1. Reference problem#

1.1. Geometry#

Consider a \(\mathrm{2D}\) square mesh:

Y

L32

P1

P2

L21

L43

P3

P4

X

L14

The sides \(\mathrm{L21}\), \(\mathrm{L32}\), \(\mathrm{L43}\), \(\mathrm{L14}\) each measure \(10\mathrm{mm}\).

1.2. Material properties#

We take: \(E=200\mathrm{GPa}\), and \(\nu =\mathrm{0,3}\).

The traction curve used is given in the following table:

\(\varepsilon\)

0.0001

0.00338

0.00338

0.03

0.03

0.04

0.07

0.10

0.15

0.2

0.2

0.2

0.2

0.2

0.3

0.4

\(\sigma\)

27.30

222.72

222.72

519.58

519.58

580.94

633.48

721.82

828.96

970.19

970.19

970.19

1084.75

1084.75

1269.57

1419.48

\(\varepsilon\)

0.5

0.7

1.0

1.0

1.5

2.0

\(\sigma\)

1547.86

1763.72

1763.72

2025.50

2025.50

The Rousselier model is used in three configurations with the following parameters:

Elasto-plastic base model ( ROUSS_PR ****) **

Elasto-plastic model (** ROUSS_PR **** ) with germination

Visco-plastic model ( **** **** )) **

Visco-plastic model (** VISCOROUSS ) and theta-method

  1. \(D=2.\)

  2. \({\sigma }_{1}=600\mathrm{MPa}\)

  3. \(\lambda =1.\)

  4. \(\mathrm{f0}=1.e-4\) (initial porosity)

  5. \(\mathrm{fc}=1.\) (critical porosity)

  6. \(A=1.\)

  1. \(D=2.\)

  2. \({\sigma }_{1}=600\mathrm{MPa}\)

  3. \(\lambda =1.\)

  4. \(\mathrm{f0}=1.e-4\)

  5. \(\mathrm{fc}=1.\)

  6. \(A=1.\)

  7. \(\mathrm{An}=0.6\)

  1. \(D=2.\)

  2. \({\sigma }_{1}=600\mathrm{MPa}\)

  3. \(\lambda =1.\)

  4. \(\mathrm{f0}=1.e-4\)

  5. \(\mathrm{fc}=1.\)

  6. \(A=1.\)

  7. \({\sigma }_{0}=27\mathrm{MPa}\)

  8. \({\varepsilon }_{0}=1.e-2\)

  9. \(\theta =0.57\)

  10. \(m=2\)

1.3. Boundary conditions and loads#

With reference to the figure in [§1.1] the boundary conditions are as follows:

  1. on the edge \(L32\) move

    _images/Object_1.svg

imposed in direction \(\mathrm{OY}\) (monotonic traction),

  1. \(\mathrm{L21}\) movements blocked following \(X\),

  2. \(\mathrm{L14}\) movements blocked following \(Y\).

The temporal evolution of elongation

_images/Object_2.svg

are shown in the following table:

+————————————————————————————————————————————————-+++ |Time :math:`[s]` ||| +————————————————————————————————————————————————-+++ |Displacement ||| + +++ | ||| + .. image:: images/Object_3.svg +++ | :width: 9 ||| + :height: 17 +++ | ||| + +++ | ||| + +++ | ||| + \([\mathrm{mm}]\) +++ | ||| +————————————————————————————————————————————————-+++

The evolution is linear between the two moments.

1.4. Initial conditions#

Zero stresses and deformations.