When F|| is negative, μs is replaced by –μs; otherwise the relations remain the same.
Unloading from the frictional state is detected by checking what the response would have been for an elastic increment (this could in principle be brought about by an increase in the normal force). If this decreases the magnitude of the tangential to normal force ratio, the next increment should be evaluated using the (unloading) elastic law. Therefore, the frictional interaction is predictive, but must always be followed by a verification.
Friction in two dimensions is covered in the literature. [Ruina, 1980] and [Ruina, 1983] discusses the sliding state once the initial friction criterion is passed. On continued motion the value of μs falls by a small amount — the friction is said to change from a static value to a kinetic value. In addition, an extra stress that is proportional to the speed of continued tangential motion needs to be introduced (this effect is sometimes known as the Ruina–Dieterich law: [Dieterich, 1979, 1981]). It should be emphasised that the measurements that underlie this law are done on blocks of rock material. In these experiments there are always many contacts at the same time, while for the present application two particles share one contact, which is approximately a point-contact, that is, a very small contact area between two convex surfaces. Direct application of the Ruina-Dieterich law may therefore not be appropriate.
While the frictional effect has been measured extensively, the actual mechanism of the contact mechanics that lead to friction is relatively unexplored. [Villagio, 1979] has put forward some interesting ideas, though they have so far not been widely followed up.
1.5.1Friction in three dimensions
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
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
1.6Contact laws in terms of material parameters
A question that is particularly of interest to the simulation community concerns the matter whether the spring constant can be related to the material properties of the particles. The prime candidate for such a theory is the Hertzian contact theory, which deals with two elastic bodies that are being compressed together — [Hertz, 1882]. The details are in [Landau and Lifschitz, 1976] and more extensively in the [Johnson, 1985] book on contact mechanics.
For two spheres pressed together by a force F⊥ the distance between the centres of two spheres with radii R and R′ is reduced by an amount D⊥
where the parameter Q contains the elastic constants (Young’s moduli E, E′ and Poisson’s ratios v, v′) of the materials of which the solid bodies are made:
An obvious aspect of this force-indentation formula is that the force-displacement relationship is non-linear. An incremental relationship is easily obtained
A relationship between the proposed spring constant and the material parameters of the particles could then be proposed as some assembly average value of k(F). For simulation purposes that would be very unsatisfactory and the vast majority of simulists code for the original relationship between D⊥ and F⊥. For analytical modelling, however, where interactive properties frequently appear as sums over nearby particles, an averaging approach may be convenient for theoretical purposes.
In two dimensions, relating a spring constant to material properties can be ascertained by looking at the compression of two cylinders along their axes. The Hertzian relationship is, see [Puttock and Thwaite, 1969] and [Williams and Dwyer-Joyce, 2001]
where ℓ is the length of the axes of the cylinders and R, R′ are the cylinder radii. Note that the expression essentially depends on the force per unit length.
This expression is also non-linear, but not easily employed, because inverting it (to give F⊥ as a function of D⊥) leads to extra numerical work.
Figure 1.5. Contact stiffness as a function of the applied compressive