.. _material-elastic-orthotropic:

*************************
Elastic Orthotropic Model
*************************

.. code-block:: sierrainput

   BEGIN PARAMETERS FOR MODEL ELASTIC_ORTHOTROPIC
       #
       # Elastic constants
       YOUNGS MODULUS = <real>youngs_modulus
       POISSONS RATIO = <real>poissons_ratio
       SHEAR MODULUS = <real>shear_modulus
       BULK MODULUS = <real>bulk_modulus
       LAMBDA = <real>lambda
       TWO MU = <real>two_mu
       #
       # required parameters
       E11 = <real>e11
       E22 = <real>e22
       E33 = <real>e33
       NU12 = <real>nu12
       NU13 = <real>nu13
       NU23 = <real>nu23
       G12 = <real>g12
       G13 = <real>g13
       G23 = <real>g23
       COORDINATE SYSTEM = <string>coordinate_system_name
       #
       # optional parameters
       ANGLE_1_ABSCISSA = <real>angle_1_abscissa
       ANGLE_2_ABSCISSA = <real>angle_2_abscissa
       ANGLE_3_ABSCISSA = <real>angle_3_abscissa
       ROTATION_AXIS_1 = <real>rotation_axis_1
       ROTATION_AXIS_2 = <real>rotation_axis_2
       ROTATION_AXIS_3 = <real>rotation_axis_3
       ANGLE_1_FUNCTION = <string>angle_1_function_name
       ANGLE_2_FUNCTION = <string>angle_2_function_name
       ANGLE_3_FUNCTION = <string>angle_3_function_name
       THERMAL_STRAIN_11_FUNCTION = 
         <string>thermal_strain_11_function_name
       THERMAL_STRAIN_22_FUNCTION = 
         <string>thermal_strain_22_function_name
       THERMAL_STRAIN_33_FUNCTION = 
         <string>thermal_strain_33_function_name
   END [PARAMETERS FOR MODEL ELASTIC_ORTHOTROPIC]

The elastic orthotropic model is a Kirchhoff linear elastic constitutive relation that is useful for modeling polymer matrix composite structures undergoing small strains.  Required inputs are

- two of the five general elastic material constants,
- directional properties, and
- the coordinate system.

The following is a brief description of each input.

- See the Sierra/SM User Manual for more information on elastic constants input.
- In each material direction, moduli :math:`E_{11}`, :math:`E_{22}`, and :math:`E_{33}` are defined with the ``E11``, ``E22``, and ``E33`` command lines, Poisson's ratios :math:`\nu_{12}`, :math:`\nu_{13}`, and :math:`\nu_{23}` are given by the ``NU12``, ``NU13``, and ``NU23`` command lines, and shear moduli :math:`G_{12}`, :math:`G_{13}`, and :math:`G_{23}` are defined using command lines ``G12``, ``G13``, and ``G23``.
- Principal material direction specification requires a user specified coordinate system given by the ``COORDINATE SYSTEM`` command line, as detailed in the Sierra/SM User Manual.  The material orientation may then be specified using **one** of the following approaches:

  - Three spatially varying, ordered Euler angle functions are given in terms of global coordinates (X, Y, Z = 1, 2, 3) for rotations about a corresponding local axis:
    
    - The rotation angle function abscissas :math:`x_1`, :math:`x_2`, and :math:`x_3`, corresponding to the global system (X, Y, Z) = (1, 2, 3), are defined with the ``ANGLE_1_ABSCISSA``, ``ANGLE_2_ABSCISSA``, and ``ANGLE_3_ABSCISSA`` command lines, respectively.
    - The axes of rotation :math:`i`, :math:`j`, and :math:`k` are defined with the ``ROTATION_AXIS_1``, ``ROTATION_AXIS_2``, and ``ROTATION_AXIS_3`` command lines, respectively.
    - The rotation angle functions :math:`\theta_{i}(x_{1})`, :math:`\theta_{j}(x_{2})`, and :math:`\theta_{k}(x_{3})` are defined with the ``ANGLE_1_FUNCTION``, ``ANGLE_2_FUNCTION``, and ``ANGLE_3_FUNCTION`` command lines, respectively.  Angles are specified in degrees. The Sierra/SM User Manual provides additional details about defining functions.

    The rotation axis and angle are applied successively in order (1, 2, 3), or (X, Y, Z); that is, each sequential Euler angle rotation defines a new coordinate system in which the subsequent rotation axis and angle are defined.

  - Alternatively, the angles and axes may be defined directly at each element integration point using ``INITIAL CONDITION`` command blocks, as described in the Sierra/SM User Manual.  Angles may be specified in degrees using the variables ``ANGLE_1``, ``ANGLE_2``, and ``ANGLE_3``, while axes are given by ``AXIS_1``, ``AXIS_2``, and ``AXIS_3``.

  - A final option is to initialize the rotation tensor to correspond to the local coordinate system defined in the ``COORDINATE SYSTEM`` command line.

  The resulting material directions may be visualized by requesting the element fields ``MATERIAL_DIRECTION_1``, ``MATERIAL_DIRECTION_2``, and ``MATERIAL_DIRECTION_3`` in the results output block.

- Functions to describe normal thermal engineering strains along the principal material directions are given by the ``THERMAL_STRAIN_11_FUNCTION``, ``THERMAL_STRAIN_22_FUNCTION``, and ``THERMAL_STRAIN_33_FUNCTION`` command lines. See the Sierra/SM User Manual for guidance on defining functions.

.. warning::

   The ``ELASTIC_ORTHOTROPIC`` model has not been tested in conjunction with the control stiffness implicit solver block.

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

.. _out-tab-elortho:

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

    1 & ``Q_XX`` & X component of the material 11 unit vector
    2 & ``Q_YY`` & Y component of the material 22 unit vector
    3 & ``Q_ZZ`` & Z component of the material 33 unit vector
    4 & ``Q_XY`` & Y component of the material 11 unit vector
    5 & ``Q_YZ`` & Z component of the material 22 unit vector
    6 & ``Q_ZX`` & X component of the material 33 unit vector
    7 & ``Q_YX`` & X component of the material 22 unit vector
    8 & ``Q_ZY`` & Y component of the material 33 unit vector
    9 & ``Q_XZ`` & Z component of the material 11 unit vector
    10 & ``ANGLE_1`` & Rotation angle about axis 1 (degrees)
    11 & ``ANGLE_2`` & Rotation angle about axis 2 (degrees)
    12 & ``ANGLE_3`` & Rotation angle about axis 3 (degrees)
    13 & ``AXIS_1`` & First axis of rotation
    14 & ``AXIS_2`` & Second axis of rotation
    15 & ``AXIS_3`` & Third axis of rotation
    16 & ``TH_STR_XX`` & Thermal stretch ratio in material 11 direction
    17 & ``TH_STR_YY`` & Thermal stretch ratio in material 22 direction
    18 & ``TH_STR_ZZ`` & Thermal stretch ratio in material 33 direction
    19 & ``MAT_STRAIN_XX`` & Green Lagrange strain in material 11 direction
    20 & ``MAT_STRAIN_YY`` & Green Lagrange strain in material 22 direction
    21 & ``MAT_STRAIN_ZZ`` & Green Lagrange strain in material 33 direction
    22 & ``MAT_STRAIN_XY`` & Green Lagrange strain in material 12 direction
    23 & ``MAT_STRAIN_YZ`` & Green Lagrange strain in material 23 direction
    24 & ``MAT_STRAIN_ZX`` & Green Lagrange strain in material 31 direction
    25 & ``MAT_STRESS_XX`` & 2nd P-K stress in material 11 direction
    26 & ``MAT_STRESS_YY`` & 2nd P-K stress in material 22 direction
    27 & ``MAT_STRESS_ZZ`` & 2nd P-K stress in material 33 direction
    28 & ``MAT_STRESS_XY`` & 2nd P-K stress in material 12 direction
    29 & ``MAT_STRESS_YZ`` & 2nd P-K stress in material 23 direction
    30 & ``MAT_STRESS_ZX`` & 2nd P-K stress in material 31 direction
    31 & ``MAT_LOG_STRAIN_XX`` & Log (Hencky) strain in material 11 direction
    32 & ``MAT_LOG_STRAIN_YY`` & Log (Hencky) strain in material 22 direction
    33 & ``MAT_LOG_STRAIN_ZZ`` & Log (Hencky) strain in material 33 direction
    34 & ``MAT_LOG_STRAIN_XY`` & Log (Hencky) strain in material 12 direction 
    35 & ``MAT_LOG_STRAIN_YZ`` & Log (Hencky) strain in material 23 direction
    36 & ``MAT_LOG_STRAIN_ZX`` & Log (Hencky) strain in material 31 direction
    37 & ``MAT_CAUCHY_STRESS_XX`` & Cauchy stress in material 11 direction
    38 & ``MAT_CAUCHY_STRESS_YY`` & Cauchy stress in material 22 direction
    39 & ``MAT_CAUCHY_STRESS_ZZ`` & Cauchy stress in material 33 direction
    40 & ``MAT_CAUCHY_STRESS_XY`` & Cauchy stress in material 12 direction
    41 & ``MAT_CAUCHY_STRESS_YZ`` & Cauchy stress in material 23 direction
    42 & ``MAT_CAUCHY_STRESS_ZX`` & Cauchy stress in material 31 direction
