2. Modeling A#
2.1. Characteristics of modeling#
The A modeling is performed on a hardware point, using SIMU_POINT_MAT.
The time step is \(\mathrm{\Delta }t=\mathrm{0,1}\mathit{sec}\), which is 300 time increments.
A slight asymmetry of \(\mathrm{ϵ}={10}^{-6}\) is introduced on the horizontal load in order to avoid an excessively marked singularity of the tangent matrix when entering plasticity:
\(\{\begin{array}{c}{\mathrm{\sigma }}_{\mathit{xx}}={\mathrm{\sigma }}_{0}\\ {\mathrm{\sigma }}_{\mathit{yy}}={\mathrm{\sigma }}_{0}\left(1+{10}^{-6}\right)\end{array}\)
This asymmetry is not taken into account in the modelling C for which the criterion is regular on a triaxial compression path.
2.2. Tested sizes and results#
2.2.1. Tested values#
The solutions are calculated at point:math: C and compared to the analytical solution at the final time:math: t=30mathit {sec} `. They are given in terms of vertical constrains:math: `{mathrm {sigma}}} _ {mathit {zz}} and horizontal:math: {mathrm {sigma}}} _ {mathit {xx}}} _ {mathit {xx}}} _ {mathit {xx}}}} _ {mathit {xx}}}, of plastic volume deformation:math: {mathrm {epsilon}}} _ {v} ^ {p}}} _ {v} ^ {p}} and of plastic deviatory deformation:math: `| {mathrm {epsilon}} _ {d} ^ {p} |=sqrt {frac {3} {2}left (mathrm {epsilon} -frac {epsilon}} -frac {{mathrm {epsilon}}} _ {v} ^ {p}} {3} Iepsilon} -frac {{epsilon} -frac {{mathrm {epsilon}}} _ {v} ^ {p}} {3} Iepsilon} -frac {mathrm {epsilon}} _ {v} ^ {p}} {3} Iepsilon} -frac thrm {:}left (mathrm {epsilon} -frac {{mathrm {epsilon}}} _ {v} ^ {p}} {3} Iright)}} `, and summarized in the following table:
\(t=30\mathit{sec}\) |
Analytical Solution |
Allowable Relative Error [%] |
\({\sigma }_{\mathit{zz}}\) |
|
|
\({\sigma }_{\mathit{xx}}\) |
|
|
\({\varepsilon }_{v}^{p}\) |
|
|
\(\mathrm{\mid }{\varepsilon }_{d}^{p}\mathrm{\mid }\) |
|
|
Table 2.2.1-1 : Validation of results for modeling A
2.2.2. Comments#
The deviation from the analytical solution is very small (less than \({10}^{-3}\text{\%}\)).