1. Reference problem#

1.1. Geometry#

We consider a cube with a side of length \(2m\) subjected to uniform traction in the vertical direction \(Z\):

_images/Shape1.gif

For reasons of symmetry, only one eighth of the structure will be considered, which will be meshed by a single cubic linear volume element.

1.2. Material properties#

The structure is assumed to be homogeneous, composed of an isotropic elastoplastic material, with linear isotropic work hardening:

  • \(E={2.10}^{4}\mathrm{MPa}\)

  • \(\nu =0.49999\)

  • \(\rho =7900\mathrm{kg}/{m}^{3}\)

  • \({\sigma }_{y}=\mathrm{0,1}\mathrm{Mpa}\) (elastic threshold SY)

  • \({E}_{T}=200\mathrm{Mpa}\) (plastic tangent module D_ SIGM_EPSI)

We therefore choose a material that always remains almost incompressible, regardless of whether we are in an elastic or plastic regime. In addition, a ratio of 100 is imposed between the elastic stiffness and the plastic tangent stiffness.

1.3. Boundary conditions#

A uniform loading of the traction type imposed according to \(Z\) is imposed on the upper face of the cube. This imposed force, which was initially zero, increases linearly with time.

The other boundary conditions are of the Dirichlet type and reflect the symmetry conditions of the problem (according to the 3 orthogonal planes \((\mathrm{xOy})\), \((\mathrm{xOz})\) and \((\mathrm{yOz})\)).

These boundary conditions are sufficient to block all rigid body movements in the system.

1.4. Initial conditions#

The first calculation being quasistatic, we just impose a zero initial displacement.