1. Reference problem#

_images/Shape1.gif

1.1. Geometry#

  • Cube size: \(\mathrm{1m}\mathrm{\times }\mathrm{1m}\mathrm{\times }\mathrm{1m}\)

  • Center of the cube: \(O:(0.\mathrm{,0}\mathrm{.}\mathrm{,0}\mathrm{.})\)

1.2. Material properties#

  • Elastic

    • \(E=5800.0\mathrm{E6}\mathrm{Pa}\) Young’s module

    • \(\nu =0.3\) Poisson’s Ratio

  • DRUCK_PRAGER or DRUCK_PRAG_N_A with linear negative work hardening

    • \(\mathrm{\alpha }=0.33\) Pressure Dependence Coefficient

    • \({p}_{\mathrm{ultm}}=0.01\) Ultimate cumulative plastic deformation

    • \({\sigma }^{Y}=2.57\mathrm{E6}\mathrm{Pa}\) Plastic constraint

    • \(h=-2.00\mathrm{E8}\mathrm{Pa}\) Work hardening module

    • \(\mathrm{\beta }=0.33\) Expansion coefficient (only for DRUCK_PRAG_N_A)

  • DRUCK_PRAGER with parabolic negative work hardening

    • \(\mathrm{\alpha }=0.33\) Pressure Dependence Coefficient

    • \({p}_{\mathit{ultm}}=0.01\) Ultimate cumulative plastic deformation

    • \({\mathrm{\sigma }}^{Y}=2.57E6\mathit{Pa}\) Plastic constraint

    • \({\mathrm{\sigma }}_{\mathit{ultm}}^{Y}=0.57E6\mathit{Pa}\) Ultimate constraint

    • \(\mathrm{\beta }=0.33\) Expansion coefficient (only for DRUCK_PRAG_N_A)

  • DRUCK_PRAG_N_Aavec exponential negative work hardening

    • \(\alpha =0.33\) Pressure Dependence Coefficient

    • \({p}_{c}=0.01\) Ultimate cumulative plastic deformation

    • \({\sigma }^{Y}=2.57\mathrm{E6}\mathrm{Pa}\) Plastic constraint

\({\sigma }_{\mathit{ultm}}^{Y}\mathrm{=}0.57\mathit{E6}\mathit{Pa}\) Ultimate constraint

  • \(\mathrm{\beta }=0.33\) Expansion coefficient

1.3. Boundary conditions and loads#

The boundary conditions and loads applied are as follows:

  • Step \(A\): \(t\in [\mathrm{0,1}\mathrm{.}]\)

Compression \(p={2.10}^{6}\mathrm{Pa}\) is progressively applied to 3 faces of the cube (top, front, right) according to the function shown in the figure below, and symmetry conditions are applied to the 3 other faces (bottom, back, left).

_images/Shape2.gif

  • Step B: \(t\in \text{]}\mathrm{1,2}\mathrm{.}\text{]}\)

Starting from the stress state at time \(t=1.s\), the following conditions are applied to the faces of the cube:

Compulsory trips:

    • the displacement varies progressively on the right face according to the function shown in the figure below:

_images/Shape3.gif

    • Symmetry conditions on all 3 sides (bottom, back, left).

Imposed loads:

A pressure of \(p={2.10}^{6}\mathrm{Pa}\) is applied on the other 2 sides (front and top).