*********
BCJ Model
*********

.. \index{PARAMETERS FOR MODEL BCJ!in BCJ material model}
.. \index{YOUNGS MODULUS!in BCJ material model}
.. \index{POISSONS RATIO!in BCJ material model}
.. \index{SHEAR MODULUS!in BCJ material model}
.. \index{BULK MODULUS!in BCJ material model}
.. \index{LAMBDA!in BCJ material model}
.. \index{C1!in BCJ material model}
.. \index{C2!in BCJ material model}
.. \index{C3!in BCJ material model}
.. \index{C4!in BCJ material model}
.. \index{C5!in BCJ material model}
.. \index{C6!in BCJ material model}
.. \index{C7!in BCJ material model}
.. \index{C8!in BCJ material model}
.. \index{C9!in BCJ material model}
.. \index{C10!in BCJ material model}
.. \index{C11!in BCJ material model}
.. \index{C12!in BCJ material model}
.. \index{C13!in BCJ material model}
.. \index{C14!in BCJ material model}
.. \index{C15!in BCJ material model}
.. \index{C16!in BCJ material model}
.. \index{C17!in BCJ material model}
.. \index{C18!in BCJ material model}
.. \index{C19!in BCJ material model}
.. \index{C20!in BCJ material model}
.. \index{DAMAGE EXPONENT!in BCJ material model}
.. \index{INITIAL ALPHA{\_}XX!in BCJ material model}
.. \index{INITIAL ALPHA{\_}YY!in BCJ material model}
.. \index{INITIAL ALPHA{\_}ZZ!in BCJ material model}
.. \index{INITIAL ALPHA{\_}XY!in BCJ material model}
.. \index{INITIAL ALPHA{\_}YZ!in BCJ material model}
.. \index{INITIAL ALPHA{\_}XZ!in BCJ material model}
.. \index{INITIAL KAPPA!in BCJ material model}
.. \index{INITIAL DAMAGE!in BCJ material model}
.. \index{YOUNGS MODULUS FUNCTION!in BCJ material model}
.. \index{POISSONS RATIO FUNCTION!in BCJ material model}
.. \index{SPECIFIC HEAT!in BCJ material model}
.. \index{THETA OPT!in BCJ material model}
.. \index{FACTOR!in BCJ material model}
.. \index{RHO!in BCJ material model}
.. \index{TEMPO!in BCJ material model}

.. code-block:: sierrainput

   BEGIN PARAMETERS FOR MODEL BCJ
     #
     # Elastic constants
     #
     YOUNGS MODULUS = <real>
     POISSONS RATIO = <real>
     SHEAR MODULUS  = <real>
     BULK MODULUS   = <real>
     LAMBDA         = <real>
     TWO MU         = <real>
     #
     #
     #
     C1 = <real>c1
     C2 = <real>c2
     C3 = <real>c3
     C4 = <real>c4
     C5 = <real>c5
     C6 = <real>c6
     C7 = <real>c7
     C8 = <real>c8
     C9 = <real>c9
     C10 = <real>c10
     C11 = <real>c11
     C12 = <real>c12
     C13 = <real>c13
     C14 = <real>c14
     C15 = <real>c15
     C16 = <real>c16
     C17 = <real>c17
     C18 = <real>c18
     C19 = <real>c19
     C20 = <real>c20
     DAMAGE EXPONENT = <real>damage_exponent
     INITIAL ALPHA_XX = <real>alpha_xx
     INITIAL ALPHA_YY = <real>alpha_yy
     INITIAL ALPHA_ZZ = <real>alpha_zz
     INITIAL ALPHA_XY = <real>alpha_xy
     INITIAL ALPHA_YZ = <real>alpha_yz
     INITIAL ALPHA_XZ = <real>alpha_xz
     INITIAL KAPPA = <real>initial_kappa
     INITIAL DAMAGE = <real>initial_damage
     YOUNGS MODULUS FUNCTION = <string>ym_function_name
     POISSONS RATIO FUNCTION = <string>pr_function_name
     SPECIFIC HEAT = <real>specific_heat
     THETA OPT = <integer>theta_opt
     FACTOR = <real>factor
     RHO    = <real>rho
     TEMP0  = <real>temp0
   END [PARAMETERS FOR MODEL BCJ]

The BCJ plasticity model is a state-variable model for describing the finite deformation behavior of metals. It uses a multiplicative decomposition of the deformation gradient into elastic, volumetric plastic, and deviatoric parts. The model considers the natural configuration defined by this decomposition and its associated thermodynamics. The model incorporates strain rate and temperature sensitivity, and damage, through a yield-surface approach in which state variables follow a hardening-minus-recovery format.

Because the BCJ model has such an extensive list of parameters, we will not present the usual synopsis of parameter names with command lines. As with most other material models, the ``thermal strain option`` is used to define thermal strains. See the Sierra/SM User Manual for further information on defining and activating thermal strains. In addition, only two of the five elastic constants are required. The user should consult [:footcite:`mat:ref:bamm1`, :footcite:`mat:ref:bamm2`, :footcite:`mat:ref:bamm3`] for a description of the various parameters. The parameters for the ``SPECIFIC HEAT``, ``THETA OPT``, ``FACTOR``, ``RHO``, and ``TEMP0`` command lines are used to accommodate changes to the model for heat generation due to plastic dissipation. For coupled solid/thermal calculations, the plastic dissipation rate is stored as a state variable and passed to a thermal code as a heat source term. For uncoupled calculations, temperature is stored as a state variable, and temperature increases due to plastic dissipation are calculated within the material model.

If temperature is provided from an external source, ``theta_opt`` is set to 1. If the temperature is calculated by the BCJ model, ``theta_opt`` is set to 1.

Output variables available for this model are listed in :numref:`bcjstvar`.

.. _bcjstvar:

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

   ``BACK_STRESS_XX`` & back stress tensor - xx component
   ``BACK_STRESS_YY`` & back stress tensor - yy component
   ``BACK_STRESS_ZZ`` & back stress tensor - zz component
   ``BACK_STRESS_XY`` & back stress tensor - xy component
   ``BACK_STRESS_YZ`` & back stress tensor - yz component
   ``BACK_STRESS_ZX`` & back stress tensor - zx component
   ``KAPPA``          & hardening scalar
   ``DAMAGE``         & damage term                  
   ``DAMAGE_RATE``    & rate of change of damage term
   ``EQPS``           & equivalent plastic strain    
   ``THETA``          & temperature for adiabatic heating
   ``HEAT``           & rate of heating due to plastic dissipation
   ``YM``             & Young's modulus
   ``PR``             & Poisson's ratio

.. warning::

   Strongly rate-dependent models may fare poorly in implicit quasistatic solution.  In implicit analyses the rate term used to evaluate the current load step is the rate seen by the model in the previous load step.  This may cause the solution to oscillate between high- and low-rate equilibrium states from step to step.

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.. footbibliography::
