2. Benchmark solution#

2.1. Calculation method used for the reference solution#

The \(\mathrm{3D}\) speed model is written as:

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

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

_images/Object_8.svg

with to \(t\mathrm{=}0\):

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

from where:

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

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

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

2.2. Benchmark results#

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

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

System \((S)\) is then solved numerically by a “Backward difference formula” using the NAG scientific library on CRAY. Reference result= \((\varepsilon ,\gamma ,\beta ,\rho )\) at nodes at \(t\mathrm{=}1\).

2.3. Uncertainty about the solution#

Library uncertainty NAG.

2.4. Bibliographical references#

  1. Library user manual NAG out of CRAY.