8. Verification of the microscale model 93
8.3 Undrained isotropic loading
In this third example, microstructure 1 is loaded from the initial configuration up to an isotropic stress state of 100 M P a (extension) and −100M P a (compression) under undrained conditions. This is done for different values of grain porosities ϕ and the pore fluid pressure as a response to this loading is studied. The influence of the porosity on the global response is through the rheological part of the fluid storage: The constant pore volume in the grains allows local storage of fluid mass with increasing fluid pressure through the compressibility of the fluid. The rela-tion between the interface normal opening and corresponding interface hydraulic opening is presented graphically in Figure 5.7. With a minimum hydraulic open-ing of ∆uminh = 0.025µm and an initial hydraulic opening of ∆u0h = 0.025µm, there is a local variation in pore volume even for slightly negative (penalized) mechanical openings, which contributes to a geometric component of the local fluid storage.
Since the applied isotropic stress state is obtained through the application of a macroscale deformation, it is possible to distinguish the two tests by a com-pressive loading (negative stresses) and extensive loading (positive stresses). The hydraulic responses to compressive loading are given in Figure 8.7 for compres-sion (left) and extencompres-sion (right) for different grain porosities ϕ. For this example, the cohesive forces at the interfaces are taken to be negligible with respect to the normal interface contact forces and the hydraulic forces on the interface bound-aries. In the case of extensional loading, the pore pressure p directly follows the enforced stress and ∂p/∂σii ≈ −1. As the interface cohesion plays an inferior role, and in extension no interface normal forces are active, practically all forces are taken by the water. In Biot’s consolidation theory, such a response corres-ponds to a Skempton coefficient βp ≈ 1, independent of the grain porosity. In case of compressive loading, the mechanical forces at the interfaces can take a more significant role in the micromechanical equilibrium. It is important to real-ize that the REV is considered globally undrained, but locally a redistribution of fluid mass is possible. This redistribution allows the fluid mass in the interface elements to enter the pore space in the grains (which is considered to be constant
8.3 Undrained isotropic loading 101
and therefore independent of the grain deformation) depending on the pore pres-sure required for the compression of the fluid mass. The larger the ratio between the grain pore volume and the initial interface volume, the lower the pore fluid pressure drives the fluid out of the interfaces and the easier the normal interface contact forces can take over the isotropic stress σii. The result is a series of pore pressure response curves starting with a one-to-one correspondence in case of zero grain porosity where the compressive loading remains fully taken by the (nearly incompressible) fluid. With increasing grain porosity, the change of pore pressure with confining stress decreases towards a −dp/dσii in the order of 0.01 for ϕ = 0.1.
The non-linearities are an effect of the non-linear nature of the normal part of the contact forces, where the non-linearities come mainly from the penaliza-tion of negative normal opening. The cohesive forces for positive interface open-ings are negligible with respect to the measured pore pressures in the experi-ment as can be observed from the quasilinear relation in case of extension. The non-linear response in case of compression shows the transition from a (quasi-)linear relation between stress and fluid pressure at relatively low pressure and a (quasi-) quadratic relation at higher pressures. This transition is a transition from a hydraulics-controlled response to a mechanics-controlled response. In the hydraulics-controlled response, the compressive load in the interfaces is mainly taken by the fluid pressure, which requires a compression of the fluid mass equal to the change in interface hydraulic volume. The (constant) grain porosity intro-duces an additional amount of water to be compressed and the total change in fluid volume (equal to the total change in interface volume) will therefore be larger in case of higher porosity ϕ. As an effect, the change in normal interface openings will be larger for higher grain porosity and a transition to a mechanics-controlled response is obtained at lower compressive stress states.
In the mechanics-controlled response, the interface contacts are penalized and the quadratic penalization leads to a change in hydraulic interface opening with the square root of the applied compressive load. The result is the 2:1 inclination in the logarithmic plot between loading and hydraulic response.
The point at which the transition from hydraulic-controlled to mechanics-controlled response takes place is determined by the relative change in fluid mass volume per change in fluid pressure. In other words, the position of the different graphs along the diagonal in Figure 8.7 is a function of the fluid bulk modulus kw and the ratio between (initial) interface hydraulic volume and the grain pore volume.
With a relatively high fluid bulk modulus of the fluid (kw∼ 1 × 109P a), the nonlinearities due to the fluid compressibility is small in the applied range of stresses (|pM| < 1 × 108P a)
Note that the smallest loading step applied here is ∆σ11= 1.0 × 10−4 M P a, which explains the choice for range of the horizontal axis. For smaller compressive loading, the ϕ = 0.1 curve will coincide with the ϕ = 0 curve to qualitatively show the same behaviour as the other curves.
compression extension
−102
−100
−10−2
−10−4 10−4 10−2 100 102
σ11[MPa]
p [MPa]
=0
=0.0001
=0.001
=0.01
=0.1 φ φ φ φ φ
1 2 1
1
0 20 40 60 80 100
−100
−80
−60
−40
−20
0
σ11[MPa]
p [MPa]
=0
=0.0001
=0.001
=0.01 φ=0.1 φ φ φ φ
Fig. 8.7: Pore fluid pressure as a reaction to undrained loading
The numerical experiments above demonstrate that even for simplified micro-structures a constant relation between macroscale stress and fluid pressure is not easily obtained. It is therefore not straightforward to compare the macroscale be-haviour with for example Biot’s theory without a strong restriction in the domain of stress/strain states to linearize different aspects of the microscale behaviour.
For example, to take away the non-linear effects in Figure 8.7, the penalization of the normal interface opening should be omitted or the hydraulic interface opening should be independent from the mechanical normal opening. Another option for linearization could be to consider fluid incompressibility.