2. Benchmark solutions#

We note \(p=-\sigma_m=-\mathrm{tr}(\boldsymbol{\sigma})/3\) and \(\varepsilon_v=\mathrm{tr}(\boldsymbol{\varepsilon})\) the pressure and the volume deformation, adopting the sign convention of the mechanics of continuous media. The relationship of these two quantities is differentiated according to the elastic or plastic regime in the first monotonic load.

2.1. Calculation method with model CAM_CLAY#

This concerns C and D models.

  • Elastic regime: When \(p\) is less than the consolidation pressure \(2P_{cr}^0-P_{trac}\):

\[p=P_0\ exp (-k_0\ varepsilon_v^e) +\ frac {K_0} {k_0}\ left (\ exp (-k_0\ varepsilon_v^e) -1\ right) -1\ right)\ right)\ right)\ longrightarrow -\ varepsilon_v^e =\ frac {1} {k_0}\ ln\ left (\ frac {k_0p+k_ {cam}}} {k_0p_0+k_ {cam}}\ right),\ quad\ text {with}\ quad k_0 =\ frac {1+e_0} {\ kappa}\]

In this case, the total deformation is equal to the elastic deformation (\(\varepsilon_v=\varepsilon_v^e\)).

  • Plastic regime: When \(p\) is greater than consolidation pressure:

\[p=2p_ {cr} ^0\ exp (-k\ varepsilon^p_v) - P_ {trac}\ longrightarrow -\ varepsilon_v^p =\ frac {1} {k} {k}\ exp (-k\ varepsilon^p_v) - P_ {trac}} {2P_ {cr} v^0}\ right),\ quad\ text {with}\ quad k=\ frac {1+e_0} {\ lambda-\ kappa}\]

In this situation, plastic deformation must be taken into account in calculating the total deformation (\(\varepsilon_v=\varepsilon_v^e+\varepsilon_v^p\)).

2.2. Benchmark solution with model MCC#

Here we treat G modeling.

  • Elastic regime: When \(p\) is below the isotropic compression limit \(2p_{c0}+\sigma_0\):

\[p=\ frac {K} {\ kappa}\ left (\ exp\ left (-\ kappa\ varepsilon_v^e\ right) -1\ right)\ Longrightarrow -\ varepsilon_v^e =\ varepsilon_v^e =\ frac {1} {\ v^e =\ frac {1} {\ kappa} {\ kappa}\ left (1+\ kappa\ frac {p} {p} {K}\ right)\]

  • Plastic regime: When \(p\) exceeds the isotropic compression limit:

\[p=2p_ {c0}\ exp\ left (-\ beta\ varepsilon_v^p\ right) +\ sigma_0\ Longrightarrow -\ varepsilon_v^p =\ frac {1}\ left (-\ frac {1}} {\ beta}\ beta}\ ln\ left (\ frac {p-\ sigma_0}} {2p_ {c0}}\ right)\]

2.3. Benchmark solution with model CSSM#

Here we treat H modeling.

  • Elastic regime: When \(p\) is below the isotropic compression limit \(2p_{c0}(1-\eta)\):

\[p=-K\ varepsilon_v^e\ Longrightarrow -\ varepsilon_v^e =\ frac {p} {K}\]

  • Plastic regime: When \(p\) exceeds the isotropic compression limit:

\[p=2p_ {c0}\ left (\ exp\ left (-\ beta\ varepsilon_v^p\ right) -\ eta\ exp\ left (2\ omega\ varepsilon_v^p\ right)\ right)\ right)\ right)\ right)\ right)\ right)\ right)\ Longrightarrow -\ varepsilon_v^p = F (p)\]

where the \(F(p)\) function does not allow simple explicit expressions in the general case where \(\eta>0\) and \(\omega>0\).

2.4. Reference quantities and results#

The test is homogeneous. The deformation component \(\varepsilon_{xx}\) and the volume plastic deformation \(\varepsilon_v^p\) are tested at various times.

2.5. Uncertainties about the solution#

None on the C, D and G models. The solution in the H modeling is approximated numerically using the fsolve command from the package. scipy.optimize (with its settings by default).