1. Reference problem#
1.1. Geometry#
According to modeling \(\mathrm{2D}\) or \(\mathrm{3D}\), we consider respectively a square or a cube with a side of 0.05 \(\text{mm}\) (see Figure. 1.1-1). Because of the symmetry, only a quarter of the geometry is shown.
Figure 1.1-a : Geometries of a square**(a) and a cube (b)**
1.2. Material properties#
Elasticity:
\(E=190000\text{MPa}\) |
Young’s module |
\(\mathrm{\nu }=0.3\) |
Poisson’s ratio |
Work hardening curve:
\(R(\mathrm{\kappa })=488.36+57.13(1-\mathrm{exp}(-8613\mathrm{\kappa }))+238.73(1-\mathrm{exp}(-10.39\mathrm{\kappa }))\)
Ductile damage law GTN:
\({q}_{1}=1.5\) |
Model parameter GTN |
\({q}_{2}=1.07\) |
Model parameter GTN |
\({f}_{0}=0.01\) |
Initial porosity |
\({f}_{n}=0\) |
Germination Parameter |
\({f}_{c}=0.05\) |
Coalescence porosity |
\(\mathrm{\delta }=3\) |
Coalescence coefficient related to coalescence |
\(c=2.22N\) |
Non-local parameter |
\(r=5000\mathit{MPa}\) |
Lagrange penalty parameter |
In particular, the non-local parameter \(c\) and the penalty parameter \(r\) are only used in the \(\mathit{GTN}\) gradient law.
In DEFI_MATERIAU, the following information should be filled in:
ELAS |
ECRO_ NL |
GTN |
NON_LOCAL |
E = 190000 |
R0= 488.361123569 |
Q1 = 1.5 |
C_ GRAD_VARI = 2.22 |
NU = 0.3 |
R1 = 57.1333673502 |
Q2 = 1.07 |
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GAMMA_1 = 8613 |
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R2 = 238.731127339 |
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GAMMA_2 =10.386585592 |
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1.3. Boundary conditions and loads#
For axisymmetric \(2D\) modeling, the vertical displacements of all the nodes are controlled: \({u}_{y}=\text{}8y\), the horizontal displacements are blocked along the axis (\(\mathit{AD}\)). The horizontal displacements are uniform for side \(\mathit{BC}\) (see Figure 1.1-1 (a) for geometry).
For modeling \(3D\), the vertical displacements are given for all the nodes: \({u}_{y}=\text{}8y\). We impose a symmetry condition along the axis \(X\) for the face \(\mathit{ADHE}\)) and along the \(Z\) axis for the for the face \(\mathit{ABCD}\). The movements along \(X\) for face \(\mathit{BCGF}\) and the movements along the \(Z\) axis for face \(\mathit{EFGH}\) remain uniform. (see Figure 1.1-1 (b) for geometry).
The boundary conditions and the loads are imposed in this way so that the axisymmetric problem and the one in \(3D\) are the same and homogeneous.
The load is imposed using 1000 identical time steps. The pseudo-calculation time is 1.