A large number of different phenomena may be coupled. It is impossible to discuss here all potential terms of coupling, and we will restrict ourselves to some basic cases often implied in environmental geomaterial mechanics. In the following paragraphs, some fundamental aspects of potential coupling are briefly described.
11.3.2.1 Hydro-Mechanical Coupling
In the case of hydro-mechanical coupling, the number of d. o.f. per node will be 3 (2 displacements +1 pore pressure) for 2D analysis and 4 (3 displacements +1 pore pressure) for 3D analysis.
Coupling mechanical deformation of soils or rock mass and water flow in pores is a frequent problem in geomechanics. The first coupling terms are related to the influence of pore pressure on mechanical equilibrium through Terzaghi’s postulate (or through any other effective stress concept or net stress use, cf. Eq. 9.20):
a = a’ + uI
with the effective stress tensor a’ related to the strain rate tensor thanks to the constitutive Eq. 11.3, and the identity tensor I.
The second type of coupling concerns the influence of the solid mechanics behaviour on the flow process, which comes first through the storage term. Storage of water in saturated media is mainly due to pores strains, i. e. to volumetric changes in the soil/rock matrix:
Another effect, which may be considered, is the permeability change related to the pore volume change, which may, for example, be modelled by the Kozeny — Carman law as a function of the porosity K = K (n).
Biot proposed an alternative formulation for rocks where contacts between grains are much more important than in soils. Following Biot, the coupling between flow and solid mechanics are much more important (Detournay & Cheng, 1991; Thimus etal., 1998).
The time dimension may cause some problems. First, an implicit scheme is used for the solid mechanics equilibrium and various solutions are possible for the pore pressure diffusion process. Consistency would imply use of fully explicit schemes for the two problems. Moreover, it has been shown that time oscillations of the pore pressure may occur for other time schemes. Associated to Terzaghi’s postulate, oscillations could appear also on the stress tensor, which can degrade the numerical convergence rate for elasto-plastic constitutive laws.
When using isoparametric finite elements, the shape functions for geometry and for pore pressure are identical. Let us consider, for example, a second order finite element. As the displacement field is of second order, the strain rate field is linear. For an elastic material, the effective stress tensor rate is then also linear. However, the pore pressure field is quadratic. Then Terzaghi’s postulate mixes the linear and quadratic fields, which is not very consistent. Some authors have then proposed to mix in one element quadratic shape functions for the geometry and linear shape functions for pore pressure. But then problems arrive with the choice of spatial integration points (e. g. should there be 1 or 4 Gauss points?).
Numerical locking problems may also appear for isoparametric finite elements when the two phase material (water plus soil) is quite incompressible, i. e. for very short time steps with respect to the fluid diffusion time scale. Specific elements have to be developed for such problems.
