ROCK STRENGTH DEPENDENCE ON SCHISTOSITY · 2015-07-28 · the PFC3D model were dependent on the...

P O S I VA O Y

T ö ö l ö n k a t u 4 , F I N - 0 0 1 0 0 H E L S I N K I , F I N L A N D

P h o n e ( 0 9 ) 2 2 8 0 3 0 ( n a t . ) , ( + 3 5 8 - 9 - ) 2 2 8 0 3 0 ( i n t . )

F a x ( 0 9 ) 2 2 8 0 3 7 1 9 ( n a t . ) , ( + 3 5 8 - 9 - ) 2 2 8 0 3 7 1 9 ( i n t . )

POSIVA 2002 -05

June 2002

Rock strength and deformationdependence on schistosity

To ivo Wanne

Simulation of rock with PFC3D

POSIVA 2002 -05

P O S I VA O Y

T ö ö l ö n k a t u 4 , F I N - 0 0 1 0 0 H E L S I N K I , F I N L A N D

P h o n e ( 0 9 ) 2 2 8 0 3 0 ( n a t . ) , ( + 3 5 8 - 9 - ) 2 2 8 0 3 0 ( i n t . )

Fa x ( 0 9 ) 2 2 8 0 3 7 1 9 ( n a t . ) , ( + 3 5 8 - 9 - ) 2 2 8 0 3 7 1 9 ( i n t . )

June 2002

Rock strength and deformationdependence on schistosity

To ivo Wanne

Saan io & R iekk o la Oy


T h e c o n c l u s i o n s a n d v i e w p o i n t s p r e s e n t e d i n t h e r e p o r t a r e

t h o s e o f a u t h o r ( s ) a n d d o n o t n e c e s s a r i l y c o i n c i d e

w i t h t h o s e o f Po s i v a .

ISBN 951 -652 -112 -6ISSN 1239 -3096

Tekijä(t) � Author(s)

Toivo Wanne Saanio & Riekkola Oy

Toimeksiantaja(t) � Commissioned by

Posiva Oy

Nimeke � Title

ROCK STRENGTH AND DEFORMATION DEPENDENCE ON SCHISTOSITY Simulation of rock with PFC3D Tiivistelmä � Abstract

The objective of the work was to study the effect of anisotropy of the rock on strength and deformation properties by simulating a standard unconfined compression test with the PFC3D-program. Particle Flow Code (PFC) was selected to be used in the simulations because of its ability to model behavior of brittle rock material including fracture propagation. The schistosity was modeled in PFC3D intrinsically by generating an anisotropic particle structure consisting of matrix particles and oriented band particles. The approach was novel and no similar studies were found for references. The model was generated and the results were compared to those of gneissic tonalite, which is a main rock type in the Research tunnel at Olkiluoto, Finland. The PFC3D simulated strength and deformation properties were found to be noticeably dependent on schistosity. The comparison to the laboratory results showed that the responses were similar. Damage formation observations made during the compression simulations indicated that the PFC3D modeling could simulate the events happening during the laboratory compression tests of rock samples by reproducing similar fracture generation and deformation. Furthermore it was noticed that the mechanical properties of the PFC3D model were dependent on the particle size and on the geometry of banding, but adding a third particle type, in addition to the previous two particle types, did not radically alter the behavior of the compression simulation. A few topics have to be studied in more detail in order to improve the simulation process and the accuracy of a material model. These include the development of the PFC3D visualization tools of the fracturing process and the optimization of the particle size and simulation times.

Avainsanat - Keywords

PFC3D simulation, anisotropy, compression test, damage formation ISBN

ISBN 951-652-112-6 ISSN

ISSN 1239-3096 Sivumäärä � Number of pages

84 Kieli � Language

English

Posiva-raportti � Posiva Report Posiva Oy Töölönkatu 4, FIN-00100 HELSINKI, FINLAND Puh. (09) 2280 30 � Int. Tel. +358 9 2280 30

Raportin tunnus � Report code

POSIVA 2002-05

Julkaisuaika � Date

June 2002

Tekijä(t) � Author(s)

Toivo Wanne Saanio & Riekkola Oy

Toimeksiantaja(t) � Commissioned by

Posiva Oy

Nimeke � Title

KIVEN SUUNTAUTUNEISUUDEN VAIKUTUS LUJUUTEEN JA MUODONMUUTOKSIINKiven mallinnus PFC3D:llä

Tiivistelmä � Abstract

Työn tavoitteena oli tutkia kiven suuntautuneisuuden vaikutusta kiven lujuuteen ja muodon-muutosominaisuuksiin simuloimalla yksiaksiaalista puristuskoetta PFC3D-ohjelmalla. Tärkeä osa sortumisen simuloinnissa on valitun mallinnusmenetelmän kyky esittää murtuman etenemistä. Particle Flow Code (PFC) ohjelmisto valittiin käytettäväksi simuloinneissa, koska menetelmä pystyy mallintamaan hauraan kivimateriaalin käyttäytymistä. Kiven suuntautuneisuutta mallinnettiin PFC3D:llä rakentamalla malli peruspartikkelimassasta ja suuntautuneista partikkeliryhmistä. Lähestymistapa mallinnukseen oli uusi eikä vastaavia tutkimuksia löydetty viitteeksi. Puristuskokeen simulointituloksia verrattiin Olkiluodossa sijaitsevasta tutkimustunnelista otettu-jen gneissimäisen tonaliitin laboratoriokokeiden tuloksiin. Sekä simulointi- että laboratoriokokeet osoittivat kiven lujuus- ja muodonmuutosominaisuuksien olevan voimakkaasti riippuvaisia suuntauksesta. Tehdyssä vertailussa todettiin laboratoriokokeiden ja simulointien tulosten olevan samankaltaisia. Havainnot murtumisen kehityksestä simulointien aikana osoittavat, että PFC3D mallinnus kykenee simuloimaan puristuskokeen aikaiset murtumistapahtumat aikaansaamalla samantapaisia rakojärjestelmiä ja muodonmuutoksia PFC3D mallissa kuin laboratoriokokeiden näytteissä. Herkkyystarkastelut osoittivat, että PFC3D mallin ominaisuudet ovat riippuvaisia mallin partikkelikoosta ja partikkeliryhmien geometriasta. Edelleen havaittiin, että mallin käyttäytyminen ei muuttunut merkittävästi lisättäessä malliin kolmas partikkelityyppi aikaisempien kahden lisäksi. Simulointimenetelmän optimointia ja materiaalimallia tulee kehittää, jotta menetelmällä pystyttäi-siin mallintamaan tehokkaasti myös normaalien kalliotilojen kokoluokkaa olevien tilojen muodon-muutoksia. Yksi ratkaisu voisi olla FLAC3D:n ja PFC3D:n yhdistäminen suurissa mallinnuksissa. Tällä hetkellä suurehko ongelma on kolmiulotteisen murtumistapahtuman visualisointi. Mallin partikkelikoon sekä kuormitusnopeuden vaikutusta simulointituloksiin tulisi myös tutkia tarkem-min. Avainsanat - Keywords

PFC3D simulaatio, anisotropia, puristuskoe, murtuman syntyminen ISBN

ISBN 951-652-112-6 ISSN

ISSN 1239-3096 Sivumäärä � Number of pages

84 Kieli � Language

Englanti

Posiva-raportti � Posiva Report

Posiva Oy Töölönkatu 4, FIN-00100 HELSINKI, FINLAND Puh. (09) 2280 30 � Int. Tel. +358 9 2280 30

Raportin tunnus � Report code

POSIVA 2002-05

Julkaisuaika � Date

Kesäkuu 2002

PREFACE

The work has been carried out by Toivo Wanne of Consulting Engineers Saanio & Riekkola during the year 2001-02 in collaboration with POSIVA OY and Svensk Kärnbränslehantering AB (SKB).

The author wishes to thank the instructor Jorma Autio of Saanio & Riekkola and Jukka-Pekka Salo of POSIVA OY who acted as contact person.

Special acknowledgement to David Potyondy of Itasca Consulting Group for his valuable contribution concerning the Particle Flow Code modeling method.

Table of contentsNext pagePrevious page

1

TABLE OF CONTENTS

ABSTRACT

TIIVISTELMÄ

PREFACE

TABLE OF CONTENTS 1

LIST OF SYMBOLS AND NOTATIONS 3

LIST OF FIGURES 4

LIST OF TABLES 6

1 BACKGROUND AND INTRODUCTION 7

1.1 General 71.2 Numerical modeling 8

1.2.1 Modeling in general 81.2.2 Numerical modeling in rock mechanics 9

1.3 Principles of probability 10

2 DESCRIPTION OF UNCONFINED COMPRESSION TEST 13

2.1 Compression test 132.2 Anisotropy 142.3 Failure development during compression test 172.4 Gneissic tonalite 19

2.4.1 Igneous and metamorphic rock 192.4.2 Properties 202.4.3 Laboratory test results 20

3 PFC THEORY 23

3.1 Particle mechanics 233.2 Distinct Element Method 233.3 Calculation cycle 243.4 Contact models 24

4 PFC3D MODEL FOR ROCK 25

4.1 Modeling with PFC3D 254.2 Model generation and compression test simulation 25

4.2.1 Isotropic model generation in PFC3D 264.2.2 Particle band generation in PFC3D 264.2.3 Compression test simulation in PFC3D 29


2

5 RESULTS OF COMPRESSION TEST SIMULATIONS 33

5.1 Quantitative results 335.2 Qualitative study 38

5.2.1 Damage formation in PFC3D 385.2.2 Failure patterns 38

6 ANALYSIS OF RESULTS 51

6.1 Crack formation 516.2 Comparison against laboratory samples 536.3 Comparison to 2D modeling results 55

7 SENSITIVITY ANALYSIS 57

7.1 Resolution of the PFC model 577.1.1 Effect of model resolution in 2D modeling 577.1.2 Effect of model resolution in 3D modeling 58

7.2 Geometry parameters 617.3 Three component model 64

8 CONCLUSION AND DISCUSSION 65

REFERENCES 67

APPENDIX 1: PFC MODEL FOR ROCK - PARAMETERS 69

Specimen genesis and material parameters 69Anisotropy installation parameters 70Unconfined compression test parameters 70Monitored parameters during testing 71

APPENDIX 2: CRACK PLOTS AND STRESS-STRAIN CURVES 72


3

LIST OF SYMBOLS AND NOTATIONS

� Angle of schistosity with respect to the vertical axis [o]�ci Crack-initiation stress [MPa] �f Unconfined / uniaxial compressive strength [MPa] �i Strain component, i=x,y,z

�i Stress component, i=x,y,z [MPa]�w Friction angle of weakness plane [o]E Young’s modulus [Pa] v Poisson’s ratio

CDF Cumulative distribution function DEM Distinct-element method FISH Built-in programming language of PFC (and other Itasca codes) OL Olkiluoto Pc Confining stress (in triaxial test) [Pa] PDF Probability density function PFC2D/3D Particle Flow Code in 2 and 3 Dimensions Strain Change in length per unit length, dimensionless Stress Constraining force applied to a material, force per unit area [Pa] UCS Unconfined / uniaxial compressive strength, peak strength [Pa]


4

LIST OF FIGURES

Figure 2-1. Schematic view of a compressional test arrangement. Uniaxial test: compression plates are placed on the bottom and top surfaces of the sample, the sides are unconstrained. In the Triaxial test, liquid is used to subject the sample to a confining pressure. (Department of Geological Sciences 2001.) 13

Figure 2-2. Left: definition of schistosity angle. Right: variation of peak strength with the angle of weakness plane (schistosity). 16

Figure 2-3. Slates, shales and sandstones exhibit strength anisotropy. Uniaxial and triaxial compression test results after Hoek & Brown (1982). 16

Figure 2-4. Failure modes for different types of rock sample stress-strain behavior. 1 – tensile splitting, 2 – shear splitting, 3 – ‘barrel’-shape failure. (Geology for Engineers 2001.) 18

Figure 2-5. Average unconfined compressive strengths (with standard deviation) of gneissic tonalite specimens with respect to schistosity angle. Three different sample sizes were tested. (Autio et al. 2000.) 21

Figure 2-6. Average Young’s modulus (with standard deviation) of gneissic tonalite specimens with respect to schistosity angle. Three different sample sizes were tested. (Autio et al. 2000.) 22

Figure 3-1. PFC calculation cycle. The velocities and accelerations are kept constant within each time step. 24

Figure 4-1. Model generation steps in PFC3D. 1-isotropic parallelpiped model, 2-anisotropic parallelpiped model, 3-anisotropic cylindrical model. 28

Figure 4-2. Band generation produces a group of five different samples for all nine schistosity angles. 28Figure 4-3. Particle band generation produces an anisotropic model. Definitions of the geometry

parameters are shown. For illustrative purposes, the schemes on the left and in the center represent schistosity oriented parallel to the longitudinal axis of the model. 29

Figure 4-4. The pre-failure region (pre-peak stress) of the stress-strain curve obtained in an unconfined compression test and definitions of Young’s modulus and peak strength. 31

Figure 5-1. Example of a stress–strain curve for a simulated unconfined compression test using PFC3D. Schistosity angle used in the model 45o, Peak strength 97 MPa, Young’s modulus 44 GPa. The unit of the vertical stress axis is Pascal [Pa]. 34

Figure 5-2. All results for simulation of Peak strength [MPa] (upper) and Young’s modulus [GPa] (lower) are plotted with respect to schistosity angle [o]. 35

Figure 5-3. Mean and standard deviations of simulation results plotted with respect to schistosity angle [o]. Peak strength [MPa] – upper, Young’s modulus [GPa] – lower. 35

Figure 5-4. Peak strength [MPa] of the laboratory tests and the PFC3D runs. The yellow transparent area encloses the laboratory results. Note that the vertical axis spans from 60 - 180 MPa. 36

Figure 5-5. Young’s modulus [GPa] of the laboratory tests and PFC3D runs. The yellow area encloses the laboratory results. Note that the vertical axis spans from 30 - 90 GPa. 36

Figure 5-6. Simulated mean values for crack initiation stress [MPa] (above) and Poisson’s Ratio (below) and their standard deviation with respect to the angle of schistosity [o]. 37

Figure 5-7. Left: Presentation of tension-induced microcrack as a octagon. Right: Vertical and horizontal sections. 39

Figure 5-8. Schematic illustration of the four phases of damage formation. 39Figure 5-9. Damage formation at schistosity angle of 11o. 40Figure 5-10. Damage formation at schistosity angle of 34o. 40Figure 5-11. Damage formation at schistosity angle of 67.5o. 41Figure 5-12. Damage formation at schistosity angle of 79o. 41


5

Figure 5-13. Model: sTx_mG_11c, schistosity angle 11o. Left: All cracks, time-dependent coloring (light = early, dark = late). Center: Cracks and displacements on the origo centered vertical section, time-dependent coloring (light = early, dark = late). Right: Cracks and modeled particles on the origo-centered vertical section, crack type coloring (red = tension, black = shear). Particle coloring: yellow = stronger matrix material, orange = weaker band material. 43

Figure 5-14. Model: sTx_mG_225d, schistosity angle 22.5o. Left: All cracks, time-dependent coloring (light = early, dark = late). Center: Cracks and displacements on the origo centered vertical section, time-dependent coloring (light = early, dark = late). Right: Cracks and modeled particles on the origo-centered vertical section, crack type coloring (red = tension, black = shear). Particle coloring: yellow = stronger matrix material, orange = weaker band material. 44

Figure 5-15. Model: sTx_mG_34d, schistosity angle 34o. Left: All cracks, time-dependent coloring (light = early, dark = late). Center: Cracks and displacements on the origo centered vertical section, time-dependent coloring (light = early, dark = late). Right: Cracks and modeled particles on the origo-centered vertical section, crack type coloring (red = tension, black = shear). Particle coloring: yellow = stronger matrix material, orange = weaker band material. 45

Figure 5-16. Model: sTx_mG_45a, schistosity angle 45o. Left: All cracks, time-dependent coloring (light = early, dark = late). Center: Cracks and displacements on the origo centered vertical section, time-dependent coloring (light = early, dark = late). Right: Cracks and modeled particles on the origo-centered vertical section, crack type coloring (red = tension, black = shear). Particle coloring: yellow = stronger matrix material, orange = weaker band material. 46


Figure 5-18. Model: sTx_mG_675e, schistosity angle 67.5o. Left: All cracks, time-dependent coloring (light = early, dark = late). Center: Cracks and displacements on the origo centered vertical section, time-dependent coloring (light = early, dark = late). Right: Cracks and modeled particles on the origo-centered vertical section, crack type coloring (red = tension, black = shear). Particle coloring: yellow = stronger matrix material, orange = weaker band material. 48


Figure 6-1. Horizontal cutting planes in two separate models. Left: schistosity angle 34o. Right: schistosity angle 45o. Red arrows show the direction of microcrack propagation. Sections shown are from the middle of the model. 52

Figure 6-2. Axial stress (uppermost black line), all cracks (black), tensile cracks (red) and shear cracks (blue) plotted against axial strain. The cracking increases after the peak strength and tensile cracking is predominant. Model sTx_mG34d, schistosity angle 34o, peak strength 99 MPa, Young’s modulus 55 GPa, number of cracks at peak strength 820pieces. 53

Figure 6-3. Laboratory samples (Autio et al. 2000) (upper) and PFC3D failure-pattern images (lower) from three different loading phases. Microfracturing of laboratory samples is superimposed in red color. PFC3D crack coloring; red = tension-induced, blue = shear-induced. Only cracks in the origo centered section are shown. 54


6

Figure 7-1. Unconfined strength (UCS) and values for Young’s modulus show a clear dependence on resolution. The relationship resembles a logarithmic dependence. 59

Figure 7-2. Damage patterns resulting from the use of different model resolutions. The numbers depict the average particle radius in the corresponding model. Crack coloring indicates the relative time when microcracks were formed [light = early, dark = late]. The models are isotropic. 60

Figure 7-3. Half of the models showing the three different geometries of the cases and the basic case (above). Cracks on the origo centered section are shown below. Yellow = matrix, red = weaker bands. In these models, the angle of schistosity is 22.5o. 62

Figure 7-4. Simulated peak strengths (upper diagram) and Young’s modulus (lower diagram) of the three cases with different geometries and results from the basic case (Normal). The fitted spline lines which connect the observations are for clarification purposes only. 63

Figure 7-5. Left: Material structure in the three-particle model. Particles on the origo centered section are shown. Yellow = old matrix, red = bands, white = new matrix. Right: Mean simulated peak strength and Young’s modulus with standard deviations. 64

LIST OF TABLES

Table 2-1. Average strengths and Young’s modulus with respect to schistosity angle. Results from three different diameter samples (41, 54, 99 mm). (Autio et al. 2000.) 21

Table 4-1. Anisotropy installation parameters and their values. 27Table 4-2. Unconfined compression test parameters for PFC3D. 30Table 4-3. List of simulated compression tests with the corresponding chapter number. 31Table 5-1. Mean values and standard deviations used for simulation of unconfined compression test. 34Table 7-1. Results of the resolution test run. The dimensions of the specimens were 42 by 42 by 120

mm. A total of 17 runs were made; five for the reference model and four for each of the other specimens. Mean values are shown (E = Young’s modulus [GPa], Sig_f = Unconfined compression strength [MPa]). 59

Table 7-2. The relationship between particle size, resolution and computing time. The dimensions of the specimens are 54 x 54 x 142 mm. (The values 51 h and 642 h are extrapolated assuming that the time-resolution relation is exponential [t=AResolution*B].) 60

Table 7-3. Geometric parameters for the three cases studied. 61


7

1 BACKGROUND AND INTRODUCTION

1.1 General

The objective of this work is to study the effect of anisotropy of the rock on strength and deformability by simulating a standard unconfined compression test with the PFC3D-program. A model is generated and the results obtained are compared to those obtained with gneissic tonalite, a main rock type in the Research tunnel at Olkiluoto, Finland (Autio et al. 1999).

The Finnish Parliament ratified in May 2001 the Government’s positive Decision in Principle on Posiva’s application to locate the repository for spent nuclear fuel at Olkiluoto. Olkiluoto is being investigated as a possible site for the final disposal of spent nuclear fuel from the Finnish nuclear power plants. The main rock types found in the area are mica gneiss and gneissic tonalite.

Assessing the stability of deep underground excavations is a subject of great importance in terms of both safety and constructability. Typical bedrock in many areas of Finland is gneissic and therefore anisotropic in nature. The anisotropy of mechanical properties such as strength and the effect of this anisotropy on the behavior and failure of rock around deep underground openings have not been studied in detail and the amount of data available is limited. The results of measurements on gneissic tonalite samples taken from the Research Tunnel at Olkiluoto showed that strength and deformation properties were significantly dependent on the orientation of schistosity.

Rock damage is produced by fracturing and this may eventually lead to instability. In crystalline rock, a significant component of progressive failure in deep underground openings is fracture propagation, and field observations have shown that in many cases this eventually reaches a stable state. For this reason, a key element in the simulation of a failure process is the ability to model fracture propagation.

While the development of computer software has made it possible to model fracture propagation in three dimensions, the number of methods available for the modeling of fracture generation and propagation is currently very limited. Particle Flow Code (PFC) was selected for use in this study because it is a well-documented and commercially available tool for modeling the behavior of brittle rock material including fracture propagation and mechanical stability in the near-field around underground openings. The PFC method has previously been used to model the behavior of gneissic tonalite in 2D (Potyondy & Autio 2001, Potyondy & Cundall 2000) and in several other studies (Li & Holt 2001, Kulatilake et al. 2001, Potyondy & Cundall 2001) with encouraging results.

Gneissic tonalite was chosen as the reference rock in this study because its anisotropy has been studied in detail and the behavior of laboratory samples has been simulated using PFC2D software.

The following chapters discuss topics of importance which concern modeling in general and the application of modeling in the field of rock mechanics. A short introduction to probability theory and its use in the study is also provided.


8

1.2 Numerical modeling

In a field such as geomechanics where data are not always available, numerical models can be useful in providing a picture of the mechanisms that may occur in particular physical systems. The following sections discuss some issues concerning modeling in general and its application in the field of rock mechanics.

1.2.1 Modeling in general

In an excellent paper on the modeling of physical systems, Oreskes et al. (1994) make the observations presented in the following paragraphs. Modeling, or the use of a numerical model, contains usually closed mathematical components such as an algorithm within a computer program. These mathematical components may be subjected to verification because they are part of closed systems that include claims which are always true. However, the models that use these components are never closed systems, because they require input parameters that are incompletely known. The lack of complete knowledge of the system being modeled forces us to make inferences and assumptions about the real world. Many assumptions can be justified based on experience, but in a new study, the degree to which our assumptions hold can never be established a priori. The additional assumptions and input parameters required to make a model work are known as ‘auxiliary hypotheses’. If verification of the problem fails, there is often no simple way to know whether the principal or auxiliary hypothesis is at fault. If we compare the response of a model with observational data and the comparison proves unacceptable, we know that something is wrong and we may or may not be able to define what it is. For this reason, the evaluation of a model should always be a step-by-step procedure. Typically, we continue to work on the model until it is an acceptable fit. When pursuing a good match, it should be remembered that more than one model construction can actually produce the same output. Another question for consideration is deciding on the point at which further modifications to the model are no longer considered acceptable.

When constructing a model we like to validate it to establish that the model or code does indeed reflect the behavior of the real world. In other words, that the model is a good representation of the actual processes that take place in a real system. The most common method of validation is a comparison of measurements from laboratory testing with the results obtained from computational models. However, an agreement between measurements and the output of numerical modeling in no way demonstrates that the model which produced the output is an exact representation of the real system. Validation is a process of building confidence in models, not provide validated models.

Numerical models are calibrated by manipulating the independent variables to obtain a match between the observed and simulated distributions of the dependent variables. As the goal of scientific theories is not truth but empirical adequacy, it could be said that a calibrated model is empirically adequate. On the other hand, as calibrated models often require refinement, this suggests that the adequacy of the models is forced. The availability of more data usually requires further adjustments. This requirement has a serious affect on the use of any calibrated model for predictive purposes such as estimating the long-term stability of an underground opening.


9

Models can confirm a hypothesis by offering evidence which strengthens what may have already been partly established by laboratory tests. Models can therefore be considered to be representations, useful for guiding further study. The philosopher Nancy Cartwright claimed that the models are like fiction. A model, like a novel, may resonate with nature but it is not a ‘real’ thing. The fundamental reason for carrying out the modeling process is a lack of full access to the phenomena of interest.

1.2.2 Numerical modeling in rock mechanics

Cundall & Starfield (1988) discuss the modeling of problems in rock mechanics. Some of the issues they handle are presented here.

The lack of detail in rock mechanics was a primary stumbling block in the early days of modeling and the possibility of including additional detail was welcomed. Some ten years ago the focus in rock mechanics moved from measurement to computation. Models are constructed because the mechanical processes involved are too complex to be fully understood. Easy access to versatile, powerful and inexpensive computer packages has increased the extent to which models are employed. The computer tools themselves are not, however, an explicit solution, but rather a means to a solution.

Guidelines for modeling are given below.

- A model is a simplification of reality. It is an intellectual tool that must be chosen for a specific task.

- The design of a model should be driven by the question that the model is expected to answer rather than the details of the system actually being modeled.

- One should aim to gain confidence in a model to modify it as one uses it. - Be clear about the reasons for building a model and the questions that it should

answer.- Examine the mechanics of the problem. - Design the simplest possible model that will allow the mechanism being examined to

occur.- Implement the model, choose your simplest experiment, and run it. Once successful

runs have been carried out, proceed to experiments that are more complex. - Only run complex models once success has been achieved with simple ones. - Visualize and anticipate solutions before actually running a model. Attempt to

visualize the deformation of the structure under load and form an approximate picture of the deformations.

Modeling carried out in a cautious and considered manner leads either to new knowledge or improved understanding.


10

1.3 Principles of probability

Hoek et al. (1995) introduced the principles of probability in rock mechanics. In a rock mass, parameters such as the uniaxial compressive strength of rock specimens, theinclination and orientation of discontinuities, and the measured in-situ stresses do not have a single fixed value but may adopt any of a number of values. It is simply not possible to establish the exact value of any one of these parameters at any given point. These parameters can therefore be said to be random variables.

In research, it is desirable to include as many samples as possible in any set of studied observations but practical considerations limit the amount of data that can be collected. It is often necessary to make estimates on the basis of experience or by comparisonswith results published by others.

Using a probability model does not allow prediction of the result of any individualexperiment, but it does enable determination of the probability that a given outcomewill fall inside a specific range of values.

A probability density function (PDF) describes the relative likelihood that a random variable will take a particular value. A PDF can be continuous (i.e. it can take all possible values) or it can be discrete. The same information can be presented in the form of a cumulative distribution function (CDF). A CDF yields the probability that thevariable concerned will have a value that is less than or equal to the selected value.

For many applications it is useful to present only the most relevant summarizingparameters from the pile of information. Widely used parameters are the sample meanvalue (x), the sample variance (s2), and the standard deviation (s).

The sample mean value (Eq. 1-1) indicates the center of gravity of a probabilitydistribution. A test which yields the results x1, x2,..., xn has a mean value of:

��

�

��

�

n

i

in x

nn

xxxxx

1

321 1..(1-1)

The sample variance is defined as the mean of the square of the difference between thevalues of xi and the sample mean value. The standard deviation (Eq. 1-2) is the positivesquare root of the variance:

2

1

)(1

1��

�

�

�

n

i

i xxn

s (1-2)

Note that for a finite number of samples it can be shown that the denominator (n-1,instead of n) gives a better and unbiased estimate for the population variance.

In a normal distribution, approximately 95% of observations will fall within the rangedefined by the mean � two standard deviations. If a result falls within these boundaries it is said that its deviation is “not significant”. In this study, the range employed is


11

limited to the mean � one standard deviation. In a normal distribution, approximately 67.5% of the observations will fall within this range.

A normal (Gaussian) distribution is the most common type of probability distribution function. Normal distributions are a family of distributions that have the same general shape (sometimes described as “bell-shaped”), i.e. symmetric with scores more concentrated in the middle than in the tails. The shape of a normal distribution can be specified mathematically by using two parameters: the mean (x) and the standard deviation (s).

In geotechnical engineering, the normal distribution is generally used for probability studies unless there are good reasons for choosing a different distribution. It is typical for variables that arise as the sum of a number of random effects to be normally distributed.


13

2 DESCRIPTION OF UNCONFINED COMPRESSION TEST

This chapter provides a brief description of the laboratory-scale unconfined compression test, the anisotropy effect, and the development of failure in a rock samplesubjected to compressive loading. Finally, gneissic tonalite and the results of laboratorytests are presented.

2.1 Compression test

The test most commonly performed on rock is uniaxial compression of cylindricalspecimens prepared from drill cores (Figure 2-1). This test is used to determine theuniaxial or unconfined compressive strength, �f, and the elastic constants Young’smodulus (E), and Poisson’s ratio (v). The response observed will depend on the nature and composition of the rock, the geometry of the test specimen and the rate of loading.For rock with similar mineralogy, compressive strength will vary with varyingproperties. For example, strength will decrease as porosity and water content increase.Anisotropy in the microstructure (e.g. schistosity) will also affect the strength propertiesof specimens. The principles of compressive testing are presented and discussed by a variety of authors such as Brady & Brown (1985), Jaeger (1972) and Obert & Duvall (1967).

To standardize the tests employed and make the results obtained comparable, theInternational Society for Rock Mechanics (ISRM) provides (ISRM Commission 1979) suggested techniques for determining the uniaxial compressive strength and deformability properties of rock material. Some features of this testing regime are givenbelow.

Figure 2-1. Schematic view of a compressional test arrangement. Uniaxial test:

compression plates are placed on the bottom and top surfaces of the sample, the sides

are unconstrained. In the Triaxial test, liquid is used to subject the sample to a

confining pressure. (Department of Geological Sciences 2001.)


14

- The test specimens should be circular cylinders having a height to diameter ratio of 2.5 – 3.0.

- The specimen diameter should be at least 10 times the size of the largest grain in therock.

- Load should be applied to the specimen at a constant stress rate of 0.5 – 1.0 MPa/s. - Axial load and axial and radial strains should be recorded throughout each test. - There should be at least five repetitions of each test.

During the test, the axial force is recorded and then divided by the initial cross-sectionalarea of the specimen to give the average axial stress, �z. This is then plotted against theoverall axial strain, �z. From this plot, it is possible to calculate a value for Young’smodulus. Corresponding equations for calculating E and v are presented below afterJaeger (1972).

Computing E and v from a triaxial compression test. For an elastic material, all stresscomponents are acting:

� �

� �

� �)(1

)(1

)(1

yxzz

xzyy

zyxx

vE

vE

vE

��

��

��

��

��

��

(2-1)

For the triaxial test:

- Strains ��x = ��y, ��z

- Stresses ��x = ��y, ��z

- The z-axis is parallel to the loading axis.

We also assume that during a triaxial compression test, the axial strain is applied with aconstant confining stress. Substituting these values in Equation (2-1) and solving for E

(2-2) and v (2-3) yields the following:

z

zE�

�

�

�

�´ (2-2)

z

xv�

�

�

�

�� (2-3)

These same equations also apply to an unconfined compression test because while in atriaxial test we assume that change in confining pressure is zero (��x=��y=0), in an unconfined test the change (and the values) are also zero.

2.2 Anisotropy

The behavior of many rocks is anisotropic because of some preferential orientation of the fabric or microstructure or the presence of bedding or cleavage. The taking of


15

samples and consequent stress release may also cause anisotropy by inducingmicrofracturing. In design, it is usual to employ only the simplest form of anisotropy,transverse isotropy. In transversely isotropic rock, peak strengths vary with respect tothe orientation of the plane of weakness.

Jaeger (Brady & Brown 1985) introduced a theory about how the strength of rock depends on the plane of weakness. The theoretical rock sample contains well-defined parallel planes of weakness. Each plane has a limiting shear strength, Equation (2-4).Slip along the weakness plane occurs when the shear stress (�) on the plane is equal to the shear strength (s). The stress transformation equations in an unconfined situation give the normal {Equation (2-5)}, and shear {Equation (2-6)} stresses on a plane.

wnwcs �� tan�� (2-4)

�

��

� 2cos22

11��n (2-5)

2

2sin1 ��

� � (2-6)

where cw is the cohesion of a plane, �w is the friction angle of the plane, � is the angle of weakness of planes with respect to the loading axis and �1 is the compression stress (seeFigure 2-2). Substituting Equation (2-5) in Equation (2-4), making s = � and rearranginggives the criterion for slip on the plane of weakness as Equation (2-7).

��

�

2sin)cottan1(

21

w

wS

c

�

� (2-7)

The compression stress required to produce slip tends to infinity as ��90o and as ��w. Between these values of � [�w ... 90o], slip is possible. The stress at which slipactually occurs varies with � according to Equation (2-7). The strength - angle-of-plane- curve is U-shaped, and the minimum strength value (Eq. 2-8) occurs when:

24w��

� �� (2-8)

For values of � which are outside the range given above, slip on the plane cannot occur and so the peak strength of the specimen must be determined by some other mechanism.

In reality, the �1 - �-curve does not take the theoretical shape as shown in Figure 2-2. Inparticular, the plateaus of constant strength in out-of-range sections of the curve are notalways present in experimental strength data. This suggests that the theoretical model isan over-simplified representation of the variation of strength in anisotropic rocks. Results concerning rock strength and orientation of the schistosity plane are mainlyavailable for sandstones, slates and shales (see Figure 2-3). These results show thatminimum strength occurs when � is between 30 and 40 degrees. (Brady & Brown1985.)


16

Schistosity angle �

Str

ess�

900

Figure 2-2. Left: definition of schistosity angle. Right: variation of peak strength with

the angle of weakness plane (schistosity).

Peak strength with respect to discontinuity angle

0

1

2

3

4

5

0 22,5 45 67,5 90Discontinuity angle [o]

Axi

al s

tren

gth

[lb

/in

2 x10e

4]

Slate 1 (unconf)

Slate 2 (unconf)

Slate 3 (conf)

Sandstone (conf)

Shale (conf)

Figure 2-3. Slates, shales and sandstones exhibit strength anisotropy. Uniaxial and

triaxial compression test results after Hoek & Brown (1982).


17

Autio et al. (2000) studied the effect of schistosity on the strength of hard crystalline rock by performing unconfined compression tests on gneissic tonalite (see Section 2.4of this report - Gneissic tonalite). The anisotropy of strength was remarkable. Autio’s conclusions concerning the test results were as follows:

- The uniaxial compressive strength, UCS, was at a maximum when the rock specimen was oriented parallel to the schistosity plane.

- The UCS was close to a minimum when orientation was perpendicular to the schistosity plane.

The main difference when comparing the results obtained by Autio to those presented in literature is that the strength of the rock specimens did not exhibit an increase when the orientation approached an angle perpendicular to the schistosity plane.

Peng and Johnson (1972) tested Chelmsford granite, mineralogically a quartz monzonite, which is homogeneous and does not exhibit schistosity or textural orientation. He concluded that the ultimate strength of the granite varied with the orientation of the specimen. Peng’s tests were made on cylindrical samples cored from three different, perpendicular, directions (three sets of orthogonal cleavages) in a granite block. The ratio between the weakest and the strongest specimens ranged from 0.84 to 0.94.

2.3 Failure development during compression test

Geology lecture notes from the University of Saskatchewan (Department of Geological Sciences 2001) introduce general deformation behavior in rock. At high rates of strain, rocks act as brittle-elastic material (see Figure 2-4). At stresses up to approximately 70% of their strength rocks deform elastically, at higher stresses crack propagation becomes dominant and eventually failure occurs as cracks coalesce to form a large fracture or failure surface. At low confining pressures, shallow depths or close to free surfaces, vertical splitting (1) is one typical mode of failure. At higher confining pressures, a single shear plane may develop (2). At even higher confining pressures, a network of inclined shear faults is formed (3). At low strain rates and very high confining pressures, the stress-strain curve does not have a distinct maximum indicating failure. Samples show the continuous deformation under load which is characteristic of ductile-plastic materials. Failed cores have a characteristic "barrel" shape.

Hazzard et al. (2000) discussed crack formation during laboratory tests. The strength of brittle rocks under compression depends on the growth of cracks and how these cracks propagate and coalesce into larger shear faults. Laboratory observations of stressed rock samples have shown that most cracks which form during compression are tensile and parallel to the maximum compression stress. Observation has also shown that shear cracks cannot propagate in their own plane. Final failure of a sample therefore occurs by interaction of the tensile cracks to form a macro shear fault.


18

Figure 2-4. Failure modes for different types of rock sample stress-strain behavior. 1 –

tensile splitting, 2 – shear splitting, 3 – ‘barrel’-shape failure. (Geology for Engineers

2001.)

Another hypothesis by Hazzard et al. (2000) proposes that as a homogeneous rock is loaded to its peak stress, cracking is randomly-located and scattered. Once the peak stress is reached, a small zone of cracks forms near the sample edge. At this point thetensile cracks interact and more microcracks form in an unstable manner. This processzone then penetrates into the unfaulted sample and a macroshear fault develops.

The behavior of rock has been extensively studied by Martin (1994) as follows. Rock is a brittle heterogeneous material that exhibits inelastic deformation because of theexistence and formation of the numerous microcracks. Under increasing load thesemicrocracks close. Once the existing cracks are closed the rock is assumed to be a linearelastic material. The elastic properties of a specimen are determined from this point. Asthe load is increased, the growth of axial cracks is dominant and the specimen expands.These axial cracks are considered to be stable since an increase in load is required tocause additional cracking. When the load is increased further, unstable crack growthstarts at an axial stress level which is 70 – 85% of the rock’s peak strength. At this point the mechanism of failure is a sliding of inclined surfaces. This is the most significantstructural change in the sample, since the density of microcracks increases seven-fold.The peak strength of the material marks the beginning of post-peak behavior. In the first part of the post-peak region, major inclined shear fractures develop.


19

Peng and Johnson (1972) studied the fracture modes of Chelmsford granite subjected to laboratory compression tests. The conclusions of his observations are as follows:

- Cracks grow near the corners and in the center of the upper and lower half of the specimen.

- Cracks are aligned parallel to the axis of the specimen. - Specimens fail either by the formation of a cone at each end or by a single inclined

fault. - Many samples showed combinations of both failure modes. - The fault surface consists of small steps, in appearance similar to a staircase.

Wong (1982) studied the interaction and coalescing of microcracks into a macroscopic fault. He observed Westerly granite samples at different stages of the compression test with a scanning electron microscope. He concluded that a localized zone in a post-failure sample consisted of cracks inclined at angles of 15o - 45o to the maximum compression direction and that the cracks followed favorably-oriented grain boundaries. It should be noted that the four minerals in Westerly granite all behaved differently during faulting and that one dominant mechanism of brittle faulting could not be isolated.

Studies made with an AE instrumentations (Heo et al. 2001) give some hints about how microcracks are distributed in granite under triaxial loading with low confinement. Firstly, the microcracks are distributed randomly, then they start to concentrate at the center of the specimen. As the peak strength of the material is approached, the cracks are about to form one or more shear bands. As the compression stress increases, these shear bands extend. As the level of stress approaches the peak rock stress, macro cracks and shear fracture zones are formed by the growth and coalescing of cracks.

The studies reviewed primarily concerned isotropic rock samples. No other studies than that performed by Autio et al. (2000) were found in which the effect of orientation on strength in clearly anisotropic granitic rocks has been studied in detail.

2.4 Gneissic tonalite

2.4.1 Igneous and metamorphic rock

Press & Siever (1986) present the three main classes of rocks: igneous, metamorphic and sedimentary. Igneous rocks result from the solidification of molten or partly-molten magma. Metamorphic rocks are the special products of geological processes acting on the solid materials of the Earth. Metamorphism is a process in which already-existing rocks are altered by temperature and pressure. The textures of metamorphic rocks are the result of re-crystallization or the conversion of one mineral to another in the solid state. The most eye-catching textural feature is a set of parallel planes. Large crystals visible to the naked eye, accompanied by some segregation of minerals into lighter and darker bands, produces schistosity. The most pronounced banding of minerals is shown by gneisses, in which coarse bands of segregated light and dark minerals are prominent throughout the rock. Foliation, lineation, and other metamorphic textures are a product of the preferential orientation of crystals related to the directions of the compressional forces of deformation that are responsible for crystallization or recrystallization.


20

The main rock type found in the Research Tunnel at Olkiluoto in Finland is gneissic tonalite, sometimes referred as anisotropic tonalite, which is slightly-foliated, medium-grained, massive and sparsely fractured. The tonalite is gneissic, i.e. oriented, with the dominant dip and dip directions being 30o and 145o respectively. (Äikäs & Sacklén 1993.) Despite the gneissic nature of tonalite, it is classified as an igneous rock based on its genesis processes.

2.4.2 Properties

Bates and Jackson (1990) describe tonalite as follows. Tonalite is a synonym for quartz diorite, a group of plutonic rocks having the composition of diorite but which contain a noticeable amount of quartz. The term gneissic describes the texture or structure typical of gneisses, with foliation that is more-widely-spaced, less-marked, and often more-discontinuous than that of rocks of a schistose texture.

Autio et al. (2000) provide following properties to tonalite. The main minerals in the gneissic tonalite from the Research Tunnel at Olkiluoto are plagioclase (45.3%), biotite (26.4%), quartz (15.6%) and hornblende (8.2%). The grain size of the plagioclase is 2 - 3 mm. Biotite occurs as clusters of flaky grains that are 1 - 3 mm in size. The size of the quartz grains is between 0.3 and 0.5 mm. The gneissic tonalite exhibits schistosity, and this is evident from the visible banding which is a product of the oriented nature of the oblong grains of biotite and hornblende. The rock is solid with an average dry density of 2810 kg/m3 and intragranular fissures are sparse. The main minerals, which represent 96% of the total mineral content, have different values for stiffness and Poisson’s ratio.

2.4.3 Laboratory test results

Tests made on samples of gneissic tonalite from the Research Tunnel at Olkiluoto (Autio et al. 2000) showed that the strength and deformation properties are significantly anisotropic and depend on the angle of schistosity in the test samples. Unconfined compression tests were performed for different schistosity angles to obtain Young’s modulus, Poisson’s ratio and strength values. The strength and modulus results are summarized in Table 2-1.

The uniaxial compressive strength as a function of specimen size and orientation is shown in Figure 2-5. Orientations were determined visually, with an estimated maximum error of � 10 degrees. A second order curve was fitted to the results.

The uniaxial strength is at a maximum when the sample is parallel to the schistosity plane (� = 0). When � is 38 - 43 degrees, measured strengths do not differ remarkably from values obtained with samples directed perpendicular to the plane (� = 90).

Measured values for Young’s modulus exhibit similar behavior with respect to the schistosity angle as strength, while the effect of orientation on Poisson’s ratio is not clear. Values for Young’s modulus are at a maximum when a sample is oriented parallel to the schistosity plane, but drops to 20% of the maximum value when orientation is close to perpendicular, see Figure 2-6.


21

Table 2-1. Average strengths and Young’s modulus with respect to schistosity angle.

Results from three different diameter samples (41, 54, 99 mm). (Autio et al. 2000.)

Orientation [degrees] Young’s modulus [GPa]

Unconfined strength[MPa]

Sample diameter[mm]

5.8 78.2 141.8 41.011.1 74.6 126.4 54.011.5 77.2 127.1 99.033.7 68.1 98.1 99.038.6 66.0 97.1 54.043.0 61.4 94.2 41.070.0 62.0 93.4 99.075.5 59.4 97.8 41.084.2 66.3 97.8 54.0

Peak strength - laboratory samplesMean with std. dev.

60

80

100

120

140

160

0,0 22,5 45,0 67,5 90,0Angle

Str

eng

th [

MP

a] o 54mmo 99mm

o 41mm

Figure 2-5. Average unconfined compressive strengths (with standard deviation) of

gneissic tonalite specimens with respect to schistosity angle. Three different sample

sizes were tested. (Autio et al. 2000.)


22

Young's modulus - laboratory samplesMean with std. dev.

40

50

60

70

80

90

100

0,0 22,5 45,0 67,5 90,0Angle

Mo

du

lus

[GP

a]

o 54mmo 99mm

o 41mm

Figure 2-6. Average Young’s modulus (with standard deviation) of gneissic tonalite

specimens with respect to schistosity angle. Three different sample sizes were tested.

(Autio et al. 2000.)


23

3 PFC THEORY

Particle Flow Code (PFC) models mechanical behavior by representing a solid as a bonded assembly of spherical particles. The modeling process is based on discrete-element (also called distinct-element) theory. PFC models are categorized as direct, damage-type numerical models in which the deformation is not a function of prescribed relationships between stresses and strains, but of changing microstructure.

The following sections describe general principles of the particle mechanics approach and the distinct element method and are based on PFC manuals published by Itasca (1999).

3.1 Particle mechanics

PFC2D and PFC3D (Particle Flow Code in 2 Dimensions and Particle Flow Code in 3 Dimensions) are programs for modeling the movement and interaction of assemblies of arbitrarily-sized circular (2D) or spherical (3D) particles. The model is composed of distinct particles that displace independently from one another and interact only at contacts between the particles. Newton’s laws of motion provide the fundamental relationship between particle motion and the forces causing the motion. Behavior that is more complex can be modeled by bonding the particles together at their contact points, and allowing the bond to break when the strength limit of the bond is exceeded. In addition to spherical particles, i.e. balls, the PFC model also includes ‘walls’. The desired model is constructed from these two entities.

3.2 Distinct Element Method

PFC models the movement and interaction of particles using the distinct element method (DEM). PFC is classified as a discrete element code because it allows finite displacements and rotations of discrete bodies and because it recognizes new contacts automatically. The program is a simplified implementation of the distinct element method because of the restriction to rigid spherical particles.

In the distinct element method, interactions between particles are treated as a dynamic process. Dynamic behavior in PFC is represented in numerical terms by a time-stepping algorithm which requires repeated application of the laws of motion to each particle, a force-displacement law to each point of contact, and a constant updating of wall position. The use of an explicit numerical scheme makes it possible to simulate the non-linear interaction of a large number of particles without the requirement for the computer equipment being used to have extensive memory.


24

3.3 Calculation cycle

The PFC calculation cycle is shown in Figure 3-1. At the start of each time step, the setof contacts is updated from the known particle and wall positions. The force-displacement law is then applied to each contact to update the contact forces, and thelaw of motion is then applied to each particle to update its velocity and position.

3.4 Contact models

A contact model describes physical behavior at each contact. The constitutive model acting at a contact consists of a stiffness model, a slip model or a bonding model. The stiffness model is an elastic relationship between force and displacement. The slipmodel introduces friction into contact behavior. Particles may also be bonded togetherat a contact (the bonding model).

Two bonding models are supported in PFC. Both bonds can be described as the ‘gluingtogether’ of two particles. The contact-bond glue is of vanishingly-small size and acts only at the contact point. The parallel-bond glue has a finite size and acts over a circularcross-section positioned between the particles. The difference between the bondingmodels is that while the contact bond can only transmit a force, the parallel bond can also transmit a moment.

Law of motion(applied to each particle)

*resultant force & moment

Force-Displacement law(applied to each contact)

*relative motion*constitutive law

contact forces

update particle + wall positions and set of contacts

Figure 3-1. PFC calculation cycle. The velocities and accelerations are kept constant

within each time step.


25

4 PFC3D MODEL FOR ROCK

Some issues of importance concerning the modeling of rock with PFC3D are discussed in this chapter. The unconfined compression test procedure, specimen genesis and anisotropy installation processes are given particular emphasis because of the weight they carry in this study

4.1 Modeling with PFC3D

The modeling of rock behavior is challenging because rock is a complex anisotropic material made up of numerous grains cemented together. The mechanical behavior of rock involves crack growth which depends on the heterogeneous nature of local stress distribution within the material.

The fundamental element in PFC3D is a spherical particle. When the problem to be modeled concerns the interaction of spherical particles, the code can be applied in a straightforward way. On the other hand, when modeling solid material such as rock, the application process is more complex as particle properties cannot be determined directly – they have to be interpreted in an iterative manner from the results of standard laboratory tests. The parameters required are called micro-properties. They dictate how the model will respond and the kind of macro-properties that it will output. In rocks, the micro-properties that produce the known macro-properties and observed behavior are not usually known. Although the behavior of the PFC3D model is found to resemble that of rock, generally the particles in a PFC3D assembly are not associated with the minerals or grains in rock. (Itasca 1999.)

Calibration is the term used to describe the iterative process of determining and modifying the micro-properties for a PFC3D model. In the calibration process, the responses of the model are compared to the responses of the rock samples in the laboratory and the micro-properties of the model are modified in an iterative way to achieve good agreement. Comparisons can be at both laboratory and field scale. The laboratory-scale properties typically chosen for comparison are the elastic modulus (E),the crack-initiation stress (�ci), and the strength envelope (�f = �f(Pc), where Pc is the confining stress (Potyondy & Cundall 2000.) In this study, the properties chosen for calibration are Young’s modulus (E) and the unconfined compressive strength (�f). The laboratory response used as a target for calibration is presented in Section 2.4 Gneissic

tonalite. In their excellent paper, Kulatilake et al. (2001) discuss general issues concerning the calibration of micro-mechanical properties for an intact model material using PFC.

4.2 Model generation and compression test simulation

The PFC3D software package comes with prepared triaxial test procedures using the program’s internal language (FISH). As this environment is somewhat limited in its scope, it was further developed to meet the requirements of this study. The primary areas of development were:

- Unconfined test procedures (rather than triaxial confined testing). - Altering the parallelpiped specimen shape used in the prepared package to a

cylindrical one.


26

- Introducing anisotropy (the prepared package environment is isotropic). - Developing the visualization tools used to visualize damage formation.

The following steps were performed when modeling with PFC3D:

1. The generation of an isotropic model that included particle generation, the compaction of these particles and internal stress reduction.

2. The introduction of anisotropy by adding banding in the specimen. 3. The simulation of a compression test.

A comprehensive description of model generation and the compression test can be found in the PFC3D manuals (Itasca 1999). The outcome of each step is illustrated in Figure 4-1.

4.2.1 Isotropic model generation in PFC3D

The procedure used to generate an isotropic model is made simpler by using a parallelpiped specimen shape. The final, cylindrical shape of the model is produced in Step 3. (Section 4.2). Firstly, bounding walls are created and the space (a parallelpiped box) is filled with spherical particles. The initial assembly is then compacted and internal isotropic stress is reduced. After additional steps, bonds between particles are introduced and the specimen is ready. Specimen dimensions are set in accordance with laboratory samples (Chapter 2.4 Gneissic tonalite). The specimen generation and material parameters are listed in Appendix 1. The specimen generation process produces an isotropic homogeneous model with given dimensions and micro-material properties.

4.2.2 Particle band generation in PFC3D

The schistosity of the gneissic tonalite is the result of oriented and clustered biotite grains. In PFC3D, an oriented rock sample is generated by running a random band generation procedure five different times at nine different schistosity angles to create bands of particles in the original isotropic packed model. This produces five different band formation for each schistosity angle, as illustrated in Figure 4-2. The final result is 45 different anisotropic models which are ready for further testing.

Potyondy & Cundall (2000) modeled anisotropy in 2-dimensions. Anisotropy was modeled here by using functions that they created. In this study, the functions were modified to be applicable in three dimensions. The following paragraphs explain the procedure used for band generation – which is partly based on the work by Potyondy & Cundall (2000).

Anisotropy is modeled by generating bands of particles within the matrix. Band particles are assigned micro-properties which are different to those possessed by the matrix particles. The procedure creates the joint-set, it marks the contacts as belonging to its joint-set. A sufficient number of joint planes are generated such that they cover the entire model. The joint planes are parallel to the schistosity planes, and their origin is at the center of the model bounding box. The thickness of each joint segment is set by first creating the joint segments via the JSET-command, and then expanding the contacts in the JSET to those that are adjacent to balls that are part of the original JSET. Input


27

parameters (with FISH-names) for establishing anisotropy are listed in Table 4-1 and in Appendix 1. The definitions used are shown in Figure 4-3. The geometry input parameters are expressed as multipliers of the average particle size used in the model.

The anisotropy geometry installation uses PFC3D JSET-command with aforementioned parameters. A joint-set is generated by assigning the joint-set ID number to all contactsbetween particles that lie upon opposite sides of each joint in the joint-set. Joints consist of a number of finite circular disks. The specified number of joint planes is generated,starting at the origin and then alternating on each side. Spacing is measured along thenormal to the mean plane orientation. The parameters an2_rmult and an2_aratio define the area which the disks will occupy in the plane. The disks are placed at random locations within a square region of the joint plane that includes the whole model. Disks are generated until the ratio of the total disk area to the total square area equals theparameter set by an2_aratio. (Itasca 1999.)

The initial joint-set thickness is one particle. The expansion function expands thenumber of contacts that are part of the given joint-set. The function finds adjacentparticles to the joint-set and adds them to the joint-set. This function is run given timesto obtain wanted joint-set thickness. The used parameter value for this function expandsthe joint-set to have thickness of two particles.

After the geometry installation the micro-properties of the joint-set particles and bonds are modified. Two functions are used to modify the ball normal and shear stiffnesses and parallel-bond stiffnesses and strengths. The functions identify the particles and bonds that are part of the joint-set and apply the given factors to reduce the particle and bond stiffnesses and strengths, as it is assumed that the bands are softer and weaker than the matrix, resembling the properties of biotite.

Table 4-1. Anisotropy installation parameters and their values.

Description FISH name Value

Number of joint-sets to be created an2 jsetnum 1Seed of random-number generator an2_random 10001...Joint spacing will equal [an2_smult]*[particle_radius]*2 an2_smult 6Number of expansions of joint-segment thickness an2_stmult 2Joint disk radius for joints will equal [an2_rmult]* [particle_radius] an2_rmult 4Joint area ration an2_aratio 0.85Schistosity angle [degrees] an2_stheta 0...90Ball stiffnesses reduction factor an2_efac 0.05Parallel bond stiffnesses reduction factor (Ef) an2_ebarfac 0.05Parallel bond strength reduction factor (Sf) an2_sigbarfac 0.2


28

1 2 3Figure 4-1. Model generation steps in PFC3D. 1-isotropic parallelpiped model,

2-anisotropic parallelpiped model, 3-anisotropic cylindrical model.

54

21

5x 5x

3

Models with schistosity angle of 22.5 degrees.Each has different band formations.

54

21

3

Models with schistosity angle of 67.5 degrees.Each has different band formations.

Original isotropic model

Band generation

5x

5x5x5x

5x

5x

5x0o

11o

90o

79o

45o

Figure 4-2. Band generation produces a group of five different samples for all nine

schistosity angles.


29

Joint disk and its radiusJoint spacing Joint thickness

Balls in joint (red)All balls in plane (red+yellow)

Joint area ratio =

�

Matrix

Band

Figure 4-3. Particle band generation produces an anisotropic model. Definitions of the

geometry parameters are shown. For illustrative purposes, the schemes on the left and

in the center represent schistosity oriented parallel to the longitudinal axis of the model.

4.2.3 Compression test simulation in PFC3D

The unconfined compression test is performed on a specimen with a circular cross-section. The cylindrical model is made out of a parallelpiped form by deleting particleswhich lie within the initial parallelpiped shaped model but outside the cylinder shape.As the test is an unconfined one, the first step in the test procedure is to remove the fourseparate side walls. The top and bottom walls act as loading platens.

At the start of the test, the platens are given a small velocity which is then graduallyincreased to the final compression-test velocity to maintain quasi-static conditionsduring the test. Induction of a compressive stress wave that will propagate through themodel and produce impact loading must be avoided. The loading velocity is controlledthroughout the test and is kept constant. Even though the velocity is quite large(0.05 m/s) the modeling environment is still in a near quasi-static condition because kinetic energy is being damped at a very high rate. Running simulations at a physicalplaten speed ~1 MPa/s (see Section 2.1 Compression test) would have taken a verylarge number of simulation steps (and time) and would have produced much the same quasi-static results. One test was performed with the platen velocity of 0.025 m/s. This showed that the modulus values were the same between the test with ‘normal’ platen


30

speed and the test with half the platen speed. The corresponding strength values deviated 2%.

Some test parameters are listed in Table 4-2, and all the parameters are listed in Appendix 1. The test is carried out until failure occurs. Both loading and the test are continued until the post-peak stress is 80% of the peak strength. During simulations of the compression test, 33 parameters are monitored and their values stored for later use. These parameters include stresses and strains on walls and within the specimen, some energy quantities, and the monitoring of microcracks. All the parameters monitored are listed in Appendix 1.

In this study, the items of quantitative interest are Young’s modulus and peak strength. The peak strength of the synthetic material is the maximum stress on the stress-strain curve. Young’s modulus is calculated from Equation (4) using the stress and corresponding strain values which are 50% of the peak strength and values of the origin as these are assumed to be within the elastic region (Figure 4-4). All the compression test simulations performed are listed in Table 4-3.

Table 4-2. Unconfined compression test parameters for PFC3D.

Test parameter Value

Final platen velocity [m/s] 0.05Test-termination criterion 0.80Model height [mm] 142.0Model diameter [mm] 54.0Number of particles ~22000


31

Strain �

Str

ess�

Peak strength

��

��

E=��/��

Figure 4-4. The pre-failure region (pre-peak stress) of the stress-strain curve obtained

in an unconfined compression test and definitions of Young’s modulus and peak

strength.

Table 4-3. List of simulated compression tests with the corresponding chapter number.

Simulation Model type Simulated schistosityangles [o]

Number of simulations

Results presentedin chapter

Basic anisotropic 0, 11, 22.5, 34, 45, 56, 67.5, 79, 90

45 Ch. 5

Resolution study isotropic N/A 17 Ch. 7.1

Geometry parameters anisotropic 0, 22.5, 45, 67.5, 90 35 Ch. 7.2

3-component model anisotropic 11, 54, 79 9 Ch. 7.3


33

5 RESULTS OF COMPRESSION TEST SIMULATIONS

The results of compression test simulations are presented in this chapter. Firstly, numerical aspects of the modeling results are presented together with the appropriate laboratory results. After the quantitative presentation, the results of visual study are shown as well as failure patterns in the models. Finally, some observations are made about crack localization and crack evolution.

5.1 Quantitative results

As described in Section 4.2, the effect of schistosity was studied by carrying out a number of simulations with different schistosity angles from 0 to 90 degrees and an interval of approximately 11 degrees. At each schistosity angle, a total of five simulations were performed, resulting in a total of 45 simulations. The height, diameter and average particle radius of the tested specimens were 142 mm, 54 mm and 1.3 mm, respectively. The vales for Young’s modulus and peak strength obtained from the simulations are shown in Table 5-1 together with their standard deviations. An example of the axial stress versus axial strain curve at a schistosity angle of 45o is shown in Figure 5-1. The shapes of all the stress-strain curves in the other simulations are similar, but the positions of the curves differ slightly. More stress-strain curves can be found in Appendix 2. The values for peak strength and Young’s modulus determined from each simulation are plotted with respect to schistosity angle in Figure 5-2. In Figure 5-3 the mean values of both peak strength and Young’s modulus are shown. The peak strengths obtained in laboratory tests are plotted in Figure 5-4 together with the PFC3D results. Figure 5-5 shows values of Young’s modulus determined from the simulations and values resulting from laboratory measurements.

The following conclusions were reached after comparing the laboratory results and the results obtained from the PFC3D simulations:

- The peak strength values obtained in simulations coincide with the values obtained from laboratory tests from 0 to approximately 65 degrees. From that point onwards, the results obtained in simulations are higher.

- If all the values obtained in simulations are shifted downwards by about 10 MPa, the two graphs (from simulations and laboratory tests) are almost similar - this is the standard calibration procedure described earlier. This suggests that the peak strength results are a qualitative match, and that if the calibration procedure were to be carried out again the model would also be quantitatively adequate.

A similar conclusion can be reached by comparing the values obtained for Young’s modulus, shown in Figure 5-5.

- The shapes of the graphs are similar, but the levels of the modulus values differ by 10 - 20 GPa. This implies that re-calibration would superimpose the results.

A better matching of the reults can be achieved by re-calibration. In the process, four micro-properties can be scaled in order to achieve the required matching. The micro-properties are the particle-based ball-ball contact modulus, the parallel-bond modulus, the parallel-bond normal strength, and the parallel-bond shear strength.


34

The minimum peak strength occurs when the schistosity angle is 30 - 40 degrees. Thisangle is the same as the angle-of-failure plane in an isotropic rock model under uniaxialcompression.

Table 5-1. Mean values and standard deviations used for simulation of unconfined

compression test.

Schistosity angle Peak strength [MPa]mean std.dev.

Young’s modulus [GPa]mean std.dev.

0 149 2.7 74 0.611 137 2.3 69 1.6

22.5 104 5.7 60 2.034 93 5.6 51 2.945 96 5.1 47 1.556 103 2.5 44 1.7

67.5 112 3.3 44 2.079 111 3.9 43 2.690 112 2.7 44 2.4

Figure 5-1. Example of a stress–strain curve for a simulated unconfined compression

test using PFC3D. Schistosity angle used in the model 45o, Peak strength 97 MPa,

Young’s modulus 44 GPa. The unit of the vertical stress axis is Pascal [Pa].


35

Peak strength and Young's modulusAll values

0

20

40

60

80

100

120

140

160

0 22,5 45 67,5 90Angle of schistosity

Str

en

gth

[MP

a]/

Mo

du

lus[G

Pa]

Figure 5-2. All results for simulation of Peak strength [MPa] (upper) and Young’s

modulus [GPa] (lower) are plotted with respect to schistosity angle [o].

Peak strength and Young's modulusMean values with std.dev.

0

20

40

60

80

100

120

140

160


Str

en

gth

[MP

a]/

Mo

du

lus[G

Pa]

Figure 5-3. Mean and standard deviations of simulation results plotted with respect to

schistosity angle [o]. Peak strength [MPa] – upper, Young’s modulus [GPa] – lower.


36

Peak strength - PFC and laboratory samplesMean values with std.dev.

60,0

80,0

100,0

120,0

140,0

160,0

180,0


Str

en

gth

[M

Pa

]

Peak strength PFC3D

Peak strength LAB

Figure 5-4. Peak strength [MPa] of the laboratory tests and the PFC3D runs. The

yellow transparent area encloses the laboratory results. Note that the vertical axis

spans from 60 - 180 MPa.

Young's modulus - PFC and laboratory samplesMean values with std.dev.

30,0

40,0

50,0

60,0

70,0

80,0

90,0


Mo

du

lus

[G

Pa

]

Young's modulus PFC3D

Young's modulus LAB

Figure 5-5. Young’s modulus [GPa] of the laboratory tests and PFC3D runs. The

yellow area encloses the laboratory results. Note that the vertical axis spans from 30 -

90 GPa.


37

The crack-initiation stress corresponds to stress at which 1% of the total number of cracks existing at peak load has been formed. Poisson’s Ratio () is calculated usingEquation (5-1), where �v is volumetric strain and �z is strain in the specimen’sz-direction.

��

�

�

��

�

�

�

�

�

z

v

�

�

� 12

1(5-1)

Strain values are determined at a compression stress of 50% of the peak strength, see Figure 4-4. Crack initiation stress depends on the ratio of the standard deviation to themean of the parallel-bond strengths. Poisson’s Ratio depends on the ratio of the shear contact stiffness to the normal contact stiffness. (Itasca 1999.) Considering that the parameters were not focused on calibrating the model, the results obtained appear to correspond well with the work done by Autio et al. (2000). Figure 5-6 shows the crackinitiation stress and values for Poisson’s Ratio with respect to schistosity angle.

Crack init. stress

0

20

40

60

80

100

0 22,5 45 67,5 9Angle

Str

eng

th [

MP

a]

0

Poisson's ratio

0,00

0,10

0,20

0,30

0,40

0,50

0 22,5 45 67,5 90

Angle

Figure 5-6. Simulated mean values for crack initiation stress [MPa] (above) and

Poisson’s Ratio (below) and their standard deviation with respect to the angle of

schistosity [o].


38

5.2 Qualitative study

Visual study of the damage formed during the compression test was carried out by looking at the damage patterns. All damage plots shown correspond to the final post-peak stage of simulations unless otherwise stated. Damage was studied in models in which the angle of schistosity angle was between 11o and 79o. Models with schistosity angles of 0o and 90o were not included in the detailed qualitative study because proper laboratory results for corresponding samples were not available.

5.2.1 Damage formation in PFC3D

The PFC3D model is constructed out of a number of particles which have been bonded together. As the model is loaded, forces build up and at some point a bond breaks. After the breaking of a bond, stress is redistributed and this may cause more cracks to form nearby. If the rock model is suitably stressed, these bond breakages may be localized into an inclined macro-fracture. Finally the model will fail. Rather than using constitutive laws in an indirect manner, deformations and fractures in the rock are modeled directly by allowing micro-mechanical damage to occur.

In the model, cracks are represented as colored octagons lying between two previously-bonded particles. The radius of the octagon is equal to the average radius of the two particles. Circular cracks are oriented perpendicular to the line joining the centers of the two particles (see Figure 5-7). Two ways of coloring the octagons are used. One way is to represent the cracks with respect to crack type as bi-colored octagons in which red represents tension-induced parallel-bond failure and black represents shear-induced parallel-bond failure. The second way is to illustrate cracks with respect to time by 16 gray-scale colors from white to black which indicate the time of crack formation; white = early, black = late.

5.2.2 Failure patterns

Study of damage was carried out by studying the damage patterns from different directions and in different sections (Figure 5-7), and by observing cracks at various phases of the simulation. In addition, an animation was made showing the generation of cracks during the simulated compression test. Based on observations made, failure patterns were divided into four groups based on the type of damage formation.

The four failure groups are as follows:

- Group A - models with a schistosity angle of 11 degrees, 5 simulations - Group B - models with a schistosity angle of 22.5 - 56 degrees, 20 simulations - Group C - models with a schistosity angle of 67.5 degrees, 5 simulations - Group D - models with a schistosity angle of 79 degrees, 5 simulations.


39

Figure 5-7. Left: Presentation of tension-induced microcrack as a octagon. Right:

Vertical and horizontal sections.

While the difference in damage formation difference between neighboring groups (e.g.A and B) is not very remarkable, the difference in damage between the end cases (A and D) is significant. The process of damage formation is divided into four phases; 1. initiation, 2. fracture growth, 3. failure (loss of strength), and 4. the collapse stage.Figure 5-8 shows cracking in each phase. The damage formations in each group are shown in the schematic Figures 5-9 to 5-12 and the related explanations are given witheach figure. The figures represent the most common types of failure pattern within thecorresponding group. All cracks are plotted in the figures, 3D cracks are projected ontoa plane (not a section).

Initiation Fracture growth Failure Collapse

Figure 5-8. Schematic illustration of the four phases of damage formation.


40

Group A, schistosity angle 11o

1. INITIATIONThe microcracks are forming evenly, mainly in thebands. It appears that there is slightly morecracking close to both ends of the model andaround its perimeter than inside the model.

2. FRACTURE GROWTHMicrocracking is, in general, concentrated in thebands and at the top of the model.

3. FAILUREThe upper corner of the model is breaking (arrow) and a major fracture is localizing into one of thebiotite bands.

4. COLLAPSEAn extensive crushing region forms at one end of the model. The formation of macro-fracturing inthe model resembles the shape of an hourglass.

Figure 5-9. Damage formation at

schistosity angle of 11o.

Group B, schistosity angle 22.5 o

- 56o

1. INITIATIONMicrocracks start to form primarily within thebiotite bands.

2. FRACTURE GROWTHCracking is concentrating intensely in a singleband. Weaker cracking is visible within parallelbands.

3. FAILUREA major fracture zone shears the model along thebiotite band and forms two separated pieces.

4. COLLAPSEA zone of crushed rock forms around the shearfault. Arrows depict the splitting of the model.

Figure 5-10. Damage formation

at schistosity angle of 34o.


41

Group C, schistosity angle 67.5o

1. INITIATIONMicrocracks start to form within the biotite bandsand at the top of the model. Most of the cracks arelocated on the outer section of the model.

2. FRACTURE GROWTHCracking increases within some biotite bands andat the top of the model.

3. FAILUREAt peak stress, a major fracture zone is developingthrough the model which only partly follows thebands of biotite. The angle of the fracture plane is about 40 degrees with respect to the vertical axiswhile the biotite band is at an angle of 67.5 degrees. The upper corner is also splitting.

4. COLLAPSEMajor crushing is taking place around the fractureand at the top of the model. Arrows show themovement of the fractured part.


at schistosity angle of 67.5o.

Group D, schistosity angle 79o

1. INITIATIONMost of the early cracking is located at both ends of the model.

2. FRACTURE GROWTHCracking is increasing around the early cracking atone end and this starts to dominate the pattern offailure.

3. FAILURECrushing at the top of the specimen. Small partsare cleaving off the model.

4. COLLAPSEFractures are still expanding around the previous cracking. The end part is totally crushed.


at schistosity angle of 79o.


42

Simulated damage patterns for seven different values of schistosity angle are shown in Figures 5-13 to 5-19. The examples shown were selected to represent the most common types of failure generation. The method used to color cracks in these plots is time-dependent except in the case of the right-hand plots where the coloring indicates the type of cracking. In each case, the figure on the left shows all the cracks, the figure in the center shows cracks and displacements that lie in the Y-Z-section, and the figure on the right shows crack types and sample structure in the same section. In all cases, the viewing direction is parallel to the X-axis. More figures resulting from simulations can be found in Appendix 2. Most of the cracking seen in the figures (about 75%) forms after the peak stress has been reached, i.e. in the collapse stage, and does not affect the failure strength.


43

Figure 5-13. Model: sTx_mG_11c, schistosity angle 11o. Left: All cracks, time-

dependent coloring (light = early, dark = late). Center: Cracks and displacements on

the origo centered vertical section, time-dependent coloring (light = early, dark = late).

Right: Cracks and modeled particles on the origo-centered vertical section, crack type

coloring (red = tension, black = shear). Particle coloring: yellow = stronger matrix

material, orange = weaker band material.


44

Figure 5-14. Model: sTx_mG_225d, schistosity angle 22.5o. Left: All cracks, time-







45

Figure 5-15. Model: sTx_mG_34d, schistosity angle 34o. Left: All cracks, time-







46

Figure 5-16. Model: sTx_mG_45a, schistosity angle 45o. Left: All cracks, time-







47








48

Figure 5-18. Model: sTx_mG_675e, schistosity angle 67.5o. Left: All cracks, time-







49








51

6 ANALYSIS OF RESULTS

This chapter contains a discussion of important observations about failure development, a comparison of the results obtained from PFC3D simulations with results of laboratory tests, and comparison with results obtained in earlier work using 2D modeling.

6.1 Crack formation

The failure patterns in the different modeling simulations have some common features. The microcracks that form in an early stage of uniaxial compression are localized randomly throughout the specimen and within the biotite bands. In spite of the fact that the model is asymmetric and anisotropic, early microcracking is located in a quite symmetrical fashion. Early microcracks are primarily formed parallel to the direction of loading. At some point in time, microcracks begin to localize within the biotite bands, cracking in one band starts to become predominant, and in most cases the band-induced fracture zone propagates through the specimen. As the bands are not continuous in their geometry but are formed from distinct biotite sections aligned in a plane, fracture zones that form within a band are forced to jump to the next band section via the stronger matrix material in order to be able to continue cracking. This phenomenon is called “bridging”. In models with schistosity angles from 11o to 67.5o, fracture propagation is more or less within a single biotite band (e.g. Appendix 2; Figures 45c and 675c).Minor fracturing also occurs parallel to the main fracture plane and this is evident in the damage plots. Major post-peak crushing regions are visible at the top and bottom of the specimens which have schistosity angles of 11o and 56o forward. In these models there is also more early-stage microcracking localized at the top and bottom ends of the model where the compressive load is exerted.

In some sections it was clear that microcracking proceeded from the outer cylindrical surface to the center of the model. In some models there was evidence that a microcrack was first formed inside a band and then spread in both directions along the band plane (Figure 6-1 is an example).


52

Figure 6-1. Horizontal cutting planes in two separate models. Left: schistosity angle

34o. Right: schistosity angle 45

o. Red arrows show the direction of microcrack

propagation. Sections shown are from the middle of the model.

Visual observation indicated that in some models there was more microcracking nearthe surface than in the center. To confirm this hypothesis radial distributions of cracks were calculated for each sample. The distributions showed that there might be slightlymore cracking in the outer periphery of the samples than inside. A possible explanationfor this could be that there is less internal cracking in the center of the model becausethe increased degree of constraint raises the strength of the matrix.

As can be seen from the time-dependent cumulative crack plot in Figure 6-2, mostfracturing takes place in the post-peak stage. This figure also shows the stress-straincurve. It is clear that the number of cracks increases rapidly after peak stress is reached. The microcracks are either tensile- or shear-induced, and tensile cracking is clearlypredominant as there is some five to ten times more tensile cracking than shear-inducedcracking. All models demonstrated this behavior. The same crack-increasingphenomenon occurs when observing laboratory compression tests on an actual rocksample with an acoustic emission instrument (see for example Autio et al. 2000). In themodel generation, the parallel bond tensile and shear strengths are set equal to one another. This produces more micro-tensile than micro-shear cracking to occur, as observed. This micro-property related effect was studied by Potyondy and Cundall (2000).

The results of simulations obtained using PFC3D clearly indicate that the method works advantageously. The results obtained show that the PFC3D results are related to corresponding laboratory results. The stress-strain curve obtained is similar in shape and the manner in which crack development takes place resembles many correspondinglaboratory observations.


53

Figure 6-2. Axial stress (uppermost black line), all cracks (black), tensile cracks (red)

and shear cracks (blue) plotted against axial strain. The cracking increases after the

peak strength and tensile cracking is predominant. Model sTx_mG34d, schistosity

angle 34o, peak strength 99 MPa, Young’s modulus 55 GPa, number of cracks at peak

strength 820 pieces.

6.2 Comparison against laboratory samples

Fracturing in 54 mm laboratory samples of gneissic tonalite and a PFC3D model arecompared in Figure 6-3. The three photographic images show damage occurring duringan unconfined laboratory compression test that was carried out in three different phases. The PFC3D images show cracking in the same phases. In spite of the relatively largeparticle size of the PFC3D model compared to actual grain size of the rock, somesimilarities can be found between the simulated failure patterns and failure in thelaboratory test specimen. In the first phase, cracking is scattered around the sample and the amount is rather small even though the level of stress is only 10% below peak strength. At peak strength, cracking is primarily focused in a couple of inclined and parallel-oriented planes. After peak strength has been passed, cracking spreads along theplanes. In the laboratory sample, post-peak cracking appears to have been spread morewidely than in the PFC3D sample. In both cases, primary cracking takes place in thebiotite (i.e. the PFC3D bands). The red dots in the photographic superposition imagesshow cracking around the black biotite grains that has been confirmed by investigatinglaboratory samples with a scanning electron microscope (SEM). As can be seen inFigures 5-13 to 5-19, most of the cracking occurs in the PFC3D biotite sections.


54

Loading 10% less than peak strength (pre-peak)

Loading at peak strength Loading 5% less than peak strength (post-peak)

Figure 6-3. Laboratory samples (Autio et al. 2000) (upper) and PFC3D failure-pattern

images (lower) from three different loading phases. Microfracturing of laboratory

samples is superimposed in red color. PFC3D crack coloring; red = tension-induced,

blue = shear-induced. Only cracks in the origo centered section are shown.


55

6.3 Comparison to 2D modeling results

Using PFC2D, Potyondy and Cundall (2000) modeled corresponding behavior of anisotropic rock. The results they obtained were compared with the results presented in this report.

Some differences between the results obtained from modeling in the two studies were observed. Most of the PFC2D models exhibited plastic-like behavior after peak loading whereas all of the PFC3D models showed brittle-like post-peak behavior (see for example Figure 6-2). Most of the PFC3D models showed secondary cracking parallel to the main cracking plane (see, for example, Figure 6-3) which was not as intense as in the PFC2D models. Some differences may arise from the fact that the PFC2D unconfined compression test were carried out using a small confinement pressure (~0.1 MPa) while the PFC3D simulations were performed without any confinement at all.

Several similarities between the results of modeling in the two studies were observed. Both methods showed a violent increase in cracking after the peak strength had been reached, with the main type of cracking being tensile-induced microcracking (see Figure 6-2). In both cases, pre-peak cracking was primarily located in biotite band segments and in bridging between these.


57

7 SENSITIVITY ANALYSIS

Three sensitivity analyses were performed: a study of the dependence of the PFC3D model on its macroscopic properties in relation to particle size, a study of the parameters related to the creation of anisotropic bands and a study of a three-particle-type model.

7.1 Resolution of the PFC model

The resolution of the model is defined as the number of particles across the model width (PFC2D) or the model diameter (PFC3D). If the dimensions of the model size are kept constant, reducing the particle size increases the model resolution and vice versa. The particle size is of interest because it has an effect on both the time required to complete simulations and the amount of computer memory required. Modeling with PFC is balancing between achieving a degree of resolution that is fine enough and the equipment resources that are available. The effect of the resolution was studied in both 2D and 3D.

7.1.1 Effect of model resolution in 2D modeling

Potyondy & Cundall (2001) examined the effect of specimen resolution on macroscopic properties in 2D models with a height to width ratio of 2:1. In the first set of tests they changed the width of the specimen while maintaining the same aspect ratio. In the second set of tests the particle sizes (i.e. radii) were changed. Specimen resolution increases when the specimen size is increased or the particle radius is reduced and vice versa. All other specimen properties were left untouched. The reference specimen for the PFC2D models was a specimen calibrated to fit the properties of Lac-du-Bonnet granite.

For both sets of tests, four different-sized specimens were generated, and for each of these specimens 10 repetitions of compression tests and Brazilian tests were performed. A total of 80 test-runs were executed.

The conclusions from the testing were as follows.

- As specimen resolution increases, �f, E and v tend towards a single mean value and the variations in value is reduced.

- Even the coarsest specimen, with 11 particles across the model width, produces average results that are very similar to those of the higher resolution specimens.

- Results achieved by altering specimen size and particle size are essentially the same.

At a macroscopic scale, strength properties can be divided into extensional or compressional conditions. The unconfined compressive strength results from compressional conditions, while the Brazilian tensile strength results from extensional conditions. While the resolution used does not affect compressional properties, it does affect extensional properties, and the Brazilian tensile strength decreased as specimen resolution increased.


58

7.1.2 Effect of model resolution in 3D modeling

The effect of particle size on the results of the PFC3D simulations was studied by performing compression test simulations with models employing different sizes of particle. Four different PFC3D specimens were constructed using different resolutions, with the alteration in resolution being achieved by changing the particle radius in each specimen. The reference material had a mean particle radius of 1.030 mm and a specimen cylinder size of 42 x 120 mm, giving a resolution (particle per width) of approximately 20 particles. The other specimens had particle sizes that were 1.5x, 2x, and 5x larger and the corresponding resolutions were 14, 11, and 4 particles respectively. A series of compression test were carried out and the strength and modulus values obtained were recorded.

The test results obtained using PFC3D were somewhat different from those obtained using PFC2D and showed a clear relationship between the resolution employed and the strength and modulus values obtained (see Figure 7-1 and Table 7-1). The effect of resolution appeared to be logarithmic in form. As the resolution increased, the values for strength and modulus obtained using the PFC3D simulation increased.

The conclusion here is that resolution affects macroscopic behavior and therefore the model has to be re-calibrated if the resolution used in the model is changed. It has more influence on the depiction of damage formation in specimens since the higher the resolution, the finer the structure of the specimen and the more detailed the damage formations that can result (see Figure 7-2). One major drawback to increasing the resolution is that models become too large to run in a standard PC (800 Mhz). Currently, an isotropic model with the resolution of 20 particles per width and the dimensions stated above (a total of 22,000 particles) requires 11 hours of computing time to running a single unconfined compression test simulation using PFC3D. Table 7-2 has values for the relationship between particle number, resolution and computer time required. Even though the specimen size used in obtaining these figures is different to that used in the resolution study, the relationship is essentially the same. As the resolution of the specimen is increased, the number of particles increases by the power of three and the computer time required to run one unconfined compression test using PFC3D increases exponentially.


59

Table 7-1. Results of the resolution test run. The dimensions of the specimens were 42

by 42 by 120 mm. A total of 17 runs were made; five for the reference model and four

for each of the other specimens. Mean values are shown (E = Young’s modulus [GPa],

Sig_f = Unconfined compression strength [MPa]).

Radiusmultiplier

Avg. particleradius [mm]

E[GPa]

Sig_f[MPa]

Resolution*[particle/width]

Number of runs

x1.0 1.03 56 146 20 5x1.5 1.54 53 126 14 4x2.0 2.04 52 116 11 4x5.0 4.94 38 61 4 4

*Resolution is defined as the number of particles across the model diameter.

Strength and modulus versus resolution

0

20

40

60

80

100

120

140

160

1 10Resolution (particle/width)

Str

eng

th[M

Pa]

/ M

od

ulu

s [G

Pa]

100

UCS

Young's modulus

Figure 7-1. Unconfined strength (UCS) and values for Young’s modulus show a clear

dependence on resolution. The relationship resembles a logarithmic dependence.


60

1.5 mm 2.0 mm 4.9 mm

Figure 7-2. Damage patterns resulting from the use of different model resolutions. The

numbers depict the average particle radius in the corresponding model. Crack coloring

indicates the relative time when microcracks were formed [light = early, dark = late].

The models are isotropic.

Table 7-2. The relationship between particle size, resolution and computing time.

The dimensions of the specimens are 54 x 54 x 142 mm. (The values 51 h and 642 h are

extrapolated assuming that the time-resolution relation is exponential [t=AResolution*B

].)

Average particleradius [mm]

Resolution*[particle/width]

Number of particles Computer time[hours]

2.0 14 6500 11.5 18 15300 41.3 21 23600 111.03 26 47400 (51)0.8 34 101100 (642)

*Resolution is defined as the number of particles across the model diameter.


61

7.2 Geometry parameters

Anisotropy was modeled by generating bands of particles as described in Section 4.2.2 Particle band generation in PFC3D. The sensitivity of the simulation results to changes in the band geometry parameters was studied by changing the band geometry but keeping all other material properties constant. Three cases were modeled and 35 simulations were performed. The parameters studied, joint area ratio and joint spacing (see Figure 4-3 for explanations), were changed so that the overall amount of band material in a model would be constant and identical to the basic case. In Case 1, the joint area ratio and joint spacing were reduced, producing a model in which the distinct band planes were no longer clearly visible. In Case 2, spacing was increased and the area of the weakness plane was filled with band particles to resemble a shale with very clear and visible continuous weakness planes. Case 3 was a modified version of Case 1, being geometrically between case 1 and the basic case. Table 7-3 is a list of the studied cases and the geometric parameters. Particle structures and damage patterns for the three cases are shown in Figure 7-3.

The results of the simulations show that cases 1 and 3 reduce the anisotropy of strength while Case 2 strengthens it. This can be seen in Figure 7-4 where peak strength and Young’s modulus are shown with respect to the angle of schistosity. The weak anisotropy effect in cases 1 and 3 is almost identical - the shape of the curves remains the same only the absolute values deviate. The strengthened anisotropy in Case 2 is clear. Case 2 resembles slate in its anisotropy effect and its microstructure as described in Chapter 2.2 Anisotropy.

According to these results, the geometry of banding has a clear effect on strength and modulus anisotropy and is therefore essential input information in the modeling process.

Table 7-3. Geometric parameters for the three cases studied.

Joint area ratio Joint spacing # schistosity # simulations

Case 1 0.40 3 5 [0o, 22.5o,...] 3 x 5 Case 2 1.00 12 5 [0o, 22.5o,...] 1 x 5 Case 3 0.50 4 5 [0o, 22.5o,...] 3 x 5 Normal 0.85 6 9 [0o, 11o,...] 9 x 5


62

Case 1 Case 2 Case 3 Basic case

Figure 7-3. Half of the models showing the three different geometries of the cases and

the basic case (above). Cracks on the origo centered section are shown below.

Yellow = matrix, red = weaker bands. In these models, the angle of schistosity is 22.5o.


63

Peak strength

0

40

80

120

160

200

0 22,5 45 67,5 9angle

Str

eng

th[M

Pa]

0

Normal

CASE1

CASE2

CASE3

Young's modulus

0

20

40

60

80

100

0 22,5 45 67,5 90

angle

Mo

du

lus[

GP

a]

Normal

CASE1

CASE2

CASE3

Figure 7-4. Simulated peak strengths (upper diagram) and Young’s modulus (lower

diagram) of the three cases with different geometries and results from the basic case

(Normal). The fitted spline lines which connect the observations are for clarification

purposes only.


64

7.3 Three component model

Gneissic tonalite consists of three main minerals: plagioclase, biotite and quartz.Simulations of the unconfined compression test were simplified by using two particletypes to reduce the number of parameters. The model employed consisted of a matrixmaterial and bands of particles that were weaker and less stiff (resembling biotite). Thebehavior of a model that contained three different particle types was modeled to assess the significance of the third component. Band geometry and material properties were the same as in the basic two-particle model. The matrix material was divided betweentwo different particle types. One of these particle types was the material already used asthe matrix component in the basic case, the new third particle type was assignedstrength and stiffness properties two times higher than the basic matrix particles. Thestructure of the third particle clusters was identified by performing the anisotropyinstallation procedure (Section 4.2.2) but with such a parameter values that no significant schistosity would occur in the model. The particle structure of this modelwith a schistosity angle of 45 degrees is shown in Figure 7-5.

Three compression tests were simulated using three different angles of schistosity, i.e.nine simulations. The simulated peak strength and Young’s modulus (see Figure 7-5),follow the results obtained with the two-particle model. In general, the strength andmodulus values obtained when simulating the three-particle model are higher than in thetwo-particle model as the additional element (the new third particle) is both stronger and stiffer. The three-particle model was not calibrated to match the laboratory resultsbecause the main item of interest was the shape of the anisotropy curve, but the resultsobtained would probably coincide with the results obtained using the two-particle modelif re-calibrated. It should be noted that the three-particle model does not radically alter the behavior of the compression simulation. According to these results, using a three-particle model would not produce results that are superior to those obtained using the two-particle model.

Peak strength and Young's modulus

0

20

40

60

80

100

120

140

160

180

0 22,5 45 67,5 90Angle

Str

en

gth

[MP

a]

/M

od

ulu

s[G

Pa]

Strength

Modulus

Figure 7-5. Left: Material structure in the three-particle model. Particles on the origo

centered section are shown. Yellow = old matrix, red = bands, white = new matrix.

Right: Mean simulated peak strength and Young’s modulus with standard deviations.


65

8 CONCLUSION AND DISCUSSION

The objective in this work was to study the effect of anisotropy in rock on strength and deformability by simulating a standard unconfined compression test with the PFC3D program. Anisotropy was modeled in 3D intrinsically by generating an anisotropic particle structure consisting of matrix particles and particles in oriented bands. The approach chosen is novel and no similar studies were found as references. The results obtained from the simulations were compared with corresponding laboratory test results. The comparison indicated that modeling using PFC3D can simulate the events which occur during laboratory compression tests of rock samples by reproducing similar fracture generation and deformation in both qualitative and quantitative terms.

Quantitative results for peak strength and Young’s modulus obtained from the simulations resembled laboratory results. The exact values were close enough to conclude that the responses are similar. Fracture patterns produced by PFC3D modeling closely resembled the fracturing seen in laboratory test samples. PFC3D failure patterns showed features typical of brittle failure. Tensile cracking was dominant and most of the early microcracks formed in a direction parallel to the maximum compression stress. Initially located in a random manner, the tensile cracks began to interact and additional microcracks formed along the macroshear fault plane. The structural development of cracks during compression in PFC3D models and their evolution over time both resembled the corresponding damage events observed in actual rock samples.

Successful simulation of the behavior of a simple geometry makes it possible to proceed to the modeling of geometries that are more complicated such as well-defined problems associated with failure around underground openings in anisotropic rock. It would however be beneficial to study some topics in greater detail to optimize the simulation process and improve the accuracy of the material model. One of the current problems in the method is visualization of the cracking process in 3D. Illustration of the spatial and temporal processes using the visualization tools that are currently available is very difficult and radical improvement of these tools is undoubtedly required. Simulation takes a considerable amout of time and is related to the number of particles in a model and the number of particles is related to the particle size in a model. The topic of interest which is closely related to optimization of the material model is the effect of particle size on the simulation results.

A shortage of current empirical data and lack of understanding of the details of the failure processes in a rock matrix are factors which hamper use of the PFC method.

In terms of accuracy, the detailed reproduction of failure processes obtained by employing PFC3D software goes well beyond earlier results obtained by modeling with other methods. The method reported here has great future potential in the analysis of well-defined and clearly-focused problems.


67

REFERENCES

Autio, J., Johansson, E. & Somervuori, P. 1999. Modelling of an in-situ failure test in the Research Tunnel at Olkiluoto. Working report 99-74. POSIVA OY. Helsinki, Finland.

Autio, J., Johansson, E., Kirkkomäki, T., Hakala, M. & Heikkilä, E. 2000. In-situ failure test in the Research tunnel at Olkiluoto. Posiva report 2000-05. POSIVA OY. Helsinki, Finland.

Bates, R. L. & Jackson, J.A. 1990. Glossary of geology. 3rd edition. American geological institute, Alexandria, USA.

Brady, B.H.G. & Brown, E.T. 1985. Rock mechanics for underground mining. George Allen & Unwin Ltd, London, Great Britain.

Cundall, P.A. & Starfield, A.M. 1988. Towards a methodology for rock mechanics modeling. International journal of rock mechanics and mining sciences &

geomechanics abstracts. 25(3). p. 99-106.

Department of Geological Sciences. 2001. Homepages for Department of Geological Sciences, University of Saskatchewan, Canada. June 26th 2001. http://www.engr.usask.ca/dept/geoe/

Hazzard, J.F., Young, R.P. & Maxwell, S.C. 2000. Micromechanical modeling of cracking and failure in brittle rocks. Journal of geophysical research. Vol 105. No. B7, p. 16 683-16 697.

Heo, J.S., Cho, H.K. & Lee, C.I. 2001. Measurement of acoustic emission and source location considering anisotropy of rock under triaxial compression. Rock mechanics – a challenge for society. Proceedings of the ISRM regional symposium Eurock 2001. A.A.Balkema. Netherlands.

Hoek, E. & Brown, E.T. 1982. Underground excavations in rock. The Institution of Mining and Metallurgy. London, England.

Hoek, E., Kaiser, P.K. & Bawden, W.F. 1995. Support of underground excavations in hard rock. A.A.Balkema. Netherlands.

ISRM. International society for rock mechanics commission on standardization of laboratory and field tests. 1979. Suggested methods for determining the uniaxial compressive strength and deformability of rock materials. International journal of rock

mechanics and mining sciences & geomechanics abstracts. 16. p. 135-140.

Itasca Consulting Group, Inc. 1999. PFC3D (Particle Flow Code in 3 Dimensions). Version 2.0. Minneapolis, USA.

Jaeger, C. 1972. Rock mechanics and engineering. Cambridge University Press. Great Britain.


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Li, L. & Holt, R.M. 2001. Simulation of granular material using particle model with non-circularly shaped super-particles. Rock mechanics – a challenge for society. Proceedings of the ISRM regional symposium Eurock 2001. Editors P. Särkkä & P. Eloranta. p. 511-516. Rotterdam: A. A. Balkema.

Kulatilake, P.H.S.W., Malama, B. & Wang, J. 2001. Physical and particle flow modeling of jointed rock block behavior under uniaxial loading. International journal of

rock mechanics and mining sciences & geomechanics abstracts. 38. p. 641-657.

Martin, D. 1994. Proceedings of TVO/SKB/AECL Workshop on rock strength. Work report TEKA-94-07. Teollisuuden voima Oy. Helsinki, Finland.

Obert, L. & Duvall, W. 1967. Rock mechanics and the design of structures in rock. John Wiley & Sons, Inc. USA.

Oreskes, N., Shrader-Frechette, K. & Belitz, K. 1994. Verification, validation, and confirmation of numerical models in the earth sciences. Science. Vol 263. p. 641-646

Peng, S. & Johnson, A. M. 1972. Crack growth and faulting in cylindrical specimens of Chelmsford granite. International journal of rock mechanics and mining sciences &

geomechanics abstracts. 9. p. 37-86. Pergamon Press. Great Britain.

Potyondy, D. & Autio, J. Bonded-particle simulations of the In-situ failure test at Olkiluoto. 2001. In Proceedings of the 38th U.S. Rock Mechanics Symposium - DC

Rocks 2001, Washington, D.C., July. 2001. Editors D. Elsworth, J.P. Tinucci & K.A. Heasley. p. 1553-1560. Rotterdam: A.A. Balkema.

Potyondy, D. & Cundall, P.A. 2000. Bonded-particle simulations of the in-situ failure test at Olkiluoto. Working report 2000-29. Posiva Oy. Helsinki, Finland.

Potyondy, D. & Cundall, P.A. 2001. The PFC model for rock: Predicting rock-mass damage at the underground research laboratory. Itasca Consulting Group, Inc. Atomic Energy of Canada Limited (AECL). Issued as Ontario Hydro Nuclear Waste Management Division Report No 06819-REP-01200-10061-R00. Toronto, Canada.

Press, F. & Siever, R. 1986. Earth, 4th edition. W. H. Freeman and Company. New York, USA.

Wong, T.-F. 1982. Micromechanics of faulting in Westerly granite. International

journal of rock mechanics and mining sciences & geomechanics abstracts. Vol. 19. p. 49-64. Pergamon Press. Great Britain.

Äikäs, K. & Sacklén, N. 1993. Fracture mapping in the research tunnel. Work report 93-01. TVO/Research tunnel. Helsinki, Finland.


69

APPENDIX 1: PFC MODEL FOR ROCK - PARAMETERS

The detailed descriptions of each parameter listed here (excluding Anisotropy

installation parameters) are found in the PFC3D manual FISH in PFC3D (Itasca 1999).

Specimen genesis and material parameters

Description FISH name ValueSpecimen-genesis control parameters Sample height [mm] et3_ylen 142Sample width [mm] et3_xlen 54Sample depth [mm] et3_zlen 54Minimum ball radius [mm] et3_rlo 2.5Ball size ratio, uniform distribution et3_radius_ratio 1.66Wall normal stiffness multiplier md_wEcfac 1.1Requested value of isotropic stress [MPa] tm_req_isostr -1.0Requested tolerance for isotropic stress tm_req_isostr_tol 0.5Stress reduction method 1 (value=0) or 2 (value=1) tm_req_isostr2 1Requested ratio of iso_max over iso_min tm_req_isostr_ratio 10.0Run stress reduction this many steps tm_req_steps 2000Min. number of contacts to be a non-floater flt_def 3Remaining floaters ratio flt_remain 0.0Particle-based material parameters Ball density [kg/m3] md_dens 2804.0Young's modulus at each particle-particle contact [Pa] md_Ec 118.0e9Ratio of particle normal to shear stiffness md_knoverks 2.5Ball friction coefficient md_fric 0.5Parallel-bond propertiesBoolean: if=1, generate parallel-bonds md_add_pbonds 1Parallel-bond radius multiplier pb_radmult 1.0Young's modulus of each parallel bond [Pa] pb_Ec 118.0e9Ratio of parallel-bond normal to shear stiffness pb_knoverks 2.5Mean value of parallel-bond normal strength [Pa] pb_sn_mean 208.0e6Std. deviation of normal strength [Pa] pb_sn_sdev 33.0e6Mean value of parallel-bond shear strength [Pa] pb_ss_mean 208.0e6Std. deviation of shear strength [Pa] pb_ss_sdev 33.0e6


70

Anisotropy installation parameters

Description FISH name ValueGeometry Seed for random-number generator an2_random ...Schistosity angle an2_stheta 0...90Joint spacing multiplier an2_smult 6Joint-segment thickness multiplier an2_stmult 2Joint disk radius an2_rmult 4Joint area ratio an2_aratio 0.85Material properties Number of contacts of balls; condition for [an2_efac] an2_plimit 2Ball stiffness reduction factor an2_efac 0.05Parallel bond modulus reduction factor an2_ebarfac 0.05Parallel bond strength reduction factor an2_sigbarfac 0.2

Unconfined compression test parameters

Description FISH name ValueLateral x-wall stiffness-reduction factor (triaxial-test) et3_knxfac N/ALateral y-wall stiffness-reduction factor et3_knyfac 1.000Lateral z-wall stiffness-reduction factor (triaxial-test) et3_knzfac N/ATarget confining xx-stress [Pa] (triaxial-test) et3_wsxx_req N/A Target vertical stress [MPa] et3_wsyy_req 0.1Target confining xx-stress [Pa] (triaxial-test) et3_wszz_req N/A Wall-servo tolerance et3_ws_tol 0.10Final platen velocity [m/s] p_vel 0.05Total cycles of platen acceleration p_cyc 400Accelerate platens in this many stages p_stages 10Test-termination criterion et3_peakfac 0.80Crack-initiation stress fraction pk_ci_fac 0.01


71

Monitored parameters during testing

Description FISH name History ID # Microcracking total number of cracks crk_num 1total number of contact bond failures in normal mode crk_num_cnf 2total number of contact bond failures in shear mode crk_num_csf 3total number of parallel bond failures in normal mode

crk_num_pnf 4

total number of parallel bond failures in shear mode crk_num_psf 5Wall- and specimen-derived strains wall-based strain in x-direction et3_wexx 10... et3_weyy 11... et3_wezz 12wall-based volumetric strain et3_wevol 16specimen-based strain in x-direction et3_sexx 210... et3_seyy 211... et3_sezz 212.. et3_sevol 216Wall-derived stresses stress in x-direction et3_wsxx 13... et3_wsyy 14... et3_wszz 15mean stress et3_wsm 17deviatoric stress et3_wsd 18Average stresses & strains from 3 measurement circles strain in x-direction et3_mexx 110... et3_meyy 111... et3_mezz 112volumatric strain et3_mevol 116stress in x-direction et3_msxx 113... et3_msyy 114... et3_mszz 115mean stress et3_msm 117deviatoric stress et3_msd 118Energy quantities work done by all walls energy boundary 30total strain energy stored in the parallel bonds energy bond 31total energy dissipated by frictional sliding at contacts

energy frictional 32

total kinetic energy; translational and rotational energy kinetic 33total strain energy stored at contacts energy strain 34increment of strain energy et3_e_delstrain 35

REFERENCES

Itasca Consulting Group, Inc. 1999. PFC3D (Particle Flow Code in 3 Dimensions). Version 2.0. Minneapolis, USA.


72

APPENDIX 2: CRACK PLOTS AND STRESS-STRAIN CURVES

In the pages 73-81 there are shown crack plots from all the simulations. The crack coloring is time-dependent. All cracks are shown in a stage that corresponds the end of the test. Viewing direction is along the x-axis. So that the plots could fit the pages they have been rotated 90o anticlockwise.

Pages 82-84 show some stress-strain and crack number curves. From each schistosity angle point one representative curve is shown. The plots are in sequence from 0 degrees to 90 degrees. The sample name and schistosity angle can be seen from THE JOB TITLE -line of the corresponding plot.

For example: Job Title: sT1_mG00a_tA00, where the number after the notation mG, 00

in this example, depicts the corresponding schistosity angle. (ex. 225=22.5o, 675 =

67.5o, etc.)


73

Specimens sT1_mG00a....e, schistosity angle 0 degrees, all cracks, time-dependent coloring

A

B

C

D

E


74

Specimens sT1_mG11a...e, schistosity angle 11 degrees, all cracks, time-dependent coloring

A

B

C

D

E


75

Specimens sT1_mG225a...e, schistosity angle 22.5 degrees, all cracks, time-dependent coloring

A

B

C

D

E


76


A

B

C

D

E


77


A

B

C

D

E


78


A

B

C

D

E


79

Specimens sT1_mG675a...e, schistosity angle 67.5 degrees, all cracks, time-dependent coloring

A

B

C

D

E


80


A

B

C

D

E


81


A

B

C

D

E


82


83


84


LIST OF REPORTS 1 (2)

POSIVA REPORTS 2002, situation 6/2002

POSIVA 2002-01 Copper corrosion under expected conditions in a deep geologic

repository

Fraser King, Integrity Corrosion Consulting Ltd, Calgary, Canada

Lasse Ahonen, Geological Survey of Finland

Claes Taxén, Swedish Corrosion Institute, Stockholm, Sweden

Ulla Vuorinen, VTT Chemical Technology

Lars Werme, Svensk Kärnbränslehantering Ab, Stockholm, Sweden

January 2002

ISBN 951-652-108-8

POSIVA 2002-02 Estimation of rock movements due to future earthquakes at four

candidate sites for a spent fuel repository in Finland

Paul La Pointe

Golder Associates Inc., Washington, USA

Jan Hermanson

Golder Associates AB, Sweden

February 2002

ISBN 951-652-109-6

POSIVA 2002-03 Fracture calcites at Olkiluoto – Evidence from Quaternary Infills for

Palaeohydrogeology

Seppo Gehör, Kivitieto Oy

Juha Karhu, University of Helsinki

Aulis Kärki, Kivitieto Oy

Jari Löfman, VTT Processes

Petteri Pitkänen, VTT Building and Transport

Paula Ruotsalainen, TUKES

Olavi Taikina-aho, Kivitieto Oy

February 2002

ISBN 951-652-110-X

POSIVA 2002-04 Structure and geological evolution of the bedrock of southern

Satakunta, SW Finland

Seppo Paulamäki, Markku Paananen, Seppo Elo

Geological Survey of Finland

February 2002

ISBN 951-652-111-8

LIST OF REPORTS 2 (2)

POSIVA 2002-05 Rock strength and deformation dependence on schistosity –


Toivo Wanne

Saanio & Riekkola Oy

June 2002

ISBN 951-652-112-6

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