The paper presents 3D DEM simulation results of undrained tests for loose assemblies with varied porosities under both triaxial compression and plane strain conditions, using a periodic cell. The undrained tests were modelled by deforming the samples under constant volume conditions, which corresponds to saturated soil samples. The undrained stress paths are shown to be qualitatively similar to physical experimental results. The triggering of liquefaction and temporary liquefaction is identified by a microscopic parameter with redundancy factor (RF) equal to unity, which defines the transition from “solid-like” to “liquid-like” behaviour. The undrained behaviour of granular soils is found to be mainly governed by the evolution of redundancy factor, and a reversal of deviatoric stress in stress path (temporary liquefaction) is found to be due to temporary loss of contacts forming a structural mechanism in the system where RF is smaller than unity during the evolution.
Interest in undrained behaviour especially for loose sand has received much attention since the pioneering experimental work [
Most DEM simulations are performed “in a vacuum” without fluid, as were done in this paper. The undrained behaviour was modeled using 2D DEM by [
Three-dimensional DEM simulations have been performed on poly disperse systems of 3600 elastic spheres using the same TRUBAL code as used in [
All the undrained simulations reported in the paper were carried out in a periodic cell. A periodic cell allows a particle that moves out of the cell to be re-mapped back into the cell at a corresponding location on the opposite face. The particle that moves out of the cell carrys all the same information as that moving into the cell, except for particle positions. An infinite lattice can be imagined by replicating one cell throughout space. Thus, the simulation can be performed free from boundaries. More information about periodic cell can be found in [
During the simulations, the ensemble average stress tensor is calculated (see [
where the summations are over the C contacts in the volume V, R1 and R2 are the radii of the two spheres in contact, N and T are the magnitudes of the normal and tangential contact forces at the contact, ni is the unit vector normal to the contact plane and ti is the unit vector parallel to the contact plane.
In this section, results are presented of undrained test simulations on all the samples and both the macroscopic and microscopic behaviouris discussed. The macroscopic behaviour is presented in terms of stress path and deviatoric stress. The microscopic behaviour is presented in terms of redundancy factor.
systems under undrained (constant volume) triaxial compression (UTC) and undrained plane strain (UPS) conditions respectively. It can be seen from the Figures that the overall trends of the stress path are not much different for the two types of strain conditions. In both cases, all the loose sample exhibits an initial increase of p’ except the loosest sample (n = 0.425). However, the loose sample (porosity, n = 0.405) does not exhibit a reversal under UPS, as it does under UTC. This indicates that the undrained behaviour is much related to strain conditions. The reason for this difference could be explained microscopically as explained later.
seen from both the figures that for the three very loose samples (n = 0.425, n = 0.419, n = 0.414), the initial peak of deviatoric stress occurs at small strains, followed by gradual decrease (with sharpest drop for the loosest sam- ple in each strain condition, but the drop in UPS is less sharp than in UTC), and with further straining the
devaitoric stress becomes more or less constant at small values (called steady state in [
In this section, microscopic description of undrained behaviour in terms of redundancy factor is presented, and the triggering of liquefaction/temporary liquefaction is identified by such a microscopic parameter with the value of unity. Both triaxial compression and plane strain simulation results are presented and compared.
For a granular assembly with spherical particles with a finite value of friction at the contacts, a microscopic parameter termed redundancy factor (RF) is defined [
where f is the fraction of sliding contacts and the average coordination number is defined as the ratio of twice number of contacts and number of particles, i.e.
system can be said to be solid-like, and for IR < 1 we have a structural mechanism and the system can be said to be liquid-like. Therefore, RF = 1 defines a limit phase transition condition from solid-like to liquid-like behaviour.
For the six systems examined, the evolutions of redundancy factor are plotted in
RF = 1 defines a phase transition from solid-like (non-liquefied) behaviour to liquid-like (liquefied or temporarily liquefied) behaviour. It can be seen that the three very loose samples do not re-establish solid-like behaviour after they drop below the phase transition line, indicating liquefaction. The medium loose samples (except the sample n = 0.405 under UPS) generally first drop below the phase transition line, and then go up until they get close to or even cross the phase transition line, indicating temporary liquefaction. In both figures, the initial intersection points of the phase trasnsition line and evolution of RF curves indicate the onset of liquefaction or temporary liquefaction. It can be easily seen that a looser sample liquefies earlier in each strain condition (UTC or UPS). Interestingly, in
A micromechanical formula for defining the triggering of soil liquefaction (temporary liquefaction) is established in terms of redundancy factor. The relation among redundancy factor, average coordination number and percentage of sliding contacts is provided. The evolution of redundancy factor is provided under undrained axisymmetric compression and plane strain conditions and compared. A phase transition is associated with redundancy factor of unity, which defines the transition from solid-like behaviour to liquid-like behaviour. This phase transition corresponds to a limit average coordination number slightly in excess of four. From comparision of simulation results under two types of strain conditions, it is found that:
1) A reversal of deviatoric stress in stress path under undrained conditions is due to the fact that the system becomes a structural mechanism (RF < 1) during part of the evolution. If RF > 1 during all the evolution, the reversal does not occur.
2) A system under undrained plane strain is generally more robust in resisting liquefaction than under UTC.
The authors acknowledge the funding by the Engineering and Physical Sciences Research Council, UK (Grant No.GR/R91588) for the work reported in the paper. The authors would also like to thank Prof. Andrew Chan from Federation University Australia and Dr. Colin Thornton from University of Birmingham UK, for their guidance and discussion on the relevant work.
Guobin Gong, (2015) DEM Simulations of Granular Soils under Undrained Triaxial Compression and Plane Strain. Journal of Applied Mathematics and Physics,03,1003-1009. doi: 10.4236/jamp.2015.38123