On the mechanical behavior of granite: Constitutive...

On the mechanical behavior of granite:

Constitutive modeling and application

to percussive drilling

Mahdi Saadati

Doctoral thesis no. 89, 2015

KTH School of Engineering Sciences Department of Solid Mechanics

Royal Institute of Technology SE-100 44 Stockholm Sweden

TRITA HFL-0567 ISSN 1104-6813 ISRN ISBN

KTH/HFL/R-15/03-SE 978-91-7595-435-6

Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan i Stockholm framlägges till offentlig granskning för avläggande av teknisk doktorsexamen fredagen den 6 mars 2015, kl. 10:00 i sal K2, Kungliga Tekniska Högskolan, Teknikringen 28, Stockholm. Fakultetsopponent är Professor Gioacchino Viggiani, 3SR lab., Université Joseph Fourier, Grenoble, Frankrike.

PhD is tough, as though as Rock!

Mahdi

Abstract

The mechanical behavior and fragmentation response of rock materials is

investigated in this work. In particular, Bohus granite is selected with application

to percussive drilling. It is well known that rock behaves totally different in

compression and tension and dynamic loading conditions and high strain rates

under the percussive drilling process makes the material behavior even more

complicated. The KST-DFH material model is shown in this work to be

appropriate in order to constitutively describe granite at dynamic fragmentation. It

consists of a plasticity model in compression and a damage model in tension. The

yield surface locus is a quadratic function of the mean pressure in the principal

stress space and the damage model is anisotropic.

Several experiments are performed in order to define the mechanical behavior and

dynamic response of granite and calibrate the KST-DFH model parameters for

this material. The material model is implemented as in a commercial finite

element program and validated based on dynamic tests such as Edge-on Impact

(EOI) and spalling test using Hopkinson bar. The numerical tool is then used to

model the rock response during the percussive drilling process. In doing so, only

one spherical tool button and just the first impact are considered for simplicity.

The anticipated fracture mechanism in percussive drilling is captured and the

penetration stiffness obtained is in agreement with practical drilling experiments.

In paper A, the experimental work is described and the granite mechanical

response is explained. In particular, the influence from pre-existing cracks and

defects is examined in great detail. In paper B, the experimental results are used to

calibrate the material model parameters. The numerical tool discussed earlier is

employed to investigate the rock fracture mechanisms at percussive drilling. In

paper C, the effect of pre-existing, or structural, cracks on dynamic fragmentation

of granite is investigated in detail. These cracks may be the result from former

impact of the drill bit, or by means of other unconventional methods such as

microwave and laser that are used to increase the effectiveness of the percussive

drilling process. In paper D, the dynamic tensile behavior of granite samples is

investigated. Spalling tests using a Hopkinson bar are performed and a strain rate

of order 102 1/s is obtained. This experimental technique involves the same order

of strain rate as present in rock materials during percussive drilling. A dynamic

tensile strength of 18.9 MPa is obtained at a strain rate of 70 1/s. This is more than

twice the tensile strength of the specimen (with the same size) at quasi-static

conditions, which is 8 MPa.

Sammanfattning

I denna avhandling har bergmaterials mekaniska beteende undersökts. Det

viktigaste tillämpningsområdet för de framtagna resultaten är bergborrning och

det har då varit naturligt att speciellt analysera granit (Bohusgranit). Granit är ett,

från mekanisk synpunkt, komplicerat material som bland annat beter sig helt olika

vid drag- och tryckbelastning och vars mekaniska beteende dessutom påverkas av

de höga töjningshastigheter som fås vid borrning. I studien används den så kallade

KST-DFH-modellen för att konstitutivt beskriva granit vid dynamisk

fragmentering. I denna modell beskrivs materialets beteende vid tryckbelastning

med hjälp av plasticitetsteori medan anisotrop skademekanik utnyttjas för att

beskriva materialbeteendet vid dragbelastning.

Ett antal olika experiment har genomförts för att kunna förstå det mekaniska

beteendet hos granit och dessutom för att bestämma parametrarna i KST-DFH-

modellen. Materialmodellen har implementerats i ett kommersiellt FEM-program

och de framtagna numeriska resultaten har jämförts med motsvarande

experimentella resultat från det så kallade EOI-testet (EOI-testet är ett dynamiskt

experiment där en aluminiumprojektil skjuts mot kanten av en granitplatta).

Denna jämförelse visade på en acceptabel överensstämmelse mellan experiment

och simuleringar, och det framtagna numeriska verktyget användes sedan för att

bland annat analysera sprickinitiering och spricktillväxt under det första inledande

skedet av borrprocessen.

I rapport A beskrivs huvudparten av det experimentella arbetet (förutom EOI-

experimenten) och det mekaniska beteendet hos granit diskuteras i detalj. I rapport

B används det ovan diskuterade numeriska verktyget för att analysera EOI-testet

och bergborrning. I rapport C undersöks i detalj effekten av redan existerande

eller strukturella sprickor, sprickor som finns i materialet före belastning, på den

dynamiska fragmenteringsprocessen. Dessa sprickor kan vara ett resultat av

tidigare påverkan från borrning eller också medvetet införda, för att underlätta

borrningen, med hjälp av metoder som mikrovågor och laser. I rapport D

undersöktes granits dynamiska draghållfasthet. Splittringstester (Hopkinson)

utfördes och töjningshastigheter av storleksordningen 102 1/s uppnåddes. Denna

experimentella teknik användes eftersom den ger ungefär samma

töjningshastigheter som vid borrning. En dynamisk draghållfasthet på 18.9 MPa

erhölls vid en töjningshastighet på 70 1/s. Detta värde är mer än dubbelt så stort

som motsvarande kvasi-statiska värde (8 MPa).

Preface

The work presented in this doctoral thesis was carried out at the Department of

Solid Mechanics, KTH Royal Institute of Technology, Stockholm between

November 2010 and March 2015. The work is fully funded by Atlas Copco which

is gratefully acknowledged.

I would like to express my sincere gratitude to my supervisor Prof. Per-Lennart

Larsson for an excellent supervision, guidance and encouragement. I would also

like to thank my supervisor at Atlas Copco Dr. Kenneth Weddfelt. We have had

many fruitful discussions and I have learned a lot from you.

Many thanks to Prof. Pascal Forquin at Grenoble ˗ Alpes University for excellent

support in performing the experiments and great technical discussions. Without

your guidance and fruitful discussion, this thesis would not have been possible.

Also many thanks to Prof. Francois Hild at LMT Cachan for initiating the

collaboration in this project and for very helpful discussions.

I am very grateful to my colleagues at Solid Mechanics Department for making

such a nice and friendly working environment. Also to my colleagues at Atlas

Copco for being so nice and supportive all the time. Thank you all!

Last but definitely not least, I would like to thank my wife Maryam for her

understanding and love during the past years. Her support and encouragement was

in the end what made this work possible. My parents receive my deepest gratitude

and love for their dedication and support in my life. Also my friends for always

being there for me. With your help and support, a lot of things became possible.

Stockholm, Jan 2015

Mahdi Saadati

List of appended papers

Paper A: On the mechanical behavior of granite material with particular

emphasis on the influence from pre-existing cracks and defects

M. Saadati, P. Forquin, K. Weddfelt, P.L. Larsson, F. Hild.

Report 545. Department of Solid Mechanics, KTH Engineering Sciences, Royal

Institute of Technology, Stockholm, Sweden.

Paper B: Granite rock fragmentation at percussive drilling - experimental and

numerical investigation


International Journal for Numerical and Analytical Methods in Geomechanics,

38.8, 2014, 828-843.

Paper C: A numerical study of the influence from pre-existing cracks on granite

rock fragmentation at percussive drilling


Accepted for publication in International Journal for Numerical and Analytical

Methods in Geomechanics, 2014 (DOI: 10.1002/nag.2331).

Paper D: On the tensile strength of granite at high strain-rates considering the

influence from pre-existing cracks

M. Saadati, P. Forquin, K. Weddfelt, P.L. Larsson.

Report 566. Department of Solid Mechanics, KTH Engineering Sciences, Royal

Institute of Technology, Stockholm, Sweden.

1A = Abstract, Ea = Extended abstract, Op = Oral Presentation, Pp = Proceeding paper

In addition to the appended paper, the work has resulted in the following

publications and presentations1:

Granite rock fragmentation at percussive drilling


Presented at International Conference on Rock Dynamics and Applicatons,

Lausanne 2013 (OP).



J. Zhao and J. Li, Eds., Rock Dynamics and Applications - State of the Art. CRS

Press, 2013 (Pp).


M. Saadati, K. Weddfelt, P.L. Larsson.

Presented at Svenska Mekanikdagar, Lund 2013 (A, OP).

Effect of pre-existing cracks on the fracture pattern of granite


Presented at International Conference on Experimental Mechanics, Cambridge

2014 (A, OP).

Contribution to the papers

The author’s contributions to the appended papers are as follows:

Paper A: Principal author. Experiments are performed together with P. Forquin.

Contributed to the theoretical analysis. Wrote the paper together with P.L.

Larsson.

Paper B: Principal author. Experiments are performed together with P. Forquin.


Larsson.

Paper C: Principal author. Experiments are performed together with P. Forquin.


Larsson.

Paper D: Principal author. Experiments are performed together with P. Forquin.


Larsson.

PhD is tough, as though as Rock!

Mahdi

Contents

Introduction .............................................................................................................. 1

Background .......................................................................................................... 1

Constitutive specification .................................................................................... 6

KST model ..................................................................................................... 11

The DFH fragmentation model ...................................................................... 12

Experimental work ............................................................................................. 13

Direct compression and tension test .............................................................. 13

Quasi-oedometric compression test ............................................................... 14

Flexural test .................................................................................................... 15

Edge-on impact test ....................................................................................... 15

Spalling test .................................................................................................... 16

Numerical modeling .......................................................................................... 16

Concluding remarks and suggestions for future work ....................................... 20

Bibliography .......................................................................................................... 23

Summary of appended papers ................................................................................ 27

Paper A

Paper B

Paper C

Paper D

1

Introduction

The work in this thesis concerns the constitutive modeling of rock fracture due to

dynamic loading with particular emphasis to percussive drilling and granite

material. Both experimental and theoretical investigations are involved in this

work. A brief review of different drilling methods is presented. In this

introduction, first a summary of the constitutive model selected for the rock

fragmentation at percussive drilling is presented. Then the experimental work to

validate and calibrate the model for granite is followed. Special attention is given

to the effect from pre-existing structural cracks on the dynamic response of the

material. Consequently some additional results from a practical drilling

experiment are compared with the numerical ones.

Background

The blocks in Egyptian pyramids were obtained by using hammers and wedges to

extract the rock in precise shape and size. From the time of the Roman Empire to

the Middle Ages, miners used a “fire setting” system to break rock. Rock was

exposed to heat followed by sudden cooling using water. This caused the rock to

crack and split along the natural seams. In the seventeenth century a drill forged

of wrought iron with an insert bit of tempered steel was employed to drill rock.

The charges were then placed in the hole and removed the rock more effectively

than those placed on or near it. Steam driven hammers were first introduced in the

nineteenth century followed by pneumatic and hydraulic drills. However, these

equipments competed with breaking the rock with hand even up to the beginning

of twentieth century [1]. The “Swedish method”, a one-man pneumatic rock drill,

was introduced in 1940s (see Fig. 1). The combination of a strong but light drill

On the mech. behavior of granite: Constitutive modeling and appl. to percussive drilling

2

with a pneumatic pusher leg and tungsten carbide drill bits was an important

achievement in this machine [2]. The drilling equipments have been constantly

evolved to this day and the drilling efficiency has increased considerably.

Fig. 1 A picture of the “Swedish method”, a one-man pneumatic rock drill [2].

Rock drilling is widely used in many industries from mining, oil and water well

drilling to civil engineering, the latest covers a large variety of different

applications. There is a number of rock drilling techniques suitable for different

needs. The most common methods are:

Percussive drilling of small to medium diameter holes using drill bits,

primarily used for blasting.

Rotary drilling of medium to large diameter holes using tricone bits,

primarily used for blasting.

Rotary drilling of small diameter holes using diamond inserts, primarily

used for core samples.

Raise Boring Machine (RBM) of large diameter holes using disc/bit

cutters, typically used for drilling shafts for installation or ventilation.

Full face Tunnel Boring Machines (TBM) using disc/bit cutters for

construction of tunnels.

3

Fig. 2 Atlas Copco Rocket Boomer WL4 C30.

Fig. 2 shows a drill rig Rocket Boomer WL4C30 with a percussive drill machine

on top of each boom. It is used for tunnel drifting underground. The principal of

percussive drilling is that a piston hits the drill rod at a velocity of about 10 m/s.

This creates a compressive stress wave with amplitude of about 200 MPa, which

propagates along the drill rod. The stress wave is then transferred into the rock

through the tool buttons. Each tool includes around ten button inserts made of

hard metal that facilitate the indentation process. Unlike the situation at quasi-

static indentation, the stress waves and rapid indentation make percussive drilling

a transient dynamic problem with high local strain rates in the rock. The rock is

then fragmented and removed by a flushing flow delivered to the drill bit via an

axial hole through the drill rod. After each blow of the piston, the drill steel is

rotated so that the inserts hit a new part of the rock next time. There are different

types of percussive drilling available depending on the applications and

conditions. The three most common are DTH (Down The Hole), COPROD and

Top Hammer Drilling, see Fig. 3.


4

Fig. 3 Percussive Rock Drilling Methods.

In order to have a good understanding of the rock-bit interaction and of the rock

fracture mechanism, the rock material constitutive behavior has to be carefully

investigated. This possibly leads to a clearer image of rock failure in front of the

tool and more effective drilling in the future. Basic defects in rocks are voids,

pores and microcracks as well as other related features [3]. Such microstructures

produce heterogeneity in the strength and stiffness of the material. It is well

known that rock fractures via initiation, growth and coalescence of microcracks,

together with sliding between individual grains and the surfaces of the

microcracks. Associated with these microscopic mechanisms, rock specimens

exhibit non-linear stress-strain behavior [3], [4]. Granite, which is one of the

hardest rocks to drill and also widely used as a testing material at the Atlas Copco

laboratory, was particularly chosen for study in this work. It has many pre-

existing cracks (e.g. [5] and [6]), but the porosity is very low (about 0.2%). Bohus

granite that was selected for experimental purpose mainly contains quartz 33 %,

plagioclase 33 %, Potassium feldspar 29 % and biotite 6 % (tested by SP, Swedish

National Testing and Research Institute). Most rocks show a transition from brittle

5

to ductile behavior when the confining pressure increases. However, silicate rocks

with low porosity are brittle at room temperature over the whole range of normal

laboratory confining pressure up to 0.5-1.0 GPa [7]. Some researchers have

reported that granite exhibits brittle fracture behavior even at confining pressures

up to 3-4 GPa [8].

In addition to the pre-existing cracks that are part of the so called intact rock, there

may be additional cracks called structural cracks hereafter. These cracks may

result from former impact of the drill bit, or by means of other exotic methods

such as microwave and laser that are used to facilitate the percussive drilling

process. The influence of these structural cracks on penetration stiffness (the slope

of the force-penetration curve of a drill bit) and the fracture pattern in the rock at

percussive drilling is investigated in this thesis. Some of the specimens used in the

experimental work were exposed to a coarser mechanical loading during the

cutting process in order to introduce such structural cracks to the material. CT

scan is performed on some of the specimens and a clear difference is captured

between the ones with structural cracks and the intact ones. For instance, the

tomography results on a specimen of size 20x20x100 mm3 that includes structural

cracks is shown for two cross sections in Fig. 4. The size of these cracks is

comparable to the size of the specimen. It should be noticed that these cracks are

mainly initiated from the surface due to the machining process.


6

Fig. 4 CT scan results on two cross sections of the specimen with structural cracks

(marked with red arrows).

Constitutive specification

Many studies have been performed during the past years to numerically simulate

rock drilling and the fragmentation process in brittle materials. Liu [9] developed

a rock and tool interaction code (R-T2D

) and studied the fragmentation process in

7

a quasi-static situation. Wang et al. [10] used an in-house numerical tool to

simulate the rock fragmentation process induced by indentation. Both Liu [9] and

Wang et al. [10] restrict their analyses to 2D plain strain conditions and do not

account for inelastic strains. Furthermore, Saksala [11]–[13] studied the impact

indentation of rocks using an isotropic damage concept for tensile loading and a

viscoplasticity consistency model for compression loading. This work is

performed under 2D plane strain conditions. More recently, the model was

extended to deal with 3D simulations [14]. Chiang et al. [15] modeled the impact

of the tool to the rock and rock fragmentation in the drilling process using a 3D

FE approach. A linear Mohr envelop together with a tension cut off plane is

employed as a criterion for the maximum strength of rock. The material behavior

is considered linear elastic before reaching final failure. Additionally, Thuro et al.

[16] used Particle Flow Code (PFC2D

) to investigate the crack pattern in drilling

and its correlation with existing foliations. PFC2D

code is based on discontinuum

mechanics approach and is built for 2D simulations.

A typical fracture system in rocks during quasi-static indentation is shown in Fig.

5. Different types of cracks are visible in this picture. Also in the region ahead of

the indenter, a crushed zone forms due to high compressive stresses. The way that

the side cracks, which are the most preferable type of cracks for removing

material, are formed during indentation has been a matter of different opinions.

Most researchers believe that these cracks are mainly created during the unloading

stage, see e.g. [17]–[19]. This idea is, however, disputed by others who believe

that these cracks are formed already during the loading stage [9], [20].


8

Fig. 5 Fracture system at rock indentation [21].

Based on classical contact mechanics and the stress states beneath a spherical

indenter, the change from radial tensile stress under purely elastic condition to

circumferential tensile stress under elastic-plastic condition is the main reason for

the change in the fracture system from ring crack (circular crack around the

indenter) with very brittle material to radial crack in semi-brittle material [22],

[23]. Porous materials behave differently and in that case crushing is the main

response to indentation [24].

One can run a simple quasi static FE analysis of elastic and elastic-plastic

indentation to look at the stress states during the loading-unloading stages. A rigid

spherical indenter of radius 5 mm is modeled using axisymmetric boundary

conditions. An elastic modulus of E = 200 GPa together with yield stress σy = 600

MPa and von Mises yield criterion is used and a simple isotropic hardening law is

employed for the post yielding behavior. Fig. 6 suggest a radial tensile stress

forms at pure elastic conditions leading to the ring cracks; while circumferential

tensile stress, present during elastic-plastic conditions, create the radial cracks.

Furthermore, the amount of axial stress, which is responsible for creation of the

side cracks, becomes considerable later in the elastic-plastic condition, during the

unloading stage (see Fig. 7). Although simplified and in absence of an accurate

constitutive model for rock material, these results are presented here to give an

idea about the indentation fracture system.

9

Loading

Elastic Elastic-Plastic

σradial

σcircumf.

σaxial

Fig. 6 FE results for indentation at elastic and elastic-plastic

conditions at the end of loading stage.


10

Unloading

Elastic Elastic-Plastic

σradial

σcircumf.

σaxial

Fig. 7 FE results for indentation at elastic and elastic-plastic

conditions at the end of unloading stage.

11

One of the aims of this thesis is to provide a constitutive model for rock material

that is also applicable at percussive drilling in a dynamic situation. It is well

known that rock behaves totally different in compression and tension loading [25].

A damage mechanism with stiffness reduction due to open cracks determines the

rock mechanical behavior in tension. In compression, rock has a plasticity-like

behavior that depends on the level of hydrostatic pressure. The combination of

these two gives a complete picture of the rock behavior when exposed to

mechanical loading such as drilling. The KST-DFH model [26]–[29] is composed

of a plasticity and damage description that is suitable to describe dynamic

fragmentation of brittle materials including rock. It is briefly discussed in the

section below.

KST model

The KST model was developed to simulate the compressive behavior of

geomaterials and provides a description of both the hydrostatic and the deviatoric

part of the behavior [26], [27]. In the deviatoric part, the radius of the yield

surface is a quadratic function of the mean pressure in the principal stress space.

20 1 2eq a a P a P (1)

where σeq is the equivalent (von Mises) stress, P the hydrostatic stress, a0, a1 and

a2 are material dependent coefficients. Moreover, it includes a piece-wise linear

equation of state linking the volumetric strain to the hydrostatic stress. At the first

stage of the hydrostatic loading, a typical brittle material behaves elastically. By

increasing the pressure, collapse of pores occurs which is modeled by an

irreversible volumetric strain. During the porosity breakage, the bulk modulus

decreases noticeably. When the pores are closed, the material exhibits a higher

bulk modulus which corresponds to the compacted material [30]. However, the

bulk modulus of granite is more or less constant in the whole hydrostatic stress

range, because the amount of porosity is very low [31].


12

The DFH fragmentation model

Single fragmentation

Under low-rate loading, the fracture process is generally the consequence of

initiation and growth of a single crack. When the stress increases, the weakest

defect is activated first. An unstable crack is then initiated and propagates very

quickly, leading to failure of the whole structure.

Defects with different sizes are randomly distributed within the material and

consequently the failure stress is random. A probabilistic approach may therfore

be employed to model this tensile behavior. Using a Poisson point-process

framework, the weakest link theory and Weibull model, the failure probability PF

is given by [32], [33]

1 exp[ ( )]tF FeffP Z (2)

where Zeff is the effective volume [34], and λt is the initiation density defined by

00

( )

m

Ft F

S

(3)

where m is the Weibull modulus, 00 /mS is the Weibull scale parameter, and σF

the maximum principal stress in the whole domain.

Multiple fragmentation

Under high strain-rate conditions, several cracks are initiated and propagate from

the initial defects leading to multiple fragmentation. The different initial defects

are assumed to be randomly distributed and activated at a random level of stress.

When such a crack is initiated, it propagates at a very high velocity (a portion of

the stress wave velocity) and relaxes the stresses in its vicinity. This prevents

activation of new defects in an obscured zone centered on this growing crack. At

the same time, the stress is increasing in the non-obscured zone and new critical

defects are activated leading to new crack openings [28], [29]. Therefore, dynamic

fragmentation corresponds to a competition between, on one hand, new critical

defects that progressively initiate cracks due to the increase of the stress level and,

13

on the other hand, obscuration of zones of potential critical defects by cracks

created before. The fragmentation process ends when the whole domain is

obscured by the opened cracks. The interaction between cracks already created

and critical defects of the material is given by the concept of probability of non-

obscuration [28]. If the initiation density is a continuous function, the probability

of non-obscuration of point M at time T in a domain Ω is described by [29]

( , ) [ ( , )]

( , )( , ) exp t

no

x t horizon of M T

x tP M T dVdt

t

(4)

where the horizon is defined by

( , ) , [ ( ) ( ( )) ]nohorizon of M T x t Z T t S kC T t (5)

where Zo is the obscured zone, k a constant parameter ( k = 0.38 may be assumed

when the crack length becomes significantly larger than the initial size), C the 1D

wave speed, T the current time and t the crack initiation time, S a shape parameter

(equal to 4π/3 when the obscuration volume is similar to a sphere in 3D) and n the

medium dimension (n = 3 in 3D).

The probability of obscuration Po = 1Pno ≡ PF corresponds to the failure

probability, expressed by Weibull and can be used as a damage variable in the

framework of Continuum Damage Mechanics. One damage variable is defined for

each principal direction.

Experimental work

In this thesis, experiments were performed on Bohus granite to obtain its

mechanical characteristics. KST-DFH model parameters are calibrated for granite

based on experiments and it is shown that this model is suitable for the granite

behavior at percussive drilling. A brief summary of the work is presented here.

Direct compression and tension test

Direct compression and tension tests are performed on the granite material in

order to investigate its mechanical properties pertinent to the DFH-KST model. In


14

the tensile test, the experimental device is composed of two socket joints to

provide a uniform stress field while the specimen is loaded by means of two

platens in the compression case. Strain gauges and LVDT sensors were used to

compare the nominal and local strains.

Fig. 8 Experimental setup for direct compression and tension tests.

Quasi-oedometric compression test

During a quasi-oedometric compression test, a cylindrical specimen tightly

enclosed in a confinement cell, is compressed axially. Both axial and radial

stresses increase during loading as the material expands in the lateral direction.

This gives an indication of the strength of the material at different levels of the

hydrostatic pressure. Both the deviatoric and volumetric behavior of the material

can be obtained from this test [35], [36].

Fig. 9 Experimental setup for quasi-oedometric compression test.

15

Flexural test

The tensile failure of brittle materials depends upon the microstructure in terms of

flaw density and failure stress distribution. To evaluate the quasi-static strength of

the rock and its distribution, 3PB tests are carried out and Weibull statistics and

size effect [32]–[34] is used to describe the strength distribution.

Fig. 10 Experimental setup for flexural test with digital image acquisition.

Edge-on impact test

In order to validate the numerical model and also to investigate the fracture

pattern after impact, Edge-on Impact (EOI) tests [37], [38] with a special

sarcophagus configuration [28] are performed. An Aluminum projectile is

impacted on the rock material at high velocity. In the experimental set-up, two

cylindrical steel confinements are used close to the impact point on both faces of

the rock specimen. This helps to increase the level of confining pressure and

accordingly the strength of the rock material in order to reduce compressive

damage immediately beneath the projectile.

Fig. 11 Experimental setup for Edge-on impact test.


16

Spalling test

The strain rate dependency of the mechanical response in brittle materials has

been widely investigated in the literature. A considerable rate dependency is

reported especially in the case of the tensile strength [39]–[43]. Spalling tests are

performed to investigate the dynamic tensile behavior of Bohus granite. Spalling

test with Hopkinson bar is a suitable technique to measure the tensile strength of

brittle materials at strain rates between 101 and 10

2 1/s [40], [42], [44]. The main

idea in this test is that the impact of the projectile induces a compressive wave

that propagates through the bar and is mainly transferred to the specimen. This

wave is reflected as a tensile wave from the free surface of the specimen that leads

to damage in the material. The results from this test are used to validate the DFH

model output for tensile strength of the material at high loading rates [45].

Fig. 12 Experimental setup for spalling test.

Numerical modeling

The numerical simulation is carried through using the KST-DFH material model

implemented as a VUMAT subroutine in Abaqus explicit as a commercial FE

modeling software. This numerical tool is used for simulation of dynamic

experiments on granite, such as the EOI and spalling tests that are performed in

this work. Pre-existing cracks are introduced in the model by considering sets of

elements with negligible tensile strength. They lead to immediate failure when

loaded in tension but can still carry compressive loads as crack closure occurs due

to compressive stresses. The numerical tool is also used to simulate percussive

17

drilling. To simplify that problem, only one hemispherical button from the bit and

only the first impact is considered [45]–[47].

In an attempt to compare the numerical results to real drilling results, the results

from a special drilling experiment with only one spherical button is used and the

incident and reflected waves in the drill rod are measured. This experiment is

performed by means of the drop hammer method and a drill string made of high

strength steel of about 3.7 m in length and 23.2 mm in diameter and elastic

modulus of E = 208 GPa is instrumented using rings of strain gauges (each ring

includes 4 gauges to capture also the bending effect) at five different positions

along the drill string. Two at a distance of 0.32 m and 0.35 m to the end of the

drill string on the piston side, and three at 0.265 m, 0.715 m and 0.745 m from the

bit-rock contact point, see Fig. 13. The piston has a length of 0.3 m and is

impacted to a shank adapter of length 0.43 mm that is attached to the drill string.

Fig. 13 Drilling experiment setup.


18

The wave data from the strain gauges close to the piston is used as an input

incident wave to the simulation and the data from the gauges closer to the rock,

which includes an interaction of incident and reflected waves, is compared to the

numerical ones. The gauges data at each ring is averaged to eliminate the bending

effect. It should be mentioned that the ring 5 gauges were not used during the

experiment. A quarter-symmetry 3D FE model of the drilling experiment is

performed and about one million 8-node linear elements with reduced integration

are used in this simulation (the mesh is shown in Fig. 14).

Fig. 14 FE mesh used in the simulation of percussive drilling experiment.

A good agreement is found between the strain data from the gauges and the

numerical results that could be seen as a validation step for the numerical

modeling of percussive drilling, see Fig. 15. The fracture pattern from the

simulation is shown in Fig. 16 and the crush zone and different types of cracks are

visible in the numerical results. This fracture system is very similar to the quasi-

static indentation results presented in Fig. 5 (see e.g. [48]). However, the impacted

rock in the drilling experiment (shown in Fig. 16) should be carefully cut and

investigated by means of CT scan or other methods to obtain the fracture pattern,

but this was not done here. Furthermore, the force-indentation data was extracted

from the simulation results, see Fig. 17.

19

Fig. 15 Strain gauges data and the FE simulation results at different positions on

the drill string.

Fig. 16 The FE simulation result for the fracture pattern and the rock surface after

the impact.


20

Fig. 17 Force-penetration data obtained from the FE simulation.

Concluding remarks and suggestions for future work

In this thesis, both experimental and theoretical investigations are performed to

build the knowledge about the fracture response of brittle material such as rock

and concrete to dynamic loading and more specifically during the drilling process.

The preliminary application of interest in this study was percussive drilling with

stress waves and rapid indentation that makes the problem a transient dynamic

problem. Bohus granite was chosen for experimental and modeling purposes

because it was already used extensively at Atlas Copco laboratories as well as

being a popular and hard material in the mining industry.

Similar to other brittle materials, granite has a different response to tensile and

compressive loads, which both exist underneath the indenter during the drilling

process. Therefore the constitutive model should include both parts to correctly

explain the material behavior in the whole range of loading. A damage

mechanism with stiffness reduction due to open cracks determines the granite

rock mechanical behavior in tension. Dynamic effects with strain rate dependency

play an important role in tension that has to be considered. In compression, rock

has a plasticity-like behavior that depends on the level of hydrostatic pressure.

The combination of these two sets of data gives a complete mechanical

characterization of rocks subjected to different loading situations, including

percussive drilling. The KST-DFH model combining a damage law in tension and

plasticity in compression is appropriate for this purpose. It is emphasized

21

structural cracks have a considerable effect on the damage mechanism in a

mechanical analysis related to granite, for example in a drilling situation. These

cracks may be the result from former impact of the drill bit, or by means of other

exotic methods such as microwave and laser heating, used to facilitate the

percussive drilling process. It is shown that pre-existing cracks help the drilling

process regardless of their orientation.

A set of quasi-static and dynamic experiments are performed on granite to

investigate the mechanical properties and to obtain the material parameters. For

instance, spalling tests are performed on granite to investigate the strain rate

dependency of the tensile strength. A considerable strain rate dependency of the

tensile strength is obtained at strain rates of about 102 1/s, this loading-rate being

pertinent to the situation of rock materials at percussive drilling. A dynamic

tensile strength of 18.9 MPa is obtained at a strain rate of 70 1/s, which is more

than twice the tensile strength of the specimen (with the same size) at quasi-static

conditions, 8 MPa. The KST-DFH has been able to model successfully fracture of

lime-stone, high strength concrete and granite. It would be a logical extension to

the present work to also investigate other types of rocks such as metamorphic

gneiss, sedimentary chert and igneous basalt and gabbro. Furthermore, ores of

different types that naturally are of great importance to the mining industry would

be considered for further work.

The numerical simulation is carried through with the KST-DFH material model

implemented as a VUMAT subroutine in the Abaqus explicit software. This

numerical tool is employed to simulate the percussive drilling problem. To

simplify the problem, only one hemispherical button from the bit is considered so

far. Another extension of the present work would be to investigate how rock

fragmentation is affected by neighboring button inserts, i.e. among others how

cracks interact on a larger scale.

It is emphasized here that the material model used in this study describes rock

fragmentation response to dynamic loading at high strain rates. It is a general

model that can be used in different dynamic applications with high strain rates. So

far, percussive drilling was mainly considred. However, it is believed that the

approach taken in this study is also applicable for the case of rock mechanical


22

excavation using cutters as there are dynamic loading and high strain rates in the

rock underneath the cutters.

23

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27

Summary of appended papers

Paper A: On the mechanical behavior of granite material with particular

emphasis on the influence from pre-existing cracks and defects.

In this paper, experiments are carried out in order to determine the mechanical

properties of Bohus granite. In particular, the influence from pre-existing cracks

and defects is examined in great detail. Direct tensile and compression tests are

performed to evaluate stiffness and strength. Quasi-oedometric tests are carried

out in order to obtain the deviatoric and volumetric behavior of the material at

different levels of hydrostatic pressure. Three point bend (3PB) tests are

performed to evaluate the quasi-static strength of the rock and its distribution.

Weibull statistics is then employed to describe the strength distribution. The intact

specimens indicate a rather low scatter in the mechanical properties such as elastic

modulus and tensile strength. However, specimens with initial large defects

behave differently. The failure mechanism in these specimens is not as brittle as

the intact ones. The crack is opened on the tensile surface of such specimens

during the 3PB test at an early stage during loading, as demonstrated by digital

image correlation results.

Paper B: Granite rock fragmentation at percussive drilling - experimental and

numerical investigation.

In this paper, the fracture system at percussive drilling is numerically modeled.

Due to the complex behavior of rock materials, a continuum approach is

employed relying upon a plasticity model with yield surface locus as a quadratic

function of the mean pressure in the principal stress space coupled with an

anisotropic damage model. In particular, Bohus granite rock is investigated and

the material parameters are defined based on previous experiments. The equation

of motion is discretized using a FE approach and the explicit time integration

method is employed. EOI (Edge-On Impact) tests are performed and the results

are used to validate the numerical model. The percussive drilling problem is then

modeled in 3D and the bit-rock interaction is considered using contact mechanics.


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Paper C: A numerical study of the influence from pre-existing cracks on granite

rock fragmentation at percussive drilling.

In this paper, the effect of pre-existing, or structural, cracks on dynamic

fragmentation of granite is investigated. These cracks may result from former

impact of the drill bit, or by means of other exotic methods such as microwave

and laser that are used to facilitate the percussive drilling process. The pre-

existing cracks are introduced in the model by considering sets of elements with

negligible tensile strength that leads to their immediate failure when loaded in

tension even though they still carry compressive loads as crack closure occurs due

to compressive stresses. Previously performed Edge-On Impact (EOI) tests are

reconsidered here to validate the numerical model. Percussive drilling is

simulated, and the influence of the presence of pre-existing cracks is studied. The

results from the analysis with different crack lengths and orientations are

compared in terms of penetration stiffness and fracture pattern. It is shown that

pre-existing cracks in all investigated cases facilitate the drilling process.

Paper D: On the tensile strength of granite at high strain-rates considering the

influence from pre-existing cracks.

In this paper, the dynamic tensile behavior of granite samples, in some of which

pre-existing cracks are introduced artificially, is investigated. Spalling tests using

a Hopkinson bar are performed and a strain rate of order 102 1/s is obtained in

both sets of specimens (with and without initial cracks). This experimental

technique is employed because it is the same order of strain rate present in rock

materials during percussive drilling. A dynamic tensile strength of 18.9 MPa is

obtained at a strain rate of 70 1/s. This is more than twice the tensile strength of

the specimen at quasi-static conditions, which is 8 MPa. The results from the

spalling tests are used to validate the model prediction of the dynamic tensile

strength of granite and also to calibrate the cohesive model parameters. On the

other hand damaged elements are numerically introduced in the finite element

(FE) calculations to simulate the spalling experiments performed on pre-damaged

samples. A good agreement is obtained between the experimental and numerical

results showing that a two-scale approach may constitute a suitable method to

simulate numerically the tensile response of pre-damaged granite.

On the mechanical behavior of granite: Constitutive...

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