7. The internal variables#

For the IT implementation, we retained the following 4 internal variables:

7.1. V1: the cumulative plastic deviatory deformation#

The work hardening variable \({\gamma }^{p}\) is proportional to the second invariant of the deviatory deformation tensor.

\({\gamma }^{p}=\sqrt{\frac{2}{3}{e}_{\text{ij}}^{p}{e}_{\text{ij}}^{p}}\)

with \({e}_{\text{ij}}^{p}={\varepsilon }_{\text{ij}}^{p}-\frac{\text{tr}({\varepsilon }_{\text{ij}}^{p})}{3}{\delta }_{\text{ij}}\)

7.2. V2: the cumulative plastic volume deformation#

Plastic volume deformation is defined by the relationship presented in paragraph [§3.2.4] on the law of evolution of the plastic mechanism: \(\dot{{\varepsilon }_{v}^{p}}=\dot{\lambda }G\)

7.3. V3: the domains of rock behavior#

Five behavior domains, numbered from 0 to 4 (cf. figure), are identified to allow a relatively simple representation of the state of damage of the rock, from intact rock to rock in residual state. These domains are a function of the cumulative plastic deviatory deformation \({\gamma }^{p}\) and the stress state. Each domain number increment defines the transition to a higher damage domain.

  • If the deviator is less than 70% of the peak deviator, then the material is in the 0 range;

  • If not:

  • If \({\gamma }^{p}=0\) then the material is in domain 1;

  1. If \(0<{\gamma }^{p}<{\gamma }^{e}\) then the material is in domain 2;

  • If \({\gamma }_{e}<{\gamma }^{p}<{\gamma }_{\text{ult}}\) then the material is in domain 3;

  • _images/Object_260.svg

If \({\gamma }^{p}>{\gamma }_{\text{ult}}\) then the material is in domain 4.

_images/Object_262.svg

7.4. V4: the state of plasticization#

It is an internal indicator of Code_Aster. It is 0 if the gauss point is in elastic charge or under discharge, and is equal to 1 if the gauss point is in plastic charge.