## Self Organized Elastic Structures

In the preceding sections we discussed the self-organization of interface films and micro topography evolution. Another type of self-organized microstructures is elastic slip waves. The mathematical formulation of quasi-static sliding of two elastic bodies (half-spaces) with a flat surface and a frictional interface is a classical contact mechanics problem. Interestingly, the stability of such sliding was not investigated until the 1990s, when, separately, Adams [2] and Martins [44] showed that the steady sliding of two elastic half-spaces is dynamically unstable, even at low sliding speeds. Steady-state sliding was shown to give rise to a dynamic instability [Adams-Matrins instabilities (AMI)] in the form of self-excited oscillations with exponentially growing amplitudes. These oscillations confine to a region near the sliding interface and can eventually lead to either partial loss of contact or to propagating regions of stick-slip motion (slip waves). The existence of AMI instabilities depends upon the elastic properties of the surfaces, however, it does not depend upon the friction coefficient, nor does it require a nonlinear contact model. The same effect was predicted theoretically by Nosonovsky and Adams [54] for the contact of rough periodic elastic surfaces.

It is well known that two types of elastic waves can propagate in an elastic medium: the shear and dilatational waves. In addition, surface elastic waves (Rayleigh waves) may exist, and their amplitude decreases exponentially with the distance from the surface. For two slightly dissimilar elastic materials in contact, the generalized Rayleigh waves (GRW) may exist at the interface zone. The instability mechanism described above is essentially one of GRW destabilization, that is, when friction is introduced, the amplitude of the GRW is not constant anymore, but exponentially grows with time.

The stability analysis involves the following scheme. First, a steady-state solution should be obtained. Second, a small arbitrary perturbation of the steady state solution is considered. Third, the small arbitrary perturbation is presented as a superposition of modes, which correspond to certain eigenvalues (frequencies). Fourth, the equations of the elasticity (Navier equations) with the boundary conditions are formulated for the modes, and solved for the eigenvalues. Positive real parts of the eigenvalues show that the solution is unstable.

For the GRW, the 2D displacement field at the interface is given by where k stands for the x- or y-component of the displacement field at the interface (y = 0), Ak is the complex amplitude, k/l is the wavenumber, l is the wavelength of the lowest wavenumber, and A is the complex frequency. It can be shown that for the frictionless case A = ± ik is purely imaginary and thus, for real Ak, the displacement is a propagating generalized Rayleigh wave, uk(x, 0, t) = Ak cos(kx ± kt). It can be shown also that if small friction with the coefficient i is present, then A = ±(ik + ai), where a is a real number, and thus one root of A always has a positive real component, leading to the instability [2, 61]. As a result, the amplitude of the interface waves grows with time. In a real system, of course, the growth is limited by the limits of applicability of the linear elasticity and linear vibration theory. This type of friction-induced vibration may be, at least partially, responsible for noise (such as car brake squeal) and other effects during friction, which are often undesirable [54].

Whereas the GRW occurs for slightly dissimilar (in the sense of their elastics properties) materials, for very dissimilar materials, waves would be radiated along the interfaces, providing a different mechanism of pumping the energy away from the interface [49]. These waves can form a rectangular train of slip pulses propagates, two bodies shift relative to each other in a "caterpillar" or "carpet-like" motion (Fig. 3.7). This microslip can lead to a significant reduction of the observed coefficient of friction, as the slip is initiated at a shear load much smaller than iW [10, 53].

The motion is observed as the reduction of the coefficient of friction (in comparison with the physical coefficient of friction i) to the apparent value of iapp = q/p, where q is applied shear force per unit area and p is the normal pressure. The slip pulses can be treated as "secondary structures'' self-organized at the interface, which result in the reduction of the observed coefficient of friction. Note that the analysis in this case remains linear and it just shows that the equations of elasticity with friction are consistent with the existence of such waves. The amplitude of the slip waves cannot be determined from this analysis, since they are dependent on the initial and boundary conditions. In order to investigate whether the slip waves will actually occur, it is important to ask the question whether it is energetically profitable for them to exist. To that end, the k k k

k uk(x, 0, t)= Re^e^V], vk(x, 0, t) = Re[Bkeikx='eAt]

Fig. 3.7 a Elastic waves radiated from the frictional interface. b Friction reduction due to propagating stick-slip zones [49]

Fig. 3.7 a Elastic waves radiated from the frictional interface. b Friction reduction due to propagating stick-slip zones [49]

energy balance should be calculated of the work of the friction force and the energy dissipated at the interface and radiated away from the interface. A stability criterion based on Eq. 3.8 can be used.

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