Читать книгу The Physics of the Deformation of Densely Packed Granular Materials - M A C Koenders - Страница 10

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 (QS) is constructed. 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.