Читать книгу The Physics of the Deformation of Densely Packed Granular Materials - M A C Koenders - Страница 13

The exposition given above is idealised in that the motion and force parameters all operate in a plane. To some extent that is a view justified by the fact that the frictional interaction takes place on the surface of two bodies in contact. The unit normal of the surface is n and if the force across the surface is F, the normal component is the inner product F_⊥ = F•n. The tangential force is then F_|| = F – (F•n)n. The sliding friction criterion may now be expressed as . This relation represents a cone, as illustrated in Fig. 1.4.

Figure 1.4. Friction cone. The opening angle is 2 tan ^–1 μs.

The procedure for obtaining the incremental interaction in the sliding state is the same as before. Basically, the force vector must be constrained to move on the surface of the cone.

The most convenient way of making progress is now to choose a coordinate frame that is aligned with the forces. One unit vector — n — is already in place; of the other two one is chosen to be aligned with F_|| and the other one normal to that (as well as normal to n). The former is called n_|| and the latter n₀ (this vector is sometimes called the binormal). In this frame the force and the force increment have components

The sliding friction criterion becomes

Expanding up to first order in the increments gives

This is exactly the same relation as for the two-dimensional case and the implications for the incremental force-displacement relation are also obtained in a similar fashion. The resulting interaction is

The question now is whether there is a coupling between the third direction and other two directions. If the third direction operates entirely independently then all coefficients with a vanish other than the diagonal term , which is probably some fraction of the normal diagonal coefficient k_⊥⊥. If, however, the tangential force influences the behaviour in the third direction then a number of extra parameters need to be taken into account. For point contacts these parameters are very difficult to measure.