Copyright 2004, ARMA, American Rock Mechanics Association

This paper was prepared for presentation at Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS): Rock Mechanics Across Borders and Disciplines, held in Houston, Texas,June 5 - 9, 2004.

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ARMA/NARMS 04-483

Modelling Dilation in Brittle Rocks

N. Cho, C.D. Martin & D.C. Sego

Dept. Civil & Environmental Engineering, University of Alberta, Edmonton, Canada, T6G 2G7

R. Christiansson

Swedish Nuclear Fuel and Waste Management Co., Stockholm, Sweden

ABSTRACT: In laboratory tests, the onset of dilation occurs at stress levels far below the peak strength but yielding of

the laboratory specimen is not synonymous with the onset of dilation, and is seldom measured or reported in traditional

laboratory testing. In field tests, the on-set of dilation is often associated with stress-induced extension fracturing. The

displacements associated with these stress-induced fractures, cannot be replicated using traditional constitutive modelling

and associated or non-associated flow rules. In this paper a methodology is developed for modeling dilation using the

Particle Flow Code (PFC) that captures many of the observations reported in conventional laboratory test results. The

findings from this research show that clumped-particle geometry provides the best agreement with laboratory test results

for both tensile and compressive loading paths.

1 INTRODUCTION

Experience with underground excavations at depth in-

dicates that one of the most significant phenomena

observed in brittle rocks is extensile fracturing. This

fracturing occurs as a result of tangential stress con-

centrations. Direct observation of brittle rock failure

around underground openings reveals that this exten-

sile fracturing exhibits significant dilation (Fig. 1). A

detailed description of the spalling process observed

around a circular test tunnel was given by Martin et al.

[1] and Lajtai [3] showed that in laboratory samples

the brittle failure process resulted in the opening of

fractures.

In materials such as metals and clays, yielding can oc-

cur without significant volume change. However, in

brittle rocks on the boundary of underground openings

overstressing results in the development of micro- and

macro-cracks. In the mining industry, the process is

often referred to as ‘spalling’ or ‘dog-earing’. In the

petroleum industry, the problem is often cast as ‘well-

bore breakouts’. One of the early descriptions in civil

engineering was given by Terzaghi [2] and referred

to as ‘popping rock’. Modeling of this process has

always been challenging and has received a lot of at-

tention in the mining, nuclear waste and petroleum

industries since the 1950’s.

With the advent of modern computers, both contin-

uum mechanics and traditional fracture mechanics

approaches have been used to model this fracturing

process [3–6]. The use of continuum mechanics to

Fig. 1: Dilation associated with stress-induced fracturing

observed in a 600-mm-diameter borehole.

simulate a fracturing process that results in an open

rough fracture, as described by Lajtai [3] and shown in

Fig. 2, is extremely problematic as the displacement

field across the fracture in a continuum must be con-

tinuous. But if the fracture is open, this requirement

cannot be satisfied. In traditional fracture mechanics,

the fracture has zero width, again suggesting that this

approach is not applicable for representing a process

that results in open fractures. In all these approaches

specific flow rules are required to capture the displace-

ment field. In continuum mechanics an associated or

non-associated flow rule is assumed.

For the fracture mechanics approach the control of

the fracture growth is related to the fracture toughness

(KIC) [4, 6–8]. In both approaches there are funda-

mental shortcomings. Using continuum mechanics,

there is no easy way to generate tension, ie., an exten-

sile crack, when the stress state is all round compres-

sion. The same restriction also applies to fracture me-

chanics codes. In fracture mechanics this shortcoming

is overcome by inclining the crack to the stress field to

generate ’wing-cracks’. By taking this approach there

is a fundamental assumption that the rock is weaker

in shear than in tension since the only way tensile

cracks can be generated is by shearing first. This is

counter to one of the most important characteristics

of rocks as noted by Neville Cook in his Müller Lec-

ture: [9], rocks are fundamentally weaker in tension

than in shear. Hence any modeling process that vi-

olates this characteristic would appear to have little

chance of modeling the spalling process.

Fig. 2: Example of the fracture surface observed in core

discing (left) and direct tension test (right). Note the rough-

ness along the fractures at the grain scale.

Since the mid 1990’s there has been growing interest

in discrete element modeling. As noted by Cundall

[10], one of the major advantages of this numerical

method is that one does not specify a flow rule. How-

ever, there are other issues associated with model cal-

ibration and micro-scale properties that need to be

resolved. In this paper the discrete element program

PFC is used to explore if it can overcome some of the

difficulties noted previously when modeling fracture

initiation and growth and the associated dialtion.

2 A BONDED PARTICLE MODEL

All rocks are heterogeneous at the micro and macro

scale. In rocks such as granite the heterogeneous na-

ture can be readily observed with the naked eye. Lab-

oratory testing of such rocks shows that when sub-

jected to sufficient deviatoric loads these rocks dilate

laterally indicating that axially aligned fractures are

forming [11]. The deviatoric load required to generate

these opening, i.e, dilatant fractures, is considerably

less than the peak strength of the rock and for uniaxial

compressive tests is often reported as 1/3 to 1/2 the

uniaxial strength. If unloaded permanent straining is

recorded in the lateral direction, while no permanent

straining is recorded in the axial direction indicating

that the dilatancy process is not elastic and not uni-

formly distributed. One of the mechanisms that can

explain this observation is illustrated in Fig. 3.

σ1

σ3

Tension Crack

(a) (b)

(c) (d)

Distortioncausedby deviatoricstress

Initiation of permanent dilation

Fracture propagation& associated dilation

Fig. 3: An illustration of the dilation process in brittle het-

erogeneous rocks.

As rock is subjected to deviatoric stresses, local ten-

sile stress will be generated because of the heteroge-

neous nature of the rock. A likely location for a crack

resulting from these tensile stresses is at the grain

boundary (Fig. 3a). Because of the irregularities at

grain boundaries, even a small amount of elastic dis-

tortion, once the tensile crack forms, could result in

dilation. This dilation may be sufficient to induce ten-

sile stresses at the crack tip and cause the crack to open

(Fig. 3b). Moreover, this elastic distortion will result

in additional tensile stresses and when distributed to

neighboring cracks could result in additional cracking

and permanent dilation (Fig. 3c). The coalescence or

alignment of individual cracks may ultimately prop-

agate in an unstable manner, i.e., because the region

of tensile stresses grows, and form a macro fracture

surface with associated dilation (Fig. 3d). The essen-

tial elements of this process are the deviatoric stresses

required to cause the distortion once the crack forms

and the nonplanar surface of the microcrack. Core

disking is an extreme example of the self propagating

fracture resulting from this process.

2.1 PFC

The bonded particle analogue used in PFC repre-

sents a rock mass as an assemblage of circular disks

connected with cohesive and frictional bonds. In this

model, the breakages of bonds between particles by

applied local stresses are expressed as cracks. PFC

does not require any flow rule for describing the post

peak region or fracture toughness to control fracture

behavior, but only requires the law of motion of parti-

cles and laws for particle bond rupture and deforma-

tion.

Diederichs [12] successfully used this approach to

simulate brittle behavior of rock under compressive

loading and noted that a bond rupture in PFC does not

create the same singular stress concentration present

at the tip of an extending micro crack within a con-

tinuum. The rupture in PFC results in stress redistri-

bution in neighboring bonds, but this redistribution

may not be adequate to rupture the adjacent contacts.

As a result the crack generating process in PFC is a

stable process, i.e., to generate new cracks the devi-

atoric stress must be increased. Therefore, PFC will

not, without modification model the dilation and crack

generation process described above.

In direct tensile loading in PFC, once cracks are ini-

tiated, they are immediately propagated in an unsta-

ble fashion because the loading boundary conditions

cause stress concentrations at the crack tips which lead

to unstable fracturing, i. e., the load does not need to be

increased once fracturing begins to get the sample to

fail. This means that once a crack is initiated, it cannot

be stabilize until the boundary stress is removed or re-

duced. Whereas in compression, crack accumulation

must occur and individual cracks must interact to cre-

ate a macroscopic rupture surfaces.This is very impor-

tant when simulating extensile fracturing around un-

derground openings. Because the stress-induced ex-

tensile fracturing which occurs at the boundary of the

underground opening where the confining stress is ap-

proximately zero, and therefore if large scale tensile

stresses can occur, extensile fracturing can propagate

in an unstable fashion similar to the tensile loading

example.

One of the disconcerting results when using PFC

to represent rock is that the uniaxial compressive

strength obtained in PFC is approximately 4 times

greater than the tensile strength. This is dispropor-

tionately low, compared to granite and most other

rocks where the ratio of σt/σci is typically reported as

0.04 to 0.03 (σt/σci = 1/24 to 1/30) [13]. Diederichs

[12] showed that calibrating PFC to the compressive

strength resulted in significantly higher relative ten-

sile strength, and a linear strength envelope without a

tension cut-off as illustrated in Fig. 4. This is clearly

not in keeping with measured laboratory results.

s1

s3

Contact

Force chain

Rupture by

tensile bond

breakage

Cracks caused by

tensile bond breakage

doesn't cross

compressive bridge

Fig. 4: Contact force chain structure in PFC and the failure

envelope when compression is used for calibration tension

[12].

Diederichs [12] concluded that the linear tensile en-

velope in PFC is attributed to the intrinsic contact

force fabric structure in PFC. Diederich’s argument

is very important when modeling the dilation process

discussed above because extensile fracturing induces

dilation and vice versa. In other words, modeling of

extensile fracturing, dilation and high tensile strength

problem in PFC are not separate issues. In the fol-

lowing section the factors that could control dilation

in PFC are explored and quantified.

3 FACTORS CONTROLLING DILATION IN

PFC

PFC is a discrete element code that represents a rock

mass as an assemblage of circular disks confined by

planar walls. In this system, the particles can move

independently of one another and interact only at con-

tacts. They are assumed to be rigid but can be over-

lapped at the contacts under compression. The parti-

cles can be bonded together be specifying the shear

and tensile bond strength. The values assigned to these

strengths influence the macro strength of the sample

and the nature of cracking and failure that occurs dur-

ing loading. The shear strength can only be mobilized

once the tensile bond strength is broken or set to zero.

Similarly, the contact shear forces are a function of

compressive normal force at the contact and coeffi-

cient of friction. After a bond breaks in PFC, stress is

redistributed and this may then cause adjacent bonds

to break. Thus, PFC only requires basic parameters

to describe contact stiffness (kn, ks), bond strength

(bn, bs), friction (µ) and does not require any plas-

tic flow rule formulation. This implies that only those

parameters and geometrical factors that related to par-

ticle structure such as particle size or particle shape

can control dilation in a discrete element model. It

is therefore, most important to understand how those

parameters may control dilation in PFC.

3.1 Bond Models

Two basic bond models are provided in PFC: (1) Con-

tact Bond and (2) Parallel bond model. The contact

bonds can be envisaged as a pair of elastic springs (or

point of glue) with constant normal and shear stiffness

acting at a point. A parallel bond approximates the

physical behavior of a cement-like substance joining

the two particles. Parallel bonds establish an elastic

interaction between particles that acts in parallel with

the slip or contact-bond constitutive models. It also

t = 1

M

V

T

L

A B

L L

2R

σ = Fn

AR+

M

I

Fn Fn

M M

Fig. 5: Illustration of parallel bond model.

can be envisioned as a set of elastic springs uniformly

distributed over a rectangular cross section with con-

stant normal bond stiffness shear bond stiffness lying

on the contact plane and centered at the contact point.

Particles in PFC are free to move in the normal and

shear direction and can also rotate between particles.

This rotation may induce a moment between particles

but the Contact Bond model cannot resist this mo-

ment. With the Parallel Bond model however, bond-

ing is activated on finite area, this bonding can resist

the moment as shown Fig. 5. The Parallel Bond model

is therefore a more realistic bond model for rock like

materials whereby the bonds break by tension rather

than by particle rotations.

3.2 Parametric Study

The PFC bond parameters were examined using uni-

axial compressive test and brazilian test simulation.

The procedure for the sample generation and load-

ing methods are illustrated elsewhere and in the PFC

manual.

To avoid issues associate with loading rate, the plate

velocity was set to 0.2 m/s for uniaxial compressive

test and 0.05 m/s for the Brazilian test. Those load-

ing rates are slow enough to ensure that the speci-

men remains in quasi-static equilibrium throughout

the test, such that no inertial effects occur (e.g., pre-

mature bond failure resulting from a stress wave pass-

ing through the specimen or platen loads in excess of

their quasi-static equilibrium values). Because of the

heterogeneity introduced in PFC (e.g. standard devi-

ation applied to micro strength and arbitrarily gener-

ated disk assembly with different ball radius), no two

PFC runs produces identical results. During this para-

metric study, each simulation was run 10 times with

same parameter and the results averaged. While this is

somewhat laborious it approximates the real variabil-

ity encountered when testing rocks and avoids getting

unique answers that may not be relevant.

The lateral strains for the compressive tests were

recorded at 80% of peak strength for the purposes

of this study. The lateral strain is used to gauge

the amount of dilation in each sample. All compres-

sive samples measured 30 mm by 60 mm and con-

sisted of 963 particles. Tensile strength was obtained

from Brazilian test simulation with the PFC specimen

taken from the uniaxial compressive test sample us-

ing the same diameter of the Brazilian sample as the

width of the compressive sample. The ratio of tensile

strength to compressive strength was determined for

each simulation. The micro parameters used during

the parametric study are were based on the Lac Du

Bonnet Granite and given in Table 1.

Table 1: Micro-parameters in PFC to represent Lac du Bon-

net granite.

Ec 49 GPa Ec 49 GPa

kn/ks 1 kn/ks 1

µ 0 σc 200 ± 50 MPa

σc/τc 1 λ 1

Rmin 0.55 mm Rmin/Rmax 1.65

3.3 Effect of Particle Friction

As parallel bond breaks either by rotation or shear in

PFC, the force acting on the particle contact are set

to a residual value that depends on the compressive

normal force at the contact and coefficient of fric-

tion. The friction between particles, when mobilized

after bond breakage, can suppress independent par-

ticle movement such as rotations and slippage. This

process could increase the shear resistance resulting

in high compressive macro strength. With increasing

friction, dilation may be suppressed provided the di-

lation process is dominated by the micro friction pa-

rameters.

In this study, the coefficient of friction was varied

from 0.0 to 0.9. Fig. 6 illustrates that as friction in-

creases dilation is suppressed. Also shown in Fig. 6 is

the ratio of tensile strength to compressive strength.

The strength ratio is not influenced by friction at the

contacts. This implies that friction between particles

does not effect the force chain structure discussed by

[12].

3.4 Effect of Bond Stiffness

Diederichs [12] showed that the normal to shear stiff-

ness ratio (kn/ks) involves Poisson’s ratio. If the stiff-

ness ratio is high, Poisson’s ratio increases and the

sample becomes brittle. For brittle material therefore,

this stiffness ratio must be set to a lower value around

1.0. In this study, the stiffness ratio was varied from

1.0 to 10.0. Fig. 7 indicates that as stiffness ratio is

increased, the amount of dilation is also increased im-

plying that more tension cracks develop.

Potyondy and Cundall [14] also found similar results

and suggested that when the stiffness ratio is relatively

high, the Poisson’s effect attracts more tensile cracks.

When the number of tensile cracks increases, the lat-

eral dilation also increases. However, the strength ra-

tio in Fig. 7 indicates that this increase in dilation does

not impact the strength ratio.

3.5 Effect of particle size

Potyondy and Cundall [15] suggested that particle

size could control tensile strength. They found a rela-

tionship between particle size and facture toughness

by using Brazilian test simulations (Fig. 8). They as-

sumed that wedge fractures occurring at the edge of

the sample could initiate a single micro fracture that

forms the rupture surface. For their example the stress

intensity factor was calculated as:

KI = α1σ√

πa (1)

At peak load, KI = KIc, σ = σt , a = βD and β is a

constant defined by the ratio of initial crack length a

and sample diameter D. Thus, the Brazilian strength

can be related to fracture toughness by:

KIc = α1σt

√

πβD (2)

0.00

0.01

0.015

0.02

0.025

0.03

0.035

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Coefficient of Friction (µ)

La

tera

l D

ilata

nt

Str

ain

(%

)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Lateral dilatant strain

Strength Ratio

σt /σ

ci

Fig. 6: Effect of friction on dilation and the ratio of tensile

strength to compressive strength.

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0 2 4 6 8 10 12

Bond Stiffness Ratio (kn / ks)

La

tera

l D

ilata

nt

Str

ain

(%

)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Lat. dilatant strain

Strength Ratio

σt / σ

ci

Fig. 7: Effect of stiffness ratio on dilation and the ratio of

tensile strength to compressive strength.

and fracture toughness can be related to micro prop-

erties by:

KIc = φ√

α2πR (3)

where φ is mean bond strength, R is mean particle

radius and α1, α2 are constants.

From Equations 2and 3, the Brazilian strength can be

related to material micro properties by: and fracture

toughness can be related to micro properties, where,

σt = α3φ

√

R

D(4)

where, α3 =√

α2/α21β and indicates all constants.

Equation 4 implies that particle size is proportional to

magnitude of tensile strength. Micro properties used

for the particle size simulation analysis are shown in

Table 2. In this case the same properties are used for

both the Brazilian and compressive test simulation.

Fig. 9 shows that as particle size decreases, the tensile

to compressive strength ratio also decreases. The ten-

sile strength to compressive strength ratio for Lac du

Bonnet granite is 10/200 = 0.05. Thus to achieve this

strength ratio the particle size would have to approach

unrealistic values as the grain size of Lac du Bonnet

F

F

a

a

σσ D

F

F

a

(a) (b)

Fig. 8: Effect of friction on dilation and the ratio of tensile

strength to compressive strength.

Table 2: Micro-parameters in PFC used in particle size

effect.

Ec 62 GPa Ec 62 GPa

kn/ks 2.5 kn/ks 2.5

µ 0.5 σc 157 ± 36 MPa

σc/τc 1 λ 1

Rmin varies Rmin/Rmax 1.66

granite is approximately 2 mm. Also as particle size

gets smaller the required calculation time exponen-

tially increases. In terms of dilation, the smaller the

particle size the lower the amount of dilation (Fig. 9).

Thus reducing the tensile to compressive strength ra-

tio in this case, does not contribute to the dilation

process.

Potyondy and Cundall [15] also revealed that particle

assemblies PFC circular particles result in macroscale

failure envelope, with unrealistically low friction an-

gles, when compared to measured values. Hence there

are two deficiencies in PFC that cannot be corrected

by simply changing the microscale contact parame-

ters: the tensile strength to compressive strength ratio

and the failure envelope.

3.6 Effect of grain shape

In brittle rock modeling using the discrete element

method (DEM) analogue, researchers often use the

circular disk elements. This circular disk element has

advantages related to reducing the calculation times,

but most microscale particles that the DEM represents

are seldom circular. Jensen et al. [16] and Thomas and

Bray [17] indicated that circular disk elements are not

adequate to model geometry dependent particles such

as irregular shaped grains. One of the solutions to

overcome this limitation is to introduce a cluster con-

0.00

0.01

0.01

0.02

0.02

0.03

0.03

0.04

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Mean particle size (mm)

La

tera

l d

ilata

nt

str

ain

(%

)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Lateral dilatant strain

Strength Ratio

σt / σ

ci

Fig. 9: Effect of particle size on dilation and the ratio of

tensile strength to compressive strength.

Crystalline rock Clustered particles

Fig. 10: Clustered particles mimicking crystalline rock.

cept. In clustered assemblies, grains are modeled as

a group of individual circular disks. The intra-cluster

bond strength can be set to different value from the

bond strength of boundary particles neighboring with

other clusters. In this study, intra-cluster bond strength

was set to infinite, thus, bond breakage can only occur

at cluster boundaries. The clustered particles with this

setting are a better representation of naturally irregu-

lar grain (Fig. 10). Moreover, this irregular boundary

modeling method may change the force chains devel-

oped in PFC.

Fig. 11 illustrates the effect of a clustered assemblage

on dilation and the strength ratio. The cluster size in

Fig. 11 represents the maximum number of circular

disks in one cluster. Slaved particle size of cluster

was determined approximately matching the cluster

diameter into previous non-clumped case. Coefficient

of friction (µ) was set to 0.0, thus friction can only be

mobilized by roughness of cluster boundaries. Based

on the previous analysis, σc/τc strength ratio was set

to 20 thus only tensile cracking is allowed. Fig. 11

clearly shows that as cluster size increases, the amount

of dilation is significantly increased and hence asso-

ciated with the irregular cluster boundary, i.e, rough-

ness. It is worth noting that this dilation process is

much more closer to the mechanical dilation describe

by [3].

0.00

0.05

0.10

0.15

0.20

0.25

0 2 4 6 8 10 12 14 16

Maximum Cluster Size

La

tera

l D

ilata

nt

Str

ain

(%

)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Lateral dilatant strain

Strength Ratio

σt / σ

ci

Fig. 11: Effect of grain (cluster) size on dilation and the

ratio of tensile strength to compressive strength.

Also shown in Fig. 11 is reduction in the strength

ratio as cluster size and dilation increase. However,

the strength ratio shown in Fig. 11 is still high com-

pared with the 0.05 for Lac du Bonnet granite and

even seems to converge to 0.1. One possible expla-

nation for this is that rotation between particles are

not suppressed enough even in the clustered particles.

Although bond strength between intra-cluster parti-

cles has been set to infinitely high (e.g. an order of

magnitude) but particle rotation cannot be completely

suppressed and this may induce unnecessary cracking

and may reduce strength. One alternative approach to

avoid this problem is to use clumped particles instead

of clustered particles.

3.7 Clustered particles and clumped particles

The clump logic provides a means to create a group

of slaved particles and they behave as a rigid body

in a clump. Particles within a clump may overlap to

any extent as a deformable body that will not break

apart, regardless of the forces acting upon it. In this

sense, a clump differs from clustered particles. The

other big difference between clustered and clumped

particle is the particle rotation mechanism. As shown

in Fig. 12, intra-cluster particles in clustered material

have rotational velocities. Whereas rotational veloc-

ities of particles in clumped material are fixed. Only

the clumped body itself can have rotational velocities.

This mechanism is much more realistic to crystalline

rock grain behavior when fracturing occurs.

Introducing clump logic to the assemblage, dila-

tion and strength were investigated again. The re-

sults are given in Fig. 13 and clearly show as clump

size increased, dilation is significantly increased and

strength ratio is close to 0.05. It is worth noting that

the amount of dilation is almost an order of magni-

tude greater than the other cases discussed above. The

strength ratio is also closer to the values measure for

+

Clustered Particles Clumped Particles

Fig. 12: Particle rotation in cluster and clump particles.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

2 3 4 5 6 7 8 9 10

Maximum clump size

La

tera

l d

ilata

nt

dtr

ain

(%

)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Lateral dilatant strain

Strength Ratio

σt / σ

ci

Fig. 13: Effect of clump on dilation and the ratio of tensile

strength to compressive strength.

Lac du Bonnet granite and many other brittle rocks.

Fig. 14 shows failed sample of clumped material.

Unlike conventional PFC biaxial test, cracks oc-

curred everywhere whereas in traditional PFC mod-

els, macro-scale shear zones always develop. In this

model large scale dilation leads to axial splitting

which is also in better agreement with laboratory ob-

servations.

3.8 Controlling factors of dilation and strength ratio

From the above discussion, it is clear that the most im-

portant factor in controlling dilation and the strength

ratio is the geometrical factors rather than micro-

contact parameters. The clustered and clumped mate-

rial all showed an order of magnitude larger amount

of dilation and lower strength ratio compared with all

other cases. Thus, it is evident that geometrical fac-

tors such as cluster or clump cases are more realistic

and effective for modeling dilation and that dilation

actually controls the strength.

PFC2D 2.00

U of A Geotechnical Engineering

Job Title: Parallel-bonded, uniaxial compression test com_cl_size11

Step 226642 03:33:02 Wed Feb 25 2004

View Size: X: -5.024e-002 <=> 5.024e-002 Y: -4.950e-002 <=> 4.950e-002

Ball

Wall

FISH function draw_sample

FISH function crk_item

Fig. 14: Sample failure in clumped assemblage.

The other interesting result is that rotation of parti-

cles in assemblage has a significant effect on mate-

rial strength. For instance, the clumped material has

more than 20 times larger compressive strength than

non-clumped material with the same micro proper-

ties. Whereas, tensile strength of the clumped mate-

rial only increased 2 to 3 times more than the non-

clumped material. Hence this kept the strength ratio

low in clumped material. In conventional PFC mod-

eling, if one adjusts the micro-parameters to match

the macro compressive strength the tensile strength

is overestimated. The clumped logic overcomes this

deficiency.

4 CALIBRATION WITH SYNTHETIC ROCK

The parametric study described above was carried out

on Lac Du Bonnet granite because it has been exten-

sively studied both in the laboratory and by other re-

searchers using PFC. In this section, we use the same

PFC logic and apply it to a brittle material that has a

peak strength that is an order of magnitude lower than

Lac du Bonnet granite.

4.1 Laboratory test properties

For the purpose of comparison and verification, con-

ventional triaxial tests and Brazilian tests were per-

formed using a synthetic rock. This synthetic rock

has brittle characteristics but a much lower compres-

sive strength. The synthetic rock chosen for this test

is ‘sulfaset’ that is generally used for setting anchor

bolt.

The strength and stiffness of our synthetic rock is

highly dependent on its initial moisture content at

mixing, and for the test reported here the initial mois-

ture content has been fixed to 50% and cured for 3

days in a constant temperature and moisture room.

To induce random heterogeneity in the sample, 10%

sand was added to all the samples. The measured sam-

ple properties are: UCS = 11.6 ± 1.0 MPa, Ten-

sile strength = 1.4 ± 0.4 MPa, and Young’s Mod-

ulus = 2.5 GPa.

Uniaxial test results for this material (75 x 150mm,

cylindrical samples) clearly showed that the material

has brittle characteristics (Fig. 15). Note that initial

nonlinearity of stress-strain curve in Fig. 15 results

from closure of pores during seating.

0

2

4

6

8

10

12

14

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6Axial Strain(%)

PFC data

Lab data

σ3= 0.0 MPa

σ1-σ

3 (

MP

a)

(a)

0

2

4

6

8

10

12

14

16

18

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0Axial Strain(%)

PFC data

Lab data

σ3= 1 MPa

σ1-σ

3 (

MP

a)

(b)

0

2

4

6

8

10

12

14

16

18

20

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Axial strain (%)

PFC data

Lab data

σ3= 2 MPa

σ1-σ

3 (

MP

a)

(c)

0

5

10

15

20

25

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0Axial Strain(%)

PFC data

Lab data

σ3= 3 MPa

σ1-σ

3 (

MP

a)

(d)

Fig. 15: Comparison of PFC and laboratory test results for each confinement.

4.2 PFC calibration to laboratory data

The PFC micro parameters were determined from the

macro-scale laboratory results. The clump size was

set to 2 and the bond strength andYoung’s modulus is

reduced to half of the macro strength. When the ten-

sile strength bonds are broken, cohesion is lost and

the strength mobilized is composed of clump friction

and particle contact friction. The coefficient of friction

has been calibrated to 0.7 and this friction can be mo-

bilized when shear cracking occurs and as confining

stress is increased.

Fig. 15 compares the biaxial test simulation results

from PFC with laboratory test results by using the

micro-scale parameters in Table 3. When compared

with the laboratory results, the initial nonlinearity is

not present which is expected because in PFC no

cracks exist prior to loading. If a random distribution

of pores or cracks were used the nonlinearity mea-

sured would also be recorded in PFC.

Fig. 15 also shows in all cases, the agreement be-

tween PFC and laboratory results is excellent. It is

interesting to note that no flow rule is required for

modeling the stress-strain behavior in post peak re-

gion in PFC. The failure envelope in Fig. 16 shows

that PFC has a slightly higher slope compared with

the measured laboratory envelope. This might be be-

cause of the two dimensional assumption defined in

PFC. Nonetheless, the agreement with the laboratory

envelope is very good. More importantly the agree-

ment with the tensile and compressive strength is also

excellent, e.g. compare the results from [12] in Fig. 4.

The shape of the failure of zone in PFC shows the typ-

ical macro shear fractures observed in the laboratory

test (Fig. 14).

Table 3: Micro parameters in PFC used to represent the

Sulfaset synthetic rock.

Ec ( GPa) 1.3 Ec (GPa) 1.3

kn/ks 2.7 kn/ks 2.5

µ 0.7 σc (MPa) 4.5 ± 0.1

σc/τc 2 λ 1

Rmin (mm) 0.25 Rmin/Rmax 1.2

Density (kg/m3) 2500 Max clump 2

5 DILATION IN LABORATORY TESTS

Numerous researchers [18–21] have reported that the

onset of dilation in uniaxial tests initiates at about 1/4

to 1/2 of the peak strength. In most cases the dila-

tion is associated with the initiation of axially aligned

microcracks. Scholz [22], Holcomb and Martin [23],

Pestman and Van Munster [24] studied the crack initi-

ation of granite, sandstone, and marble using acoustic

emission and they showed that the onset of dilation

could be approximated by:

σ1 = 0.4σci + 1.5 to 2σ3 (5)

A series of PFC simulations were carried out to deter-

mine this dilational boundary in PFC. Fig. 17 shows

the onset of dilation in a typical PFC model for 3 MPa

confining stress. Fig. 18 shows the dilational boundary

for all the calibrated PFC models and this boundary

can be approximated by:

σ1 = 0.38σci + 2.1σ3 (6)

Note that the measured dilational boundary measured

in the laboratory specimens for various rocks and

given by Equation 5 is in good agreement with the

dilational boundary determine from PFC for the syn-

thetic rock and given by Equation 6.

Currently the calibration between real materials and

PFC is usually made via the traditional laboratory

compression tests. In order to know if PFC can be

used to model engineering problems such as slopes

and tunnels more complex loading paths are required.

Fig. 19 shows bending of a rock beam confined at the

5

10

15

20

25

30

-2.0 -1.0 0.0 1.0 2.0 3.0 4.0σ3 (MPa)

σ1 (MPa)

PFC best fitLab data best fitPFC dataLab data

c = 2.7 MPa, φ=40o

c = 2.7 MPa, φ=37o

Fig. 16: Strength envelope from PFC compared with en-

velope from laboratory test results.

σ3 = 3.0MPa-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 0.2 0.4 0.6 0.8 1 1.2

Axial Strain(%)

V /

V (

%)

3

8

13

18

23

28

Dilation Volumetric Strain

Total Volumetric Strain

Axial Stress Strain

Onset of Dilation

σ1 (M

Pa)

Fig. 17: Onset of dilation in PFC for a sample confined by

3 MPa.

ends. This complex loading path also results in ax-

ial fractures at the top of the beam. Proof testing has

been completed and samples are being prepared for

the final testing. During the testing extensive monitor-

ing will be used to measure the dilation. The dilation

obtained from these tests will then be compared with

PFC results.

6 CONCLUSIONS

Modeling of brittle failure has been attempted by vari-

ous approaches, i.e., continuum and fracture mechan-

ics, over the past 50 years. In nearly all cases these

approaches have provided poor agreement with ob-

servations of brittle failure in the field. In laboratory

tests calibration is often made with the peak strength

yet the onset of dilation in uniaxial compression test

is observed far below the peak strength.

During the past ten years discrete elements, particu-

5

10

15

20

25

-2 -1 0 1 2 3 4σ3 (MPa)

σ1 (MPa)

Peak Strength

Dilational boundaryσ1= 0.38 σci + 2.1 σ3

Fig. 18: The dilational boundary in the PFC models of

synthetic rock.

Frame

Frame

Ball

Ball

Beam

Wiremesh

Roller

Notch

Loading plateHydraulic ram

Fig. 19: Test equipment for measuring dilation induced by

extensile fracture.

larly the software PFC, have been used to model brit-

tle failure. Three significant deficiencies have been

identified when doing conventional PFC modeling:

(1) the tensile strength to compressive strength ratio is

considerably greater than that measured in laboratory

tests, (2) the failure envelop gives very low friction

angles compared to measured laboratory values, and

(3) the onset of dilation occurs much closer to the peak

strength compared to measured values. Adjusting the

micro parameters appears to have little effect on these

deficiencies. The findings from this research indicate

that by introducing clumped-particle geometry these

deficiencies are minimized or eliminated.

Using the clumped logic excellent agreement is found

between laboratory tests results and PFC for the Sul-

faset synthetic rock. Testing of more complex stress

paths and different rock types is continuing before

applying this logic to engineering problems such as

tunnels and slopes.

ACKNOWLEDGEMENTS

This work was supported by the Natural Sciences

and Engineering Research Council of Canada and the

Swedish Nuclear Fuel and Waste Management Co.

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