2 The simulation method
3.6 Strain distribution under sand piles
In this section, we focus on the sensitivity of the strain distribution (total strain) to the preparation of sand piles. Before interpreting the results for the strain tensor, first, we present simulation results for the movement of each individual grain inside the sand pile under gravity reduction. The result obtained from the simulation is represented in Fig.
3.16. Each arrow shown in the figure corresponds to the movement of an individual particle. The arrow is drawn from the initial position of the centre of mass
(
x yi, i)
of the particle i at the ambient gravity level of a sand pile at g=9.81 /m s2 towards the final point of the centre of mass(
x yi′ ′, i)
at the new state of the sand pile obtained by reducing gravity slowly by about 50%.As expected, the range of movement of a particle decreases towards the bottom layers of the sand pile, and increases towards the surface and the tip of the sand pile.
87 Simulation results
A B
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.5
1 1.5 2 2.5 3 3.5 4
x 10-3
x(m) -u yy
4.5cm 9cm 13.5cm 18cm 22.5cm 27cm
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0
0.5 1 1.5 2 2.5 3 3.5
4x 10-3
x(m) -u yy
4.5cm 9cm 13.5cm 18cm 22.5cm 27cm
Figure.3.17: Vertical normal strain distribution at different heights of simulated sand pile. Left: sand piles constructed from a point source (Average of 11 sand piles and height of the pile is 33.5 cm), Right: sand piles poured from a line source (Average of 11 sand piles and height of the pile is 31.5 cm).
The vertical normal strain tensor component obtained from DEM simulations is displayed in Fig. 3.17 for two types of sand piles that were constructed using the two different pouring protocols. The averaged strain tensor was evaluated throughout the sand pile; we represent it via a plot of tensor components as a function of the lateral coordinate x of the pile for layers of given heights y1, y2, ... yn.
We give this component of the strain tensor to obtain a qualitative picture, although the foregoing discussion in Section 2.6.2 shows that it is not a rigorously determined quantity. While it has the correct scaling with gravity level, vertical and horizontal strains are of course coupled, so the errors produced by the method in the horizontal direction will also affect the vertical direction. The topmost curve in the graph shows the strain tensor result at the bottom layer of the corresponding sand pile, whereas the bottom curve corresponds to the top layer. Heights are given as function of the total height of the pile to its apex. An interesting feature of the vertical normal strain tensor for various heights is that the vertical normal strain changes with the layer position in the sand piles like the stress tensor. The vertical normal strain shows a dip (Fig. 3.17.A) near the centre of the piles that are poured from a point source. It can be seen that the strain dip appears not only at the bottom layer but also exists up to the certain height inside the sand pile. On the other hand, the vertical normal strain increases towards the centre and towards the bottom layer of sand piles poured from a line source, i.e., a strain dip does not occur in sand piles constructed from a line source, see in figure 3.17.B.
88 Simulation results
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
1 1.5 2 2.5 3 3.5 4
x 10-3
-u yy
x(m) Cambou Cundall Liao diff
Figure 3.18: Vertical normal strain tensor at the bottom layer of the sand pile constructed from a line source obtained using four different approaches.
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
1 2 3 4 5 6
x 10-3
-u yy
x(m) Cambou Cundall Liao diff
Figure 3.19: Vertical normal strain tensor at the bottom layer of the sand pile constructed from a point source obtained using four different approaches.
89 Simulation results
3.6.1 Comparison of strain tensor
Here, we are interested in determining the strains (total strain) using four different versions of strain tensors (three best-fit strains and derivative strain) using the equations (2.57), (2.66), (2.73), and (2.76) by numerical investigation, and to compare the results for the vertical normal strain tensor both qualitatively and quantitatively with each other.
The numerical simulations results were obtained using our code. The results of the average (negative) vertical normal strain at the bottom layer of the sand pile constructed from a line source are illustrated in Fig. 3.18. All the strain tensors were measured via imposing 10% of reduction of gravity from the actual state of the gravity level.
For the case of a line source, we have averaged results over seven sand piles in order to reduce fluctuations. We observe from Fig. 3.18 that the best-fit strains of Cambou et al.
and Cundall et al. are close to each other with a deviation of few percent, while the best fit strain of Liao et al significantly differs from the Cambou strain and Cundall strain, the deviation went up to 30-40%. The reason for this large deviation may be the inclusion of particle rotations in the calculation of the Liao strain, instead of consideration of only the translation of the particle centre. Presumably, the Liao strain might be useful in theories employing micro-polar continua and involving couple stresses in addition to force stresses. Then micro rotation effects may partially compensate for the excess strains of Liao et al. As long as we assume a symmetric stress tensor, the other strain definitions are more useful.
On the other hand, the vertical normal (negative) strain obtained using the differentiation method shows a different behaviour than the other strains especially in the vicinity of the surface of the sand pile, but, shows a similar behaviour towards the centre of the sand pile. It is in good agreement quantitatively with the best-fit strains of Cambou et al. and Cundall et al. Clearly, numerical differentiation should be avoided whenever possible and the deviations near the extremities of the sand pile are artifacts of the procedure.
Furthermore, we compared the results of the four types of strains quantitatively for sand piles constructed from a point source. Fig. 3.19 gives the simulation results of average negative vertical normal strains at the bottom layer of the point source sand pile. For the point source case, we averaged the strains over seven sand piles each. All four methods produce a strain dip under piles constructed from a point source, as expected. Again, quantitative comparison indicates strong deviations for the Liao strain, which should not be used in our context, and exhibits the deficiencies of the differentiation method.
90 Simulation results