Nonetheless, it is possible to do this. [Koenders and Stefanovska, 1993] show an approximation method, based on a least-squares approach of the force and moment equilibrium equations for an elasto-frictional material in two dimensions. The result for a biaxial cell test are very similar to the ones measured by, for example, [Konishi, 1978]. The latter is an experiment on photo-elastic discs – see Fig. 1.1. The statistics of the micro-mechanical variables are faithfully reproduced. These include the mean contact distribution and the distribution of the slipping contacts as the test progresses. Macroscopic features, such as the stress ratio reaching a maximum and the occurrence of dilatancy are also found.
Despite the relative success of this method, it has not been pursued by many other researchers, who have preferred dynamic methods.
These are obviously attractive if, in addition to slow changes to an assembly in the high contact number régime, faster changes and granular flow also need to be studied. To accommodate the dynamics, a particle mass and moment of inertia terms need to be introduced to the equilibrium equations, so that a full Newtonian set of equations is processed. To solve Newton’s equations simultaneously with evolving contact properties, such as detecting new contacts and deleting old ones, for all particles in a large assembly (say, N > 1000) requires a massive computer effort. In a molecular dynamics method, called the Discrete Element Method (DEM), a sequential approach is taken, using a small time step and moving and rotating the particles in the assembly one at a time and after that updating the contact properties. If the time step is small enough, this would be equivalent to a simultaneous solution. The method was first introduced by [Cundall and Strack, 1979] — a two-dimensional version of the DEM. Since its inception it has been developed further and has been expanded to three dimensions, more complicated contact laws and extensions to include more general boundary conditions, including periodic ones. More complicated particle shapes with rough boundaries have been included in an attempt to model realistic, natural conditions. The method has had a tremendous influence on the development of the subject, not least because proposed theoretical models in which micro-mechanical assertions are put forward could be tested against computer simulations.
Free software and many informative documents are available, so researchers can run their own simulations [Yade, 2019].
It is fair to say that reporting on the results of the method has not always been entirely complete. It is also the case that in some instances the reporters have been arrogant in asserting that the simulation results are superior to physical experiments, though it is true that in the computer certain boundary conditions can be simulated that are very difficult to realise with a laboratory apparatus, see for example [Thornton, 2000]. Consistent examples of papers on simulations that use the method (and discuss some of the difficulties with it) are by [Thornton and Antony, 1998] and a very informative paper by [Thornton and Sun, 1993]. Further useful papers, showing the potential and increased subtlety of the method, are by [Ferellec and McDowell, 2010], [Macaro and Utili, 2012] and [McDowell and Li, 2016]. This little list is illustrative only and does by no means justice to the extent to which papers on this subject have been published. There must be many thousands.
A computer method that lies somewhere between QS and the DEM is the Contact Dynamics (CD) method. The background to this is the following. The time step in a fully dynamic implementation of the equations of motion needs to be so small that it is adequate to follow the changes in the contact laws. The latter allow for a small indentation in what are essentially rigid particles. The problem with the contact laws is that they are highly non-linear and therefore a large number of time steps is required to model, say, a collision between two particles. In the CD method the accelerations are not calculated, but the particles are subject to a velocity field. The latter can change abruptly, both in direction and magnitude. This is so-called non-smooth motion. For the method to work, the motion during the encounter between two particles is integrated, taking into account the non-linear contact laws. The inputs to any collision encounter are the velocities of the two participating particles, while the output consists of the velocities after the encounter has taken place. The actual integration cannot be done exactly, but certain estimates have to be made. These have been elevated to a high art by the CD community and it is generally assumed that the method is no less accurate than the DEM method. More specifically, the propagating error introduced by the exceedingly small time step in a fully dynamic program may well be of the same order of magnitude as the error incurred in the approximations in the integration method in CD. Any computational method is approximate in some sense. However, CD is much faster than DEM, as rather larger time steps can responsibly be taken. Relevant references for this method are by the inventors of the method [Jean and Moreau, 1992] and [Jean, 1999], as well as an informative introductory paper by [Radjai, 2008]. Again, as with the treatment of the DEM before, there are many more papers that could be quoted, especially as the method has gained in popularity in recent times.
1.4Contact laws
When drawing up a suitable constitutive law for contact relating contact displacement and contact force the first thought should be ‘what is it meant to achieve?’ In molecular studies and studies of small particles in liquids very sophisticated interactive relations have been put forward that account for surface potential effects and quantum mechanical interactions. These relations are highly non-linear and allow for both repelling and attractive phenomena. However, in dealing with larger particles a simple law that just ensures that the particles only overlap by a very small amount would appear to be sufficient. The difficulty with increasing sophistication is that it requires more and more parameters, which may be difficult to measure. Also, the benefit of more complex laws is marginal. The need for a contact law arises from the existence of a statically indeterminate state. The first goal is to fix this problem by simple means and get some insight in the properties of such systems. Added complexity can be inserted later as a refinement.
Any two surfaces that touch one another could in a first approximation be assumed to repel one another as springs. This gives a relation for the normal force between the surfaces that is characterised by merely the spring constant k. The latter will generally be a function of the contact force itself. The non-linearity that is associated with that gives rise to the need to introduce incremental contact laws (the need for incremental laws will be discussed in more detail below). So, if the normal displacement D⊥ is related to the normal force F⊥ via a spring constant
Then the incremental law reads
The function k(F⊥) may contain a number of features. In addition to the non-linearity the incremental spring constant may be either assumed to be entirely elastic or reflect certain plastic effects (that is, have different values for loading and unloading).
1.5The frictional interaction
One effect that is without doubt very important in the constitutive contact law is the effect of friction and to introduce that the normal force alone is insufficient; a tangential force-displacement rule must be added to the description.
The friction effect is obviously plastic. When the force ratio (that is the magnitude of the tangential force to the normal force) reaches a certain value μs, persistent further motion will not change it; a constraint has become active that keeps the force ratio constant. This was established by [Coulomb, 1785] (based on measurements by [Amontons, 1699], see [Heyman, 1972] for the history of the subject and many more references). The concern here is essentially with dry friction. [Bowden and Tabor, 1956] treat the subject from an engineering standpoint and also extend their treatment to include effects of
