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

.. code-block:: sierrainput

   BEGIN PARAMETERS FOR MODEL ELASTIC_ORTHOTROPIC_SHELL
     #
     # required parameters
     #
     YOUNGS MODULUS RR = <real>
     YOUNGS MODULUS SS = <real>
     YOUNGS MODULUS TT = <real>
     POISSONS RATIO RS = <real>
     POISSONS RATIO ST = <real>
     POISSONS RATIO TR = <real>
     SHEAR MODULUS RS  = <real>
     SHEAR MODULUS RT  = <real>
     SHEAR MODULUS ST  = <real>
   END [PARAMETERS FOR ELASTIC_ORTHOTROPIC_SHELL]

The ELASTIC ORTHOTROPIC SHELL model describes the linear elastic response of an orthotropic material where the planar orientation of the principal material directions can be arbitrary.  This material uses the shell section orthotropic alignment commands described in the Sierra/SM User Manual to define the local RST coordinate system.

The general 3D response of an orthotropic material is given above. For the shell model, the ABC coordinate system is replaced with the RST coordinate system. For plane stress the stiffness is calculated with the constraint that :math:`\sigma_{TT} = 0`. From this constraint the thickness strain, which is used to calculate the thickness change for the shell, is

.. math::

   \varepsilon_{TT} = -\frac{1}{1-\nu_{RS}\nu_{SR}}\left[
     \left(\nu_{RT}+\nu_{RS}\nu_{ST}\right) \varepsilon_{RR}
   + \left(\nu_{ST}+\nu_{SR}\nu_{RT}\right) \varepsilon_{SS}
   \right]

and results in the following stiffness

.. math::

   \begin{Bmatrix}
     \sigma_{RR} \\
     \sigma_{SS} \\
     \sigma_{RS} \\
     \sigma_{ST} \\
     \sigma_{TR}
   \end{Bmatrix} =
   \begin{bmatrix}
   \bar{\mathbb{C}}_{RRRR} & \bar{\mathbb{C}}_{RRSS} & 0 & 0 & 0 \\ \\
   \bar{\mathbb{C}}_{RRSS} & \bar{\mathbb{C}}_{SSSS} & 0 & 0 & 0 \\ \\
   0 & 0 & 2G_{RS} & 0 & 0 \\ \\
   0 & 0 & 0 & 2G_{ST} & 0 \\ \\
   0 & 0 & 0 & 0 & 2G_{TR}
   \end{bmatrix}
   \begin{Bmatrix}
     \varepsilon_{RR} \\
     \varepsilon_{SS} \\
     \varepsilon_{TT} \\
     \varepsilon_{RS} \\
     \varepsilon_{ST} \\
     \varepsilon_{TR}
   \end{Bmatrix}

where

.. math::

   \bar{\mathbb{C}}_{RRRR} = \frac{E_{R}}{1-\nu_{RS}\nu_{SR}} \;\;\; ; \;\;\;
   \bar{\mathbb{C}}_{SSSS} = \frac{E_{S}}{1-\nu_{RS}\nu_{SR}} \;\;\; ; \;\;\;
   \bar{\mathbb{C}}_{RRSS} = \frac{\nu_{SR}E_{R}}{1-\nu_{RS}\nu_{SR}} = \frac{\nu_{RS}E_{S}}{1-\nu_{RS}\nu_{SR}}

In the above command blocks, all the following are required inputs.

- Young's modulus of the orthogonal R, S, and T axes are defined with the ``YOUNGS MODULUS RR``, ``YOUNGS MODULUS SS`` and ``YOUNGS MODULUS TT`` command lines.
- ``POISSONS RATIO RS`` defines the strain in the S direction when the material is pulled in the R direction
- ``POISSONS RATIO ST`` defines the strain in the T direction when the material is pulled in the S direction
- ``POISSONS RATIO TR`` defines the strain in the R direction when the material is pulled in the T direction
- The shear moduli in each of shear directions are defined with the ``SHEAR MODULUS RS``, ``SHEAR MODULUS RT``, and ``SHEAR MODULUS ST`` command lines.

Error messages for the ELASTIC ORTHOTROPIC SHELL model concern input that results in a non-positive definite stiffness. The error messages, and their meanings, are

``Model parameters chosen so that determinant of stiffness < 0``

.. math::

   1
   - \nu_{RS}^{2}\frac{E_{S}}{E_{R}}
   - \nu_{ST}^{2}\frac{E_{T}}{E_{S}}
   - \nu_{TR}^{2}\frac{E_{R}}{E_{T}}
   - 2\nu_{RS}\nu_{ST}\nu_{TR} < 0

``Model parameters chosen so that RS sub-determinant of stiffness < 0``

.. math::

   1 - \nu_{RS}^{2}\frac{E_{S}}{E_{R}} < 0

``Model parameters chosen so that ST sub-determinant of stiffness < 0``

.. math::

   1 - \nu_{ST}^{2}\frac{E_{T}}{E_{S}} < 0

``Model parameters chosen so that TR sub-determinant of stiffness < 0``

.. math::

   1 - \nu_{TR}^{2}\frac{E_{R}}{E_{T}} < 0

.. warning::

   In previous releases the ``ELASTIC_ORTHOTROPIC_SHELL`` model required input parameter ``POISSONS RATIO SR``, which could have led to an inconsistent set of parameters.  The model also previously did not require ``YOUNGS_MODULUS_TT``, ``POISSONS RATIO ST``, or ``POISSONS RATIO TR``, which are now required parameters in the current version. For backward compatibility \emph{only}, the original input syntax remains valid.  However, the new behavior of the model is to ignore any input value of :math:`\nu_{SR}` and compute it automatically as :math:`\nu_{SR} = \nu_{RS}(E_{SS}/E_{RR})`.  If :math:`E_{TT}` is not input, it is computed as :math:`E_{TT} = (E_{RR} + E_{SS})/2` by default. If no value is input for :math:`\nu_{ST}` or :math:`\nu_{TR}`, it will default to zero. For best results \emph{all required values should be input} in future usages of this model.
