e_{3}` axes aligned with the :math:`x,y,z` axes.

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

The Hill plasticity model is tested in uniaxial tension along the three orthogonal principal material directions.
The tests looks at the stress, the strain, and the equivalent plastic strain and compares these values against analytical results for the same problem.
In this verification problem only the normal stresses are needed, and the shear terms are not exercised.
Therefore, the parameters :math:`R_{12}`, :math:`R_{23}`, and :math:`R_{31}` are not used in the problem and a separate verification test will be needed for shear response.

The model is tested in uniaxial stress in the :math:`x`, :math:`y`, and :math:`z` directions, giving three test problems.
Each problem can be formulated exactly the same.
For the description of the test we will only look at loading in the :math:`x` direction (:math:`x_{1}` direction).

For the uniaxial stress problem, the only non-zero stress component is :math:`\sigma_{11}`.
In the analysis that follows :math:`\sigma_{11} = \sigma`.
There are three non-zero strain components, :math:`\varepsilon_{11}`, :math:`\varepsilon_{22}`, and :math:`\varepsilon_{33}`.
In the analysis that follows :math:`\varepsilon_{11} = \varepsilon`.
Furthermore, the axial elastic strain, :math:`\varepsilon_{11}^{\text{e}} = \sigma/E` will be denoted by :math:`\varepsilon^{\text{e}}`.

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

The uniaxial stress calculated by the model in Adagio is compared to analytical solutions.
For uniaxial loading in the :math:`e_{1}` direction, the effective stress is

.. math::

   \phi = \frac{\sigma}{R_{11}}

If the stress state is on the yield surface, then :math:`\phi = \bar{\sigma}\left(\bar{\varepsilon}^{p}\right)`, so the axial stress, as a function of the hardening function, is

.. math::
   :label: eq:hillver1

   \sigma = R_{kk}\bar{\sigma}\left(\bar{\varepsilon}^{p}\right)

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

To evaluate the axial stress we need the equivalent plastic strain as a function of the axial strain.
If we equate the rate of plastic work we get

.. math::

   \bar{\sigma} \dot{\bar{\varepsilon}}^{p} = \sigma \left( \dot\varepsilon - \dot\varepsilon^{\text{e}} \right)
   \;\;\; \rightarrow \;\;\;
   \dot{\bar{\varepsilon}}^{p} = R_{11} \left( \dot\varepsilon - \dot\varepsilon^{\text{e}} \right)

which, when integrated, gives us an implicit equation for the equivalent plastic strain

.. math::
   :label: eq:verhilleqps

   \bar{\varepsilon}^{p} = R_{11} \left( \varepsilon - R_{11} \frac{\bar{\sigma}(\bar{\varepsilon}^{p})}{E} \right)

The equivalent plastic strain can then be used in :eq:`eq:hillver1` to find the axial stress, :math:`\sigma`.

The axial stresses for loading in the other directions can be found the same way.
The axial stresses for loading in the :math:`e_{1}`, :math:`e_{2}`, and :math:`e_{3}` directions are shown in :numref:`fig-hillver0`.

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

For the lateral strains we need the plastic strains and therefore the normal to the yield surface.
The components of the normal to the yield surface are

.. math::

   \frac{\partial\phi}{\partial \sigma_{11}} = \frac{1}{R_{11}}
   \;\;\; ; \;\;\;
   \frac{\partial\phi}{\partial \sigma_{22}} = - H R_{11}
   \;\;\; ; \;\;\;
   \frac{\partial\phi}{\partial \sigma_{33}} = - G R_{11}

The elastic axial and lateral strain components are 

.. math::

   \varepsilon^{\text{e}}_{11} = \frac{\sigma}{E} = \varepsilon^{\text{e}}
   \;\;\; ; \;\;\;
   \varepsilon^{\text{e}}_{22} = \varepsilon^{\text{e}}_{33} = -\nu\frac{\sigma}{E} = -\nu\varepsilon^{\text{e}}

The plastic axial strain component is

.. math::

   \varepsilon^{\text{p}}_{11} = \varepsilon_{11} - \frac{\sigma}{E} = \varepsilon - \varepsilon^{\text{e}}

which comes from the additive decomposition of the strain rates.
Using the equivalent plastic strain :eq:`eq:verhilleqps` we can find the lateral plastic strain components 

.. math::

   \varepsilon^{\text{p}}_{22} = - \left( \varepsilon - \varepsilon^{\text{e}} \right) H R_{11}^{2}
   \;\;\; ; \;\;\;
   \varepsilon^{\text{p}}_{33} = - \left( \varepsilon - \varepsilon^{\text{e}} \right) G R_{11}^{2}

The lateral *total* stain components prior to yield are :math:`\varepsilon_{22} = \varepsilon_{33} = -\nu \varepsilon`.
After yield they are

.. math::

   \varepsilon_{22} & = -\nu \varepsilon^{\text{e}} - H R_{11} \bar{\varepsilon}^{p}
   \nonumber \\
   \\
   \varepsilon_{33} & = -\nu \varepsilon^{\text{e}} - G R_{11} \bar{\varepsilon}^{p}
   \nonumber

where :math:`\varepsilon^{\text{e}} = \sigma/E`.

For loading in the :math:`y` direction, a similar analysis leads to the lateral strains, after yield

.. math::

   \varepsilon_{33} & = -\nu \varepsilon^{\text{e}} - F R_{22} \bar{\varepsilon}^{p}
   \nonumber \\
   \\
   \varepsilon_{11} & = -\nu \varepsilon^{\text{e}} - H R_{22} \bar{\varepsilon}^{p}
   \nonumber

For loading in the :math:`z` direction, a similar analysis leads to the lateral strains, after yield

.. math::

   \varepsilon_{11} & = -\nu \varepsilon^{\text{e}} - G R_{33} \bar{\varepsilon}^{p}
   \nonumber \\
   \\
   \varepsilon_{22} & = -\nu \varepsilon^{\text{e}} - F R_{33} \bar{\varepsilon}^{p}
   \nonumber

Results for all three loadings are shown in :numref:`fig-hillver1`, :numref:`fig-hillver2`, and :numref:`fig-hillver3`.

.. _fig-hillver0:

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

   Stresses when loading in the :math:`e_{1}`, :math:`e_{2}`, and :math:`e_{3}`-directions using the Hill model with a Voce hardening law.

.. _fig-hillver1:

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

   Lateral strain as a function of axial strain for the Hill model of 2090-T3 aluminum. Loading is in the :math:`e_{1}`-direction and the hardening law is a Voce law.

.. _fig-hillver2:

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

   Lateral strain as a function of axial strain for the Hill model of 2090-T3 aluminum. Loading is in the :math:`e_{2}`-direction and the hardening law is a Voce law.

.. _fig-hillver3:

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

   Lateral strain as a function of axial strain for the Hill model of 2090-T3 aluminum. Loading is in the :math:`e_{3}`-direction and the hardening law is a Voce law.

.. _models-hill-verification-pure-shear:

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

The shear stress calculated by the Hill plasticity model in Adagio is compared to analytical solutions.
Without loss of generality we will look at solutions for pure shear with respect to the :math:`e_{1}`-:math:`e_{2}` axes.
Solutions for shear with respect to the other axes will be similar.
In what follows, the only non-zero shear stress will be :math:`\sigma_{12}`, and the only non-zero shear strain will be :math:`\varepsilon_{12}`
In general, for pure shear with respect to the :math:`e_{1}`-:math:`e_{2}` axes, the effective stress is

.. math::

   \phi = \sqrt{3} \, \frac{\sigma_{12}}{R_{12}}

If the stress state is on the yield surface, then :math:`\phi = \bar{\sigma}\left(\bar{\varepsilon}^{p}\right)`, so the shear stress is

.. math::
   :label: eq:hillvershear1

   \sigma_{12} = \frac{R_{12}}{\sqrt{3}}\bar{\sigma}\left(\bar{\varepsilon}^{p}\right)

This shows that the pure shear stress state can be calculated from the hardening law and the anisotropy parameters.

To evaluate the shear stress we need the equivalent plastic strain as a function of the shear strain.
If we equate the rate of plastic work we get

.. math::

   \bar{\sigma}\dot{\bar{\varepsilon}}^{p} = 2 \sigma_{12} \left( \dot{\varepsilon}_{12} - \dot{\varepsilon}_{12}^{\text{e}} \right)
   \;\;\; \rightarrow \;\;\;
   \dot{\bar{\varepsilon}}^{p} = \frac{2 R_{12}}{\sqrt{3}} \left( \dot{\varepsilon}_{12} - \dot{\varepsilon}_{12}^{\text{e}} \right)

which, when integrated, gives us an implicit equation for the equivalent plastic strain

.. math::

   \bar{\varepsilon}^{p} = \frac{2 R_{12}}{\sqrt{3}} \left( \varepsilon_{12} - \frac{R_{12}}{\sqrt{3}} \frac{\bar{\sigma}\left(\bar{\varepsilon}^{p}\right)}{2\,G} \right)

If we define :math:`\hat{R}_{12} = R_{12}/\sqrt{3}` then we get a form similar to what we had for uniaxial stress

.. math::

   \bar{\varepsilon}^{p} = 2 \hat{R}_{12} \left( \varepsilon_{12} - \hat{R}_{12} \frac{\bar{\sigma}\left(\bar{\varepsilon}^{p}\right)}{2\,G} \right)

The equivalent plastic strain can now be used to find the shear stress.

Boundary Conditions for Pure Shear
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The deformation gradient that gives pure shear for loading relative to the :math:`e_{1}`-:math:`e_{2}` axes is

.. math::

   \left[ {\bf F} \right] = \begin{bmatrix}
   \frac{1}{2} \left( \lambda + \lambda^{-1} \right) & \frac{1}{2} \left( \lambda - \lambda^{-1} \right) & 0 \\ \\
   \frac{1}{2} \left( \lambda - \lambda^{-1} \right) & \frac{1}{2} \left( \lambda + \lambda^{-1} \right) & 0 \\ \\
   0 & 0 & 1
   \end{bmatrix}
   \;\;\; \rightarrow \;\;\;
   \left[ {\boldsymbol{\varepsilon}} \right] = \begin{bmatrix}
   0 & \varepsilon & 0 \\ \\
   \varepsilon & 0 & 0 \\ \\
   0 & 0 & 0
   \end{bmatrix}
   \;\;\; ; \;\;\;
   \varepsilon = \ln\lambda

For loading relative to the :math:`e_{2}`-:math:`e_{3}` axes and the :math:`e_{3}`-:math:`e_{1}` axes the boundary conditions are modified appropriately.

Results
^^^^^^^

The results for the Hill plasticity model loaded in pure shear are shown in :numref:`fig-hillver4`.
We see that the stress strain curves in pure shear as calculated by Adagio follow the expected stress strain curves.
All other stress and strain components for the three problems are zero.

.. _fig-hillver4:

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

   Shear stress versus shear strain using the Hill model with a Voce hardening law. Results are for shear in the three orthogonal planes of the material coordinate system.

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 not 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 Hill yield surface.  Additionally, the rate *dependent* models are tested 
for a wide range of strain rates (over five decades) and 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 various rate *dependent* and rate *independent* hardening coefficients are found in :numref:`tab-hill-rateDepVerProps` 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) will first be considered and then the rate *dependent* forms (Johnson-Cook, power-law breakdown).

.. _tab-hill-rateDepVerProps:

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

To examine the performance of the rate *independent* linear hardening model, the verification exercises from :ref:`Appendix A <appendix-common-constant-eqps-rate>` are used. In this case, as the Hill yield surface is being considered, the responses are determined numerically and analytically in the uniaxial stress case with loading in three different principal material directions and three different shear planes for the pure shear case. These results are presented in :numref:`fig-hill-verLinHard`. From these responses, superb agreement between the analytical and numerical results is noted. Additionally, the constant linear stress-strain response during plastic deformations clearly demonstrates the behavior giving this model its name.

.. _fig-hill-verLinHard:

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

   .. image:: ../../_static/figures/hill/hill_normal_linear.png
      :alt: Uniaxial Stress
      :scale: 35 %

   .. image:: ../../_static/figures/hill/hill_shear_linear.png
      :alt: Pure Shear
      :scale: 35 %

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

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

The rate *independent* power-law hardening model is verified by using the uniaxial stress and pure shear problems of :ref:`Appendix A <appendix-common-constant-eqps-rate>`.  
Results of these endeavors determined analytically and numerically are presented in :numref:`fig-hill-verPLHard` in which the uniaxial 
stress problem is presented for loading aligned with the three different principal material directions and three different shear planes for the 
pure shear case.  From these results, outstanding agreement is noted between both numerical and analytical results sets verifying the model.  Also, the initially stiff 
hardening decreasing to a lower linear tangent modulus characteristic of power-law hardening models is clearly evident in the various result sets of 
:numref:`fig-hill-verPLHard`.

.. _fig-hill-verPLHard:

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

   .. image:: ../../_static/figures/hill/hill_normal_power_law.png
      :alt: Uniaxial Stress
      :scale: 35 %

   .. image:: ../../_static/figures/hill/hill_shear_power_law.png
      :alt: Pure Shear
      :scale: 35 %

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

Voce
^^^^

Verification of the rate *independent* Voce hardening model is pursued by considering both the uniaxial stress and pure shear approaches of 
:ref:`Appendix A <appendix-common-constant-eqps-rate>`.  The results of these investigations determined analytically and numerically are shown in :numref:`fig-hill-verVoceHard`.  
For the uniaxial stress cases, loadings in each of the three principal material directions is presented while complementary results from the three 
shear planes are shown for the pure shear case.  In each of these six instances, exemplary agreement is observed between the different results sets.  
Additionally, such stress-strain results also show the saturation behavior associated with Voce models in which at some equivalent plastic strain 
the material no longer hardens. 

.. _fig-hill-verVoceHard:

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

   .. image:: ../../_static/figures/hill/hill_normal_voce.png
      :alt: Uniaxial Stress
      :scale: 35 %

   .. image:: ../../_static/figures/hill/hill_shear_voce.png
      :alt: Pure Shear
      :scale: 35 %

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

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

As noted in :ref:`Appendix A <appendix-common-uniaxial-stress>`, the uniaxial stress response depends on the yield surface anisotropy coefficients (for the Hill model the :math:`R's`).  
The respective coefficients are given in the aforementioned appendix while :numref:`fig-hill-verJCUniStress0` and :numref:`fig-hill-verJCUniStress` present the results of forty-five different 
verification exercises corresponding to different combinations of the three material principal directions (:math:`\hat{e}_1,~\hat{e}_2,` and :math:`\hat{e}_3`), 
five equivalent plastic strain rates(:math:`1\times10^{-3},~1\times10^{-2},~1\times10^{-1},~1\times10^{0}` and :math:`1\times10^1~\text{s}^{-1}`), and 
three rate *independent* hardening models (linear, power-law, and Voce).  For each combination, the analytical and numerical results match 
to within acceptably small numerical differences.

.. _fig-hill-verJCUniStress0:

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

   .. image:: ../../_static/figures/hill/normal_jc_lh_xx.png
      :alt: Linear Hardening -- 11
      :scale: 35 %

   .. image:: ../../_static/figures/hill/normal_jc_lh_xx.png
      :alt: Linear Hardening -- 22
      :scale: 35 %

   .. image:: ../../_static/figures/hill/normal_jc_lh_xx.png
      :alt: Linear Hardening -- 33
      :scale: 35 %

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

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

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

   Uniaxial stress-strain response of the Hill plasticity model 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-hill-verJCUniStress:

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

   .. image:: ../../_static/figures/hill/normal_jc_vh_xx.png
      :alt: Voce Hardening -- 11
      :scale: 35 %

   .. image:: ../../_static/figures/hill/normal_jc_vh_xx.png
      :alt: Voce Hardening -- 22
      :scale: 35 %

   .. image:: ../../_static/figures/hill/normal_jc_vh_xx.png
      :alt: Voce Hardening -- 33
      :scale: 35 %

   Uniaxial stress-strain response of the Hill plasticity model 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 problem discussed in :ref:`Appendix A <appendix-common-pure-shear>` is considered.  The results still depend on the Hill :math:`R` coefficients 
and forty-five different loadings are presented in :numref:`fig-hill-verJCShear0` and :numref:`fig-hill-verJCShear`.  In this instance, three different shearing planes 
are used in lieu of the principal directions.  Nonetheless, for these results the key result remains the same -- analytical matches numerical further 
verifying rate dependent capabilities.

.. _fig-hill-verJCShear0:

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

   .. image:: ../../_static/figures/hill/shear_rd_lh_xy.png
      :alt: Linear Hardening -- 11
      :scale: 35 %

   .. image:: ../../_static/figures/hill/shear_rd_lh_yz.png
      :alt: Linear Hardening -- 22
      :scale: 35 %

   .. image:: ../../_static/figures/hill/shear_rd_lh_zx.png
      :alt: Linear Hardening -- 33
      :scale: 35 %

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

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

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

   Stress-strain response of the Hill plasticity model 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-hill-verJCShear:

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

   .. image:: ../../_static/figures/hill/shear_rd_vh_xy.png
      :alt: Voce Hardening -- 11
      :scale: 35 %

   .. image:: ../../_static/figures/hill/shear_rd_vh_yz.png
      :alt: Voce Hardening -- 22
      :scale: 35 %

   .. image:: ../../_static/figures/hill/shear_rd_vh_zx.png
      :alt: Voce Hardening -- 33
      :scale: 35 %

   Stress-strain response of the Hill plasticity model 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
^^^^^^^^^^^^^^^^^^^

For the power-law breakdown model, the same forty-five cases discussed in the previous section (three directions, five rates, three hardening models) 
are again solved via the approach of :ref:`Appendix A <appendix-common-uniaxial-stress>` in :numref:`fig-hill-verPLBUniStress0` and :numref:`fig-hill-verPLBUniStress`.  Although the impact of rate 
on the responses differs due to the assumed representation of the rate-dependent hardening, excellent agreement is still noted between 
analytical and numerical results.

.. _fig-hill-verPLBUniStress0:

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

   .. image:: ../../_static/figures/hill/normal_plb_lh_xx.png
      :alt: Linear Hardening -- 11
      :scale: 35 %

   .. image:: ../../_static/figures/hill/normal_plb_lh_xx.png
      :alt: Linear Hardening -- 22
      :scale: 35 %

   .. image:: ../../_static/figures/hill/normal_plb_lh_xx.png
      :alt: Linear Hardening -- 33
      :scale: 35 %

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

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

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

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

.. _fig-hill-verPLBUniStress:

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

   .. image:: ../../_static/figures/hill/normal_plb_vh_xx.png
      :alt: Voce Hardening -- 11
      :scale: 35 %

   .. image:: ../../_static/figures/hill/normal_plb_vh_xx.png
      :alt: Voce Hardening -- 22
      :scale: 35 %

   .. image:: ../../_static/figures/hill/normal_plb_vh_xx.png
      :alt: Voce Hardening -- 33
      :scale: 35 %

   Uniaxial stress-strain response of the Hill plasticity model with rate *dependent*, power-law breakdown type hardening in with (a-c) Voce rate *independent* hardening. Solid lines are analytical results while open symbols are numerical.

To expand on the uniaxial stress results, the response through pure shear is also probed via the method of :ref:`Appendix A <appendix-common-pure-shear>`.  
Again forty-five different cases are investigated and their results are presented in :numref:`fig-hill-verPLBShear0` and :numref:`fig-hill-verPLBShear`.  Once again, 
the results aligning thereby verifying the capability of the model and producing additional credibility.

.. _fig-hill-verPLBShear0:

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

   .. image:: ../../_static/figures/hill/shear_plb_lh_xy.png
      :alt: Linear Hardening -- 11
      :scale: 35 %

   .. image:: ../../_static/figures/hill/shear_plb_lh_yz.png
      :alt: Linear Hardening -- 22
      :scale: 35 %

   .. image:: ../../_static/figures/hill/shear_plb_lh_zx.png
      :alt: Linear Hardening -- 33
      :scale: 35 %

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

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

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

   Stress-strain response of the Hill plasticity model 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-hill-verPLBShear:

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

   .. image:: ../../_static/figures/hill/shear_plb_vh_xy.png
      :alt: Voce Hardening -- 11
      :scale: 35 %

   .. image:: ../../_static/figures/hill/shear_plb_vh_yz.png
      :alt: Voce Hardening -- 22
      :scale: 35 %

   .. image:: ../../_static/figures/hill/shear_plb_vh_zx.png
      :alt: Voce Hardening -- 33
      :scale: 35 %

   Stress-strain response of the Hill plasticity model 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 HILL_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>
     R11 = <real> (1.0)
     R22 = <real> (1.0)
     R33 = <real> (1.0)
     R12 = <real> (1.0)
     R23 = <real> (1.0)
     R31 = <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 HILL_PLASTICITY]

In the command blocks that define the Hill 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 ratio of the normal yield stress in the :math:`\bar{\bf e}_{1}\bar{\bf e}_{1}` material direction is defined with the ``R11`` command line. The default is 1.0.

- The ratio of the normal yield stress in the :math:`\bar{\bf e}_{2}\bar{\bf e}_{2}` material direction is defined with the ``R22`` command line. The default is 1.0.

- The ratio of the normal yield stress in the :math:`\bar{\bf e}_{3}\bar{\bf e}_{3}` material direction is defined with the ``R33`` command line. The default is 1.0.

- The ratio of the shear yield stress in the :math:`\bar{\bf e}_{1}\bar{\bf e}_{2}` material direction is defined with the ``R12`` command line. The default is 1.0.

- The ratio of the shear yield stress in the :math:`\bar{\bf e}_{2}\bar{\bf e}_{3}` material direction is defined with the ``R23`` command line. The default is 1.0.

- The ratio of the shear yield stress in the :math:`\bar{\bf e}_{3}\bar{\bf e}_{1}` material direction is defined with the ``R31`` command line. The default is 1.0.

.. include:: hardening/user_output.rst

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

.. _out-tab-hillstvar:

.. csv-table:: State Variables for HILL 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::
