Friction of Shear-Fracture Zones.

A shear fracture of brittle solids under compression undergoes a substantial evolution from the initial microcracking to a fully formed powder-filled shear zone. Experiments covering the entire process are relatively easy to conduct, but they are very difficult to investigate in detail. Numerically, the large strain limit has remained a challenge. An efficient simulation model and a custom-made experimental device are employed to test to what extent a shear fracture alone is sufficient to drive material to spontaneous self-lubrication. A “ weak shear zone ” is an important concept in geology, and a large number of explanations, specific for tectonic conditions, have been proposed. We demonstrate here that weak shear zones are far more general, and that their emergence only demands that a microscopic, i

Fragmentation and fracture of materials is a class of fundamental material-physics processes that still have inadequate aspects of established theory.The formation of single cracks and tensile fragmentation, which is the result of many interacting single cracks, are reasonably well understood.The most difficult type of material fracture, fully developed compressive shear fragmentation at large strains, is also perhaps the most important.This type of fracture includes, apart from the initial crack propagation and the formation of a crack network and fragments, a grinding phase in which the early fragments are slowly ground to a powder.
Fracture in shear zones has been the subject of a large number of experimental [1][2][3] and numerical investigations [4][5][6][7][8][9][10][11].These investigations are typically focused on small strains and the effective friction that appears near the early peak in shear stress or shortly thereafter.If the objective is to evaluate the shear strength of, e.g., a rock joint, this approach is appropriate, but it gives little or no information of the large-strain mechanical behavior.Reported results in the large-strain limit are also inconsistent.For example, in [3], higher confining pressure did not have an effect to friction coefficient, whereas in [2], higher confining pressure reduced friction.Some of the most important applications of shear fracture can be found within the fields of geophysics and mining technology.The grinding of minerals to powder-comminution-in order to extract metals from minerals, consumes a significant fraction of the global yearly energy production, and any optimization of this is of huge economical and environmental significance [12,13].In geophysics, the triggering failure in landslides and snow avalanches are typically shear fractures, and perhaps the most magnificent fractured shear zones of all are tectonic faults.Some of these faults have a surprisingly low effective friction (0.1) [14].There are many suggested explanations for this, of which most refer to specific tectonic conditions [15], or mineral or gouge geology [16].An exception would be, e.g., statically strong but dynamically weak faults [17,18].From a physics point of view, a relevant question is whether weak shear zones are specific to tectonic environments, or if they appear under more general conditions.
In order to test if spontaneous self-lubrication, induced by fragmentation, is a generic phenomenon, we constructed a minimalistic, scale-invariant, computationally efficient numerical model.The model must be minimalistic in the sense that it contains only the essential physics with no reference to any specific material or environment.Scale invariance means that the model length scale needs to be rescalable to prove that observed phenomena may appear at many length scales.The numerical model must be computationally efficient to collect enough statistics, and the results are finally compared to experiments and observations.
The numerical model [19,20] material used here is constructed by numerically sedimenting two-dimensional, random-sized inelastic discs, which are fused together by massless inelastic beams.Discs in contact also interact via kinetic friction.Material samples in rectangular shear cells with periodic boundary conditions in the shear direction are broken by a forced shear strain.The beams stretch, bend, and break if deformed beyond a fracture threshold in elastic energy.The numerical algorithm is described in detail in Ref. [19].To model the large strain limit, strains of up to 200% were simulated using ∼10 7 time steps in each of the hundreds of simulations containing ∼10 4 discs (Fig. 1).
Experiments on granite were performed with equipment designed exclusively for this purpose.The design is displayed in Fig. 1, together with a schematic illustration of its working principle.Four different types of granite drill cores from the Olkiluoto region in Western Finland were used.They were prepared by making a shallow circular cut around the cylindrical surface of the drill core, and then breaking the sample in half with a hammer.This produced a natural crack surface.The grinding of the shear zone was then done by twisting the lower half while the upper half was pressed against it.Very large strains could be obtained in this way.The only stopping criteria used was attainment of reasonably constant sliding friction.
As mentioned above, shear-zone formation typically begins with small cracks that grow and coalesce to form a crack network and initial fragments.As shear strain increases, more cracks are formed and the initial fragments break up into smaller pieces that eventually turn into a fine powder.This process is composed of a large number, n b , of localized fracture events.To construct a theory for effective macroscopic friction, μ M , we need to define a few parameters.The tangential shear stress is τ x , and the compressive stress across the shear zone is σ y .Effective friction is then μ M ¼ τ x =σ y .If the fracture process is generic, we would expect that n b ðσ y Þ behaves roughly similar to, e.g., acoustic emission counts observed during the compression of granular material [21].This would correspond to our experiments in the limit τ x → 0. As pressure is increased on a fragment-filled shear zone, the initial dilatancy due to cracking is again reduced by grinding.First, the most vulnerable grains break, and the new fragments move to fill up local pore space.As a pore space is filled up, a region will become much harder to break, and it will thereafter deform, mainly by elastic compression.As a larger fraction of local pore space is filled up, an ever increasing force is needed to squeeze grain fragments into the vanishing pore spaces.The exact relation between pressure and fracture is likely to be rather complicated and vary between different materials and environments.It may, nevertheless, be useful to postulate the simplest possible relation.This would be linear: dσ y =dn b ∝ σ y , resulting in an exponential function, σ y ðn b Þ, or inversely, a logarithmic function n b ðσ y Þ.Such behavior has been observed, e.g., by measuring acoustic emission counts during the compressive compaction of granules [22].
It is reasonable to set the unit of σ y such that σ y ¼ 1 at the uniaxial compression strength (UCS) of a material.(I.e., σ y is defined as the dimensionless pressure P=P UCS , where P UCS is the pressure at UCS.) When pressure is increased above UCS, the material will begin to disintegrate, and we would expect a transition to a more rapid increase in n b : dσ y =dn b ∝ const.This UCS point typically marks the beginning of the transition zone from a brittle solid with a well-defined kinetic friction to the ductile and rate dependent friction of a powder, for which measured effective friction often decrease as strain rate is reduced [23,24].
The first spanning crack in the emerging shear zone typically has a very pronounced roughness.In order for shear motion to be possible without any fracture of the fragile crack surfaces, opposite sides of the shear zone would have to rise over the roughness asperities, which implies significant dilation; i.e., σ y must be low for dilation to occur rather than fracturing.As shear motion is forced under higher pressure, asperities and fragments begin to break, beginning at the locations that resist shear motion the most, and which therefore have the highest shear stress concentration.As these shear frustrations are broken, the grinding would continue with less significant shear pinning points.If, as above, we postulate a linear relation, we get dμ M =dn b ∝ μ M .
The above can be summarized as: and where ci and c i , ði ¼ 1-4Þ, are constants.These constants are not arbitrary: in the limit n b → 0, we have μ M ¼ c 3 .This would represent the very early phase when the shear zone is only a rough crack.We thus expect, c 3 ∼ 1.0.Furthermore, if n b is measured as the fraction of broken beams in our numerical model material, or the fraction of grain contact separation in granite, we expect c 2 ∼ 1, as c 2 ¼ n b at the brittle-ductile transition, at which point the material is already largely ground to a powder.There are not many general limitations on the constant c 4 .If it is negative, friction will be reduced by a fracture, otherwise it is the opposite.If its absolute value is large, changes will be small, if not, changes in friction will be large with a fracture.Finally, we expect expð−c 2 =c 1 Þ to be very small because n b ¼ 0 only for very small σ y .
In order to test the above, we investigated numerically different types of brittle material models.Based on the results below, these materials can be divided into three different categories.(i) A material of spherical particles where the smallest possible fragments are single discs with a friction μ m .Microscopic shear relaxation in this material is dominated by sliding for very small μ m , and by fragmenton-fragment rolling for large μ m .(ii) A material of "angular" particles and small μ m .This material can be fragmented only so far that any disc remains connected to at least one other.With a small μ m , shear stress can still be relaxed fairly easily by sliding.(iii) The third category is angular in the same way, but with a large μ m .For this material there is no longer a mechanism to relax shear stress at the microscopic scale.In this case, μ M is, more or less, independent of n b until the beginning of the brittle-ductile transition regime is reached.I.e., as long as σ y < 1.This would imply a larger absolute value for c 4 compared to the other two cases, for which we would expect that c 4 ∼ −10 −1 .In essence, these models together capture three significant deformation mechanisms for shear zones formed below UCS: fracture, rotation, and sliding [25].
To extend the numerical model to and above UCS, creep, plasticity, and viscoelasticity would need to be incorporated.
The values of the theoretical constants can be estimated based on numerical results.Best fits of such an exercise were: c 3 ≈ 1.02; 2.03; 1.56, for (i) to (iii), respectively, which is consistent with the theoretical c 3 ∼ 1.0.Best fits for the second constant were c 2 ≈ 0.424; 0.528; 1.515, which is also consistent with its theoretical estimate c 2 ∼ 1, even though the last value is obviously too large to be the fraction of broken beams, which means n b ∼ 1 in case (iii) already prior to UCS.We expect c 4 ∼ −10 −1 , except for (iii) in which case we expect a larger absolute value.Best fits gave: c 4 ≈ −0.251; −0.318; −4.686.Finally, according to theory, c 1 should be significantly smaller than c 2 .Best fits gave: c 1 ≈ 0.037; 0.045; 0.157, respectively.Numerical results are shown in Fig. 2.
Above, μ M is a function of σ y and n b , but macroscopic friction, μ M , should also depend on microscopic friction μ m .The effect of μ m on μ M ðσ y Þ can be reasonably well accounted for by the rescaling of parameters: Equations ( 1) and ( 2) can then be written as: By substituting Eq. ( 5) with Eqs. ( 3) and ( 4), we get In the cross-over region between brittle and emerging ductility, μ M should be described by a smooth function, which sets constraints on the constants.We can then arrive at an approximate single function for μ M ðσ y Þ: where c 5 ∼ 1.The parameters for the three materials can be summarized as (i) 13, β ≈ −0.5, and c 1 =c 4 ≈ −0.034.
For material (i), β is almost zero, which means that macroscopic friction has hardly any dependence on the friction of the fragments, μ m .The reason for this is revealed in Tables I and II.Table I shows the ratio of average slip over average rotation for spherical particles.Only for large σ y and μ m ¼ 0 does sliding of fragments dominate over rotation as stress relaxation mechanism.For larger μ m , rotations dominate completely.This means that there are mechanisms for relaxing microscopic, i.e., fragments scale, shear stress for all μ m .This leads to a small c 3 e c 2 =c 4 , and a nonzero c 1 =c 4 resulting in a very low effective shear zone friction, μ M , as pressure, σ y , is increased.For material (ii), microscopic shear stress can still be efficiently relaxed by rotation as long as σ y is very small, but for large σ y , sliding always dominates.This means that shear stress can be relaxed only for small μ m , i.e., material (ii).In this case, the absolute value of β is considerably increased, but c 1 =c 4 is still nonzero, which means that μ M decreases with increasing σ y .Finally, for material (iii), fracture can no longer induce relaxation of microscopic shear stress, and fragments have to slide, even though μ m is high.This leads to c 1 =c 4 ∼ 0. I.e., this material is not self-lubricating.
In order to compare Eq. ( 7) to the data, we need to know μ m and UCS.For the simulations these are easily determined.For the experiments on granite, we used μ m ¼ 0.5 [26,27] and UCS ¼ 119 MPa [28].
In order to further test the theory, check its scale invariance, and its relevance for tectonic faults, we tested how well Eq.( 7) fits with what is observed at the San Andreas Fault Observatory at Depth (SAFOD).SAFOD reports that the fault gouge found in a creeping part of the fault contains quartz, corrensite, saponite, feldspar, and calcite [29].The UCS values for these rock types have a relatively large range.The weakest is saponite, which has a UCS ∼ 10 MPa [30], while for, e.g., granular quartzite, the UCS ¼ 200-500 MPa [31].The exact value of the UCS for a composite material depends in a nontrivial way on its structure, and the exact value is not know to us.We therefore simply make the crude estimate that a reasonable value is UCS ¼ 100 MPa, which is close to the estimated pressure inside the fault, 122 MPa [29].For microscopic friction, we used μ m ¼ 0.2 [26,32].SAFOD data were extracted from references [21,33], and they are marked as SAFOD 1 and SAFOD 2 in Fig. 3, respectively.In addition, we compared the theory to friction experiments conducted on smectite, illite, and quartz powders under high relative pressures reported in Ref. [34].For these data, we used the parameters: UCS ¼ 30, 100, 200 MPa, and μ m ¼ 0.2, 0.34, 0.35, respectively.The results are shown in Fig. 3.It is interesting to note that for these powders, the exponent c 1 =c 4 is typically very small, which is consistent with the fact that powders are highly fragmented to begin with.One would then expect little or no increase in n b with increasing σ y below the UCS.This means c 1 ∼ 0, consistent with the result.
In conclusion, we have demonstrated that there is selflubrication by fragmentation within shear-fracture zones if there is a mechanism for stress relaxation at the microscopic scale of single grains or fragments.In the numerical model used here, this mechanism is either low-friction sliding or rotation.If neither of these mechanisms are present, no selflubrication is observed in the investigated model.In the generic case, it is possible to imagine additional stress relaxation mechanisms that only need to appear locally at significant stress carrying contacts between fragments.Such mechanisms could be: partial melting, plastic deformations, creep, vibrations, ductile deformations, etc.In particular, for clay's frictional weakness, it is associated with interactions between the pore fluid and the chemically-charged clay surfaces [35].In all cases, the arguments leading to Eq. ( 7) are still valid.Near and above the UCS, macroscopic friction decays fast and should become rate dependent.Here, we have been able to demonstrate the cross-over to a fast decay of μ M , but not the exact dependence on rate.It should also be noted that our investigation only considers a constant confining pressure.A variable or cyclic loading may modify the friction behavior.

FIG. 1 .
FIG. 1.The experimental equipment is shown in (a) together with a schematic presentation of the machine (b).In (c) the simulation setup is shown with the initial crack and in (d) after shearing.

3 FIG. 2 .
FIG. 2. Fraction of broken beams as a function of normal pressure (a) and macroscopic friction coefficient as a function of fraction of broken beams (b).The data has been rescaled with the parameters presented in the main text and fitted with Eqs.(3) and (5) in (a) and (b), respectively.

TABLE I .
Total slip displacement divided by total angular rotation of the discs during a simulation with spherical particles.Italic entries slip/rot < 0.4, and bold entries slip/rot > 1.0.FIG.3.Friction parameter μ M μ βm =ðc 3 e c 2 =c 4 Þ as function of pressure σ y for the three material classes (i) (red), (ii) (green), (iii) (blue).The data is compared to theoretical functions.