1. Reference problem#

1.1. Geometry#

_images/Object_1.svg

Fig. 1.1 Height: \(h=1\) m, width: \(l=1\) m, thickness: \(e=1\) m.#

1.2. Material properties#

1.2.1. Behaviour model of CAM_CLAY#

The parameters specific to the elastoplastic law CAM_CLAY [r7.01.14], with respect to models A, B and C, are:

  • \(\mu =6\) MPa, \(PORO=0.66\), \(\lambda =0.25\), \(\kappa =0.05\),, \(M=0.9\),, \(P^0_{cr}=300\) kPa, \(K_{cam}=0\) Pa, \(P_{trac}=0\) Pa.

The saturating fluid is assumed to have an infinite \(K_w\) modulus of compressibility. The initial specific mass is \(\rho_{w}^0=1000\text{kg/m}^3\), and the initial porosity \(\varphi_0=PORO=0.66\). The poromechanical coupling uses a Biot coefficient \(b=1\).

1.2.2. Behaviour model of MCC#

The parameters specific to the elastoplastic law MCC [r7.01.10] are given in the Next v7.31.122-table_parametres_CSSM regarding D modeling.

The compressibility module of the saturating fluid is \(K_w=10^3\) GPa, its initial specific mass \(\rho_{w}^0=1000\text{ kg/m}^3\), and the initial porosity \(\varphi_0=0.3\). The poromechanical coupling uses a Biot coefficient \(b=1\).

1.2.3. Behaviour model CSSM#

Parameters specific to model CSSM [r7.01.44] are grouped in The v7.31.122-table_parametres_CSSM concerning E modeling.

The compressibility module of the saturating fluid is \(K_w=10^3\) GPa, its initial specific mass \(\rho_{w}^0=1000\text{ kg/m}^3\), and the initial porosity \(\varphi_0=0.3\). The poromechanical coupling uses a Biot coefficient \(b=1\).

Note:

The Biot coefficient b, the compressibility module of solid phase \(K_s\), and the drained compressibility module of skeleton \(K_0\), the one calculated by the ELAS keyword, are assumed to be linked by the following relationship:

So \(b=1\rightarrow K_0/K_s=0\), and regardless of the value entered in \(K_0\) which, a priori, is a parameter independent of the non-linear elasticity to be entered in the keywords CAM_CLAY or MCC. More generally, reference may be made to the influence of parameters \(b,K_s,K_0\) on the evolution of porosity [r7.01.10] in the more general case where \(b<1\).

1.3. Boundary conditions and loads#

For models A, B and C, concerning model CAM_CLAY, the first loading path is an isotropic compression, under drained condition (\(PRE1=0\)), up to pressure \({P}_{sup}\). For the second path, the pressure \({P}_{sup}\) is kept constant on the lateral faces and, simultaneously, an imposed vertical compression displacement is imposed. This second charge is not drained, which corresponds to a zero hydrostatic flow on all sides.

  • For modeling A: \({P}_{sup}={P}_{conso}=2{P}^0_{cr}-P_{trac}=600\) kPa (contracting final state);

  • For B modeling: \({P}_{sup}={P}^0_{cr}=300\) kPa (critical final state with zero volume variation);

  • For C modeling: \({P}_{sup}=220\) kPa \(<P^0_{cr}\) (dilating final state).

For modeling D, concerning model MCC, the first load is also an isotropic compression, under drained condition, up to \({P}_{sup}=2\) MPa. The second load is identical to that described previously in models B, C and D.

For modeling E, concerning model CSSM, the first load is an isotropic compression, under drained condition, up to \(P_{sup}=100\) kPa The second load is identical to those described in the previous models.

Note:

For models A, B and C, for model CAM_CLAY, it is required that in the initial state the hydrostatic stress is strictly greater than zero (see [r7.01.14]). For this, a purely elastic calculation by changing the pressure from \(0\) to \(1\) Pa is carried out. From this calculation the stress field at the gauss points is extracted. This field of constraints resulting from elastic calculation is considered to be the initial state of the hydrostatic stress necessary for law CAM_CLAY of the following calculation (isotropic compression under drained conditions up to \({P}_{sup}\)).