Granular Matter Theory Research Articles

Constitutive modeling for undrained shear behavior of gassy sand considering energy dissipation at the mesoscopic level

The gas phase in the gassy sand is in the form of occluded bubbles which are surrounded by water within the void space. Due to the high compressibility of the gas bubbles, it has a significant impact on the shear strength, volume change and other engineering properties of gassy sand. Based on the thermodynamic theory of granular matter, combined with the dilatancy equation considering the state parameter and the relationship between pore gas pressure and pore gas volume change, a thermodynamic constitutive model is developed to describe the undrained shear behavior of gassy sand. The model considers the dissipation at the mesoscopic level, and the concept of granular temperature is introduced to describe the irreversible deformation of gassy sand. The relationship between the dissipation mechanism of gassy sand and the macroscopic mechanical behavior is established through the migration coefficients and energy density functions. The ability of the model to describe the undrained shear behavior of gassy sand is verified by the numerical simulation of the undrained triaxial shear tests of the remolded Ottawa sand with different degrees of saturation.

Existence and uniqueness of solutions for a boundary value problem arising from granular matter theory

We consider a system of PDEs of Monge–Kantorovich type that, in the isotropic case, describes the stationary configurations of two-layers models in granular matter theory with a general source and a general boundary data. We propose a weak formulation which is consistent with the physical model and permits us to prove existence and uniqueness results.

Granular statistical mechanics – a personal perspective

The science of granular matter has expanded from an activity for specialised engineering applications to a fundamental field in its own right. This has been accompanied by an explosion of research and literature, which cannot be reviewed in one paper. A key to progress in this field is the formulation of a statistical mechanical formalism that could help develop equations of state and constitutive relations. This paper aims at reviewing some milestones in this direction. An essential basic step toward the development of any static and quasi-static theory of granular matter is a systematic and useful method to quantify the grain-scale structure and we start with a review of such a method. We then review and discuss the ongoing attempt to construct a statistical mechanical theory of granular systems. Along the way, we will clarify a number of misconceptions in the field, as well as highlight several outstanding problems.

Isaac Goldhirsch: a pioneer of granular matter theory

About 20years ago, Isaac Goldhirsch was one of the first to recognize that granular matter, behaving neither as a normal fluid nor as a normal solid, was not only an exotic terra incognita for basic science, but is paradigmatic for a whole class of complex systems with irreversible interactions. This field became his passion at a time when he was already a very successful, internationally renowned theoretical physicist. Certainly this helped to establish granular matter theory as a lively new branch of science. The very existence of a prominent new scientific journal “Granular Matter”, entirely devoted to this field speaks for itself. The present special issue of “Granular Matter” is dedicated to the memory of Isaac Goldhirsch (1949–2010). It is a tribute of colleagueswhodo research in this field, in return for the profound inspiration and deep insight they have received from Isaac. It also demonstrates Isaac’s lasting impact. Isaac Goldhirsch contributed immensely to the transformation of the field of granular matter from a largely phenomenology driven description to a mathematically well founded theory. Cornerstones that will remain connected to his name are the investigation of the clustering instability in granular gases, the elaboration of kinetic theory, and the development of the foundations of granular elasticity, just to name themostprominent ones. Isaac Goldhirsch had the rare capability of integrating seemingly disparate scientific approaches: The point of view of an engineer was as familiar to him as that of a mathematician. Isaac combined atoms, grains, complex mesoscopic

A nonhomogeneous boundary value problem in mass transfer theory

We prove a uniqueness result of solutions for a system of PDEs of Monge–Kantorovich type arising in problems of mass transfer theory. The results are obtained under very mild regularity assumptions both on the reference set $${\Omega\subset \mathbb{R}^n}$$ , and on the (possibly asymmetric) norm defined in Ω. In the special case when Ω is endowed with the Euclidean metric, our results provide a complete description of the stationary solutions to the tray table problem in granular matter theory.

An existence result for the sandpile problem on flat tables with walls

We derive an existence result for solutions of a differential system whichcharacterizes the equilibria of a particular model in granular matter theory, theso-called partially open table problem for growing sandpiles.Such result generalizesa recent theorem of [6] established for the totally open table problem.Here,due to the presence of walls at the boundary, the surface flow density at theequilibrium may result no more continuous nor bounded, and its explicit mathematicalcharacterization is obtained by domain decomposition techniques.At the same time weshow how these solutions can be numerically computed as stationary solutions of adynamical two-layer model for growing sandpiles and we present the results of somesimulations.

Hölder regularity of the normal distance with an application to a PDE model for growing sandpiles

Given a bounded domain Ω \Omega in R 2 \mathbb {R}^2 with smooth boundary, the cut locus Σ ¯ \overline \Sigma is the closure of the set of nondifferentiability points of the distance d d from the boundary of Ω \Omega . The normal distance to the cut locus, τ ( x ) \tau (x) , is the map which measures the length of the line segment joining x x to the cut locus along the normal direction D d ( x ) Dd(x) , whenever x ∉ Σ ¯ x\notin \overline \Sigma . Recent results show that this map, restricted to boundary points, is Lipschitz continuous, as long as the boundary of Ω \Omega is of class C 2 , 1 C^{2,1} . Our main result is the global Hölder regularity of τ \tau in the case of a domain Ω \Omega with analytic boundary. We will also show that the regularity obtained is optimal, as soon as the set of the so-called regular conjugate points is nonempty. In all the other cases, Lipschitz continuity can be extended to the whole domain Ω \Omega . The above regularity result for τ \tau is also applied to derive the Hölder continuity of the solution of a system of partial differential equations that arises in granular matter theory and optimal mass transfer.

On a system of partial differential equations of Monge–Kantorovich type

We consider a system of PDEs of Monge–Kantorovich type arising from models in granular matter theory and in electrodynamics of hard superconductors. The existence of a solution of such system (in a regular open domain Ω ⊂ R n ), whose construction is based on an asymmetric Minkowski distance from the boundary of Ω, was already established in [G. Crasta, A. Malusa, The distance function from the boundary in a Minkowski space, Trans. Amer. Math. Soc., submitted for publication]. In this paper we prove that this solution is essentially unique. A fundamental tool in our analysis is a new regularity result for an elliptic nonlinear equation in divergence form, which is of some interest by itself.

Representation of equilibrium solutions to the table problem of growing sandpiles

In the dynamical theory of granular matter the so-called table problem consists instudying the evolution of a heap of matter poured continuously onto a bounded domain \Omega\subset \R^2 . The mathematical description of the table problem, at an equilibrium configuration, can be reduced to a boundary value problem for a system of partial differential equations. The analysis of such a system, also connected with other mathematical models such as the Monge-Kantorovich problem, is the object of this paper. Our main result is an integral representation formula for the solution, in terms of the boundary curvature and of the normal distance to the cut locus of \Omega .

Overshoot effect in the Janssen granular column: A crucial test for granular mechanics

We present experimental results on the mechanical behavior of static granular assemblies confined in a vertical column. Our measurements confirm the universal Janssen scaling for the stress saturation curve. We show consequently, in the context of isotropic elasticity, a relation between the Poisson ratio and the granular packing fraction. Moreover, using a systematic study of the overshoot effect created by a top mass equal to the saturation mass, we show behaviors that are reproduced qualitatively by isotropic elastic materials but of a spectacular amplitude in the case of a granular assembly. These experimental results are strong tests for any theory of granular matter.

Ping-pong ball avalanche experiments

AbstractPing-pong ball avalanche experiments have been carried out for the last 3 years at the Miyanomori ski jump in Sapporo, Japan, to study three-dimensional granular flows. Up to 550 000 balls were released near the top of the landing slope. The balls then flowed past video cameras positioned close to the flow, which measured individual ball velocities in three dimensions, and air-pressure tubes at different heights. The flows developed a complicated three-dimensional structure with a distinct head and tail, lobes and “eyes”. “Eyes” have been observed in laboratory granular flow experiments, and the other features are similar not only to snow avalanches” but also to other large-scale geophysical flows. The velocities attained showed a remarkable increase with the number of released balls. A power law for this relation is derived by similarity arguments. The air-pressure data are used to deduce the structure of the airflow around the avalanche and, in conjunction with the kinetic theory of granular matter, to estimate the balance of forces in the avalanche head.