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


Calculation method used for the reference solution
--------------------------------------------------------

The :math:`\mathrm{3D}` speed model is written as:

:math:`\mathrm{\{}\begin{array}{cccc}\dot{\sigma }\mathrm{-}\dot{\rho }\Lambda {\varepsilon }_{e}\mathrm{-}\rho \Lambda \dot{{\varepsilon }_{e}}& \mathrm{=}& 0& (\Lambda \text{tenseur élasticité isotrope linéaire})\\ \dot{\beta }\mathrm{-}\dot{p}D\mathrm{exp}(\frac{{\sigma }_{H}}{{\sigma }_{1}\rho })& \mathrm{=}& 0& \\ \dot{\varepsilon }\mathrm{-}\dot{{\varepsilon }_{e}}\mathrm{-}\rho \dot{p}\frac{\mathrm{\partial }f}{\mathrm{\partial }\sigma }& \mathrm{=}& 0& \\ \dot{f}& \mathrm{=}& 0& \end{array}`

which, in the case of an imposed tension-shear load :math:`(\sigma (t)\mathrm{=}\alpha (t)\left[\begin{array}{ccc}{\sigma }_{0}& {\tau }_{0}& 0\\ {\tau }_{0}& 0& 0\\ 0& 0& 0\end{array}\right])`, leads to the integration of a system of 6 ordinary differential equations in :math:`y\mathrm{=}(\varepsilon ,\gamma ,{\varepsilon }_{e},{\gamma }_{e},\beta ,p)` of the form :math:`A(y,t)\dot{y}\mathrm{=}G(y,t)`.


.. image:: images/Object_8.svg
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.. _RefImage_Object_8.svg:

with to :math:`t\mathrm{=}0`:

:math:`f\mathrm{=}0`, :math:`\rho (0)\mathrm{=}1`, :math:`\beta (0)\mathrm{=}0`

from where:

:math:`\alpha (0){\sigma }_{\mathit{eq}0}\mathrm{-}R(0)+D{\sigma }_{1}{F}_{0}\mathrm{exp}(\frac{\alpha (0){\sigma }_{0}}{3{\sigma }_{1}})`

which is solved by a method of NEWTON for :math:`\alpha (0)`:

:math:`\mathrm{\{}\begin{array}{ccccc}\varepsilon (0)& \mathrm{=}& \frac{1}{E}\alpha (0){\sigma }_{0}& \mathrm{=}& {\varepsilon }_{e}(0)\\ \gamma (0)& \mathrm{=}& \frac{1}{2\mu }\alpha (0){\tau }_{0}& \mathrm{=}& {\gamma }_{e}(0)\\ p(0)& \mathrm{=}& 0& & \end{array}`


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

We impose :math:`\alpha (t)\mathrm{=}\alpha (0)+t` with :math:`{\sigma }_{0}\mathrm{=}{\tau }_{0}\mathrm{=}150\mathit{MPa}`.

We get :math:`\alpha (0)\mathrm{=}1.73138` and :math:`\alpha (1)\mathrm{=}2.73138`.

System :math:`(S)` is then solved numerically by a 'Backward difference formula' using the NAG scientific library on CRAY. Reference result= :math:`(\varepsilon ,\gamma ,\beta ,\rho )` at nodes at :math:`t\mathrm{=}1`.


Uncertainty about the solution
---------------------------

Library uncertainty NAG.


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


1. Library user manual NAG out of CRAY.