Analysis of wave propagation in dry granular soils using DEM simulations

Abstract

In this paper, a three-dimensional particle-based technique utilizing the discrete element method (DEM) is proposed to study wave propagation in a dry granular soil column. Computational simulations were conducted to investigate the soil response to sinusoidal motions with different amplitudes and frequencies. Three types of soil deposits with different void ratios were employed in these simulations. Different boundary conditions at the base such as rigid bedrock, elastic bedrock, and infinite medium were also considered. Analysis is done in time domain while taking into account the effects of soil nonlinear behavior. The computational approach is able to capture a number of essential characteristics of wave propagation including motion amplification and resonance. Dynamic soil properties were then extracted from conducted simulations and used to predict the response of the soil using the widely used equivalent linear method program SHAKE and compare its predictions to DEM results. Generally, there was a good agreement between SHAKE and DEM results except when the exciting frequency was close to the resonance frequency of the deposit where significant discrepancy in computed shear strains between SHAKE predictions and DEM results was observed.

Appendix 1: Averaged stresses and strains

Stress is a continuum characteristic that requires averaging and homogenization procedures to be used to evaluate stress fields consistent with particle interaction forces. The average stress tensor $\bar{\varvec{\upsigma}}$ for particles whose centers lie within a certain control volume was evaluated using [13]:

$$ \bar{\varvec{\upsigma}}=-\left(\frac{1-n}{\sum_{p=1}^{N_p} V_p}\right)\sum_{p=1}^{N_p} \sum_{c=1}^{N_c} {\varvec{\ell}}_{c,p}{\bf f}_{c} $$

(10)

where the indices p and c refer respectively to particles and inter-particle contacts within the control volume, N _p is total number of particles within this volume, N _c is corresponding total number of contacts, V _p is volume of particle p, ${\varvec{\ell}}_{c,p}$ is vector connecting the position of contact c and centroid of particle p, and f _c is inter-particle force at contact c. The strain tensor at a location x was evaluated through time integration of the strain rate tensor:

$$ \dot{\varvec{\upvarepsilon}}=\frac{1}{2}\left({\bf G}+{\bf G}^T\right) $$

(11)

where G is an estimate of the velocity gradient. This gradient was obtained based on a least squares regression using velocities of particles within a control volume centered at x. The components of G are evaluated so that to minimize the following measure of mismatch [30]:

$$ \theta=\sum_{p=1}^{N_p}{\left|\tilde{{\bf v}}^{(r)}_p-{\bf v}^{(r)}_p\right|}^2 $$

(12)

in which v ^(r) _p is a relative velocity vector of particle p with respect to mean velocity of particles within the control volume:

$$ {\bf v}^{(r)}_p={\bf v}_p-\frac{1}{N_p}\sum_{p=1}^{N_p} {\bf v}_{p} $$

(13)

where v _p is the velocity of particle p. The vector $\tilde{{\bf v}}^{(r)}_p$ provides an estimate of particle velocity which is consistent with the gradient G:

$$ \tilde{{\bf v}}^{(r)}_p={\bf G}{\bf x}_p^{(r)} $$

(14)

where x ^(r) _p is vector of relative location of particle p with respect to the centroid of particles within the control volume:

$$ {\bf x}_p^{(r)}={\bf x}_{p}-\frac{1}{N_p}\sum_{p=1}^{N_p} {\bf x}_{p} $$

(15)

in which x _p is position vector of particle p. The components of G are obtained by solving the following system of equations [30]:

$$ \frac{\partial {\theta}}{\partial {\bf G}}=0 $$

(16)