Super Administrateur

2D channel on inclined plane

1 mm glass beads

GENERAL CONDITIONS OF THE EXPERIMENT

– The two lateral walls are parallel and frictionless

– Free bottom exit

– Beads are initially displayed in a mono layer all along the full channel

Following my stay in the Department of Civil Engineering and Applied Mechanics from McGill University (Montreal, Canada) and my collaboration with SB Savage, we continued our studies on a numerical model of granular media subjected to cohesive forces (Model molecular dynamics). This model allows to describe exhaustively the micro interactions taking place between the grains. We used the laws of mechanical behavior typically allowed in these models. The grains may intersect and thus oppose a force proportional to the normal overlap but delayed by a carefully chosen shock (hence the name « Spring-Dashpot ») to avoid the effects of oscillations such as sound propagation .Similarly, the tangential forces are defined by a coefficient depending on sliding tangential grain and similarly depreciated. In addition, the tangential forces are governed by the Coulomb criterion when the contact is purely sec. The addition of cohesion forces can generate the tangential greater.

This model has thus to simulate the behavior of ice floes (ice block wide but thin) subject to the currents. By extension, in this model we can introduce elements consist of several grains which initially glued break under the action of external loads (due to shear currents, Coriolis, wind, etc. …..) but also elements that weld is under compression (agglomeration of ice blocks mechanically or thermally). These results are analyzed and modified according to parameters obtained from experiments « size » made in the centers of ice of Canada (NRC Newfoundland). Preliminary results on two-dimensional piles subjected to biaxial compression have showed that the overall coefficient of friction of a pile was almost constant irrespective of the value of the coefficient of friction between grains. This is in contradiction with the generally accepted theories in soil mechanics (Caquot, Bishop Rowe, etc …) have proposed a linear dependence between these two parameters. Similarly, the overall friction coefficient is never zero even if the value of the coefficient of friction between the grain is the geometric dimensions playing the role of low shear stresses.

Figure: Force displacement during a biaxial test with different values of confinement (between 20 and 200kPa).

We have shown that the friction coefficient increased with the overall growth of the distribution of grain sizes. . Indeed, the appearance of areas more or less geometrically ordered causes the transmission of contact forces over large correlation lengths and thus distorts the overall coefficient of friction (decrease). The anisotropic distribution of contact forces due to biaxial compression is not compensated by a uniform distribution of directions of these forces.

Granular media real 2D and 3D show the characteristic of being difficult to analyze in terms of contacts between grains because it is often imprecise (contact time is not even a numerical point of view). These neighborhood problems can be solved by our programs Voronoi tessellation For many years we have developed algorithms to analyze the spatial geometry of stacks of grain in two dimensions. We can completely paved space by polygons whose sides are defined by the radical axis between two neighboring grains.This paving is unique and each polygon contains only one grain. All the topological and metric properties of these structures has been analyzed in two dimensions. We recently extended this study to three dimensions.

This achievement enables us to study the topological and metric properties of granular media « real » and the 3D foam. The achievement of homogeneous stacks in three dimensions application to take into account a number of considerations such as steric effects or exclusion, the periodic conditions and more generally the history of the development (individually or collectively). We studied beaches compact easily achievable according to the numerical methods used. Algorithms Poisson, RSA (Random Sequential Adsorption), Powell and de Visscher and Bölsterli place a sphere after another permanently; algorithm Jodrey and Tory or modified by Jullien allows densification by rearrangement almost static and collectively. SPH algorithms, PIC, MD and ED are dynamic and take into account interactions between the spheres collisionnelles. We can scan the whole range of porosity (0 to 0.7405).

As a first step, we analyzed the properties of disordered stacks of spheres monotailles. We confirmed the validation of three-dimensional relationships Aboav-Weaire on average faces of the cells. To find the maximum possible stacks of our disorderly, we analyzed the structures slightly disturbed crystalline stacks of classics (in Faces Centered Cubic and Hexagonal Compact). A weak disorder (about 0.1%) in the structure increased the average number of neighbors of 12 to 14 directly.

We analyzed the dependencies of the distributions of numbers of faces and neighbors in terms of compactness of the pile. To do this, we had to generate different types of stacks (different « stories »), such as RSA (C = 0 to 30%), « dense packing » type or Powell and Bölsterli Visscher (C = 60-64%), Dynamics Molecular & Event-Driven (C = 0 to 60%) or Jullien (C = 40 to 64%). We have seen that this could influence some geometric parameters such as average number of faces and its distribution.This has the advantage of power used as criteria of disorder or order from existing stacks (by measuring the difference in solving random type RSA).Our models have been extended to cases where the binary and polydisperse. Figure 13 shows a difference of behavior for compactness greater than or less than 0.5 and according to the original configuration of the pile. The spheres of stacks originating with a CFC or HC have suffered a « thermal agitation » Event-Driven Model. This gave them a disorderly character for small compact car similar to the results obtained from the stacks type Jullien. As against over 0.5, the order inherent in the original structure is partially preserved and thus affects the outcome.

Figure: Evolution of average number of faces of Voronoi cells based on compactness. The initial (before reduction of the radii and agitation) are stacks of face-Centered Cubic (FCC) Hexagonal Compact (HC) or disordered (Jullien).

We then sought to understand how moving a stack initially disordered when subjected to thermal agitation. To do this we analyzed the importance of an order parameter (Q 6) which takes into account the spatial orientation of the various links connecting the sphere to its neighbors as defined by the Voronoi cell, we have also highlighted the evolution of the phase transitions between disordered (C <49.5%), a transition zone (49.5% 54.5%). The separation of different contributions (polyhedrons from a disordered crystal of HC or CFCs) has allowed us to confirm the presence of the two structures during the crystallization of an initially disordered stacking. We highlighted that the CFC structure becomes dominant for high densities or crystallizations.

As the dominant structure is the CFC, we felt it appropriate to apply the basic principles of percolation to see how this structure evolved during the crystallization. So we followed the emergence and growth of CFC cell clusters at stirring of a disordered stacking of spheres initially monotailles compactness with more than 0545. As expected, there appeared a large number of mini cells CFCs that are organized to create a cluster of percolation through the entire structure.

(a)

(b)

Figure: Crystallisation of a disordered stacking of spheres monotailles (a) subject to a uniform thermal agitation after 10 binary collisions 9 (b). The colors represent the areas being quasi-crystalline environment.

It is very easy to study numerically the three-dimensional stacks of objects but it is much more difficult to carry out similar studies experimentally. Few practical techniques used to obtain information experimental studies on the surface, measurements tomography, magnetic resonance imaging, liquid iso-index without being destructive. The most conventional is to practice cutting, serial or random 3D structure in order to obtain relevant information. We therefore sought to verify that it was the reliability of certain interpretations of measurements made on a cut based on the true three-dimensional measurement. Thus we compared the results of conventional 2D Voronoi tessellations and 3D obtained from a 3D stack cut by a plane or from a cut made in the 3D tessellation of the original structure.

Figure: 2D Voronoi tessellations obtained from a 3D stack cut by a plane and a cut made in the 2D tessellation 3D initial structure.