Читать книгу The Physics of the Deformation of Densely Packed Granular Materials - M A C Koenders - Страница 9

A static packed assembly of grains in contact confined by a compressive stress is equivalent to a network of forces. As it is static, force and moment equilibrium will hold. The question being addressed in this section is: how many forces in the network can be specified in such a way that force and moment equilibrium alone are sufficient to determine them, given the detailed geometry of the conformation?

A few conditions need to be laid down to come to a non-trivial answer. The first is that a regular packing is excluded from the analysis; an assembly in a regular packing satisfies certain symmetry rules which need to be imposed in addition to the equilibrium equations. Thus, a medium that consists of identical spherical particles is not accounted for at this stage. Rather, a polydisperse grain-size distribution is envisaged, making for a random packing. Alternatively, rough particles may make up the assembly. No isotropic condition imposition is necessary, though this is often (sometimes tacitly) assumed in the literature. Furthermore, it is assumed that the assembly is very large, so that the number of forces on the perimeter of the sample is small compared to the number of forces in the network. Basically, any condition that somehow constrains the forces in the network is excluded for the moment, implying that the equilibrium equations alone are sufficient to do the analysis. Specific constraining assumptions are discussed below.

In a random packing with N interacting particles in d dimensions there are Nd force equilibrium equations, as each particle that participates in the network is in equilibrium. The force moment equilibrium for each particle provides d(d – 1)/2 equations, so for N particles there are Nd + Nd(d – 1)/2 = Nd(d + 1)/2 equations. Each contact force will have d components and is shared by two particles. Equating the two gives the result that it is possible to calculate N(d + 1) forces, or an assembly coordinate number, that is the number of contacts per particle, of Nc,iso = d + 1 forces on average (the subscript ‘iso’ refers to the isostatic case). Note that this average pertains to particles that participate in the force network. It is well possible that a fair percentage of particles have no contact and these obviously do not contribute to the evaluation of the isostatic coordinate number.

When there are more force-carrying contacts, the equilibrium equations alone will not be able to permit the calculation of the forces. The system is then statically indeterminate. When there are fewer than d + 1 contact forces per particle there are more equations than unknowns and the system cannot be in static equilibrium. The isostatic state is therefore a very precarious, marginally stable state. The slightest disruption that results in the loss of even one contact will make the structure change until the number of force-carrying contacts is at least equal to the required number.

The number d + 1, which equals 3 in two dimensions and 4 in three dimensions, compared to any experimental result for a densely packed material shows that for practical purposes the statically indeterminate state is much more relevant. However, the analysis changes somewhat when constraints are imposed. So, the assumption of randomness is still maintained, but a constraint may follow from the fact that certain contacts slip. In that case the nature of friction must be considered.

Particles interact via their surfaces and these need not be smooth. As long as the surfaces are ‘infinitely sticky’ the force component that is tangential to the surface is free to take any value. In cases where slip is relevant, a Coulomb-type constraint reigns in which the magnitude of the tangential force remains equal to μs times the normal force. Contact forces must then be classified according to those that stick and those that slip. Let the ratio of slipping contacts in the assembly be given as a fraction fμ of all the contact forces, then the number of sticking forces populates a fraction 1 – fμ. The number of equations and unknowns now stack up as follows:

Nd force equilibrium equations

Nd(d – 1)/2 moment equilibrium equations

fμNNc,iso/2 slipping conditions

NdNc,iso/2 unknown contact force components

Equating the number of equations with the number of unknowns gives the result that the coordinate number per particle is

The implication is that as the fraction of slipping contacts increases, the number of contacts that need to be accommodated in the assembly will go up. When all contacts slip (fμ = 1) in both two and three dimensions the value of Nc,iso is 6.

A very special case occurs when there is no friction and the particles are perfectly spherical or discs. In that case all forces are normal to the contact surfaces and the moment equations become redundant: Nc,iso = 2d.

Again it is emphasised that these considerations only pertain to the particles that participate in the force network.

An experiment may be envisaged in which the particle assembly starts of as very dilute; it is then compacted (say, isotropically). There comes a point in this process when the particles begin to touch. When the number of particles that touch is sufficient for the medium to be on the edge of static equilibrium the assembly is said to ‘jam’. Compressing the assembly further will involve the compression of enduring contacts and therefore the development of a stress. The packing density at which the jamming transition takes place may be determined in numerical simulations. The outcome depends on assumptions on polydispersity (for spheres and discs), the details of the simulation method (gravity on or off, for example) and — indeed — the precise definition of the jamming density. Therefore, the concept of a ‘jamming transition density’ may only have approximate meaning.

Moreover, the analysis above shows that the number of contacts that can be supported in the isostatic state depends strongly on the fraction of the contacts that slip. In numerical simulations parameters can be tightly controlled to set the value of inter-particle friction (infinite and zero are popular choices), as well as the shape of the particles that participate in the simulation and the strain path that is employed. In any physical experiment with natural or manufactured particles, however, these parameters are not so easily controlled. The inter-particle friction coefficient, for example, may exhibit natural variation and therefore take a range of values; furthermore, particles tend to be rough and only approximately spherical.

A further question is whether an assembly of particles can be ‘partly isostatic’, that is that regions within the assembly can be distinguished for which the numbers of equilibrium equations equals the number of forces while there are also regions where there are fewer. Doubtlessly conditions can be found, involving factors such as closeness to the jamming condition and nature of the particle interaction (for example rough or smooth), where this is the case. In the references the relevant literature is highlighted. One aspect that comes to the fore in these papers is the need to distinguish fluctuations in the local geometry. For dense assemblies, where the intention is to obtain a stress-strain relation, the most convenient approach is to introduce an inter-particle interaction and to develop the theory further taking account of the fluctuations in that context.

An interesting feature of the present discussion is an historical perspective. The conditions for isostaticity were originally laid out by [Maxwell, 1864]. In fact, Maxwell’s text employs identical arguments as the one at the beginning of this section. A fully elaborated theory of static indeterminacy was produced by Mohr in 1874, see [Mohr, 1906]. Not until a century later did these concepts find their way into the literature of granular mechanics. In the early 2000s a more rounded view of the subject became available and the notion that sliding friction may influence the theory. A great help has been the development of simulation methods so that the jamming transition may be studied ‘experimentally’. Jamming under non-isotropic conditions has been included more recently.

An extensive overview of the jamming transition is described by [Liu and Nagel, 2010]. Stresses in an isostatic assembly are derived by [Blumenfeld, 2007] and in this paper some other problems regarding the concept of isostaticity are also highlighted. Non-isotropic compression and jamming (with physical experiments) is discussed by [Bi et al., 2011]. An exhaustive list of publications relevant to this subject is somewhat outside the scope of this text, however most relevant ones are in the references mentioned.