Arch. Min. Sci., Vol. 52 (2007), No 3, p

Arch. Min. Sci., Vol. 52 (2007), No 3, p. 387 8 387 DANUTA KRZYSZTOŃ* FRICTIONAL RESISTANCE IN THE POST-CRITICAL FAILURE OF ROCK SAMPLES CAUSED BY TRIAXIAL COMPRESSION OPÓR TARCIA W POKRYTYCZNYM NISZCZENIU PRÓBEK SKALNYCH WYWOŁANYM TRÓJOSIOWYM ŚCISKANIEM Triaxial compression tests in a stiff testing machine at a confining pressure of 0 70 MPa were performed on Carboniferous rock samples collected from the Upper Silesian Coal Basin. The results showed that the post- critical failures of waste rocks were congruent with smooth curves; post-failure behaviour of coal, however, were characterised by a stick-slip (Fig. 1). The normal and shear stresses at the sample slip plane of a known slope were calculated for the determined values of critical stress and residual stress in a uniaxial state of stress, and for a given confining pressure. The coefficients of maximum and residual frictions were also calculated (Tables 1 5). The obtained results, shown in Figs. 6 and 7, were compared with the Byerlee law, which describes the frictional shear strength. At high values of confining pressure the frictional shear strength can reach the value of the fracture strength. Then the transition from brittle fracture to ductile flow takes place. A knowledge of the conditions in which stick-slip occurs has a practical meaning with regard to the prediction of mining tremors. Keywords: brittle fracture, confining pressure, ductile flow, frictional resistance, stable sliding, stick-slip Pokrytyczne niszczenie próbek skalnych w warunkach konwencjonalnego trójosiowego ściskania zależy od wartości ciśnienia bocznego (okólnego) symulującego głębokość zalegania skały. W badaniach laboratoryjnych pokrytyczne niszczenie próbki skalnej przedstawione jest opadającą częścią charakterystyki naprężeniowo-odkształceniowej, na podstawie której określa się moduł spadku (osłabienia) i naprężenie resztkowe. Wieloletnie badania nad własnościami typowych skał karbońskich Górnośląskiego Zagłębia Węglowego wykazały, że kształt pokrytycznej charakterystyki znacznie różni się w przypadku skał płonnych (zlepieniec, piaskowce średnio- i gruboziarniste, mułowiec, iłowiec) i węgli (półbłyszczący, matowy). Dla skał płonnych charakterystyka w sposób gładki opada do naprężenia resztkowego, natomiast dla węgla, przy wyższych ciśnieniach okólnych, występuje tzw. poślizg przerywany ( stick-slip ) charakteryzujący * FACULTY OF MINING AND GEOENGINEERING, AGH UNIVERSITY OF SCIENCE AND TECHNOLOGY, AL. MICKIEWI- CZA 30, 30-059 KRAKÓW, POLAND

388 się cyklicznymi spadkami i wzrostami naprężenia (Sanetra,1994; Krzysztoń et al., 1998, 2002). Na rysunku 1 przedstawiono wybrane wyniki badań uzyskane dla piaskowca i węgla przy różnych ciśnieniach okólnych w przedziale 0 70 MPa. Ze studiów literatury wynika, że poślizg przerywany jest obserwowany w badaniach różnych typów skał w warunkach działania wysokich ciśnień okólnych (Brace i Byerlee, 1966; Byerlee, 1967, 1975; Byerlee i Brace, 1968; Shimada, 2000). Na rysunku 2 przedstawiono za Patersonem (1978) przykłady. Efekt poślizgu przerywanego jest przedmiotem zainteresowania wielu badaczy gdyż uważa się, że może być on przyczyną trzęsień ziemi (Brace i Byerlee, 1966; Byerlee i Brace, 1968; Brace, 1972) i wstrząsów górniczych (Gibowicz, 1989; Dyskin et al., 1998). Przyczyną pojawiania się poślizgu przerywanego jest zmiana oporu tarcia wzdłuż powierzchni poślizgu. Tarcie między powierzchniami skalnymi zależy od szorstkości powierzchni i od naprężenia normalnego do powierzchni poślizgu. W rozdziale 2 opisano typowy eksperyment dla określenia siły tarcia w skałach, na podstawie którego uzyskuje się wykres siły tarcia jako funkcji przemieszczenia (Byerlee, 1978; Earthquake Hazards Program, 2002). Na wykresie wyróżniono początkową, maksymalną i resztkową siłę tarcia (rys. 4). Współczynnik tarcia określany jest według wzoru: µ = τ/σ n, gdzie τ jest naprężeniem stycznym a σ n naprężeniem normalnym, które działają w płaszczyźnie ścinania. Przy określaniu współczynnika tarcia należy rozróżnić wartości współczynnika wyznaczone dla początkowej, maksymalnej i resztkowej siły tarcia (Byerlee, 1978). Badania eksperymentalne prowadzone dla różnych typów skał wykazały, że dla przedziału naprężenia normalnego 5 MPa σ n 100 MPa zależność między naprężeniem ścinającym a naprężeniem normalnym przedstawia linia prosta o równaniu: τ = 0,85σ n (Byerlee, 1978). Zależność tę zastosowano przy określaniu współczynnika tarcia dla początkowej i maksymalnej siły tarcia różnych rodzajów skał. Dla początkowej siły tarcia uzyskano duże rozrzuty punktów eksperymentalnych względem przyjętej linii prostej, co uzasadniano trudnością określania punktu występowania początkowej siły tarcia (punkt C na rys. 4). Natomiast dla maksymalnej siły tarcia prosta o równaniu: τ = 0,85σ n dobrze aproksymuje wyniki badań eksperymentalnych, przeprowadzonych dla różnych rodzajów skał (rys. 5). Badania własne dotyczyły wyznaczania współczynników tarcia dla maksymalnej i resztkowej siły tarcia. W tym celu wykorzystano wyniki badań nad własnościami wytrzymałościowymi i odkształceniowymi typowych skał karbońskich GZW (piaskowce: średnio- i drobnoziarniste, iłowiec, węgle: półbłyszczący i matowy) uzyskane w ramach projektu badawczego KBN (Krzysztoń et al., 2002). Niektóre badania obejmowały również pomiar kątów nachylenia płaszczyzny ścinania próbek. Okazało się, że kąty ścinania badanych skał wzrastają wraz ze wzrostem ciśnienia okólnego (w przedziale od 0 do 50 MPa), przyjmując bliskie wartości dla poszczególnych typów skał. W związku z tym do obliczeń przyjęto taki sam kąt ścinania dla wszystkich badanych skał, zależny tylko od ciśnienia okólnego. Znając średnie wartości naprężenia krytycznego i resztkowego, wyznaczone dla 4-6 próbek badanej skały przy stosowanym ciśnieniu okólnym oraz kąt zniszczenia próbki skalnej, obliczono naprężenia normalne i styczne w płaszczyźnie ścinania oraz odpowiadające współczynniki tarcia zarówno w próbce zwięzłej (maksymalna siła tarcia), jak i spękanej (resztkowa siła tarcia). Wyniki obliczeń zestawiono w tablicach 1 5. Uzyskane wyniki naniesiono na rysunkach 6 i 7 i porównano z zależnością Byerlee ego. Zarówno wytrzymałość na ściskanie kruchych skał, jak i wytrzymałość tarciowa (prawo Byerlee ego dla maksymalnej siły tarcia) są funkcjami ciśnienia okólnego. Na podstawie studiów nad zagadnieniami pękania i tarcia ( fracture and friction ) przypomniano, że punkt przecięcia charakterystyki wytrzymałości na ścinanie badanej skały i tarciowej wytrzymałości określa naprężenie normalne (ciśnienie okólne), przy którym następuje przejście ze stanu kruchego w stan ciągliwy. Poznanie warunków występowania poślizgu przerywanego ma duże znaczenie praktyczne ze względu na możliwość przewidywania wstrząsów górniczych. Słowa kluczowe: ciągliwe płynięcie, ciśnienie okólne, kruche pękanie, opór tarcia, poślizg przerywany, stabilny poślizg

389 1. Introduction The post-critical failure of rock samples in the conditions of conventional triaxial compression depends on the value of lateral pressure (confining pressure) simulating the depth of rock deposition. In laboratory investigations the post-critical failure of a rock sample is represented by the descending part of the stress-strain characteristic, on the basis of which the drop modulus (modulus of softening) and the residual stress are determined. Many years of investigations on the influence of confining pressure on the properties of typical Carboniferous rocks from the Upper Silesian Coal Basin have shown that the shapes of post-critical characteristics differ considerably in the case of waste rocks (mudstone, claystone, fine- and medium-grained sandstones) and coals (dull, bright). For waste rocks the characteristic descends in a continuous way to the residual stress; for coals, however, cyclic changes of stress occur (Sanetra, 1994; Krzysztoń et al., 1998, 2002; Krzysztoń & Sanetra, 2003). For the presentation of the influence of confining pressure on the form of stress-strain characteristics of waste rocks and coals, the computer printouts registered during a triaxial compression of fine-grained sandstone and coal samples at an increasing value of confining pressure in the range 0 70 MPa are shown in Figure 1 (Sanetra, 2004). A visual observation of the characteristics reveals that the shape of the ascending part is similar for the investigated rocks and has a form of a continuous curve with different increase rates. However, the post-critical part of a characteristic showing the process of rock failure has a different run for sandstone (waste rock) and for coal. In waste rocks the characteristic descends in a continuous way to the residual stress; for coal, however, the cyclic oscillations of the stress characteristic of a stick-slip occur. The amplitude of these oscillations as well as the length of a cycle (time) increases with the increase of confining pressure (Byerlee & Brace, 1968; Jaeger, 1971). Stick-slip oscillations have also been observed in tests of triaxial compression of various types of rocks under the conditions of a high confining pressure (Brace & Byerlee, 1966; Byerlee, 1967, 1975; Byerlee & Brace, 1968; Sanetra, 1994; Krzysztoń et al., 1998; Shimada, 2000). Stick-slip oscillations occurring on the ground slip surface obtained by saw cut of granite sample under an angle corresponding to the natural failure angle of the granite sample (Fig. 2a) and on the natural failure in a gabbro sample (Fig. 2b) are presented in Figure 2 after Paterson (1978). In the case of a slip along the natural failure surface (Fig. 2b), the increase in the confining pressure has an effect on the form of post-critical curves, which at lower confining pressures are continuous curves and at higher confining pressure reveal oscillations of a stick-slip. A great deal of foreign literature exists on the stick-slip as the cause of earthquakes (Brace & Byerlee, 1966; Byerlee & Brace, 1968; Brace, 1972) and seismic events induced by mining (Gibowicz, 1989; Dyskin et al., 1998). A stick-slip is induced by a change of friction resistance along the surface of sliding.

390 a) Sandstone 100 60 20 5 10 15 20 sample 8a/4 confining pressure p = 0 MPa 120 80 5 10 15 20 sample 8b/5 confining pressure p = 5 MPa 150 100 50 10 20 30 sample 8c/3 confining pressure p = 10 MPa 180 120 60 10 20 30 sample 8d/3 confining pressure p = 15 MPa

391 180 120 60 10 20 sample 8e/1 confining pressure p = 20 MPa 30 300 200 100 10 20 30 sample 8f/2 confining pressure p = 30 MPa 300 200 100 10 20 30 sample 8g/4 confining pressure p = 50 MPa 0 200 10 20 30 sample 8h/2 confining pressure p = 15 MPa

392 b) Coal 12 8 4 5 10 15 sample 14a/2 confining pressure p = 0 MPa 60 20 10 20 30 sample 14b/1 confining pressure p = 5 MPa 60 20 20 60 sample 14c/3 confining pressure p = 10 MPa 80 60 20 20 60 80 sample 14d/3 confining pressure p = 15 MPa

393 120 80 20 60 sample 14e/1 confining pressure p = 20 MPa 80 120 80 20 60 sample 14f/3 confining pressure p = 30 MPa 80 120 80 80 120 sample 14g/1 confining pressure p = 50 MPa 120 80 80 120 160 sample 14h/4 confining pressure p = 70 MPa Fig. 1. Axial stress axial strain characteristics at different values of confining pressure (after Sanetra, 2004); a) Sandstone, b) Coal Rys. 1. Charakterystyki σ 1 = f (ε 1 ) dla różnych wartości ciśnienia okólnego (wg Sanetry, 2004); a) Piaskowiec, b) Węgiel

394 a) b) Differential Stress 1 3, MPa 1500 1000 500 Westerly Granite with sawcut 1 3= 210 MPa Differential Stress 1 3, MPa 1500 1000 0 1 2 0 2 4 6 8 Axial Displacement, mm Axial Displacement, mm 500 San Marcos Gabbro, fractured F F F 510MPa 83MPa 42MPa Fig. 2. Stick-slip oscillations observed during frictional sliding in triaxial apparatus at confining pressures shown: a) On a ground saw-cut surface in granite (after Brace and Byerlee, 1966), b) On previously induced shear fracture in gabbro, F denoting the point of fracture (after Byerlee and Brace, 1968) (after Paterson, 1978) Rys. 2. Oscylacje poślizgu przerywanego obserwowane podczas tarciowego poślizgu w aparacie trójosiowym przy podanych ciśnieniach okólnych: a) na wyszlifowanej powierzchni granitu powstałej w wyniku przecięcia piłą (wg Brace a i Byerlee ego, 1966), b) na uprzednio wywołanym pęknięciu poślizgowym w gabro, F oznacza punkt pęknięcia (wg Byerlee ego i Brace a, 1968) (wg Patersona, 1978) 2. Experimental determination of friction force Friction between rock surfaces depends on the roughness of the surfaces and on the stress normal to a surface of sliding. The measurement of friction force in rocks may be carried out by different methods (Hoskins et al., 1960; Jaeger & Cook, 1971; Singh, 1988). A typical experiment for the determination of friction force is shown in Figure 3. A rider of mass m is free to slide on a rigid flat base. The tangential force required to move the rider is applied through a spring AB by moving the point B slowly to the right at a velocity v. The force acting in the spring is equal to the friction force resisting the movement. m A B v Fig. 3. Schematic diagram of a typical friction experiment (after Byerlee, 1978) Rys. 3. Schemat typowego eksperymentu dla określenia siły tarcia (wg Byerlee ego, 1978)

395 If the force in the spring is plotted as a function of the displacement of point A then the dependence between the friction force and displacement of the rider is obtained (Fig. 4). At the beginning the dependence is linear up to point C, after which a non-linear dependence occurs in the range C D. This indicates that there is a relative displacement between the rider and the flat base or that the rider or the flat base is deforming nonelastically. D F G C Force E Displacement Fig. 4. Schematic diagram of the frictional force plotted as a function of the displacement of the rider (after Byerlee, 1978) Rys. 4. Wykres siły tarcia jako funkcji przemieszczenia suwaka (wg Byerlee ego, 1978) At point D a maximum force is reached and then a displacement of the rider may follow in a jerky or smooth way. In the case of a jerky movement of the rider, the force in the spring suddenly drops to point E and then increases to point F at which a sudden slip takes place once more. This sudden jerky type of movement is known as a stick-slip (Brace & Byerlee, 1966; Byerlee & Brace, 1968; Byerlee, 1970; Demirel & Granick, 1996; Paterson, 1978). The alternative way of movement is a stable sliding when the movement between the rider and flat base takes place smoothly and the force displacement curve is continuous as shown schematically by the dotted line in Figure 4. The force at points C, D and G is known as the initial, maximum and residual friction force, respectively (Byerlee, 1978). Experimental investigations carried out for different types of rocks have shown that the dependence between the shear stress and the normal stress has the form of a straight line: τ = 0.85σ n for the range of normal stress 5 MPa σ n 100 MPa (Byerlee, 1978). This dependence was applied for determining the friction coefficient for the initial and the maximum friction of different types of rocks. For the initial friction a wide scatter

396 of experimental points relative to the straight line was obtained (Byerlee, 1978). This was explained by the difficulty in precisely defining when a movement between the sliding surfaces commences (point C in Figure 4). However, for the maximum friction the straight line: τ = 0.85σ n well approximates the experimental data obtained for different rock types (Fig. 5). 100 90 80 70 EXPLANATION Symbol Reference Rock type 3 6F 6S 7 14 16 20 26 9 13 Limestone, Gabbro Weber Sandstone, faulted Weber Sandstone, saw cut Greywacke, Sandstone, Quartzite, Granite Granite, Gabbro Plaster in joint of Quartz Monzonite Quartz Monzonite joints Granite Granodiorite Gneiss and Mylonite L L L Q = 0.85 n G Q, [MPa] Shear stress 60 50 L L Q G G SS Q G G 30 G Q GW 20 G G 10 SS GW G SS G GW Q G SS SS SS SS 0 0 10 20 30 50 60 70 80 90 100 Normal stress n, [MPa] Fig. 5. Shear stress plotted as function of normal stress at the maximum friction for a variety of rock types at normal stresses up to 100 MPa (after Byerlee, 1978) Rys. 5. Naprężenie ścinające jako funkcja naprężenia normalnego przy maksymalnym tarciu dla różnych typów skał przy naprężeniu normalnym do 100 MPa (wg Byerlee ego, 1978)

397 3. Determination of the normal and shear stresses at the shear plane of a rock sample Brittle rocks subjected to triaxial compression undergo a failure by a single shear fracture (Gustkiewicz & Nowakowski, 2004). The angle of failure, i.e. the angle between the shear plane and the direction of stress σ 1 (the axis of a sample) increases with an increase of the confining pressure from the value of a dozen or so degrees for the samples compressed uniaxially (p = 0) to more than forty degrees for the samples compressed at the confining pressure p 60 MPa (Kwaśniewski, 1983). The normal stress and the shear stress at the shear plane are defined by the following formulae (Kwaśniewski, 2002): 1 1 n ( 1 3) ( 1 3) cos2 2 2 1 ( 1 3) sin 2 2 (1) where: σ 1 the major principal stress σ 3 = σ 2 = p confining pressure θ the acute angle between the direction of major principal stress and the shear plane (the fracture angle). When the values of the normal stress σ n and shear stress τ at the shear plane are known, the coefficient of internal friction µ (and the angle of internal friction φ) can be calculated: µ = tg φ = τ/σ n for the maximum friction resisting a slip in a solid rock. For determining the coefficient of friction in the fractured rock, the residual stress is substituted into the formula (1) instead of the critical stress. Then, the calculated coefficient of friction and the angle of friction correspond to the residual friction (Byerlee, 1978). When the values of the normal stress σ n and of the fracture angle θ are known, then the corresponding confining pressure p can be calculated from the formula (Shimada, 2000): 2 1 p n 1( 1 cos2 ) 1 cos2 (2) 2

398 4. Determination of the coefficients of friction for the maximum and residual forces of friction During the years 1999 2002 studies on the influence of confining pressure on the stress-strain properties of typical Carboniferous rocks from the Upper Silesian Coal Basin were conducted. In some tests measurements of the fracture angle were performed on rock samples that faulted under triaxial compression conditions at a confining pressure in the range of 0 50 MPa (Krzysztoń et al., 2002). The triaxial compression tests of rock samples carried out at a longitudinal strain rate equal to 10 4 s 1 have shown that the average fracture angle obtained for different samples of the same type of rock in the range of confining pressure 0 50 MPa undergoes changes from 13 to 28 for sandstones, from 23 to 28 for claystone and from 8 to 26 for coal (Wadas, 2002). The calculations of the coefficients of friction for solid rocks and for fractured rocks were conducted for the Carboniferous rock samples from the Upper Silesian Coal Basin that had been made from the following rock cores: core no. 2 medium-grained sandstone (Coal Mine Piast, roof of seam 207) for p = 0: σ cr = 21.6 MPa, σ res = 1.42 MPa core no. 8 fine-grained sandstone (Coal Mine Polska Wirek, roof of seam 502/II) for p = 0: σ cr = 123.2 MPa, σ res = 2.72 MPa core no. 13 claystone (Coal Mine Polska Wirek, roof of seam 502/II) for p = 0: σ cr = 30.7 MPa, σ res = 4.98 MPa core no. 14 semi-bright coal (Coal Mine Polska Wirek, seam 502/I) for p = 0: σ cr = 14.5 MPa, σ res = 0.11 MPa core no. 15 semi-bright and dull coal (Coal Mine Polska Wirek, seam 502/2) for p = 0: σ cr = 24.4 MPa, σ res = 0.71 MPa core no. 16 dull coal (Coal Mine Niwka Modrzejów, seam 510) for p = 0: σ cr = 48.0 MPa, σ res = 0. MPa core no. 17 semi-bright coal (Coal Mine Grodziec, seam 816) for p = 0: σ cr = 36.7 MPa, σ res = 0.19 MPa. These critical stress and residual stress values are the mean values obtained from the investigations of 4 6 samples of the same rock performed in a uniaxial state of stress (p = 0). The shear angles of the investigated rock samples increased with an increase of confining pressure, assuming similar values for the particular rocks in the range of confining pressure p = 10 50 MPa. Therefore, the calculations for all the investigated rocks were carried out for the same shear angle dependent only on the confining pressure simulating the depth of rock deposition.

399 For the applied values of confining pressure p the calculation of the coefficients of friction for maximum and residual force were conducted for the following values of shear angles θ: p = 5 MPa θ = 10 (Table 1) p = 10 MPa θ = 18 (Table 2) p = 20 MPa θ = 22 (Table 3) p = 30 MPa θ = 26 (Table 4) p = 50 MPa θ = 27 (Table 5). Calculation of the friction coefficients in compact (µ) and fractured (µ res ) samples for p = 5 MPa, θ = 10 Obliczanie współczynników tarcia w próbce zwięzłej (µ) i spękanej (µ res ) dla p = 5 MPa, θ = 10 TABLE 1 TABLICA 1 p = 5 MPa, θ = 10, sin2θ = 0.342, cos2θ = 0.9 Rock σ 1 σ 3 = p 1 2 (σ 1 σ 3 ) 1 2 (σ 1 + σ 3 ) τ σ n core Rock type No. MPa MPa MPa MPa MPa MPa µ φ 2 sandstone * 8 sandstone 157.00 76.00 81.00 25.99 9.56 2.719 69 48 13 claystone 60.30 27.65 32.65 9.46 6.66 1.420 54 50 5 14 coal 71.90 33.45 38.45 11.44 7.01 1.633 58 30 15 coal 63.90 29.45 34.45 10.07 6.77 1.488 56 06 16 coal 74.00 6 34.00.00 11.63 8.04 1.446 55 20 17 coal Rock σ res σ 3 = p 1 2 (σ res σ 3 ) 1 2 (σ res + σ 3 ) τ res σ n res core Rock type No. MPa MPa MPa MPa MPa MPa µ res φ res 2 sandstone 8 sandstone 57.79 26.39 31.39 9.03 6.58 1.371 53 53 13 claystone 33.92 14.46 19.46 4.94 5.87 0.843 07 5 14 coal.71 17.85 22.85 6.11 6.07 1.010 45 09 15 coal 31.91 13.45 18.45 4.60 5.81 0.792 38 23 16 coal.80 6 17. 23. 5.95 7.04 0.845 11 17 coal *) Blank places in the tables denote that the rock was not investigated at the given confining pressure.

0 Calculation of the friction coefficients in compact (µ) and fractured (µ res ) samples for p = 10 MPa, θ = 18 Obliczanie współczynników tarcia w próbce zwięzłej (µ) i spękanej (µ res ) dla p = 10 MPa, θ = 18 TABLE 2 TABLICA 2 p = 10 MPa, θ = 18, sin2θ = 0.588, cos2θ = 0.809 Rock σ 1 σ 3 = p 1 2 (σ 1 σ 3 ) 1 2 (σ 1 + σ 3 ) τ σ n core Rock type No. MPa MPa MPa MPa MPa MPa µ φ 2 sandstone 74.90 32.45 42.45 19.08 16.20 1.178 49 8 sandstone 158.30 74.15 84.15 43.60 24.16 1.804 61 00 13 claystone 84.80 37. 47. 21.99 17.14 1.283 52 03 14 coal 82.10 10 36.05 46.05 21.20 16.89 1.255 51 27 15 coal 81.60 35.80 45.80 21.05 16.84 1.250 51 20 16 coal 91.30.65 50.65 23.90 17.76 1.345 53 22 17 coal 103. 46.70 56.70 27.46 18.92 1.451 55 26 Rock σ res σ 3 = p 1 2 (σ res σ 3 ) 1 2 (σ res + σ 3 ) τ res σ n res core Rock type No. MPa MPa MPa MPa MPa MPa µ res φ res 2 sandstone 58.87 24.43 34.43 14.37 14.67 0.980 44 24 8 sandstone 64.17 27.08 37.08 15.93 15.17 1.050 46 23 13 claystone 36.69 13.34 23.34 7.85 12.55 0.625 32 00 14 coal 57.95 10 23.97 33.97 14.10 14.58 0.967 44 02 15 coal 33.41 11.70 21.70 6.88 12.24 0.562 29 21 16 coal 68.41 29.20 39.20 17.17 15.58 1.102 47 47 17 coal 56.76 23.88 33.38 14.04 14.06 0.999 44 57 Calculation of the friction coefficients in compact (µ) and fractured (µ res ) samples for p = 20 MPa, θ = 22 Obliczanie współczynników tarcia w próbce zwięzłej (µ) i spękanej (µ res ) dla p = 20 MPa, θ = 22 TABLE 3 TABLICA 3 p = 20 MPa, θ = 22, sin2θ = 0.695, cos2θ = 0.719 Rock σ 1 σ 3 = p 1 2 (σ 1 σ 3 ) 1 2 (σ 1 + σ 3 ) τ σ n core Rock type No. MPa MPa MPa MPa MPa MPa µ φ 2 sandstone 104.30 42.15 62.15 29.29 31.84 0.920 42 36 8 sandstone 219.50 99.75 119.75 69.33 48.03 1.443 55 17 13 claystone 107.80 43.90 63.90 30.51 32.33 0.943 43 20 14 coal 125.20 20 52.60 72.60 36.56 34.78 1.051 46 25 15 coal 122.20 51.10 71.10 35.51 34.36 1.033 45 57 16 coal 120.10 50.05 70.05 34.78 34.06 1.021 45 35 17 coal 132.20 56.10 76.10 38.99 35.76 1.090 47 28

1 Rock σ res σ 3 = p 1 2 (σ res σ 3 ) 1 2 (σ res + σ 3 ) τ res σ n res core Rock type No. MPa MPa MPa MPa MPa MPa µ res φ res 2 sandstone 84.95 32.47 52.47 22.57 29.12 0.775 37 46 8 sandstone 117.80 48.90 68.90 33.98 33.74 1.007 45 12 13 claystone 78.57 29.38 49.28 20.35 28.23 0.721 35 47 14 coal 90.97 20 35.48 55.48 24.66 29.97 0.823 39 26 15 coal 93.51 36.75 56.75 25.54 30.33 0.842 06 16 coal 80.16 30.08 50.08 20.91 28.45 0.735 36 18 17 coal 91.19 35.59 55.59 24.74 30.00 0.825 39 30 Calculation of the friction coefficients in compact (µ) and fractured (µ res ) samples for p = 30 MPa, θ = 26 Obliczanie współczynników tarcia w próbce zwięzłej (µ) i spękanej (µ res ) dla p = 30 MPa, θ = 26 TABLE 4 TABLICA 4 p = 30 MPa, θ = 26, sin2θ = 0.788, cos2θ = 0.616 Rock σ 1 σ 3 = p 1 2 (σ 1 σ 3 ) 1 2 (σ 1 + σ 3 ) τ σ n core Rock type No. MPa MPa MPa MPa MPa MPa µ φ 2 sandstone 121.70 45.85 75.85 36.13 47.61 0.759 37 11 8 sandstone 310.00 1.00 170.00 110.3 83.76 1.317 52 47 13 claystone 143.10 55.00 86.55 43.34 52.67 0.823 39 27 14 coal 141.30 30 55.65 85.65 43.85 51.37 0.854 29 15 coal 124.30 47.15 77.15 37.15 48.11 0.772 37 16 coal 17 coal 152.90 61.45 91.45 48.42 53.60 0.903 42 05 Rock σ res σ 3 = p 1 2 (σ res σ 3 ) 1 2 (σ res + σ 3 ) τ res σ n res core Rock type No. MPa MPa MPa MPa MPa MPa µ res φ res 2 sandstone 109.20 39.60 69.60 31.20 45.21 0.690 34 37 8 sandstone 127.72 48.86 78.86 38.50 48.76 0.790 38 17 13 claystone 111.80.90 70.90 32.23 45.71 0.705 35 11 14 coal 120.58 30 45.29 75.29 35.69 47.39 0.753 36 58 15 coal 83.04 26.52 56.52 20.90.18 0.520 27 28 16 coal 17 coal 102.07 36.03 66.03 28. 43.84 0.648 32 55

2 Calculation of the friction coefficients in compact µ and fractured µ res samples for p = 50 MPa, θ = 27 TABLE 5 TABLICA 5 Obliczanie współczynników tarcia w próbce zwięzłej µ i spękanej µ res dla p = 50 MPa, θ = 27 p = 50 MPa, θ = 27, sin2θ = 0.809, cos2θ = 0.588 Rock σ 1 σ 3 = p 1 2 (σ 1 σ 3 ) 1 2 (σ 1 + σ 3 ) τ σ n core Rock type No. MPa MPa MPa MPa MPa MPa µ φ 2 sandstone 8 sandstone 386.70 168.35 218.35 136.2 119.4 1.141 48 46 13 claystone 198.30 74.15 124.15 59.99 80.55 0.745 36 14 coal 165.70 50 57.85 107.85 46.80 73.83 0.634 32 22 15 coal 147.50 48.75 98.75 28.66 70.08 0.9 22 14 16 coal 17 coal Rock σ res σ 3 = p 1 2 (σ res + σ 3 ) 1 2 (σ res + σ 3 ) τ res σ n res core Rock type No. MPa MPa MPa MPa MPa MPa µ res φ res 2 sandstone 8 sandstone 221.51 85.75 135.75 69.38 85.33 0.813 39 06 13 claystone 159.46 54.73 104.73 44.28 72.55 0.610 31 23 14 coal 148.49 50 49.24 99.24 39.84 70.29 0.567 29 32 15 coal 127.71 38.85 88.85 31.43 66.01 0.476 25 27 16 coal 17 coal The obtained relationships between the shear stress and the normal stress of the investigated rock samples are indicated alternately by circles (for p = 5, 20 and 50 MPa) and crosses (for p = 10 and 30 MPa) in Figures 6 and 7. The sets of points for the successive pressures create the straight lines, the slope of which decreases with an increase of the confining pressure. The solid line represents the relationship τ = 0.85σ n known as Byerlee s law (Shimada, 2000). On the whole, the Byerlee equation approximates the experimental results for solid rocks well (Fig. 6). However, it overestimates the results for fractured rocks under conditions where the normal residual stresses are higher than MPa. The analysis of the results reveals that the friction coefficients depend on the type of rock and that they decrease with an increase of the confining pressure. The coefficients of friction for fractured rocks, however, are smaller than those for solid rocks. The same results have been obtained when determining the angles of internal friction for different types of rocks from the Upper Silesian Coal Basin on the basis of tangents to the parabolic envelope of Mohr s circles (Sanetra, 2004). However, the assumption of the same relationship for all types of rocks in the range of normal stress investigated is justifiable considering practical applications.

3 80 60 = 0.85 n 20 0 0 20 60 80 100 120 n [MPa] Fig. 6. Shear stress plotted as a function of normal stress in a shear plane inclined at an angle conditional upon confining pressure in solid samples of different rock types Rys. 6. Naprężenie ścinające jako funkcja naprężenia normalnego w płaszczyźnie ścinania nachylonej pod kątem uwarunkowanym ciśnieniem okólnym w zwięzłych próbkach różnych typów skał 100 80 60 = 0.85 n 20 0 0 10 20 30 50 60 70 80 90 n_res [MPa] Fig. 7. Shear stress plotted as a function of normal stress in a shear plane inclined at an angle conditional upon confining pressure in fractured samples of different rock types Rys. 7. Naprężenie ścinające jako funkcja naprężenia normalnego w płaszczyźnie ścinania nachylonej pod kątem uwarunkowanym ciśnieniem okólnym w spękanych próbkach różnych typów skał 5. Sliding friction in the failure plane At the limiting shear stress, the process of post-critical failure along a shear plane takes place (Jaeger, 1969). This process is complicated and depends on many factors, the most important of which are the surface conditions of macro-fracture and the pressure

4 normal to the slip plane (Rabinowicz, 1965). The surface conditions change in the failure process as the roughness of sliding surfaces in general decreases during displacement, which has an influence on the value of the kinetic coefficient of friction. The kinetic coefficient of friction, called the dynamic coefficient of friction or the coefficient of sliding friction (Scholz et al., 1972), has a smaller value than the static coefficient of friction determined at the maximum friction. A variation in frictional resistance along the sliding surfaces has an influence on their relative motion. It is assumed that rough surfaces are composed of asperities and the resistance to sliding is determined by the strength of these asperities (Byerlee, 1967). A stick-slip motion occurs due to high asperities on sliding surfaces which momentarily lock together, then suddenly release and slide forward, and then lock again (Byerlee & Brace, 1968). During sliding the shearing stress drops; during locking the stress increases. The stress drop in a jerky motion is accompanied by a sudden release of strain energy in rock material that is already fractured. The released energy can be the cause of an outburst of the coal material and an occurrence of a bump. In the conventional triaxial compression of cylindrical samples, the stress drops and rises can be observed on the post-critical stress-strain characteristic. Considering the cyclical character of these stress variations, the recurrence of a coal material bump phenomenon can also be predicted. Many years of investigations on the nature of sliding have shown that a stick-slip sliding occurs in rocks with low porosity during relative motion of smooth as well as rough fracture surfaces. The increase in the stress normal to the sliding plane enhances the possibility of stick-slip occurring and the increase in stiffness of the testing machine reduces the drop of stress in stick-slip and the duration of sliding. On the other hand, an increase in temperature and in the displacement rate causes the transition of stick-slip into stable sliding (Blanpied et al., 1992; Brace & Martin, 1968; Jaeger, 1969, 1971; Paterson, 1978). When the stress in a rock reaches the limiting shear strength value, then the fracture occurs followed by sliding and the friction force that opposes the sliding. The friction force depends on the normal stress and on the coefficient of friction. Byerlee (1978) showed that at low normal stress the coefficient of friction is strongly dependent on surface roughness, however at high normal stress the coefficient of friction is nearly independent of the rock type. The maximum friction is a linear function of the normal stress in the form of Byerlee s law (1975) which is applied to all rock types (Shimada, 2000). This law determines the frictional shear strength. The fracture shear strength as well as the frictional shear strength are functions of the confining pressure. 6. Fracture shear strength and frictional shear strength The compression strength of brittle rocks increases with an increase of confining pressure according to the curve with the convex directed to the differential stress axis.

5 The frictional resistance opposite to sliding is the rectilinear function of normal stress, the slope of which depends on the coefficient of internal friction. In the case of natural sliding the increase of confining pressure has an influence on the form of the post-critical curve, which is a continuous curve at small values of confining pressure and reveals a stick-slip at high values of confining pressure. It is important to determine the normal stress at which the transition from stable sliding to stick-slip takes place for each type of rock. In order to ascertain this, the fracture and friction processes should be considered (Lyakhovsky, 2004). Figure 8 shows the shear stress at fracture (fracture shear strength) for a granite rock and the frictional shear stress for sliding (frictional shear strength) as a function of normal stress across the sliding plane (Byerlee, 1967). These two curves intersect when the normal stress is approximately 1750 MPa which, taking into consideration the angle of shear plane, corresponds to a confining pressure of approximately 1000 MPa. This indicates that at about 1000 MPa pressure the axial stress required to create a fracture surface in a granite is equal to the axial stress required to cause sliding on the newly created surface. The intersection point of fracture shear strength of the investigated rock with the frictional shear strength determines the normal stress (confining pressure) at which brittle ductile transition occurs. The resistance caused by internal friction during the deformation of rock material subjected to triaxial compression has an influence on the shape of pre-critical stress-strain characteristic. In Figure 9 the compressive strength of several low-porosity rocks is shown as a function of confining pressure. The frictional strength, according to Byerlee,, [GPa] Shear stress 1.0 0.5 Fracture shear strength Frictional shear strength 0.5 1.0 1.5 Normal stress n, [GPa] Fig. 8. Fracture shear strength and frictional shear strength versus normal stress for Westerly granite (after Byerlee, 1967) Rys. 8. Wytrzymałość na ścinanie i tarciowa wytrzymałość ścinająca jako funkcja naprężenia normalnego dla granitu Westerly (wg Byerlee ego, 1967)

6 is indicated by the dashed line. In the part of the graph representing the range of a low confining pressure, the compressive strength increases non-linearly with increasing confining pressure. However, in the part of the graph representing the range of a high confining pressure, the compressive strength increases linearly with increasing confining pressure. Changes in the rate of increase of compressive strength for all rocks occur close to the intersection points of frictional strength with the compressive strength of a given rock. Experimental investigations on shear strength and on frictional strength have led to the friction hypothesis for brittle-ductile transition. According to this hypothesis the rocks become ductile at the confining pressure at which the compressive strength is equal to the shear strength (Shimada, 2000; Kwaśniewski, 2002). 5 Compressive strength 1 3, [GPa] 4 3 2 1 Frictional strength Akaishi eclogite Horoman dunite Man-nari granite Murotomisaki gabbro 0 0 1 2 3 Confining pressure 3 Fig. 9. Compressive strength of low porosity dry silicate rocks at room temperature (Shimada & Cho, 1990). The dashed line is the frictional strength from Byerlee (1978) (after Shimada, 2000) Rys. 9. Wytrzymałość na ściskanie suchych skał krzemianowych o niskiej porowatości w temperaturze pokojowej (Shimada & Cho, 1990). Przerywana linia oznacza wytrzymałość tarciową wg Byerlee ego (1978) (wg Shimady, 2000)

7 7. General conclusion In experimental studies on the displacement of rocks under loading conditions, the initial, maximum and residual friction should be taken into consideration when determining the influence of the friction force that opposes motion (Byerlee, 1978). In triaxial conventional compression of rocks the initial resistance of friction, which is a linear function of confining pressure, has an influence on the shape of the pre-critical stress-strain characteristic (Fig. 9). However, the maximum resistance of friction determines the mode of failure of rock samples in the slip plane. When the maximum friction resistance is smaller than the limiting shear strength of the investigated sample, stick-slip may occur which is characterised by cyclical drops and rises of shear stress. When friction along the fracture surface exceeds the shear strength of the rock, the brittle ductile transition occurs. The conditions in which the stick-slip phenomenon can occur and the ways to eliminate it have been presented on the basis of the results of foreign investigations. This is important in the practical application for the prediction and prevention of rock tremors. This research work was performed as part of the statutory investigations no. 11.11.100.588, financed by the Committee for Scientific Research. REFERENCES Blanpied M.L., Lockner D.A., Byerlee J.D., 1992. An earthquake mechanism based on rapid sealing of faults. Nature, Vol. 358, pp. 574-576. Brace W.F., 1972. Laboratory studies of stick-slip and their application to earthquakes. Tectonophysics, Vol. 14, No. 3/4, pp. 189-200. Brace W.F., Byerlee J.D., 1966. Stick-slip as a mechanism for earthquakes. Science, Vol. 153, No. 3739, pp. 990-992. Brace W.R., Martin R.J., 1968. A test of the law of effective stress for crystalline rocks of low porosity. Int. J. Rock Mech. Min. Sci., Vol. 5, pp. 415-426. Byerlee J.D., 1967. Frictional characteristics of granite under high confining pressure. J. Geophys. Res., Vol. 72, No. 14, pp. 3630-3648. Byerlee J.D., 1970. The mechanics of stick-slip. Tectonophysics, Vol. 9, pp. 475-486. Byerlee J.D., 1975. The fracture strength and frictional strength of Weber sandstone. Int. J. Rock Mech. Min. Sci., Vol. 12, pp. 1-4. Byerlee J.D., 1978. Friction of rocks. Pure Appl. Geophys. Vol. 116, pp. 615-626. Byerlee J.D., Brace W.F., 1968. Stick slip, stable sliding and earthquakes effect of rock type, pressure, strain rate and stiffness. J. Geophys. Res., Vol. 73, No. 18, pp. 6031-6037. Demirel A.L., Granick S., 1996. Friction fluctuations and friction memory in stick-slip motion. Physical Review Letters, Vol. 77, No. 21, pp. 4330-4333. Dyskin A.V., Galybin A.N., Brady B.H., 1998. Catastrophic sliding over a fault caused by accumulation of dilation zones. Mechanics of Jointed and Faulted Rock (MJFR-3), H.-P. Rossmanith (ed.). Balkema, Rotterdam, pp. 69-74. Earthquake Hazards Program, 2002, U.S. Geological Survey, http://quake.wr.usgs.gov/research/physics/lsab/ primer.html

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