Averaging procedure

One can obtain detailed information about measurable quantities such as forces, stresses, and displacements of an individual particle from the discrete element simulation.

However, the behaviour of an individual particle is not significant for the behaviour of the whole system, as most of the measurable microscopic quantities in granular material vary strongly as a function of position.

In this regard, one common example is the stress tensor, which is not constant across grains (microscopic level), but usually it shows its largest values for particles with a large number of contacts. Moreover, the microscopic stress tensor would not be a convenient means to describe the macroscopic sand pile, as it fluctuates widely within a volume containing a few sand grains. In fact, it is zero in the voids between grains. Hence for a continuum description, we need to average microscopic stresses over suitable domains, which will reduce the relative fluctuations. In order to suppress the fluctuations, we need to perform averages over sufficiently many particles in an averaging volume element.

But, the question is how many particles are actually required to determine the average macroscopic tensorial quantities, which means, one has to determine the number of particles (or appropriate size of the volume element) providing realistic results for the macroscopic stress tensor and also for other macroscopic tensorial quantities including fabric, strain, and volume fraction of the sand pile.

In our work, averages are performed by introducing a representative volume element (RVE) via the requirement that the average becomes size independent, if the volume is taken equal to this value or larger. The RVE averaging strategy is used by many engineers to obtain scalar and tensorial quantities, especially in problems that deal with particulate materials.

1 ^{P P},

ij ij

p

F V F

V =V

∑

68 Simulation method Averaging over different volumes gives different results, as long as the volume element is

too small. As we increase the size of the volume element in the computation of the average, the latter converges to a certain value. A size of the volume element near but above the minimum needed for convergence gives the representative volume element to be used in evaluations of stress and other fields.

We have taken into account those particles whose centres of mass lie inside the averaging volume element to determine the macroscopic stress tensor. Sometimes, this method is referred to as the particle centre averaging technique. It is noted that the results of the macroscopic stress tensor were obtained by taking averages over many particles that correspond to the middle region of the sand pile. The simulation results for the individual components of the stress tensor against the number of particles are displayed in Fig. 2.8.

The number of particles shown in the graph corresponds to the size of the volume element. The blue curve connecting square symbols represents the vertical normal stress tensor, whereas the red and black curves represent the horizontal and shear stresses, respectively. It can be seen in the figure that all the components of the stress tensor are converged approximately at the same number of particles. We find that a size of the volume element containing 100-200 particles is sufficient to serve as RVE.

On the other hand, we determine the size of the RVE for the strain tensor as well as for the fabric tensor. Fig. 2.9 displays the RVE for the strain tensor and the one for the fabric tensor is illustrated in Fig. 2.10. We find from the figure that the size of the volume element is the same as for the stress tensor. We have not determined individually the size of RVE for the inertia tensor fields. It should be noted that the size of the volume element that we consider for the calculation of the stress, fabric and strains is same as for the inertia, elastic constants and volume fraction of the sand pile.

69 Simulation method

0 50 100 150 200 250

0 2000 4000 6000 8000 10000 12000 14000

stress tensor[N/m]

number of particles

σ σ σ σyy σ σ σ σxx σ σ σ σxy

Figure 2.8: Representative volume element (RVE) for stress tensor. The volume element was located in the centre and near the bottom of the sand pile.

0 50 100 150 200 250

-5 0 5 10 15 20x 10^-3

strain tensor

number of particles uyy

u_xx uxy

Figure 2.9: Representative volume element (RVE) for strain tensor.

70 Simulation method

0 50 100 150 200 250

-0.5 0 0.5 1 1.5 2

fabric tensor

number of particles

f_yy f_xx fxy

Figure 2.10: Representative volume element (RVE) for fabric tensor.

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.