STUDY OF THE MECHANICAL BEHAVIOR OF INTACT AND JOINTED ... · Study of the mechanical behavior of intact and jointed rocks in laboratory with particular emphasis on dilatancy Ph.D

Number of words of this Ph.D. Thesis summary: 7522

THESIS SUMMARY Javier Arzúa, Ph.D.

STUDY OF THE MECHANICAL BEHAVIOR OF INTACT AND

JOINTED ROCKS IN LABORATORY WITH

PARTICULAR EMPHASIS ON DILATANCY

Study of the mechanical behavior of intact and jointed rocks in laboratory with particular emphasis on dilatancy Ph.D. Thesis summary by Javier Arzúa

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THESIS SUMMARY

Excavation in rock masses has developed during last years a boom, primarily due to improved

technology, excavation equipment and treatment processes. Examples of major projects are found

all around the world, for instance in the Swiss Alps, where the 57 km‐long Gotthard Base tunnel has

recently been put in full service; large underground mines as some large caving projects in Sweden

or South America; large open pit mines like Chuquicamata in Chile or Bingham Canyon in the United

States of America. Also urban use of underground space is being promoted, this promotion mainly

comes from the coldest countries in Northern Europe —where the existing rock mass beneath the

cities has good quality— or countries with a limited surface space (like Japan). It is also required to

go underground when discussing physics laboratories such as CERN —a particle accelerator in

Switzerland— or the U.S. Fermilab.

In addition to the excavation of rock masses for mining or civil purposes, sometimes it is only

necessary to fracture the rock mass in order to obtain an improvement of the properties required

for a purpose, for instance, in the case of oil and gas, to cause fracturing of the rock to enhance the

conductive capacity of the rock mass is a common (although socially controversial) practice, and,

thus, obtain a higher output in less time; or in the case of caving mining methods, the rock is required

to flow into the draw point, so it is necessary for the rock mass being sufficiently (whether or not

naturally or by means of pre‐conditioning) fractured.

One may realize that the best way to optimize the excavation or the breaking of the rock mass is to

best know its behavior under the conditions —particularly the stress state— in which rock mass will

be. This task is easy to say, but difficult to perform due primarily, but not only, to the natural

heterogeneity of the rock mass. It should be noted here that a rock mass is a three‐dimensional set

or net of rock blocks crossed by geological discontinuities of different origin, so the rock mass

behavior depends not only on the constituting rock, but also on the discontinuities that intersect it.

Both constituents also usually do not behave the same way all along a singular rock mass, because

there will be areas which may have suffered certain phenomena (weathering, metamorphism,

failures…) that may have locally modified their properties and have not affected nearby areas.

Since the early rock mechanics developments during the 60’s and 70’s of last century, much effort

has been expended to studying and modelling the elastic —or prior‐to‐failure— behavior of rock

masses, and, thus, there are models able to reasonably represent the actual pre‐failure behavior of

rock masses. However, post‐failure behavior has been much less studied, due to the inherent

difficulties of such study and because, typically, the primary objective of an engineer is avoiding

failure. But, as it has been discussed in previous paragraphs, the fracturing of the rock mass is a

requirement for some applications in mining industry. And even knowing how a rock mass will

behave in its plastic (“broken”) state is a key design factor in the prediction of the correct installation

moment —distance to the face— of the support of a tunnel. This knowledge of post‐failure rock

mass behavior can also be a key when analyzing damaged pillars into an underground mine and

proposing corrective measures.

To fully characterize a rock mass —i.e. to obtain the complete behavior of a rock mass under

different conditions of stress— regardless of whether its behavior is elastic‐brittle, strain softening

or elastic‐perfectly‐plastic, one must know the elastic parameters (Young’s modulus and Poisson’s


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ratio), the peak failure criterion (typically Hoek‐Brown or Mohr‐Coulomb), the residual failure

criterion (which has the same shape as the peak failure criterion but different parameters), the

evolving failure criterion from peak to residual state, the post‐failure parameters linking the

relationship between stress and strain, and the post‐failure relationship between strains. A correct

characterization of post‐peak behavior can be achieved if one knows, for instance (a) the dilation

angle and the drop modulus —the slope of the axial stress‐axial strain curve during softening— or

(b) the dilation angle and the plastic parameter values for which dilatancy and evolving failure

criterion are achieved.

This Ph.D. thesis aims to improve our understanding of post‐failure behavior of rock masses.

Unfortunately, the in situ deformability tests of rock masses are often unaffordable for various

reasons (scale of the required equipment; difficulty in isolating the factors; or the high cost, just to

mention a few), so it is necessary to resort to laboratory testing where one can more easily control

the variables that come into play. Yet there are inherent difficulties in the study of post‐failure

behavior of rocks that need to be addressed when proposing an experimental program:

A sufficiently large number of samples is needed to obtain a reliable result. The process of

obtaining and preparing the specimens is long and must be done accurately to meet

International Standards (as proposed in the Suggested Methods of the International Society

for Rock Mechanics, 2007).

The press frame shall be stiffer than the tested rock, in order to be able to handle and to

record the transition from peak to residual strength (softening phase). In the opposite case

(the tested rock is stiffer than the press frame) the specimen will explosively or abruptly fail,

making it impossible to collect reliable data on the softening phase.

Moreover, appropriate servo‐control for regulating the force exerted by the press on the

specimen during the test is required, just to, again, gain control over the softening process

between peak and residual strength.

The sensors used to measure the strains during the tests shall be accurate enough for the

required appreciation (between thousandths and tenths of a millimeter) and shall have a

sufficiently broad range of measure, in order to capture the large deformations in the

residual state. The latter consideration about strain sensors precludes the use of strain

gauges due to their limited range of measurement as well as other problems associated

with their use, such as the inability to capture localized strains (shear bands).

Since strain gauges cannot be used, it becomes impossible to measure the radial strain of

the specimen in triaxial tests without a specifically adapted Hoek’s cell. But in order to

obtain the complete stress‐strain curves of the tested specimen, one needs to measure, at

least, another strain (lateral or volumetric) apart from the axial one. We could not measure

lateral strain, so we measured the amount of hydraulic fluid that must be entered or

removed from the triaxial cell to maintain the confining stress during such tests. This

amount of displaced fluid is assumed to be the lateral component of the volumetric strain,

in such a way that the volumetric strain can be related to the volumetric strain of the

specimen and, thus, using both volumetric and axial strain, one can obtain the lateral strain.

Therefore, it is also necessary to have control on the confining stress and to be able to

measure the amount of displaced fluid. So, one needs another servo‐control for the

confining stress.


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The further processing of the data collected during the tests requires the identification of

the elastic (“reversible”) and plastic (“irreversible”) components of strains. While this

distinction between elastic and plastic deformation is theoretically simplistic, since it does

not consider inelastic phenomena (neither elastic nor plastic in the yield sense, but

irreversible in any case), or the flaws (micro‐cracks, pores…) initial closure —that gives the

initial concave shape to the axial stress‐ axial strain curve— it is valid for the level of detail

in which the research of post‐failure behavior of rocks and rock masses is. The approach

used in this Ph.D. thesis is sufficient, since it is assumed that the plastic strain does not begin

until it reaches the stress level named Crack Damage.

To make this distinction between elastic and plastic strains, is advisable to perform

unloading‐reloading cycles during the tests, especially in the softening and residual phases.

These cycles allow to obtain the locus known as Irrecoverable Strain Locus, which links axial

and volumetric plastic strains. Thanks to the Irrecoverable Strain Locus one can obtain one

of the most important parameters of post‐failure behavior of rocks and rock masses, the

dilation angle, . These cycles will also help to control and record the transition from peak

to residual strength.

Still speaking of data processing, note that it is not an easy task and it is also difficult to

automate because of the natural variability of the tests results. This process involves some

not‐so‐logical decisions that a computer is not always able to consider, so each one of the

tests must be carefully analyzed. Moreover, the automation of the process may potentially

present some associated loss of control and monitoring of the data that may lead to

difficult‐to‐detect errors.

All these reasons make the process of obtaining post‐failure behavior real data being long and costly,

if one also adds the already commented thought about avoiding failure as the main objective of

most engineers, it is not difficult to understand the lack of data related to post‐failure behavior of

rocks and rock masses. However, over the years, as underground excavations have grown in size

and numerical simulations become more and more important, easily available and more realistic, it

has been shown that it is necessary to improve knowledge of post‐failure behavior of rocks and rock

masses in order to understand the mechanisms that take place, and achieve the objectives of the

numerical simulation.

In order to deepen on our knowledge of post‐failure behavior of rocks, this Ph.D. thesis initially

(Chapters 3 and 4) proposed an experimental programme aimed at obtaining dilation angle in rocks

from laboratory tests on standard NX‐sized specimens (54.74 mm in diameter). More than 200

unconfined and confined strength tests were performed on specimens from eight different intact

rocks (Figure 1). These tests included the previously mentioned unloading‐reloading cycles in order

to be able to identify the plastic strains that are necessary to calculate the dilation angle. Some other

laboratory tests aimed at characterizing the rocks were also performed.

Complete stress‐strain curves of each of these tests were obtained (Figure 2). The most relevant

parameters needed to fully characterize the behavior of intact rock were also computed for each

test: Young’s modulus and Poisson’s ratio; peak and residual strength, the latter only when possible,

it is known that the residual strength of an unconfined strength test is usually null; drop modulus,

also when it was possible to estimate; and the evolution of the dilation angle with increasing strain.


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From these individual data, the evolution of the parameters of each of the studied rocks with

confining stress was also studied, showing the already known relationships, namely, when

increasing confinement, Young’s modulus, peak strength and residual strength increase, while

Poisson’s ratio does not seem to be affected and drop modulus reduces (the rock becomes more

ductile).

Figure 1. Different pictures of the eight studied rocks (Amarelo Pais granite, Blanco Mera, granite, Vilachán

granite, Indiana limestone, Toral de los Vados limestone, Carrara marble, Touro amphibolite and Noia

gneiss).


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Figure 2. Complete stress‐strain curve resulting from a confined compressive strength test on Amarelo País

granite specimen with unloading‐reloading cycles. Also shown is how key parameters were obtained.

The most used failure criteria in rock mechanics (Mohr‐Coulomb and Hoek‐Brown) were fitted to

peak and residual strength results, obtaining very good fits in both cases (Figure 3). It is to remark

that using the generalized Hoek‐Brown criterion (Hoek et al., 2002) to fit the results of residual

strength significantly improved the fit.

Figure 3. Peak and residual strength tests results and Mohr‐Coulomb and Hoek‐Brown failure criteria fits for

the three studied granites.


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Finally, in this part of the Ph.D. thesis, and after obtaining the evolution of the dilation angle with

plastic strain for each test, all obtained data for this parameter were pooled, thus, revealing

dependencies as targeted by Alejano & Alonso (2005), namely, the dilation angle decreases with

increasing both confining stress and plastic strain.

The obtained dilation angle was fitted to the, at that time, two existing models of evolution of

dilatancy. First one (Alejano & Alonso, 2005), although very easy to implement, was not able to

compute the peak dilation angle obtained in the tests, though it reasonably captured the evolution

of the dilation angle with plastic strain. Second model (Zhao & Cai, 2010) fitted the data fairly well

and the behavior of the dilation angle was precisely captured (Figure 4). However, this model

involves many difficulties due to the nine parameters —with difficult physical correlation— that are

required to characterize the model, so its application to everyday engineering work is questionable.

Alejano & Alonso (2005) also noted that the dilation angle should also depend on scale. The problem

of scale in rock mechanics, although known, is not fully solved due to, again, the natural variability

of the rock materials and the occurrence of discontinuities at all scales (from pores or micro‐cracks

to regional faults).

The problem of scale involves at least two causes with a similar effect:

First cause of scale effect is the variation of strength when changing size of intact specimen,

this can be named as sample scale effect. It is commonly assumed that the strength of a

specimen reduces monotonically as size of the sample increases —until the Elementary

Representative Volume (REV) is achieved— due to the increasing occurrence of flaws as size

increases. Recent researches (Masoumi et al., 2015) show that there is a double size effect

for sedimentary rocks, as was also observed by Nishimatsu et al. (1969) from the uniaxial

compressive tests on a number of different igneous rocks.

This double size effect implies that UCS (Uniaxial Compressive Strength) increases with

increasing sample size up to 40‐60 mm in diameter and, above this size, strength decreases

as sample size increases (again until REV is achieved). But, although there is still much to

research about size effect of intact rock, our laboratory equipment did not allow us to

perform triaxial tests on a sufficient variety of sizes in order to obtain reliable results, so we

decided to explore the second cause of scale effect, as explained in following paragraphs.

Second cause of scale effect is the variation of strength when considering the change from

intact rock at laboratory scale to a jointed rock mass at the engineering work scale. This can

be named as scale effect at large or structure effect. This Ph.D. thesis has attempted to

address the problem of scale at large (Chapters 5 and 6) creating two series of rock

specimens crossed by two artificial joint sets (the first series of specimens has two sub‐

horizontal and one sub‐vertical joints, second series has three sub‐horizontal and two sub‐

vertical joints). Cutting and preparing these specimens was not simple since the author does

not know any physical experiments as the tests presented here, so some previous tests were

needed before reaching a methodology aimed to obtain and test jointed specimens with a

reasonably degree of quality.


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Figure 4. Dilation angle vs. plastic shear strain for various confining levels and laboratory results and Zhao &

Cai (2010) dilatancy model fits for the three studied granites.


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As said before, the author does not know an experimental study with the same characteristics as

the one presented here, but, of course, there are several studies reflected in the literature on jointed

rock mass physical model behavior, even if many of them do not seem to have been adopted in

engineering practice. These studies encompass both persistent (most of references provided below)

and non‐persistent (Prudencio & Van Sint Jan, 2007; Fan et al., 2015) differently oriented joints

under uniaxial (Kulatilake et al., 1997; Singh et al., 2002), biaxial (John, 1969; Ladanyi &

Archambault, 1972; Kulatilake et al., 1997), triaxial (Brown & Trollope, 1970; Einstein & Hirschfield,

1973) and true triaxial stress (Tiwari & Rao, 2004, 2006) conditions.

A brief summary of the above cited references is presented in next paragraphs in order to show

main characteristics and main findings of these studies, as well as to note the differences with the

current study.

Physical model experiments on the mechanics of jointed rocks to study their strength and

deformability appeared in the literature during the sixties, with the study by John (1969). This author

carried out a large number of plain strain biaxial tests on schematized model rock systems with

variable parameters of joints and rock elements made of barite, zinc oxide and paraffin oil. He

presented results of strength and deformability in terms of ratios, trying to correlate results with

joint orientations. John (1969) identified the two basic mechanisms of failure associated to sliding

through existing joints —when they were favorably oriented— and to failure through intact material

more or less combined with discontinuities. In the current Ph.D. thesis, the mechanism associated

to sliding has been avoided by appropriately orienting joints.

Einstein and co‐workers (1969, 1973) carried out a similar study with gypsum plaster jointed blocks

submitted to standard triaxial stresses. They advanced in the study of possible failure mechanisms

based on the representation of Mohr‐Coulomb failure criteria of intact material, joints and jointed

models. Disregarding sliding phenomena where statically possible, they differentiated two mixed

failure mechanisms. At low stresses, that is, below the modified Griffith criterion (McClintock &

Walsh, 1967) failure occurs in a more brittle manner along surfaces (axial splitting) that passed

through intact material and intersected pre‐existing joints. At higher stresses —over the modified

Griffith criterion— failure occurs through the intact material, although the strength was more

difficult to estimate. These authors stated that a transition from brittle to ductile behavior is

observed with increasing confinement, a fact also clearly observed in the results presented in this

Ph.D. thesis.

Brown & Trollope (1970) carried out a series of triaxial compression tests on an idealized model of

jointed rock made up of gypsum plaster one inch cubes, to produce square prisms of dimensions

10x10x20 cm. Four directions of the joint sets were tested under five different confinements. They

concluded that the strengths observed were always far lower than that of the intact material, except

for the vertical‐horizontal case. They also remarked that linear strength laws were not applicable to

the jointed rock models.

Ladanyi & Archambault (1969, 1972) built models made of compressed brick rods, with a square

cross‐section of 12.7 mm in side and a length of 63.5 mm. Some of their models with blocks were

configured in a brick wall manner, that is to say, they presented continuous joints in one direction

and equally persistent and spaced joints with the continuity and spacing of a block side in the

perpendicular direction. Large size tests of these models involving around one thousand blocks were


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carried out for seven possible orientations of joints (from 0 to 90) and for four levels of confinement

(0.35, 0.7, 1.4, and 2.8 MPa). Results showed lower and more regular strength levels than those

computed considering the combined theoretical response of joints and intact rock, as pointed out

by Hoek (1983) (Figure 5).

Figure 5. Comparison between (a) predicted and (b) observed strength of brickwall model tested by Ladanyi

& Archambault (1972). Taken from Hoek (1983). This figure explains how the sharply defined predicted

transitions between different failure modes do not occur in practice.

Interpretation of these results by Hoek (1983) points to a number of relevant facts. Similar trends

in theoretically predicted and observed strength values with joint orientations can be derived, even

if observed values are systematically lower than predicted ones. This means that intact material

peak strength is not attained. The sharply defined predicted transitions between different failure

modes do not occur in practice, they are largely smoothed out due to rotation and crushing of blocks

(Figure 5). Moreover, Hoek (1983) suggests that the degree of interlocking of the model blocks is

extremely relevant in the strength response of the models, since it will control the freedom of the

blocks to move and rotate. In the light of the mentioned results, the author concludes that the

behavior of jointed rock elements or rock masses with some joint sets with similar strength

characteristics would tend to that of a homogeneous isotropic system.

Although the author of this Ph.D. thesis do not fully agree with the latter statement, it is clear that

in practice there is not usually a single crystal clear simple mechanism explaining the observed

response of jointed rock systems, but a combination of mechanisms producing a strength response.

In this sense, the author considers that the results presented in this Ph.D. thesis need not to be

assigned to a particular failure mechanism (in fact, it is observed that they combine various

mechanisms) and they are valuable by themselves in that they can be interpreted from different

scopes.


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Ramamurthy and Arora (1994) carried out a large number of unconfined and triaxial tests on intact

and jointed (one set of notched joints) sandstone and plaster of Paris samples. Based on these data,

they proposed a joint factor accounting for (i) the number of joints per meter length, (ii) inclination

of the sliding joint, and (iii) shear strength along this joint. This factor is found to be related to the

ratio of compressive strength of jointed rock to that of the intact rocks for the material tested.

Singh et al., (2002) under uniaxial compression and Tiwari & Rao (2004, 2006) under standard and

true triaxial compression, studied the deformability and strength behavior of sand‐lime brick jointed

models. The first authors identified failure mechanisms in uniaxial compression, including sliding

through joints, splitting, shearing and rotation of blocks. In the extended true triaxial studies, the

authors mapped failure mechanisms, showing that the strain‐softening is the most common

behavior under triaxial stresses when sliding is not statically possible, for instance when, such in the

case showed in this Ph.D. thesis, the main set of joints dips around 20. For horizontal and vertical jointing, Tiwari & Rao (2006) also remarked that all stress‐strain curves tend to meet at a single point

on failure, corresponding to the ultimate strength of intact rock.

Kulatilake and his co‐workers (Kulatilake et al., 1997, 2001, 2003; Fan et al., 2015) also carried out

some jointed rock physical models. They further developed a non‐linear rock mass failure criterion

for biaxial loading conditions incorporating the fracture tensor component.

Although the here presented tests only include two sets of persistent joints cut in particular

orientations and tested only in one direction, this study differentiates from previous approaches in

various aspects: blocks are made up of intact hard rock instead of plaster or other engineered

typically weak material; samples present a standard size, that makes them comparable to intact rock

tests; various unloading‐reloading cycles are performed in every test to reflect post‐failure

evolution; and, finally, various tests were carried out for every confining stress in order to ensure

representativeness.

The idea behind this Ph.D. thesis is to study how discontinuities not producing structurally controlled

failures affect the response of jointed samples. It is sometimes important to single out behavior

modes to try to understand them, avoiding other existing mechanisms. This is why the approach has

been developed as it is, and the strength, elastic and post‐failure parameters have been investigated

in a single direction. The followed methodology is in line with modelling philosophy as stated by

Baran and Sweezy (1968): “The purpose of models is not to give a mirror image of reality, not to

include all its elements in their exact proportions, but rather to single out and make available for

intensive investigation of those elements which are decisive. We abstract from non‐essentials, we

blot out the unimportant to get an unobstructed view of the important, we magnify in order to

improve the range and accuracy of our observation.”

As commented before, two sets of joints were cut in each specimen and in two series with different

amount of joints. These joints were oriented in samples in order to avoid slip through them.

Accordingly, sub‐horizontal joints were cut forming and angle of roughly 22 with the specimen

bases, a smaller value than the tested friction angle —which was slightly over 30—. This angle was

selected to make sliding through joints statically impossible, accounting for the response of a sample

with a joint as described for instance, in Jaeger & Cook (1969) or Brady & Brown (2006). The idea

was to study how joints affect the response of samples, but avoiding the so‐called structurally

controlled failure or sliding through pre‐existing joints.


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22 and 20 of these “small scale rock masses” were obtained from one of the previously studied rocks

(Blanco Mera granite) and triaxial compression strength tests were performed. It should be noted

that due to the characteristics of the specimens, it was neither possible to perform unconfined

compression strength nor indirect tensile strength (Brazilian) tests. Tensile strength is assumed to

be zero, since there is no cohesion in the joints. The unconfined strength is also considered null

since, although these specimens could reach some UCS in these conditions, the value would be very

small compared to the obtained triaxial strength, so it could be assumed zero.

The same procedures and analyses as in previous chapters were performed for the studied

parameters. The results were again compared with the confining stress, revealing the same

dependencies as for the intact specimens. But these results were also compared with those of the

intact specimens, revealing very interesting trends that shall be confirmed with more intensive

experimentation.

Young’s modulus of jointed specimens is significantly reduced compared to the intact specimens

(Figure 6). This observation is logical if one takes into account the lower stiffness of the

discontinuities relative to the rock matrix. In addition, it is the expected/observed behavior when

the stiffness of the rock is compared to that of the rock mass.

It was also observed that Young’s modulus grows with confinement, but it does not grow linearly as

hitherto believed, but rather it seems to be a logarithmic relationship: there is a strong increase of

Young’s modulus for small initial increments of confinement and, then, for successive increments of

confinements, Young’s modulus growth slows down.

It is well known the dependence of the elastic modulus of a jointed rock mass on its geotechnical

quality (Serafim & Pereira, 1983), for instance, and considering the last relevant reference, the

formula of Hoek & Diederichs (2006) may allow us to obtain a value of GSI (Geological Strength

Index) for each tested jointed specimen. Acting this way, mean values of GSI = 55 and GSI = 50 are

obtained for each series of jointed specimens. I will further take up these values.

Poisson’s ratio hardly changed its value (0.16 in intact specimens to 0.16 and 0.19 in jointed

specimens) so it seems reasonably to think that the general approach —in which, for good quality

rock masses, one can consider the same Poisson’s ratio as for the rock— is realistic.

Peak strength has been collected for each test as the ultimate strength of the specimen and showed

its dependency on confinement, as previously observed in intact specimens. When comparing peak

strength results of the intact specimens with those of the jointed cores, one can observe a reduction

in strength as jointing increases. The most used failure criteria in rock mechanics (Mohr‐Coulomb

and Hoek‐Brown) were fitted to the results.

Mohr‐Coulomb failure criterion fitted relatively well to the results of all types of specimens. Figure 7

shows the results of peak strength fits in p‐q axes and in 1‐3 axes. A noteworthy decrease in peak

strength is observed when increasing jointing, although the slope of the fits remains similar. This

suggests that the loss in strength mainly occurs in the cohesive component.


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Figure 6. Graphical representation of the apparent (a) tangent (or average) and (b) secant elastic Young’s

moduli for the intact, (1+2) and (2+3) jointed specimens.


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Figure 7. Peak strength test results and Mohr‐Coulomb failure criterion fit for intact, (1+2) and (2+3) jointed

specimens in terms of (a) p‐q line and (b) results in 1‐3 axes.

Hoek‐Brown failure criterion could not be used in its simpler form (a=0.5; s=1) because the fits for

jointed specimens gave out of range values of m parameter. So, in order to estimate jointed sample

strength, and since the jointed specimens are not “intact rock”, it seems more reasonable to

consider the generalized Hoek‐Brown failure criterion (Hoek et al., 2002) and obtain a GSI value for

the tested specimens. Fitting this failure envelope to the triaxial tests results (Figure 8) mean GSI

values of 83 and 67 are obtained for the (1+2) and (2+3) series of jointed specimens respectively.

These values are larger than those computed using the elastic approach. But, let us now consider a

reduction of strength in rock associated to sample scale up to a 70% (as commented by Martin et

al., 2014; and Hoek & Brown, 1980). In this case, the mean obtained GSI of the jointed specimens

are 65 and 50 respectively, which are near the values obtained with the elastic approach

Figure 8. Peak strength test results and generalized Hoek‐Brown failure criterion fit for intact, (1+2) and (2+3)

jointed specimens in terms of (a) (1‐3)2‐3 axes and (b) results in 1‐3 axes.

Residual strength has also been estimated for each test and the results show that all residual

strength values of all unjointed and jointed specimens tend to coincide for each confinement.


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As for the peak strength, Mohr‐Coulomb (Figure 9) and generalized Hoek‐Brown (Figure 10) failure

criteria were fitted to the residual strength fits obtaining similar results for all types of specimens.

It is to remark that the obtained residual GSI of the jointed specimens featured a confinement

dependence (Figure 10.b), which is in line with Bahrani & Kaiser comments (2013), who indicated

that in a jointed rock mass, the generalized Hoek‐Brown failure criterion tend to systematically

underestimate the strength of the rock mass at high levels of confinement, when assuming a

constant value of GSI.

Figure 9. Residual strength tests results and Mohr‐Coulomb failure criterion fit for intact, (1+2) and (2+3)

jointed specimens and all tests together in terms of a) p‐q line, and b) results in 1‐3 axes.

Figure 10. a) Residual strength tests results and generalized Hoek‐Brown failure criterion fit for intact, (1+2)

and (2+3) jointed specimens. b) residual strength GSI results for intact, (1+2) and (2+3) jointed specimens,

suggesting a relevant confinement effect in jointed samples.

Drop modulus was also collected from each test and showed its dependency on confinement, as

previously observed on intact specimens. When comparing this parameter results of jointed

specimens with intact ones a more ductile behavior is also observed in jointed specimens. The same

explanation like for the Young’s modulus may be valid here.


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Summarizing and tanking up GSI values, empirical based correlations (Hoek & Diederichs, 2006) of

elastic moduli suggest that (1+2) and (2+3) jointed samples behave elastically similar to rock masses,

presenting average values of GSI around 56 and 50 respectively. The application of the Hoek &

Brown approach to peak strength results suggests that jointed and even more jointed samples’

strength would correspond to rock masses presenting average values of GSI around 82 and 67,

respectively. But if strength results are reduced to 70 % of the recorded values —in order to remove

the scale‐effect— the GSI value corresponding to (1+2) jointed samples becomes 65 and that for

(2+3) jointed samples 50. Additionally, post‐failure response indicates that the jointed samples

behave as rock masses showing decreasing drop moduli and less dilation. In the case of (2+3) jointed

samples a value of GSI of 40 can be roughly estimated from approaches regarding post‐failure

response of rock masses. In conclusion, the response of (2+3) jointed samples may represent a rock

mass with average rock mass quality (GSI=50‐40), whereas that of (1+2) jointed samples represent

a somewhat better quality rock mass (55‐65).

If we think now in a standard 4 m diameter tunnel excavated in a granitic rock mass showing a

regular pattern of normal discontinuities with fair behavior and presenting a spacing of 0.9 m, an

equivalent GSI of around 60 would be obtained. In the case of a spacing of 0.4 m, the estimated GSI

would be around 50 (Figure 11). If we could test a 1 m diameter and 2 m high sample in both cases,

the presented structure would be homothetic to that presented in our jointed samples. The stress

strain response of these samples would be representative of the rock mass at the scale of the tunnel.

Since the rock structure in these samples would be homothetic to our samples, the response of our

samples should be the same of the rock mass, once corrected the sample scale effect. We have

corrected the sample scale effect (by considering 70% of peak strength) and we have shown that, in

a rough way, the response of our samples is similar to that expectable for the equivalent rock mass.

Figure 11. An approach to estimate rock mass parameters based on tested jointed samples representative of

rock mass structure and comparison with the traditional approach based on GSI.

This suggests that once reduced the sample scale effect, by considering a 70 % of the computed

value of peak strength for every test in jointed samples, the stress‐strain response could be similar


16

to that expectable in equivalent rock masses at the scale of the engineering work. Therefore, tests

in jointed samples as presented can be of interest to investigate rock mass behavior. Remark that,

in the same way as in rock joints, peak strength seems to be affected by scale but residual strength

does not seem to be (Barton & Bandis, 1982).

Even if the jointed samples present very regular fracture patterns, and the showed results are

limited, the above indicated approaches suggest that the behavior of rock masses is largely

controlled by their structure, and this structure is associated to the geological stress history of the

rock mass. Therefore, a continuous behavior in terms of the complete stress‐strain curve can be

derived as presented in Figure 12. This is in line with Archambault et al. (1993) suggestion on the

fact that patterns of shear and tension discontinuities and the resultant shear strength scale effects

in rocks and rock masses are the end result of a progressive material and mass softening mechanism

associated to progressive shear deformation and failure, and to rotational simple shear complexity,

which takes place at all scales.

Figure 12. Deduced behavior of intact and increasingly fissured rock samples in parallel with decreasing

quality rock masses in terms of a) axial stress‐axial strain, b) strength, c) volumetric strain response and d)

elastic moduli.

In Figure 12, the evolution of axial stress‐axial strain behavior for three different levels of

confinement is represented (Figure 12.a). The strength weakening is associated either to the

occurrence of joint in a sample or to that of a joint structure in a rock mass. As far as the rock is


17

more jointed or the rock mass more fractured, the peak strength diminishes even if residual strength

seems to keep constant (Figure 12.b), the material is more deformable (Figure 12.d), and the

capability of dilating is more limited (Figure 12.c).

In this line we have also noted that the irrecoverable strain loci of jointed specimens superpose that

of the intact specimens if they are moved downwards and rightwards (Figure 13). This seems to

suggest that during the process of rock fissuring in the lab or during the process of rock failure in

nature to produce a jointed rock mass, an irrecoverable strain takes place in such a way that when

the material is reloaded, it retakes the volumetric strain process, stopped at a particular stage in the

past.

Figure 13. Volumetric strain vs. axial strain response of three characteristic samples, where also the

irrecoverable strain loci is delineated in dotted lines. Observe that when moving the irrecoverable strain loci

of fissured samples downwards and rightwards, these irrecoverable strain loci tend to superpose over the

locus corresponding to intact rock sample.

Finally, in this part of the study, obtained dilation angle values were fitted to the mobilized dilation

angle model by Zhao & Cai (2010) (Figure 14) and to the newly proposed dilatancy model by Walton

& Diederichs (2015) (Figure 15) and compared to the previously obtained model of the intact

specimens. The dilation angle showed its dependence on degree of jointing, resulting in smaller

dilation angles for the jointed specimens than for intact ones at lower confinements, but with a

decreasing gap between intact and jointed specimens as confining stress increases, becoming

almost equal for confinements from 6 MPa onwards.


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Figure 14. Zhao & cai (2010) dilatancy model fits for different confining stresses and for the three types of

specimens —intact, (1+2) and (2+3) jointed specimens—. Please, note the different vertical scale.


19

Figure 15. Walton & Diederichs (2015) dilatancy model fits for different confining stresses and for the three

types of specimens —intact, (1+2) and (2+3) jointed specimens—. Please, note the different vertical scale.

Note that where the dilatant behavior of rock masses may have significance for engineering works

is on the free sides of an excavation —i.e. where the confinements are low— so that this discovery

in the dilation angle difference between intact and jointed specimens at low confinement may have

a relevant impact on rock engineering.


20

Although the dilation angle is an important parameter in the post‐failure behavior of rocks and rock

masses, itself alone is not able to define this behavior, so hereafter (Chapter 7) a model able to

represent the strain‐softening behavior is proposed. For this purpose —from the stress‐strain curves

of the first tests carried out on the three granitic rocks— the relations between the axial stresses

and the plastic strains before, during and after the softening phase (transition from peak to residual

strength) were obtained (Figure 16).

Figure 16. Estimate of relevant 1‐p points from stress‐strain complete curves.

Subsequently, an empirical approach was applied and a mathematical equation was fitted to the

obtained pairs of points (axial stress and plastic strain). Combining this model of evolution of the

relationship between axial stresses and plastic strains with the mobilized dilation angle model of

Zhao & Cai (2010) previously fitted, allowed the author to simulate using FLAC (Itasca, 2011) the

dilatant softening behavior of the studied rocks.

Since it is known that mesh size of the numerical model when considering strain‐softening behavior,

affects the results (Varas et al., 2005), we also tried different mesh sizes to try to check this

possibility.

While there are still differences between the model and actual behavior, the obtained approach is

much better than existing models (Figure 17).


21

Figure 17. Complete stress‐strain curve (σ1 ‐ 1, σ1 ‐ 3 and v ‐ 1) for two triaxial tests (2 and 6 MPa) in

Amarelo País granite. Actual tests and FLAC results with mesh size 15 x 60 corresponding to such confining

pressures.

Finally (Chapter 8), to illustrate the differences between considering or not a variable dilatancy

model, a numerical model of a deep tunnel using FLAC (Itasca, 2011) was created (Figure 18) and

different situations were compared by considering three rock mass qualities (three different post‐

failure behaviors) with both a constant and a variable dilation angle approaches.

Figure 18. FLAC mesh used to perform the tunnel simulations to investigate face behavior


22

The results show that for average rock masses (those presenting strain‐softening behavior) the

consideration of a variable dilation angle increases calculated strains on the face and walls of the

tunnel, while it does not seem to affect the extent of the yielded zone (Figure 19).

Figure 19. Displacement distributions around the tunnel face for different behaviors of the average quality

rock mass.


23

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STUDY OF THE MECHANICAL BEHAVIOR OF INTACT AND JOINTED ... · Study of the mechanical behavior of intact and jointed rocks in laboratory with particular emphasis on dilatancy Ph.D