FRACTURE AND FRICTION: A REVIEW

Transcription

1 A C T A G E O P H Y S I C A P O L O N I C A Vol. 52, No FRACTURE AND FRICTION: A REVIEW Panayiotis VAROTSOS Solid Earth Physics Institute, Department of Physics University of Athens Panepistimiopolis, Zografos, Athens, Greece pvaro@otenet.gr Abstract Earlier reviews on fracture and friction, as well as relevant monographs (e.g., Theory of Earthquake Premonitory and Fracture Processes by R. Teisseyre), presented the knowledge accumulated at that time. By putting emphasis on the physics of these phenomena, the present review summarizes the experimental and theoretical advances that have been achieved during the last years. This is preceded, in the description of each phenomenon, by a careful inspection of the problems envisaged in the frame of the earlier aspects. As a first example, we refer to the recent experiments which demonstrate that the cracks cannot accelerate up to the Rayleigh wave speed predicted by classic theories. Secondly, contrary to what had been thought previously, the most recent theoretical models of the crack propagation are intrinsically unstable against Yoffe-like deflections (i.e., deviations of the crack motion from a straight line) at all speeds. Recent laboratory experiments shed light on what happens during individual slip events in stick-slip phenomena (e.g., the frictional force is larger for increasing than for decreasing velocity within individual events due to memory effects). Furthermore, measurements on laboratory scale reveal that the frequency of precursor events increases dramatically just before a major slip event (even when studying the friction between relatively smooth solids like). The latter effect is primarily due to defects (or disorder, in general), which are shown to play a major role in both phenomena investigated. Key words: friction constitutive law, weakening effect, fracture, precursory effects.

2 106 P. VAROTSOS 1. FRACTURE How things break. The strength of solids calculated on the basis of the perfect lattice properties comes out completely wrong. (In order to prove it, Varotsos and Alexopoulos, 1995, for example, present two such approaches.) The calculations lead to values of the critical stress equal to Y/4 or Y/6 (where Y is Young s modulus) which are about two orders of magnitude larger than the practical strength of a material. The experimental values of the strength of materials can only be explained if we move from an ideal material to a real one, which contains defects. The edge dislocations can satisfactorily explain the slip in a monocrystal. These dislocations may be charged (mainly in ionic crystals, as well as in geophysically interesting materials) and their role on the generation of electrical precursory phenomena (e.g., see the model developed by Teisseyre and coworkers, e.g., Teisseyre, 1996; 1997; 2001a, b,c,d; Teisseyre and Nagahama, 1998; 2001) has been discussed in detail elsewhere (Varotsos, 2003). A concept complementary to edge dislocations is that of the cracks. The fracture in heterogeneous systems, which seem to be relevant to geological systems such as earthquake faults, has been an area of active study in recent years mainly due to its importance in practical engineering materials such as polycrystals and fiber composites, biological systems like bone, etc. Elastic stresses are long ranged; thus, regions of high local damage produce high local stresses which start an avalanche of breaks across the entire macroscopic system. In general, one has to combine the statistical evolution of damage initiating around weaker heterogeneities with the associated stress redistributions to accurately predict the point of instability. The localized nature of fracture has, among others, the following consequences: (a) The strength is inherently statistically distributed; identical materials (i.e., same geometry, underlying distribution of element strengths but with different statistical realization) can have different strengths; (b) A large system can be formally considered as composed of a collection of independent subsystems coupled in series so that failure in the weakest subsystem causes failure across the entire system. Recent work on the heterogeneous fracture problem has drawn some analogies between fracture and first order phase transitions (e.g., Buchel and Sethna, 1996) and also related the scaling of some subcritical damage events to predictions from meanfield theory for special distributions of heterogeneity (Hansen and Hemmer, 1994; Zapperi et al. 1997). However, Curtin (1998) raised doubts whether such relationships can be rigorously extended to predict the size-dependent failure strength, since phase transitions and mean-field theory models do not exhibit size scaling of any features such as the critical field. Curtin found that a large system of n elements can be viewed as a collection of n/n c smaller systems of a critical size n c. The system fails as if it is composed of a series collection of n/n c bundles, the weakest of which causes failure.

3 FRACTURE AND FRICTION: A REVIEW Introduction to cracks The important role of cracks of amplifying the stress, which is analogous to that of the edge effects that lead to appreciably high electric fields (e.g., Varotsos et al., 1998; 2000), has been recognized as follows: In 1913, English first studied a large plate of material, with an elliptical hole and upon pulling the plate with a uniform stress σ (far from the hole), he found the following: the stress near the narrow edge of the hole exceeds σ by a factor 2(l/ρ) 1/2, where l is the length of the hole and ρ is the radius of the curvature. Thus, taking ρ 1Å and l = 1 µm, we find (l/ρ) 1/2 100; this explains the practical strength of brittle solids, since it is quite a challenge to prepare materials without micrometer-sized flaws at the surface, ready to spring into action at stresses smaller than expected. Notice that there is no requirement of a critical density of flaws. A single one will do (Marder and Fineberg, 1996). Griffith (1920) first analyzed the stability of an isolated crack in a solid subjected to an applied stress by minimizing the total free energy of the system (cracked rock + loading system). It can be shown that the maximum stress that can be applied to a rock containing one crack of length l is approximately: γ Y σ1 2 π l, (1) where γ is the thermodynamic surface energy. This equation shows that the mechanical strength in traction depends (beyond the intrinsic physical parameters Y, γ ) on an 1/ 2 σ τ σ τ τ τ Fig. 1. The three fracture models: opening mode (I), sliding mode (II), tearing mode (III). The last figure summarizes the geometry of a crack propagating through a heterogeneous medium and is taken from Ramanathan et al. (1997); the crack front, free crack surfaces, and the local normal, forward and tangent vectors are shown as they are the directions of tensile (I), shear (II), and tearing (III) loading, respectively. The crack front is oriented along the z-axis and moves in the positive x-direction. As the crack moves, it leaves behind free surfaces S + and S on either side of the crack, located in body coordinates at y = h(x, z) for x < f(z, t). The x (inplane) coordinate of the front at time t, x c (z, t) = f(z, t) is assumed to be single valued in z. The instantaneous position of the front is given by the curve r CF (z, t) = f(z, t) ˆx + h[f(z, t), z] ŷ + z ẑ.

4 108 P. VAROTSOS extrinsic parameter, the crack length l. The largest crack controls the mechanical response. Griffith s result was generalized by considering any configuration resulting from the combination of the three possible fracture propagation modes depicted in Fig. 1 (labelled I, II, and III); similar relations with the above Griffith s result for mode I fractures can be derived for modes II and III. In reality, the surface energy γ does not adequately describe the fracture resistance. Crack propagation is not reversible and dissipative forces (friction forces) should also be considered, leading to the conclusion that the fracture surface energy Γ (or fracture toughness, the true crack surface energy, see also Observations in Section 1.4) should be used instead of γ. (Concerning the difference between these two quantities, see, for example, Gueguen and Palciauskas, 1994.) σ 0 C σe β γ elastic domain closure of cracks failure starts opening propagation Fig. 2. Typical stress-strain plot for a cylindrical rock under uniaxial compression σ. ε In a real rock, its mechanical properties are controlled by a population of cracks and not by a single crack. The typical behaviour of the stress strain curve of a rock sample submitted to a uniaxial or triaxial compression is given in Fig. 2, where ε denotes the axial strain. The lowest part of the plot correspond to the closure of cracks oriented perpendicular to the compression stress σ. The second part, labelled β, is just the elastic region, the slope of which, in a uniaxial compression, is the static Young s modulus. The (upper) third part corresponds to the opening and propagation of cracks oriented parallel to σ. Failure starts at c 0 (the uniaxial compressive strength) and continues beyond this point (cf. the dilatancy becomes evident in this region when the cracks are opened). 1.2 Brittle and ductile materials There is no yet completely satisfactory answer to the question of why some materials are brittle and others are ductile (Marder and Fineberg, 1996). This is, in reality, the question of what makes a crack grow. In other words, it could be described as follows: Take a slab of material, make a saw cut in it, and pull. In a brittle material, the tip of the saw cut spontaneously sharpens down to atomic dimensions, and like a knife blade, one atom wide, slices its way forward; on the other hand, in a ductile material, the tip of the saw cut blunts, broadens and flows, so that great effort is required to make it progress. A usual attack on the problem considers atomically sharp cracks, in

5 FRACTURE AND FRICTION: A REVIEW 109 an otherwise perfect crystal, and asks what happens when slowly increasing stress is applied; estimations have been published on whether the crack will propagate forward in response to such a stress, or whether instead a crystal dislocation will pop out of the crack tip, causing the tip to become blunt. Very large computer simulations of a ductile material recently appeared (e.g., see Marder and Fineberg, 1996, and references therein). For example, Fig. 3 refers to a 35 million atoms simulation, and shows what happens when an elliptical crack attempts to propagate in a 0.1 µm thick copper sheet; the tip of the crack spawns clouds of dislocations which provide strong impediments to further motion. Fig. 3. Simulation, with 35 million atoms, showing what happens in a ductile material. An elliptical crack (shown in the center) in a 0.1 µm thick copper sheet is placed under tension in the vertical direction. As the crack attempts to propagate horizontally, it emits clouds of dislocations (white) which provide strong impediments to further motion (Marder and Fineberg, 1996). Brittle-ductile transition. This transition is usually not abrupt and occurs over a finite temperature and pressure range. The stress-strain plots depicted in Fig. 4 show that at room temperature, the transition is very progressive for granite (at around 300 MPa), but abrupt for limestone (around MPa) (e.g., see Gueguen and Palciauskas, 1994). Most solids have a temperature at which they make the transition from brittle to ductile behaviour, e.g., for silicon this is around 500 C. This transition is not well understood yet.

6 110 P. VAROTSOS σ (a) ε σ (b) Fig. 4. Brittle-ductile transition on a stress-strain plot for granite (a) and limestone (b). ε The above results could be, in principle, understood, if we simultaneously consider the two microscopic mechanisms: crack propagation and dislocation propagation (dislocations are always present near cracks). The mobility of point defects, through which dislocations move, depends drastically on temperature; thus, at low temperatures, dislocations are practically frozen, but at higher temperatures they become mobile. At sufficiently high temperatures, the dislocations, which lie in the neighbourhood of a crack, are put into motion by the stress field due to the mobile crack. Since the dissipative processes increase with temperature, they reflect an increase with increasing temperature of the fracture surface energy Γ leading (eq. 1) to the increase with temperature of the critical value of σ 1 above which crack propagation occurs and fracture develops. An alternative model accounting for the brittle-ductile transition was suggested by Renshaw and Schulson (2001). In summary, both processes, i.e., crack propagation and dislocation propagation (which are, in fact, interdependent) are affected by increasing temperature: it makes crack propagation less likely, but it favours dislocation propagation. There exists a domain, called the transition domain, when both processes are simultaneously active. 1.3 Crack dynamics Recent advances revealed several surprising features of the behaviour of cracks. We first review the conventional theory and the problems envisaged, and then proceed to the current theoretical aspects, as well as to those concerning the basic features of the crack propagation. Conventional theory and the problems envisaged. Mott (e.g., see Freund, 1990 and references therein) made the first calculations and achieved a scaling theory, which stood up remarkably well to increasingly sophisticated mathematical improvement. This scaling theory, which is based on energy balance considerations, could be roughly summarised as follows (Marder and Fineberg, 1996): Consider a crack of length l growing at rate υ in a plate (Fig. 5a). For very slowly moving cracks, we can disregard kinetic energy and consider only potential and fracture energies. The potential energy decreases as the crack extends and (since the size of the region where this happens scales as l 2 ) its release scales as ~ l 2. As for the fracture energy, we consider that making the crack moving forward it requires breaking bonds, creating new surfaces and generating heat, and hence the energy required scales as the crack length l. Thus, for very small cracks, since the potential energy decreases as l 2 and the fracture energy increases as l, the fracture energy (for small l) is always larger leading to a total

7 FRACTURE AND FRICTION: A REVIEW 111 l V ENERGY Slowly growing crack l c CRACK LENGTH, l Fig. 5. The left figure shows how a crack of length l grows at a rate υ in a plate according to the calculation of Mott. Above the Griffith point, i.e., l >l c, the crack extension is rapid (see the right figure) and spontaneous (see the text) (Marder and Fineberg, 1996). energy (sum of the potential and fracture energies) which increases with l. This is depicted by the increasing branch of the curve in Fig. 5b for small l, which shows the total energy versus l. It is a fortunate fact, because otherwise all solids would be completely unstable upon application of a slightest mechanical stress. At a critical crack length l c (i.e., the so-called Griffith point), however, the potential energy exceeds the fracture energy, and from here on, more energy is released than consumed by crack extension, resulting in a rapid and spontaneous crack extension. Then the role of the kinetic energy becomes also important and since the amount of mass that moves as the crack opens scales as l 2, the kinetic energy should scale as l 2 υ 2. Considering that the energy is conserved by converting potential to kinetic energy, and recalling that for l > l c the sum of fracture and potential energies decreases as (l l c ) 2, we find that the velocity of the crack is given by l c υ() l = υmax 1. l (2) In the rigorous formulation (Freund, 1990), υ max is the speed of Rayleigh wave the speed of sound travelling over a flat surface, or of seismic waves over the Earth s surface. In spite of the great success of the scaling in the theory above, experiments showed that the cracks cannot accelerate up to the Rayleigh wave speed. One possibility was that the motion of cracks might be more complicated than that in straight lines.

8 112 P. VAROTSOS In the early calculations of dynamical fracture by Yoffe (1951) see also Freund (1990) it was shown that at around 60% of the speed of sound, the stress field surrounding the crack develops lobes that could force crack motion to deviate from a straight line (Yoffe deflections). Furthermore, Ching et al. (1996) showed (see below) that cracks should always move perpendicular to themselves, and stable motion (hereafter, for reason of brevity, called stable cracks ) should be impossible. Thus, if we work in the frame of classical elasticity and assume that cracks are stable, first, we get an equation of motion that cracks do not obey and second, recent theoretical insight makes it puzzling that they are able to propagate at all. As for the recent experiments (e.g., Fineberg et al., 1991; 1992; Gross et al., 1993), they show that cracks in brittle materials suffer a dynamical instability, which makes them unable to accelerate up to high velocities predicted by the classic theories of dynamic fracture. The origin of the dynamical instability. Yoffe s (1951) analysis, as mentioned above, had shown that the stresses in the neighbourhood of a crack tip might favour out of plane deflections, but at high speeds only. Ching et al. (1996) argued that, contrary to what had been thought previously, the most familiar models in fracture mechanics are intrinsically unstable against Yoffe-like deflections at all speeds. Ching et al. (1996) studied a so-called cohesive-zone model, in which an isotropic, ideally brittle solid obeys linear elasticity everywhere outside sharply defined fracture surfaces, and a finite-ranged cohesive stress opposes the separation of these surfaces near the crack tip. Such a model differs from the previous ones in which the crack extension was studied by examining only the singular stress fields in the neighbourhood of a geometrically sharp crack tip. The shape of the cohesive zone is a dynamic entity, which moves in response to the stresses in its neighbourhood, and is adjusted in such a way that the stresses are nonsingular and continuous at all times. Ching et al. (1996) investigated, as an example, a mode I crack propagating at steadystate speed υ along the centerline the x axis ahead of the crack (see Fig. 1). They showed that near the tip of this crack, the tangential stress is larger than the normal stress by a significant amount, because the tangential stress is amplified by the relativistic contraction (cf. as it approaches not the speed of the light, but that of sound) of length in the direction of motion. In simple words, Ching et al. (1996) showed that, if one looks out in front of a crack moving at any speed and asks in what direction the stresses act most strongly to tear the material apart, the answer is that the largest stresses are straight ahead of the crack, but at right angles to its direction of propagation; thus, cracks should always propagate perpendicular to themselves, and stable motion should be impossible (Marder and Fineberg, 1996). In summary, Ching et al. (1996) found that propagating cracks are strongly unstable against deflection. Furthermore, they suggested that this instability is governed by detailed mechanisms of deformation and decohesion at crack tips.

9 FRACTURE AND FRICTION: A REVIEW 113 Basic features of the motion of cracks. The experimental results have been reviewed by Marder and Fineberg (1996) (see also Sharon et al., 1995; 1996). Figure 6 shows, as an example, the behaviour of two cracks in plexiglas. They travel differently, depending on the force with which they are pulled. Figure 6a depicts crack velocity versus time: the lower curve shows that, for relatively gentle forces, the velocity increases smoothly and slowly with time; the latter smooth variation vanishes, however, in the upper curve, where the crack propagates beyond a critical velocity. Figure 6b depicts what the cracks leave behind them, in the two cases of Fig. 6a. The lower image shows that slowly propagating cracks tend to leave smooth surfaces; on the other hand, the upper image shows that when the propagating cracks exceed a critical velocity, they leave a thicket of small branches penetrating the surface behind them. The inability of cracks to accelerate to the predicted limiting speed could now be described as follows: Once instability sets in (upper curve in Fig. 6a), pulling more on a crack simply makes it dig in its heels harder, generating much more subsurface damage but scarcely leading to any more acceleration. Fig. 6. Cracks in plexiglas: (a) velocity versus the time (see the text for details on the two curves shown); (b) image of what the cracks leave behind them when they propagate at velocities smaller (lower image) or faster (upper image) than the critical velocity (Marder and Fineberg, 1996). The basic behaviour of crack propagation which can be understood, to some extent, by calculations at the atomic scale exhibits the following features (Marder and Fineberg, 1996): (1) Birth. There is a range of velocities (starting at zero and lasting until around 20% of the velocity of the sound) at which steady crack propagation is forbidden. (2) Childhood. Above the forbidden (velocity) band, there is a range of velocities for which steady crack propagation is allowed and perfectly stable. At exactly the same externally applied stress, a stationary crack could also be stable. (3) Crisis. Above a critical velocity, steady crack propagation becomes unstable and the crack might build tree-like patterns of subsurface cracks.

10 114 P. VAROTSOS 1.4 Laboratory observations related to crack propagation We first present a notion on the fractal dimension, which might be useful for understanding the suggested scaling invariance in the geometry of fracture surfaces, and then proceed to the presentation of recent experimental results. Introductory note on fractal dimension. We follow, for example, Varotsos (2004). Assume a reference volume V which consists of N smaller elementary volumes r D, i.e., N = V/r D. The smaller volume r D is the reference volume used for measurement. For D = 1, a segment of unit length can be decomposed into N smaller segments of length r, i.e., N = 1/r. In two dimensions, i.e., D = 2, a surface of unit area can be decomposed into N smaller areas 1/r 2. Generalizing this procedure, a dimension D can be defined through the relation: D = log N log(1/ r ). This definition allows a noninteger dimension, or fractional dimension. N is the number of elementary elements necessary to cover the unit surface, curve or volume, N = 1/r D. The latter can be equivalently formulated as r = N (1/D), which is the similarity variable between the elementary element and the whole. Therefore, when the fractal curve has length L and is measured with a ruler of length ε : and hence D = log( L/ ε) log(1/ ε), =. 1 L( ε) ε D Obviously, if D = 1 (Euclidean dimension), L is a constant independent of ε. On the other hand, if D 1 the length of the curve depends on the choice of ε. An example is the so-called von Koch s curve, which is obtained by a process of repeated dissection. A segment AB is dissected into four new segments, each being one-third the original length in the example of Fig. 7. This is repeated at the next stage. Therefore, if L(ε) and L(ε /3) denote the total lengths at the stages n and n+1, respectively, we should have L(ε /3) = 4/3L(ε). If L(ε) = ε 1 D, we obtain (ε /3) 1 D = (4/3)ε 1 D with D = log 4/log 3 = Von Koch s curve is a fractal of dimension D = It is a self-similar curve, i.e., a curve invariant on a change of scale. The ratio of selfsimilarity is 4/3. In general, we may say the following: Self-similarity of an object is equivalent to the invariance of its geometrical properties under isotropic rescaling of lengths (see also Varotsos et al., 2002, for recent applications to Seismic Electric Signals). In many physically relevant cases, however, the structure of the objects is such that it is invariant under dilation transformation, only if the lengths are rescaled by direction dependent factors. These anisotropic fractals are called self-affine (Vicsek, 1989). (For fur-

11 FRACTURE AND FRICTION: A REVIEW 115 ther details, see Varotsos, 2004.) Such examples are the fracture surfaces as it will be explained below. Α Β Α Α Fig. 7. The example of a von Koch s curve. Here, the fractal dimension is D = log 4/log 3 = 1.26 and the ratio of self-similarity is 4/3 (see the text). The geometry of fracture surfaces. In summary, a fracture surface z(x, y) is said to be a self-affine object in the sense that it remains invariant under the transformation (x, y, z) (αx, αy, α ζ z), where ζ is the so-called roughness exponent. Recent investigations reveal that fracture surfaces exhibit self-affine scaling (but the origin of the long spatial correlations leading to self-affinity has not yet been fully understood). Mandelbrot et al. (1984) and Mandelbrot (1985) first showed that the fracture surfaces of many heterogeneous materials are self-affine with a roughness exponent ζ in most cases close to 0.8. The idea of a possible universal ζ-value for heterogeneous materials has also been suggested. However, this idea was first questioned by Milman et al. (1993) on the basis of scanning tunnelling microscopy experiments, where fracture surfaces of metallic materials were investigated at the nanometer scale. They found a roughness exponent close to 0.5, i.e., significantly smaller than the earlier reported universal value of ζ 0.8. It was later conjectured (Bouchaud et al., 1993; Daguier et al., 1996) that the following might happen: the roughness exponent is 0.75 at large length scales and 0.5 at short length scales. The two regimes are separated by a cross-over length ξ c, and the regime of the short length scales corresponds to the vicinity of the depinning transition (Narayan and Fisher, 1992a, b; 1993; see also Section 1.6), where the crack front is just able to free itself from the pinning microstructural obstacles. If we study a planar crack front, that is aligned along x and propagates along y, we may say the following (Schmittbuhl et al., 1995): Two regimes appear successively. First, the crack develops some roughness, which increases with the crack mean position <y>; then the roughness is limited by the system size and becomes statistically independent of <y>. In order to model completely the scaling properties of the roughness, one needs two exponents. The first is the dynamic exponent z, which accounts for the time development of the crack front fluctuations, and the other is the roughness exponent in the steady state.

12 116 P. VAROTSOS Observations Observations with microscope and high resolution digital camera. The propagation of a crack through a transparent plexiglas block was studied experimentally by Schmittbuhl and Maloy (1997). The crack front (in mode I, see Fig. 1) was observed optically with a microscope and high resolution digital camera. A value ζ = 0.55±0.05 was found when the crack front advances at a very slow mean speed (10-7 to m/s). Observations with an atomic force microscope and a scanning electron microscope. The fracture surfaces of an intermetallic alloy and the stress corrosion fracture surfaces of a silicate glass were investigated as a function of crack velocity by Daguier et al. (1997). In the alloy, the fracture surfaces were observed, for four different crack velocities spanning a wide range, using both atomic force microscopy (AFM) and standard scanning electron microscopy (SEM). This simultaneous use of SEM and AFM allowed for an observation of the fracture surfaces over five or six decades of length scales. In the glass, the crack velocity was measured by imaging the crack tip with AFM at different times. The results, obtained on materials as different as an intermetallic alloy and a glass, showed that there are two self-affine fracture regimes: At smaller length scales, the roughness exponent is close to ζ c 0.50, while at large enough length scales the universal roughness index ζ = 0.78 was obtained. The crossover length ξ c separating these regimes strongly depends on the material (e.g., at υ 10-7 m/s, the length ξ c is of the order of 10-3 and 10 µm for the glass and the alloy, respectively), and exhibits a power law decrease with the measured crack velocity υ, i.e., ξ c υ -φ with φ 1. Laboratory measurements indicating interaction of sound with fast crack propagation. The influence of ultrasounds on crack dynamics in brittle materials was experimentally studied by Boudet and Ciliberto (1998) by using both natural sound emitted by the propagating crack and artificially generated ultrasound bursts. The main conclusion was that, in spite of the weak energy of the natural sound emitted by the crack (which is only 5% of the energy needed to propagate the crack), sound interacts with the crack tip and the crack velocity is strongly modified by the sound wave. Laboratory measurements of the crack velocity as a function of the energy flow to the crack tip. Hauch et al. (1999) measured the velocity υ of a crack propagating in a silicon single crystal as a function of the energy flux to the crack tip (fracture energy G). Samples were loaded in a thin (e.g., along the xz plane) strip configuration by displacing the edges of the wafer by a constant amount δ. In such a tensile loading configuration, G is simply the elastic strain energy stored per unit area (xz plane) ahead of the crack, and can be written as G 1 E = v 2 δ, W

13 FRACTURE AND FRICTION: A REVIEW 117 where W stands for the sample width, with extension δ, E is Young s modulus and ν is Poisson s ratio. The results showed an initial sharp rise in velocity, followed by slowly increasing crack velocities as fracture energy increases. However, the experimental details could not be reproduced quantitatively by simulations. 1.5 Theoretical studies of crack propagation An Einstein model of brittle crack propagation. The simplest model of brittle crack propagation considers only the motion of the crack-tip atom. This is a one atom nonlinear model. In spite of its simplicity, the model captures many essential features of steady-state crack velocity and is in excellent quantitative agreement with many body dynamical simulations. This minimal, one atom, non-linear model is called Einstein ice-skater (EIS) model, because it is reminiscent of the Einstein model for solids, in which a monoatomic solid consisting of N atoms is simplified by 3N non-interacting harmonic oscillators with the same frequency ω E (Einstein frequency). Fig. 8. An Einstein model of brittle crack propagation: Initial atomic coordinates for crack propagation in a triangular lattice-strip, four close-packed (vertical) lines wide; the inner two lines are mobile, in contrast to the outer two lines which are fixed. The crack tip atom is indicated by the large open circle, which moves initially approximately in the direction of the arrow (the light dashed line is a just-broken bond with neighbour No. 6 stretching the bond with neighbour No. 1 until breakage, then heads toward its final equilibrium position (small circle). Heavy lines indicate equilibrium (nearest-neighbour) bonds of length r 0 = 1, while heavy dashed lines are slightly stretched, nearly vertical bonds. The light lines are bonds elastically stretched to length r 1 = 1 + 3ε/4 by the uniaxial strain ε in the x-direction (Holian et al., 1997). The model (Holian et al., 1997; Holian and Thomson, 1997) speculates that the steady-state velocity of a brittle crack could be approximated by a single-particle Einstein cell model, where the mobile crack-tip atom (the EIS atom in Fig. 8) moves in the field of six immobile neighbours (the sixth, with whom the bond has just been

14 118 P. VAROTSOS broken, is assumed to be beyond the range of interaction). The bond-breaking event launches the EIS along the bonding direction (see the light dashed line in Fig. 8). This compressive non-linear event, results in a shearing motion along the transverse pair of close-packed lines at ±60 to the propagation direction, and gives rise to the local vibrational excitations that move coherently with the crack tip (Holian and Ravelo, 1995; Zhou et al., 1996; Gumbsch et al., 1997). Assume now that t break denotes the time elapsed, since the last bond-breaking event, until the EIS atom reaches a point, where the strains are sufficiently large and stretches the next bond to breaking. The pattern then repeats to the other side of the ice skating phase and the crack has then advanced by one nearest-neighbour spacing r 0 along the forward direction in the time 2t break. Thus, the crack velocity υ crack is given by υ crack = r 0 / 2t break. A crude estimate for t break can be obtained by assuming that the EIS starts at the turning point of its motion in the final harmonic equilibrium well. Approximating the time t break with the one-half the period (from one turning point to the other at bond breaking), we finally get: υ crack / cs = 2/ π = 0.45, where c s is the triangular-lattice shear-wave speed, which is very close to the Rayleigh or surface wave speed. Holian et al. (1997) went beyond this crude estimate of t break by investigating two kinds of attractive snapping bond potentials: harmonic and anharmonic. This investigation finally showed that: (a) EIS model reveals the lattice-trapping phenomenon, because the crack starts moving only when the strain exceeds a certain value; (b) When considering the more realistic anharmonic interactions, the EIS model gives steady crack-tip velocities that never exceed 0.4 of the Rayleigh wave speed, in excellent agreement with experiments (e.g., see Sharon et al., 1995). However, we note that as dislocation emission and real crack branching are completely absent in this one-particle model, it cannot explain the observed dynamical instabilities of cracks. Molecular dynamics simulations. Several large-scale molecular-dynamics (MD) simulations have investigated crack propagation. We report below two examples of recent simulations in order to show that, in spite of the large difference in the physical properties among the materials studied, a consistency seems to have been established for the results concerning the exponent for fracture surfaces, in the following sense: There are two regimes: Initially, as crack propagates slowly, the crack front profile is characterised by a roughness exponent around 0.4, but above a certain speed (or exceeding a crossover length) the exponent becomes around MD study of silicon nitride. The crack propagation in amorphous Si 3 N 4 film was investigated by Nakano et al. (1995). These MD-simulations show a crossover from slow to rapid fracture. The morphology of the crack surfaces correlates with the crack dynamics: the slow propagation creates smooth crack surfaces with a roughness exponent around 0.4, while the rapid propagation creates rough surfaces with an exponent of around 0.8.

15 FRACTURE AND FRICTION: A REVIEW 119 MD simulation in a graphite sheet. The crack propagation in a graphite sheet was investigated with a million atom MD simulation by Omeltchenko et al. (1997). For certain crystalline orientations, multiple crack branches with nearly equal spacing sprout as the crack tip reaches a critical speed of 0.6υ R, where υ R is the Reyleigh wave speed. This results in a fracture surface with secondary branches and overhangs. It was found that the morphology of crack branches is well characterized by two values of the roughness exponent ζ : Below a certain crossover length and within the same local branch ζ = 0.41±0.05 ( intrabranch correlations); on the other hand, for interbranch correlations (correlations no longer restricted to the same branch) exceeding the crossover length ζ 0.71± Review of the dynamics of planar crack fronts in heterogeneous media The dynamics of cracks in heterogeneous media involves much physics that is yet to be understood. Even in situations in which the path of the crack is predetermined, e.g., by a preweakened fault, its dynamics can still be complicated (Ramanathan and Fisher, 1998). In this paragraph, we restrict ourselves to the simplest situation, which is, a crack confined to a plane. The general behaviour is as follows: For small loads across such a planar crack, the crack front will be at rest. As the load is gradually increased, the crack front may undergo some transient motion but then be arrested again. However, when the load exceeds a critical load, the crack front will begin to propagate through the sample. In this paragraph we focus our attention on the behaviour near the onset of propagation of (tensile) planar cracks. For the system of a planar crackfront moving through a heterogeneous medium, the bulk degrees of freedom lead to effective long-range interactions between the points of the front (e.g., Gao and Rice, 1989). Thus, when a point on the crack front moves ahead, the stress at all other points on the front increases due to the elastic interactions tending to pull them forward. In addition, elastic waves are emitted as the crack front moves non-uniformly. When one point moves ahead, these waves result in stresses elsewhere on the front, which, for a while, are greater than the stresses due to just the static elastic deformations which will remain long after the waves have passed. These stress overshoots along with the long range interactions have been shown (e.g., Ramanathan and Fisher, 1998) to play an important role in the dynamics of the crack front, when it is moving with a non-zero mean velocity. However, we only summarize below the simple case of the quasistatic approximation, in which the sound waves are neglected and the stress transfer is instantaneous. Quasistatic limit with instantaneous stress transfer. The crack front exhibits a dynamic critical phenomenon (see Varotsos, 2004; and references therein), with a second-order-like transition from a pinned to a moving phase as the applied load is increased through a critical value G c qs (i.e., there are two phases separated by a unique critical load):

16 120 P. VAROTSOS (a) When the applied load G is small, there is no steady-state motion and the crack front is pinned by the random toughness in one of many locally stable configurations. (b) As the load G is gradually increased, there are a series of local instabilities of the crack front which lead to avalanches, i.e., segments of the crack front will overcome the local toughness and jump forward, perhaps causing other segments to jump and thereby trigger an avalanche which will eventually be stopped by tougher regions. The avalanches show a power law size distribution up to a characteristic length ξ with the larger avalanches being much rare. The cut-off length ξ defines the correlation length below threshold. (c) As the threshold load qs G c is approached, the correlation length diverges as qs ξ ~ ( Gc G ν ) with ν = 1.52±0.02. Furthermore, we find: κ = 1 1/ν (for κ, see below). (d) At the threshold, there is no characteristic length scale and the distribution of the avalanche sizes is a pure power law. The crack front is self-affine with correlations: 2 2 < [ f( z, t) f( z+ r, t)] > ~ r ζ, where < > denotes the average over the randomness, and the roughness exponent is ζ = 0.34±0.02. The dynamic exponent is found from the duration of avalanches as a function of size l; they typically last for τ l l z, with z = 0.74±0.03. qs (e) As the load increases above the critical load G c, the crack front depins and begins to move, albeit very jerkily, with a nonzero, mean steady-state velocity υ; this velocity, just above the critical load, scales as υ (G qs G ) c β with β = 0.68±0.06 and the fluctuations in velocity are correlated up to a distance ξ +, which diverges as the threshold is approached from above as ξ + (G qs G c ) ν+, with ν + = ν = ν = 1.52±0.02, if there is only one divergent length scale in the problem. The following exponent identities predicted from the scaling and renormalization-group (RG) analysis (e.g., Ertas and Kardar, 1992; Narayan and Middleton, 1994). β = (z ζ)ν, ν = 1/(1 ζ) are found to hold, so that there are only two independent exponents, z and ζ, characterizing the transition (i.e., the correlation length exponent and the roughness exponent, respectively).

17 FRACTURE AND FRICTION: A REVIEW 121 We now recapitulate the scaling relations for the distribution of avalanche sizes below the threshold loading following Ramanathan and Fisher (1998), Narayan and Fisher (1992a, b) and Narayan and Middleton (1994): The probability density distribution of avalanche size roughly the extent along the crack front of an avalanche has the form: Prob(size of avalanche > l) 1/l κ ρ(l/ξ ). In other words, this is the fraction of avalanches with size larger than l, as G is slightly increased: G G + dg. The cumulative probability of the size of an avalanche being greater than l, as the load is swept slowly from zero to the critical load scales as 1 1 d G ~, k l ρ(/ l ξ) l (where the integration is made from G = 0 to G qs = G c ). In other words, the fraction of all avalanches with size larger than l is 1/l; in order to obtain this result, the aforementioned scaling relation κ = 1 1/ν (which stems from the increase in the mean position as Gc is approached) was used along with the observation that the rate of avalanche production, n A (G ), per increase in G goes to a constant at G c. Thus, for the probability density we have Prob(size of a given avalanche = l) 1/l The scaling relation w A 0.32±0.03 for the width w of fracture as a function of the fracture area A Walmann et al. (1996) studied fault fracture patterns in slabs of clay during extensional deformations. These laboratory measurements showed that fractures nucleate and grow on many scales, but certain scaling relations are obeyed. Specifically, Walmann et al. (1996) found empirically for the length l of a fracture as a function of the (fracture) area A, the relation: L A β with β = 0.68 ± 0.03 (3) for many deformation types. The average width w of a fracture scales with the area (wl = A) as w A 1 β, i.e., w A 0.32±0.03. (4) The scaling laws l A β and w A 1 β imply that the large fractures are relatively thinner than the small fractures, i.e., the width of a fracture scales with the length as W l (1 β)/β = l 0.47±0.03. (5) The above scaling relations are compatible with a physically based network model (see the lower curve in Fig. 9), i.e., a spring network that was shown to repro-

18 122 P. VAROTSOS duce the fault fracture pattern both visually and statistically. The determined exponents were robust: independent of the clay type, moisture content, the type of extensional deformation, and the deformation speed within certain limits. The exponent 0.32±0.03 in relation (4) might be related to the slope when plotting the logarithm of the amplitude of Seismic Electric Signals versus the magnitude of the earthquakes (for a given epicentral distance); the latter slope has been experimentally found, more than 20 years ago by Varotsos and Coworkers, to lie between 0.3 and 0.4 (see Varotsos, 2004 and references therein). Fig. 9. The upper curve depicts the plot of length l as a function of the area A(= lw) for 3000 fractures observed experimentally, while the lower curve shows the results of a simulation by a network of elastic springs. The experimental data have been shifted upwards by a factor of The shaded area indicates one standard deviation of the observed length l for given A, to each side of the average length. The exponents resulting from the experimental data and from simulation are similar if we consider the statistical uncertainties (the fit was performed over the region indicated by the solid part of the fitted lines) (Walmann et al., 1996). 1.8 The fracture as a critical phenomenon. Laboratory acoustic emission experiments Microfractures occur before the final break up and it has been noticed (Lockner et al., 1991) that as the applied stress is increased, microfractures tend to nucleate and form a major fault, eventually causing the failure of the material. One way of monitoring these microfractures is the recording of their acoustic emissions (AE). Energy from AE released by fractures has been studied in many different situations, e.g., in granite (Lockner et al., 1991), in chemically induced fracture (Cannelli et al., 1993), in plaster samples cracked by piercing through them (Petri et al., 1994). Here, we restrict ourselves to those of the recent experiments in which they load the sample in such a way that the pressure P is imposed and slowly increased. For example, Garcimartin et al.

19 FRACTURE AND FRICTION: A REVIEW 123 (1997) studied AE of microfractures before the break-up in composite inhomogeneous materials, such as plaster, wood or fiberglass. These studies provided, for mode I fracture, a list of positions of microfractures, the strain of the samples, and the energy released as a function of P. The instantaneous energy ε released by a sample versus the pressure P had a form similar to that reported by Hadjicontis and Mavromatou (1995; 1996), i.e., it consists of bursts, which correspond to microcracks. The cumulative energy E, i.e., the total energy released up to a pressure P, was found to be: E = E 0 (1 P/P c ) α, with α = 0.27±0.05; this value corresponds to wood but it was found to depend only slightly on the material, e.g., for fiberglass α = 0.22±0.05. The power law (close to P c ) can be considered as an indication that fracture can be described as a critical phenomenon. Garcimartin et al. (1997) also studied the distribution of microfractures that occur within a given pressure interval. They found that, when formulating a measure of the clustering of microfractures, this yields information about the critical load. Constructing the histogram of the instantaneous energy ε from the events registered in 11 wood samples, it was found that the relation lnn = lnε 0 γ lnε is obeyed with γ = 1.51±0.05; the value of γ differs a little from one material to another, e.g., for fiberglass γ = 2±0.1. The power law behaviour, i.e., N ε γ, strongly suggests critical dynamics. 2. FRICTION The reason why several attempts have been made to model EQs by spring-block systems (with many degrees of freedom) is the following. Assume that a block resting on a surface is attached to a spring, the other end of which is pulled at a constant velocity. It has been observed that, at sufficiently slow velocities, the sliding process is not a continuous one, but the motion proceeds by jerks; the contact surfaces stick together until (as a result of the gradually increasing pull) there is a sudden break with a consequent very rapid slip. This behaviour has been termed stick-slip motion and is reminiscent of fault seismicity (e.g., Brace and Byerlee, 1966; Scholz, 1998). Concerning the justifications of the classical laws of friction (e.g., the static friction force is proportional to the load), since the early attempts (Coulomb, 1785) the role played by interactions between asperities at the surfaces of solids was emphasized. Furthermore, the idea was used (Bowden and Tabor, 1950; 1964) that the real contact area is very small and thus involves such large stresses that a significant plastification occurs. In simple words, one may view a material surface as being rough, consisting of asperities of different sizes which will deform under pressure. Thus, for static friction, in the frame of the so called adhesion model, the friction results from the intermolecular adhesion between two surfaces at the points of contact (e.g., see

20 124 P. VAROTSOS Bowden and Tabor, 1950; 1964). The basic assumption of this model is that when placing one surface on top of another, the deformation will cease when the total yield pressure of the asperities becomes equal with the load of the upper surface divided by the total contact area A c. This area is usually several orders of magnitude smaller than the apparent area A, e.g., A c /A 10-6 (Johansen et al., 1993); thus, even though the apparent area A may be macroscopic, the actual contact area A c can be small to such an extent that microscopic randomness may not simply average out. This is the stochastic element that can result in strong fluctuations of the static friction. Hence the aforementioned classical friction law, stating that the static friction force is proportional to the load, holds only in an average sense. In the case of dynamic friction, as the velocity increases, there will be more momentum transfer into the normal direction, producing an upward force on the upper surface. This results in an increase in the separation between the two surfaces, thus leading to a decrease in the contact area. In the frame of the adhesion model, the decrease of the contact area reflects a reduced adhesion, which qualitatively explains the experimental results. An alternative model, which uses the collisions between the asperities as the dissipation mechanism, was also suggested. However, in spite of their partial success, these two models cannot fully account for the observed (non-linear hysteretic) phenomena. Both models lead to the following picture: If the fluctuations in the static friction are indeed determined by surface area and contact area, then the dynamic friction should fluctuate for the same reasons. This reflects that the deterministic friction velocity relations suggested hold only in an average sense. Furthermore, there are puzzles even for fully understanding very simple cases. For example, take a coin and launch it across your desk, recording the distance travelled. Now spin the coin about the axis perpendicular to its surface as you launch it; the coin will travel farther, even if we launch it with the same initial velocity. The coin will stop moving and stop spinning at exactly the same instant (Farkas et al., 2003). The explanation of such experiments (e.g., see Halsey, 2003 and references therein) may be achieved on the basis of the suggestion that the coupled equations, describing how the spinning motion of the coin and its velocity both decrease with time, are highly non-linear. It is recently shown (Farkas et al., 2003) that even when considering a coin sliding on a table is not a simple system. The frictional mechanics of the coin is governed by material phenomena at scales much smaller than the size of the coin. The well-known laws of friction (i.e., the so-called Amonton s laws) ignore the existence of these complex phenomena at smaller length scales entirely, at the cost of introducing a highly non-linear description of the macroscopic scale of the coin (Hasley, 2003). 2.1 Hysteresis and precursors in the stick-slip friction of a granular system The response of a granular medium to shear forces plays a major role in EQ dynamics. Slipping events occur along faults, which are often separated by a gouge filled with