Arches and contact forces in a granular pile

Assemblies of granular particles mechanically stable under their own weight contain arches. These are structural units identified as sets of mutually stable grains. It is generally assumed that these arches shield the weight above them and should bear most of the stress in the system. We test such hypothesis by studying the stress born by in-arch and out-of-arch grains. We show that, indeed, particles in arches withstand larger stresses. In particular, the isotropic stress tends to be larger for in-arch-grains whereas the anisotropic component is marginally distinguishable between the two types of particles. The contact force distributions demonstrate that an exponential tail (compatible with the maximization of entropy under no extra constraints) is followed only by the out-of-arch contacts. In-arch contacts seem to be compatible with a Gaussian distribution consistent with a recently introduced approach that takes into account constraints imposed by the local force balance on grains.Comment: 7 pages, 7 figures, major revisio

Tail of the contact force distribution in static granular materials (AGGREGATION CH 2)

We numerically study the distribution P(f) of contact forces in frictionless bead packs, by averaging over the ensemble of all possible force network configurations. We resort to umbrella sampling to resolve the asymptotic decay of P(f) for large f, and determine P(f) down to values of order 10-45 for ordered and disordered systems in two (2D) and three dimensions (3D). Our findings unambiguously show that, in the ensemble approach, the force distributions decay much faster than exponentially: P(f)~ exp(-cfα), with α = 2.0 for 2D systems, and α = 1.7 for 3D systems

Entropy Maximization in the Force Network Ensemble for Granular Solids (THESIS VERSION)

A long-standing issue in the area of granular media is the tail of the force distribution, in particular whether this is exponential, Gaussian, or even some other form. We demonstrate that conservation of the total area of a reciprocal tiling, a direct consequence of local force balance, is crucial for predicting the local stress and force distribution. Maximizing entropy while conserving the tiling area and total pressure leads to a distribution of local pressures with a generically Gaussian tail that is in excellent agreement with numerics, both with and without friction and for two different contact networks (triangular lattice and square lattice)

Force network ensemble for granular solids (AGGREGATION DATA)

Granular materials such as sand have both liquid-like and solid-like properties similar to both liquids and solids. Dry sand in an hour-glass can flow just like water, while sand in a sand castle closely resembles a solid. Because of these interesting properties granular matter has received much attention from numerous physicists. Part of the research on these materials focuses on the statistics of contact forces between individual particles and how these statistics can be used to understand and predict material properties. Contact forces in granular materials are organized in so-called forces networks. A common quantity to characterize these force variations is the probability distribution of the contact force, P(f). A long-standing issue is the asymptotic behavior of this distribution. In particular, one discusses whether the tail of P(f) is exponential, Gaussian, or has a different form. Furthermore, its relation with material and system properties is unclear. In view of the technical difficulty to measure contact forces, especially in the bulk of the material, we used computer simulations in the so-called force network ensemble of Snoeijer et al. (Phys. Rev. Lett., 2004, 92, 054302). Unfortunately, the estimation of P(f) for large contact forces f is inefficient. The reason is that by far the largest fraction of generated force networks contains only small forces. For unambiguous conclusions on the asymptotic behaviour of P(f), extremely long (of the order of years or even centuries) computer calculations are needed. To obtain better statistics for large contact forces, we developed an umbrella sampling method for the force network ensemble. In Chapter 1, an overview is given of the different methods to study the statistics of force networks. We also explained the umbrella sampling method that we developed to obtain excellent statistics for large forces. In Chapter 2, we applied this method to study the tail of the force distribution P(f) for different systems. The average number of contacts of a particle and the packing configuration are shown not to be important for the asymptotic behavior of P(f). Only the dimensionality of the system has a significant influence: P(f)~exp[-c f a] with a?2.0 for two-dimensional systems, a?1.7 for three-dimensional systems and a?1.4 for four-dimensional systems. In Chapter 3, a possible explanation is presented for the Gaussian decay of large contact forces in two-dimensional systems. It was found that mechanical balance on each particle is essential for the tail of the contact force distribution, which throws serious doubts on the statement that exponential statistics are a generic property of static granular materials. In Chapter 4, we focused on several details of the contact forces and their distribution. We also investigated how well the force network ensemble describes systems with “real” interactions

Tail of the contact force distribution in static granular materials (THESIS VERSION)

We numerically study the distribution P(f) of contact forces in frictionless bead packs, by averaging over the ensemble of all possible force network configurations. We resort to umbrella sampling to resolve the asymptotic decay of P(f) for large f, and determine P(f) down to values of order 10-45 for ordered and disordered systems in two (2D) and three dimensions (3D). Our findings unambiguously show that, in the ensemble approach, the force distributions decay much faster than exponentially: P(f)~ exp(-cfα), with α = 2.0 for 2D systems, and α = 1.7 for 3D systems

Statistics of large contact forces in granular matter

Granular materials such as sand have both liquid-like and solid-like properties similar to both liquids and solids. Dry sand in an hour-glass can flow just like water, while sand in a sand castle closely resembles a solid. Because of these interesting properties granular matter has received much attention from numerous physicists. Part of the research on these materials focuses on the statistics of contact forces between individual particles and how these statistics can be used to understand and predict material properties. Contact forces in granular materials are organized in so-called forces networks. A common quantity to characterize these force variations is the probability distribution of the contact force, P(f). A long-standing issue is the asymptotic behavior of this distribution. In particular, one discusses whether the tail of P(f) is exponential, Gaussian, or has a different form. Furthermore, its relation with material and system properties is unclear. In view of the technical difficulty to measure contact forces, especially in the bulk of the material, we used computer simulations in the so-called force network ensemble of Snoeijer et al. (Phys. Rev. Lett., 2004, 92, 054302). Unfortunately, the estimation of P(f) for large contact forces f is inefficient. The reason is that by far the largest fraction of generated force networks contains only small forces. For unambiguous conclusions on the asymptotic behaviour of P(f), extremely long (of the order of years or even centuries) computer calculations are needed. To obtain better statistics for large contact forces, we developed an umbrella sampling method for the force network ensemble. In Chapter 1, an overview is given of the different methods to study the statistics of force networks. We also explained the umbrella sampling method that we developed to obtain excellent statistics for large forces. In Chapter 2, we applied this method to study the tail of the force distribution P(f) for different systems. The average number of contacts of a particle and the packing configuration are shown not to be important for the asymptotic behavior of P(f). Only the dimensionality of the system has a significant influence: P(f)~exp[-c f a] with a?2.0 for two-dimensional systems, a?1.7 for three-dimensional systems and a?1.4 for four-dimensional systems. In Chapter 3, a possible explanation is presented for the Gaussian decay of large contact forces in two-dimensional systems. It was found that mechanical balance on each particle is essential for the tail of the contact force distribution, which throws serious doubts on the statement that exponential statistics are a generic property of static granular materials. In Chapter 4, we focused on several details of the contact forces and their distribution. We also investigated how well the force network ensemble describes systems with “real” interactions

Entropy Maximization in the Force Network Ensemble for Granular Solids (AGGREGATION CH 3)

A long-standing issue in the area of granular media is the tail of the force distribution, in particular whether this is exponential, Gaussian, or even some other form. We demonstrate that conservation of the total area of a reciprocal tiling, a direct consequence of local force balance, is crucial for predicting the local stress and force distribution. Maximizing entropy while conserving the tiling area and total pressure leads to a distribution of local pressures with a generically Gaussian tail that is in excellent agreement with numerics, both with and without friction and for two different contact networks (triangular lattice and square lattice)

Krachtenbalans op zandkorrels

Granulaire materialen zoals zand en graankorrels hebben eigenschappen overeenkomstig met zowel vloeisto en als vaste sto en. Zo kan droog zand in een zandloper net als water stromen, terwijl als we op het strand lopen zand juist meer op een vas- te stof lijkt. Door deze bijzondere eigenschappen staan granulaire materialen in de belangstelling bij natuurkundigen. Een deel van het onderzoek naar deze materialen richt zich op zogenoemde krachtennetwerken

Krachtenbalans op zandkorrels

Granulaire materialen zoals zand en graankorrels hebben eigenschappen overeenkomstig met zowel vloeisto en als vaste sto en. Zo kan droog zand in een zandloper net als water stromen, terwijl als we op het strand lopen zand juist meer op een vas- te stof lijkt. Door deze bijzondere eigenschappen staan granulaire materialen in de belangstelling bij natuurkundigen. Een deel van het onderzoek naar deze materialen richt zich op zogenoemde krachtennetwerken

Numerical study of the force network ensemble

The force network ensemble of Snoeijer et al. (Force network ensemble: a new approach to static granular matter, Phys. Rev. Lett. 92 (2004), 054302) is a convenient model to study networks of contact forces that are typically present in granular matter. Recently, we have shown that it is possible to extremely accurately determine the probability distribution of contact forces in the framework of this ensemble (van Eerd et al., Tail of the contact force distribution in static granular materials, Phys. Rev. E 75 (2007), 060302(R); Tighe et al., Entropy maximisation in the force network ensemble for granular solids, Phys. Rev. Lett. 100 (2008), 238001). In this work, we review several important details of these computations. In particular, we study in detail the angle-resolved contact force distribution, finite-size effects, the maximum allowed shear stress and the effect of walls. In addition, we investigate how well the force network ensemble resembles systems with 'real' interactions