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,
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
In many instances the increase in the tangential force increment for purely tangential motion is negligible, implying that k|||| = 0. The frictional state is then entirely described by two parameters, k⊥⊥ and μs.
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
