.. _models-orthotropic_crush:

***********************
Orthotropic Crush Model
***********************

Theory
======

The orthotropic crush model in LAMÉ is designed to model the energy
absorbing capability of crushable orthotropic materials, e.g. aluminum
honeycomb, and is empirically based.  The formulation follows that used
for metallic honeycomb materials in LS-DYNA [:footcite:`mat:ref:whir`].  Three response
regimes are assumed for this material: (*i*) orthotropic elastic, (*ii*)
crush, and (*iii*) complete compaction (fully crushed).  
During the elastic regime, the model exhibits the response of an elastic, orthotropic
material with *all Poisson's ratio equal to zero*.  After full compaction, the response is taken to be 
that of an isotropic, perfectly plastic material and the response between these two stages is tailored
to smoothly transition between the two extremes.  Crushing, incorporating both nonlinear elastic and 
plastic-like behaviors, is taken to begin as soon as
volumetric contraction is noted (:math:`J=\det\left(F_{ij}\right)<1`).  As such, the purely elastic response 
is primarily seen during cyclic loadings in which the material is unloaded.  
An internal state variable, :math:`J_c`, is introduced to track the crushed state of the material and is defined as the 
minimum :math:`J` over the entire deformation history such that,

.. math::

   J_c=\min_{t>0}\left[J\left(t\right)\right].

The crushing process manifests through two distinct behaviors: (*i*) the elastic properties scale linearly with the 
crush state from the initial orthotropic state to the of the final isotropic completely compacted material; and (*ii*) a 
plastic-like response is observed associated with corresponding crush curves (analogous to hardening curves).  

Before complete compaction, the incremental constitutive relation may be written in terms of the rate of deformation tensor, 
:math:`D_{ij}`, as,

.. math::
   :label: oc:theory1

   \left\{ \begin{matrix}
          \stackrel{\circ}{\sigma}_{11} \\
          \stackrel{\circ}{\sigma}_{22} \\
          \stackrel{\circ}{\sigma}_{33} \\
          \stackrel{\circ}{\sigma}_{12} \\
          \stackrel{\circ}{\sigma}_{23} \\
          \stackrel{\circ}{\sigma}_{31}
          \end{matrix} \right\}
   = \left[ \begin{matrix}
          \hat{E}_{11} & 0 & 0 & 0 & 0 & 0  \\
          0 & \hat{E}_{22} & 0 & 0 & 0 & 0  \\
          0 & 0 & \hat{E}_{33} & 0 & 0 & 0  \\
          0 & 0 & 0 & 2\hat{G}_{12} & 0 & 0 \\
          0 & 0 & 0 & 0 & 2\hat{G}_{23} & 0 \\
          0 & 0 & 0 & 0 & 0 & 2\hat{G}_{31}
          \end{matrix} \right]
   \left\{ \begin{matrix}
          D_{11} \\
          D_{22} \\
          D_{33} \\
          D_{12} \\
          D_{23} \\
          D_{31}
          \end{matrix} \right\}

where :math:`\hat{E}_{11}`, :math:`\hat{E}_{22}`, and :math:`\hat{E}_{33}` are the normal stiffness and
:math:`\hat{G}_{12}`, :math:`\hat{G}_{23}`, and :math:`\hat{G}_{31}` are the shear stiffness.  A clear decoupling 
between the different directional components is evident in :eq:`oc:theory1`.  All six
stiffness components are assumed to be functions of the current compaction level which 
may be defined as :math:`1-J_c` and the evolution of these terms is responsible for crushing 
behavior :math:`(*i*)` alluded to previously.

The functional forms of the stiffness are given by,

.. math::
   :label: oc:th:eqn1

   \hat{E}_{\beta} & = E_{\beta}+\alpha\left(E-E_{\beta}\right)\qquad\qquad \beta=11,~22,~33 \nonumber \\
   \hat{G}_{\gamma} & = G_{\gamma}+\alpha\left(G-G_{\gamma}\right)\qquad\qquad \gamma=12,~23,~31,

where :math:`E` and :math:`G` are the Young's and shear moduli, respectively, of the
fully compacted material while :math:`E_{\beta}` and :math:`G_{\gamma}` are the 
input orthotropic elastic stiffness components of the virgin, uncompacted material.  It is assumed 
that these stiffness vary linearly
between the pre- and post-compacted material such that,

.. math::

   \alpha=\frac{\left(1-J_c\right)}{V_{min}},


with :math:`V_{min}` being the minimum relative volume (or maximum compaction).

With respect to the second behavior observed during crushing, a plastic-like response governed by crush curves is observed.  
Given the decoupling between the different stresses and deformations, a crush curve needs 
to be defined for each of the six normal and shear stresses.  An example of such a curve 
is presented in :numref:`oc_crush`, and three distinct regions are evident.  
Initially, at low compaction levels, a plateau is observed.  This plateau is essentially 
an initial crush strength and prior to this stress level all nonlinear deformations associated with 
material compaction manifest through changes in the respective moduli.  When the stress reaches 
the specified levels, however, the curves play a role analogous to the hardening curve and the 
material stress follows the curve.  Physically, the plateau is associated with crushing the internal honeycomb 
or foam structure of the material.  
As the material approaches full compaction and microstructural contact effects become important, 
a sharp rise in the stress is noted (see :math:`\approx 0.6\leq 1-J_c\leq0.7=V_{min}` in :numref:`oc_crush`).  
After complete compaction another plateau corresponding to perfect plasticity is evident.

.. _oc_crush:

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

   An example of an input crush curve for an aluminum honeycomb.

Above some value of compaction (:math:`1-J_c=V_{min}`), the material will be fully compacted
and behave as an elastic, perfectly plastic material. The fully compacted
response is given by the Young's modulus, :math:`E`, Poisson's ratio,
:math:`\nu`, and the yield stress, :math:`\sigma_{y}`. 
Details of this response may be found in previous sections on the various elastic-plastic models (e.g. :numref:`models-elastic_plastic-theory`).

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

Implementation of the orthotropic crush model involves addressing two cases: before and after complete 
compaction.  When the material is fully crushed, the model reduces to that of an isotropic perfectly 
plastic response.  As corresponding isotropic elastic-plastic models with various hardenings have 
been extensively explored in prior sections, this response will not be discussed here and the reader is referred
to those sections (e.g. :numref:`models-elastic_plastic-implementation`).  The two cases are distinguished by 
the previous compaction state variable, :math:`J_c^{n}`, where :math:`J_c^{n+1}=\min\left[J_c^n,J^{n+1}\right]` with 
:math:`J^{n+1}=\det\left(F_{ij}^{n+1}\right)=\det\left(V_{ij}^{n+1}\right)`.  If :math:`J_c^{n}>1-V_{min}`, 
the material has not yet fully crushed and the response is evaluated as discussed in the following.

To determine the material state prior to complete compaction, the current values of orthogonal stiffness 
must be determined via :eq:`oc:th:eqn1` noting

.. math::

   \alpha^{n+1}=\frac{1-J_c^{n}}{V_{min}}.
 



By assuming completely elastic deformation, trial stresses may then be computed as,

.. math::

   \sigma^{tr}_{11} & = \sigma^n_{11}+\Delta t\hat{E}_{11}\left(\alpha^{n+1}\right)d^{n+1}_{11}, \nonumber \\
   \sigma^{tr}_{22} & = \sigma^n_{22}+\Delta t\hat{E}_{22}\left(\alpha^{n+1}\right)d^{n+1}_{22}, \nonumber \\
   \sigma^{tr}_{33} & = \sigma^n_{33}+\Delta t\hat{E}_{33}\left(\alpha^{n+1}\right)d^{n+1}_{33}, \\
   \sigma^{tr}_{12} & = \sigma^n_{12}+2\Delta t\hat{G}_{12}\left(\alpha^{n+1}\right)d^{n+1}_{12}, \nonumber \\
   \sigma^{tr}_{23} & = \sigma^n_{23}+2\Delta t\hat{G}_{23}\left(\alpha^{n+1}\right)d^{n+1}_{23}, \nonumber \\
   \sigma^{tr}_{31} & = \sigma^n_{31}+2\Delta t\hat{G}_{31}\left(\alpha^{n+1}\right)d^{n+1}_{31}, \nonumber

with :math:`d_{ij}^{n+1}` being the unrotated rate of deformation tensor.  Given the decoupling between the different 
stress components, the various trial stresses are considered individually. 
Specifically, each trial stress must be compared to the crush stress for the current compaction level.  Denoting 
:math:`\sigma^{crush}_{\beta}=\hat{\sigma}_{\beta}\left(1-J_c^{n+1}\right)` (with :math:`\beta= 11,~22,~33,~12,~23,` or :math:`31`) 
to be the current crush stress specified by the crush curve, the current stress of interest is,

.. math::

   \sigma_{\beta}^{n+1} = \left\{\begin{array}{cc}\sigma^{tr}_{\beta}, & |\sigma^{tr}_{\beta}| \leq \sigma^{crush}_{\beta} \\
                               \text{sgn}\left(\sigma^{tr}_{\beta}\right) \sigma^{crush}_{\beta}, & |\sigma^{tr}_{\beta}| > \sigma^{crush}_{\beta}, \end{array}\right.

where :math:`\text{sgn}\left(x\right)` returns the sign of the argument and is used as :math:`\sigma^{crush}_{\beta}` is entered as a positive number.

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

The orthotropic crush model was verified through a series of uniaxial compression tests.  
Given the lack of coupling between the different directions, 
such a variety of tests were performed to test each loading component.
One set of material properties was used for all tests and they are given in :numref:`tab-oc-verProps`.

.. _tab-oc-verProps:

.. csv-table:: The material properties for the orthotropic crush model used for the uniaxial crush tests.
   :align: center
   :delim: &

   :math:`E_{11}`    & 50.0 ksi & :math:`E`  & 1000.0 ksi
   :math:`E_{22}` & 220.0 ksi  & :math:`\nu` & 0.25
   :math:`E_{33}`    & 10.0 ksi & :math:`\sigma_y` & 2.0 ksi
   :math:`G_{12}`    & 110.0 ksi &  &
   :math:`G_{23}` & 5.0 ksi & :math:`V_{min}` & 0.7
   :math:`G_{31}` & 25.0 ksi &  &

The crush curves used as input for these tests are given in :numref:`fig-oc-crushCurves`.

.. _fig-oc-crushCurves:

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

   Input crush curves used for uniaxial crush analysis.

To test this model, both the anisotropic nature and different deformation regimes need to be tested.  
Therefore, given the decoupled directional nature prior to complete compaction, each component will be 
tested.  For the diagonal stress components, a simple uniaxial displacement of the form,

.. math::

   u_{i}=-\lambda\delta_{i\beta},
 
where :math:`\beta=1,~2,` or :math:`3` corresponding to the directional component being tested is applied.  In such cases (with a 
monotonically increasing :math:`\lambda`), :math:`J_c = 1-\lambda`.  The model described in the prior to sections
can be easily evaluated analytically under such conditions, and the corresponding analytical
and numerical results are presented in :numref:`fig-oc-crushResults`.

.. _fig-oc-crushResults:

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

   Analytical and numerical results for uniaxial crush cases.

User Guide
==========

.. code-block:: sierrainput

   BEGIN PARAMETERS FOR MODEL ORTHOTROPIC_CRUSH
     #
     # Elastic constants - Post lock-up
     YOUNGS MODULUS = <real>
     POISSONS RATIO = <real>
     SHEAR MODULUS  = <real>
     BULK MODULUS   = <real>
     LAMBDA         = <real>
     TWO MU         = <real>
     #
     # Orthotropic Elastic properties - Pre-Crush 
     #
     EX             = <real>
     EY             = <real>
     EZ             = <real>
     GXY            = <real>
     GYZ            = <real>
     GZX            = <real>
     #
     # Crush properties
     #   
     CRUSH XX       = <string>
     CRUSH YY       = <string>
     CRUSH ZZ       = <string>
     CRUSH XY       = <string>
     CRUSH YZ       = <string>
     CRUSH ZX       = <string>
     VMIN           = <real>
     #
     # Post lock-up yield properties
     #
     YIELD STRESS   = <real>
     #
   END [PARAMETERS FOR MODEL ORTHOTROPIC_CRUSH]

In the above command blocks:


- The ``EX``, ``EY``, ``EZ``, ``GXY``, ``GYZ``, and ``GZX`` command lines define, respectively, the initial, pre-crush directional moduli :math:`E_{xx}`, :math:`E_{yy}`, :math:`E_{zz}`, :math:`G_{xy}`, :math:`G_{yz}`,  and :math:`G_{zx}` from :eq:`oc:theory1`.
- ``CRUSH`` ``XX``, ``YY``, ``ZZ``, ``XY``, ``YZ``, and ``ZX`` inputs require the name of a function defined via a ``FUNCTION`` command line in the SIERRA scope.  These functions describe the directional crush characteristics of the material and give the current stress value (in a direction) as a function of the current compaction (:math:`1-J_c`).
- The command ``VMIN`` defines the minimum relative volume of the material that is achieved when the material is  completely crushed.  This parameter may also be considered as the maximum compaction.
- The elastic constant commands refer to the *post lock-up, fully compacted* isotropic response of the material.
- ``YIELD STRESS`` refers to the plateau stress of the material after lock-up when the response is perfectly plastic.

Output variables available for this model are listed in :numref:`out-tab-orthcrushstvar`. For information about the orthotropic crush model, consult [:footcite:`mat:ref:whir`].

.. _out-tab-orthcrushstvar:

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

   ``CRUSH`` & current (unrecoverable) compaction/relative volume

.. raw::
   html

   <hr>

.. footbibliography::
