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:

resat@hacettepe.edu.tr.

& Xia-Ting Feng

xtfeng@whrsm.ac.cn; xia.ting.feng@gmail.com

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

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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)

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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)

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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.

References

Akai K (1968) Failure mechanism of rock materials under combined

compressive stresses. J Soc Mater Sci Jpn 17:848–855 (inJapanese)

Akai K, Mori H (1967) Study on the failure mechanism of a sandstone

under combined compressive stresses. Proc Jpn Soc Civil Eng

147:11–24 (in Japanese)Alexeev AD, Revva VN, Alyshev NA, Zhitlyonok DM (2004) True

triaxial loading apparatus and its application to coal outburst

prediction. Int J Coal Geol 58:245–250. doi:10.1016/j.coal.2003.

09.007

Atkinson R, Ko H-Y (1973) A fluid cushion, multiaxial cell for testing

cubical rock specimens. Int J Rock Mech Min 10:351–361.

doi:10.1016/0148-9062(73)90043-0

Boker R (1915) Die mechanik der bleibenden formanderung in

kristallinish aufgebauten Korpern. Verhandl Deut Ingr Mitt

Forsch 175:1–51

Cai M (2008) Influence of intermediate principal stress on rock

fracturing and strength near excavation boundaries—insight

from numerical modeling. Int J Rock Mech Min 45:763–772.

doi:10.1016/j.ijrmms.2007.07.026

Colmenares LB, Zoback MD (2002) A statistical evaluation of intact

rock failure criteria constrained by polyaxial test data for five

different rocks. Int J Rock Mech Min 39:695–729. doi:10.1016/

S1365-1609(02)00048-5

Descamps F, Ramos da Silva M, Schroeder C, Verbrugge J,

Tshibangu J (2012) Limiting envelopes of a dry porous

Fig. 13 Typical experimental results: a differential stress (r1 - r3)–strain relationships for a sandstone; b photograph and sketch of the broken

specimen

Fig. 14 Intermediate principal stress effect

X. Zhang et al.

123

RETRACTEDART

ICLE

limestone under true triaxial stress states. Int J Rock Mech Min

56:88–99. doi:10.1016/j.ijrmms.2012.07.013

Du K, Li X, Li D, Weng L (2015) Failure properties of rocks in true

triaxial unloading compressive test. T Nonferr Metal Soc

25:571–581. doi:10.1016/S1003-6326(15)63639-1

Esaki T, Kimura T, Aoki K, Nishida T (1988) True triaxial test of

rock under stress and strain rate control. In: Donaghe RT,

Chaney RC, Silver ML (eds) Advanced triaxial testing of soil

and rock. ASTM International, Philadelphia, pp 834–843

Fairhurst CE, Hudson JA (1999) Draft ISRM suggested method for

the complete stress–strain curve for intact rock in uniaxial

compression. Int J Rock Mech Min Sci 36:279–289. doi:10.

1016/S0148-9062(99)00006-6

Feng X-T, Hudson JA (2010) Specifying the information required for

rock mechanics modelling and rock engineering design. Int J

Rock Mech Min 47:179–194. doi:10.1016/j.ijrmms.2009.12.009

Feng X-T, Zhang X, Kong R, Wang G (2016) A novel Mogi type true

triaxial testing apparatus and its use to obtain complete stress–

strain curves of hard rocks. Rock Mech Rock Eng

49(5):1649–1662. doi:10.1007/s00603-015-0875-y

Feng X-T, Zhang X, Yang C, Kong R, Liu X, Peng S (2017)

Evaluation and reduction of the end friction effect in true triaxial

tests on hard rocks. Int J Rock Mech Min. doi:10.1016/j.ijrmms.

2017.04.002

Foppl A (1900) Abhangigkeit der Bruchgefahr von der Art des

Spannungszustandes. Mitth Mech Tech Lab K Tech Hochsch

Munchen 27:1–35

Frash LP, Gutierrez M, Hampton J (2014) True-triaxial apparatus for

simulation of hydraulically fractured multi-borehole hot dry rock

reservoirs. Int J Rock Mech Min 70:496–506. doi:10.1016/j.

ijrmms.2014.05.017

Haimson B (2006) True triaxial stresses and the brittle fracture of

rock. Pure appl Geophys 163:1101–1130. doi:10.1007/s00024-

006-0065-7

Haimson B, Bobet A (2012) Introduction to suggested methods for

failure criteria. Rock Mech Rock Eng 45:973–974. doi:10.1007/

s00603-012-0274-6

Haimson B, Chang C (2000) 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 37:285–296. doi:10.1016/S1365-1609(99)00106-9

He M, Miao J, Feng J (2010) Rock burst process of limestone and its

acoustic emission characteristics under true-triaxial unloading

conditions. Int J Rock Mech Min 47:286–298

Ingraham MD (2012) Investigation of localization and failure

behavior of castlegate sandstone using true triaxial testing.

PhD Thesis, Clarkson University

Kaiser PK, Kim BH (2015) Characterization of strength of intact

brittle rock considering confinement-dependent failure pro-

cesses. Rock Mech Rock Eng 48(1):107–119

Karman von T (1911) Festigkeitsversuche unter allseitigem Druck.

Ztg d. Vereins Deutscher lngenieure, Jg. 55

Kawamoto T, Tomita K (1970) Variation in the characteristics of

deformation of marble and mortar due to transition of stress from

isotropic state to anisotropic state (in Japanese). J Jpn Mater Sci

Soc 20(209):50–57

Kawamoto T, Tomita K, Akimoto K (1970) Characteristics of

deformation of rock-like materials under triaxial compression.

In: Proceedings of the 2nd congress of the international society

of rock mechanics, Beograd, vol 1, 2-2, pp 287–293

King M (2002) Elastic wave propagation in and permeability for

rocks with multiple parallel fractures. Int J Rock Mech Min

39:1033–1043

King MS, Chaudhry NA, Shakeel A (1995) Experimental ultrasonic

velocities and permeability for sandstones with aligned cracks.

Int J Rock Mech Min 32:155–163. doi:10.1016/0148-

9062(94)00033-Y

Kwasniewski M, Takahashi M, Li X (2003) Volume changes in

sandstone under true triaxial compression conditions. In:

Proceeding of the 10th ISRM congress technology roadmap for

rock mechanics, Sandton, September 8–12, 2003, vol 1. The

South Africa Institute of Mining and Metallurgy, Johannesburg,

pp 683–688

Labuz J, Biolzi L (1991) Class I vs class II stability: a demonstration

of size effect. Int J Rock Mech Min 28:199–205

Labuz J, Biolzi L (2007) Experiments with rock: remarks on strength

and stability issues. Int J Rock Mech Min 44:525–537

Labuz JF, Bridell JM (1993) Reducing frictional constraint in

compression testing through lubrication. Int J Rock Mech Min

30:451–455. doi:10.1016/0148-9062(93)91726-Y

Labuz J, Dai S-T, Papamichos E (1996) Plane-strain compression of

rock-like materials. Int J Rock Mech Min Sci Geomech Abstr

33:573–584

Li XC, Shi L, Bai B, Li Q, Xu D, Feng X (2012) True-triaxial testing

techniques for rocks–state of the art and future perspective. In:

Kwasniewski M, Li X, Takahashi M (eds) True triaxial testing of

rocks. CRC Press, Boca Raton, pp 3–18

Makhnenko R, Labuz J (2014) Plane strain testing with passive

restraint. Rock Mech Rock Eng 47:2021–2029

Mogi K (1970) Effect of the triaxial stress system on rock failure.

Rock Mech Jpn 1:53–55

Mogi K (1971) Fracture and flow of rocks under high triaxial

compression. J Geophys Res 76:1255–1269. doi:10.1029/

JB076i005p01255

Mogi K (2007) Experimental rock mechanics. CRC Press, London

Murrell SAF (1963) 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) Rock mechanics: proceedings

of the fifth rock mechanics symposium. Pergamon Press, Oxford,

pp 563–577

Nasseri MHB, Goodfellow SD, Lombos L, Young RP (2014) 3-D

transport and acoustic properties of Fontainebleau sandstone

during true-triaxial deformation experiments. Int J Rock Mech

Min 69:1–18

Okubo S, Nishimatsu Y (1985) Uniaxial compression testing using a

linear combination of stress and strain as the control variable. Int

J Rock Mech Min Sci 22:323–330

Protodyakonov MM (1962) Mechanical properties and drillability of

rocks. In: Proceedings of the fifth symposium on rock mechan-

ics, University of Minnesota, Minneapolis, MN, pp 103–118

Rice J, Rosengren GF (1968) Plane strain deformation near a crack tip

in a power-law hardening material. J Mech Phys Solids 16:1–12

Rudnicki JW, Rice JR (1975) Conditions for the localization of

deformation in pressure-sensitive dilatant materials. J Mech Phys

Solids 23(6):371–394

Secq J (2010) Triaxial cell for the testing of geomaterials in

compression and in shear. France Patent application CA

2730046, US8770038

Sriapai T, Walsri C, Fuenkajorn K (2013) True-triaxial compressive

strength of Maha Sarakham salt. Int J Rock Mech Min

61:256–265. doi:10.1016/j.ijrmms.2013.03.010

Su G, Jiang J, Zhai S, Zhang G. (2017) Influence of tunnel axis stress

on strainburst: an experimental study. Rock Mech Rock Eng.

doi:10.1007/s00603-017-1181-7

Takahashi M, Koide H (1989) Effect of the intermediate principal

stress on strength and deformation behavior of sedimentary rocks

at the depth shallower than 2000 m. In: Maury V, Fourmaintraux

D (eds) Rock at great depth 1. Balkema, Rotterdam, pp 19–26

The National Standards Compilation Group of People’s Republic of

China (2014) GB/T50218–2014 standard for engineering

ISRM Suggested Method for Determining Stress–Strain Curves for Rocks Under True Triaxial…

123

RETRACTEDART

ICLE

classification of rock masses. China Planning Press, Beijing,

p. 84 (in Chinese)Wawersik WR (1968) Detailed analysis of rock failure in laboratory

compression tests. Ph.D. thesis, University of Minnesota, p. 165

Wawersik WR, Fairhurst C (1970) A study of brittle rock fracture in

laboratory compression experiments. Int J Rock Mech Min

7:561–575. doi:10.1016/0148-9062(70)90007-0

Wawersik WR, Carlson LW, Holcomb DJ, Williams RJ (1997) New

method for true-triaxial rock testing. Int J Rock Mech Min 34:

330.e1–330.e14. doi:10.1016/S1365-1609(97)00049-X

Young RP, Nasseri MHB, Lombos L (2013) Imaging the effect of the

intermediate principal stress on strength, deformation and

transport properties of rock using seismic methods. In:

Kwasniewski M, Li X, Takahashi M (eds) True triaxial testing

of rocks. Taylor & Francis, London, pp 167–179

Zhang X, Mavroulidou M, Gunn MJ, Sutton J, Cabarkapa Z, Kichou

Z (2014) Application of a novel laser sensor volume measure-

ment system to the triaxial testing of an unsaturated lime-treated

soil. Acta Geotech 9(6):945–957

X. Zhang et al.

123

RETRACTEDART

ICLE