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

Transkrypt

1 Arch. Min. Sci., Vol. 52 (2007), No 3, p MAREK KWAŚNIEWSKI* MECHANICAL BEHAVIOUR OF ROCKS UNDER TRUE TRIAXIAL COMPRESSION CONDITIONS VOLUMETRIC STRAIN AND DILATANCY ZACHOWANIE SIĘ SKAŁ W WARUNKACH PRAWDZIWIE TRÓJOSIOWEGO ŚCISKANIA ODKSZTAŁCENIE OBJĘTOŚCIOWE I DYLATANCJA It has long been accepted that dilatancy is a phenomenon that can be treated as an important precursor of the brittle failure of rocks. Dilatancy is understood as an inelastic increase in volume during deformation under applied differential stress and is interpreted as being due to microcracking within the rock with a concomitant increase in void space. Dilatancy in rocks is of considerable practical importance since it is associated with possible premonitory signs of earthquakes, mining-induced rockbursts and mine roof collapses. The fundamentals of dilatancy in rocks are introduced at the beginning of the present paper. These are followed by a brief explanation of what the dilatancy-related precursory phenomena to brittle failure are. The introductory part is completed by a review of the studies of dilatancy under true triaxial compression conditions to-date. The main part of the paper includes a presentation of the results of true triaxial compression tests on samples of a foliated, anisotropic crystalline schist and a Coal-Measure, medium- to coarse-grained sandstone. The main aim of the tests was to reveal the effect of, separately, confining pressure and intermediate principal stress on the volumetric deformation mode and the dilatant behaviour of rocks. Results of the experimental studies show that the effect of dilatancy is suppressed or inhibited by the intermediate principal stress. While important negative, dilatant strain corresponds to the peak differential stress under axisymmetric stress conditions, under true triaxial compression, where intermediate principal stress is much higher than the minimum stress, the rock material apparently undergoes microcracking to a much lesser extent and behaves in an increasingly brittle manner. Faulting of rock samples in the post-peak region is much more violent and is accompanied by a strong acoustic effect associated with a release of elastic strain energy. Keywords: Anisotropy, dilatancy, true triaxial compression, volumetric strain Dylatancja niesprężysty wzrost objętości w warunkach działania przyłożonych naprężeń różnicowych jest zjawiskiem poprzedzającym kruche zniszczenie ośrodka skalnego, może zwiastować zbliżające * FACULTY OF MINING AND GEOLOGY, SILESIAN UNIVERSITY OF TECHNOLOGY, AKADEMICKA STREET 2, GLIWICE, POLAND

2 410 się trzęsienie ziemi lub wstrząs górniczy. Pojawiając się przy naprężeniach bliskich granicy wytrzymałości dylatancja stanowi także o wielu innych obserwowanych w laboratorium i in situ efektach ostrzegawczych poprzedzających zniszczenie. Należą do nich m.in. spadek ciśnienia porowego (i towarzyszący mu efekt wzmocnienia dylatancyjnego), anomalne zmiany aktywności sejsmoakustycznej i sejsmicznej, zmiany prędkości przebiegu fal sprężystych, zmiany siły ciężkości, zmiany pola magnetycznego i elektrycznego, obniżenie poziomu wód gruntowych, wzmożona emisja radonu i innych gazów. Pionierskie badania nad efektem dylatancji w skałach, prowadzone w latach 1960-tych i 1970-tych, stymulowane były hipotezą o pierwszorzędnej roli tego efektu w mechanizmie trzęsień ziemi. Powstało kilka modeli dylatancyjnych służących objaśnieniu obserwowanych w naturze zjawisk poprzedzających kruche pękanie mas skalnych. Jednak praktyczne wykorzystanie tych modeli jest ograniczone ze względu na to, że wyniki pomiarów polowych jest bardzo trudno interpretować, szczególnie jeśli ma się do czynienia ze środowiskiem skalnym wysoce niejednorodnym i/lub anizotropowym, poddanym działaniu sił rozłożonych nierównomiernie i zmieniających się. Tak więc dalsze badania są niezbędne, żeby można było lepiej poznać właściwości odkształceniowe i zrozumieć zachowanie się skał w warunkach złożonego stanu naprężenia. Szczególnie mało wiadomo jest o dylatancji w skałach poddanych tzw. prawdziwie trójosiowemu stanowi naprężeń ściskających, a więc stanowi gdzie wszystkie trzy składowe główne tensora naprężenia są różne. Badania nad odkształceniami objętościowymi skał w takich warunkach prowadzono do tej pory jedynie w Instytucie Trzęsień Ziemi Uniwersytetu Tokijskiego (K. Mogi), w Japońskim Instytucie Geologicznym w Tsukubie (M. Takahashi i H. Koide) i w Uniwersytecie stanu Wisconsin w Madison (C. Chang i B. Haimson). Badania te omówione szczegółowo w rozdziale 3 tego artykułu wykazały, że liniowe odkształcenie poprzeczne w kierunku działania pośredniego naprężenia głównego (σ 2 ) maleje ze wzrostem tego naprężenia, a efekt dylatancji ulega osłabieniu (rys. 4, 6 i 7). Te godne najwyższego uznania osiągnięcia badawcze daleko jednak nie wyczerpują całości zagadnienia. Dotyczą przede wszystkim progu dylatancji względnej, podczas gdy dla prognozowania zniszczenia skał znacznie ważniejsza jest znajomość progu dylatancji właściwej (rys. 1). Poza tym wciąż bardzo niewiele jest danych ilościowych o odkształceniach objętościowych na obu progach dylatancji i na granicy wytrzymałości. Brakuje ponadto podstawowych przynajmniej danych na temat tego, jaką część odkształceń objętościowych na różnych poziomach naprężenia w stanie przed- i pokrytycznym stanowią odkształcenia sprężyste, a jaką odkształcenia nieodwracalne, trwałe. Próby na prawdziwe trójosiowe ściskanie ciągle jeszcze nastręczają wielu trudności technicznych, nie jest też łatwo mierzyć odkształcenia objętościowe. To właśnie dlatego próby obciążania i odciążania próbek skalnych w warunkach prawdziwego trójosiowego ściskania w ogóle jeszcze do tej pory nie były wykonywane, zresztą nie tylko w Polsce, gdzie nie ma odpowiedniej aparatury, ale i w tych zagranicznych ośrodkach badawczych, w których aparaty do badań próbek skalnych na prawdziwe trójosiowe ściskanie zostały zbudowane. W rozdziale 4 tego artykułu przedstawione są wyniki badań nad efektem dylatancji w próbkach skały anizotropowej metamorficznego łupku krystalicznego Chichibu o wyraźnej, gęstej foliacji. Prostopadłościenne próbki tej skały (rys. 10 i 11), o wymiarach 15 mm 15 mm 30 mm, badane były za pomocą aparatu Mogiego (rys. 9) na prawdziwe trójosiowe ściskanie przy naprężeniu najmniejszym (σ x ) równym 50 MPa i naprężeniach pośrednich (σ y ) sięgających 168 MPa. Naprężenia te były zorientowane względem płaszczyzn foliacji pod takim samym kątem 45 ; kierunek naprężenia największego (σ z ) tworzył z płaszczyznami foliacji kąt 30 (rys. 12). Podczas gdy w warunkach tzw. konwencjonalnego trójosiowego ściskania (σ x = σ y = 50 MPa) odkształcenia poprzeczne ε x i ε y tej skały były niemal takie same (tak, jak w przypadku skał izotropowych), w warunkach prawdziwie trójosiowego ściskania odkształcenia poprzeczne w kierunku pośredniego naprężenia pośredniego (ε y ) były znacznie mniejsze od odkształceń ε x. W rezultacie dylatancja wywoływana była głównie przez odkształcenie poprzeczne (duże wydłużenia) w kierunku naprężenia najmniejszego (rys. 14). W rozdziale 5 przedstawiono wyniki badań nad wpływem, oddzielnie, ciśnienia okólnego i pośredniego naprężenia głównego na właściwości objętościowego odkształcania się średnio-gruboziarnistego piaskowca karbońskiego z KWK Śląsk. Prostopadłościenne próbki tej skały (rys. 16), o wymiarach 35 mm 35 mm 70 mm, poddawano próbom obciążania do i odciążania od różnych poziomów naprężenia różnicowego (σ 1 σ 3 ) w stanie przed- i pokrytycznym w warunkach konwencjonalnego trójosiowego ściskania i prawdziwego trójosiowego ściskania za pomocą trójosiowej maszyny wytrzymałościowej Koidego i Takahashiego (rys. 15 i 17). Próby na konwencjonalne trójosiowe ściskanie przeprowadzono

3 411 przy ciśnieniu okólnym równym 12,5, 25,0 i 37,5 MPa. W próbach na prawdziwe trójosiowe ściskanie najmniejsze naprężenie główne (σ 3 ) wynosiło 25,0 MPa, a pośrednie naprężenie główne (σ 2 ) było 1,5, 2 i 2,5 razy większe od σ 3. Z przeprowadzonych badań wynika, że ciśnienie okólne silnie zahamowuje procesy dylatancyjne w próbkach badanych w warunkach konwencjonalnego trójosiowego ściskania; zwiększanie ciśnienia wywoływało rosnącą kompakcję materiału skalnego (rys. 21). Dylatancja był także znacząco tłumiona przez pośrednie naprężenie główne. Podczas gdy znaczne ujemne odkształcenie objętościowe odpowiadało granicznemu naprężeniu różnicowemu przy niskich pośrednich naprężeniach głównych, w warunkach działania wysokich naprężeń pośrednich materiał skalny ulegał uszkodzeniu w daleko mniejszym stopniu (rys. 22). W rezultacie, pękanie próbek skalnych następowało w bardzo wczesnym stadium pokrytycznym; było przy tym znacznie bardziej gwałtowne i towarzyszył mu silny efekt akustyczny powodowany wyzwoleniem się dużej ilości energii odkształcenia sprężystego. Wyniki badań nad niesprężystymi (trwałymi) odkształceniami objętościowymi piaskowca Śląsk w warunkach prawdziwie trójosiowego ściskania przedstawione zostaną w oddzielnej pracy. Słowa kluczowe: Anizotropia, dylatancja, odkształcenie objętościowe, prawdziwe trójosiowe ściskanie 1. Introduction In this paper the subject of the mechanical behaviour of rocks under true triaxial compression conditions will be addressed. In particular, volumetric strain and dilatancy, an effect of inelastic increase in volume during deformation under applied differential stress, will be discussed in detail. The phenomenon of dilatancy in rocks is of considerable practical importance. Much of pioneering research on dilatancy in the 1960s and 1970s was stimulated by the hypothesis of its central role in the mechanism of earthquakes. The phenomenon of dilatancy in rocks is also associated with possible premonitory signs for mining-induced rockbursts and mine collapses. In Polish deep coal and copper mines, which are greatly in danger of coal bumps and rock bursts, advanced microseismic, gravimetric and ground deformation measurements have been taken for decades to assess the danger of these dynamic manifestations of rock pressure. However, the measurement results are, in general, very difficult to interpret, especially if one has to deal with a highly inhomogeneous and anisotropic rock environment, subjected to non-uniform and changing stress fields. It is believed that a better understanding of the deformational properties and behaviour of rocks under triaxial stress conditions may lead to the alleviation of the problem. Dilatancy has long been known to be an important aspect of the deformational behaviour of granular aggregates such as soils. In such materials, the dilatancy can usually be attributed to changes in the packing of the grains during their relative movement. In hard rocks, however, dilatancy occurs due to microfracturing induced at the tips of preferentially oriented intragranular cracks and/or due to the microfracturing that occurs along the grain boundaries.

4 Basics of dilatancy If we consider the differential stress volumetric strain relationship under, say, axisymmetric compressive stress conditions and assume that compaction has a positive sign, in the case of an ideal, linear material, we would expect that strain increments would result in proportional increments in stress. As a consequence, the stress strain plot would follow a straight line (the dashed line in Figure 1), meaning if we assume that compaction has a positive sign that compaction increases, or the volume of the elastic body decreases, at a constant rate. In the case of rocks, however, a totally different behaviour pattern is normally observed. Very early, that is already at low differential stress levels, the stress strain plot starts to deviate from the straight line. This stress level is commonly referred to as the onset of dilatancy OD or C as was first proposed by Brace et al. (1963). Deviation of this stress strain curve from the straight line towards the stress axis means that the volumetric, compactant strain increases at a lower and lower rate. Finally, at a stress level termed the threshold of absolute dilatancy, the compactant (positive) volumetric strain attains a maximum value, that is the volume of the rock body reaches minimum and the incremental Poisson s ratio assumes a value equal to 0.5. With a further increase of the differential stress the volumetric strain normally starts to decrease and the volume of the rock becomes greater and greater, that is the rock undergoes absolute dilatancy. Let s denote the initial volume of a rock body as V 0. With an increase in volumetric strain under compressive applied stress conditions, the volume decreases to assume, at the threshold of absolute dilatancy (TD), the minimum value (V min ). Then, as the volumetric strain starts to decrease, the volume starts to increase. It may happen, although this is not an effect of a universal character, that the stress strain curve crosses the stress axis, which means that volume of the rock body returns to its initial value (V 0 ). Then, it may increase even further, becoming larger than in the initial or non-loaded state. To sum up: It should be noted that three stages can be distinguished in this plot: the stage of compaction, the stage of relative dilatancy and the stage of absolute dilatancy. These are separated by two characteristic stress levels: the onset of dilatancy or the threshold of relative dilatancy (OD) and the threshold of absolute dilatancy (TD). In general, dilatancy-related precursory phenomena include: crustal deformation (rapid uplift rates, anomalous ground tilts), changes in the density of the earth and in the gravity field, changes in the velocity of seismic waves, lowering of the groundwater level, emission of radon and other gases, changes in the magnetic field and electrical resistivity of the earth, and changes in pore pressure and flow rate. As it has already been mentioned, all these dilatancy-related premonitory anomalies are of considerable practical importance as possible precursors or forerunners of earthquakes, mining-induced rockbursts and mine roof falls.

5 413 Fig. 1. Differential stress volumetric strain curve with the threshold of relative dilatancy (OD), threshold of absolute dilatancy (TD) and ultimate strength (F) Rys. 1. Charakterystyka naprężenie różnicowe odkształcenie objętościowe z zaznaczeniem progu dylatancji względnej (OD), progu dylatancji bezwzględnej (TD) i granicy wytrzymałości (F) When discussing the features of the volumetric deformational behaviour of rocks under triaxial compression conditions, the axisymmetric and the true triaxial states of compressive stresses that are normally generated in solid cylindrical and cuboidal rock samples subjected to, respectively, conventional triaxial compression and true triaxial compression will be taken into account in this paper. In the CTC tests, the intermediate and the minimum principal stresses (σ 2 and σ 3, respectively) are equal to each other and equal to the confining pressure (Fig. 2a). In the TTC tests, σ 2 and σ 3 can be varied independently (Fig. 2b). 3. Previous work The early findings on the basic qualitative and quantitative features of dilatancy in rocks under uniaxial and axisymmetric triaxial stress conditions have been discussed and summarised by Paterson in his famous book on experimental rock deformation (Paterson, 1978; Paterson & Wong, 2005). Mogi was the first to study, in detail, the volumetric changes in rocks under true triaxial compression conditions. The development of an ingenious, novel triaxial compression technique in the late 1960s enabled him to measure all principal stresses and strains of rocks under general triaxial stress states (Mogi, 1971a, b; 1972a, b).

6 414 Fig. 2. Loading schemes in conventional triaxial compression (CTC) and true triaxial compression (TTC) tests on rock samples Rys. 2. Schematy obciążania próbek skalnych w próbach na konwencjonalne trójosiowe ściskanie (CTC) i prawdziwie trójosiowe ściskanie (TTC) Koide and Takahashi of the Geological Survey of Japan, having built a new, servocontrolled triaxial testing machine in the late 1970s, studied the effect of the intermediate principal stress on the strength and deformational behaviour of sedimentary rocks (Takahashi, 1984; Takahashi & Koide, 1989). A few years ago Haimson and Chang from the University of Wisconsin in Madison built another, highly sophisticated version of a Mogi-type triaxial apparatus and investigated samples of KTB amphibolite and Westerly granite under high σ 3 and σ 2 stress conditions (Chang & Haimson, 2000; Haimson and Chang, 2000; Chang, 2001). The major findings of these earlier investigations will be reviewed briefly below. In Figure 3, the effect of σ 2 on lateral strains is illustrated based on the triaxial compression tests conducted by Mogi on samples of Yamaguchi marble. The graph on the left shows the case of an axisymmetric stress state and the general triaxial stress state. In the case where σ 2 is equal to σ 3, the two principal lateral strains are equal and the rock deforms in a ductile manner. When σ 2 was much higher (105 MPa) than σ 3 (25 MPa), the corresponding lateral strain ε 2 was much lower than ε 3, and the rock behaved in a more brittle manner. The effect of the intermediate principal stress on the lateral strain is shown even more distinctly in Figure 4. It can be clearly seen from the graph presented in this figure that lower values of the lateral strain ε 2 correspond to higher values of the intermediate principal stress (in the range from 25 to 154 MPa).

7 415 Fig. 3. Differential stress axial strain curves and lateral strains of Yamaguchi marble samples tested under axisymmetric (left) and asymmetric (right) compressive stress conditions (Mogi, 1977, 2006) Rys. 3. Charakterystyki naprężenie różnicowe odkształcenie podłużne oraz odkształcenia poprzeczne próbek marmuru Yamaguchi w warunkach osiowo-symetrycznego (na lewo) i asymetrycznego (na prawo) stanu naprężeń ściskających (Mogi, 1977, 2006) Mogi showed that in true triaxial tests, as σ 2 is increased from σ 3 to σ 1, the dilatant lateral strain in the direction of σ 2 is progressively suppressed until eventually all dilatancy occurs by expansion in the σ 3 direction (Fig. 5a). This effect, called anisotropic dilatancy and explained as the opening of microcracks perpendicular to the minimum principal stress, inspired Mogi to propose a mechanism for the development of different types of faults, depending on the orientation of principal stresses (Fig. 5b). A reverse faulting occurs when the minimum principal stress is oriented in a vertical direction and lateral faults develop when the minimum principal stress is oriented horizontally. The significant influence of σ 2 on the dilatant behaviour of rocks was also observed by Takahashi and Koide (1989) in their true triaxial compression tests on Shirahama sandstone samples. They found that the onset of dilatancy (OD) increases and the dilatant volumetric strain decreases with an increase in the intermediate principal stress from 50 to 190 MPa (Fig. 6).

8 416 Fig. 4. Lateral strain as a function of axial strain for the minimum principal stress equal to 25 MPa and different values of the intermediate principal stress in Yamaguchi marble (Mogi, 1977, 2006) Rys. 4. Odkształcenie poprzeczne marmuru Yamaguchi w funkcji odkształcenia podłużnego dla najmniejszego naprężenia głównego równego 25 MPa i różnych wartości naprężenia pośredniego (Mogi, 1977, 2006) a) b) Fig. 5. Anisotropic dilatancy in rocks under true triaxial compression conditions: a Differential stress (σ 1 σ 3 ) versus axial strain (ε 1 ), lateral strain (ε 2 and ε 3 ) and volumetric (ε V ) strain characteristics of Inada granite, b Formation of a thrust fault (left) and a strike-slip fault (right) (Mogi, 1977, 2006) Rys. 5. Anizotropowa dylatancja w skałach w warunkach prawdziwie trójosiowego ściskania: a Charakterystyki naprężenie różnicowe (σ 1 σ 3 ) odkształcenie podłużne (ε 1 ), poprzeczne (ε 2 i ε 3 ) i objętościowe (ε V ) granitu Inada, b Formowanie się uskoku odwróconego (na lewo) i uskoku przesuwczego (na prawo) (Mogi, 1977, 2006)

9 417 A similar observation was made by Haimson and Chang (2000) for Westerly granite from Rhode Island. A gradual increase in the dilatancy onset (OD) relative to the limiting or peak major principal stress was observed as the intermediate principal stress was increased. The dilatancy onset was equal to 0.48 and 0.65 of the peak major principal stress for σ 2 equal to 60 MPa and 249 MPa, respectively. Fig. 6. Differential stress volumetric strain curves of Shirahama sandstone at a minimum principal stress (σ 3 ) equal to 50 MPa for different values of the intermediate principal stress (σ 2 ) (Koide & Takahashi, 1989) Rys. 6. Charakterystyki naprężenie różnicowe odkształcenie objętościowe piaskowca Shirahama przy najmniejszym naprężeniu głównym (σ 3 ) równym 50 MPa i różnych wartościach pośredniego naprężenia głównego (σ 2 ) (Koide i Takahashi, 1989) The strong effect of intermediate principal stress on the volumetric strain was also revealed by Chang and Haimson (2000) on the KTB amphibolite, a rock extracted from a superdeep hole drilled near Windischeschenbach in Northern Bavaria within the framework of the German Continental Deep Drilling Program. Dilatancy onset in this rock generally increased with a magnitude of σ 2. For instance, for σ 3 = 100 MPa the onset of dilatancy increased steadily from 0.45 of the peak major principal stress for σ 2 = σ 3 to 0.86 of the peak major principal stress for σ 2 = 600 MPa (Fig. 7). 4. Dilatant behaviour of an anisotropic rock under true triaxial compression conditions The main reason for undertaking studies on the mechanical behaviour of anisotropic rocks under true triaxial compression conditions was, that in axisymmetric tests it is not possible to determine the effect of either the magnitude or the orientation of the interme-

10 418 Fig. 7. Differential stress volumetric strain curves of KTB amphibolite at a minimum principal stress (σ 3 ) equal to 100 MPa for different values of the intermediate principal stress (σ 2 ) (Chang & Haimson, 2000) Rys. 7. Charakterystyki naprężenie różnicowe odkształcenie objętościowe amfibolitu KTB przy najmniejszym naprężeniu głównym (σ 3 ) równym 100 MPa i różnych wartościach pośredniego naprężenia głównego (σ 2 ) (Chang i Haimson, 2000) diate principal stress on the behaviour of anisotropic rocks (Fig. 8). True triaxial compression tests were, therefore, carried out with the aim of gaining a better understanding of the effect of intermediate principal stress on the ultimate strength, the deformability, the mode of failure and the pattern of faults developed in the samples of anisotropic rock (Kwaśniewski & Mogi, 1990, 2000; Kwaśniewski, 2002; Mogi, 2006). a) b) Fig. 8. Samples of a stratified, anisotropic rock tested under axisymmetric (a) and true triaxial (b) compressive stress conditions Rys. 8. Próbki warstwowanej, anizotropowej skały badanej w warunkach osiowo-symetrycznego (a) i prawdziwie trójosiowego (b) stanu naprężeń ściskających

11 419 The rock selected for testing was a metamorphic, homogeneous, green crystalline schist with a distinct, dense foliation. The rock came from Chichibu on Honshu island, Japan. Its uniaxial compressive strength in the direction perpendicular to the foliation planes (β = 90 ) was equal to about 150 MPa. The coefficient of anisotropy of the uniaxial compressive strength, being a ratio of the maximum and minimum values of the strength (with the minimum one corresponding to an angle β of 30 ) was equal to A plane view of the general triaxial compression apparatus that was designed by Mogi and built at the Earthquake Research Institute of the University of Tokyo in the late 1960s is shown in Figure 9. In the triaxial cell of the apparatus, three principal stresses can be generated independently in a rectangular prismatic rock sample. Solid pistons Fig. 9. Schematic diagram of a triaxial vessel of Mogi s apparatus for testing rock samples (right) under true triaxial stress conditions; 1 triaxial vessel, 2 rock sample, 3 vertical piston (σ 1 ), 4 lateral piston (σ 2 ), 5 load cells, 6 silicone oil, 7 to jack 1 (σ 1 ), 8 to jack 2 (σ 2 ), 9 fixed platen (after Mogi, 1971a, b, 2006) Rys. 9. Schemat komory trójosiowej aparatu Mogiego do badania prostopadłościennych próbek skalnych (na prawo) na prawdziwe trójosiowe ściskanie (wg Mogiego, 1971a, b, 2006)

12 420 apply loads to the sample in directions 1 and 2, while the minimum principal stress is induced in the sample by oil pressure. In Figure 9, the sample (2) is shown inside the triaxial vessel (1), subjected to loading by oil pressure, vertical piston (3) and horizontal piston (4). Load cells (5) that were used to measure loads acting in directions 1 and 2 can also be seen. The test specimen design is schematically shown in Figure 10. The sample was a rectangular prism 15 mm square by 30 mm long. Note the cylindrical steel end pieces on the top and bottom of the sample, and the lateral side pieces attached to the sample in the direction of the intermediate principal stress. The walls of the sample in the direction of minimum principal stress are covered with a silicone rubber jacket that prevents the intrusion of the confining pressure medium (oil) into the rock. Fig. 10. Method of measuring lateral strain ε 2 Rys. 10. Sposób pomiaru odkształceń poprzecznych ε 2 The axial strain and lateral strain in direction 2 were measured by strain gauges cemented onto these accessible walls of the sample. The lateral strain in the direction of the minimum principal stress was measured using a special displacement transducer. The transducer consists of strain gauges mounted on

13 421 curved metal sheets of which one end was fixed to the surface of the sample and the other end was fixed to the ring-shaped base which is movable in the vertical direction. The two photographs on the upper part of Figure 11 show a sample assembly in two mutually perpendicular directions. Note the foil strain gauges used to measure axial strain ε 1 and lateral strain ε 2. These two metal pins were used to attach the displacement transducer employed to measure lateral strain ε 3. On the right, a plug is shown, ready to be installed in the triaxial vessel, with a vertical piston and a sample attached to the piston. Fig. 11. Sample assembly ready for testing in Mogi s triaxial apparatus Rys. 11. Próbka przygotowana do badania w aparacie trójosiowym Mogiego Rock samples were cut from large blocks in such a way that their vertical axis was oriented perpendicularly to the planes of foliation (which is the direction of the highest compressive strength) or at angle β = 30 (which is the direction of the lowest compressive strength). In the latter case the samples were prepared so that the angle (ω) between the direction of the intermediate principal stress and the strike of the foliation planes was 0, 45 or 90 (Fig. 12).

14 422 Fig. 12. Samples of foliated Chichibu schist prepared for testing under true triaxial compression conditions; four different orientations of the maximum (σ z ), intermediate (σ y ) and minimum (σ x ) applied stress relative to the foliation planes Rys. 12. Próbki blaszkowatego łupku Chichibu przygotowane do badań na prawdziwe trójosiowe ściskanie; cztery różne orientacje największego (σ z ), pośredniego (σ y ) i najmniejszego (σ x ) naprężenia przyłożonego względem płaszczyzn foliacji Figure 13 shows the effect of intermediate principal stress on the ultimate strength of Chichibu schist for different orientations of the maximum and the intermediate principal stress to the weak planes of foliation. When the maximum principal stress was perpendicularly oriented to the planes of foliation, the effect of the intermediate stress was very similar to that observed in isotropic rocks: with an increase in σ 2, the ultimate strength markedly increased. However, when the stress σ 1 was oriented to the foliation planes at angle β equal to 30, the effect of σ 2 on strength was clearly dependent on the orientation of this stress to the strike of the foliation planes, that is on angle ω. And thus, it was negligible for ω = 0, moderate for ω = 45 and very strong for ω = 90. Strains in three directions were measured using samples of Chichibu schist with the vertical axis oriented at an angle of 30 and horizontal axes oriented at an angle of 45 to the foliation planes. As a result, the volumetric strain, including dilatancy, was determined. Under conventional triaxial compression conditions, where the confining pressure was equal to 50 MPa, lateral strains ε x and ε y were almost the same, as is normally the case in isotropic rocks. In the true triaxial compression state, however, the lateral strain in the direction of the intermediate principal stress (ε y ) became less and less important with the increase of this stress (Fig. 14). Consequently, the dilatancy was caused mainly by the ever increasing strain in the direction of the minimum principal stress (ε x ). This effect is very well illustrated by this plot which was obtained for σ x equal to 50 MPa and σ y equal to 168 MPa.

15 423 Fig. 13. Effect of the magnitude and orientation of the intermediate principal stress relative to the foliation planes on the ultimate strength of Chichibu schist Rys. 13. Wpływ wielkości i orientacji pośredniego naprężenia głównego względem płaszczyzn foliacji na wytrzymałość graniczną łupku Chichibu 5. Dilatant behaviour of a sandstone under true triaxial compression conditions As has already been mentioned, the main objective of true triaxial tests on samples of Coal-Measure sandstone was to elucidate the role of dilatancy in the mechanism of rockbursts and its association with possible premonitory signs of dynamic manifestations of rock pressure in mines. Experimental studies have been undertaken with the aim of determining the effect of the intermediate principal stress on the threshold of absolute dilatancy, on the volu-

16 424 Fig. 14. Stress strain curves of Chiibu schist samples oriented at an angle β = 30 and ω = 45, tested under true triaxial compression conditions at σ x = 50 MPa and different σ y values Rys. 14. Charakterystyki naprężenie odkształcenie próbek łupku Chichibu o orientacji β = 30 i ω = 45, badanych na prawdziwe trójosiowe ściskanie przy σ x = 50 MPa i różnych wartościach σ y metric strain at the threshold of dilatancy and at the peak stress, and on the permanent volumetric strain induced in rock samples loaded to different differential stress levels (Kwaśniewski et al., 2003, 2005). The experimental program included loading and unloading tests to and from different levels of differential stress in the pre- and post-peak stages under conditions of uniaxial compression, conventional triaxial compression at confining pressures up to 37.5 MPa and true triaxial compression at σ 3 equal to 25 MPa and σ 2 being 1.5-, 2- and 2.5-times higher than σ 3. The rock tested was a massive, homogeneous, medium- to coarse-grained sandstone taken from a depth of about 975 m in the Śląsk coal mine in the Upper Silesian Coal Basin, Poland. It was composed of angular, moderately sorted quartz grains and an illite and dolomite matrix. The bulk density of the rock was equal to 2.44 g/cm 3. According to mercury intrusion porosimetry, pores of a radius of 0.4 µm were dominant (vol. ratio of 10%) in the rock. The average pore radius was equal to 0.19 µm and the total poro-

17 425 sity was equal to 5.9%. The uniaxial compressive strength of the rock (σ C ) was equal to 90.4 MPa, while the threshold of absolute dilatancy (σ TD ), Young s modulus (E) and Poisson s ratio (ν), determined under uniaxial compression conditions, were equal, on average, to 0.68σ C, 30.2 GPa and 0.28, respectively. A servo-controlled true triaxial testing machine developed by Koide and Takahashi at the Geological Survey of Japan in the late 1970s was used for the tests (Fig. 15). In this machine samples are loaded in the same manner as in Mogi s triaxial apparatus (see Fig. 9), that is the maximum and the intermediate principal stresses are applied by axial and lateral pistons, respectively, and the minimum principal stress is applied by oil-confining pressure. The high-pressure chamber of the machine can accommodate samples of remarkably large dimensions: 50 mm 50 mm 100 mm. In addition, pore pressure can be generated in the samples. Fig. 15. Simplified scheme of the triaxial testing machine used in the present study: 1 fixed upper and lower platen, 2 hydraulic power supply, 3 control unit, 4 pressure vessel, 5 load cells for axial (σ 1 ) and lateral (σ 2 ) directions, 6 pistons for axial (σ 1 ) and lateral (σ 2 ) loads, 7 servovalve, 8 displacement transducers for axial (1) and lateral (2) directions, 9 lateral (σ 2 ) steel end piece, 10 axial (σ 1 ) steel end piece, 11 pore fluid inlet, 12 pore fluid outlet, 13 rock sample (after Takahashi & Koide, 1989) Rys. 15. Uproszczony schemat trójosiowej maszyny wytrzymałościowej, z której korzystano w opisywanych badaniach (wg Takahashiego i Koidego, 1989)

18 426 The rock sample assembly ready for testing consisted of a rectangular prismatic rock specimen of the dimensions 35 mm 35 mm 70 mm, with steel end pieces attached to the top, to the bottom and to those sides of the specimen which were to be loaded in direction 2. The sample was covered with a silicone rubber jacket and equipped with strain-gauged displacement transducers to measure the vertical and lateral strains in two directions (Fig. 16). a) b) c) Fig. 16. Schematic view of the specimen assembly used in true triaxial tests: a view in direction 3 (σ 3 ), b view in direction 2 (σ 2 ), c view in direction 1 (σ 1 ); 1 rock sample, 2 strain-gauged displacement transducers for measuring axial strain (ε 1 ), 3 strain-gauged displacement transducers for measuring lateral strain ε 2, 4 strain-gauged displacement transducer for measuring lateral strain ε 3, 5 silicone rubber jacket Rys. 16. Schematyczny widok próbki przygotowanej do próby na prawdziwe trójosiowe ściskanie: a widok w kierunku 3 (σ 3 ), b widok w kierunku 2 (σ 2 ), c widok w kieurnku 1 (σ 1 ) These are views of the rock sample, ready for testing, in three mutually perpendicular directions: direction 3 (Fig. 16a), direction 2 (Fig. 16b) and direction 1 (Fig. 16c). Please note the strain-gauged displacement transducers used to measure axial strain and lateral strains in two directions. A rock sample assembly shortly before being enclosed inside the thick-walled cylinder of the triaxial vessel is shown in Figure 17a. Figures 17b and 17c show the triaxial vessel inside the load frame of the true triaxial compression machine. Please note the two solid pistons which, actuated by hydraulic jacks, apply a load to the sample in direction 2 and two steel bars that are normally used to prevent excessive deflection of the load-bearing frame of the triaxial testing machine during loading.

19 427 a) b) c) Fig. 17. Final stages of the preparations for a true triaxial compression test Rys. 17. Ostatnie stadia przygotowań do próby na prawdziwie trójosiowe ściskanie The two plots presented in Figure 18 summarise the experimental test results on the triaxial strength of the sandstone studied. The open circles correspond to the axisymmetric stress conditions where σ 2 is equal to σ 3, while the crosses correspond to true triaxial stress conditions (σ 2 σ 3 ) with σ 3 being equal to 25 MPa. All the data shown in Figure 18a were fitted using an empirical strength criterion which in the form F ( ) n 1 3 oct A (1) 2 F oct f ( 3) (2) 1 was proposed by Mogi (1971a, b) as the so-called generalised von Mises criterion (Fig. 18b). In Equation 1 F τ oct is the octahedral shear stress at failure, F (σ 1 + σ 3 )/2 is the mean

20 428 normal stress at failure, and A and n are certain empirical material parameters. It should be noted that Mogi consciously did not include the intermediate principal stress (σ 2 ) into the expression for the mean normal stress (Mogi calls it the effective mean pressure on faulting ) that opposes the creation of the fracture plane in order to account for the observation that fault planes that develop in rock samples tested under true triaxial compression conditions are normally oriented parallel to the direction of σ 2. a) b) Fig. 18. Results of experimental studies on the ultimate strength of Śląsk sandstone under uniaxial compression, conventional triaxial compression and true triaxial compression conditions (a), plotted as the octahedral shear stress versus mean normal stress at failure and fitted using Mogi s power failure criterion (b) Rys. 18. Wyniki badań nad wytrzymałością graniczną piaskowca Śląsk w warunkach jednoosiowego, konwencjonalnego trójosiowego i prawdziwego trójosiowego ściskania (a), przedstawione w formie zależności pomiędzy granicznym oktaedrycznym naprężeniem ścinającym a naprężeniem średnim i aproksymowane potęgowym warunkiem wytrzymałościowym Mogiego (b) For the medium- to coarse-grained Śląsk sandstone tested, values of parameters A and n in Equation 1 are equal to and 0.817, respectively. The two plots in Figure 19 show examples of the stress strain characteristics of Śląsk sandstone under uniaxial compression conditions. The curves on the left were obtained based on a single-run test, while the curves on the right characterise the behaviour of the sandstone in repeated loading/unloading tests. Please note the solid circles that indicate the threshold of absolute dilatancy that is equal to about two-thirds of the ultimate strength. Examples of the stress strain curves obtained from the conventional and true triaxial compression tests on samples of Śląsk sandstone are shown in Figure 20. Under

21 429 Fig. 19. Examples of stress strain curves of Śląsk sandstone tested under uniaxial compression conditions Rys. 19. Przykłady charakterystyk naprężenie odkształcenie piaskowca Śląsk w warunkach jednoosiowego ściskania a) b) Fig. 20. Examples of stress strain curves of Śląsk sandstone tested under conventional triaxial compression conditions (a) and true triaxial compression conditions (b) Rys. 20. Przykłady chrakterystyk naprężenie odkształcenie piaskowca Śląsk w warunkach konwencjonalnego trójosiowego ściskania (a) i prawdziwego trójosiowego ściskania (b)

22 430 axisymmetric stress conditions lateral strains in two directions ε 2 and ε 3 were the same (Fig. 20a). The results of true triaxial compression tests, where σ 2 was 2.5-times higher than σ 3, show that lateral strain ε 2, that is strain in the direction of the intermediate principal stress, is much smaller than lateral strain in direction 3 (Fig. 20b). The conventional triaxial compression tests revealed a typical monotonic increase in the strength (F) of Śląsk sandstone as the confining pressure was increased from 0 to 37.5 MPa. The influence of confining pressure on the threshold of dilatancy (TD) was similar, with the ratio of TD to ultimate strength increasing, at a gradually decreasing rate, from 0.68 for unconfined conditions to 0.87 for confining pressure equal to 37.5 MPa (Fig. 21a). Higher values of the volumetric strain at the threshold of absolute dilatancy also correspond to higher confining pressures. These are positive values, which means that greater compaction of rock material corresponds to higher confining pressures, thus at higher and higher thresholds of dilatancy, the tested samples had a smaller and smaller volume (Fig. 21b). a) b) Fig. 21. Results of conventional triaxial compression tests on samples of Śląsk sandstone; The effect of confining pressure (p) on the threshold of absolute dilatancy (TD) and ultimate strength (F) (a), and on the volumetric strain at the threshold of dilatancy (b) Rys. 21. Wyniki badań próbek piaskowca Ślask na konwencjonalne trójosiowe ściskanie; Wpływ ciśnienia okólnego (p) na próg dylatancji bezwzględnej (TD) i wytrzymałość graniczną (F) (a) oraz na odkształcenie objętościowe na progu dylatancji (b) Both ultimate strength and the threshold of absolute dilatancy increase as the intermediate principal stress increases. However, this effect is considerably weaker than that caused by the increasing confining pressure. In particular, ultimate strength does not seem to increase any more at intermediate principal stresses higher than 50 MPa (Fig. 22a).

23 431 a) b) Fig. 22. Results of true triaxial compression tests on samples of Śląsk sandstone; The effect of intermediate principal stress (σ 2 ) on the threshold of absolute dilatancy (TD) and ultimate strength (F) (a), and on the volumetric strain at the threshold of dilatancy and at the peak differential stress (b) Rys. 22. Wyniki badań próbek piaskowca Ślask na prawdziwe trójosiowe ściskanie; Wpływ pośredniego naprężenia głównego (σ 2 ) na próg dylatancji bezwzględnej (TD) i wytrzymałość graniczną (F) (a) oraz na odkształcenie objętościowe na progu dylatancji i na granicy wytrzymałości (b) Similarly, the volumetric deformability of the rock tested was affected by intermediate principal stress to a much lesser extent. Values of volumetric strain at the threshold of absolute dilatancy increased slightly with increasing σ 2. As these were compactant strains, increasing the intermediate principal stress resulted in a small decrease in the volume of the tested samples at the (slightly increasing) threshold of absolute dilatancy. However, at differential stresses higher than the threshold of dilatancy an important increase in the volume of the rock occurred, particularly at low values of the intermediate principal stress (Fig. 22b). Increasing the intermediate principal stress led to a significant weakening of the effect of dilatancy; in the case of samples tested at intermediate principal stress equal to 62.5 MPa, values of the volumetric strain at peak differential stress were close to zero. It can be inferred from this observation that at high intermediate principal stresses, the process of microcracking and deterioration of the rock was arrested to some extent. Suppression of dilatancy under high intermediate principal stresses was additionally evidenced by an auditory observation: faulting of rock samples in the post-peak region was much more brittle and violent, and was accompanied by a strong acoustic effect. To sum up: The threshold of absolute dilatancy increases appreciably with the magnitude of σ 2. Thus the intermediate principal stress appears to extend the elastic

24 432 range of the stress strain behaviour for a given σ 3 and hence to retard the onset of the failure process. The dilatant volumetric strain decreases with an increase of σ 2 and the rock behaves in an increasingly brittle manner. 6. Conclusions In the 1970s Mogi was the first to study the effect of, independently, confining pressure, intermediate principal stress (σ 2 ) and minimum principal stress (σ 3 ) on the differential stress axial strain characteristics of rocks. Using samples of a marble, a dolomite and a limestone, Mogi (1971b, 1972a, b) showed that the effects of σ 2 and σ 3 on the stress strain characteristics are very different. For example (see Figure 23), the ductility of Yamaguchi marble increased with an increase in σ 3 but decreased with an increase in σ 2. The yield stress and the coefficient of strain-hardening in the post-yield Fig. 23. Differential stress axial strain curves of Yamaguchi marble subjected to triaxial compression: a curves for different values of confining pressure (p = σ 2 = σ 3 ), b curves for different values of intermediate principal stress (σ 3 = 55 MPa), c curves for different values of minimum principal stress (σ 2 = 108 MPa) (Mogi, 1973, 2006) Rys. 23. Krzywe naprężenie różnicowe odkształcenie podłużne marmuru Yamaguchi w warunkach trójosiowego ściskania: a krzywe dla różnych wartości ciśnienia okólnego (p = σ 2 = σ 3 ), b krzywe dla różnych wartości pośredniego naprężenia głównego (σ 3 = 55 MPa), c krzywe dla różnych wartości najmniejszego naprężenia głównego (σ 2 = 108 MPa) (Mogi, 1973, 2006)

25 433 region increased with an increase in σ 2 but was nearly independent of σ 3, that is, the effect of σ 2 on yielding and strain-hardening under true triaxial compression conditions was similar to that of confining pressure in conventional triaxial compression tests. The ultimate goal of the present studies was to reveal the effect of confining pressure and the effect of intermediate principal stress on the volumetric deformation mode and the dilatant behaviour of rocks. There is no doubt that volumetric strain is a quantity that provides much more information about the mechanical behaviour of rocks than the axial strain alone. The general features of the volumetric deformation mode of rock in the brittle field are presented in Figure 24. These are: onset or threshold of relative dilatancy (OD), threshold of absolute dilatancy (TD), strength failure (F), which may occur either in the compaction or in the dilation quadrant of the stress strain plane, faulting (f), which normally is accompanied by a stress drop (isd in Figure 24 indicates the immediate stress drop), and residual stress (RS). Fig. 24. General features of the volumetric deformation mode of rock in the brittle field Rys. 24. Wielkości charakteryzujące proces objętościowego odkształcania się skały w polu zachowań kruchych The results of experimental studies on the effect of confining pressure and the effect of intermediate principal stress have been presented in this paper. The major findings of these studies will be summarised here using the plots shown in Figure 25.

26 434 Please note that under true triaxial compression conditions dilatancy was strongly hampered by the intermediate principal stress. In addition, rock samples faulted in a brittle manner at a very early stage, that is immediately after entering the post-peak region. The faulting was violent and accompanied by a huge stress drop (Fig. 25b). a) b) Fig. 25. Volumetric deformation mode of Śląsk sandstone under triaxial compression conditions: a axisymmetric state of stress, σ 3 = σ 2 = 25 MPa; b true triaxial state of stress, σ 3 = 25 MPa, σ 2 = 62.5 MPa Rys. 25. Charakter objętościowego odkształcania się piaskowca Śląsk w warunkach trójosiowego ściskania; a osiowo-symetryczny stan naprężenia, σ 3 = σ 2 = 25 MPa; b prawdziwie trójosiowy stan naprężenia, σ 3 = 25 MPa, σ 2 = 62,5 MPa The behaviour of the sandstone tested under axisymmetric stress conditions was completely different. After strength failure, the rock deformed extensively, undergoing a significant increase in volume. Faulting occurred in a well-advanced post-peak region. It was accompanied by a small immediate stress drop (Fig. 25a). No acoustic effect was heard. In general, under axisymmetric stress conditions the sandstone tested behaved in a less brittle and highly dilatant manner. ACKNOWLEDGEMENTS I wish to extend my thanks to Professor Kiyoo Mogi of the Earthquake Research Institute, the University of Tokyo and to Dr Manabu Takahashi of the Research Core for Deep Geological Environments, the National Institute of Advanced Industrial Science and Technology (AIST) in Tsukuba, Japan, with whom I had the privilege and pleasure to work at their laboratories, and who offered me invaluable help in my experimental studies of the mechanical behaviour of rocks under true triaxial compression conditions.

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