1. Reference problem

1.1. Geometry

The model consists of \(N+1\) elements in total: it involves placing \(N\) soil layers (\(N=10\) in the test case) on an infinitely rigid porous elastic substrate represented by a \({0}^{\mathrm{ème}}\) layer. Each layer consists of a mesh element (quadratic), each with a height of \(\mathrm{2m}\). The floor column once built therefore measures \(\mathrm{20m}\) in total. In principle, the problem is two-dimensional (plane deformations occur in a vertical plane): in fact, we assume that the ground is invariant by horizontal translation, which requires that mechanical deformations and hydraulic flows be zero in the horizontal direction (oedometric model).

1.2. Material properties

The elastic, anelastic and hydraulic properties of the layers are given below:

	Settings		Values
PROPRIÉTÉS AND LASTIQUES	\(E\)	Young’s modulus (for the substratum, we take \(100\times E\))	\(100\mathrm{MPa}\)
	\(\nu\)	Poisson’s ratio	0.3
	\({\rho }_{h}\)	homogenized density	\(2105\mathrm{kg}/{m}^{3}\)
PROPRIÉTÉS HUJEUX	\(n\)	Exponent of the elastic law in power	0.89
	\(d\)		1.7
	\(b\)		1
	\(\alpha\)	expansion coefficient	1
	\(\varphi\)	angle of friction	21°
	\(\psi\)	angle of dilatance	21°
	\({P}_{\mathrm{co}}\)	critical or consolidation pressure	\(25\mathrm{kPa}\)
	\({P}_{\mathrm{réf}}\)	reference pressure	\(1\mathrm{MPa}\)
	\({a}_{\mathrm{mon}}\)		0.005
	\({a}_{\mathrm{cyc}}\)		0.005
	\({c}_{\mathrm{mon}}\)		0.18
	\({c}_{\mathrm{cyc}}\)		0.18
	\({r}_{\mathrm{dév}}^{m}\)	monotonic deviatory elastic ray	0.025
	\({r}_{\mathrm{iso}}^{m}\)	monotonic isotropic elastic radius	0.01
	\({r}_{\mathrm{dév}}^{c}\)	cyclic deviatory elastic ray	0.025
	\({r}_{\mathrm{iso}}^{c}\)	cyclic isotropic elastic ray	0.01
	\({r}_{\mathrm{hys}}\)		0.1
	\({r}_{\mathrm{mob}}\)		0.5
	\({x}_{m}\)		2
PROPRIÉTÉS HYDRAULIQUES	\(\phi\)	porosity	0.35
	\({\rho }_{e}\)	Density of water	\(1000\mathrm{kg}/{m}^{3}\)
	\(B\)	Biot coefficient	1
	\({K}^{-1}\)	The inverse of the compressibility of water	\(\mathrm{9,35}\times {10}^{-8}{\mathrm{Pa}}^{-1}\)
	\({K}_{\text{int}}\)	intrinsic permeability of water	\({10}^{-12}\)
	\(v\)	Viscosity of water	\(\mathrm{0,001}\mathrm{Pa.s}\)
	\(\mathit{Dv}\mathrm{/}\mathit{DT}\)	Derived from viscosity by temperature	\(0{\mathrm{Pa.s.K}}^{-1}\)

1.3. Boundary conditions and loads

In the model under consideration, the boundary conditions apply to the \(n+1\) layers present in the calculation step \(n\). They are the same as for an oedometer (the soil column is a sample of an infinite space by horizontal translation):

Horizontal invariance conditions:

• \({u}_{x}=\mathrm{0 }\) on the side stitches;

A condition for blocking the 0th layer (supposed to be rigid):

• \({u}_{y}=\mathrm{0 }\) on the bottom mesh;

A condition of zero hydraulic pressure at the free surface of the column:

• \({\mathrm{PRE}}_{1}=0\) on the top mesh of the \(n+{1}^{\mathrm{ème}}\) layer (the last layer applied);

An initial isotropic and non-zero stress state in each layer placed, due to the aversion of Hujeux’s law for stress states close to zero:

• \({\sigma }_{\mathrm{xx}}\text{'}={\sigma }_{\mathrm{yy}}\text{'}={\sigma }_{\mathrm{zz}}\text{'}={\sigma }_{0}\text{'}=-{20.10}^{+3}\mathrm{Pa}\) in the \(n+{1}^{\mathrm{ème}}\) layer (the last layer applied);

Loading conditions:

• the entire column is subject to gravity (with acceleration \(g=\mathrm{9,81}m/{s}^{2}\) and directed according to \(-{\overrightarrow{e}}_{y}\));

The construction of the column is carried out respecting a period of time \(\Delta t={10}^{+6}\) seconds between the start of stage \(n\) and that of stage \(n+1\). During this period of time, there is diffusion of the fluid and consolidation of the column under the effect of its own weight (compaction). It is important to ensure that this period of time is sufficient, by relating it to the permeability value of the porous material [1] _ . In particular, the product of \(\Delta t\) with this permeability gives a diffusion distance of the fluid that must be sufficient (here of the order of \(10m\)) with respect to the dimension of the column (\(20m\) in height).

	Model elements	Values
Boundary conditions	BAS	\(\mathrm{DY}=0\); zero hydraulic flow
	FACES LATERALES	\(\mathrm{DX}=0\); zero hydraulic flow
	HAUT	\(\mathrm{PRE1}=0\)
Initial conditions	COUCHE \(n+1\) (at stage \(n\))	\(\mathrm{SIXX}=\mathrm{SIYY}=\mathrm{SIZZ}=20\mathrm{kPa}\)
Loading	TOUT	\(\mathrm{PESANTEUR}=\mathrm{9,81}m/{s}^{2}\)

[1]

The conductivity of the fluid is first calculated from the intrinsic conductivity: \(\lambda =\frac{{K}^{\text{int}}}{\upsilon }={10}^{-9}{\mathrm{kg}}^{-1}\mathrm{.}{m}^{3}\mathrm{.}s\); this finally gives for the permeability of the porous material: \(K={\rho }_{e}\times g\times \lambda =\mathrm{9,81}\times {10}^{-6}{\mathrm{m.s}}^{-1}\)