Modeling A
==============


Characteristics of modeling
-----------------------------------

The **A** modeling is performed on a hardware point, using SIMU_POINT_MAT.

The time step is :math:`\mathrm{\Delta }t=\mathrm{0,1}\mathit{sec}`, which is 300 time increments.

A slight asymmetry of :math:`\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:

:math:`\{\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.


Tested sizes and results
-------------------------------


Tested values
~~~~~~~~~~~~~~~~

The solutions are calculated at point:math: `C` and compared to the analytical solution at the final time:math: `t=30\ mathit {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} I\ epsilon} -\ frac {{\ epsilon} -\ frac {{\ mathrm {\ epsilon}}} _ {v} ^ {p}} {3} I\ epsilon} -\ frac {\ mathrm {\ epsilon}} _ {v} ^ {p}} {3} I\ epsilon} -\ frac thrm {:}\ left (\ mathrm {\ epsilon} -\ frac {{\ mathrm {\ epsilon}}} _ {v} ^ {p}} {3} I\ right)}} `, and summarized in the following table:


.. csv-table::

    ":math:`t=30\mathit{sec}` ", "Analytical Solution", "Allowable Relative Error [%]"
    ":math:`{\sigma }_{\mathit{zz}}` "," :math:`\mathrm{-}\mathrm{1,732895416041E+5}` "," :math:`\mathrm{1,E-1}`"
    ":math:`{\sigma }_{\mathit{xx}}` "," :math:`50000.` "," :math:`\mathrm{1,E-1}`"
    ":math:`{\varepsilon }_{v}^{p}` "," :math:`\mathrm{1,6784502547224E-4}` "," :math:`\mathrm{1,E-1}`"
    ":math:`\mathrm{\mid }{\varepsilon }_{d}^{p}\mathrm{\mid }` "," :math:`\mathrm{3,3099447556585E-4}` "," :math:`\mathrm{1,E-1}`"


**Table** 2.2.1-1 **: Validation of results for modeling A**


Comments
~~~~~~~~~~~~

The deviation from the analytical solution is very small (less than :math:`{10}^{-3}\text{\%}`).