experimental and numerical....pdf

ORIGINAL PAPER

Experimental and Numerical Studies on Determinationof Indirect Tensile Strength of Rocks

Nazife Erarslan • Zheng Zhao Liang •

David John Williams

Received: 6 September 2011 / Accepted: 15 November 2011 / Published online: 25 November 2011

� Springer-Verlag 2011

Abstract Indirect tension tests using Brisbane tuff Bra-

zilian disc specimens under standard Brazilian jaws and

various loading arcs were performed. The standard Brazilian

indirect tensile tests caused catastrophic, crushing failure of

the disc specimens, rather than the expected tensile splitting

failure initiated by a central crack. This led to an investiga-

tion of the fracturing of Brazilian disc specimens and the

existing indirect tensile Brazilian test using steel loading arcs

with different angles. The results showed that the ultimate

failure load increased with increasing loading arc angles.

With no international standard for determining indirect

tensile strength of rocks under diametral load, numerical

modelling and analytical solutions were undertaken.

Numerical simulations using RFPA2D software were con-

ducted with a heterogeneous material model. The results

predicted tensile stress in the discs and visually reproduced

the progressive fracture process. It was concluded that

standard Brazilian jaws cause catastrophic, crushing failure

of the disc specimens instead of producing a central splitting

crack. The experimental and numerical results showed that

20� and 30� loading arcs result in diametral splitting fractures

starting at the disc centre. Moreover, intrinsic material

properties (e.g. fracture toughness) may be used to propose

the best loading configuration to determine the indirect

tensile strength of rocks. Here, by using numerical outcomes

and empirical relationships between fracture toughness and

tensile strength, the best loading geometry to obtain the most

accurate indirect tensile strength of rocks was the 2a = 30�loading arc.

Keywords Indirect tensile strength � Brazilian test �RFPA method � Diametral loading of rock discs

Abbreviations

ISRM International Society for Rock Mechanics

UCS Uniaxial compressive strength test

BTS Brazilian tensile strength

1 Introduction

The indirect Brazilian test is widely used in engineering

practice to indirectly determine the tensile strength of

rocks. In 1978, the Brazilian test was officially proposed by

the International Society for Rock Mechanics (ISRM) as a

suggested method for determining the tensile strength of

rock materials (ISRM 2007). The Brazilian test (splitting

tension test) is performed by applying a concentrated

compressive load across the diameter of a disc specimen.

The standard Brazilian test results in a reasonably nar-

row measured data scatter. In addition, the biaxial stress

state in a test specimen subjected to the Brazilian test is

considered approximately similar to that which appears in a

rock mass when tensile failure occurs in situ. However,

there are a few unsolved problems concerning this test; for

example, how to guarantee crack initiation at the centre of

the specimen (beneath the concentrated load), how to

determine the true stress state in a test specimen and how to

locate the point from which the main crack initiates.

Although the Brazilian test has been studied extensively,

N. Erarslan (&) � D. J. Williams

Golder Geomechanics Centre, School of Civil Engineering,

The University of Queensland, Brisbane, QLD 4072, Australia

e-mail: [email protected]

Z. Z. Liang

School of Civil and Hydraulic Engineering,

Dalian University of Technology, Dalian 116024, China

123

Rock Mech Rock Eng (2012) 45:739–751

DOI 10.1007/s00603-011-0205-y

both experimentally and theoretically, relatively little

attention has been directed towards researching the validity

of the test. Fairhurst (1964) first discussed the important

issue of the validity of the Brazilian test. He stated that

‘failure may occur away from the centre of the test disc for

small angles of loading contact area’ and indicated that the

calculated tensile strength from a Brazilian test is lower

than the true value of the tensile strength. Hudson et al.

(1972) observed that failure always initiated directly under

the loading points if flat steel plates were used to load the

specimen, an approach that actually invalidates the test for

the determination of tensile strength.

It is possible to estimate stress distribution in a disc

specimen theoretically if some assumptions are made.

Although various methods have been proposed for this

purpose, the common fundamental assumptions are: (1)

rock behaves elastically in both tension and compression;

(2) breakage of the loading zone does not influence the

stress distribution at the centre of the specimen; and (3)

high compressive stress exists around the two loading

points and, therefore, the tensile crack never initiates from

this region. Based on those assumptions, classical theory

(ISRM 2007) assumed that the concentrated load was

applied over an infinitesimally small width as a line load,

but clearly, this would lead to stresses of very high inten-

sity. The actual loads are not concentrated, but are dis-

tributed over a finite arc of the disc. The tensile strength of

a rock disc specimen, rt, is calculated using:

rt ¼2P

pdtð1Þ

where P is failure load and d and t are the diameter and

thickness, respectively, of the rock disc. According to

Jaeger and Cook (1976), if a circular cylinder of radius R is

compressed across its diameter between flat surfaces that

apply concentrated loads of W per unit axial length of the

cylinder, then the stresses on this diameter are:

rx ¼ �W

pR

ry ¼Wð3R2 þ y2ÞprðR2 � y2Þ :

ð2Þ

By symmetry, these are principal stresses and the major

and minor principal stresses are along and across the y-axis,

respectively. If the load is applied to the circumference of

the cylinder as a pressure p distributed over an arc 2a by

using shaped platens, so that W = 2paR, then equal biaxial

compression exists near the contacts with a value of

p (Jaeger and Cook 1976).

However, the actual loads in an experiment are not

concentrated, but are distributed over finite portions of the

disc. The distributed load applied to a disc under diametral

compression is more difficult to analyse than that of the

concentrated load. Hondros (1959) analysed the Brazilian

test for the case of a thin disc loaded by a uniform pressure

applied radially over a short strip of the circumference at

each end of the disc. He obtained the full-field stresses in a

series solution by using the series expansion technique and

applied these solutions to evaluate the Young’s modulus

and Poisson’s ratio in concrete, by measuring strain as

follows:

rr ¼ �2p

paþ

Xn¼1

n¼1

1� 1� 1

n

� �r

R

� �2� �

r

R

� �2n�2

sin 2na cos 2nh

( )

rh ¼ �2p

pa�

Xn¼1

n¼1

1� 1þ 1

n

� �r

R

� �2� �

r

R

� �2n�2

sin 2na cos 2nh

( )

srh ¼ �2p

p

Xn¼1

n¼1

1� r

R

� �2� �

r

R

� �2n�2

sin 2na cos 2nh

( )

ð3Þ

where p is applied pressure, R is the radius of the disc,

r and h are the polar coordinates of a point in disc and a is

the half central angle of the applied distributed load

(Fig. 1). It can be seen from (3) that the magnitude of aaffects directly the stress distribution within the disc.

In addition to analytical solutions that assume the rock

material to be isotropic and homogenous, recently many

researchers have included the effect of rock heterogeneity in

their experimental and numerical studies (Van de Steen 2001;

Cai and Kaiser 2004; Van de Steen et al. 2005; Lanaro et al.

2009; Liu 2004). Van de Steen et al. (2005) experimentally and

numerically studied the irregular, granular nature of rock as

well as the effect of the presence of defects and weaknesses on

Fig. 1 A disc specimen subjected to diametric distributed

compression

740 N. Erarslan et al.

123

the stress distribution in a specimen. They successfully simu-

lated the deformation in disc specimens, including defects, by

using a boundary element code DIGS, and they proposed that

fracturing in the Brazilian test is initiated in shear near one of

the platens and it subsequently grows in tension.

Liu (2004) used a heterogeneous material model

implemented into the R-T2D code in his numerical studies.

He showed the effect of rock heterogeneity on the stress

distribution and fracturing process in tested specimens by

using a variety of numerical tests such as the uniaxial

compressive strength (UCS) test, the Brazilian tensile

strength (BTS) test, diametral compression testing of a

notched Brazilian disc as well as three-point bending and

four-point shearing tests.

Recently, the effect of pre-existing cracks in terms of

heterogeneity was examined numerically by Lanaro et al.

(2009). They found that the propagation of initiated cracks

produces a stress field that is very different from that

assumed when considering the rock material as continuous,

homogeneous, isotropic and elastic, as in many numerical

models as well as analytical solutions. They concluded that

stress concentrations at the bridges between the cracks

could reach tensile stresses much higher than the direct

tensile strength of the intact rock that was used as input in

their numerical models. This was due to the development

of large stress gradients between the cracks.

According to the Griffith criterion (Griffith 1020), the

exact centre of the disc is the only point where the conditions

for tensile failure at a value equal to the uniaxial strength are

met. However, in some experimental studies carried out

using standard Brazilian jaws, cracks initiated just under the

loading points instead of causing a central crack (Hudson

et al. 1972; Markides et al. 2010; Yu et al. 2009). Never-

theless, in the tests described herein, catastrophic crushing

failure developed on standard Brazilian testing of brittle

Brisbane tuff disc specimens. In contrast, central cracks were

obtained, corresponding to the location of the maximum

tensile stress, for loading of Brisbane tuff specimens over an

arc length. Thus, the objective of this paper is to criticise the

standard Brazilian test by comparing the results obtained

using the test with those obtained using a loaded arc, together

with a comparison of the experimental results and the ana-

lytical and numerical modelling results.

2 Experimental Study

2.1 Indirect Tension Tests

A series of Brazilian disc tests was carried out using

specimens prepared from Brisbane tuff, a host rock of

Brisbane, Australia’s first motorway tunnel, CLEM7, from

which core samples were obtained. The test specimens

were standard Brazilian discs with a diameter of 52 mm

and thickness of 26 mm (a diameter to thickness ratio of

0.5). The test load was applied by a stiff hydraulic Instron

loading frame with an ISRM-suggested loading rate of

200 N/s (ISRM 2007).

Four series of indirect tension tests were conducted: (1)

standard Brazilian jaws, (2) 15� steel loading arcs, (3) 20�steel loading arcs and (4) 30� steel loading arcs. Up to four

repetitions were carried out. The steel loading arcs were

machined from standard mild steel, as recommended by

ISRM (2007) (Fig. 2).

The tensile strength of the rock specimens tested using

standard Brazilian jaws were calculated using the formula

given by ISRM (2007). Since loading boundaries of steel

loading arcs are different from standard Brazilian jaws, the

formula given by ISRM (2007) cannot be used to calculate

indirect tensile strength of specimens under loading arcs.

The tensile strength of the samples tested under angled

loading arcs was calculated at the centre of the disc by

using the analytical solutions given in (3).

The details of the test results are given in Table 1. The

maximum recorded failure load was obtained by using

loading arcs with 2a = 30�. The second highest failure

load was obtained with the standard Brazilian jaws, which

also produced the highest standard deviation among the

ultimate loads.

In our tests on Brisbane tuff, loading with Brazilian jaws

caused catastrophic crushing failure of the disc specimens

(Fig. 3a). However, a single failure plane diverged from

the load axis and secondary cracks occurred under 15�loading arcs (Fig. 3b). In contrast to the results from the

15� loading arcs, one axial splitting failure plane through

the diametral loading axis was obtained with the samples

tested by using the 20� loading arcs (Fig. 3c). Unlike the

results obtained in the standard Brazilian and the 15� and

20� loading arc tests, arrested and vertical aligned central

cracks were obtained with 30� loading arcs (Fig. 3d).

As an alternative to the uniaxial tensile test, which is

difficult to perform to acceptable standards for brittle

materials, diametral compression of Brazilian disc speci-

mens has strong practical appeal. It is clear that a diametral

compression test can never adequately replace a direct

uniaxial tensile test due to their different loading bound-

aries and specimen geometries. However, conducting

direct tests are still too difficult and expensive for routine

application to a large numbers of specimens, and diametral

compression tests appear to offer the most desirable alter-

native. On that basis, this study focuses on the failure in

standard Brazilian discs under diametral compression to

determine the most representative indirect tensile strength

of rocks. Thus, the Brazilian test under 30� loading arcs is

preferable to such a test under Brazilian jaws, in terms of

central tensile splitting. However, it is not easy to conclude

Determination of Indirect Tensile Strength of Rocks 741

123

that the obtained failure load under 30� loading arcs rep-

resents the real value of the indirect tensile strength of

rocks. To investigate further, numerical and analytical

analyses are used for the determination of the indirect

tensile strength of rocks in this research.

3 Numerical Modelling of Heterogeneous Rock

Fracturing Under Indirect Tensile Stresses

3.1 Numerical Method

Rock or rock-like materials have composite or heteroge-

neous microstructures with various scales. The behaviour of

rock fractures and the growth of cracks or micro-cracks in

rocks are strongly influenced by these heterogeneous

microstructures (Griffith 1920; Liu 2004; Cai and Kaiser

2004; Lanaro et al. 2009). Consequently, the numerical

methods used for studying the fracture behaviour of rocks

should take heterogeneity into consideration. Among the

numerical methods developed in the past, some researchers

have included heterogeneity in their programs (Schlangen

and Garboczi 1997; Tang et al. 1998; Blair and Cook 1998).

A numerical approach called Rock Fracture Process

Analysis (RFPA) was used in this study to elucidate the

fracturing process in a Brazilian disc under different

boundary loading conditions. Liu (2004) has listed three

main advantages of the RFPA model over other models: (1)

by introducing the heterogeneity of rock properties into the

model, RFPA can simulate the non-linear deformation of a

quasi-brittle rock with an ideal brittle constitutive law for

the local material; (2) by introducing the reduction of the

Fig. 2 a Brazilian jaws and

steel loading arcs, b disc

between standard Brazilian jaws

and c disc between loading arcs

Table 1 Results of indirect

tensile tests on Brisbane tuff

disc specimens

Recorded maximum load (kN)

Standard Brazilian jaws 15� loading arcs 20� loading arcs 30� loading arcs

Replicate 1 25.00 12.50 17.06 21.10

Replicate 2 16.77 16.39 19.82 24.60

Replicate 3 15.43 15.65 20.23 21.13

Replicate 4 21.00 14.70 19.41 22.17

Average 19.60 14.81 19.20 22.30

Standard deviation 4.34 1.69 1.64 1.64


123

material parameters after element failure, RFPA can sim-

ulate strain-softening and discontinuum mechanics prob-

lems in a continuum mechanics mode; and (3) by recording

the event rate of failed elements, RFPA can simulate

micro-seismicity associated with progressive fracture

process.

To deal with real random microstructures in a numerical

simulation, rock heterogeneity can be characterized better

with statistical approaches (Liu 2004; Tang et al. 1998). In

the RFPA2D software, since the numerical specimens

consist of elements with the same shape and size, there is

no priority, geometrically, in any orientation in the speci-

men. Disorder can be obtained by means of random dis-

tributions of the mechanical properties of the elements. The

statistical distribution of the elastic modulus can be

described by the Weibull distribution function, even dis-

tribution function or normal distribution function. In this

study, only the Weibull distribution function:

WðxÞ ¼ m

x0

x

x0

� �m�1

exp � x

x0

� �m� �ð4Þ

is used. In (4), x is the elemental mechanical parameter,

such as uniaxial compressive strength, elastic modulus,

Poisson ratio or specific weight, x0 is the expected value of

x and m is the shape of the Weibull distribution function.

According to the Weibull distribution and the definition of

homogeneity index, a larger m implies that there are more

elements with mechanical properties approximated to the

mean value, thus a more homogeneous rock specimen.

Even though the mechanical parameters in each element

differ, which makes the specimen heterogeneous, the

mechanical properties within a single element are assumed

to be homogeneously distributed.

In RFPA2D, each element follows an elastic-brittle

constitutive law during the loading process. Until the stress

of the element satisfies the strength criterion, the elastic

modulus is a constant with the same value as before

loading. When the stress increases to a value leading to the

failure of the element, in elastic damage mechanics, the

elastic modulus of the element may degrade gradually as

damage progresses.

The elastic modulus of the damaged element is defined

as follows:

E ¼ ð1� DÞE0 ð5Þ

where D represents the damage variable, and E and E0 are

the elastic modulus of the damaged and undamaged ele-

ments, respectively. It must be noted that the element and

its damage are assumed to be isotropic and, therefore, E, E0

and D are all scalar.

Here, it was assumed that each element might fail in

either tensile failure or shear failure modes. If the ele-

mental stress state satisfies both the tensile failure criterion

and the shear failure criterion, the tensile failure mode has

the higher priority. If the element is subjected to uniaxial

tensile stress, before the tensile stress (the minimal prin-

cipal stress) of the element reaches its tensile strength, the

element remains linear elastic. When the minimal principal

Fig. 3 Failed Brisbane tuff

specimens under a standard

Brazilian jaws, b 15� loading

arcs, c 20� loading arcs and

d 30� loading arcs


123

stress increases beyond the tensile strength, the element

fails, the elastic modulus changes to a small value and the

element’s strength falls to a small value, which we can call

residual tensile strength. When the tensile stress increases

to a larger value, which we call the ultimate tensile

strength, the element loses its capability of loading.

Accordingly, the element stays linear elastic before the

uniaxial compressive stress reaches the uniaxial compres-

sive strength. If the elemental stress level meets the shear

failure criterion, the element will damage gradually. The

damage variable D when the element is subjected to uni-

axial tension can be described as:

D ¼0 ðe3 [ er0Þ1� rrt

e3E0ðer0� e3� eutÞ

1 ðe3� eutÞ

8<

:

9=

; ð6Þ

where rrt is the residual strength of the element (defined as

rrt ¼ �kjrtj), et0 is the tensile strain at the point of failure,

e3 is the principal strain, eut is the ultimate tensile strain

(described as eut ¼ get0 where g is called the ultimate

tensile strain coefficient) and k is coefficient of residual

tensile strength.

The damage evolution function mentioned above only

considers the tensile failure mode of the mesoscopic ele-

ments. Compressive softening induced by shear damage at

the mesoscopic level is also assumed to exist when the

mesoscopic element is under compressive and shear stress.

In shear failure mode, the damage variable D can be

described as follows:

D ¼ 0 ðe1\ec0Þ1� rrc

e1E0ðe1� ec0Þ

� ð7Þ

where rrc is the peak strength of the element subjected to

uniaxial compression, ec0 is the compressive strain at the

elastic limit and e3 is the principal strain.

Variation in the damage variable is obtained when the

element is subjected to uniaxial tensile stress or uniaxial

compressive stress. If the specimens are subjected to

complex stress loading, we can extend the analysis from a

one-dimensional damage model to a three-dimensional

model by using an equivalent strain e instead of the uni-

axial tensile strain or the compressive strain in (6) and (7).

When the equivalent strain of an element reaches the

ultimate tensile strain, the damaged elastic modulus is zero,

which would make the system of equations ill conditioned.

In order to keep the physical continuum of the numerical

model, the element is not removed from the model and a

relatively small number, i.e. 1.0 9 10-6 is specified for the

elastic modulus for this consideration. Therefore, contin-

uum mechanics can be applied to resolve discontinuous

problems.

However, it is impossible to obtain the parameters of the

mesoscopic elements. We can only obtain the mechanical

parameters of a macroscopic specimen for a specific rock

specimen in laboratory experiments. For Weibull distri-

bution determination, a parametric study can be performed

to obtain the relationships between the macroscopic

parameters (compressive strength rc and elastic modulus

E0) of the specimen and the seed parameters (mean value

of compressive strength rc and elastic modulus E0) of the

mesoscopic elements by using a linear least squares

technique:

rc ¼ a1 lnðmÞ þ b1½ �rc

E0 ¼ a2 lnðmÞ þ b2½ �E0

ð8Þ

where a1, b1, a2 and b2 are the constants in the linear least

squares technique.

3.2 Numerical Brazilian Tensile Strength (BTS) Test

This section describes our use of RFPA in numerical simu-

lations. Here, RFPA code (Tang et al. 1998) was used to

simulate Brazilian disc behaviour under various diametral

loading conditions. To model the typical brittle failure of

Brisbane tuff, the following characteristic parameters are

used in the numerical test. Based on experiments and a series

of numerical tests, the heterogeneity index for the simulated

rock is 4.0. The mean value of the uniaxial compressive

strength and elastic modulus for the rock are 202.5 MPa and

26 GPa, respectively, and the Poisson ratio value and friction

angle are 0.25 and 35�, respectively. With regard to the

loading steel arcs (plates), the heterogeneity index is a suf-

ficiently high value to consider the material to be almost

homogenous. The mean strength and elastic modulus are

1,500 MPa and 100 GPa, respectively. The stress is applied

by using a constant displacement movement of 0.0025 mm

per step on the top of the steel arc (plate), while the bottom

arc is fixed in the vertical direction. The numerical model can

be assumed as a plain stress problem.

In this study, a numerical model for the BTS test is

based on the geometry used in our experiments, i.e. the

diameter of the disc is 52 mm. The three simulated loading

arcs and Brazilian jaws are shown in Fig. 4. The degree of

greyness as shown in Fig. 4 represents the distribution of

the elastic modulus. The lighter the grey colour, the higher

is the value of the elastic modulus.

3.3 Numerical Simulation Results

Figure 5 shows the progressive fracture process in heter-

ogeneous Brazilian disc samples under standard Brazilian

jaws and three different loading arcs. As the loading

increases using standard Brazilian jaws, cracks start at

points close to the upper contact along the vertical diameter

of the disc (Fig. 5a). Subsequently, cracks unstably


123

propagate radially outward in the upper half of the disc,

resulting in a diametral fracture plane with small crack

coalescence.

When the angle of the steel arcs is increased, it was

difficult for the fractures to open just under the loading

point, due to the constraint stress in the horizontal direction

under the loading arcs. It is also interesting to mention that

any crushed zone just below the loading arcs, due to

the high compressive loading, was not observed in the

numerical simulations. In all loading arc simulations, the

first failure and/or crack initiation occurred at the centre of

the disc. In contrast, primary crack initiation occurred very

close to the loading point in standard Brazilian jaw simu-

lations. Central primary crack initiation took place at the

centre of the disc with the 15�, 20� and 30� loading arc

simulations (Fig. 5b, c, d). At the end of all fracture pro-

cess simulations, crack coalescence and failure took place

along the vertical diameter of the discs.

In addition to the importance of determining the central

crack initiation point, the propagation paths of the primary

cracks are also important. In the simulations there are some

secondary failure planes beside the main splitting failure

plane in rock, just next to the contact points between the

15� loading arc and the disc specimen (Fig. 5b-3). This

result supports the experimental findings shown in Fig. 3.

Recorded peak loads obtained during simulations are

given in Table 2. The obtained ultimate loads during

experiments are also provided in Table 2 to allow com-

parison between simulation and experimental results.

Similar to the trend in the experimental results, the

numerical ultimate load obtained from simulations done by

using RFPA2D was found to increase with increasing

loading arc angles.

An analytical solution for a disc loaded by a uniform

pressure that is applied radially over a short strip of the

circumference at each end of the disc is given by Hondros

(1959). Under plane stress (disc) conditions, the theoretical

horizontal tensile stress along the vertical diameter, as

shown in Fig. 1, is given by:

rh ¼2p

psin 2a

1� 2 rR

�2cos 2aþ r

R

�41� r

R

� �2� �(

� arctanrR

�2sin 2a

1� rR

�2cos 2a

" #� a

)ð9Þ

where p is the applied stress, R is the radius of the speci-

men and t its thickness and r is the polar coordinate.

A comparison between the analytical solution obtained

by using (9) for 20� loading arc angle (b = 20�) and the

numerical simulation results of all loading geometries are

given in Fig. 6. The theoretical horizontal stress distribu-

tion is reasonably consistent with the numerical results,

although there are clear discrepancies towards the bound-

aries. The reason for these discrepancies may come from

differences in the assumed boundary conditions for the two

types of analyses.

As shown in Fig. 6, tensile stresses reach a maximum

at the centre of the disc (r = 26 mm) and persist over

more than half of the diameter of the specimen. In

general, the analytically determined tensile stress distri-

bution moves to the centre of the disc at a greater

increasing rate than the numerically calculated distribu-

tion. On the other hand, the increment in the tensile

stress rate of the Brazilian jaw simulation is the lowest

when approaching the centre of the disc in all simula-

tions and under all analytical values. The results indicate

that concentration of the tensile stress under Brazilian

jaw loading geometry through the centre of the disc is

lower than that using loading arcs. Details of the

obtained stress values from simulations and analytical

results are given in Table 3.

As shown in Fig. 6, the rx stress distribution inside the

numerical model changes dramatically because of the

Fig. 4 Numerical models for the BTS test under a standard Brazilian jaws, b 15� loading arcs, c 20� loading arcs and d 30� loading arcs


123

heterogeneity effect (Liu 2004). For example, at the posi-

tions of crack initiation, the tensile stress drops to zero,

while at the bridges between the cracks, the tensile stress

might rise to values larger than the tensile strength of the

rock material.

Figures 7 and 8 show simulation-derived fingerprint

patterns of the maximum principal stress and minimum

principal stress induced in disc specimens subjected to the

same displacement loading, but under different loading

geometries. The stress distribution near the centre of disc

under diametral loading is quite uniform for both Brazilian

jaws and diametral loading arc cases. The stress distribution

in the y-axis direction is more uniform than that in the x-axis

direction. When the angle of the loading arc increases, the

maximum principal stress distribution (compressive stress)

becomes more complex along the loading direction com-

pared with the standard Brazilian jaw results (Fig. 7). On

the other hand, the minimum principal stress (maximum

tensile stress) is focused more at the centre of the disc as the

angle of the loading arcs increases (Fig. 8). Thus, the

obtained fingerprint patterns are of great significance for

elucidating the crack initiation point in non-transparent rock

material under different diametral loading boundaries. The

fingerprint patterns show that the crack initiation point

Fig. 5 Progressive fracture

process induced between

a standard Brazilian jaws, b 15�loading arcs, c 20� loading arcs

and d 30� loading arcs


123

moves to the centre of the disc under increasing loading arc

angles, similar to the results for the progressive fracture

process shown in Fig. 5.

To obtain a proper loading condition to determine the

most accurate indirect tensile strength of rocks, some

relationships between intrinsic material properties, such as

fracture toughness, crack initiation and propagation in a

sample, were considered. There are many suggested

methods to determine fracture toughness of rocks. Here, it

was decided to use the method of Guo et al. (1993) to

determine fracture toughness value from Brazilian tests,

since it uses Brazilian discs without a notched crack. To

determine the mode I fracture toughness (KIC) from Bra-

zilian tests, crack propagation in a Brazilian disc was

studied analytically and experimentally by Guo et al.

(1993). Their fracture toughness value is determined by:

KIC ¼ P� B� Uða=RÞ ð10Þ

where, KIC is fracture toughness ðMPffiffiffiffimpÞ, P is the mini-

mum load in a load–displacement curve (MN), U(a/R) is a

dimensionless stress intensity factor, B ¼ 2=ðp32 � R

12

�t � aÞ, R is the radius of the disc (m), a is the half loading

arc angle (radian) and t is disc thickness (m).

Some of the fracture toughness values determined by

using the Brazilian test are close to the results obtained by

using the Chevron Bend (CB) method, which is the method

suggested by ISRM (Guo et al. 1993; Alkilicgil 2010).

According to the assumptions of Guo et al. (1993), the

crack initiation point cannot be guaranteed to locate at the

disc centre, and the stress distribution on the loading arc is

assumed to be uniform. On that basis, Wang and Xing

(1999) modified the Brazilian disc method of Guo et al.

(1993) by flattening the loading ends of the disc. According

to the solution given by Wang and Xing (1999), the frac-

ture toughness value is determined by:

KIC ¼PminffiffiffiRp� t

Umax ð11Þ

where Pmin is the local minimum load determined from

recorded force–displacement curve, R and t are the radius

and thickness of the specimen, respectively, and Umax is the

maximum dimensionless stress intensity factor.

The maximum mode I stress intensity factor (KImax)

values calculated from the numerical analysis given by

Guo et al. (1993) and Alkilicgil (2010) are used in the

calculation of the maximum dimensionless stress intensity

factors ðUmaxÞ for loading angles from 2a = 5� to

2a = 50� (Fig. 9). As expected, the maximum dimen-

sionless stress intensity factor decreases with increasing

loading angles.

Figure 10 shows the recorded force–loading displace-

ment curves in diametral compression of the standard

Brazilian jaws (Fig. 5a) and the three different loading arc

angles (Fig. 5b, c, d). Initially, the rock specimen behaves

as an intact elastic material with increasing load resulting

in an increasing diametral displacement until the diametral

crack initiates and propagates unstably. Subsequently, the

load reaches its local minimum corresponding to the

Table 2 Experimental and numerical results of failure loads for

Brisbane tuff

Loading mode Experimental ultimate

load (kN) (average

of 4 repeats)

Normalized

numerical ultimate

load (kN)

Standard Brazilian 19.6 17.1

15� loading arcs 14.8 18.6



Fig. 6 Horizontal stress distribution along the vertical diameter (BY)

of the disc (-rx: tension; -ry: compression)

Table 3 Experimental, numerical and analytical results of failure

loads for Brisbane tuff

Loading

mode

Experimental

ultimate load

(kN)

Theoretical

tensile stress at

the centre of the

disc (MPa)

Numerical tensile

stress at the

centre of the disc

(MPa)

Standard

Brazilian

jaws

19.60 9.20 7.91

15� loading

arcs

14.81 6.61 8.04

20� loading

arcs

19.20 8.21 8.29

30� loading

arcs

22.30 8.85 8.70


123

maximum dimensionless stress intensity factor. After that

time, an increase in the applied load application is required

to make the crack propagate further. At that time, Pmax can

be used to calculate the splitting stress (indirect tensile

strength) of the Brazilian disc; however, determination of

fracture toughness is possible by using the local minimum

load (Pmin) during unstable crack propagation, derived

from (11).

Fracture toughness values calculated by using (11) and

the numerical simulations of the standard Brazilian jaws,

15� loading arcs, 20� loading arcs and 30� loading arcs

are: 0.85 MPaffiffiffiffimp

, 0.7 MPaffiffiffiffimp

, 0.81 MPaffiffiffiffimp

and 1.24

MPaffiffiffiffimp

, respectively. In contrast, the mode I fracture

toughness of Brisbane tuff was found experimentally to

be 1.18 MPaffiffiffiffiffiffimp

(Erarslan 2011), using the Crack

Chevron Notched Brazilian Disc (CCNBD) method

suggested by ISRM (1995). The CCNBD specimens have

the same diameter and thickness as standard Brazilian

disc specimens; thus, a relationship between fracture

toughness and indirect tensile strength of rocks is both

helpful and meaningful. The numerical stress intensity

value closest to the experimental fracture toughness of

1.18 MPaffiffiffiffimp

, was obtained with the 2a = 30� loading

arc simulations (1.24 MPaffiffiffiffimp

). To support that result, a

well-known empirical relationship between the mode I

fracture toughness (KIC) and the tensile strength of rocks

(rt) is used (Whittaker et al. 1992). According to this

Fig. 7 Fringe patterns of maximum principal stress induced between a standard Brazilian jaws, b 15� loading arcs, c 20� loading arcs and d 30�loading arcs

Fig. 8 Fringe patterns of minimum principal stress induced between a standard Brazilian jaws, b 15� loading arcs, c 20� loading arcs and d 30�loading arcs

Fig. 9 Maximum dimensionless stress intensity factors for various

loading angles


123

relationship, KIC can be calculated using (Whittaker

et al. 1992):

KICðMPaffiffiffiffimpÞ ¼ 0:27þ 0:107rtðMPaÞ: ð12Þ

The indirect tensile strength was found to be 8.55 MPa

by using the experimental mode I fracture toughness value

as 1.18 MPaffiffiffiffimp

in (12). The closest indirect tensile stress

to this value is obtained by 2a = 20� and 2a = 30� loading

arc simulations.

4 Discussion

The main objective of this study was to analyse the tensile

stress distribution and failure of disc specimens under

loading arcs of different angles and compare them with the

results of the standard Brazilian test. This was tested

experimentally, as well as numerically and analytically.

In the experiments described herein, the disc specimens

shattered into several pieces under the standard Brazilian

jaws. It is likely that these multiple fractures were caused by

the release of energy stored in the testing machine after the

specimen split into two hemi-cylinders, making the standard

Brazilian test very sensitive to the stiffness of the testing

machine being used. In contrast, applying distributed

diametral loads by using loading arcs causes single failure

planes along the loaded diameter in disc specimens tested.

According to Jaeger and Cook (1976), shear fracture at the

contacts can be decreased if the load is applied to the cir-

cumference of the cylinder. They stated that experiments

with distributed loads over a narrow arc, usually about 15�,

yielded tensile strength values little different from those

obtained with line loads and gave rise to similar diametral

extension fractures (Jaeger and Cook 1976).

The indirect tensile strength of Brisbane tuff was cal-

culated as 9.2 MPa using the ISRM (ISRM 2007) sug-

gested formula (Table 3). According to the analytical

solutions of Hondros (1959), when 2a\ 11� and r/R tends

to zero, the tangential stress along the vertical diameter

may be calculated using the same formula. This indicates

that the formula proposed by ISRM is not just for line

loading, but is also suitable for diametral loading over an

arc of up to 2a = 11�. This outcome is also supported by

the work of Jaeger and Cook (1976). Based on experiments

with distributed loads over a narrow arc, usually up to

about 2a = 15�, they reported that tensile strength varied

little from that obtained using line loads and gave similar

diametral extension fractures.

Analytically and numerically calculated tensile stresses at

the centre of the disc specimen are given in Table 3. Although

the highest failure load was found experimentally with a 30�loading arc, the highest tensile stress was found with Brazilian

tests after analytical calculations. This result shows that fur-

ther theoretical studies are needed to formulate the relation-

ship between line and diametral loading on disc specimens.

On the other hand, no large difference between obtained

tensile stresses for Brazilian and loading arc tests were found

among the numerical simulations (Table 3). The reason may

be related to material heterogeneity, since the horizontal stress

distribution changes dramatically along the vertical diameter.

To account for heterogeneity, the maximum tensile stress at

the centre of the disc was calculated by averaging values from

5 to 10 tests. Although a difficulty related to heterogeneity did

arise, there was agreement between numerical and experi-

mental results, presumably because we included material

heterogeneity in the numerical model.

In numerical simulations, rock heterogeneity was found as

a stress concentrator, similar to the presence of pre-existing

cracks under loading. In front of the crack propagation

direction, if the strength of the elements is not too high, the

crack propagates in a straight manner, which may be con-

sidered as the trans-granular failures observed in experiments.

On the other hand, if the strength of the elements is very high,

the crack will propagate around elements, and a tortuous crack

propagation path is observed, which can be considered as the

inter-granular failures observed in experiments.

Hondros’s (1959) solutions indicate that the simple Bra-

zilian formula for principal tensile stress at the centre of the

Fig. 10 Load–displacement curves by a standard Brazilian jaws and

b loading arcs


123

sample is in error by less than 2% for contact arcs of up to 15�.

Jaeger and Hoskins (1966) indicate that the ring stress con-

centration factors for line loads differ from those for loads

distributed over 15� arcs by an error of less than 2% (Jaeger

and Hoskins 1966; Mellor and Hawkes 1971). On that basis,

it is concluded that 15� is an acceptable upper limit for

analytical solution agreements between line and diametral

distributed loading. However, although the highest failure

load was obtained experimentally, the tensile stress for 30�loading arcs was smaller than that given for line loading by

analytical solutions. This lack of agreement indicates that a

detailed analytical solution is needed to determine the

applicability of Hondros’s (1959) equations for calculations

of tensile stresses along the loading diameter of disc.

According to the Griffith criterion (Griffith 1920) used for

failure in solid discs, the exact centre of the disc is the only

point at which the conditions for tensile failure at a value

equal to the uniaxial strength are met. This result is also

indicated by analytical solutions given by Hondros (1959).

Fairhurst (1964) generalized the Griffith criterion to account

for variation of the compression/tension strength ratio

(n) from the theoretical value, n = 8, so that the conditions

for failure according to the generalized Griffith criterion

become available. His results indicate that to assure fracture

initiation near the centre of a homogeneous specimen, it is

necessary to spread the applied load over an appreciable

contact arc (20� or more). He suggests that with a narrow

contact strip (a = 5�), there will be a pronounced tendency

for off-centre fracture initiation in rocks that have a low

compression/tension strength ratio. He also showed that if

the compression/tension strength ratio is relatively small and

the contact strip is narrow, there will be a systematic

underestimation of tensile strength, if the fracture occurs

near the centre of the specimen. Our experimental and

numerical results support Fairhurst’s results (1964).

Guo et al. (1993) indicate that the standard Brazilian test

has some disadvantages: crack initiation and propagation

cannot be guaranteed to be located at the centre, arc

loading is hard to apply and stress distribution on the

loading arc cannot be assumed to be uniform. By consid-

ering these disadvantages, a modified Guo et al. (1993)

method was used in fracture toughness calculations by

using a flattened Brazilian disc method (Wang and Xing

1999). However, extensive analytical, numerical and

experimental validation of the method is needed.

5 Conclusions

Indirect tensile tests, which induce non-uniform stress

fields controlled partly by the properties of the rock

material and loading geometry, can never fully substitute

for a direct uniaxial tensile test. However, a Brazilian disc

under diametral compression is capable of giving a very

good measure of uniaxial tensile strength for Griffith-type

and/or heterogeneous materials when it is carefully per-

formed, with special attention paid to control of contact

stresses. In reality, there is no perfect source creating direct

tensile stresses for rock materials; thus, the use of indi-

rectly produced tensile stresses and indirect tension tests

are much more applicable to rock mechanics applications.

From the results of the experimental studies described

herein using Brisbane tuff disc specimens, it was concluded

that indirect tensile strength of rocks is strongly dependent

on the type of diametral loading. A general increase in

failure load was obtained with an increasing angle of

loading arc for all disc specimens tested. The maximum

recorded failure load, corresponding to the finest and most

central-located crack, was obtained with 2a = 30� loading

arcs. The highest failure load standard deviation was

obtained with standard Brazilian jaws, which resulted in

catastrophic specimen failure.

Numerical simulations show that rock can be specified

by using characteristic parameters. Numerical simulations

indicated that the maximum tensile stress appears at the

centre of the disc under 30� loading arcs, while the lowest

maximum tensile stress appears at the centre of the disc

under standard Brazilian jaws.

The RFPA numerical model successfully simulated rock

fracturing in Brazilian disc specimen under line and

diametral loading. Since fracture formation and the

mechanism of fracture propagation were monitored as a

progressive process with R-T2D code, it was possible to

analyse crack initiation and propagation as a continuous

process in a Brazilian disc under various diametral loading.

Based on the agreement between experimental and

numerical results, the RFPA method is deemed reasonable

and applicable for research into the fracture process of

heterogeneous materials, such as rocks.

Experimentally obtained fracture toughness values of

Brisbane tuff were compared with the fracture toughness

values obtained from numerical simulations. It was con-

cluded that the best loading geometry to find indirect tensile

strength of rock material was obtained from the 2a = 20�and 2a = 30� loading arc simulations. To support that result,

well-known empirical relationships between fracture

toughness and tensile strength of rocks were applied.

Acknowledgments Acknowledgement is made to Leighton Con-

tractors, who provided core samples of Brisbane tuff from the

CLEM7 Project, and to Professor Ted Brown AC, Les McQueen,

Mark Funkhauser and Rob Morphet of Golder Associates Pty Ltd for

their assistance and advice. The work described forms part of the first

author’s PhD research carried out at the Golder Geomechanics Centre

at The University of Queensland. The first author was supported by an

Australian Postgraduate Award/UQRS and the Golder Geomechanics

Centre.


123

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