8. G modeling#
8.1. Characteristics of modeling#
Modeling \(C\) is*tridimensional* and*static nonlinear*.
We first perform an elastic preconsolidation (\(\mathrm{ELAS}\)) of the sample up to \({p}_{c}=200\mathrm{kPa}\) (1st phase of the calculation). This preconsolidation takes place in \(1\), a time step between \(t=-2\) and \(t=0\).
The vertical displacement imposed on the upper facet varies between \(0\) and \(-0.2\) (2nd phase of the calculation) in \(100\) time steps between \(t=0\) and \(t=10\). The modeling considers large deformation kinematics with the operator GDEF_LOG (DEORMATION =” GDEF_LOG “).
In the integration of equilibrium equations, an update of the tangent matrix is required, which is provided by the routines of Hujeux’s law and significantly accelerates convergence. We also require the subdivision of the time step (command DEFI_LIST_INST) to deal with situations where local integration fails due to excessively large load increments. This feature is highly recommended.
8.2. Tested sizes and results#
8.2.1. Tested values#
The solutions are calculated at point \(C\) and compared to both the results of the C modeling and the reference values of GEFDYN. They are given in terms of equivalent stress \(Q\), total volume deformation \({\varepsilon }_{v}\)/\(J=\text{det}\left(\underline{\underline{F}}\right)\), and isotropic work-hardening coefficients \(({r}_{\text{ela}}^{\text{iso},m}+{r}_{\text{iso}}^{m})\) and deviation \(({r}_{\text{ela}}^{d,m}+{r}_{\text{dev}}^{m})\), and summarized in the following tables:
\(Q=\sqrt{\frac{1}{2}\underline{\underline{s}}\mathrm{:}\underline{\underline{s}}}[\mathit{Pa}]\)
\({\epsilon }_{\mathit{zz}},{E}_{\mathit{zz}}\) |
Code_Aster (DEFORMATION =” PETIT) |
Code_Aster (DEFORMATION =” GDEF_LOG) |
relative error |
GEFDYN |
relative error |
-1% |
|
|
|||
-2% |
|
|
|||
-5% |
|
|
|||
-10% |
606137 |
|
|
||
-20% |
|
|
\({\varepsilon }_{V}\mathrm{=}\text{trace}(\varepsilon )\), \(J=\text{det}\left(\underline{\underline{F}}\right)\)
\({\epsilon }_{\mathit{zz}},{E}_{\mathit{zz}}\) |
Code_Aster (DEFORMATION =” PETIT) |
Code_Aster (DEFORMATION =” GDEF_LOG) |
relative error |
GEFDYN |
relative error |
|
-1% |
-7.389E-3 |
-0.00725 |
-0.00725 |
|
-7.47E-3 |
|
-2% |
-1.001E-2 |
-0.00979 |
-0.00979 |
|
-1.005E-2 |
|
-5% |
-1.229E-2 |
-0.01197 |
|
2.730% |
-1.227E-2 |
|
-10% |
-1.096E-2 |
-0.01063 |
|
-1.092E-2 |
|
|
-20% |
-4.88E-3 |
-0.00467 |
-0.00467 |
|
-4.88E-3 |
|
\(({r}_{\text{ela}}^{d,m}+{r}_{\text{dev}}^{m})\)
\({\epsilon }_{\mathit{zz}},{E}_{\mathit{zz}}\) |
Code_Aster (DEFORMATION =” PETIT) |
Code_Aster (DEFORMATION =” GDEF_LOG) |
relative error |
GEFDYN |
relative error |
|
-1% |
0.642 |
0.643 |
0.643 |
|
0.648 |
|
-2% |
0.761 |
0.762 |
0.762 |
|
0.765 |
|
-5% |
0.877 |
0.877 |
|
0.074% |
0.878 |
|
-10% |
0.931 |
0.931 |
0.931 |
|
0.932 |
|
-20% |
0.964 |
0.964 |
0.964 |
7.82ND -03% |
0.964 |
|
\(({r}_{\text{ela}}^{\text{iso},m}+{r}_{\text{iso}}^{m})\)
\({\epsilon }_{\mathit{zz}},{E}_{\mathit{zz}}\) |
Code_Aster (DEFORMATION =” PETIT) |
Code_Aster (DEFORMATION =” GDEF_LOG) |
relative error |
GEFDYN |
relative error |
|
-1% |
0.102 |
0.102 |
0.102 |
|
0.102 |
|
-2% |
0.107 |
0.107 |
|
0.376% |
0.108 |
|
-5% |
0.115 |
0.115 |
0.115 |
|
0.115 |
|
-10% |
0.125 |
0.125 |
0.125 |
|
0.126 |
|
-20% |
0.147 |
0.147 |
0.147 |
|
0.147 |
|
8.2.2. Comments#
It is observed that the result of integration into large deformations with the operator GDEF_LOG makes it possible to reproduce, with enriched kinematics, the behavior prescribed by the law of behavior. In this case a greater variation is observed for high deformations (maximum variation of the order of \(5\text{\%}\)). See Figures and.