Numerical Modeling of Liquefaction

Abstract

The response of a saturated sand deposit to seismic motion is a very significant and challenging problem of soil dynamics, and a completely satisfactory generalized solution does not yet exist. A qualitative and quantitative prediction of the phenomena leading to permanent deformation or unacceptably high buildup of pore pressure is therefore essential to guarantee the safe behavior of engineering structures under transient consolidation and dynamic conditions. The significant approaches to model the behavior of a two-phase porous medium are usually categorized as uncoupled and coupled approaches. In the uncoupled analysis, the response of saturated soil is modeled without incorporating the interaction between soil and fluid, and then the pore pressure is accounted separately through a pore pressure generation model. In the coupled analysis, a mathematical framework is developed for computation of displacements and pore pressures at each time step. A comparison has been performed considering both approaches based on the consideration of soil non-linearity. The coupled analysis resembles closely with the liquefaction phenomena as compared to uncoupled approach. Hence, the usual decoupled and factor of safety approach may not be considered as most appropriate in the analysis of such dynamic behavior.

Appendix

P-Z Mark III model takes into account the linear distribution of the stress ratio for approximating sand dilatancy (Nova & Wood, 1982).

$${d}_{g}=\frac{d{\in }_{v}^{p}}{d{\in }_{s}^{p}}=\left(1+{\alpha }_{g}\right)\left({M}_{g}-\eta \right)$$

where $d{\in }_{v}^{p}$ and $d{\in }_{s}^{p}$ are plastic volumetric and deviatoric strains increments, respectively. M_g is correlated with the angle of friction $(\varnothing$) as follows:

$${M}_{g}=6Ssin\varnothing /(3S-sin\varnothing )$$

The plastic potential surface relationship is evaluated as follows:

$$g=\left\{q-{M}_{g}p\left(1+\frac{1}{\alpha }\right)\left[1-{\left(\frac{p}{{p}_{g}}\right)}^{\alpha }\right]\right\}$$

The bounding or the yield surface is given as follows:

$$f=\left\{q-{M}_{f}p\left(1+\frac{1}{\alpha }\right)\left[1-{\left(\frac{p}{{p}_{c}}\right)}^{\alpha }\right]\right\}$$

where p and q are the mean effective and deviatoric stress, respectively; α and M_f are constants; M_g is slope of the critical state line and p_g and p_c are size parameters. Fig. shows the plastic potential and yield surface for the loose and dense sand, respectively.

Plastic Flow for Loading and Unloading

The loading plastic flow vector n_gL and unloading plastic flow vector n_gu are given as follows:

$${\eta }_{gL}=\frac{1}{\sqrt{1+{d}_{g}}}\left\{\begin{array}{c}{d}_{g}\\ 1\end{array}\right\}\, and\, {\eta }_{gU}=\frac{1}{\sqrt{1+{d}_{g}}}\left\{\begin{array}{c}-\left|{d}_{g}\right|\\ 1\end{array}\right\}$$

The absolute sign is used in such a way that constant densification occurs during unloading and modeling of the liquefaction is done.

Plastic Modulus for Loading and Unloading

During loading phase, Pastor and Zienkiewicz (1986) have given the relationship for obtaining the plastic modulus as follows:

$${H}_{L}={H}_{0}p{H}_{f}\left\{{H}_{v}+{H}_{s}\right\}{H}_{DM}$$

where ${H}_{0}$ is the intial plastic modulus for loading and other parameters are defined as follows:

$${H}_{f}={\left(1-\frac{\eta }{{\eta }_{f}}\right)}^{4} { H}_{v}=1-\frac{\eta }{{M}_{g}}$$

$${H}_{s}={\beta }_{0}{\beta }_{1}{e}^{{-\beta }_{0}\xi } \xi =\int d\xi =\int \left|{d\epsilon }_{q}^{p}\right|$$

where ${d\epsilon }_{q}^{p}$ is the plastic shear strain.

Pastor et al. proposed the following relationship for the plastic unloading modulus H_U0:

$${H}_{u}={H}_{u0}{\left(\frac{{M}_{g}}{{\eta }_{u}}\right)}^{{\gamma }_{u}} for \left|\frac{{M}_{g}}{{\eta }_{u}}\right|>1$$

$${H}_{u}={H}_{u0} for \left|\frac{{M}_{g}}{{\eta }_{u}}\right|<1$$

where η_u is the unloading stress ratio.