Method of preparation of the sand pile

71 Simulation results

Simulation results

In this chapter, we present the numerical results on effective material properties of two dimensional sand piles of soft convex polygonal particles using the discrete element method (DEM). We focus primarily on discussing the simulation results of the micro- scopic force distribution, and then show how the shape of the particles and construction history of the piles affects the pressure distribution under a sand pile. In addition, other measurable macroscopic tensorial quantities including strain, fabric distributions, and orientation of the particles inside a sand heap obtained from simulations are discussed.

We first give in the following a short description how the sand pile is constructed from two different types of procedures. We then measure averaged stress and strain, the latter via imposing a 10% reduction of gravity, as well as the fabric tensor. Then, we compare the vertical normal strain tensor between sand piles qualitatively and show how the construction history of the piles affects their strain distribution. The simulation results of volume fraction of sand piles are compared qualitatively with the existing experimental results in the literature. In the next step, the elastic constants are measured assuming Hooke’s law to be valid in relating incremental stress and strain tensors to each other. We then determine correlation between the measured elastic material constants and the trace of the fabric tensor, and between invariants of the incremental stress and strain tensors for a small change in gravity.

72 Simulation results

A.

B.

Figure 3.1.A-B: Schematic diagrams for possible construction procedures of the sand pile. A. Particles are dropped onto the apex of the pile, B. Particles are dropped in a uniform distribution on each layer of the pile.

tributed way, from a so-called ‘‘line source’’. Note that the line source is reduced in length during the procedure so as to have a length equal to or slightly smaller than that of the top plateau of the pile at any instant of time. When a pile is constructed from a point source, the particles are dropped always onto the apex of the sand pile and roll down the slopes of the pile, as shown in Fig. 3.1.A. On the other hand, when a sand pile is constru- cted from a line source procedure, the particles are dropped in a layer wise manner, and

73 Simulation results slowly from a height of slightly greater than one particle diameter onto the already

present layer, as shown in Fig. 3.1.B.

In order to investigate the microscopic force structure and macroscopic distributions of quantities such as stress, strain, inertia and fabric under a sand pile, and to check how the construction history of the granular aggregates influences their pressure, and strains distribution, we perform numerical simulations in which sand piles are constructed from either a point source or a line source. In the followings we explain in detail how we construct two different types of sand piles from two different qualitative procedures.

3.1.1 Sand piles from a point source

We constructed a sand pile from several thousands of convex polygonal particles with varying shapes, sizes and edge numbers where the particles were dropped from a point source. The particles were dropped on to the system from 50 cm height with initial velocity of 0.2 m sand the time for the generation of the new particle is 0.1s at the same position. We have constructed 14 sand piles using a point source. The individual simu- lations differ only by the initialization of the random number generator for the generation of particles. The number of polygon edges varies from six to eight for each simulation.

The particle corners are placed randomly on an ellipse (RAND1), see in ref. [103]. The radius of the particles is 2.75 mm and degree of poly-dispersity of the particles is about 30%.

The average angle of repose for 14 sand piles was approximately 28by taking the ave- rage over the left and right base angles. A snapshot of one of the simulated sand piles constructed from a point source is illustrated in Fig. 3.2. Different colour corresponds to particles dropped at different times. We used a flat bottom ground plate. The walls and the funnel are made of immobile specially shaped particles and the bottom ground plate is fixed in shape as well, hence effects that may spoil real-world experiments such as ground plate bending [125] or compression [20], when the grains are deposited are excluded. The characteristic properties of the ground plate, side walls and funnel are equal to the particles properties, which mean the Young’s modulus of elasticity of the bottom ground wall is the same as for the particles, and the static friction coefficient between the bottom ground wall and the particles is the same as between the particles.

3.1.2 Sand piles from a line source

Next, we constructed sand piles from a line source that consisted of around 6500 particles. The dropping height of each new layer from the already present layer of the system is 8 mm and particles are dropped uniformly onto the already present layer. The size, shape of the particles and the time for the generation of the new particle we used for

74 Simulation results constructing a sand pile from a line source are the same as that for the point source case.

The corner number of each particle is 7, and the degree of poly-dispersity of the particle is 10%. The positions of corner are placed randomly on an ellipse (RAND1). The measured average angle of repose of 11 sand piles is about 27 . Fig. 3.3 displays one of the simulated sand piles that was constructed from a line source. The simulation parameters and their values used for the construction of sand piles are given in the following table.

Name of variable Value

Static friction 0.54

Dynamic friction 0.54

Particle density 5000 kg m²

Young’s modulus 10⁷N m

Damping coefficient 0.75

Time step 2 10× ⁻⁶s

Size of poly-dispersity (point source) 30%

Size of poly-dispersity (line source) 10%

Initial velocity of the particles 0.2m s/

Particle radius 2.75 mm

Cohesion coefficient 0

Acceleration 9.81m s²

Position of particle corner RAND1 Number of corners(point source) (6,8) Number of corners(line source) 7

Table 2: Parameters used for the DEM simulation of the sand pile.

75 Simulation results

Figure 3.2: Snapshot for a simulated sand pile constructed from a point source with 30%

poly-disperse mixture of particles.

Figure 3.3: Snapshot for a simulated sand pile constructed from a line source with 10%

poly-disperse mixture of particles.

76 Simulation results