.. _models-hosford_plasticity:

************************
Hosford Plasticity Model
************************

Theory
======

Like other elastic-plastic models in LAMÉ, the Hosford plasticity model is a rate-independent hypoelastic formulation.  Unlike the Hill and other 
more complex plasticity models, it is isotropic.  In a similar fashion to those models, 
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}.

The Hosford plasticity model utilizes a yield surface first put forth by W. F. Hosford in the 1970's [:footcite:`mat:ref:hosford`] that is isotropic 
but non-quadratic.  This specific form was proposed due to experimental observations of biaxial stretching in which neither the Tresca 
or :math:`J_2` yield surfaces could describe the results.  In contrast to many of the yield surfaces proposed for similar purposes, only 
two parameters are utilized.  Even with these limited terms, the developed model is quite versatile and can 
be reduced to von Mises or Tresca conditions as well as capturing responses in between.  This yield surface 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 Hosford 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 Hosford effective stress is a non-quadratic function of the principal stresses (:math:`\sigma_i,~~i=1,2,3`) 
and is given as 

.. math::

   \phi\left(\sigma_{ij}\right)=\left[\frac{|\sigma_1-\sigma_2|^a+|\sigma_2-\sigma_3|^a+|\sigma_1-\sigma_3|^a}{2}\right]^{1/a}

in which :math:`a` is the yield surface exponent.  Interestingly, if :math:`a=2` or :math:`4` the yield surface reduces to that of a :math:`J_2` von Mises surface 
while :math:`a=1` or as :math:`a\rightarrow\infty` produces a Tresca like shape.  If the value of :math:`a` is above 4 the yield surface takes a position between 
the Tresca and :math:`J_2` limits.  Typical values are :math:`a=6` or :math:`a=8` for *bcc* and *fcc* metals, respectively [:footcite:`mat:ref:graf`].  
To highlight this variability the yield surface is plotted below in :numref:`fig-hp-theory-yield` for three values of :math:`a` -- :math:`a~=~4,~8,` and 100.

.. _fig-hp-theory-yield:

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

   Example Hosford yield surfaces, :math:`f\left(\sigma_{ij},\bar{\varepsilon}^p=0;a\right)`, presented in the deviatoric :math:`\pi`-plane. The presented surfaces correspond to the different yield exponents :math:`a~=~4,~8,` and :math:`100`.

For the hardening function, :math:`\bar{\sigma}\left(\bar{\varepsilon}^p\right)`, a variety of forms including linear, power law, or a 
more general user defined function may be used.

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

.. math::

   \dot{D}_{ij}^{\text{p}}=\dot{\gamma}\frac{\partial\phi}{\partial\sigma_{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`.

For details on the plasticity model, please see [:footcite:`mat:ref:wmsch4`].  Additional details on 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.

.. include:: hardening/hardening.rst

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

The Hosford plasticity model is implicitly integrated using a closest point projection (CPP) return mapping algorithm (RMA).  The resulting 
nonlinear equations are solved via a line search augmented Newton-Raphson method and the stress update routine is very similar to that 
of the Hill plasticity model.  The key difference between the two is the isotropy.  Specifically, given that the Hosford yield surface 
is isotropic and the functional form is given in terms of principal stresses, the stress update routine is performed in *principal stress* 
space and then converted to global Cartesian values.

For a loading step, a trial stress state, :math:`T^{tr}_{ij}`, may be computed by knowing the rate of deformation, :math:`d_{ij}`, and time step as,

.. math::

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

The principal stresses, :math:`T_{i}^{tr}`, may then be used to determine the trial yield function value, :math:`\phi^{tr}=\phi\left(T_i^{tr}, \bar{\varepsilon}^{p\left(n\right)}\right)`.  If :math:`\phi^{tr}<0`, the elastic trial solution is acceptable.  On the other hand, 
if the trial solution is inadmissible, the aforementioned CPP-RMA problem is solved in principal stress space.  The crux of 
this algorithm is the simultaneous solution of two nonlinear equations -- (*i*) the flow rule and (*ii*) consistency 
condition.  The former leads to a residual, :math:`R_i`, of the form (again in principal stress space), 

.. math::

   R_{i}=\Delta d^p_{i}-\Delta \gamma\frac{\partial\phi}{\partial T_{i}}=0,

while the latter is enforced by the yield function,

.. math::

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

and its derivative (:math:`\dot{f}`) being zero.  This system is solved via a Newton-Raphson type approach in which the state variables 
(stress, :math:`T_{i}`, and consistency multiplier, :math:`\gamma`) 
are iteratively corrected until the residuals are satisfied.  Using :math:`\left(k+1\right)` and :math:`\left(k\right)` to denote the next 
and current iterations, this updating takes the form,

.. math::
   :label: hos:eqn:increment1

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

in which :math:`T^{\left(0\right)}=T_i^{tr}` and :math:`\Delta \gamma^{\left(0\right)}=0`.  Consistent linearization of the two equations 
can be solved to give correction increments of the form,

.. math::

   \Delta\left(\Delta\gamma\right) & = \frac{f^{\left(k\right)}-R^{\left(k\right)}_i\mathcal{L}_{ij}^{\left(k\right)}
             \frac{\partial\phi^{\left(k\right)}}{\partial T_j}}{\frac{\partial\phi^{\left(k\right)}}{\partial T_i}
             \mathcal{L}_{ij}^{\left(k\right)}\frac{\partial\phi^{\left(k\right)}}{\partial T_j}+H^{'\left(k\right)}} ,\nonumber \\
   \Delta T_i & = -\mathcal{L}_{ij}^{\left(k\right)}\left(R_j^{\left(k\right)}+\Delta\left(\Delta\gamma\right)\frac{\partial\phi^{\left(k\right)}}
            {\partial T_j}\right),

with :math:`\mathcal{L}_{ij}^{\left(k\right)}` being the Hessian of the CPP-RMA problem and :math:`H^{'\left(k\right)}` is the 
slope of the hardening curve.

Previous studies have indicated that the Newton-Raphson method alone may be insufficient to guarantee convergence with arbitrary stress 
states in the case of non-quadratic yield surfaces [:footcite:`mat:ref:armero:02`, :footcite:`mat:ref:perezFoguet:02`, :footcite:`mat:ref:wmsch4`].  To 
address this, a *line search* method is adopted.  In such an approach, the incrementation rule :eq:`hos:eqn:increment1` is modified 
such that,

.. math::
   :label: hos:eqn:increment2

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

where :math:`\alpha\in\left(0,1\right]` is the step magnitude.  This parameter enforces that the solution be converging and is determined 
via various convergence criteria.  The :math:`\alpha=1` case corresponds to the Newton-Raphson method.  Utilization of this approach 
has been shown to greatly increase the robustness of this algorithm under large trial stresses [:footcite:`mat:ref:wmsch4`].

Finally, upon convergence of the algorithm, the Cartesian stress are found from the principal stresses via,

.. math::

   T_{ij}^{n+1}=\sum_{k=1}^3T_k^{n+1}\hat{e}_i^k\hat{e}_j^k,

in which :math:`\hat{e}_i^k` is the eigenvector of the :math:`k^{th}` principal stress.

Details of this implementation and the line search algorithm may be found in the work of Scherzinger [:footcite:`mat:ref:wmsch4`].

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

The Hosford plasticity material model is verified through a variety of loading and material conditions.  
For these cases, the elastic properties corresponding to 2090-T3 aluminum [:footcite:`mat:ref:barlat:05`] given in :numref:`models-hill-verification` 
are 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.

The elastic properties are :math:`E=70` GPa and :math:`\nu = 0.25` while a linear hardening law of the form,

.. math::

   \bar{\sigma}\left(\bar{\varepsilon}^p\right) = \sigma_{y} + K\bar{\varepsilon}^p,

with :math:`\sigma_{y}=200` MPa and :math:`K=E/200` is assumed.  For these studies, two different yield surface exponents will be used, :math:`a=4,~8`.  The former 
corresponds to the :math:`J_2` surface while the latter is a common value for aluminum. 

.. _sec-ver-hos-uniStress:

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

In the case of uniaxial stress (:math:`\sigma`), it is trivial to note that the corresponding principal stress state is simply 
:math:`\sigma_1=\sigma,~\sigma_2=\sigma_3=0`.  As such, regardless of :math:`a`, 

.. math::

   \phi=|\sigma_1|.

With the aforementioned linear hardening, this case reduces to that discussed in :numref:`sec-ep-ver-uniaxialStress`.  Corresponding analytical 
and numerical results (both with :math:`a=4` and :math:`8`) of the axial stress and lateral strain are presented in 
:numref:`hp-ver-uniaxialStress`\ (a) and :numref:`hp-ver-uniaxialStress`\ (b), respectively.  In these figures, the invariance of response 
on yield surface exponent through this loading is clearly observed.

.. _hp-ver-uniaxialStress:

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

   .. image:: ../../_static/figures/uniaxialStress_Stress.png
      :alt: Uniaxial Stress
      :scale: 25 %

   .. image:: ../../_static/figures/uniaxialStress_Strain.png
      :alt: Lateral Strain
      :scale: 25 %

   Axial stress-strain (a) and lateral strain (b) results of the Hosford plasticity model determined analytically and numerically for the case of yield surface exponents :math:`a=4,8`.

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

To explore the impact of the yield exponent :math:`a`, the case of pure shear is considered.  Specifically, the only shear component shall 
be in the Cartesian :math:`e_1-e_2` direction such that :math:`\sigma_{12}=\tau` and :math:`\varepsilon_{12}` are the only non-zero components.  
Noting that the three principal stresses are :math:`\tau,0,-\tau`, the yield condition simplifies to 

.. math::

   \phi=\left[1+2^{a-1}\right]^{1/a}\tau.

The equivalent plastic strain may then be found as a function of :math:`\varepsilon_{12}` in the same way as presented in :numref:`models-hill-verification-pure-shear`.  
Shear stress-strain results for both :math:`a=4,~8` are presented in :numref:`fig-hp-ver-pureShear` as determined both by adagio and analytically.  
The boundary conditions for this loading are given in :ref:`Appendix A <appendix-common-pure-shear>`.  In these results, 
the effect of the yield surface exponent, :math:`a`, may clearly be seen.

.. _fig-hp-ver-pureShear:

.. figure:: ../../_static/figures/pureShearResults.png
   :align: center
   :scale: 15 %

   Shear stress-strain results of the Hosford plasticity model determined analytically and numerically for the case of yield surface exponents :math:`a=4,~8`. 

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 different yield surface shapes.  In the current Hosford case, multiple yield surface exponents, :math:`a`, are considered to probe this effect.  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-hosford-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-hosford-rateDepVerProps:

.. csv-table:: The model parameters for the hardening verification tests used with the Hosford 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
^^^^^^
 
The aforementioned verification exercises from :ref:`Appendix A <appendix-common-constant-eqps-rate>` are used to investigate the numerical implementation of the rate *independent* 
linear hardening model. Results from uniaxial stress and pure shear exercises determined analytically and numerically are given in :numref:`fig-hosford-verLinHard` 
for three different exponents :math:`a=~4,~8,` and :math:`20`. The first exponent produces a :math:`J_2` like response with the latter increasing the curvature of the 
yield surface. As discussed in :numref:`sec-ver-hos-uniStress`, a purely uniaxial response is independent of exponent thus producing the collapsed 
results in :numref:`fig-hosford-verLinHard`. In both the uniaxial stress and pure shear cases, clear agreement is noted between the two sets of results.  
The linear slope (tangent modulus) giving the model its name is also observable in the results of :numref:`fig-hosford-verLinHard`.

.. _fig-hosford-verLinHard:

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

   .. image:: ../../_static/figures/hosford/hosford_normal_linear.png
      :alt: Uniaxial Stress
      :scale: 35 %

   .. image:: ../../_static/figures/hosford/hosford_shear_linear.png
      :alt: Pure Shear
      :scale: 35 %

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

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

To consider the performance of the common power-law hardening model with the Hosford yield surface, the uniaxial stress and pure shear 
exercises of :ref:`Appendix A <appendix-common-constant-eqps-rate>` are solved analytically and numerically.  These results are presented in :numref:`fig-hosford-verPLHard` 
for three different Hosford exponents -- :math:`a=~4,~8` and :math:`20`.  As expected from previous discussions the uniaxial stress results in :numref:`fig-hosford-verPLHard`\ (a) 
are independent of :math:`a`.  For both the uniaxial stress and pure shear results, the desired agreement between analytical and numerical solutions is apparent.  
These results also highlight the initial curved response during plastic-deformation eventually transitioning into a more linear type response.

.. _fig-hosford-verPLHard:

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

   .. image:: ../../_static/figures/hosford/hosford_normal_power_law.png
      :alt: Uniaxial Stress
      :scale: 35 %

   .. image:: ../../_static/figures/hosford/hosford_shear_power_law.png
      :alt: Pure Shear
      :scale: 35 %

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

Voce
^^^^

For the rate *independent* Voce hardening model, the problems of :ref:`Appendix A <appendix-common-constant-eqps-rate>` are used to verify the model response.  
Specifically, results for the uniaxial stress and pure shear analyses are presented in :numref:`fig-hosford-verVoceHard` as determined 
analytically and numerically for three different values of :math:`a` -- :math:`a=~4,~8,` and :math:`20`.  From these results, clear agreement 
is noted between the two sets of results; including the invariance of the uniaxial stress case to :math:`a` (:numref:`fig-hosford-verVoceHard`\ a). 
Additionally, the results of :numref:`fig-hosford-verVoceHard` also exemplify the saturation nature of the Voce hardening model as the 
stress-strain response eventually asymptotes.

.. _fig-hosford-verVoceHard:

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

   .. image:: ../../_static/figures/hosford/hosford_normal_voce.png
      :alt: Uniaxial Stress
      :scale: 35 %

   .. image:: ../../_static/figures/hosford/hosford_shear_voce.png
      :alt: Pure Shear
      :scale: 35 %

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

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

As noted in :numref:`sec-ver-hos-uniStress`, the uniaxial stress response is independent of :math:`a`.  This is also reflected :ref:`Appendix A <appendix-common-uniaxial-stress>` in which the stress weighting coefficients (:math:`\Gamma`) for the Hosford uniaxial case are one.  As such in :numref:`fig-hosford-verJCUniStress` the results of the constant equivalent plastic strain rate uniaxial stress test are presented with :math:`a=8` and using the linear (:numref:`fig-hosford-verJCUniStress`\ a), power-law (:numref:`fig-hosford-verJCUniStress`\ b), and Voce (:numref:`fig-hosford-verJCUniStress`\ c) rate *independent* hardening models for 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}`. In all cases in :numref:`fig-hosford-verJCUniStress` excellent agreement is observed between the results.

.. _fig-hosford-verJCUniStress:

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

   .. image:: ../../_static/figures/hosford/normal_jc_lin_a8.png
      :alt: Linear Hardening
      :scale: 25 %

   .. image:: ../../_static/figures/hosford/normal_jc_pl_a8.png
      :alt: Power-Law Hardening
      :scale: 25 %

   .. image:: ../../_static/figures/hosford/normal_jc_voce_a8.png
      :alt: Voce Hardening
      :scale: 25 %

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

Unlike the uniaxial stress case, for pure shear the response depends on the exponent :math:`a`.  Therefore, in addition to the three hardening models, 
results are also presented for three different exponent values -- :math:`a=4,~8,` and :math:`20`.  The results for all nine cases are presented in :numref:`fig-hosford-verJCShear0` and :numref:`fig-hosford-verJCShear` 
and again excellent agreement is noted in all instances.

.. _fig-hosford-verJCShear0:

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

   .. image:: ../../_static/figures/hosford/shear_jc_lin_a4.png
      :alt: Linear Hardening a=4
      :scale: 35 %

   .. image:: ../../_static/figures/hosford/shear_jc_lin_a8.png
      :alt: Linear Hardening, a=8
      :scale: 35 %

   .. image:: ../../_static/figures/hosford/shear_jc_lin_a20.png
      :alt: Linear Hardening, a=20
      :scale: 35 %

   .. image:: ../../_static/figures/hosford/shear_jc_pl_a4.png
      :alt: Power-Law Hardening, a=4
      :scale: 35 %

   .. image:: ../../_static/figures/hosford/shear_jc_pl_a8.png
      :alt: Power-Law Hardening, a=8
      :scale: 35 %

   .. image:: ../../_static/figures/hosford/shear_jc_pl_a20.png
      :alt: Power-Law Hardening, a=20
      :scale: 35 %

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

.. _fig-hosford-verJCShear:

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

   .. image:: ../../_static/figures/hosford/shear_jc_voce_a4.png
      :alt: Power-Law Hardening, a=4
      :scale: 35 %

   .. image:: ../../_static/figures/hosford/shear_jc_voce_a8.png
      :alt: Power-Law Hardening, a=8
      :scale: 35 %

   .. image:: ../../_static/figures/hosford/shear_jc_voce_a20.png
      :alt: Power-Law Hardening, a=20
      :scale: 35 %

   Stress-strain response of the Hosford 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
^^^^^^^^^^^^^^^^^^^

As mentioned in the previous Johnson-Cook section, for the Hosford model under uniaxial stress the response is independent of yield surface exponent, :math:`a`. Therefore, :numref:`fig-hosford-verPLBUniStress` presents the results of the constant equivalent plastic strain rate verification test of :ref:`Appendix A <appendix-common-uniaxial-stress>` for strain rates spanning five decades -- :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}`. The tests are performed for each rate- *independent* hardening model. In all fifteen cases excellent agreement is noted between numerical and analytical results. 

.. _fig-hosford-verPLBUniStress:

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

   .. image:: ../../_static/figures/hosford/normal_plb_lin_a8.png
      :alt: Linear Hardening
      :scale: 25 %

   .. image:: ../../_static/figures/hosford/normal_plb_pl_a8.png
      :alt: Power-Law Hardening
      :scale: 25 %

   .. image:: ../../_static/figures/hosford/normal_plb_voce_a8.png
      :alt: Voce Hardening
      :scale: 25 %

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

Similarly, :numref:`fig-hosford-verPLBShear0` and :numref:`fig-hosford-verPLBShear` gives the results of the pure shear variant of the constant equivalent plastic strain rate 
verification test of :ref:`Appendix A <appendix-common-pure-shear>`.  The same five rates used in the uniaxial stress case are again utilized although in 
this instance as the pure shear response does depend on :math:`a` the results are given for three yield surface exponents -- :math:`a=~4,~8` and :math:`20`.  
In the forty-five cases shown in :numref:`fig-hosford-verPLBShear0` and :numref:`fig-hosford-verPLBShear` quite acceptable agreement is noted verifying the capabilities of 
the rate *dependent* Hosford implementation.

.. _fig-hosford-verPLBShear0:

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

   .. image:: ../../_static/figures/hosford/shear_jc_lin_a4.png
      :alt: Linear Hardening, a=4
      :scale: 35 %

   .. image:: ../../_static/figures/hosford/shear_plb_lin_a8.png
      :alt: Linear Hardening, a=8
      :scale: 35 %

   .. image:: ../../_static/figures/hosford/shear_plb_lin_a20.png
      :alt: Linear Hardening, a=20
      :scale: 35 %

   .. image:: ../../_static/figures/hosford/shear_plb_pl_a4.png
      :alt: Power-Law Hardening, a=4
      :scale: 35 %

   .. image:: ../../_static/figures/hosford/shear_plb_pl_a8.png
      :alt: Power-Law Hardening, a=8
      :scale: 35 %

   .. image:: ../../_static/figures/hosford/shear_plb_pl_a20.png
      :alt: Power-Law Hardening, a=20
      :scale: 35 %

   Stress-strain response of the Hosford 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-hosford-verPLBShear:

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

   .. image:: ../../_static/figures/hosford/shear_plb_voce_a4.png
      :alt: Power-Law Hardening, a=4
      :scale: 35 %

   .. image:: ../../_static/figures/hosford/shear_plb_voce_a8.png
      :alt: Power-Law Hardening, a=8
      :scale: 35 %

   .. image:: ../../_static/figures/hosford/shear_plb_voce_a20.png
      :alt: Power-Law Hardening, a=20
      :scale: 35 %

   Stress-strain response of the Hosford 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 HOSFORD_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> 
     A            = <real> (4.0)

.. 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 HOSFORD_PLASTICITY]

In the command blocks that define the Hosford plasticity model:

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

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

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

.. include:: hardening/user_output.rst

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

.. _out-tab-hosfordstvar:

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