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.
1.3The statically indeterminate case and computer simulations
The next problem must be how the contact forces are going to be solved in the statically indeterminate state. In this case there are more force variables than force and moment balance equations (and more contacts per particle than Nc,iso). A solution is possible when a constitutive equation is introduced. Such an equation gives the relation between force and displacement difference between particles (particles may also rotate and this too needs to be incorporated in the constitutive equations). It necessarily implies that the particles are deformable. This may be counterintuitive as particles are frequently thought of as rigid (sand grains, for example, would appear to be very stiff). More precisely, a rigid limit can be thought of when the stiffness of the particles is very much greater than the pressure associated with the stress in the assembly. However, allowing for small indentations during particle contact resolves the issue of static indeterminacy. Here is a list of unknowns and equations for all the particles that participate in the force network.
Unknowns
Nd particle displacements
Nd(d – 1)/2 particle rotations
NdNc/2 contact forces
Equations
Nd force equilibrium equations
Nd(d – 1)/2 moment equilibrium equations
NdNc/2 contact force — relative particle displacement and rotation relations (the contact laws)
The number of unknowns (that is, the displacements and rotations) is equal to the number of equations and (assuming no mathematical pathologies) a solution may be constructed. The reader may now be surprised that there is no mention of a torque constitutive equation. There is an underlying assumption here (which is similar to the rigidity assumption) that the contacts may be thought of as point contacts. A point contact cannot transmit a torque. So, unless the particles are very deformable — and the contact area may acquire an appreciable value — this aspect may be neglected. A problem would arise when the grains in the force network are so irregularly shaped that two neighbouring particles may share more than one contact. In that case, of course, a torque may be transmitted. In principle the theory can be easily amended to account for a complication like that by introducing a particle contact(s) torque in addition to the contact forces and an extra set of constitutive equations relating particle rotation to the transmitted torque. This is not followed up here, where it is assumed that the particles are hard (though slightly deformable) and share at most one contact.
The set of equations, as outlined above, can be solved using computer simulations and in that way displacements and rotations of the particles in an assembly may be determined under suitably chosen boundary conditions.
In the literature it is only very rarely that a procedure is encountered in which a quasi-static solution
