Molecular dynamics simulations of complex-shaped particles using Voronoi-based spheropolyhedra

The spheropolyhedra method has been used earlier for efficient and accurate molecular dynamics simulations of granular matter with particles with complex shapes. Also the Voronoi construction is a tool of proved utility for the virtual representation of powders and grains. In this paper a technique that combines the two methods and provides a framework for the study of the three-dimensional mechanical behavior of granular matter is proposed. In order to understand the capabilities of the new method, a number of computer simulations of the cubic (true) triaxial test, measuring the mechanical behavior of packing of particles, is carried out. Results from tests with packing of complex-shaped particles represented by “Voronoi particles” are compared with corresponding results of packing of spherical particles. Features such as the saturation value for the macroscopically observed coefficient of friction, as reported in the literature, are compared for the packing of spheres and for the packing of “Voronoi particles,” showing that the difference in shape strongly affects the results. The proposed technique and simulation results can be used to help understand how the individual shape of grains affects the macroscopic mechanical behavior of granular matter such as cohesionless soils.

Figure 1

Dilation of a square by a disk. The disk sweeps the profile of the square producing a larger square with rounded corners.

Figure 2

Erosion of a square by a disk. In the erosion the square is shrunk by moving its sides inward (the opposite direction of the normal vector $\vec{n}$) by a distance $R$. This distance $R$ corresponds here to the radius of the spheropolyhedra desired.

Figure 3

Opening of a square by a disk. First the left square is eroded resulting in a smaller square. Then this smaller square is dilated. Both operators are carried out using the same spheroradius $R$.

Figure 4

The opening of a 3D particle: initially the cube is eroded, and then is dilated by a sphere.

Figure 5

Some examples of spheropolyhedra. Left: sphero-“rice” (edge $\oplus$ sphere). Center: spherotetrahedron (tetrahedron $\oplus$ sphere). Right: sphero-“star” (three perpendicular edges $\oplus$ sphere).

Figure 6

The Voronoi construction explained. The points are randomly distributed in a plane and then their corresponding Voronoi polygons are built by bisecting the joining lines between the given points and its closer neighbors until a convex and closed polygon is obtained.

Figure 7

The opened Voronoi grid. For visual aid the radius of the dilating disk is smaller than the eroding one. In this study the two radii are equal an there are not void spaces between the grains except for the rounded corners.

Figure 8

(Color online) A three-dimensional array of Voronoi spheropolyhedra.

Figure 9

Cumulative distribution of volumes for the Voronoi ensemble. The volume is given in length units, where one length unit is the cell side of the original cubic grid used for the generation.

Figure 10

The three different overlapping distances $\delta$ and their possible normal vectors $\vec{n}$ (a) In the case of the interaction of a spherovertex with a spheroplane the overlapping distance is always the penetration of the first into the second. (b) The collision of two spheroedges is taken as the collision of two cylinders. (c) The special case of two spheres considered as the collision of two spherovertices.

Figure 11

The value of the tangential force $F_t$ grows proportionally to the displacement $\epsilon$ but when the Coulomb threshold $\left(\mu F_n\right)$ is reached, it immediately takes this value.

Figure 12

(Color online) The test setup, the Voronoi array is put inside the rectangular box composed by six planes. The stresses, lateral ${\sigma }_1$, frontal ${\sigma }_2$ (not shown but applied to the front and back planes) and vertical ${\sigma }_3$, are applied to the planes and a triaxial test can be simulated. To measure the strain in the $x$ direction, ${\epsilon }_1$, the change in the initial length $\Delta L_x$ is divided by the initial length $L_x$.

Figure 13

In a $q$ vs $p$ plane the TTT has two stages for both the compression and extension tests: (1) an isotropic compression is applied by setting ${\sigma }_1={\sigma }_2={\sigma }_3$; (2a) by applying a constant strain rate in the $z$ direction compressing the volume, ${\sigma }_3$ increases and therefore $p$ and $q$ values also increase proportionally until the failure is reached; and (2b) The strain rate in the $z$ direction now extends the volume and $q$ increases while $p$ decreases. The dashed lines are the failure envelopes, and join the origin with the failure points.

Figure 14

Typical response of the stress ratio in a true triaxial test.

Figure 15

The results with one particular triaxial test with a confining pressure of $1.0e^{-4}{K}_n$ on a compression path.

Figure 16

Results from the true triaxial experiments for both compression ($p$ increasing) and extension ($p$ decreasing) showing the deviatoric stress $q$ versus the isotropic stress (or pressure) $p$. The dashed lines join the failure points of each experiment and their slopes give the macroscopic friction angle. These lines however do not intersect at the origin giving a nonlinear effect.

Figure 17

(Color online) A similar test setup was made for spheres with the same number of particles, average mass and parameters.

Figure 18

Results for several compression tests on packing of spheres showing the failure points for both spheres (circles) and Voronoi spheropolyhedra (squares).

Figure 19

Results for several true triaxial experiments in which the microscopic friction coefficient ${\mu }_{micro}$ is changed for both the Voronoi (squares) and the spheres (circles) arrays.

Figure 20

Volumetric strain ${\epsilon }_v$ versus the deviatoric strain ${\epsilon }_d$ for both Voronoi grains (squares) and spheres (circles). Straight lines indicate the slope or dilatancy $\psi$ at the plastic deformation regime. For the Voronoi packing the dilatancy value is $\psi =12.53°$ and for the sphere packing is $\psi =7.52°$.

Figure 21

Sliding contact fraction $s_f$ versus the microscopical friction coefficient ${\mu }_{micro}$ for the Voronoi array (circles) and the sphere array (squares).