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ISRM SUGGESTED METHOD
ISRM Suggested Method for Determining Stress–Strain Curvesfor Rocks Under True Triaxial Compression
Xiwei Zhang1 • Xia-Ting Feng1 • Xiaochun Li2 • Bezalel Haimson3 •
Kenichiro Suzuki4
� Springer-Verlag GmbH Austria 2017
1 Introduction
The stress states r1[ r2 = r3 and r1[ r2[ r3 (where
r1, r2, and r3 are the principal stresses) are defined as the
conventional triaxial stress state and the true triaxial stress
state, respectively. Conventional triaxial compression
testing has been widely used since Karman (1911) devel-
oped the first conventional triaxial apparatus, which offered
appealing usability and operability due to the intermediate
principal stress being equal to the minor principal stress,
r2 = r3. Murrell (1963) confirmed the effect of the inter-
mediate principal stress r2 on rock strength by analysing
the experimental data from Boker (1915) for a triaxial
extension stress state (r1 = r2[ r3). Some geotechnical
engineering studies in Japan during 1960s extended to the
true triaxial framework (Akai and Mori 1967; Akai 1968;
Kawamoto and Tomita 1970; Kawamoto et al. 1970),
which prompted Mogi (1970, 1971) to design and build a
true triaxial apparatus. Mogi’s experimental work first
demonstrated the effect of r2 on the yield and failure
characteristics of rocks. Since then, various types of true
triaxial apparatuses have been developed to test the
mechanical behaviours of rocks under the general stress
true triaxial state (r1[ r2[r3) to meet a variety of
research demands.
The design of a true triaxial apparatus begins with the
experimental requirements, such as the size and shape of
the specimen and the loading and control methods. How-
ever, some critical technical difficulties have yet to be
solved; thus, the effectiveness of the existing true triaxial
apparatuses varies.
True triaxial apparatuses can be used to study rock
behaviour under specific 3D boundary stress conditions,
such as the plane strain problem, in which the deformation
on the plane on which the intermediate principal stress is
applied is kept constant (Rice and Rosengren 1968; Labuz
et al. 1996; Makhnenko and Labuz 2014); the violent
fracture condition that occurs during rapid unloading on
one surface of the specimen (Alexeev et al. 2004; He et al.
2010; Du et al. 2015; Su et al. 2017); hydraulic fracturing
(Frash et al. 2014); and permeability (King et al. 1995).
Furthermore, true triaxial apparatuses have been exten-
sively used to determine the constitutive behaviour of
rocks, which includes the stress path, deformation,
strength, and post-peak behaviour (Mogi 1970; Esaki et al.
1988; Takahashi and Koide 1989; Haimson and Chang
2000; Kwasniewski et al. 2003; Ingraham 2012; Young
et al. 2013; Nasseri et al. 2014; Feng et al. 2016). The
loading methods used by true triaxial apparatuses, i.e. rigid
plate (Type-I), flexible medium (Type-II), and mixed type
Please send any written comments on this ISRM Suggested Method to
Prof. Resat Ulusay, President of the ISRM Commission on Testing
Methods, Hacettepe University, Department of Geological
Engineering, 06800 Beytepe, Ankara, Turkey. Email:
[email protected] .
& Xia-Ting Feng
[email protected] ; [email protected]
1 Key Laboratory of Ministry of Education for Safe Mining of
Deep Metal Mines, Northeastern University,
Shenyang 110819, Liaoning, China
2 State Key Laboratory of Geomechanics and Geotechnical
Engineering, Institute of Rock and Soil Mechanics, Chinese
Academy of Sciences, Wuhan 430071, Hubei, China
3 Geological Engineering Program, Department of Materials,
Science and Engineering, University of Wisconsin, 225
Materials Science and Engineering Building, 1509 University
Avenue, Madison, WI 53706, USA
4 Geotechnical Department, Obayashi Corporation Technical
Research Institute, Tokyo 204-8558, Japan
123
Rock Mech Rock Eng
DOI 10.1007/s00603-017-1282-3
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(Type-III), were summarized by Takahashi and Koide
(1989) and Mogi (2007). The different loading methods
lead to structural diversity in the apparatuses.
The length (width)-to-height ratios of the specimens used
in the true triaxial tests described above were approximately
equal to the diameter-to-height ratios (1:2) of the specimens
used in conventional triaxial tests. This similarity allows the
mechanical behaviours of the two test methods to be com-
pared. As mentioned by Wawersik and Fairhurst (1970),
specimens that exhibit Class II behaviour tend to respond in a
brittle fashion to axial loading. The Classes I and II failure
behaviours of rocks were often observed in true triaxial
compression tests as well. The unconfined compressive
strength of hard rock is in a wide range of 30–100 MPa
(Protodyakonov 1962; The National Standards Compilation
Group of People’s Republic of China 2014; Kaiser and Kim
2015), as hard rock failure has significant brittle characteristic,
it is worth to investigate the Class II failure under true triaxial
condition. True triaxial apparatuses based on stress loading by
rigid platens and flexible media are more popular for deter-
mining the stress–strain curves for rocks under true triaxial
compression.1
The true triaxial apparatus developed by Mogi (1970) is
advantageous for loading tests with high stress and volume
change measurements. Since the end friction cannot be
neglected when employing the rigid and flexible mixed
loading methods, an appropriate anti-friction agent must be
used to obtain high-quality data. Atkinson and Ko (1973)
proposed the design of a fluid cushion, multiaxial cell for a
cubic rock specimen and used coal specimens for verifi-
cation. Because the friction between the fluid cushion and
the specimen was much lower than that of the steel-rock
interface, two types of seal materials were used: an injec-
tion moulded polyurethane seal for lower pressures (up to
20.7 MPa) and a leather-vinyl combination seal for higher
pressures (up to 82.7 MPa). However, this type of design is
not appropriate for measuring the strength of hard rocks,
which are subjected to much higher minor and intermediate
principle stresses. For example, the true triaxial compres-
sive strength of Dunham dolomite can reach 900 MPa
under r3 = 145 MPa and r2 = 400 MPa (Haimson 2006).
Haimson and Chang (2000) developed true triaxial appa-
ratuses that captured the post-peak behaviour of rocks; Li
et al. (2012) and Feng et al. (2016) developed true triaxial
apparatuses for larger specimens and removed the loading
gap that existed in previous designs.
The research on the influence of r2 on rock strength
prompted a new round of development for true triaxial testing
machines and expanded their application in rock mechanics
(Colmenares and Zoback 2002; Kwasniewski et al. 2003;
Haimson 2006; Cai 2008; Descamps et al. 2012). Brittle
fracturing of hard rock under true triaxial compression is
unstable and usually creates a localized feature; therefore,
specific methods are needed to achieve stable control.
The post-peak behaviour is not the intrinsic constitutive
behaviour of hard rock materials. However, the fracture
process captured by a stiff, servo-controlled true triaxial
apparatus still provides valuable information, even con-
sidering localization effects. The post-peak curve with
decreasing stress can be used to analyse the energy released
during crack propagation and to evaluate the brittle evo-
lution, which is referred to as the rock burst mechanism in
deep rock engineering (Feng and Hudson 2010).
The failure criterion proposed by Haimson and Bobet
(2012) has been verified using experimental data under the
general stress state. The true triaxial testing approach can
be used to study current engineering and scientific prob-
lems in deep tunnelling and mining, such as the fracture
mode and deformation characteristics related to different
loading and unloading paths, the influence of the geo-stress
on hydraulic fracturing, and the mechanical behaviour on
the deviatoric plane perpendicular to the hydrostatic axis.
To promote and regulate true triaxial testing, this sug-
gested method describes some key technical issues related to
suppressing off-centre movement, reducing the end friction
effect, removing the loading gap of the specimen, and
accurately measuring the volume change of the specimen
during the course of the test. The suggested method is
intended to clarify some influencing factors and standardize
the testing methodology to produce high-quality data that
can be used to understand failure under real geo-stress fields.
2 Scope
This suggested method focuses on using a true triaxial
apparatus to determine the stress–strain curves for rocks,
especially hard rocks, under true triaxial compression. This
1 ‘‘Force–displacement’’, ‘‘load–deformation’’ and ‘‘stress–strain’’
relationships have been widely used to characterize the mechanical
behaviour of rock materials (Fairhurst and Hudson 1999; Labuz and
Biolzi 2007). The term ‘‘stress–strain’’ refers to calculated data,
whereas ‘‘force–displacement’’ refers to measured data. The local-
ization of deformation into a shear band in the post-peak stage of
brittle rocks, many be considered a result of an instability of
homogeneous deformation (Rudnicki and Rice 1975). In this case, the
cross-sectional area and the displacement increments are not properly
calculated during localized failure. The ‘‘stress–strain’’ definition in
the post-peak stage for the localization mode is not completely
accurate as it may include the opening of cracks. Thus, the term
‘‘force–displacement’’ might be more appropriate, but accurately
reflecting the relative motion of the slip surface is still difficult due to
the localized failure. To maintain consistency in the definitions of the
pre- and post-peak behaviours of brittle rocks and by following the
recommendations of other ISRM Suggested Methods and the
consensus of the ISRM Testing Method Commission, the term
‘‘stress–strain’’ is still used in this ISRM Suggested Method.
However, the stress and strain in the post-peak region must be
carefully clarified as calculated values.
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suggested method is limited to rock specimens subjected to
stress loadings with a negligible creep effect.
3 Apparatus
The output force and stiffness of a true triaxial apparatus
must match the size and strength of a typical hard rock
specimen. Three principal stresses are independently loa-
ded using servo-control techniques. In addition, some
special issues need to be considered: establishing a fixed
reference point at the centre of the specimen during load-
ing, measuring the three principal strains, reducing the end
friction effect, and removing the loading gap. Geophysical
measurements (e.g. acoustic emission and p and s wave
velocities) and permeability measurements can also be
introduced into a true triaxial apparatus system.
3.1 General Structure of Mixed Rigid and Flexible
Loadings
The operation of the apparatus consists of two stresses that
are applied by rigid platens, while the third stress is applied
using a flexible medium, which uses oil pressure that is
applied to a membrane sealed onto the specimen. This
system prevents interference from a third pair of rigid
platens.
The next step is to ensure that the centre of the specimen
maintains a constant position during compression. Two
approaches can be used to suppress off-centre specimen
movement. The first approach uses four or six actuators in a
biaxial or true triaxial apparatus (three loading frames) to
simultaneously compress the specimen; in this case, the
true triaxial apparatus uses two modes of force and dis-
placement control on three independent pairs of actuators.
Young et al. (2013) and Nasseri et al. (2014) adopted this
technique in a multi-axis loading frame. The second
method is to use two actuators with mobile or floating
loading frames (Mogi 1971, 2007). In this method, two
orthogonal loading frames are arranged horizontally (Li
et al. 2012) or vertically (Feng et al. 2016). Regardless of
how the loading frames are arranged, the major principal
stress is applied along the longitudinal direction of the
specimen.
3.2 Loading System Stiffness
Wawersik and Fairhurst (1970) used a high-stiffness test
machine to obtain the complete stress–strain curves of
brittle rocks under uniaxial compression, the typical Clas-
ses I and II curves are shown in Fig. 1. The draft ISRM
Suggested Method for obtaining the complete stress–strain
curve for intact rock under uniaxial compression (Fairhurst
and Hudson 1999) includes both a high-stiffness loading
frame and a closed-loop servo-control system. This method
indicates the importance of using a high-stiffness loading
frame to capture the post-peak failure behaviour of brittle
rocks because a high-stiffness loading frame can reduce the
stored elastic energy under very high stress conditions. At a
high loading frame stiffness, a decreasing amount of pos-
itive work acts on the specimen when failure occurs.
Labuz and Biolzi (1991) analytically showed that in
addition to the stiffness of the testing machine, the size and
geometry of the specimen are related to the Class II brittle
failure mode. The shape, side length ratio, and size of the
specimen are suggested for comparing the mechanical
behaviours of different types of true triaxial apparatuses, as
discussed in the following sections.
This suggested method proposes a loading system
stiffness concept to supersede the original stiffness of the
loading frames, which is critical but not unique. The
compressibility of the hydraulic oil plays an important role
in releasing the hydraulic elastic energy in the loading
system and affects the post-peak failure behaviour.
The stiffness of the loading system depends on various
aspects of the loading system, which consists of a loading
frame, hydraulic oil, hydraulic actuator, accumulator, and
oil tube. These factors must be considered in order to
match the stiffness of the test machine to that of the rock
specimens. The stress–strain curves for hard rocks that are
obtained from true triaxial tests will contribute to a better
understanding of the mechanical behaviours of these rocks
under the general stress condition. The standard configu-
ration that is recommended for true triaxial tests is a high-
stiffness system with a closed-loop servo-controlled sys-
tem. As a result of improved production technologies, the
stiffness of the loading frames is recommended to be 5–10
times greater than that of the hard rock specimen.
Medium viscosity hydraulic oil with a viscosity range
from 40 mm2/s at 40 �C to 7 mm2/s at 100 �C is suggested
Fig. 1 Classification of Classes I and II behaviour of rock failure in
uniaxial compression. Replot from Wawersik (1968)
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for use. The volume of oil used in the actuator should be
minimized, which can be achieved most effectively by
reducing the stroke of the actuator to approximately
50 mm. When selecting the high-pressure oil tube, the
length, tube stiffness and matching the inner diameter of
the hydraulic circuits need to be considered. The diameter
and cross section of the tubes should ensure a fast reaction
at the actuator when the servo valve is opening or closing
to optimize the oil volume versus response characteristics.
3.3 Hydraulic System
The hydraulic loading system is composed of a hydraulic
manifold, an accumulator and a high-speed, high-fre-
quency–response servo valve. Fairhurst and Hudson (1999)
suggested various specifications for the hydraulic system,
and the following issues were highlighted: the static actu-
ator should be sealed using a low-friction sealing material,
and the accumulator should be fixed for a short duration
using a fluid pulse absorber, and the nitrogen pressure
range should be from 4 to 8 MPa. The servo valve is a key
part of the hydraulic system, and its actual flow is depen-
dent on an electrical command signal and the valve pres-
sure drop. To control the post-peak behaviour, the
suggested response time is in the range of 6–10 ms, and the
overshoot of the servo valve should be tuned both in static
and dynamic control prior to the formal tests.
The confining cell is an important component of a
hydraulic system and should receive substantial attention in
the design of a four-piston structure. An auto-compensation
structural design (Secq 2010) is used to ensure that the
pistons in a triaxial apparatus remain stationary as the
confining pressure increases, which is convenient during
specimen installation. Accordingly, the output force of the
loading frames is a differential stress (r1 - r3 or r2 - r3)that is calculated based on the cross-sectional area of the
specimen. In contrast, when using pistons without an auto-
compensation structure, the piston diameter must match the
area of the specimen and the piston when applying the
confining pressure. Furthermore, the output force from the
loading frames is the total stress, i.e. r1 or r2, which is
calculated based on the cross-sectional area of the
specimen.
3.4 Servo-Control System
Servo-control systems are employed in most true triaxial
apparatuses, and a typical system consists of a feedback
signal (from specified transducers), a controller (that
compares deviations between the measured and target
values), and a performing unit (servo valve or servo
motor). To capture the post-peak failure of hard rocks
under true triaxial compression, the concept of closed-loop
control must be characterized in detail. The closed-loop
transfer function relies on the stability and monotonicity of
the control system, which are related to the rock and
machine stiffnesses. The instability of the brittle fracture
process in hard rocks is a significant challenge for con-
trolling the post-peak failure stage. In general, the mono-
tonous measured value should be compared with the target
value in real time, and the difference between them (the
error signal) is used as a feedback variable that is an input
for the transfer function.
The control system uses a negative feedback algorithm
and generates an actuating signal to drive the servo valve
and to achieve stable closed-loop control. For a Class I
failure, the displacement in the direction of the major
principal stress can be used as a feedback variable because
of its monotonicity. For a Class II failure, the average
lateral displacement is widely used as a feedback variable
due to the dilation that occurs during compression of the
rock material. However, the displacement in the major
principal stress direction is non-monotonic (called snap-
back) during the rapid extraction of energy from the rock
specimen. Therefore, the constant minor principal strain
rate control method is recommended for Class II failure.
Okubo and Nishimatsu (1985) proposed a linear combi-
nation of stress and strain as the control variable with a
fixed modulus value that is between the tangential Young’s
modulus before and after the peak strength.
A 5-kHz sampling rate for each transducer is recom-
mended to ensure that the proportional-integral-derivative
(PID) controller sends a command every 0.2 ms (which is
equivalent to a sampling rate of 5 kHz = 0.0002 s). The
servo valve performs an action based on the feedback
command. As mentioned in Sect. 3.3, the performance of
the servo valve should match the sampling rate and accu-
racy of the controller and transducer.
The feedback signal selection and proper tuning of the
closed-loop system are critical. The PID parameters must
be readjusted according to the strength, stiffness, and
brittleness of the specimen. The fitness of the PID param-
eters and tuning of the system should be optimized, but
details of this optimization are beyond the scope of this
suggested method.
3.5 Measurement Transducers and Data Logger
To determine the stress–strain curves of rocks under true
triaxial compression, a minimum of three principal stresses
and three principal strains should be measured. In addition,
some other transducers are used to control and monitor the
piston movement. In the major principal stress loading
subsystem (Fig. 2), two load cells, which are denoted 01
and 02, are used to monitor the forces on both sides of the
specimens. The load cells can be arranged inside and
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outside of the confining cell, as shown in Fig. 2a, b. When
using the outside arrangement, the friction on the pistons of
the confining cell is measured. This force can also be
obtained by measuring the hydraulic pressure in each
actuator with pressure transducers and by using a correc-
tion function for the friction effects, as shown in Fig. 2c. At
low load levels, the accuracy of this method is somewhat
reduced by the friction from all the relevant sealing inter-
faces, but this measurement technique is much cheaper
than using load cells.
Three Linear Variable Differential Transformers
(LVDTs) are used to measure the movement of the actuator
piston and the two pistons in the confining cell. A similar
design plan is selected for the transducers in the interme-
diate principal stress loading subsystem. Pressure trans-
ducers are used to measure the confining pressure in the
subsystem for the minor principal stress loading, and the
LVDTs are used to monitor the movement of the pressure
intensifier.
Because deformation plays a crucial role in a constitu-
tive model, the requirements for the volume change mea-
surement in true triaxial tests must be stated in detail. The
major principal strain (e1) and the intermediate principal
strain (e2) can be measured using strain gauges (Haimson
and Chang 2000), strain-gauge-type displacement trans-
ducers (Kwasniewski et al. 2003; Li et al. 2012; Mogi
1971), or LVDTs (Feng et al. 2016). Young et al. (2013)
used nine independent LVDTs to monitor deformation
along the three axes (with three LVDTs in each direction)
in a true triaxial apparatus cell. Deformation in the minor
principal strain (e3) direction was recorded as a computed
feedback signal in the servo-control system to capture the
stress–strain curves. Measuring deformation in the e3direction is very difficult due to the positioning of the
sensor in 3D space; furthermore, the limited space in the
confining cell and the separation of the e3 sensor from the
other sensors need to be considered. e3 can be measured
using a split cantilever beam sensor (Feng et al. 2016) or
the flexible side of a beryllium-copper strain-gauged beam
(Haimson and Chang 2000) (Figs. 3, 4 5). The beam sensor
can measure the deformation of the centre point of the
specimen in the e3 direction.The data logger is an electronic device that records data
over time with a built analogue-to-digital converter. In
general, advanced controllers have an ancillary function for
acquiring data; hence, the data logger is a sub-module of
the advanced controllers. The servo-control feedback sig-
nal is produced at a sampling rate of 5 kHz; however, this
is too high a rate for recording experimental data, and some
information may become useless due to noise. The data
should be stored at a specified sampling rate.
3.6 End Friction Effect Reduction
The mismatch of the elastic parameters (Young’s modulus
and Poisson’s ratio) between the metal platens and the rock
specimen produces interface friction during loading, which
results in a non-uniform stress distribution at the specimen
ends, which is defined as the end friction effect.
To demonstrate the end friction effect on the true triaxial
test results, Fig. 6 shows the differential stress (r1 - r3)and strain (e1, e2, and e3) relationships from a typical
dataset for two granite specimens that underwent com-
pression at r3 = 50 MPa and r2 = 200 MPa. These results
Fig. 2 Examples of force measurement in the major principal stress
loading subsystem, for simplicity, the associated part on the
intermediate principal stress is removed: a direct measurement using
load cells that are installed in the cell; b direct measurement using
load cells that are installed outside the cell; c indirect measurement
using a pressure sensor
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show a pronounced end friction effect. The stress–strain
curves for rocks under true triaxial compression that con-
sider the end friction effect show higher strengths and
constrained deformations (Feng et al. 2017).
Various end friction reduction methods, such as apply-
ing stearin (stearic acid) (Foppl 1900); Teflon sheets, which
are also called PTFE (Mogi 1971, 2007); or a mixture of
stearic acid and Vaseline, can be used to reduce the end
friction effect (Labuz and Bridell 1993; Haimson and
Chang 2000). The mean friction coefficients of the Teflon
and the mixture of stearic acid and Vaseline are 0.043 and
0.018, respectively (Feng et al. 2017). Using the mixture of
stearic acid and Vaseline as an anti-friction agent is rec-
ommended because it results in a lower friction coefficient
than using Teflon sheets. This difference could be related
to the super fluidity and heat conductivity properties of
Fig. 3 Deformation
measurement in the minimum
principal stress direction: a after
Mogi (1971); b after Haimson
and Chang (2000), where
a strain gauges, b rock
specimen, c strain-gauged
beam, and d fixed pins for
strain-gauged beam
Fig. 4 Volume change measurement with strain-gauged transducers,
after Kwasniewski et al. (2003). Schematic view of the specimen
assembly used in true triaxial tests: a view in the r3 direction; b and
c view the in r2 direction; 1 top steel end piece, 2 rock sample, 3 thin
copper foil, 4 thin Teflon foil, 5 silicone rubber jacket, 6 and 11 lateral
(r2) steel end pieces, 7 bottom steel end piece, 8 strain-gauge
displacement transducers, which are seated in sockets fixed to the top
and bottom end pieces to measure axial strain e1, 9 strain-gauge
displacement transducer, which is seated in coned sockets cemented
onto the specimen to measure lateral strain e3, 10 strain-gauge
displacement transducers, which are seated in sockets fixed to the
lateral end pieces to measure lateral strain e2
Fig. 5 Volume change measurement transducers attached to the
overlapping platens, after Feng et al. (2016)
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Teflon, which promote the formation of a hydro-dynamic
film between two sliding surfaces.
3.7 Loading Gap Removal
Typically, four high-strength alloy steel platens are used to
transfer loads from the circular pistons to the rock specimen
in the directions of the major and intermediate principal
stresses. To avoid mutual interference during specimen
compression, a majority of the shorter platens in true triaxial
experimental set-ups exhibit a loading gap (Fig. 7a) or a
bevelled design (Fig. 7b) in the 2D projection plane; both of
these cases are also called blank corners. Because the area of
the loading platens does not completely cover the specimen,
the decrease in r2 due to this small gap may lead to a 10 to
15% decrease in the peak strength (Feng et al. 2016).
A structure for overlapping platens (Fig. 8) was sug-
gested to overcome the loading gap effect (Li et al. 2012;
Feng et al. 2016). During specimen compression, the pla-
tens fully cover the four surfaces of the specimen by
driving the platen with the edge of another platen, thus
forming an overlapping structure.
The dimensions of the metal platen should be slightly
greater than those of the rock specimen. In general, a 0.5-
mm margin in r3 direction is maintained on each side to
cover the swelled specimen during compression. A Rock-
well hardness (HRC) of 60 is suggested.
4 Specimens
4.1 Preparation
The specimens used in true triaxial tests are rectangular
prisms. First, a slightly oversize specimen is obtained from a
large block of rock using a digital rock sawingmachine; then,
a grinding machine is used to process the specimen to the
required geometric dimensions and tolerance. The dimen-
sional tolerance is referred to the International Tolerance
(IT) grades table (ISO 286). The recommended grade is IT6,
which specifies tolerances for associated manufacturing
processes and a given dimension, such as 50 ± 0.009 and
100 ± 0.011 mm. The recommended perpendicularity tol-
erance is 0.025 mm for each side as a datum plane. The
recommended surface roughness is Ra = 1.6.
Because a specimen is subjected to stresses in the three
principal directions during a true triaxial test, the orienta-
tion of the specimens with respect to any geology textures
present should be consistent. Consequently, the three
directions of each specimen should be marked so as to be
distinguishable. Furthermore, methods such as ultrasonic
velocity measurement can be used to determine the ani-
sotropy of each specimen.
4.2 Size
Different specimen sizes, such as 15 9 15 9 30 mm3
(Mogi 1971), 19 9 19 9 38 mm3 (Haimson and Chang
2000), 35 9 35 9 70 mm3 (Takahashi and Koide 1989),
51 9 51 9 51 mm3 (King 2002), 80 9 80 9 80 mm3
(Young et al. 2013), 50 9 50 9 100 mm3 (Feng et al.
2016), 54 9 54 9 108 mm3 (Sriapai et al. 2013), and
76 9 76 9 178 mm3 (Wawersik et al. 1997), have been
used. Most tests use rectangular prismatic specimens with
an approximate length–width–height ratio of 1:1:2.
To achieve similar sizes and ratios of the side length in
the ISRM Suggested Method for true triaxial tests and
conventional triaxial tests (Fairhurst and Hudson 1999), a
rectangular prismatic specimen with a length–width–height
ratio of 1:1:2 is recommended to allow the results of both
tests to be compared.
At the same length and width, the apparent strength of a
cubic specimen is higher than that of a specimen with a
length–width–height ratio of 1:1:2. Therefore, using cubic
specimens for determining stress–strain curves under true
triaxial compression is not recommended because their
strength increases with decreasing height. The short side
length of the specimen should be at least 10 times larger
than the largest grain size in the rock microstructure.
4.3 Conditions
To become better acquainted with the rock mechanical
behaviour, basic physical properties, such as density, porosity,
andmoisture content, should also bemeasured and tested state
(e.g. oven dry, saturated) noted prior to testing. Furthermore,
the recommended sample test number is at least six. The
number of specimens tested under the same conditions should
Fig. 6 Effect of the end friction on the stress–strain relationship in
true triaxial tests with and without an anti-friction agent. M–R direct
metal–rock contact; M-MSV-R anti-friction agent (mixture of stearic
acid and Vaseline) applied at the metal–rock contact
ISRM Suggested Method for Determining Stress–Strain Curves for Rocks Under True Triaxial…
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be sufficient to adequately represent the rock sample; a min-
imumof three successful true triaxial compression tests per set
of testing conditions is recommended.
5 Testing Procedure
True triaxial testing is a complicated process. The operator
must have a good understanding of the working principle
of the test machine and good operation skills. Therefore,
each step including specimen assembly, installation, sensor
calibration and adjustment, and the loading method should
be carefully evaluated to ensure that the test is performed
successfully.
5.1 Specimen Assembly and Sealing
As mentioned above, considering the influence of the
loading blank gap, mutually overlapping platens are used to
fix the specimen (Fig. 8). The platens are connected and
interlocked by four clamp screws. If using another design
for the specimen assembly structure, uniformity in the
stress distribution should be carefully considered.
Prior to fixing the specimen to the platens, an anti-
friction agent is applied to the surfaces between the platen
and the specimen to reduce friction. The exposed surfaces
in the minor principal stress direction and all junctions
between the platens need to be sealed using a sealant. The
sealant has a similar function as the heat-shrink Viton
jacket in conventional triaxial tests. In general, silicone or
polyurethane sealant with a thickness of approximately
5 mm is recommended.
After sealing, the specimen should be exposed to air for
at least three days to age the sealant, and the specimen
should be checked and repaired during this period.
5.2 Sensor Calibration and Adjustment
To ensure the accuracy of the measurement system, all the
utilized deformation transducers should be periodically
Fig. 7 Demonstration of the
loading gap: a short platen
design; b bevelled design
Fig. 8 Working principle of the
overlapping platens: a before
deformation; b after
deformation (after Feng et al.
2016)
X. Zhang et al.
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calibrated to a precision of 0.1% of full scale. In addition,
the calibration should be verified using an elastic dummy
specimen, such as an aluminium specimen with glued
strain gauges. To determine whether the sensor is accurate,
a 0.5-mm-thick standard gauge block is used. Furthermore,
the entire deformation range of the rock needs to match the
calibration range, and the initial value of the transducers
must be known in the calibration range. If using strain
gauge, the preliminary testing on steel or aluminium
specimen needs to perform to determine the correct elastic
parameters so that the operation skills are fully grasped.
5.3 Loading Test for Determining the Stress–Strain
Curve
After attaching the sensors along the three principal stress
directions, the sealed specimen is placed in a confining cell.
To avoid off-centre loading, four pistons are used to adjust
and fix the specimen at the centre of the confining cell. An
anti-friction agent must be placed between the pistons and
the specimen platens.
The window of the confining cell must be covered; then,
hydraulic oil is provided by a pump. In the next step, the
two loading frames are placed in an orthogonal state to
ensure that the centre of each loading frame coincides with
the centre of the specimen. A small preload (5 kN) that is
sufficient to overcome the friction of the piston should be
applied to the specimen through force control.
The stress path is used to represent the successive states
of stress in a test specimen during loading or unloading in
the three-dimensional principal stress space. The succes-
sive change in the locus is called the stress path, which can
be in the form of a straight line or a curve. The influences
of the stress path on the strength, plastic hardening, and
pore water have been studied in soil mechanics; however,
in rock mechanics, stress paths related to the excavation
and supporting behaviour have received more attention.
Complicated stress paths can be designed in true triaxial
tests based on the research goals, and an unlimited number
of stress paths can be used. In this suggested method, the
load failure behaviour is the fundamental means of deter-
mining the stress–strain curve; hence, two representative
loading paths for determining the stress–strain curve are
described below.
5.3.1 Representative Loading Path 1
The hydrostatic pressure is monotonically increased by
increasing the fluid pressure in the confining cell until a
target value of r1 = r2 = r3 is obtained. Then, r3 is heldconstant, while r1 and r2 are simultaneously and gradually
increased to the desired value for r2. In the final stage, r2
and r3 are held constant, and loading is applied in the r1
direction using either stress control or strain control until
failure occurs (Fig. 9). To determine the stress–strain
curves for brittle hard rocks, a combined control mode may
be adopted. At approximately 70% of the peak force, which
could correspond to a dilation point where the volumetric
strain increases from its minimum value, the control mode
is switched to deformation control in the minor principal
stress direction until a complete force–displacement curve
is obtained. In the elastic deformation stage, stress control
is used to increase r1 at a rate of 0.5 to 1 MPa/s. After the
dilation point, the stress control is switched to minor
principal strain control; the corresponding strain rates
range from approximately 1 to 10 9 10-6/s.
5.3.2 Representative Loading Path 2
The application of the hydrostatic pressure is the same as
that under loading path 1; in the following stage, the major
principal stress and the intermediate principal stress are
exchanged, that is, a principal stress rotation is performed.
The stress produced via the rigid loading platens in axis 2
(red line in Fig. 10; the definitions of the principle stress
are based on the final stress state) is increased to the target
stress level. Because this stress is higher than the other
stresses, it becomes the temporary major principal stress,
and the stress is applied by the other pair of loading platens
in axis 1 (see the black line in Fig. 10). In the third loading
stage, the stress in axis 1 is applied using either stress
control or strain control until failure occurs. Because this
stress is greater than the temporary major principal stress, it
becomes the final major principal stress.
5.4 Safety and Health
This suggested method does not purport to address all
safety concerns. The safety and health of the personnel
must be ensured when performing the experiment; thus,
Fig. 9 Representative loading path 1
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protection measures should be taken. For example, when
sealing the specimen using silicone or polyurethane sea-
lant, a mask and glasses should be worn, a fan should be
used, and a window or door should be open. All experi-
ments should be conducted in accordance with safety
regulations.
6 Calculation
The true triaxial stresses are obtained from the built-in load
cell data, and the strains can be directly recorded from
strain gauges or calculated from displacements depending
on the type of sensor used.
6.1 Calculation of the Principal Stresses
The major and the intermediate principal stresses are cal-
culated as
ri ¼ Fi=Ai ð1Þ
where i = 1, 2, r1 is along the longitudinal axis of the
specimen, and r2 is along the transverse axis of the spec-
imen, F1 is the force in the direction of the major principal
stress, A1 is the initial cross-sectional area perpendicular to
F1, F2 is the force in the direction of the intermediate
principal stress, and A2 is the initial cross-sectional area
perpendicular to F2.
It should be noted that the initial cross-sectional area
perpendicular to each principal stress is assumed to be
constant in the stress calculation because of the small
deformation (Fairhurst and Hudson 1999), for general hard
rock prior to peak strength, the calculated stress is
0.5–1.5% greater than the actual stress. In the post-peak
stage of brittle rocks, the correction on the stress is of
uncertainty because the volume change measurement
transducer becomes unreliable if the localizable shear plane
forms across the specimen (Zhang et al. 2014).
The minor principal stress r3 equals the pressure of the
fluid in the confining cell.
In this test procedure, the stress is defined such that a
positive value indicates compression.
6.2 Calculation of the Principal Strains
The principal strains, ei, are calculated as
ei ¼ Dli=li ð2Þ
where i = 1, 2, 3; Dli is the change in the measured length,
which is defined as the original length minus the current
length in the direction of each principal stress; and li is the
original dimension of the specimen in the direction of each
principal stress.
In this test procedure, the strain is defined such that a
positive value indicates compression.
6.3 The Modulus of Deformation
The Young’s modulus, E, of an isotropic rock is defined as
the ratio of the change in the maximum principal stress to
the change in the major principal strain in the linear range,
while r2 and r3 remain constant.
E ¼ Dre1=Dee1 ð3Þ
where Dre1 and Dee1 are the elastic changes in the maximum
principal stress and major principal strain, respectively.
6.4 Volumetric Strain
The volumetric strain is an important parameter for
assessing the deformation characteristic and is the sum of
the three principal strains under small deformation condi-
tion; for a given stress level,
ev ¼ e1 þ e2 þ e3 ð4Þ
7 Reporting of Test Results
The type of true triaxial apparatus used and the principal
stress application method should be reported; furthermore,
the end friction effect reduction and loading gap removal
should be mentioned in the end platen design.
The output of the test data includes the stress–strain
curves, strength, deformation modulus, stress path, and
failure mode of the rock specimen under true triaxial
compression. In addition, other information such as Lode
angle, octahedral shear stress, and parameters related to the
strength should be included to the greatest extent possible.
Fig. 10 Representative loading path 2 (colour figure online)
X. Zhang et al.
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7.1 Rock Information
Information about the source of the specimen includes the
following:
1. Project name
2. Location where the sample was obtained; the location
is frequently specified in terms of the borehole number
and depth or may be a block sample for drilling in the
laboratory
3. Lithological description of the rock, including grain
size
4. Specimen texture (bedding planes, foliation, flow
banding)
5. Time between obtaining the rock sample and the
beginning of testing.
7.2 Specimen Information
The specimen information should include the following:
1. The ID of the tested specimen
2. Any other observable or available physical data, such
as specific gravity, porosity, and permeability; the
method used to determine each property should be
cited
3. Orientation of the three loading axes with respect to
rock anisotropy
Fig. 11 Typical experimental results: a differential stress (r1 - r3)–strain relationships for a granite; b photograph and sketch of the broken
specimen
Fig. 12 Typical experimental results: a differential stress (r1 - r3)–strain relationships for a marble; b photograph and sketch of the broken
specimen
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4. Rate of loading, deformation, or strain
5. Specimen dimensions (width, length, and height)
6. Water content and degree of saturation of the specimen
at the time of testing if relevant (or whether oven dry)
7. Date of testing and test duration.
7.3 Typical Experimental Results
1. Stress–strain plots. Typical stress–strain curves for
granite, marble, and sandstone are shown in Figs. 11,
12 and 13, respectively.
2. Failure mode and fracture angle. The fracture angle, h,is defined as the angle between the normal to the
fracture plane and the r1 direction. A photograph and a
sketch of the broken specimen should be provided.
3. The intermediate principal stress effect. The influence
of r2 on rock strength is shown in Fig. 14.
Acknowledgements The authors acknowledge the five reviewers
(Professors Joe Labuz, Paul Young, Ming Cai, Heinz Konietzky, and
Manabu Takahashi) for their critical reviews and their constructive
suggestions for the manuscript. Their valuable discussion on the use
of the stress–strain curve term in the ISRM Commission on Testing
Methods is also acknowledged.
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