Partition of dry-friction forces among multiple contacts: extending de Gennes' 1997 asperity-recoil analysis to elastic solids, impact transients, two-dimensional anisotropic asperities, and sequential silo filling

When a solid is held by several simultaneous contacts, statics cannot fix how the dry-friction (tangential) forces distribute among them: equilibrium supplies fewer equations than unknowns, and Coulomb coefficients give only inequality bounds. In a 1997 note, P.-G. de Gennes addressed this indeterminacy microscopically, using the Caroli–Nozières independent-asperity model of dry friction as a hysteretic elastic response to derive how the tangential force partitions between two frictional regions of one sliding solid, and applied the result to a single grain settling against a silo wall. The note itself listed what the toy calculation leaves open: the elastic deformation of the object (which would make the two contacts' recoils differ), the impact transient of a grain meeting a wall (sharply peaked normal force, evolving Hertzian — elastic-sphere — contact area, active-asperity density not proportional to the transient load), the extension from one-dimensional to two-dimensional anisotropic asperity friction with its multistability, and the passage from one grain to the sequential filling of a silo with its cumulative stress distributions. No recorded follow-up continues that specific program, and adjacent literatures touch its physical targets by other means. The problem as stated — a quantitative, history-resolved theory of multi-contact friction partition covering the four named extensions — remains open.

Background

Amontons–Coulomb dry friction states that the tangential force a contact can sustain is bounded by $|F|\le\mu N$ ($\mu$ the friction coefficient, $N$ the normal force), but for a rigid body held by three or more contacts this bound does not select the actual forces: the force- and torque-balance equations are fewer than the unknown normal and tangential components, so the force distribution is statically indeterminate (the assembly is 'hyperstatic'). Granular materials make this indeterminacy unavoidable. In a silo — a tall container filled with grains — the vertical stress saturates with depth instead of growing hydrostatically (the Janssen effect), because wall friction redirects part of the grain weight to the walls; the classical Janssen analysis must however assume that wall friction is fully 'mobilized' (acting at its threshold $\mu N$), although nothing imposes this for a freshly poured packing.

P.-G. de Gennes' 1997 note attacked the indeterminacy at the asperity level in the simplest nontrivial setting. Its microscopic ingredient is the Caroli–Nozières model of dry friction as a hysteretic elastic response: each asperity (a microscopic protrusion of a contacting surface) is an independent elastic element pinned in a potential well of the opposing surface; when the slider is displaced, each asperity is dragged until it slips and repins, so the contact obeys a force–displacement curve with memory over a characteristic displacement scale, the capture range $\Delta$. For a solid whose sliding face has two distinct frictional regions A and B with densities $\nu_A,\nu_B$ of active asperities, de Gennes derived the total tangential force as a function of the common recoil $\delta$ (the tangential elastic displacement of the solid), $F_T(\delta)=F_{ST}-\nu_A\Sigma_A(\delta)-\nu_B\Sigma_B(\delta)$, where $F_{ST}$ is the static threshold and $\Sigma_{A,B}(\delta)$ are areas under the individual regions' force–displacement curves. This fixes the partition of tangential force between the contacts — including regimes in which one contact is pushed forward while the other is pulled back — and, applied to a circular grain rolling on a sloping granular bed (contact A) until it touches a vertical wall (contact B), yields closed forms for the final recoil and the common tangential force when the $\Sigma$ functions are linearized and strong damping is assumed. The framework rests on a one-dimensional quasistatic asperity picture, which Tanguy and Nozières (1996) generalized to two-dimensional sliding: they showed that anisotropy of the pinning potential changes the dissipative behavior of an asperity qualitatively and quantitatively, although a one-dimensional description survives over a large parameter range.

The note closes by stating what this formal exercise does not achieve, and those limitations define the present problem: the common-recoil assumption ignores the elastic deformation of the object itself (estimated to matter only at atomic scale for typical stiff grains, but not for larger or softer objects); the dynamics of a grain bouncing against a wall is reduced to a linearized treatment with constant normal force $N_B$, constant contact area, and an active-asperity density assumed proportional to $N_B$ during the impact; the whole analysis is one-dimensional; and the silo application treats a single grain, not the sequential filling in which grains roll and stop against already-fixed grains while others are deposited above. Later work established that the motivating concern is real — silo stresses depend on construction history, and wall friction is far from fully mobilized during filling (the Janssen 'overshoot') — but no later work executed the four extensions within (or as a replacement of) the source's asperity-dynamics framework.

Problem Statement

Replace the toy assumptions of de Gennes' 1997 analysis by a quantitative treatment of the partition of dry-friction (tangential) forces among the simultaneous contacts of a solid — within, or as a documented replacement of, the hysteretic asperity-dynamics framework — to the point of quantitative predictions for realistic granular materials. Concretely, four linked extensions of the 1997 analysis are requested:

(1) Elastic body. Allow the recoils of the distinct contact regions to differ, $\delta_A\ne\delta_B$, coupled through the elastic deformation of the object itself; determine when the common-recoil assumption breaks down (the source estimates the difference to be of atomic scale for typical stiff grains) and how the partition law $F_T(\delta)=F_{ST}-\nu_A\Sigma_A(\delta)-\nu_B\Sigma_B(\delta)$ — the total tangential force as a function of the common recoil $\delta$ (the tangential elastic displacement of the solid), with $F_{ST}$ the static threshold, $\nu_{A,B}$ the densities of active asperities at the two contact regions, and $\Sigma_{A,B}(\delta)$ the areas under the individual regions' asperity force–displacement curves — is corrected for larger or softer objects.

(2) Impact dynamics. Treat the arrival of a grain at a wall as a genuine transient collision: a normal force $N_B(t)$ that is not constant and may be sharply peaked, a contact area that evolves during the impact (Hertzian, i.e. elastic-sphere contact with area growing as a power of the load, or otherwise), and a density of active asperities that need not remain proportional to the instantaneous normal force under extreme transient loading; derive the resulting post-impact tangential-force state and partition.

(3) Two-dimensional friction. Extend the analysis to two-dimensional sliding in which the asperity pinning potential is anisotropic, so that the asperity landscape is multistable (several metastable pinned configurations coexist for the same load); determine how this modifies the multi-contact partition and the hysteresis loops under oscillating tangential loads. (Refs. [2] and [6] already treat the two-dimensional single-asperity dissipation and memory; what remains open is how two-dimensional multistability propagates to the multi-contact partition and its hysteresis loops.)

(4) Sequential filling. Extend the single-grain analysis to the cumulative filling of a silo — grains rolling and stopping against already-fixed grains, further grains deposited above — and derive the resulting wall-stress distributions and the degree of wall-friction mobilization as a function of filling history, without assuming that wall friction acts at its threshold.

The common objective uniting the four components is a microscopic, history-resolved account of the tangential-force partition that removes the static indeterminacy of multi-contact friction without threshold assumptions. The treatment must reduce to the 1997 results in the toy limits (rigid object with a common recoil, quasistatic loading, one-dimensional isotropic asperities, single grain, linearized Σ functions) and identify the domain of validity of each new ingredient. The four extensions (1)-(4) are jointly mandatory: an answer that treats only a subset of them does not resolve the task, and a documented replacement of the framework for a component discharges that component only if it yields the same microscopic, history-resolved partition and reduces to the 1997 results in the stated toy limits.

References

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