6. D modeling

6.1. Characteristics of modeling

The 3D modeling is carried out on a 3D hardware point with STAT_NON_LINE. The difference compared to modeling B is the calculation of the initial state by a thermal loading. To bring the sample to the initial isotropic stress \({P}_{0}=10\mathit{kPa}\), the sample is brought from 20° to 30° Celsius. The movements of the sample are blocked, so that thermal expansion brings the sample into compression. We get:

\({\mathrm{\sigma }}_{0}=\frac{E}{9\left(1-2\mathrm{\nu }\right)}\mathrm{\alpha }\mathrm{\Delta }T={P}_{0}\)

Let the following value for the coefficient of thermal expansion be: \(\mathrm{\alpha }=\frac{9\left(1-2\mathrm{\nu }\right){P}_{0}}{E\mathrm{\Delta }T}\)

6.2. Tested sizes and results

6.2.1. Tested values

The solutions are post-treated at point \(C\), in terms of:

• vertical constraint \({\mathrm{\sigma }}_{\mathit{zz}}\);

• horizontal deformation \({\mathrm{ϵ}}_{\mathit{xx}}\);

• \({e}^{P}=\Vert {e}^{P}\Vert\) deviatoric plastic deformation norm

They are compared to an analytical solution (described in the next paragraph) in terms of maximum difference between \(t=0\) and \(t=20\). The results are summarised in the following tables:

\(Q=\sqrt{\frac{1}{2}\underline{\underline{s}}\mathrm{:}\underline{\underline{s}}}[\mathit{Pa}]\)

Variable	Absolute deviation | Code_Aster − Analytics |
\({\mathrm{\sigma }}_{\mathit{zz}}\)	0
\({\mathrm{ϵ}}_{\mathit{xx}}\)	3.10-5
\({e}^{P}\)	1.333 10-5

6.2.2. Comments

The difference with the analytical solution is very small.