.. _appendix-common-constant-eqps-rate:

*******************************************
A.5 Constant Equivalent Plastic Strain Rate
*******************************************

Typically, for a given boundary value problem it is desirable to know either the stress or deformation (strain) state and solve for the complementary response functions.  In the case of rate-dependent hardening, or often rate-independent, it is preferable to prescribe a constant *equivalent plastic strain rate*, :math:`\dot{\bar{\varepsilon}}^p`. Knowing, and controlling, this variable is often essential to finding and solving analytical solutions to verify hardening models.

As the equivalent plastic strain, :math:`\bar{\varepsilon}^p`, is the internal (hidden) state variable corresponding to isotropic hardening, it is counterintuitive to think of prescribing it's value. Nonetheless, for many plasticity models such a case is not only possible but relatively simple. Details of this approach may be found in [:footcite:`mat:ref:ostien:2017`], but are repeated here for convenience and completeness. In the following, two cases are treated -- uniaxial stress and pure shear.  For either problem, it is assumed that the stress state is initially at yield :math:`\phi[\sigma_{ij}(t=0,\bar{\varepsilon}^p=0)]=\sigma_y^0` and a constant equivalent plastic strain rate is prescribed such that,

.. math::

   \bar{\varepsilon}^p\left(t\right)=\dot{\bar{\varepsilon}}^pt.

Furthermore, it is recalled that the yield surface, :math:`f`, is written as,

.. math::

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

where,

.. math::

   \bar{\sigma}\left(\bar{\varepsilon}^p,\dot{\bar{\varepsilon}}^p\right) = \tilde{\sigma}_y\left(\bar{\varepsilon}^p\right)\hat{\sigma}_y\left(\dot{\bar{\varepsilon}}^p\right).

Note, throughout this section function forms for :math:`\tilde{\sigma}_y` and :math:`\hat{\sigma}_y` are not given.  For the purposes of the developed problem, 
the specific forms are unnecessarily as long as :math:`\tilde{\sigma}_y` depends only on :math:`\bar{\varepsilon}^p` and :math:`\hat{\sigma}_y` on the corresponding rate.

=====================
A.5.1 Uniaxial Stress
=====================

During uniaxial stress, the state of stress reduces to, 

.. math::

   \sigma_{ij}=\sigma\delta_{i\eta}\delta_{j\eta}~~~\left(\text{no sum on }\eta\right)
 
where :math:`\eta` is the direction of loading (taken to align with one of the material principal axes) and 

.. math::
   :label: bvp-eqn-uniSig

   \sigma=\Gamma_{\eta\eta}\bar{\sigma}\left(t\right),

with :math:`\Gamma_{\eta\eta}` being a constant associated with and dependent on the model parameters of the plasticity model.  Specific forms for the various 
yield surfaces are given later in this section.  Given this stress state, the axial elastic strain is simply,

.. math::

   \varepsilon^{\text{el}}_{\eta\eta}=\frac{\Gamma_{\eta\eta}\bar{\sigma}\left(t\right)}{E}.

To determine the plastic state of the material, the equivalency of plastic work (:math:`\sigma_{ij}\dot{\varepsilon}^p_{ij}=\bar{\sigma}\dot{\bar{\varepsilon}}^p`) 
is invoked enabling the axial plastic strain to be given as,

.. math::

   \dot{\varepsilon}^p_{\eta\eta}=\frac{\bar{\sigma}}{\sigma}\dot{\bar{\varepsilon}}^p=\frac{1}{\Gamma_{\eta\eta}}\dot{\bar{\varepsilon}}^p.

Integrating,

.. math::

   \varepsilon_{\eta\eta}^p\left(t\right)=\frac{1}{\Gamma_{\eta\eta}}\bar{\varepsilon}^p.

The total strain is found simply as the sum of elastic and plastic components,

.. math::
   :label: bvp-eqn-totStrain

   \varepsilon_{\eta\eta}\left(t\right)=\varepsilon^{\text{el}}_{\eta\eta}+\varepsilon^{p}_{\eta\eta}=\frac{\Gamma_{\eta\eta}\bar{\sigma}\left(t\right)}{E}+\frac{1}{\Gamma_{\eta\eta}}\dot{\bar{\varepsilon}}t.

For this boundary value problem, only the axial displacement need be prescribed as zero traction conditions are required on the remaining surfaces to achieve 
the uniaxial stress state. As the equivalent plastic strain rate is constant, the flow stress, :math:`\bar{\sigma}\left(t\right)`, is known and the total strain 
of :eq:`bvp-eqn-totStrain` is only a function of time. Therefore, the desired displacement boundary condition may be prescribed as a function of time 
alone and is simply,

.. math::

   u_{\eta}\left(t\right)=\exp\left(\varepsilon_{\eta\eta}\left(t\right)\right)-1.

J\ :sub:`2` Plasticity
----------------------

In the case of an isotropic J\ :sub:`2` effective stress definition, for a uniaxial state of stress,

.. math::

   \Gamma_{\eta\eta}=1~~\left(\text{no sum on }\eta\right).

Hosford Plasticity
------------------

As the Hosford effective stress definition is isotropic, for a uniaxial state of stress the coefficients :math:`\Gamma_{\eta\eta}` are simply,

.. math::

   \Gamma_{\eta\eta}=1~~~\left(\text{no sum on }\eta\right).

Hill Plasticity
---------------

For a Hill effective stress definition, by inspection of :eq:`bvp-eqn-uniSig` it is clear that,

.. math::

   \Gamma_{\eta\eta}=R_{\eta\eta}~~~~\left(\text{no sum on }\eta\right).

Barlat Plasticity
-----------------

With a Barlat effective stress definition, the anisotropy coefficients are,

.. math::

   \Gamma_{\eta\eta} = \frac{1}{\omega_{\eta}}~~~\left(\text{no sum on }\eta\right),

where

.. math::

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

.. math::

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

.. math::

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

================
A.5.2 Pure Shear
================

To produce a pure shear stress state, the pure shear conditions discussed in Section A.3 are utilized.  
In this case, for pure shear in the :math:`\hat{e}_{\eta}-\hat{e}_{\zeta}` plane, a deformation gradient of the form,

.. math::

   F_{ij}=\frac{1}{2}\left(\lambda+\lambda^{-1}\right)\left(\delta_{i\eta}\delta_{j\eta}+\delta_{i\zeta}\delta_{j\zeta}\right)
         +\frac{1}{2}\left(\lambda-\lambda^{-1}\right)\left(\delta_{i\eta}\delta_{j\zeta}+\delta_{i\zeta}\delta_{j\eta}\right)
         +\delta_{i\theta}\delta_{j\theta},~~~\left(\text{no sum on }\eta,~\zeta,~\theta\right),


with :math:`\hat{e}_{\theta}` being the cross-product of :math:`\hat{e}_{\eta}` and :math:`\hat{e}_{\zeta}`.  With such a deformation, 

.. math::

   \varepsilon_{ij}=\ln\lambda\left(\delta_{i\eta}\delta_{j\zeta}+\delta_{i\zeta}\delta_{j\eta}\right),

meaning the appropriate displacement boundary conditions may be applied if the total shear strain is known.

For the pure shear strain case, the stress tensor is simply :math:`\sigma_{ij}=\tau\left(\delta_{i\eta}\delta_{j\zeta}+\delta_{i\zeta}\delta_{j\eta}\right)` 
and may be equated to the shear stress as,

.. math::

   \tau=\Gamma_{\eta\zeta}\bar{\sigma}\left(t\right)~~~~\left(\eta\neq\zeta\right).

The elastic strain may then simply be written as 

.. math::

   \varepsilon_{\eta\zeta}^{\text{el}}=\frac{\tau}{2\mu}=\frac{\Gamma_{\eta\zeta}}{2\mu}\bar{\sigma}\left(t\right).

To find the plastic strain rate, the plastic work equivalency is recalled such that,

.. math::

   \sigma_{ij}\dot{\varepsilon}_{ij}^p=2\sigma_{\eta\zeta}\dot{\varepsilon}^p_{\eta\zeta}=\bar{\sigma}\dot{\bar{\varepsilon}}^p,~~~\left(\eta\neq\zeta\right)

which produces an expression for the plastic strain rate as,

.. math::
   :label: bvp-eqn-shearStrainRate

   \dot{\varepsilon}^p_{\eta\zeta}=\frac{1}{2\Gamma_{\eta\zeta}}\dot{\bar{\varepsilon}}^p,~~~\left(\eta\neq\zeta\right).

Integrating :eq:`bvp-eqn-shearStrainRate` yields,

.. math::

   \varepsilon^p_{\eta\zeta}\left(t\right)=\frac{1}{2\Gamma_{\eta\zeta}}\bar{\varepsilon}^p,~~~\left(\eta\neq\zeta\right)

leading to a total strain of the form,

.. math::

   \varepsilon_{\eta\zeta}\left(t\right)=\frac{\Gamma_{\eta\zeta}}{2\mu}\bar{\sigma}\left(t\right)+\frac{1}{2\Gamma_{\eta\zeta}}\dot{\bar{\varepsilon}}^pt,~~~\left(\eta\neq\zeta\right)

and

.. math::

   \lambda\left(t\right)=\exp\left(\varepsilon_{\eta\zeta}\left(t\right)\right),~~~\left(\eta\neq\zeta\right).

J\ :sub:`2` Plasticity
----------------------

In the case of a isotropic J\ :sub:`2` effective stress, the pure shear coefficients are,

.. math::

   \Gamma_{\eta\zeta} = \frac{1}{\sqrt{3}}~~~\left(\eta\neq\zeta\right).

Hosford Plasticity
------------------

Although isotropic, the Hosford effective stress definition is non-quadratic leading to a stress multiplier of,

.. math::

   \Gamma_{\eta\zeta} = \frac{1}{\left[1+2^{a-1}\right]^{1/a}}~~~\left(\eta\neq\zeta\right).

Hill Plasticity
---------------

Like the uniaxial case, for the pure shear response a direct connection may be made between the :math:`R` stress ratios and :math:`\Gamma_{\eta\zeta}` such that,

.. math::

   \Gamma_{\eta\zeta} = \frac{R_{\eta\zeta}}{\sqrt{3}}~~~\left(\eta\neq\zeta\right).

Barlat Plasticity
-----------------

The Barlat effective stress definition produces stress relationships of the form,

.. math::

   \Gamma_{\eta\zeta} = \frac{1}{\xi_{\eta\zeta}}~~~\left(\eta\neq\zeta\right),

where,

.. math::

   \xi_{12}=\left\{\frac{1}{2}\left[|c_{44}^{\prime}-c_{44}^{\prime\prime}|^{a}+|c_{44}^{\prime}+c_{44}^{\prime\prime}|^a+|c_{44}^{\prime}|^a+|c_{44}^{\prime\prime}|^a\right]\right\}^{1/a},
   \\
   \xi_{23}=\left\{\frac{1}{2}\left[|c_{55}^{\prime}-c_{55}^{\prime\prime}|^{a}+|c_{55}^{\prime}+c_{55}^{\prime\prime}|^a+|c_{55}^{\prime}|^a+|c_{55}^{\prime\prime}|^a\right]\right\}^{1/a},
   \\
   \xi_{31}=\left\{\frac{1}{2}\left[|c_{66}^{\prime}-c_{66}^{\prime\prime}|^{a}+|c_{66}^{\prime}+c_{66}^{\prime\prime}|^a+|c_{66}^{\prime}|^a+|c_{66}^{\prime\prime}|^a\right]\right\}^{1/a}. 
