Benchmark solution
=====================


Calculation method
------------------

The calculation is analytical.

We used the computer calculation program Mathematica to do it.

We know that the field of movement is:

:math:`\mathrm{dX}=\mathrm{2x}+\mathrm{3y}+\mathrm{4z}`

:math:`\mathrm{dY}=\mathrm{3x}+\mathrm{5y}+\mathrm{6z}`

:math:`\mathrm{dZ}=\mathrm{4x}+\mathrm{6y}+\mathrm{7z}`

The deformation field :math:`{\varepsilon }_{G}` in the global coordinate system is therefore constant and equal to:


.. image:: images/Object_6.svg
    :width: 98
    :height: 76


.. _RefImage_Object_6.svg:

Let :math:`P` be the transition matrix allowing a vector to pass from the global coordinate system :math:`(A,X,Y,Z)` to the local coordinate system :math:`(A,L,N,T)`.

Either

.. image:: images/Object_7.svg
    :width: 98
    :height: 76


.. _RefImage_Object_7.svg:

the deformation tensor in the local coordinate system. We have: :math:`{\varepsilon }_{L}\mathrm{=}P\mathrm{\cdot }{\varepsilon }_{G}\mathrm{\cdot }{P}^{T}`

The Hooke tensor

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    :width: 98
    :height: 76


.. _RefImage_Object_9.svg:

is known in the local coordinate system, or

.. image:: images/Object_10.svg
    :width: 98
    :height: 76


.. _RefImage_Object_10.svg:

the stress tensor in this coordinate system. We have:


.. image:: images/Object_11.svg
    :width: 98
    :height: 76


.. _RefImage_Object_11.svg:

We get the tensor

.. image:: images/Object_12.svg
    :width: 98
    :height: 76


.. _RefImage_Object_12.svg:

constraints in the global frame of reference by:

.. image:: images/Object_13.svg
    :width: 98
    :height: 76


.. _RefImage_Object_13.svg:

In the case where a temperature field is applied, the equations above are modified as follows:

The field of deformations

.. image:: images/Object_14.svg
    :width: 98
    :height: 76


.. _RefImage_Object_14.svg:

in the global frame of reference is always the same:


.. image:: images/Object_15.svg
    :width: 98
    :height: 76


.. _RefImage_Object_15.svg:

Either

.. image:: images/Object_16.svg
    :width: 98
    :height: 76


.. _RefImage_Object_16.svg:

the deformation tensor in the local coordinate system. We have:

.. image:: images/Object_17.svg
    :width: 98
    :height: 76


.. _RefImage_Object_17.svg:

The tensor of mechanical deformations in the local coordinate system is therefore equal to:


.. image:: images/Object_18.svg
    :width: 98
    :height: 76


.. _RefImage_Object_18.svg:

with

.. image:: images/Object_19.svg
    :width: 98
    :height: 76


.. _RefImage_Object_19.svg:

, the other components being zero

The Hooke tensor

.. image:: images/Object_20.svg
    :width: 98
    :height: 76


.. _RefImage_Object_20.svg:

is known in the local coordinate system. Either

.. image:: images/Object_21.svg
    :width: 98
    :height: 76


.. _RefImage_Object_21.svg:

the stress tensor in this coordinate system. We have:


.. image:: images/Object_22.svg
    :width: 98
    :height: 76


.. _RefImage_Object_22.svg:

We get the tensor

.. image:: images/Object_23.svg
    :width: 98
    :height: 76


.. _RefImage_Object_23.svg:

constraints in the global frame of reference by:

.. image:: images/Object_24.svg
    :width: 98
    :height: 76


.. _RefImage_Object_24.svg:


Benchmark results
----------------------

They are obtained by performing the operations described above with Mathematica.


Uncertainties about the solution
----------------------------

The uncertainty is zero because the solution is analytical.


Bibliographical references
---------------------------

For the description of Hooke matrices for transverse isotropic and orthotropic materials for 3D models, plane stresses and plane deformations, the reference chosen was:

*'Hooke matrix for orthotropic materials* '. Internal report applications in Mechanics No. 79‑018 by Jean-Claude Masson CISI.