1. Reference problem#
1.1. Material properties#
Elastic:
\(E=3000\mathrm{MPa}\) Young’s module
\(\nu =\mathrm{0,25}\) Poisson’s ratio
DRUCK_PRAGER**linear (A modeling) :
\(\alpha =\mathrm{0,20}\) Pressure dependence coefficient
\({p}_{\mathrm{ultm}}=\mathrm{0,04}\) Ultimate cumulative plastic deformation
\({\sigma }_{Y}=6\mathrm{MPa}\) Plastic constraint
\(h=100\mathrm{MPa}\) Work hardening module
DRUCK_PRAGER**parabolic (B modeling) :
\(\alpha =\mathrm{0,20}\) Pressure dependence coefficient
\({p}_{\mathrm{ultm}}=\mathrm{0,04}\) Ultimate cumulative plastic deformation
\({\sigma }_{Y}=6\mathrm{MPa}\) Plastic constraint
\({\sigma }_{Y}^{\mathrm{ult}}=10\mathrm{MPa}\) Ultimate plastic constraint
1.2. Loads and boundary conditions#
Volume deformation \({\varepsilon }_{v}=\text{tr}(\varepsilon )\) is imposed. The loading is not monotonic: we first charge up to the volume deformation \({\varepsilon }_{v\text{1}}\), by exceeding the plastification threshold, then we discharge at a zero deformation level; then we charge again at the deformation \({\varepsilon }_{v\text{2}}\), thus exceeding the ultimate cumulative plastic deformation \({p}_{\mathrm{ultm}}\), thus exceeding the ultimate cumulative plastic deformation, beyond which we find perfect plasticity; we discharge again at zero stress (deformation equal to the plastic deformation), thus exceeding the ultimate cumulative plastic deformation, beyond which perfect plasticity is restored; we discharge again at zero stress (deformation equal to the plastic deformation) \({\varepsilon }_{v\text{2}}^{p}\)) and recharging is carried out by plasticizing subsequently until deformation \({\varepsilon }_{v\text{3}}\). The loading time (see) is fictional because plastic laws are independent of time.
\(t\) |
|
0 |
0 |
10 |
|
14 |
0 |
26 |
|
30 |
|
40 |
|
Table 1.2-1 : imposed volume deformation.
1.3. Initial conditions#
All stress and strain components are zero at the start of loading.