PHYSICAL REVIEW E VOLUME 51, NUMBER 5 MAY 1995 Nonlinear analysis of the stick-slip bifurcation in the creep-controlled regime of dry friction We perform an experimental study of the amplitude and frequency shift of the stick-slip (SS) oscilla- tions of a paper-on-paper block and spring system close to the SS — steady-sliding Hopf bifurcation. The confrontation of the experimental results with a weakly nonlinear analytical analysis of the dynamics and with direct numerical calculations yields very satisfactory agreement with the model of creep- controlled dry friction proposed by Heslot et al. [Phys. Rev. E 49, 4973 (1994)]. PACS number(s): 05. 45. + b, 46.30. pa, 62.20. Hg, 91. 30. Px (3) I. INTRODUCTION This leads to defining a characteristic "memory" length [5] Do — found to lie in the micrometer range — such that In a recent article [1] we reported the results of an ex- tensive study of the dry friction dynamics of a paper-on- paper block and spring system (Fig. 1), performed over a wide range of relevant control parameters, namely, the imposed velocity V at which the "free" end of the pulling spring is being driven, the spring stiffness K, and the slid- er mass M. One of the main results emerging from this study is the existence, for V less than a few tens of pms, of low velocity regime in which the friction dy- namics of paper exhibits the same characteristic features as already observed on metal [2] and rock [3] systems. This strongly suggests that, in this regime, the nature of the dynamics is material independent and thus suscepti- ble to a generic description. Its main features can be summarized as follows. (i) It is in this small-V range that the dynamic friction coefficient pd, measured in the steady-sliding regime, ex- hibits velocity weakening. It is well fitted by the func- tional form (4) p (r) =pd(D /or) (iii) The sliding motion changes from oscillatory (the so-called stick-slip oscillations) at low spring stiffnesses to steady sliding at large K. The bifurcation between these two regimes shows the qualitative features of a direct (su- percritical) Hopf bifurcation controlled, at constant V, by the ratio E/M of the stiFness K and the slider mass M (Fig. 2). (iv) Once the system is rapidly loaded to a level p;„&p„pulling then being stopped, the shear stress is ob- served to relax via a slow nonexponential creeplike motion, the amplitude of which increases as p;„ap- 1x10 pd(V)=a, — b, lnV . Sx1 0 Here and in all the following velocities and times will be understood to be respectively expressed in pm/s and s. Thus lengths are expressed in pm. (ii) On the other hand, due to plastic aging of the mi- crocontacts under the normal load [4], the static friction coefficient p, increases with the time of the stick t„as 6x10 4x10 ax10 p, (t„) =a, +b, ln(t„) (2) 0 0 0 and it is found [1, 3] that 0. 1 10 100 V (Nms ) K V FIG. 2. Phase diagram in control parameter space ( V, K /M). Circles indicate the location of the bifurcation from stick-slip to steady sliding. The straight full line corresponds to a logarith- mic fit of the data in the creep-dominated regime. For driving velocities V~50 pms ', the data markedly deviate from this line, indicating the crossover to the inertia-dominated regime (see Ref. [1]) symbolized by a dashed line. slider FICx. 1. Schematic block and spring arrangement. The mass of the slider is M. The stiffness of the spring is K and its "free" end is driven at constant velocity V. 1063-651X/95/51(5)/4005 (6) f$06. 00 1995 The American Physical Society 51 T. Baumberger, ' C. Caroli, B. Perrin, ' and O. Ronsin' Laboratoire de Physique de la Mati'ere Condensee de l'Ecole Xormale Superieure, 24 rue Lhomond, 75231 Paris Cedex 05, France Groupe de Physique des Solides, Universites Paris VI et VII, 2 place Jussieu, 75251 Paris Cedex 05, France (Received 6 December 1994)