Slip Waves Research Articles

Subsonic slip waves along the interface between two anisotropic elastic half-spaces in sliding contact with separation

The Stroh sextic formalism, together with the Fourier analysis and singular integral equation technique, is used to study propagation of possible slip waves in presence of local separation at the interface between two contact anisotropic solids. The existence of such waves is discussed in details. It is found that such waves may exist if at least one medium admits a Rayleigh wave below the minimum limiting speed of the two media. The wave speed is not fixed. It can be of any value in some regions between the lower Rayleigh wave speed and minimum limiting speed, depending on the existence of the first and second slip-wave solutions without interfacial separation studied by Barnett et al. [Proc. Roy. Soc. Lond. A 415 (1988) 389]. The waves have no free amplitude directly, but involve arbitrary size of the separation zone which depends on the intensity of the motion. The interface normal traction and the particle velocities involve square-root singularities at both ends of the separation zones.

Ondes adhérence–glissement–décollement sous contact unilatéral frottant

A numerical study on the dynamic response of a mechanical system composed of two coaxial cylinders in relative rotation and unilateral contact with Coulomb friction is presented. This discussion complements the semi-analytical results on stick–slip waves given in Moirot, Nguyen, C. R. Acad. Sci. Paris Sér. IIb 328 (2000) 663–669, by some numerical simulations on the dynamic transition of the system from a given initial state to a surface wave. The case of stick–slip–separation waves is considered. To cite this article: A. Oueslati et al., C. R. Mecanique 331 (2003).

518 塑性変形下の交差すべり量と超音波速度の相関

518 塑性変形下の交差すべり量と超音波速度の相関

An example of stick–slip and stick–slip–separation waves

The dynamical problem of a brake-like mechanical system composed of an elastic cylindrical tube with Coulomb's friction in contact with a rigid and rotating cylinder is considered. This model problem enables us to give an example of non-trivial periodic solutions in the form of stick-slip or stick-slip-separation waves propagating on the contact surface. A semi-analytical analysis of stick-slip waves is obtained when the system of governing equations is reduced by condensation to a simpler system involving only the contact displacements. This reduced system, of only one space variable in addition to time, can be solved almost analytically and gives some interesting informations on the existence and the characteristics of stick-slip waves such as the wave numbers on the circumference, stick and slip proportions, wave celerities, tangential and normal forces. It is shown in particular that the stick-slip-separation solutions would occur for small normal pressures or high rotational speeds. Since the analytical discussion becomes cumbersome in this case, a second approach based on numerical analysis by the finite element method is performed. The existence and the characteristics of stick-slip and stick-slip-separation waves are discussed numerically.

On the theory of wave propagation in anisotropic plates

A theoretical framework is developed which describes wave propagation in infinite homogeneous elastic plates of unrestricted anisotropy. The approach exploits the propagator matrix which is the exponential of the fundamental elasticity matrix underlying Stroh9s formalism of anisotropic elastodynamics. The matrices of plate impedance and admittance are introduced, and their analytical properties are established, which appear fruitful for computing and analysing the plate wave spectra. On this basis, the dispersion equation can be cast into the form of a real equation involving the monotonic function, whose zeros and poles are the wave velocities in a given plate subjected to different boundary conditions (traction–free or clamped faces). It is proved that three fundamental wave branches exist for any orientation of wave propagation in an anisotropic plate with traction–free faces, and that those branches are missing if one or both faces are clamped. The intrinsic symmetry of the wave motion in an arbitrary plate is revealed. The general formalism is applied to elaborate the long–wavelength low–frequency approximation for a (thin) free plate of unrestricted anisotropy. The frequency–dispersive wave velocities, displacements and tractions at the onset of the fundamental wave branches are derived explicitly. The conditions for the extreme velocity values and some other useful universal connections are determined for an arbitrary thin plate, and exemplified for specific anisotropy cases.

An example of stick–slip waves

An analytical solution representing a family of stick–slip waves is obtained in a simple example modelling the dynamic behaviour of an elastic cylindrical tube in contact with Coulomb's friction with a rigid and rotating cylinder. This family of waves, representing the non-trivial periodic responses of a continuous system of one space variable, is not classical in the literature.

Mathematical modeling of slip in a rock along the uneven fracture

A mathematical model is presented for the unstable deformation of rock along an existing fracture in it. It is shown that this process is essentially nonlinear and can be described by the Gordon equation. An analysis is made for the influence of geometric irregularities and friction of the contact surfaces on the evolution of the velocity of solitary slip waves which are caused by local deformation effects and propagate along the fracture.

Steady Sliding of Two Elastic Half-Spaces With Friction Reduction due to Interface Stick-Slip

The sliding of two perfectly flat elastic half-spaces with a constant interfacial coefficient of friction is investigated. Previous work has demonstrated that this configuration is dynamically unstable due to the destabilization of frictional slip waves. It was speculated that this dynamic instability could lead to stick-slip motion at the sliding interface. It is shown here that stick-slip motion at the interface can exist with a speed-independent interface coefficient of friction. Steady motion persists sufficiently far from the interface and thus gives the impression of uniform sliding. This type of stick-slip motion is due to interfacial slip waves and allows the bodies to slide with an apparent coefficient of friction which is less than the interface coefficient of friction. Furthermore it is shown that the apparent friction coefficient decreases with increasing speed even if the interface friction coefficient is speed-independent. Finally, it is shown that the presence of slip waves may make it possible for two frictional bodies to slide without a resisting shear stress and without any interface separation. No distinction is made between static and kinetic friction.

Existence of second slip waves in anisotropic elastic media

Existence of subsonic second slip waves, a variant of interfacial (Stoneley) waves, is investigated using the Stroh sextic formalism. We have examined the second slip wave propagating along an interface between two identical half-infinite anisotropic media which have the same orientation. Special attention is paid to exceptional transonic states where the appearance of a certain bulk wave may prevent the existence of the second slip wave. By examining the impedance tensor at various transonic states, we formulate an existence theorem for second slip waves. Numerical examples are shown for materials with various symmetries, which give convincing evidence for the existence of second slip waves.

Properties of mass‐loading shocks, 2. Magnetohydrodynamics

The one‐dimensional magnetohydrodynamics of shocked flows subjected to significant mass loading are considered. Recent observations at comets Giacobini‐Zinner and Halley suggest that simple nonreacting MHD is an inappropriate description for active cometary bow shocks. The thickness of the observed cometary shock implies that mass loading represents an important dynamical process within the shock itself, thereby requiring that the Rankine‐Hugoniot condition for the mass flux possess a source term. In a formal sense, this renders mass‐loading shocks qualitatively similar to combustion shocks, except that mass loading induces the shocked flow to shear. Nevertheless, a large class of stable shocks exist, identified by means of the Lax conditions appropriate to MHD. Thus mass‐loading shocks represent a new and interesting class of shocks, which, although found frequently in the solar system, both at the head of comets and, under suitable conditions, upsteam of weakly magnetized and nonmagnetized planets, has not been discussed in any detail. Owing to the shearing of the flow, mass‐loading shocks can behave like switch‐on shocks regardless of the magnitude of the plasma beta. Thus the behavior of the magnetic field in mass‐loading shocks is significantly different from that occurring in nonreacting classical MHD shocks. It is demonstrated that there exist two types of mass‐loading fronts for which no classical MHD analogue exists, these being the fast and slow compound mass‐loading shocks. These shocks are characterized by an initial deceleration of the fluid flow to either the fast or the slow magnetosonic speed followed by an isentropic expansion to the final decelerated downstream state. Thus these transitions take the flow from a supersonic to a supersonic, although decelerated, downstream state, unlike shocks which occur in classical MHD or gasdynamics. It is possible that such structures have been observed during the Giotto‐Halley encounter, and a brief discussion of the appropriate Halley parameters is therefore given, together with a short discussion of the determination of the shock normal from observations. A further interesting new form of mass‐loading shock is the “slow‐intermediate” shock, a stable shock which possesses many of the properties of intermediate MHD shocks yet which propagates like a slow mode MHD shock. An important property of mass‐loading shocks is the large parameter regime (compared with classical MHD) which does not admit simple or stable transitions from a given upstream to a downstream state. This suggests that it is often necessary to construct compound structures consisting of shocks, slip waves, rarefactions, and fast and slow compound waves in order to connect given upstream and downstream states. Thus the Riemann problem is significantly different from that of classical MHD.

Slip waves along the interface between two anisotropic elastic half-spaces in sliding contact

A variant of the Stoneley-wave problem, namely slip waves between two homogeneous elastic half-spaces whose interface is incapable of supporting shear stresses, is studied. For two isotropic half-spaces there is either no or one slip-wave mode. In the case of anisotropic half-spaces, the possibility of a new slip-wave mode, called the second slip-wave mode, arises. The case of two identical anisotropic half-spaces of the same orientation is discussed in detail; criteria for the existence of a second slip-wave mode in terms of the nature of the transonic state are developed. It is concluded that for many anisotropic media a second slip-wave mode will exist within certain ranges of orientation of the slip-wave geometry. Numerical computations for iron (cubic symmetry) demonstrate that second slip-wave modes indeed exist in this material.

PLANE WAVE PROPAGATION IN ANISOTROPIC JOINTED MEDIA

The propagation of plane waves through a rock mass containing a major set of regularly spaced parallel joints is analyzed for wavelengths large compared with joint spacing, using a continuum description. The intact rock is isotropic elastic and the joint motion is restricted to tangential slip governed by a Coulomb failure criterion and subsequent ideal slip. Plane strain is assumed in a plane containing the joint normal and the propagation direction, with the joint normal inclined to the propagation direction. During slip the response is analogous to that of an anisotropic elastic-plastic model with a non-associated flow rule. The three physical parameters governing the response are the ratio of the bulk and shear elastic moduli, the joint friction coefficient, and the joint inclination. A detailed analysis is made of the motion initiated by a purely longitudinal pulse. Conditions for real characteristics in both elastic and slip regions are discussed, and also conditions for the interleaving of the elastic and slip wave speeds (two of each) necessary for proper matching conditions across an elastic-slip interface. Finally, the possible wave motions initiated by a monotonic normal pressure over a plane boundary are discussed.

38—ROLLER SLIP AND THE IRREGULARITY OF COTTON AND RAYON–STAPLE DRAWFRAME SLIVERS PART 2. THE NATURE OF ROLLER SLIP AND THE THEORY OF DRAFT DISTRIBUTION

The preceding Memoir has described the experiments on roller slip at the drawframe, and discussed their practical application. This Memoir first gives a simple account of the fundamental principles of roller slip and an explanation of the formation of the roller–slip wave. This explanation is applied to the calculation of the critical roller weighting required to prevent roller slip, and to the effect of draft distribution on the critical weighting. The predictions of the theory are shown to agree excellently with the experimental results.