The Physics of the Deformation of Densely Packed Granular Materials. M A C Koenders

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 lubrication. On unloading the contact may recover its elastic properties, though not necessarily with the same elastic constant as the loading curve. The process is illustrated in Fig. 1.3, where F_||/F_⊥ is shown as a function of the tangential contact displacement D_||. In this figure the spring constants for loading and unloading are taken as constants; when a nonlinearity is taken into account the straight loading and unloading lines become curved.

      Figure 1.3. Illustration of the Coulomb friction principle.

      It is clear that when behaviour like this is encountered an incremental formulation is necessary. The normal and tangential motion become coupled, so a general form for the incremental contact response relates the force increment to a displacement increment

      The elements of the matrix are the spring constants. Some properties of these are easily determined.

      In the elastic state there is an incremental potential, , from which the force increment is obtained by partial differentiation. It follows immediately that for this case the matrix must be symmetric: k_⊥|| = k_||⊥. If, in addition it is assumed that normal and tangential increments are uncoupled, it follows that the off- diagonal elements vanish. This does not preclude anisotropy with respect to the direction of the contact surface, so k_⊥⊥ need not be equal to k_||||. If they are assumed to be equal then the one-parameter model k = k_⊥⊥δ follows, which has the pleasant property that it is invariant under rotation, so the direction of the contact is irrelevant (the Kronecker delta δ is formally introduced in Section 2.2). For contacts that only interact through the normal force (frictionless) k_// = 0.

      In the frictional sliding state an additional force increment added to the state (F_⊥, F_//) should leave the ratio F_///F_⊥ constant at the value of μs. Taking F_⊥ and F_|| both positive, leads to the following

      In other words

      This constrains the elements of the matrix k by the additional relation

      which must hold for arbitrary displacements, hence

      So, for this case the matrix k takes the form