1. Reference problem#

The problem comes from an article by J.C. Simo and*F. Armero [1].

1.1. Geometry#

The geometry can be visualized in the figure and will be used for all models.

_images/10000201000002100000022AE3CEFA7E1D6B4F0C.png

Figure 1.1-1: Membrane geometry

Coordinate of Node A (48.60) and Node C (24.30).

1.2. Material properties#

The material is elastic:

  • Modulus of elasticity: \(E=\mathrm{206,9}\mathit{GPa}\)

  • Poisson’s ratio: \(\mathrm{\nu }=\mathrm{0,29}\)

The plasticity model is a Von Mises model with linear isotropic nonlinear work hardening. The work-hardening curve is of the form:

(1.1)#\[ R (p) = {\ mathrm {\ sigma}}} _ {y\ mathrm {\ sigma}} _ {\ mathrm {\ sigma}} _ {y,\ mathrm {\ infty}}} - {\ mathrm {\ infty}}} - {\ mathrm {\ sigma}}} _ {y\ mathrm {\ sigma}} _ {y\ mathrm {\ sigma}}} _ {y\ mathrm {\ sigma}} _ {y\ mathrm {e}}} - {\ mathrm {\ infty}}} - {\ mathrm {\ infty}}} - {\ mathrm {\ infty}}} - {\ mathrm {\ infty}}} - {\ mathrm {\ sigma}} mathrm {\ delta} p})\]

  • Elastic limit: \({\mathrm{\sigma }}_{y\mathrm{,0}}=450\mathit{MPa}\)

  • Work hardening limit: \({\mathrm{\sigma }}_{y,\mathrm{\infty }}=715\mathit{MPa}\)

  • Saturation setting: \(\mathrm{\delta }=16.93\)

  • Work hardening parameter: \(H=129.2\)

The work hardening curve has been finely tabulated in order to correctly describe the curve with the keyword TRACTION. The deformation model used is GDEF_LOG.

1.3. Boundary conditions and loads#

We embed the left side of the membrane (\(\mathit{DX}=\mathit{DY}=0\)), and we apply a vertical surface force on the right side such as \(\mathit{FY}=\mathrm{0,3125}\mathit{kN}/\mathit{mm}\mathrm{²}\)

1.4. Extension to 3D#

The 3D extension is obtained by extruding in the \(z\) direction (\(1\mathit{mm}\) thickness). In addition, the movements along DZ are blocked in order to be reduced to the solution in plane deformations.