A study of the geogrid–subballast interface via experimental evaluation and discrete element modelling

Abstract

This paper presents a study of the interface of geogrid reinforced subballast through a series of large-scale direct shear tests and discrete element modelling. Direct shear tests were carried out for subballast with and without geogrid inclusions under varying normal stresses of $\sigma _n =6.7$ to $45\hbox { kPa}$. Numerical modelling with three-dimensional discrete element method (DEM) was used to study the shear behaviour of the interface of subballast reinforced by geogrids. In this study, groups of 25–50 spherical balls are clumped together in appropriate sizes to simulate angular subballast grains, while the geogrid is modelled by bonding small spheres together to form the desired grid geometry and apertures. The calculated results of the shear stress ratio versus shear strain show a good agreement with the experimental data, indicating that the DEM model can capture the interface behaviour of subballast reinforced by geogrids. A micromechanical analysis has also been carried out to examine how the contact force distributions and fabric anisotropy evolve during shearing. This study shows that the shear strength of the interface is governed by the geogrid characteristics (i.e. their geometry and opening apertures). Of the three types of geogrid tested, triaxial geogrid (triangular apertures) exhibits higher interface shear strength than the biaxial geogrids; and this is believed due to multi-directional load distribution of the triaxial geogrid.

Appendix: mathematical framework of DEM modelling

At a given time, the force vector $\vec {{\varvec{F}}}$ that represents the interaction between two particles is resolved into normal ($\vec {F}_N$) and the shear component ($\vec {F}_T$) with respect to the contact plane [39]:

$$\begin{aligned} {\vec {F}_N}= & {} {K_N}{U^n} \end{aligned}$$

(1)

$$\begin{aligned} \delta {\vec {F}_T}= & {} - {K_T}\cdot \delta {U^s} \end{aligned}$$

(2)

where, $K_{N}$ and $K_{T}$ are normal and tangential stiffness at the point of contact; $U^{n}$ is the normal penetration between two particles (Fig. 13a); $\delta U^{s}$ is the incremental tangential displacement; and $\delta \vec {F}_T$ is the incremental tangential force. The new shear contact force is determined by summing the old shear force at the start of the time-step with the increment of elastic shear force.

$$\begin{aligned} {\vec {{{\varvec{F}}}}_T} \leftarrow {\vec {{{\varvec{F}}}}_T} + \delta {\vec {{{\varvec{F}}}}_T} \le \mu ~{\vec {{{\varvec{F}}}}_N} \end{aligned}$$

(3)

where, $\mu$ is the coefficient of friction.

Shear stresses in a given volume V are calculated by the summation of discrete contact forces as:

$$\begin{aligned} \sigma _{ij} =\left( {\frac{1-n}{\mathop \sum \nolimits _{N_p } V}} \right) \mathop \sum \nolimits _{N_p } \mathop \sum \nolimits _{N_c } \left| {x_i^{\left[ c \right] } -x_i^{\left[ p \right] } } \right| n_i^{\left( {c,p} \right) } f_j^{\left( c \right) } \end{aligned}$$

(4)

where $N_{p}$, $N_{c}$ are the number of particles and the number of contacts of these particles, respectively; n is the porosity within the given volume; $x_{i}^{[p]}$ and $x_{i}^{[c]}$ are the positions of a particle centroid and its contact, respectively; $n_{i}^{(c,p)}$ is the unit normal vector; and $f_{j}^{(c)}$ is the force acting at contact (c) arising from a particle.

Contact forces are characterised by the probability density distribution of inter-particle contact orientation $\bar{E}(\Omega )$ proposed by Ouadfel and Rothernburg [46] as:

$$\begin{aligned} \bar{E}\left( \mathrm{{\Omega }} \right) = \frac{1}{{4\pi }}\left[ {1 + {F_{ij}}{n_i}{n_j}} \right] \end{aligned}$$

(5)

where, $F_{ij}$ is a second order fabric tensor which represents the distribution of contact orientations in the volume of interest, and is determined by:

$$\begin{aligned} F_{ij} =\frac{1}{N_c }\mathop \sum \nolimits _{k=1}^{N_c } n_i^k n_j^k \end{aligned}$$

(6)

Note that $F_{ij}$ is symmetrical (i.e. $F_{ij}=F_{ji})$ with the three principal values $F_{11} ,F_{22} ,F_{33}$ where their sum is unity; $n_{i}^{k}$ is a unit vector representing the orientation of the k contact (Fig. 13b); and the components of a unit vector are ($\cos \gamma ,\sin \gamma \cos \beta ,\sin \gamma \sin \beta$). The probability density function of all contacts satisfies:

$$\begin{aligned} \mathop \smallint \nolimits _0^{2\pi } \mathop \smallint \nolimits _0^\pi E\left( {\Omega } \right) \text {sin}\gamma d\gamma d\beta =1 \end{aligned}$$

(7)

The principal direction of contact forces, $\theta _r$ can be described by the following Fourier series approximation introduced by Rothenburg and Bathurst [47], as given below:

$$\begin{aligned} E\left( \theta \right) =\frac{1}{2\pi }\left[ {1+a^{r}\text {cos}2\left( {\theta -\theta _r } \right) } \right] \end{aligned}$$

(8)

where, $a^{r}$and $\theta _r$ are coefficients of anisotropy of contact and the corresponding major principal directions, respectively. By comparing the contact force orientations obtained in DEM simulations with those determined by Eq. 8, the principal direction of contact forces, $\theta _r$ can then be estimated at a given shear strain during the DEM analysis.