Reference problem
=====================


Material properties
----------------------


*
    * *Elastic*:

:math:`E=3000\mathrm{MPa}` Young's module

:math:`\nu =\mathrm{0,25}` Poisson's ratio

*DRUCK_PRAGER**linear (A modeling) :*

:math:`\alpha =\mathrm{0,20}` Pressure dependence coefficient

:math:`{p}_{\mathrm{ultm}}=\mathrm{0,04}` Ultimate cumulative plastic deformation

:math:`{\sigma }_{Y}=6\mathrm{MPa}` Plastic constraint

:math:`h=100\mathrm{MPa}` Work hardening module

*DRUCK_PRAGER**parabolic (B modeling) :*

:math:`\alpha =\mathrm{0,20}` Pressure dependence coefficient

:math:`{p}_{\mathrm{ultm}}=\mathrm{0,04}` Ultimate cumulative plastic deformation

:math:`{\sigma }_{Y}=6\mathrm{MPa}` Plastic constraint

:math:`{\sigma }_{Y}^{\mathrm{ult}}=10\mathrm{MPa}` Ultimate plastic constraint


Loads and boundary conditions
-------------------------------------

Volume deformation :math:`{\varepsilon }_{v}=\text{tr}(\varepsilon )` is imposed. The loading is not monotonic: we first charge up to the volume deformation :math:`{\varepsilon }_{v\text{1}}`, by exceeding the plastification threshold, then we discharge at a zero deformation level; then we charge again at the deformation :math:`{\varepsilon }_{v\text{2}}`, thus exceeding the ultimate cumulative plastic deformation :math:`{p}_{\mathrm{ultm}}`, thus exceeding the ultimate cumulative plastic deformation, beyond which we find perfect plasticity; we discharge again at zero stress (deformation equal to the plastic deformation), thus exceeding the ultimate cumulative plastic deformation, beyond which perfect plasticity is restored; we discharge again at zero stress (deformation equal to the plastic deformation) :math:`{\varepsilon }_{v\text{2}}^{p}`) and recharging is carried out by plasticizing subsequently until deformation :math:`{\varepsilon }_{v\text{3}}`. The loading time (see) is fictional because plastic laws are independent of time.



.. csv-table::

    ":math:`t` "," :math:`{\varepsilon }_{v}`"
    "0", "0"
    "10"," :math:`{\varepsilon }_{v\text{1}}\mathrm{=}\mathrm{0,018}`"
    "14", "0"
    "26"," :math:`{\varepsilon }_{v\text{2}}\mathrm{=}\mathrm{0,045}`"
    "30"," :math:`{\varepsilon }_{v\text{2}}^{p}\mathrm{=}\mathrm{0,03667}`"
    "40"," :math:`{\varepsilon }_{v\text{3}}\mathrm{=}\mathrm{0,06}`"


**Table** 1.2-1 **: imposed volume deformation.**


Initial conditions
--------------------

All stress and strain components are zero at the start of loading.