1. Reference problem#

1.1. Geometry#

_images/10000000000000E50000010ECCE334BD207B5CD4.jpg

Point coordinates

\(X\)

\(Y\)

\(A\)

0

0

\(B\)

0.5

0

\(C\)

0.5

1.0

\(D\)

0

1,0

The geometry is rectangular in shape.

1.2. Material properties#

For modeling A, the mass consists of an elasto-plastic material with linear negative work-hardening.

For modeling B, the mass consists of an elasto-plastic material with parabolic negative work hardening.

The elastic parameters of the material are as follows:

  • Young’s module: \(E=5800\mathrm{MPa}\)

  • Poisson’s ratio: \(\nu =\mathrm{0,3}\)

  • Real constant density: \(\rho \mathrm{=}2764\)

  • Isotropic thermal expansion coefficient: \(\alpha =0\)

The characteristics of work hardening are then given by:

  • Pressure dependence coefficient: \(\alpha =\mathrm{0,33}\)

  • Ultimate cumulative plastic deformation: \({P}_{\mathrm{ULT}}=\mathrm{0,01}\)

  • Elastic limit: \({\sigma }_{y}\mathrm{=}2.11{10}^{6}\)

  • Elastic limit for the mesh concerned: \({\sigma }_{y}\mathrm{=}2.\mathrm{\times }{10}^{6}\)

For modeling A:

  • Work hardening module for the entire mesh: \(H\mathrm{=}\mathrm{-}200.\mathrm{\times }{10}^{6}\)

For modeling B:

  • Ultimate constraint: \({\sigma }_{\mathit{yULT}}\mathrm{=}0.47\mathrm{\times }{10}^{6}\)

  • Ultimate constraint for the mesh concerned: \({\sigma }_{\mathit{yULT}}\mathrm{=}0.44\mathrm{\times }{10}^{6}\)

1.3. Boundary conditions and loads#

A drained biaxial test (D_ PLAN) is modelled. The trips normal to the study plan are therefore zero. A vertical displacement is imposed on \(\mathrm{[}\mathit{DC}\mathrm{]}\) while maintaining the constant lateral pressure (\(2\mathit{MPa}\)) in the study design. The boundary conditions are therefore as follows:

\({u}_{y}\mathrm{=}0\) out of \(\mathrm{[}\mathit{AB}\mathrm{]}\) (mesh group \(\mathit{BAS}\))

\({u}_{x}\mathrm{=}0\) out of \(\mathrm{[}\mathit{AD}\mathrm{]}\) (mesh group \(\mathit{GAUCHE}\))

\({\sigma }_{n}\mathrm{=}2.\mathrm{\times }{10}^{6}\) out of \(\mathrm{[}\mathit{BC}\mathrm{]}\) (mesh group \(\mathit{EXTREM}\))

A vertical displacement is then imposed on \(\mathrm{[}\mathit{DC}\mathrm{]}\) (mesh group \(\mathit{HAUT}\)) to apply vertical deformation up to \(\text{3\%}\).

1.4. Results#

The solutions are post-treated in order to control the evolution of the quantities of INDL_ELGA and PDIL_ELGA in non-regression.