5.2 Modelling of Liquefaction Phenomena
5.2.3 Undrained Sand Behaviour under Cyclic Loading
the soil skeleton breaks down. The relation between the pore pressure and the volumetric deformations can be explained based on the effective stress principle, viz.:
(TS + TF)·I = TSE ·I − pI·I −→ trT = trTSE − 3p . (5.5) Therefore, the interplay between the mean effective stress and the excess pore pressure for cases (3) and (4) is depicted in Figure 5.9 . For case (3) of moderate-dense speci-mens, Figure 5.9, left, shows a phase transformation state between the contractive and the dilative phases, where the excess pore pressure and the effective stress reverse their behaviours. In the dilative tendency stage, the applied load is increasingly carried by the effective stress, whereas the pore pressure dissipates. Figure 5.9, right, shows how the effective stress reduces and the pore pressure increases until the flow liquefaction takes place. When the effective stress becomes equal to zero, the whole applied stress is carried by the pore fluid.
5.2 Modelling of Liquefaction Phenomena 99
Under cyclic loading, saturated sand undergoes liquefaction with patterns similar to that observed under monotonic loading. Using experimental results of anisotropically consoli-dated undrained triaxial tests, Ishihara [97] showed that the cyclic behaviour of saturated sand (for the sake of seismic-induced liquefaction modelling) can be well understood by comparison with the behaviour under monotonic loading conditions, cf. Box (5.6).
In Box (5.6), the investigation of undrained sand considers two cases of initial density:
A low-dense sand, which undergoes flow liquefaction, and a moderate-dense sand that exposes a cyclic mobility behaviour. Under cyclic loading, saturated sand undergoes liquefaction with patterns similar to that observed under monotonic loading.
Employing the elasto-viscoplastic constitutive model as introduced in Section 3.2 with isotropic hardening and in analogy to the works by Ishihara [97] and Ishihara et al. [98], it is possible to follow the excess pore-pressure development and the onset of liquefac-tion events under cyclic loading with irregular amplitudes. If the unloading-reloading process is carried out inside the yield surface of the elasto-viscoplastic model, then the response is governed by the hyperelastic material law. In this case, the reloading process follows the same path as the unloading process and only oscillatory but not accumulative pore-pressure behaviour can be detected. Whenever a plastic yielding occurs, a perma-nent change in the pore pressure and the effective stress takes place. In particular, the accumulation of the pore pressure and the onset of liquefaction phenomena are mainly governed by the volumetric strains in the plastic range.
In the present work, the flow liquefaction and the cyclic mobility are numerically figured out using an initial-boundary-value problem (IBVP) that represents the triaxial test.
Here, the material parameters are chosen for a sand with two different initial densities: A very loose sand as in case (4), Table 5.1, and a sand with moderate density as in case (3), Table 5.1 . The geometry of the IBVP and the other material parameters are given in Appendix B, whereas the applied loads are illustrated in Figure 5.10, left, and Figure 5.12, left. The Neumann boundary conditions in this investigation are chosen in analogy to those in Box (5.6), viz.:
0−A consolidated drained step,
A−B drained deviatoric step (anisotropic consolidation), B−C undrained monotonic step for the monotonic CU test, B−D undrained cyclic step for the cyclic CU test.
(5.7)
The excess pore pressure in case of a very loose sand is depicted in Figure 5.10, right.
Here, when the unloading-reloading step crosses the yield surface, an accumulation of the pore pressure accompanied by a volumetric plastic deformation can be observed.
Figure 5.11 shows the flow liquefaction behaviour under irregular cyclic loading, where the last loading cycle leads to the same pattern as for the monotonic loading, see, Figure 5.8, case (4). The latter results are comparable with what has been presented in Box (5.6) for a low-dense sand.
In this connection, very loose sand is a highly contractive material under deviatoric stress, which leads to a continuous pore-fluid pressure accumulation until collapse. This happens
-0.1 0 0.1 0.2
0 2000 4000 6000 8000
A B
C
D
(σ1−σ3)[MN/m2]
time [s]
cyclic monotonic
-0.1 0 0.1 0.2 0.3
0 2000 4000 6000 8000
pore-pressure[MN/m2]
time [s ]
cyclic monotonic
Figure 5.10: Loading path with boundary conditions according to (5.7) (right) and pore-pressure build-up during the cyclic triaxial test (left) for a very loose sand
-0.15 -0.1 -0.05 0 0.05 0.1 0.15
-0.2 0 0.2 0.4 0.6 0.7
A B
C D
(σ1−σ3)[MN/m2 ]
−Iσ[MN/m2]
cyclic monotonic
-0.15 -0.1 -0.05 0 0.05 0.1 0.15
0 0.2 0.4 0.6 0.8
0.12 0.14
0.05 0.1
(σ1−σ3)[MN/m2 ]
ε1 % [−]
cyclic monotonic
Figure 5.11: Mean effective stress versus deviatoric stress (left) and axial strain versus deviatoric stress (right) under cyclic and monotonic loading for a loose sand
when the stress state is found on the yield surface in both the compression and the extension sides. Here, before the deviatoric stress reaches a peak value, the pore pressure can still be relatively low. After the deviatoric stress attains its maximum value, the pore pressure and the axial strain rapidly increase, which corresponds to a collapse of the solid skeleton.
The pore pressure behaviour for the case of moderate-dense sand under undrained condi-tions is depicted in Figure 5.12, right. Here, the tendency to contraction under deviatoric stress is limited, which prevents the occurrence of the flow liquefaction. Moreover, Fig-ure 5.12, right, shows that the last cycle causes the pore pressFig-ure to increase and then to decrease, which refers to a contractive and then to a dilative behaviour.
Similar to the laboratory-based observations ofIshiharain Box (5.6), the numerical results of the considered elasto-viscoplastic model (Figure 5.13) show that at a relatively low stress ratio3, the granular material exposes a contractive tendency allowing for a reduction in the effective stress and a limited plastic strain. Such a behaviour happens when the
3stress ratio = (σ1−σ3)/Iσ(t0).
5.2 Modelling of Liquefaction Phenomena 101
-0.1 0 0.1 0.2 0.3 0.4 0.5
0 2000 4000 6000 8000 10000
A B
C
D
(σ1−σ3)[MN/m2 ]
time [s]
cyclic monotonic
-0.1 0 0.1 0.2
0 2000 4000 6000 8000 10000
C D
pore-pressure[MN/m2 ]
time [s]
cyclic monotonic
Figure 5.12: Loading path with boundary conditions according to (5.7) (right) and pore-pressure build-up during triaxial shear test (left) for the case of a moderate-dense sand
-0.2 -0.1 0 0.1 0.2 0.3 0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 A
B
C D
(σ1−σ3)[MN/m2]
−Iσ [MN/m2]
cyclic monotonic
-0.2 -0.1 0 0.1 0.2 0.3 0.4
0 0.2 0.4 0.6 0.8
0.08 0.12 0.16
0.05 0.1
(σ1−σ3)[MN/m2]
ε1 % [−]
cyclic monotonic
Figure 5.13: Mean effective stress versus deviatoric stress (left) and axial strain versus deviatoric stress (right) under cyclic and monotonic loading for a moderate-dense sand stress state is found in the extension or the compression sides of the yield surface due to cyclic loading. At a higher stress ratio, the behaviour switches from contractive to dilative.
This corresponds to the phase transformation state (Box (5.2)), where significant shear strain occurs at almost constant deviatoric stress.
Remark: The realisation of the unloading-reloading behaviour of saturated porous media under undrained conditions is an important step towards the modelling of seismic-induced liquefaction events, cf. Section 6.4 for an example.