Strength and deformability of a low-porosity sandstone under true triaxial compression conditions

International Journal of Rock Mechanics & Mining Sciences 127 (2020) 104204

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Strength and deformability of a low-porosity sandstone under true triaxial compression conditions Shihuai Zhang a, b, 1, Shunchuan Wu a, b, *, Guang Zhang a a b

Land Resources Engineering, Kunming University of Science and Technology, Kunming, Yunnan, 650093, China Key Laboratory of Ministry for Efficient Mining and Safety of Metal Mines, University of Science and Technology Beijing, Beijing, 100083, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Brittle failure ⋅ Modified Hoek-Brown failure criterion ⋅ Failure plane angle ⋅Intermediate principal stress ⋅ Brittle-ductile prediction

True triaxial tests (σ1 � σ2 � σ3 ) were conducted on the Zigong sandstone with a relatively low porosity of ~6.5% with the aim of assessing the influence of the intermediate principal stress σ 2 on the strength and deformability. Three series of true triaxial tests (σ 3 ¼ 0, 20, and 60 MPa) were performed. Within each series, σ 2 was planned to be varied from σ2 ¼ σ3 to σ 2 ¼ σ 1 from test to test. For each test, σ 1 was raised monotonically to failure while σ 2 and σ 3 were kept constant and the post-peak behavior has been captured. It is found that the strain in the σ 3 -direction is always larger than that in the σ2 -direction under true triaxial stresses, which is attributed to the stress-induced tensile microcracks aligned with the σ1 -σ 2 plane. For a constant σ 3 , the dilatant strain in the σ2 -direction is progressively suppressed and becomes compactive as σ2 raises from σ 2 ¼ σ3 to σ2 ¼ σ1 . Based on a modified Hoek-Brown failure criterion with three stress invariants, the best fitting brittle strength envelope reveals that the σ 2 effect on rock strength, i.e., the ascending-then-descending trend for a given σ3 , is the combined effect of mean stress and Lode angle. In the series of biaxial tests (σ 3 ¼0 MPa), macro tensile fractures that parallel to the σ1 -σ2 plane develop regardless of σ2 , displaying a failure plane angle of approxi­ mately 90� . As σ 3 is raised, failure plane angle generally decreases. When σ 3 ¼ 20 MPa, the failure plane angle generally increases with σ2 while there is no evident dependence on σ 2 when σ 3 ¼ 60 MPa. Comparisons with the other two sandstones with higher porosity reveal that porosity exerts critical control on the brittle strength and that the σ 2 effect on deformability is also the result of two competing effects induced by mean stress and Lode angle. It is also found that the residual strength data facilitate the prediction of brittle-ductile transition pressure when σ 2 ¼ σ 3 in spite of relatively low σ3 level, which matches well with the empirical relation proposed by Mogi.1

1. Introduction Brittle fracture in rocks can be observed over a broad range from laboratory scale to a large scale such as earthquake. In particular, geotechnical applications, such as underground mining, deep tunneling, and borehole drilling, are intimately associated with the failure of the widespread sandstone formations. Commonly, the brittle failure process is related to crack initiation, propagation, and coalescence in the prepeak stage as well as the development of tabular failure planes (shear bands or shear fractures), with a concomitant increase in volume. In rare cases, however, a combination of very porous sandstone and very high stresses brings about a decrease in volume in a delocalized manner (compaction bands).2,3 Both deformation bands forming in the host

sandstones can affect local and regional permeability and reduce the engineering efficiency. Thus, it is of fundamental importance to under­ stand failure conditions and characteristics of sandstone. Key to facilitating this understand is the laboratory experiments. Historically speaking, extensive conventional triaxial tests have been conducted on cylindrical specimens of different sandstones,4 which are subjected to axisymmetric stress states where the intermediate principal stress σ2 equal to either the minimum principal stress σ3 or the maximum principal stress σ1 . These tests have revealed that the me­ chanical behavior of sandstone depends on the first two stress invariants related to the mean stress and the differential principal stress,5–8 respectively. Little is known about the mechanical response to the in­ termediate principal stress σ2 or the third stress invariant, except for the

* Corresponding author. Land Resources Engineering, Kunming University of Science and Technology, Kunming, Yunnan, 650093, China. E-mail addresses: [email protected] (S. Zhang), [email protected] (S. Wu), [email protected] (G. Zhang). 1 Present address: Department of Earth Sciences, Chair for Geothermal Energy and Geofluids, ETH Zürich, Zürich, 8092, Switzerland https://doi.org/10.1016/j.ijrmms.2019.104204 Received 12 October 2019; Received in revised form 28 December 2019; Accepted 29 December 2019 Available online 21 January 2020 1365-1609/© 2020 Elsevier Ltd. All rights reserved.

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International Journal of Rock Mechanics and Mining Sciences 127 (2020) 104204

limited comparison between conventional triaxial compression (CTC: σ 1 > σ 2 ¼ σ 3 ) and conventional triaxial extension (CTE: σ1 ¼ σ 2 > σ 3 ).9–11 Moreover, in situ stress measurements at depths indicate a state of stress in the earth’s crust that is totally non-uniform (σ1 6¼ σ 2 6¼ σ 3 ),12–15 implying a nonnegligible role played by σ2 in the rock behavior. Since the seminal work of Mogi,10 various Mogi-type true triaxial apparatus (TTA), i.e., adding the load of the third direction to the CTC device, have been developed and revealed significant effect of σ 2 on the deformability and strength of different rock types.16–22 With regard to sandstone, there are only a few true triaxial tests performed system­ atically,23–26 which, however, have the following limitations: (1) the end effect has not been given enough attention, leading to a potential increase in strength16,27; (2) the strength characteristics cannot been depicted in the three-dimensional principal stress space, failing to give an intuitive description by a single failure criterion; (3) the complete post-peak behavior are not available due to the insufficiency of the loading frame stiffness and stability in strain control mode; (4) speci­ mens are often small, giving rise to potential size effect.23 To avoid these key issues, we employed a Mogi-type TTA21 to carry out a comprehensive series of true triaxial tests on the low-porosity (~6.5%) Zigong sandstone. By improving the volume change measure­ ment, decreasing the end friction, removing the loading gap on the specimen, and integrating a servo control system with high stiffness loading frames, we captured accurate pre-peak and post-peak behaviors of the Zigong sandstone in the brittle regime. Further, to characterize the strength characteristics of the Zigong sandstone, a modified Hoek-Brown failure criterion was adopted to formulate its brittle strength envelope in the three-dimensional principal stress space. At the conclusion of tests, dip angles of main failure planes were determined based on failed specimen. Typical microscopic characteristics of some main failure planes under different stress states were revealed by SEM study. Together with other two studies,28,29 focusing on the brittle failure under tension and uniaxial compression conditions, respectively, this paper provides a comprehensive understanding of mechanical behavior of the low-porosity Zigong sandstone under general stress states.

σ 2 . The minimum principal stress σ3 is applied by the hydraulic oil filling

in the triaxial vessel. In addition to inheriting the advantages of the original Mogi-type design, the machine has been greatly improved in terms of loading capacity, convenience for operation and capability for capturing the complete stress-strain curves of hard rock. As shown in Fig. 1, there are two pairs of end spacers placed between the pistons of the triaxial vessel and the corresponding specimen sur­ faces. Similar to the design of Haimson and Chang,19 the pair corre­ sponding to the σ1 -direction has identical area to that of the specimen faces while the other pair in the σ 2 -direction has the length of 97 mm, shorter than the length of specimen. Thus, contact of the spacers in the σ 1 - and σ 2 -direction can be prevented during compression of the spec­ imen. Polyurethane was painted on both surfaces of the specimen sub­ jected to σ3 and on four blank edges to form a coating to prevent the intrusion of hydraulic oil. During the test, one can measure the principal stresses directly by pressure transmitters. Two principal strains, ε1 and ε2 , in theσ1 - and σ2 -direction can be measured by the linear displace­ ment transducers (LVDTs with less than 0.4% measurement accuracy) mounted onto the two pairs of spacers. The third principal strain ε3 in the σ 3 -direction can be measured by a beam-type strain gauge attached to the center points of both free surfaces in the σ 3 -direction. In such an elaborate manner, one can obtain the post-peak strains precisely, elim­ inating the drawback that only local strains can be measured by attaching two strain gauges directly to one of the surfaces in the σ 3 -direction.19,25 To reduce end effect, which exists ubiquitously in rock laboratory tests and plays a significant role in affecting the rock strength,31,32 a copper foil (0.05 mm in thickness) and a polytetrafluoroethylene sheet (0.05 mm in thickness) with grease fortified with MoS2 smeared be­ tween them were inserted between each pair of specimen surface and spacer. Note that it is preferable to place copper foil next to the surface of the specimen to prevent the intrusion of the lubricants. On the basis of such anti-friction measures, the coefficient of friction (COF) between the specimen and the spacer can be reduced to 0.006 under a normal stress between 20 and 120 MPa,21 which is lower than almost all published values using different fiction reducers.20,33–35 The end effect caused by such a low COF is small and thus can be negligible.27

2. Experimental methodology

2.3. True triaxial testing procedure

2.1. Experimental material and specimen preparation

Fig. 2 shows a typical loading path adopted in the true triaxial compression experiments. Firstly, a stress of 4 MPa and 2 MPa was preloaded in the σ 1 - and σ2 -direction, respectively. The confining pres­ sure was then raised monotonically at a constant rate of 0.1 MPa/s up to the desired σ3 level and kept constant subsequently. Thereafter σ1 and σ2 were increased simultaneously at a rate of 0.2 MPa/s until the preset σ2 was reached. Finally, σ 1 was further increased monotonically by a stress rate of 0.2 MPa/s and a strain control in the σ3 -direction was used

A red sandstone block, taken from a quarry in Zigong, Sichuan Province, China, was selected to carry out a comprehensive series of true triaxial tests. Referred to as Zigong sandstone herein, the selected sandstone is mainly composed of silicate minerals (60% feldspar and 35% quartz) in addition to 5% magnesium oxide and clay filling in the void and acting as cementation. It has an average grain size of about 0.2 mm and a porosity of about 6.5%, representing a relatively low porosity. Rectangular prism specimens were cut with the long axes perpendicular to the sedimentary bedding and further machined and polished to a dimension of 50 � 50 � 100 mm3, reaching a dimensional tolerance of �0.01 mm and a perpendicularity tolerance of 0.02 mm. Note that, no obvious sedimentary sequences can be observed at the sample scale, facilitating the assumption of isotropic samples. 2.2. True triaxial apparatus and test preparation A true triaxial apparatus (TTA) for hard rock21 was adopted in this study, which emulates the original Mogi-type design30 with novel modifications to the structure and triaxial vessel. The machine has a loading capacity of 1000 � 1000 � 100 MPa in the σ 1 -, σ2 - and σ 3 -di­ rection, respectively, and accommodates a 50 � 50 � 100 mm3 rect­ angular prism specimen. With the help of two moveable stiffer frames with low moving resistance in a horizontal layout, orthotropic loads can be provided as the maximum and intermediate principal stresses, σ 1 and

Fig. 1. Schematic diagram of the assembly of specimen, spacers and sensors from the top view. 2

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International Journal of Rock Mechanics and Mining Sciences 127 (2020) 104204

particular, the uniaxial compression strength (UCS), the Young’s modulus and the Poisson’s ratio obtained on the basis of the uniaxial compression test are 100.1 MPa, 18.27 GPa and 0.20, respectively. As for the true triaxial tests, three σ 3 levels were applied, i.e., 0, 20 and 60 MPa. For each given σ 3 , the magnitude of σ 2 was designed to cover a full range from σ 2 ¼ σ 3 to σ2 ¼ σ1 . Profiting from the strain control mode, complete stress-strain relationships were obtained in the true triaxial tests. An example of the relationship between the axial differential stress (σ 1 -σ3 ) versus all three principal strains (ε1 ,ε2 and ε3 ) and the volumetric strain εv ( ¼ ε1 þ ε2 þ ε3 ) when σ3 ¼ 60 MPa and σ2 ¼ 180 MPa is shown in Fig. 3 (b). It is easy to find that strains including ε1 , ε2 and εv are compactive throughout the test although they becomes dilatant in the post-peak stage, while ε3 is always dilatant. The loading path, as described in Fig. 2, is partly reflected on the stress-strain curves. The start point of each curve represents the start of the triaxial extension loading stage (CTE:σ1 ¼ σ 2 > σ3 ) after σ 3 reaching its desired value (60 MPa), i.e., the hydrostatic loading stage is not shown here. ε1 and ε2 are almost the same in the triaxial extension loading stage, the end of which is marked as a hard inflection point, indicating that only σ 1 is increased monotonically henceforth. In the next stage before peak, the lateral strain ε2 starts to dilate and the axial strain ε1 continues to contract with a more moderate slope. In particular, on the (σ1 -σ3 ) -εv curve, there exists a quasi-linear part (dashed line) suggesting an elastic mechanical response approximately. Under an elevated axial stress, this curve starts to deviate from linearity, exhibiting the onset of dilatancy, until the peak

Fig. 2. An example of loading path adopted in true triaxial tests (σ3 ¼60 MPa and σ2 ¼ 180 MPa). Note that there are two sharp drops of σ 1 in the postpeak stage.

instead (strain rate: 5 � 10 6 s 1) when the rate of ε3 approached 5 � 10 6 s 1. Thus, loading can be well controlled even in the post-peak stage, during which σ 1 decreased to a residual strength when fric­ tional sliding dominates. Note that, the post-peak behavior could not be captured for the CTE tests due to the stress control mode rather than strain control mode was used. At the end of each test, failed specimen was extracted from the true triaxial cell. After removing the polyurethane on each σ 3 surface, the specimen was then impregnated with a thin, penetrative epoxy to pre­ serve failure conditions for microscopic observation. Planes orthogonal to σ 2 -direction or σ 1 -direction were surface grounded flat and then carefully micro-polished to 0.05 μm. Thin sections were coated with an approximately 0.06 μm thick layer of gold, which were further inspected using a backscattered scanning electron microscope (SEM model: Phenom XL) at an accelerating voltage of 15 kV.

f

stress σ1 is reached. During the post-peak stage, due to the strain control in ε3 , the specimen exhibits the Class-II behavior36 in the σ1 - direction when σ3 ¼ 60 MPa and σ2 ¼ 180 MPa. It is believed that macro rupture is forming during this stage, which is a mixed process between the shear fracture of intact rock and the frictional sliding on the failure planes.37 After the coalescence of macro shear band, the specimen turns to the residual stage, when the axial stress σ 1 is constant and the frictional sliding takes place completely. In this paper, deformability and brittle strength of the Zigong sand­ stone under different stress states are the focuses. The brittle strength data are obtained from the peak of the stress-strain curves for both the CTC tests and the true triaxial tests, and are listed in Table 1. It should be noted that, during some true triaxial tests, especially the test series of σ3 ¼ 0 MPa, the fracture invalidated the ends of beam-type transducer on the free surfaces in the σ 3 -direction, leading to an unsuccessful post-peak close-loop control since ε3 is a critical feedback. More detailed information of deformability of Zigong sandstone is provided by the stress-strain curves in terms of the σ 2 effect. Taking the σ 3 ¼ 60 MPa series as an example for its sufficient quantity of σ 2 , the relationships between the differential stress (σ1 -σ 3 ) versus all three principal strains (ε1 ε2 , ε3 ) and volumetric strain εv are shown in Fig. 4, in

3. Results 3.1. Deformability of Zigong sandstone Along with true triaxial tests, a separate series of CTC tests were conducted on the cylindrical sandstone specimen (Diameter: 50 mm, Height: 100 mm) using MTS815 Rock Mechanics Test System. As shown in Fig. 3 (a), near-vertical stress drop occurs in all tests, indicating a brittle failure in the post-peak stage. As σ3 rises from 0 MPa to 80 MPa, both peak strength and residual strength increase monotonically. In

Fig. 3. (a) Complete stress-strain curves of the CTC tests conducted on cylindrical specimens. (b) A typical example of complete stress-strain curve obtained from true triaxial test (σ3 ¼60 MPa and σ 2 ¼ 180 MPa), including all three principal strain (ε1 ,ε2 and ε3 ) and the volumetric strain εv ¼ ε1 þ ε2 þ ε3 . 3

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Table 1 Brittle strength data for the Zigong sandstone obtained from both triaxial tests and true triaxial tests. σ3 (MPa)

σ2 (MPa)

σf1 (MPa)

Conventional triaxial compression tests on cylindrical specimens 0 0 100.1 10 10 182.5 20 20 240.3 30 30 283.0 40 40 326.2 50 50 363.9 60 60 383.7 70 70 414.7 80 80 441.5 True triaxial tests on rectangle prism specimens 0 24.9 123.3 0 49.8 135.1 0 74.7 139.3 0 94.8 94.8 20 20 226.4 20 30 259.2 20 50 263.0 20 74.8 260.9 20 99.8 285.8 20 149.6 272.3 20 223.5 223.5 60 99.9 405.7 60 140 433.5 60 159.7 474.6 60 179.6 471.9 60 219.5 471.3 60 261.9 449.9 60 298.3 467.6 60 339.4 484.6 60 381.6 463.1

σr1 (MPa)

I1 (MPa)

pffiffiffiffiffif J2 (MPa)

pffiffiffiffiffi f ð J2 Þp (MPa)

Relative error (%)

NA 67.9 106.4 139.2 174.4 224.7 251.8 282.3 326.8

100.1 202.5 280.3 343 406.2 463.9 503.7 554.7 601.5

57.8 99.6 127.2 146.1 165.2 181.2 186.9 199.0 208.7

56.4 95.3 121.1 140.3 158.3 174.0 184.3 197.1 208.4

2.50 4.30 4.76 3.97 4.17 4.01 1.39 0.97 0.14

NA NA NA NA NA 111.7 119.5 119.3 127.7 129.1 NA 241.4 NA 283.5 267.5 NA 273.4 263.4 283.3 279.5

148.2 184.9 214.0 189.7 266.4 309.2 332.9 355.7 405.6 441.9 461.9 565.6 633.5 694.3 711.5 750.8 771.8 825.9 884 904.7

65.2 68.3 69.7 54.8 119.2 135.3 132.5 126.3 136.4 126.2 116.4 189.1 196.6 216.4 211.9 207.4 195 204.8 215.8 213.1

66.6 69.0 69.6 56.1 116.7 129.6 132.8 130.8 137.5 128.6 120 195.7 203 213.9 211.8 208.6 199.2 204 209.5 206.6

2.14 0.94 0.15 2.50 2.06 4.24 0.21 3.54 0.84 1.96 2.16 3.50 3.24 1.16 0.03 0.58 2.16 0.39 2.91 3.05

f

NA: The residual strength cannot be obtained due to either the stress control mode or unusual fractures. Superscript “f”: values at failure. Superscript “r”: values at residual stage. pffiffiffiffi Subscript “p”: values of J2 at failure predicted by the three-invariant strength criterion as described by Eq. (1).

which σ2 varies from 60 to 381.6 MPa. Similar to Fig. 3 (b), the slope of the (σ 1 -σ3 ) - ε1 curve in the triaxial extension stage is larger than that in the subsequent stage until peak, shown in Fig. 4 (a), indicating that damage accumulates at an accelerated rate when only σ1 is increased monotonically. As σ 2 increases from σ2 ¼ σ 3 to σ 2 ¼ σ1 , ε2 displays a change from dilatancy at σ2 ¼ σ 3 , through negligible dilatancy at σ2 ¼ 219.5 MPa, to compaction at σ2 ¼ 381.6 MPa, which is particularly obvious in the post-peak stage (Fig. 4 (b)). For a high magnitude of σ 2 (298.3, 339.4 and 381.6 MPa), it should be noted that the residual strength listed in Table 1 is less than the magnitude of σ 2 , suggesting a rotation of the stress σ1 and σ 2 in the post-peak stage. Therefore, compaction dominates in the post-peak stage in the original σ2 -direc­ tion. Moreover, the dilatant later strain ε3 is always larger than ε2 when σ 2 is truly intermediate. Since ε1 is always compactive, the volume expansion is completely resulted from the dilatancy of ε3 finally. This phenomenon, called anisotropic dilatancy, had been observed in true triaxial tests on Mizuho trachyte, Inada granite and Yamaguchi marble and can be explained as the opening of microcracks perpendicular to σ 3 -direction.38 Of all curves, a quasi-linear elastic mechanical response can be easily identified on the (σ1 -σ 3 ) - εv curve. By finding the change in the deriv­ ative of (σ1 -σ 3 ) with respect to εv , the onset of dilatancy is determined and marked as open diamond in Fig. 4 (d). One can observe that the stress at the onset of dilatancy increases with σ 2 , indicating that the dilatancy behavior is suppressed or the elastic behavior is extended by the increasing σ2 . This observation is in good accordance with previous finds in different rocks subjected either to conventional triaxial stress39 or to true triaxial stress,16,23,25,40 which can be attributed to the retar­ dation and suppression effect of σ2 on the microcracking process. Dilatancy preceding failure is evident in this test series but the one when σ 2 ¼ 99.9 MPa, which may be due to the leakage of hydraulic oil in the

post-peak region. Beyond the peak stress, the post-peak behavior un­ dergoes a gradual transition from dilatancy to compaction. As mentioned before, this may be due to the compaction in the σ 2 -direction on account of the stress rotation. Further, with the onset of dilatancy determined, the amount of in­ elastic volumetric strain εv; ​ in can be calculated by subtracting the elastic component (dashed line in Fig. 3 (b)) from the total volumetric strain, which can be used to quantify the degree of dilatancy upon failure. Two sets of test data (σ 3 ¼20, 60 MPa) are used for the calcula­ tion of εv; ​ in upon failure. As shown in Fig. 5, dilatancy can be noticed for all test data of Zigong sandstone, the degree of which, however, de­ creases generally with σ2 . This phenomenon may be attributed to that compaction due to the increased mean stress surpasses the shearinduced dilatancy due to the rise in the Lode angle, which will be dis­ cussed in section 4.2 in detail. 3.2. Three-dimensional characteristics of brittle strength In this section, a modified Hoek-Brown failure criterion41 is adopted to depict the brittle strength characteristics of Zigong sandstone in the three-dimensional principal stress space. The modified Hoek-Brown failure criterion is a combination of a new deviatoric functiongðθσ Þ42 pffiffiffiffi and the meridian function ð J2 Þmax of the original Hoek-Brown failure 43 criterion, which has three stress invariants and can be expressed as follows:

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Fig. 4. Relationships of between the differential stress (σ1 -σ 3 ) versus (a) ε1 , (b) ε2 , (c) ε3 and (d) εv of Zigong sandstone in the σ3 ¼ 60 MPa series. For each test, the end of triaxial extension loading stages is marked as square and the onset of dilatancy is marked as diamond.

� pffiffiffiffi J 2 ; I1 ; θ σ pffiffiffiffi pffiffiffiffi � J2 max gðθσ Þ ¼ J2

F

¼

pffiffiffiffi J2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mi σ c þ m2i σ2c þ 12mi σc I1 þ 36σ2c pffiffi 6 3

� � π 1 sin 1 A sin α 3 3 � � π 1 sin α þ sin 1 ðA sin 3θσ Þ 3 3

¼0 (1) where I1 ¼ σ ii is the first invariant of the stress tensor and often asso­ ciated with the mean stress p ¼ σ ii =3, J2 ¼ sij sij =2 is the second invariant of the stress deviator sij ¼ σ ij σkk δij =3. θσ is the Lode angle, which can be taken as the third invariant by: ! pffiffi 1 3 3 J3 1 (2) θσ ¼ sin 3 2 J 3=2 2

Fig. 5. Variation of inelastic volumetric strain upon failure with the normalized intermediate principal stress.

5

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compression state (θσ ¼-30� ). If materials are not dependent on the Lode angle, both curves should be coincident and the failure curves in the deviatoric plane should be circular as indicated by the Drucker-Prager failure criterion.

in which J3 ¼ detðsij Þ is the third invariant. In this paper, Lode angle θσ defines a 60� sector in the deviatoric plane where p is constant. The limiting values of θσ correspond to axisymmetric compression ( 30� ) and axisymmetric extension (30� ), respectively. The modified Hoek-Brown failure criterion has four strength pa­ rameters, i.e., mi , σ c , A, and α. The first two are the ones used in the original Hoek-Brown failure criterion while the latter two are dependent on the hydrostatic stress, controlling the curvature and aspect ratio of the deviatoric shapes, respectively. Following the procedure for deter­ mination of strength parameters described by Wu et al.,41 strength pa­ rameters mi and σ c can be gained by the plot of the octahedral shear qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 f f f stressτoct ¼ ðσ 1 σ 2 Þ þ ðσ2 σ3 Þ2 þ ðσ 1 σ3 Þ =3 versus the mean f

f

As shown in Fig. 7 (c), six sets of failure curves, corresponding to I1 ¼ 139, 311, 483, 655, 828 and 1000 MPa, respectively, along with test data are projected in the deviatoric plane, which is a plane normal to the 0 hydrostatic axis. The CTC data are on the σ 1 axis and the CTE data are on 0 the negative of the σ 3 axis. The other data of deviatoric stress state are in between. On the assumption that the Zigong sandstone is isotropic, the fitting curves are projected in all six sectors of the deviatoric plane, forming triangular shapes with smoothly rounded corners. A more f

intuitive Lode dependence can be obtained that, for a given I1 , the maximum strength occurs at θσ ¼ -30� and decreases to the minimum when the Lode angle θσ increases up to 30� , varying monotonically be­

σf þσ

stress without the intermediate principal stress σ m;2 ¼ 1 2 3 as shown in Fig. 6. Agreement of test data obtained from the CTC tests and the true triaxial tests illustrates the consistency of the Zigong sandstone regardless of specimen shape or test apparatus. Additionally, the determination method of the strength parameters A and α has been detailed by Wu et al.41 With the determined strength parameters, i.e., σ c , mi , A and α, the brittle strength envelope of Zigong sandstone can be constructed in the three-dimensional principal stress space. As shown in Fig. 7 (a), the failure surface is shaped as a pointed bullet with smoothly rounded triangular cross-sections (bold curves, known as deviatoric planes, also pffiffiffiffi f shown in Fig. 7 (c)). For a quantitative comparison, the predicted J2 by this criterion and relative errors with respect to the true values are also listed in Table 1. Among all data, the maximum relative error is 4.76%, indicating that the test data are well fitted by the three-invariant failure criterion. Visual inspection of the envelope also reveals a remarkable nonlinear feature. To check this nonlinearity, the con­ pffiffiffiffi f f structed envelope is then projected onto the J2 - I1 plane, deviatoric

f

tween the upper and lower boundaries as delineated in Fig. 7 (b). As I1 increases, in addition, the triangular shape becomes more circular, which has been observed for many geomaterials.44,45 f

Finally, inspection of the test data in the σ1 -σ 2 plane reveals a typical ascending-then-descending trend of strength. That is, for a given σ3 , strength increases with σ 2 at first until a plateau is reached, beyond which failure occurs at a lower stress level. This so-called σ2 effect has been widely observed during most laboratory tests25,46 except for some rocks (such as hornfels and metapelite, Chang and Haimson).47 Note f

that, from Fig. 7 (b) and Fig. 7 (c), both I1 (or mean stress) and Lode angle θσ increase as σ 2 increases from σ3 to σ1 for a given σ 3 . As mentioned above, strength in the brittle regime is proportional to the mean stress while inversely proportional to the Lode angle. Therefore, it can be hypothesized that the σ2 effect is actually the competing process between the mean stress effect and the Lode angle effect. In other word, when σ2 is moderate, if the strengthening effect due to the increased mean stress surpasses the weakening effect due to the rise in the Lode angle, rock strength increases with σ 2 . And when σ 2 is larger than some value, the Lode angle effect starts to prevail, leading to a decreasing strength.

f

plane and σ1 -σ 2 plane, respectively. pffiffiffiffi f f As shown in Fig. 7 (b), in the J2 -I1 plane, the best fitting curves corresponding to triaxial compression state and triaxial extension state act as boundaries, keeping all the test data within the restrained region. pffiffiffiffi f f J2 continuously rises with I1 (or mean stress) at a decreasing rate for both triaxial compression state (θσ ¼-30� ) and triaxial extension state (θσ ¼30� ). This trend has been recorded in porous sandstones deformed under CTC tests5 and true triaxial tests25 in the brittle regime and has generally been fitted by parabolic equations in terms of the first two stress invariants. In this way, one has to fit a best fitting curve for every Lode angle if the true triaxial test data are available, which is inconve­ nient for the analysis of mean stress effect. In the meanwhile, Lode dependence of the Zigong sandstone is obviously indicated as the failure curve for triaxial extension state (θσ ¼30� ) falls below the one for triaxial

3.3. Failure planes The failure process is always accompanied by the development of failure planes. In the brittle regime, failure plane generally takes the form of shear fracture (or shear band). In CTC tests, the influence of confining pressure on the failure plane angle can be investigated. As shown in Fig. 8, the failure plane angle is defined as the angle between the normal of the failure plane and σ 1 -direction. When σ 3 ¼ 0 MPa, axial splitting or axial cleavage is the most prominent feature, displaying a near vertical fracture. The axial splitting probably stems from local tensile stresses due to the preexisting microcracks or the heterogeneities of the sandstone itself. Then as σ 3 rises, axial splitting is suppressed and the failure plane angle tends to decrease, which is consistent with those reported by many researchers experimentally48–50 or theoretically.51,52 This trend can also be seen as the mean stress effect, since the Lode angle is invariably kept at 30� while the mean stress increases mono­ tonically. The decreasing failure plane angle has also been observed in true triaxial tests of other sandstones subjected to an increasing mean stress and a constant Lode angle.24–26 Under true triaxial stress states, shear band generally displays a different pattern, dipping in the σ3 -direction and striking in the σ 2 -di­ rection.1,53 In the series of σ3 ¼ 0 MPa, which is actually biaxial test, more than two macro tensile fractures can be observed densely spaced, subparallel and adjacent to the surfaces subjected to σ3 , as shown in Fig. 9. Thin sections along the σ 2 -σ3 and σ1 -σ 3 planes were prepared for microstructural observation by a scanning electron microscope (SEM). SEM micrographs reveals that the stress-induced open microcracks are aligned with the σ1 -σ 2 plane and transgranular, as indicated by arrows,

Fig. 6. Relation of the octahedral shear stressτfoct and the mean stress without the intermediate principal stressσfm;2 for Zigong sandstone at failure.

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International Journal of Rock Mechanics and Mining Sciences 127 (2020) 104204

Fig. 7. (a) Best fitting brittle strength envelope plotted in the three-dimensional principal stress space. Both the test data and fitting envelope are projected onto the pffiffiffiffiffif f f (b) J2 -I1 plane, (c) deviatoric plane and (d) σ1 -σ2 plane. Test data of triaxial compression test conducted on cylindrical specimens are marked as red while the true triaxial test data are marked as blue. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

explaining the anisotropic dilatancy shown in stress-strain curves. The two splitting parts of each grain shows no shear displacement, sug­ gesting that these microcracks are predominantly tensile. In addition, it can be seen that the propagation of microcracks is disturbed remarkably

by the texture and structure, e.g., cleavages and grain boundaries, forming tortuous propagation paths locally. The macro tensile fractures resemble the so-called “onion-skin” fractures or slabs near the under­ ground excavation boundary54–56 or the breakouts around

Fig. 8. Variation of failure plane angle with σ3 in the CTC tests. 7

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International Journal of Rock Mechanics and Mining Sciences 127 (2020) 104204

Fig. 9. A typical macro tensile fracture pattern and SEM micrographs shown in the (a) σ 2 -σ 3 plane and (b) σ 1 -σ3 plane (in the case of biaxial tests, σ3 ¼ 0 MPa and σ 2 ¼ 50 MPa).

boreholes,57,58 typically characterized by tensile fractures parallel to the excavation surface. In this series, it seems that the magnitude of σ2 could hardly influence the failure plane pattern (thus not shown here), i.e., tensile fractures that densely spaced, subparallel and adjacent to the σ 3 surfaces. Even for the CTE test, the specimen explodes into several slabs, showing consistency. In the series of σ 3 ¼ 20 MPa, as shown in Fig. 10 (a), there is an unforeseen failure plane feature when σ 2 ¼ 20 MPa. Two subparallel shear bands are found within the specimen dipping in the σ 2 -direction and striking in the σ 3 -direction, contrary to the attitude of other shear bands when σ2 > σ3 . To find out the reason, cross sections were cut along the σ 1 -σ2 and σ 2 -σ 3 plane for microscopic observation. As shown in Fig. 11 (a), the stress-induced microcracks are parallel or subparallel to σ 1 -direction and extend though a multitude of preexisting transverse fissures aligned with the σ3 -direction. These tensile and transgranular microcracks terminate at or are offset by the grain boundaries (see ar­ rows), also displaying tortuous characteristics. Note that the opening of these stress-induced microcracks aligned with the σ 1 -σ 2 plane is the reason why macro shear bands have a different strike along the

σ 3 -direction. On the other hand, irregular fragments can be seen in the vicinity of macro shear band in the σ 2 -σ 3 plane, shown in Fig. 11 (b),

indicating that uniform stress state allows the stress-induced micro­ cracks to initiate and propagate in random directions. Therefore, it may be inferred that there were substantial preexisting microcracks aligned with the σ 3 -direction within the untested specimen, and the rock frac­ turing process was mainly controlled by these intrinsic heterogeneities due to uniform stress state. Furthermore, the failure plane angle is about 82� , 12.3% larger than that observed on the cylindrical specimen under the same stress state. This is probably attributed to different specimen geometries and boundary conditions. In the same series (σ 3 ¼ 20 MPa), when σ2 > σ3 , there are generally two failure planes forming asymmetrical V-shape fractures within each specimen, which are closely related to two times of steep stress drops experienced in the post-peak stage. As summarized in Fig. 12, the measured failure plane angles are marked, which are in the range of 72–79� for the series of σ 3 ¼ 20 MPa and in the range of 62–75� for the series of σ 3 ¼ 60 MPa. Both show fluctuations with the increasing σ 2 , not showing the widely observed σ 2 effect on the failure plane angle in other

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International Journal of Rock Mechanics and Mining Sciences 127 (2020) 104204

Fig. 10. Variations of failure plane angle with σ 2 in the series of (a) σ3 ¼ 20 MPa and (b) σ 3 ¼ 60 MPa. All specimens are shown in the σ1 -σ 3 plane except for the one at σ2 ¼ σ 3 ¼ 20 MPa, which is shown in the σ1 -σ 2 plane for its unusual failure plane strike.

Fig. 11. SEM micrographs of sections along the (a) σ1 -σ 2 and (b) σ2 -σ3 plane in the case of σ2 ¼ σ3 ¼ 20 MPa.

sandstones25 or crystalline rocks.19,59,60

stress state. To this end, results of true triaxial tests on Coconino and Bentheim sandstone25 are selected here for comparison. The Coconino sandstone has a mean grain size of 0.1 mm and a porosity of 17.5%, and the Bentheim sandstone has a mean grain size of 0.3 mm of and a porosity of 24%. All three sandstones are siliciclastic rocks, mainly containing quartz and feldspar, and can be representative for low-, medium- and high-porosity sandstone, respectively. Moreover, effects of cementation and clay content are less dominant for these three sand­ stones,25,28,29 thus facilitating the comparison. Similar to the method described in Section 3.2, the brittle strength data of Coconino and Bentheim sandstone are used to determine the strength parameters of the modified Hoek-Brown failure criterion. As

4. Discussion 4.1. Effects of porosity on the brittle-ductile transition pressure Porosity is an important property of sedimentary rocks. Numerous CTC tests had been conducted on sedimentary rocks such as sandstones with different porosities, revealing significant influences of porosity on peak stress, failure mode, deformability and brittle-ductile tran­ sition.5,61–63 However, the extent is still obscure to which the porosity can influence sandstone brittle-ductile transition under true triaxial 9

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curves corresponding to θσ ¼ �30� are also plotted to depict the strength features. For a given σ 3 , e.g., 0 MPa, the brittle strength of Zigong sandstone is much larger than that of Coconino sandstone, even larger than that of Bentheim sandstone under σ3 ¼ 15 MPa. For Bentheim pffiffiffiffi f sandstone, the brittle-ductile transition occurs when J2 is beyond 100 MPa, where Zigong and Coconino sandstone is still in the brittle regime. The brittle-ductile transition of Bentheim sandstone is also accompanied by the evolution of failure mode from shear bands, through shearpffiffiffiffi f enhanced compaction bands, to pure compaction bands. As J2 is raised up to about 208 MPa, Coconino sandstone starts to enter the brittle-ductile transition, exhibiting several parallel and conjugate shear bands. Although the applied σ3 is not large enough for Zigong sandstone to reach the brittle-ductile transition, it can be inferred that there is still large brittle range before the transition for its low porosity. As normally expected, without the clay weakening effect, a less porous and finegrained sandstone tends to be more brittle, and has a higher peak strength and transition pressure, which was also found in carbonate rocks.64 Furthermore, in Fig. 14, a dash line is plotted with each sandstone

Fig. 12. Variation of failure plane angle with σ 2 in each constant σ3 series.

shown in Fig. 13, the brittle strength envelopes of three kinds of sand­ stone are compared in the three-dimensional principal stress space. Common to all three kinds of sandstone is that the brittle strength en­ velope is shaped like an asymmetric bullet with the pointed apex and that the brittle strength rises with mean stress at a decreasing rate as the curvature of brittle strength envelope deceases. The brittle strength envelope for sandstone with lower porosity is remarkably lager, indi­ cating a larger brittle range. For comparison, the failure stress data of Coconino and Bentheim sandstone corresponding to the brittle-ductile transition are plotted as open symbols. Note that we use the term “failure stress” rather than “strength” since strain hardening is dominant in the transition. For Zigong sandstone, the mechanical behavior is still brittle when σ3 is raised up to 60 MPa, since the stress drop in the postpeak region is large. However, the failure stress for Coconino sandstone starts to level off when σ3 is beyond 80 MPa, signifying the brittle-ductile transition. Furthermore, when σ3 varies in the range of 60–150 MPa, the failure stress for Bentheim sandstone continuously drops after reaching a peak, mapping out a cap qualitatively. Since failure is dominantly due to compaction, the cap has been referred to as “compactive yield cap” and generally formulated by an elliptical cap model.4,5 pffiffiffiffi f f Clearer insight can be gained based on the J2 -I1 plane. As shown in pffiffiffiffi f f Fig. 14, all test data are projected onto the J2 -I1 plane. The best fitting

f

based on Mogi’s (2007) empirical relation (σ 1 -σ3 ¼ 3.4σ3 ) delineating the brittle-ductile boundary of silicate rocks only under CTC stress state. For both Coconino and Bentheim sandstone, it is found surprisingly that, when σ 2 ¼ σ3 , the intersection of the brittle strength envelope and this boundary line coincides well with the brittle-ductile transition, implying the applicability of this empirical relation to porous sandstone. Such intersection for the case of Zigong sandstone indicates a transition at σ 2 ¼ σ 3 ¼ 137 ​ MPa, which is much larger than the confining pressure in the experiment. To verify this predicted transition stress level, Oro­ wan’s frictional hypothesis is further utilized,65 assuming that the brittle-ductile transition occurs when the frictional resistance on the fault surface becomes higher than the shear strength. Benefiting from our elaborated experiments, residual/frictional strength data are plotted additionally (open triangles for CTC tests, filled triangles for true triaxial tests). Then it is found that the frictional strength data from CTC tests can be fitted rather well with a linear relation (dot dash line), giving a coefficient of 4.24, a little smaller than 4.4. The transition stress level is hence predicted as σ2 ¼ σ 3 ¼ 150 ​ MPa, which is close to the value predicted by Mogi’s relation. Since the transition pressure is higher when σ 1 ¼ σ 2 than when,σ2 ¼ σ3 24 this prediction method provides a least transition pressure for a given σ3 .

Fig. 13. Brittle strength envelopes of Zigong, Coconino, and Bentheim sandstone in the brittle regime, as well as failure stress data (open symbols) corresponding to the brittle-ductile transition. 10

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International Journal of Rock Mechanics and Mining Sciences 127 (2020) 104204

pffiffiffiffi f f Fig. 14. Comparison of failure stress of Zigong, Coconino, and Bentheim sandstone under true triaxial stress state on the J2 -I1 plane. The dash lines represent 1 Mogi’s empirical pressure boundary for brittle-ductile transition in silicate rocks under CTC tests, while the dot dash line for Zigong sandstone gives the best fitting frictional sliding trend when σ 2 ¼ σ 3 .

4.2. Comparison of the σ 2 effect on deformability

The enhanced effect of mean stress on compaction can be easily verified by visual inspection of the failure mode developed in specimens. For both Coconino and Bentheim sandstone under CTC tests, i.e., constant Lode angle, failure mode can be observed changing from a single shear band to compaction bands as mean stress rises. As for the Lode angle effect, an extensive suite of true axial tests had been performed in Coconino and Bentheim sandstone by applying a novel loading path, i.e., maintaining constant Lode angle throughout each test.26 Since mean stress was not a constant, a limited number of specimens coincidentally failed at almost the same mean stress had been selected from the true triaxial tests by Ma.66 It was found that, when σ2 ¼ σ 3 ¼ 120 MPa, there are several conjugate failure planes appearing close to one end of the Coconino sandstone specimen. In the increasing process of Lode angle θσ , a clearer and steeper single shear band gradually emerges, indicating the increase in Lode angle can embrittle the sandstone and make it prone to dilatancy. Moreover, a more evident trend can be observed in Ben­ theim sandstone, showing a transition from pure compaction band to shear band as Lode angle increases. This also demonstrates that sand­ stone tested under higher Lode angle is more brittle, suggesting a higher brittle-ductile transition pressure when Lode angle θσ is closer to 30� . Similar conclusion was also obtained by Ingraham et al.,24 who con­ ducted true triaxial tests on Castlegate sandstone with porosity of 26% by keeping mean stress constant. Consequently, it can be concluded that, as σ2 increases from σ3 to σ1 , if compaction due to the increased mean stress surpasses the shear-induced dilatancy due to the rise in the Lode angle, the degree of dilatancy upon failure decreases or the degree of compaction increases. On the contrary, a transition from compaction to dilatancy will occur as with the case of Coconino sandstone under σ 3 ¼ 100 and 150 MPa.

Further, deformability is investigated in terms of σ2 by comparing the abovementioned three sandstones. The amount of inelastic volu­ metric strain εv; ​ in upon failure is used as an index for estimating the volume change. For comparison, values of εv; ​ in upon failure of Coco­ nino and Bentheim sandstone are adopted as supplements to Fig. 5, which is replotted in Fig. 15. Dilatancy is observed in Zigong sandstone under σ 3 ¼ 20 and 60 MPa, which is expected given the rock porosity and stress levels. The degree of dilatancy decreases generally with σ2 . Similar trend can also be found in Coconino sandstone under σ 3 ¼ 50 MPa and in Bentheim sandstone under σ3 ¼ 30, 60 and 150 MPa, some of which even change from dilatancy to compaction. However, Coconino sandstone under higher σ3 (100 and 150 MPa) experiences an opposite trend, changing from compaction to dilatancy, as σ 2 rises from σ3 to σ1 . Both variations are probably attributed to the two competing pro­ cesses, i.e., the shear-induced dilatancy due to the rise in Lode angle and the compaction due to the rise in mean stress. In order to confirm this speculation, one should isolate the mean stress and Lode angle effect.

5. Conclusion True triaxial experiments were conducted with the aim of deter­ mining the deformability and strength on a low-porosity sandstone, Zigong sandstone. Using a Mogi-type TTA, the experiments contains three series of true triaxial tests in which σ 3 was maintained at 0, 20, and 60 MPa, respectively, while σ 2 was varied from σ 2 ¼ σ3 to σ 2 ¼ σ 1 from

Fig. 15. Variations of inelastic volumetric strain εv; ​ in upon failure with σ 2 for three sandstones. 11

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International Journal of Rock Mechanics and Mining Sciences 127 (2020) 104204

test to test. The experimental results indicate that the mechanical response of Zigong sandstone depends not only on σ3 but also on σ2 , which are summarized as follows:

Foundation of China (51774020) and Beijing Training Project for the Leading Talent in S & T (Z151100000315014). The authors thank Dr. Lu Shi at the Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, for his kind help with the true triaxial tests. We are grateful to Dr. Xiaodong Ma for providing his experimental data and for helpful comments on the manuscript. Special thanks to the editor Prof. Zim­ merman and anonymous reviewers for their critical comments that significantly improved the manuscript. Test data and Matlab code for 3D failure envelope are available from the first author [email protected] upon request.

1. A modified three-invariant Hoek-Brown failure criterion provides a general method to characterize the three-dimensional strength fea­ tures, avoiding individual fitting for every Lode angle as is done by Ma and Haimson.25 The modified Hoek-Brown failure criterion has been proved to be applicable to low-, medium-, and high-porosity sandstone in the brittle regime, extending its range of application. The quantification of brittle strength of three sandstones reveals a typical ascending-then-descending trend as σ 2 raises from σ 3 to σ1 , which is the combined effect of mean stress and Lode angle. Since strength in the brittle regime is proportional to the mean stress while inversely proportional to the Lode angle, when σ 2 is moderate, the strengthening effect due to the increased mean stress surpasses the weakening effect due to the rise in the Lode angle, leading to an increase in rock strength. And when σ 2 is larger than some value, the Lode angle effect starts to prevail, leading to a decreasing strength. Curiously, it is found that the frictional strength seems to be inde­ pendent of σ2 , which may be directly related to the fracture plane morphology and the stress state. 2. It is found that the failure plane angle decreases with σ3 in the CTC tests. As for the true triaxial tests, in the series of σ3 ¼ 0 MPa, vertical tensile fractures are found for all values of σ 2 , resembling the onionskin fractures, spalling, and slabbing near the underground excava­ tion boundary. When σ3 ¼ 20 MPa, it is found that the fracture plane angle generally increases with σ2 while there is no evident depen­ dence on σ2 when σ 3 ¼ 60 MPa. In particular, when σ2 ¼ σ3 , two parallel failure planes emerge within the rectangular prism specimen dipping in the σ 2 -direction and striking in the σ 3 -direction, which are unforeseen and totally different from other cases. After the SEM in­ spection, it is concluded that the rock fracturing process is mainly controlled by intrinsic heterogeneities when subjected to uniform stress state, while the differential stress dominates when σ2 is raised. 3. A less porous sandstone tends to be more brittle and less compactive, and has a higher brittle strength and transition pressure. The pre­ dicted brittle-ductile transition pressure based on frictional strength matches well with the value from the Mogi’s relation, which gives a least transition pressure for a given σ3 . Similar to the σ2 effect on peak strength, the dependence of deformability on σ2 also can be regarded as the combined effect of mean stress and Lode angle. As σ 2 increases from σ 3 to σ 1 , if compaction due to the increased mean stress surpasses the shear-induced dilatancy due to the rise in the Lode angle, the degree of dilatancy upon failure decreases or the degree of compaction increases. On the contrary, a transition from compaction to dilatancy occurs.

References 1 Mogi K. Experimental Rock Mechanics. London: Taylor & Francis; 2007. 2 Mollema PN, Antonellini MA. Compaction bands: a structural analog for anti-mode I cracks in aeolian sandstone. Tectonophysics. 1996;267(1- 4):209–228. 3 Eichhubl P, Hooker JN, Laubach SE. Pure and shear-enhanced compaction bands in Aztec Sandstone. J Struct Geol. 2010;32(12):1873–1886. 4 Wong TF, Baud P. The brittle-ductile transition in porous rock: a review. J Struct Geol. 2012;44:25–53. 5 Wong TF, David C, Zhu W. The transition from brittle faulting to cataclastic flow in porous sandstones: mechanical deformation. J Geophys Res. Solid Earth. 1997;102 (B2):3009–3025. 6 Gu�eguen Y, Dormieux L, Bout�eca M. Fundamentals of poromechanics. In: Gu�eguen Y, Bout� eca M, eds. Mechanics of Fluid-Saturated Rocks. Amsterdam: Elsevier; 2004:1–54. 7 Grueschow E, Rudnicki JW. Elliptic yield cap constitutive modeling for high porosity sandstone. Int J Solids Struct. 2005;42(16-17):4574–4587. 8 Baud P, Vajdova V, Wong TF. Shear-enhanced compaction and strain localization: inelastic deformation and constitutive modeling of four porous sandstones. J Geophys Res. Solid Earth. 2006;111(B12):401. 9 Murrell S. A criterion for brittle fracture of rocks and concrete under triaxial stress, and the effect of pore pressure on the criterion. In: Fairhurst C, ed. Proceedings of the 5th Symposium on Rock Mechanics. Minneapolis, MN: University of Minnesota; 1963: 563–577. 10 Mogi K. Effect of the intermediate principal stress on rock failure. J Geophys Res. 1967;72(20):5117–5131. 11 Jimenez R, Ma X. A note on the strength symmetry imposed by Mogi’s true-triaxial criterion. Int J Rock Mech Min Sci. 2013;64:17–21. 12 Haimson BC. The hydrofracturing stress measuring method and recent field results. Int J Rock Mech Min Sci Geomech Abstr. 1978;15(4):167–178. 13 McGarr A, Gay NC. State of stress in the Earth’s crust. Annu Rev Earth Planet Sci. 1978;6(1):405–436. 14 Brace WF, Kohlstedt DL. Limits on lithospheric stress imposed by laboratory experiments. J Geophys Res. Solid Earth. 1980;85(B11):6248–6252. 15 Vernik L, Zoback MD. Estimation of maximum horizontal principal stress magnitude from stress-induced well bore breakouts in the Cajon Pass scientific research borehole. J Geophys Res. Solid Earth. 1992;97(B4):5109–5119. 16 Takahashi M, Koide H. Effect of the intermediate principal stress on strength and deformation behavior of sedimentary rocks at the depth shallower than 2000 m. In: ISRM International Symposium. Pau, France; 30 August-2 September. 1989:19–26. 17 Xu D, Xing Z, Li X, Zhang G, Wei M. Development of RT3 type rock high pressure true triaxial machine. Rock Soil Mech. 1990;11(2):1–14. 18 Wawersik WR, Carlson LW, Holcomb DJ, Williams RJ. New method for true-triaxial rock testing. Int J Rock Mech Min Sci. 1997;34(3-4), 330. e1- 330. e14. 19 Haimson B, Chang C. A new true triaxial cell for testing mechanical properties of rock, and its use to determine rock strength and deformability of Westerly granite. Int J Rock Mech Min Sci. 2000;37(1-2):285–296. 20 Feng X, Zhang X, Kong R, Wang G. A novel mogi type true triaxial testing apparatus and its use to obtain complete stress–strain curves of hard rocks. Rock Mech Rock Eng. 2016;49(5):1649–1662. 21 Shi L, Li X, Bing B, Wang A, Zeng Z, He H. A Mogi-type true triaxial testing apparatus for rocks with two moveable frames in horizontal layout for providing orthogonal loads. Geotech Test J. 2017;40(4):542–558. 22 Browning J, Meredith PG, Stuart CE, Healy D, Harland S, Mitchell TM. Acoustic characterization of crack damage evolution in sandstone deformed under conventional and true triaxial loading. J Geophys Res. Solid Earth. 2017;122(6): 4395–4412. 23 Oku H, Haimson B, Song SR. True triaxial strength and deformability of the siltstone overlying the Chelungpu fault (Chi-Chi earthquake), Taiwan. Geophys Res Lett. 2007; 34:L09306. 24 Ingraham M, Issen K, Holcomb D. Response of Castlegate sandstone to true triaxial states of stress. J Geophys Res. Solid Earth. 2013;118(2):536–552. 25 Ma X, Haimson BC. Failure characteristics of two porous sandstones subjected to true triaxial stresses. J Geophys Res. Solid Earth. 2016;121(9):6477–6498. 26 Ma X, Rudnicki JW, Haimson BC. Failure characteristics of two porous sandstones subjected to true triaxial stresses: applied through a novel loading path. J Geophys Res. Solid Earth. 2017;122(4):2525–2540. 27 Xu YH, Cai M, Zhang XW, Feng X. Influence of end effect on rock strength in true triaxial compression test. Can Geotech J. 2017;54(6):862–880. 28 Zhang S, Wu S, Zhang G, Guo P, Chu C. Three-dimensional evolution of damage in sandstone Brazilian discs by the concurrent use of active and passive ultrasonic techniques. Acta Geotech. 2018:1–16.

Our experiments are fundamental to a more rational characterization of strength and deformability of low-porosity sandstones under general stress states. In addition, both the intact rock strength and frictional strength under true triaxial stresses are provided, giving the upper and lower bound of crustal stress in this sedimentary layer and facilitating scientific research and industrial activities in the near future. Declaration of competing interest We declare that we do not have any commercial or associative in­ terest that represents a conflict of interest in connection with the manuscript entitled “Strength and deformability of a low-porosity sandstone under true triaxial compression conditions”, and that this manuscript is approved by all listed authors for publication. Acknowledgement The research is supported by the National Natural Science 12

S. Zhang et al.

International Journal of Rock Mechanics and Mining Sciences 127 (2020) 104204 48 B� esuelle P, Desrues J, Raynaud S. Experimental characterisation of the localisation phenomenon inside a Vosges sandstone in a triaxial cell. Int J Rock Mech Min Sci. 2000;37(8):1223–1237. 49 Baud P, Klein E, Wong T. Compaction localization in porous sandstones: spatial evolution of damage and acoustic emission activity. J Struct Geol. 2004;26(4): 603–624. 50 Tembe S, Baud P, Wong TF. Stress conditions for the propagation of discrete compaction bands in porous sandstone. J Geophys Res. Solid Earth. 2008;113:B9409. 51 Rudnicki JW, Rice JR. Conditions for the localization of deformation in pressuresensitive dilatant materials. J Mech Phys Solids. 1975;23(6):371–394. 52 Challa V, Issen KA. Conditions for compaction band formation in porous rock using a two-yield surface model. J Eng Mech. 2004;130(9):1089–1097. 53 Haimson B. True triaxial stresses and the brittle fracture of rock. Pure Appl Geophys. 2006;163(5-6):1101–1130. 54 Martin CD. The strength of massive Lac du Bonnet granite around underground openings. PhD Thesis. Winnipeg: University of Manitoba; 1993. 55 Read RS, Martin CD. Technical Summary of AECL’s Mine-By Experiment Phase I: Excavation Response. Pinawa, MB (Canada): Atomic Energy of Canada Ltd; 1996. 56 Read RS. 20 years of excavation response studies at AECL’s Underground Research Laboratory. Int J Rock Mech Min Sci. 2004;41(8):1251–1275. 57 Lee M, Haimson B. Laboratory study of borehole breakouts in Lac du Bonnet granite: a case of extensile failure mechanism. Int J Rock Mech Min Sci Geomech Abstr. 1993; 30(7):1039–1045. 58 Song I. Borehole Breakouts and Core Disking in Westerly Granite: Mechanisms of Formation and Relationship in Situ Stress. PhD thesis. Madison, Wisconsin: University of Wisconsin- Madison; 1998. 59 Chang C, Haimson B. True triaxial strength and deformability of the German Continental Deep Drilling Program (KTB) deep hole amphibolite. J Geophys Res. Solid Earth. 2000;105(B8):18999–19013. 60 Lee H, Haimson BC. True triaxial strength, deformability, and brittle failure of granodiorite from the San Andreas fault observatory at depth. Int J Rock Mech Min Sci. 2011;7(48):1199–1207. 61 Logan JM. Porosity and the brittle-ductile transition in sedimentary rocks. In: AIP Conference Proceedings. New York. 1987:229–242. 62 Scott TE, Nielsen KC. The effects of porosity on the brittle-ductile transition in sandstones. J Geophys Res. Solid Earth. 1991;96(B1):405–414. 63 Baud P, Wong T, Zhu W. Effects of porosity and crack density on the compressive strength of rocks. Int J Rock Mech Min Sci. 2014;67:202–211. 64 Vajdova V, Baud P, Wong TF. Compaction, dilatancy, and failure in porous carbonate rocks. J Geophys Res. Solid Earth. 2004;109:B5204. 65 Orowan E. Chapter 12: mechanism of seismic faulting. Mem Geol Soc Am. 1960;79(1): 323–345. 66 Ma X. Failure Characteristics of Compactive, Porous Sandstones Subjected to True Triaxial Stresses. PhD Thesis. Madison, Wisconsin: The University of Wisconsin-Madison; 2014.

29 Zhang S, Wu S, Chu C, Guo P, Zhang G. Acoustic emission associated with selfsustaining failure in low-porosity sandstone under uniaxial compression. Rock Mech Rock Eng. 2019;52(7):2067–2085. 30 Mogi K. Fracture and flow of rocks under high triaxial compression. J Geophys Res. 1971;76(5):1255–1269. 31 Brady BT. Effects of inserts on the elastic behavior of cylindrical materials loaded between rough end-plates. Int J Rock Mech Min Sci Geomech Abstr. 1971;8(4): 357–369. 32 Choi S, Thienel K, Shah SP. Strain softening of concrete in compression under different end constraints. Mag Concr Res. 1996;48(175):103–115. 33 Hawkes I, Mellor M. Uniaxial testing in rock mechanics laboratories. Eng Geol. 1970; 4(3):179–285. 34 Glomb J, Patas P. Experimental analysis of the determination of a test specimen as a method of verifying the compressing conditions while testing concrete in multiaxial stress conditions. In: Proceedings of the RILEM Symposium on the Deformation and the Rupture of Solids Subjected to Multiaxial Stress. Cannes; 4- 6 October. 1971:79–91. 35 Labuz JF, Bridell J. Reducing frictional constraint in compression testing through lubrication. Int J Rock Mech Min Sci Geomech Abstr. 1993;30(4):451–455. 36 Wawersik WR, Fairhurst CH. A study of brittle rock fracture in laboratory compression experiments. Int J Rock Mech Min Sci Geomech Abstr. 1970;7:561–575. 37 Ohnaka M. A constitutive scaling law and a unified comprehension for frictional slip failure, shear fracture of intact rock, and earthquake rupture. J Geophys Res. Solid Earth. 2003;108(B2):2080. 38 Mogi K. Dilatancy of rocks under general triaxial stress states with special reference to earthquake precursors. J Phys Earth. 1977;25(Suppl):203–217. 39 Schock RN, Heard HC, Stephens DR. Stress-strain behavior of a granodiorite and two graywackes on compression to 20 kilobars. J Geophys Res. 1973;78(26):5922–5941. 40 Kwaniewski M, Takahashi M, Li X. Volume changes in sandstone under true triaxial compression conditions. In: Proceedings of the ISRM 2003, Technology Roadmap for Rock Mechanics. South African Institute of Mining and Metallurgy; 2003:683–688. 41 Wu S, Zhang S, Zhang G. Three-dimensional strength estimation of intact rocks using a modified Hoek-Brown criterion based on a new deviatoric function. Int J Rock Mech Min Sci. 2018;107:181–190. 42 Wu S, Zhang S, Guo C, Xiong L. A generalized nonlinear failure criterion for frictional materials. Acta Geotech. 2017;12(6):1353–1371. 43 Hoek E, Brown ET. Empirical strength criterion for rock masses. J. Geotech. Geoenviron. 1980;106:1013–1035. 44 Lade PV. Modelling the strengths of engineering materials in three dimensions. Mech Cohesive-Frict Mater. 1997;2(4):339–356. 45 Yu M, Zan Y, Zhao J, Yoshimine M. A unified strength criterion for rock material. Int J Rock Mech Min Sci. 2002;39(8):975–989. 46 Colmenares LB, Zoback MD. A statistical evaluation of intact rock failure criteria constrained by polyaxial test data for five different rocks. Int J Rock Mech Min Sci. 2002;39(6):695–729. 47 Chang C, Haimson B. Non-dilatant deformation and failure mechanism in two Long Valley Caldera rocks under true triaxial compression. Int J Rock Mech Min Sci. 2005; 42(3):402–414.

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