Fracture and friction: Stick-slip motion

We discuss the stick-slip motion of an elastic block sliding along a rigid substrate. We argue that for a given external shear stress this system shows a discontinuous nonequilibrium transition from a uniform stick state to uniform sliding at some critical stress which is nothing but the Griffith threshold for crack propagation. An inhomogeneous mode of sliding occurs when the driving velocity is prescribed instead of the external stress. A transition to homogeneous sliding occurs at a critical velocity, which is related to the critical stress. We solve the elastic problem for a steady-state motion of a periodic stick-slip pattern and derive equations of motion for the tip and resticking end of the slip pulses. In the slip regions we use the linear friction law and do not assume any intrinsic instabilities even at small sliding velocities. We find that, as in many other pattern forming system, the steady-state analysis itself does not select uniquely all the internal parameters of the pattern, especially the primary wavelength. Using some plausible analogy to first-order phase transitions we discuss a soft selection mechanism. This allows to estimate internal parameters such as crack velocities, primary wavelength and relative fraction of the slip phase as functions of the driving velocity. The relevance of our results to recent experiments is discussed.     

Self-healing slip pulses and the friction of gelatin gels

We present an extensive experimental study and scaling analysis of friction of gelatin gels on glass. At low driving velocities, sliding occurs via propagation of periodic self-healing slip pulses whose velocity is limited by collective diffusion of the gel network. Healing can be attributed to a frictional instability occurring at the slip velocity V = V _c. For V > V _c, sliding is homogeneous and friction is ruled by the shear-thinning rheology of an interfacial layer of thickness of order the (nanometric) mesh size, containing a solution of polymer chain ends hanging from the network. In spite of its high degree of confinement, the rheology of this system does not differ qualitatively from known bulk ones. The observed ageing of the static friction threshold reveals the slow increase of adhesive bonding between chain ends and glass. Such structural ageing is compatible with the existence of a velocity-weakening regime at velocities smaller than V _c, hence with the existence of the healing instability. Received: 7 March 2003 / Accepted: 2 May 2003 / Published online: 11 June 2003 RID="b" ID="b"e-mail: ronsin@gps.jussieu.fr     

The mobility of the amorphous phase in polyethylene as a determining factor for slow crack growth

Polyethylene (PE) pipes generally exhibit a limited lifetime, which is considerably shorter than their chemical degradation period. Slow crack growth failure occurs when pipes are used in long-distance water or gas distribution though being exposed to a pressure lower than the corresponding yield stress. This slow crack growth failure is characterized by localized craze growth and craze fibril rupture. In the literature, the lifetime of PE pipes is often considered as being determined by the density of tie chains connecting adjacent crystalline lamellae. But this consideration cannot explain the excellent durability of the recent bimodal grade PE for pipe application. We show in this paper the importance of the craze fibril length as the determining factor for the pipe lifetime. The conclusions are drawn from stress analysis. It is found that longer craze fibrils sustain lower stress and are deformed to a lesser degree. The mobility of the amorphous phase is found to control the amount of material that can be sucked in by the craze fibrils and thus the length of the craze fibrils. The mobility of the amorphous phase can be monitored by dynamic mechanical analysis measurements. Excellent agreement between the mobility thus derived and lifetimes of PE materials as derived from FNCT (full notch creep test) is given, thus providing an effective means to estimate the lifetime of PE pipes by considering well-defined physical properties.     

A thin-film equation for viscoelastic liquids of Jeffreys type

We derive a novel thin-film equation for linear viscoelastic media describable by generalized Maxwell or Jeffreys models. As a first application of this equation we discuss the shape of a liquid rim near a dewetting front. Although the dynamics of the liquid is equivalent to that of a phenomenological model recently proposed by Herminghaus et al. (S. Herminghaus, R. Seemann, K. Jacobs, Phys. Rev. Lett. 89, 056101 (2002)), the liquid rim profile in our model always shows oscillatory behaviour, contrary to that obtained in the former. This difference in behaviour is attributed to a different treatment of slip in both models.     

Effect of interfacial slippage in peel test: Theoretical model

Peel test is an efficient method to assess the performance and characteristics of materials such as adhesives and adhesive tapes. Recent experiments evidenced that the measured adhesive strength is closely related to the shear-induced interfacial slippage near the delamination front due to the concomitant Poisson contraction effect of the adhesive. Based on the experimental observations, a theoretical model is presented in this paper to examine the effect of the shear-induced interfacial slippage in the peel test. The influence of the interfacial slippage, represented by the shear displacement in the cohesive zone, on the fracture energy of decohesive zone is analyzed. An implicit expansion method with a Gauss-Chebyshev quadrature scheme is used to derive the solution. It is found that the length of the slippage zone and the receding contact angle of adhesives are the two most significant contributors to the total fracture energy of the decohesive zone. These results demonstrate that the mechanism of interfacial slippage plays a significant role in the adhesion and peeling behaviors of adhesives.     

Scaling of the size and temporal occurrence of burst sequences in creep rupture of fiber bundles

We present a detailed statistical analysis of the size and temporal occurrence of burst sequences in the creep rupture of a proposed linear viscoelastic fiber bundle model. According to the model, the burst sequences of fiber breaks display a power law asymptotic behavior analogous to that of the static-fracture [Kloster et al., Phys. Rev. E 56, 2615, (1997)]. Moreover, power law asymptotics apply to inter-arrival times between successive bursts with a universal exponent close to unity.     

A new handshaking of Tight-Binding and Molecular Dynamics in multi-scale simulation

A new handshake scheme is presented for tight-binding (TB) and molecular dynamics (MD) for multi-scale simulation of covalent crystals. In the present scheme, when calculating the forces on MD atoms in the handshake region, the TB atoms in close proximity to the MD atoms are treated as MD atoms. The scheme is thus seamless for calculation of MD atoms. When determining the electronic states of the TB subsystem, instead of the four basic atomic orbitals, hybrid orbitals are employed as bases in TB method and also as representing the action of MD atoms on TB atoms. The present handshaking methodology has several advantages. Firstly, it avoids determining the physical parameters required by introducing a new orbital model. Secondly, the “seam” almost decreases by one order of magnitude compared to that of Silogen model. Thirdly, the whole scheme is stable for dynamic simulation.     

Experimental model of cracking induced by drying shrinkage

Drying induced shrinkage in materials may yield the formation of surface crack patterns. We report on various experimental observations of the geometry of the crack array and the kinetics of crack formation on a model system consisting of a layer of a paste made of clay, sand, and water deposited on a rigid substrate. We investigate in detail the influence of the layer geometry (size and thickness). Received 19 April 1999     

Why is nacre strong? Elastic theory and fracture mechanics for biocomposites with stratified structures

Nacre, stratified ceramic layers surrounded by organic matrix, is a tough material found inside certain seashells. We construct a coarse-grained elastic energy for such an anisotropic system and present an analytic solution for a notch crack normal to the stratified sheets. This analysis proves the reduction in stress concentration which was announced in our earlier work (P. G. de Gennes and K. Okumura, C. R. Acad. Sci. Paris 1, Ser. IV, 257 (2000)) and the related increase in toughness. Received 18 March 2000     

Shear-induced adhesive failure of a rigid slab in contact with a thin confined film

A rigid-glass prism (square or rectangular base, rectangular cross-section) is sheared off a thin film of silicone elastomer bonded to a glass plate by applying a tangential force at various distances above the prism/elastomer interface. At a given tangential force, the prism starts to slide on the elastomeric film. As the sliding velocity, thus the frictional force, is progressively increased, an elastic instability develops at the interface that results in the formation of numerous bubbles. These bubbles, the lateral dimension of which is comparable to the thickness of the film, move across the interface with speeds 1000 times faster than the overall sliding speed of the glass prism against the PDMS film. It is found that the glass prism continues to slide on the elastomeric film as long as the applied shear stress is less than a critical value. During sliding, however, a normal stress is developed at the interface that decays from the front (i.e. where the force is applied) to the rear end of the prism. When the normal stress reaches a critical value, the prism comes off the film. The critical shear stress of fracture increases with the modulus of the film, but decreases with the thickness following a square root relationship, as is the case with the removal of rigid punches from thin elastomeric films by normal pull-off forces.     

Impact of gel balls beyond the Hertzian regime

We study experimentally the impact of spherical gel balls on a rigid substrate, where the balls largely deform like a pancake at high impact velocities. In our previous study (Y. Tanaka, Y. Yamazaki, K. Okumura, Europhys. Lett. 63, 149 (2003)), we measured the contact time τ_f and maximally deformed size versus impact velocity and explained the behaviors at the scaling level. In this study, we further measure τ_m, the time required to reach the maximum deformation (from the initial contact), and the restitution coefficient e. We also make a static experiment where we obtain the force-deformation curve of the gel balls up to fairly large deformations to explain the data on the impact. We propose two phenomenological treatments going beyond the scaling argument, one for intermediate impact velocities and the other for large velocities; the former is based on the static experiment while the latter on a Lagrangian constructed from appropriate constraints. Results from these treatments reproduce the experimental behavior of τ_m.     

Morphology transition and slow dynamics in the collapse of amphiphilic monolayers at the air-water interface

The morphology on collapsed monolayers at the air-water interface has been studied using phase contrast microscopy. It is found that the transition from randomly distributed to quasi-one dimensional crack pattern takes place, depending on the pH value of the subphase and the presence of specific divalent metal ions. In these macroscopic patterns, the former exhibits a surface roughening due to a monolayer buckling while the latter becomes more smooth and uniform. The occurrence of the former is instantaneous and the latter follows a slow dynamics, i.e., the crack propagation in monolayers occurs with a delay for crack nucleation. Thus the change of pattern indicates the existence of a dynamic transition. The transition is discussed with the scenario of a crack instability in brittle materials. The changes of viscous nature and of ion binding, and the compression direction probably operate for the observed behavior effectively. Received 11 January 1999     

Very low friction for natural diamond in water of different pH values

Very high reductions in the friction coefficient are reported for natural diamond sliding upon natural diamond when water is introduced at the interface of contact. This reduction is found to depend on the pH value of the water, the load and the sliding velocity. The results are interpreted in terms of the reduction of adhesion due to adsorption of the liquid on the surface, and of graphitisation occurring during sliding, with graphite acting as a lubricant. Received 15 September 1999     

Nonlinear spring model for frictional stick-slip motion

Frictional stick-slip dynamics is discussed using a model of one oscillator pulled by a nonlinear spring force. We focus our attention on the nonlinear spring parameter k₀. The dynamics of the model is carefully studied, both numerically and analytically. Our numerical investigation, which involves bifurcation diagrams, shows a rich spectrum of dynamical behavior including periodic, quasi-periodic and chaotic states. On the other hand, and for a good selection of parameters , the motion of the particle involves periodic stick-slip, erratic and intermittent motions, characterized by force fluctuations, and sliding. This study suggests that the transition between each of motion strongly depends on the nonlinear parameter k₀. The system also displays resonance at fractional frequencies of the oscillator.     

Experimental study of cracking induced by desiccation in 1-dimensional systems

We designed a simple experiment to study both the dynamical and statistical properties of cracking that occurs in a one-dimensional system composed of wet clay (or similar material) exposed to shrinkage induced by desiccation. We study both the dynamical formation of cracks and the statistical characteristics of the final cracks pattern. We observe that the drying rate has a strong influence on the way cracks appear and grow. We find that the final crack width is related to the order of apparition of the cracks. We discuss the statistical distributions of cracks width and separation between two adjacent cracks. We also study the correlations between these two quantities. Our results are compared to the predictions of existing models. Finally, a comparison with another kind of clay is made. Received 6 May 2002 and Received in final form 5 July 2002     

Antiferromagnetism in a doped spin-Peierls model: Classical and quantum behaviors

We investigate the sequential cracking of thin brittle coatings attached adhesively to substrates. The focus of our study are uniaxial tensile loading conditions, where we monitor the behavior in a continuous picture and the coating strength follows a two-parameter Weibull distribution. For fracture well under way we derive an approximate analytical expression for the distribution of fragment lengths. We recover that the distribution scales with the average fragment length and that is a power function of the applied strain , i.e. , where depends on the distribution of the strength of the coating and on the adhesive's nonlinearity. Furthermore we compare our approximate analytical expression with numerical solutions and with simulations' findings. Received 16 February 2000     

Generalization of a nonlinear friction relation for a dimer sliding on a periodic substrate

An atomic cluster moving along a solid surface can undergo dissipation of its translational energy through a direct mode, involving the coupling of the center-of-mass motion to thermal excitations of the substrate, and an indirect mode, due to damping of the internal motion of the cluster, to which the center-of-mass motion can be coupled as a result of surface potential. Focussing only on the less well understood indirect mode, on the basis of numerical solutions, we present, departures from a recently reported simple relationship between the force and velocity of nonlinear friction. A generalization of the analytic considerations that earlier led to that relationship is carried out and shown to explain the departures satisfactorily. Our generalization treats for the system considered (dimer sliding over a periodic substrate) the complete dependence on several of the key parameters, specifically internal dissipation, natural frequency, substrate corrugation, and length ratio. Further predictions from our generalizations are found to agree with new simulations. The system analyzed is relevant to nanostructures moving over crystal surfaces.     

Local friction at a sliding interface between an elastomer and a rigid spherical probe

This paper reports on spatially resolved measurements of the shear stress distribution at a frictional interface between a flat rubber substrate and a glass lens. Silicone rubber specimens marked close to their surface by a colored pattern have been prepared in order to measure the surface displacement field induced by the steady-state friction of the spherical probe. The deconvolution of this displacement field then provides the actual shear stress distribution at the contact interface. When a smooth glass lens is used, a nearly constant shear stress is achieved within the contact. On the other hand, a bell-shaped shear stress distribution is obtained with rough lenses. These first results suggest that simple notions of real contact area and constant interface shear stress cannot account for the observed changes in local friction when roughness is varied.     

Why is nacre strong? II. Remaining mechanical weakness for cracks propagating along the sheets

In our previous paper (Eur. Phys. J. E 4, 121 (2001)) we proposed a coarse-grained elastic energy for nacre, or stratified structure of hard and soft layers found in certain seashells . We then analyzed a crack running perpendicular to the layers and suggested one possible reason for the enhanced toughness of this substance. In the present paper, we consider a crack running parallel to the layers. We propose a new term added to the previous elastic energy, which is associated with the bending of layers. We show that there are two regimes for the parallel-fracture solution of this elastic energy; near the fracture tip the deformation field is governed by a parabolic differential equation while the field away from the tip follows the usual elliptic equation. Analytical results show that the fracture tip is lenticular, as suggested in a paper on a smectic liquid crystal (P.G. de Gennes, Europhys. Lett. 13, 709 (1990)). On the contrary, away from the tip, the stress and deformation distribution recover the usual singular behaviors ( and 1/, respectively, where x is the distance from the tip). This indicates there is no enhancement in toughness in the case of parallel fracture. Received 16 November 2001