Multi-scale simulation of wave propagation and liquefaction in a one-dimensional soil column: hybrid DEM and finite-difference procedure

Abstract

The paper describes a multi-phase, multi-scale rational method for modeling and predicting free-field wave propagation and the weakening and liquefaction of near-surface soils. The one-dimensional time-domain model of a soil column uses the discrete element method (DEM) to track stress and strain within a series of representative volume elements (RVEs), driven by seismic rock displacements at the column base. The RVE interactions are accomplished with a time-stepping finite-difference algorithm. The method applies Darcy’s principle to resolve the momentum transfer between a soil’s solid matrix and its interstitial pore fluid. Different algorithms are described for the dynamic period of seismic shaking and for the post-shaking consolidation period. The method can analyze numerous conditions and phenomena, including site-specific amplification, down-slope movement of sloping ground, dissolution or cavitation of air in the pore fluid, and drainage that is concurrent with shaking. Several refinements of the DEM are described for realistically simulating soil behavior and for solving a range of propagation and liquefaction factors, including the poromechanic stiffness of the pore fluid and the pressure-dependent drained stiffness of the grain matrix. The model is applied to four sets of well-documented centrifuge studies. The verification results are favorable and highlight the importance of the pore fluid conditions, such as the amount of dissolved air within the pore water.

A Poromechanics of dry soil

The poroelastic characteristics of dry soil are rarely considered in geomechanics, due to the wide difference in the stiffnesses of air and of the solid matrix. However, just as with water, the paper’s algorithm requires the stiffnesses of all pore fluids, whether water, air, or quasi-saturated water. The air of a dry soil is assumed to obey the ideal gas law,

$$\begin{aligned} pV_{\text {v}} = \bar{N}RT \end{aligned}$$

(38)

where $\bar{N}$ is the gas substance (moles) of air in the void volume $V_{\text {v}}$; T is the absolute temperature; and R is the ideal gas constant. Assuming that temperature is constant, the initial and current values of quantity $pV_{\text {v}}/\bar{N}$ are equal, so that the current pressure is

$$\begin{aligned} p = p_{\text {o}} \frac{\bar{N}}{\bar{N}_{\text {o}}} \frac{V_{\text {vo}}}{V_{\text {v}}} \end{aligned}$$

(39)

where $p_{\text {o}}$ is the initial air pressure; $V_{\text {v}}$ and $V_{\text {vo}}$ are the current and initial void volumes; and $\bar{N}$ and $\bar{N}_{\text {o}}$ are the current and initial moles of air. The ratio $\bar{N}/\bar{N}_{\text {o}}$ is equal to the ratio $\text {det}(\mathbf {F})/\text {det}(\mathbf {W})$, as described near the beginning of Sect. 3. Multiplying and dividing Eq. (39) by total soil volume $V_{\text {o}}$ and noting that $V_{\text {vo}}/V_{\text {o}}$ is the initial porosity $n_{o}$, we have

$$\begin{aligned} p = p_{\text {o}} n_{\text {o}} \frac{\text {det}(\mathbf {F})}{\text {det}(\mathbf {W})} \frac{V_{\text {o}}}{V_{\text {v}}} \end{aligned}$$

(40)

The current void volume $V_{\text {v}}$ is the difference of the total volume V and solid volume $V_{\text {s}}$,

$$\begin{aligned} \begin{aligned} V_{\text {v}}&= V - V_{\text {s}} \\&= V_{\text {o}}\left[ \text {det}(\mathbf {F}) - (1-n_{\text {o}}) \left( 1 + \frac{\sigma _{kk}^{\prime }-\sigma _{\text {o},kk}^{\prime }}{3K_{\text {s}}} - \frac{p-p_{\text {o}}}{K_{\text {s}}} \right) \right] \end{aligned} \end{aligned}$$

(41)

where $K_{\text {s}}$ is the bulk modulus of the solid grains; $\text {det}(\mathbf {F})=V/V_{\text {o}}$ is the determinant (Jacobian) of the deformation gradient (i.e., the ratio of the current and initial soil volumes); and $1-n_{\text {o}}=V_{\text {so}}/V_{\text {o}}$ is the initial solid fraction. This equation accounts for the combined changes in the volumes of the grains due to changes in the air pressure and in the contact forces, by assuming that the grains are composed of a linear-elastic material [8, 46, 82].

Substituting Eq. (41) into Eq. (40), we have the following quadratic equation for pressure p:

$$\begin{aligned}&Ap^{2} + Bp + C = 0\end{aligned}$$

(42)

$$\begin{aligned}&A = (1-n_{\text {o}}) / K_{\mathbf {s}}\end{aligned}$$

(43)

$$\begin{aligned}&B = \text {det}(\mathbf {F}) - (1-n_{\text {o}}) \left[ 1 + (\sigma _{kk}^{\prime }-\sigma _{\text {o},kk}^{\prime }) / 3K_{\text {s}} \right] \end{aligned}$$

(44)

$$\begin{aligned}&C = p_{\text {o}} n_{\text {o}} \text {det}(\mathbf {F}) / \text {det}(\mathbf {W}) \end{aligned}$$

(45)

and the following equation for fluid outflow $\text {det}(\mathbf {W})$:

$$\begin{aligned} \begin{aligned} \text {det}(\mathbf {W}) &=\frac{p_{\text {o}} n_{\text {o}}\text {det}(\mathbf {F})}{p}\\&\Big /\left[ \text {det}(\mathbf {F}) - (1-n_{\text {o}}) \left( 1 + \frac{\sigma _{kk}^{\prime }-\sigma _{\text {o},kk}^{\prime }}{3K_{\text {s}}} - \frac{p-p_{\text {o}}}{K_{\text {s}}} \right) \right] \end{aligned} \end{aligned}$$

(46)

In a one-dimensional setting, $\text {det}(\mathbf {W})=1+du_{3}/dX_{3}+dw_{3}/dX_{3}$ and $\text {det}(\mathbf {F})=1+du_{3}/dX_{3}$, where $w_{3}$ is the pore fluid displacement relative to the matrix displacement $u_{3}$, and the $X_{3}$ is the original position within the soil column.