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Modified 12/06/2007

TWO DIMENSIONAL FRACTURE PROPAGATION CODE

(VERSION 2.3)

USER’S MANUAL

FRACOM [email protected]

FRACOD2D A F RAC OM Product

FRACOD V2.3 User’s Manual – Modified 12/06/2007

SUMMARY

FRACOD2D is a two-dimensional boundary element code which was developed to simulate fracture initiation and propagation in an elastic and isotropic rock medium. The current version of the code is fully Window based and user friendly. The code can simulate up to 10-15 non-symmetrical, and randomly distributed fractures.

FRACOD2D Version 2.3 can simulate the following problems:

(1) Complex fracture propagations in jointed rock mass;(2) Multiple region problems;(3) Time-dependent problems;(4) Gravitational problems

This manual provides: (a) the basic theoretical background of the FRACOD code; and (b) a detailed instruction on how to use the code. A number of simple examples are provided at the end of the manual to demonstrate the applicability of the code. FRACOD2D code was developed based on a Ph.D. research (Shen, 1993). Further work on the code was conducted during 1998-2007, supported by SKB, Fracom Ltd, Tekes, JAEA and Hazama Corporation.

i


TABLE OF CONTENTS

ABSTRACT i

1 INTRODUCTION 1

2 THEORETICAL BACKGROUND 2

2.1 DISPLACEMENT DISCONTINUITY METHOD (DDM) 22.1.1 DISPLACEMENT DISCONTINUITY METHOD IN AN INFINITE SOLID 22.1.2 NUMERICAL PROCEDURE 42.2 SIMULATION OF ROCK DISCONTINUITIES 72.3 FRACTURE PROPAGATION CRITERION 92.4 DETERMINATION OF FRACTURE PROPAGATION USING

DDM 112.5 FRACTURE INITIATION CRITERION 133 MULTIPLE REGION PROBLEMS 18

3.1 INTRODUCTION 183.2 THEORETICAL FORMULATION FOR MULTI-REGION

FUNCTION 193.3 CODE IMPLEMENTATION 254 ITERATION PROCESS 26

4.1 ITERATION FOR JOINT SLIDING 264.2 ITERATION FOR FRACTURE PROPAGATION 285 TIME-DEPENDENT MODELLING 32

5.1 THEORETICAL BACKGROUND 325.1.1 SUBCRITICAL FRACTURE MODEL FOR A MODE I FRACTURE UNDER PURE TENSION 325.1.2 SUBCRITICAL FRACTURE MODEL FOR SHEAR AND COMPRESSION 355.2 CODE IMPLEMENTATION 366 GRAVITATIONAL PROBLEMS 38

6.1 THEORETICAL BACKGROUND 386.2 CODE IMPLEMENTATION 407 FRACOD COMMAND LIST 44

8 CONDUCT AND MONITOR THE CALCU-LATION 58

9 FRACOD VERIFICATION TESTS 68

9.1 SINGLE FRACTURE SUBJECTED TO NORMAL TENSILE STRESS 68

9.2 SINGLE FRACTURE SUBJECTED TO PURE SHEAR STRESS 719.3 MULTIPLE REGION MODEL 749.4 SUBCRITICAL CRACK GROWTH - CREEP 779.5 GRAVITY PROBLEMS 83

ii


REFERENCES 99

Appendix I – How to use the preprocessor to set up models 104

iii

FRACOD User’s Manual

1 INTRODUCTION

Fracture propagation code (FRACOD) is a two-dimensional computer code that was designed to simulate fracture initiation and propagation in elastic and isotropic rock mediums. The code employs the Boundary Element Method (BEM) principles and a newly proposed fracture propagation criterion for detecting the possibility and the path of a fracture propagation, Shen and Stephansson (1993).

The current version of the FRACOD code provides the basic functions needed for studying rock fracture propagation in a rock mass subjected to far-field stresses. The code is created for running on PCs with a MS Windows platform. It provides an easy-to-use user’s interface that enables users to monitor and interrupt the calculation. It also provides an independent pre-processor to help users in preparing the input file for a given problem.

The capacity of the current version of the FRACOD code is limited to about 10-15 fractures, depending upon the complexity of the fracture system and the excavation. As a general estimate, a fracture system with 10 non-symmetrical fractures will requires about 24 hours of calculation on a PC/3GHz to get a reasonably accurate prediction of fracture propagation.

This user’s manual provides some basic theoretical background of the code in Chapters 2 - 6, and a detailed instruction on how to use the code in Chapters 7- 8. Chapter 9 provides several verification tests cases. Appendix I describes a pre-processor of the FRACOD code. For those who may be only interested in knowing how to use the code rather than the theory, it is recommended to ignore Chapters 2 - 6 and start reading from Chapter 7.

1


2 THEORETICAL BACKGROUND

The FRACOD code is based on the Boundary Element Method principals. It utilises the Displacement Discontinuity Method (DDM), one of the three commonly used boundary element methods. In the FRACOD code, a newly proposed fracture criterion, the modified G-criterion (Shen and Stephansson, 1993), is incorporated into the numerical method for simulating fracture propagation. This section describes in detail the numerical method DDM as well as the modified G-criterion.

2.1 DISPLACEMENT DISCONTINUITY METHOD (DDM) A crack or fracture has two surfaces or boundaries, one effectively coinciding with the other. Conventional boundary element methods, such as the Direct Integration Method, therefore become inefficient in simulating this problem. The Displacement Discontinuity Method (DDM) was developed by Crouch (1976) to cope with problems of this type. The DDM is based on the analytical solution to the problem of a constant discontinuity in displacement over a finite line segment in the x, y plane of an infinite and elastic solid. Physically, one may imagine a displacement discontinuity as a line crack whose opposing surfaces have been displaced relative to one another (see Figure 2-1.)

2.1.1 Displacement Discontinuity Method in an infinite solid

The problem of a constant displacement discontinuity over a finite line segment in the x, y plane of an infinite elastic solid is specified by the condition that the displacements be continuous everywhere except over the line segment in question. The line segment may be chosen to occupy a certain portion of the x-axis, say the portion |x|a, y=0. If we consider this segment to be a line crack, we can distinguish its two surfaces by saying that one surfaces is on the positive side of y=0, denoted y=0+ , and the other is on the negative side, denoted y=0- . In crossing from one side of the line segment to the other, the displacement undergoes a constant specified change in value Di = (Dx, Dy).

We will define the displacement discontinuity Di as the difference in displacement between the two sides of the segment as follows:

2


)0,()0,()0,()0,(

xuxuDxuxuD

yyy

xxx 2-1

Because ux and uy are positive in positive x and y co-ordinate direction, it follows that the Dx and Dy are positive as illustrated in Figure 2-1.

Figure 2-1. Constant displacement discontinuity components Dx and Dy.

The solution of the subject problem is given by Crouch (1976) and Crouch and Starfield (1983). The displacement and stresses can be written as:

yyyyxyxxy

xyxyxxyxx

yffDyffDu

yffDyffDu

,,,,

,,,,

)1(2)21(

)21()1(2

2-2

and

xyyyyyyyyxxy

yyyyyyxyyxyy

yyyyyyxyyxyxxx

yfGDyffGD

yffGDyfGD

yffGDyffGD

,,,

,,,

,,,,

22

22

222

2-3

where f,x represent the derivative of function f(x,y) against x, similarly as for f,y, f,xy, f,xxy etc. Function f(x,y) in these equations is given by:

3

+Dx

+Dy

2a

x

y


2222 )(ln)()(ln)(

)arctan(arctan)1(4

1),(

yaxaxyaxax

axy

axyyyxf

2-42.1.2 Numerical procedure

For a crack of any shape, such as curved, we assume it can be represented with sufficient accuracy by N straight segments, joined end by end. The positions of the segments are specified with reference to the x, y co-ordinate system shown in Figure 2-2. If the surface of the crack are subjected to stress (for example, a uniform fluid pressure – p), they will displace relative to one another. The displacement discontinuity method is a means of finding a discrete approximation to the smooth distribution of relative displacement that exits in reality. The discrete approximation is found with reference to the N subdivisions of the crack depicted in Figure 2-2a. Each of the subdivisions is a boundary element and represents an elemental displacement discontinuity.

Figure 2-2. Representation of a crack by N elemental displacement discontinuities.

The elemental displacement discontinuities are defined with respect to the local co-ordinates s and n indicated in Figure 2-2. Figure 2-2b depicts a single elemental displacement discontinuity at jth segment of the crack. The components of discontinuity in the s and n directions at this segment are

donated as s

j

D and n

j

D . These quantities are defined as follows:

4

1 2 3

i

j

N

s

n

1 2 3

n=0-

N

s

nn=0+

x

y

(x,y)

(a) (b)


j

n

j

n

j

n

j

s

j

s

j

s

uuD

uuD

2-5

In these definitions, s

j

u and n

j

u refer to the shear (s) and normal (n)

displacement of the jth segment of the crack. The superscripts ‘+’ and ‘-‘ denote the positive and negative surfaces of the crack with respect to local co-ordinate n.

The local displacements s

j

u and n

j

u form the two components of a vector.

They are positive in the positive direction of s and n, irrespective of whether we are considering the positive or negative surface of the crack. As a consequence, it follows from Equation 2-5 that the normal component of

displacement discontinuity n

j

D is positive if the two surfaces of the crack

displace toward one another. Similarly, the shear component s

j

D is positive

if the positive surface of the crack moves to the left with respect to the negative surface.

The effects of a single elemental displacement discontinuity on the displacements and stresses at an arbitrary point in the infinite solid can be computed from the results for section 2.1.1, provided we suitably transform the equations to account for the position and orientation of the line segment in question. In particular, the shear and normal stresses at the midpoint of the ith element in Figure 2-2b can be expressed in terms of the displacement discontinuity components at the jth element as follows:

j

n

ij

nn

j

s

ij

ns

i

n

j

n

ij

sn

j

s

ij

ss

i

s

DADA

DADA

i=1 to N 2-6

where ss

ij

A ,etc., are the boundary influence coefficients for the stresses. The

coefficient ns

ij

A , for example, gives the normal stress at the midpoint of the

ith element (i.e. n

j

) due to a constant unit shear displacement discontinuity

over the jth element (i.e. s

j

D =1).

5


Returning now to the crack problem depicted in Figure 2-2b, we place an elemental displacement discontinuity at each of the N segments along the crack and write, from Equation 2-6,

N

j

j

n

ij

nn

N

j

j

s

ij

ns

i

n

N

j

j

n

ij

sn

N

j

j

s

ij

ss

i

s

DADA

DADA

11

11

i=1 to N 2-7

If we specify the values of the stress s

j

and n

j

for each element of the

crack, then Equation 2-7 is a system of 2N simultaneous linear equations in 2N unknowns, namely the elemental displacement discontinuity components

s

j

D and n

j

D . We can find the displacements and stresses at designated

points in the body by using the principle of superposition. In particular, the displacements along the crack of Figure 2-2a are given by expressions of the form

N

j

jn

ijnn

N

j

js

ijns

in

N

j

jn

ijsn

N

j

js

ijss

is

DBDBu

DBDBu

11

11 i=1 to N 2-8

where ss

ij

B , etc., are the boundary influence coefficients for the

displacements. The displacements are discontinuous when passing from one side of the jth element to the other, so we must distinguish between these two sides when computing the influence coefficients in Equation 2-8. The diagonal terms of the influence coefficients in these equations have the values

);0(21);0(

21

0

nnBB

BBij

nn

ij

ss

ij

ns

ij

sn2-9

The remaining coefficients (i.e. the ones for which ij) are continuous and they can be obtained by using Equations 2-1, 2-2 and 2-3 in Section 2.1.1.

Displacements s

iu and n

iu in Equation 2-8 will exhibit constant

discontinuities s

iD and n

iD , as required.

6


2.2 SIMULATION OF ROCK DISCONTINUITIES

For a rock discontinuity (crack, joint, etc.) in an infinite elastic rock mass, the system of governing equations 2-7 can be written as

N

j

i

n

j

n

ij

nn

N

j

j

s

ij

ns

i

n

i

s

N

j

j

n

ij

sn

N

j

j

s

ij

ss

i

s

DADA

DADA

10

1

011

)(

)(

i=1 to N 2-10

where i

s and i

n represent the shear and normal stresses of the ith element

respectively; 00 )(,)(

i

n

i

s are the far-field stresses transformed in the crack

shear and normal directions. ij

ssA , ... , ij

nnA are the influence coefficients, and j

n

j

s DD , represent displacement discontinuities of jth element which are

unknowns in the system of equations.

A rock discontinuity has three states: open, in elastic contact or sliding. The system of governing equations 2-10, developed for an open crack, can be easily extended to the case for cracks in contact and sliding. For different crack states, their system of governing equations can be rewritten in the

following ways, depending on the shear and normal stresses (i

s and i

n )

of the crack.

For an open crack i

s = i

n = 0, therefore the system of governing

equations 2-10 can be rewritten as:

N

j

i

n

j

n

ij

nn

N

j

j

s

ij

ns

i

n

i

s

N

j

j

n

ij

sn

N

j

j

s

ij

ss

i

s

DADA

DADA

10

1

011

)(0

)(0

i=1 to N 2-11

When the two crack surfaces are in elastic contact, the magnitude of i

s and i

n will depend on the crack stiffness (Ks, Kn) and the displacement

discontinuities (j

n

j

s DD , )

7


i

nn

i

n

i

ss

i

s

DK

DK

2-12

where Ks and Kn are the crack shear and normal stiffness, respectively. Substituting Equation 2-12 into Equation 2-10 and carrying out the simple mathematical manipulation, the system of governing equations then becomes:

N

j

i

nn

i

n

j

n

ij

nn

N

j

j

s

ij

ns

i

ss

i

s

N

j

j

n

ij

sn

N

j

j

s

ij

ss

DKDADA

DKDADA

10

1

011

)(0

)(0

i=1 to N 2-13

For a crack with its surfaces sliding i

nn

i

n DK

tantani

nn

i

n

i

s DK 2-14

where is the friction angle of the crack surfaces. The sign of i

s depends

on the sliding direction. Consequently, the system of equations 2-10 can be presented as:

N

j

i

nn

i

n

j

n

ij

nn

N

j

j

s

ij

ns

i

nn

i

s

N

j

j

n

ij

sn

N

j

j

s

ij

ss

DKDADA

DKDADA

10

1

011

)(0

tan)(0

i=1 to N 2-15

The displacement discontinuities (j

n

j

s DD , ) of the crack are obtained by

solving the system of governing equations using conventional numerical techniques, e.g. Gauss elimination method. If the crack is open the stresses

(i

s ,i

n ) on the crack surfaces are zero, otherwise if the crack is in contact

or sliding, they can be calculated by Equations 2-12 or 2-14.

The state of each crack (joint) element can be determined using the Mohr-Coulomb failure criterion:

(1) open joint: n > 0

(2) elastic joint: n < 0, |s| < c + |n|tan

8


(3) sliding joint: n < 0, |s| ³ c + |n|tan

where a compressive stress is taken to be negative and c is cohesion. If the joint has experienced sliding, c = 0.

Many joints have dilatancy during shear movement. As a result, the joint tends to open during shearing if there is no restriction in joint normal displacement. With confinement in normal direction, however, the tendency of open movement will be absorbed by the normal stiffness of the joint, leading to a high normal stress but very little change in normal displacement.

When the dilation angle of a joint (d) is considered, the additional normal stress caused by the dilation is calculated by

For dilasive joint, Equation (2-15) becomes

A joint with higher dilation angle is more difficult to shear because any shear movement will be transformed into an increase in joint normal stress and hence high friction resistance.

2.3 FRACTURE PROPAGATION CRITERION

In modelling fracture propagation in rock masses where both tensile and shear failure are common, a fracture criterion for predicting both mode I and mode II fracture propagation is needed. The exiting fracture criteria in the macro-approach can be classified into two groups: the principal stress (strain)-based criteria and the energy-based criteria. The first group consists of the Maximum Principal Stress Criterion and the Maximum Principal Strain Criterion; the second group includes the Maximum Strain Energy Release Rate Criterion (G-criterion) and the Minimum Strain Energy Density Criterion (S-criterion). The principal stress (strain)-based criteria are only applicable to the mode I fracture propagation which relies on the principal tensile stress (strain). To be applied for the mode II propagation, a fracture criterion has to consider not only the principal stress (strain) but

9


also the shear stress (strain). From this point of view, the energy based criteria seem to be applicable for both mode I and II propagation because the strain energy in the vicinity of a fracture tip is related to all the components of stress and strain.

Both the G-criterion and the S-criterion have been examined for application to the mode I and mode II propagation (Shen and Stephansson, 1993), and neither of them is directly suitable. In a study by Shen and Stephansson (1993) the original G-criterion has been improved and extended. The original G-criterion states that when the strain energy release rate in the direction of the maximum G-value reaches the critical value Gc, the fracture tip will propagate in that direction. It does not distinguish between mode I and mode II fracture toughness of energy (GIc and GIIc). In fact, for themost

of the engineering materials, the mode II fracture toughness is much higher than the mode I toughness due to the differences in the failure mechanism. In rocks, for instance, GIIc is found in laboratory scale to be at least two orders of magnitude higher than GIc (Li, 1991). Applied to the mixed mode I and mode II fracture propagation, the G-criterion is difficult to use since the critical value Gc must be carefully chosen between GIc and GIIc.

A modified G-criterion, namely the F-criterion, was proposed (Shen and Stephansson, 1993). Using the F-criterion the resultant strain energy release rate (G) at a fracture tip is divided into two parts, one due to mode I deformation (GI) and one due to mode II deformation (GII). Then the sum of their normalized values is used to determine the failure load and its direction. GI and GII can be expressed as follows (Figure 2-3): if a fracture grows an unit length in an arbitrary direction and the new fracture opens without any surface shear dislocation, the strain energy loss in the surrounding body due to the fracture growth is GI. Similarly, if the new fracture has only a surface shear dislocation, the strain energy loss is GII. The principles of the F-criterion can be stated as follows:

G = GI+ GII

Originalsurface

New Growthsurface

(a) (b) (c)Figure 2-3. Definition of GI and GII for fracture growth. (a) G, the growth has both open and shear displacement; (b) GI, the growth has only open displacement; (c) GII, the growth has only shear displacement.

10


(1). In an arbitrary direction () at a fracture tip there exists a F-value, which is calculated by

F GG

GG

I

Ic

II

IIc

( ) ( ) ( ) 2-16

(2). The possible direction of propagation of the fracture tip is the direction (=0) for which the F-value reaches its maximum.

F ( ) max. 0

2-17

(3). When the maximum F-value reaches 1.0, the fracture tip will propagate, i.e.

0.1)(0F 2-18

The F-criterion is actually a more general form of the G-criterion and it allows us to consider mode I and mode II propagation simultaneously. In most cases, the F-value reaches its peak either in the direction of maximum tension (GIc = maximum while GIIc=0) or in the direction of maximum shearing (GIIc = maximum while GIc=0). This means that a fracture propagation of a finite length (the length of an element, for instance) is either pure mode I or pure mode II. However, the fracture growth may socialite between mode I and mode II during an ongoing process of propagation, and hence form a path which exhibits the mixed mode failure in general.

2.4 DETERMINATION OF FRACTURE PROPAGATION USING DDM

The key step in using the F-criterion is to determine the strain energy release rate of mode I (GI) and mode II (GII) at a given fracture tip. As GI and GII

are only the special cases of G, the problem is then how to use DDM to calculate the strain energy release rate G.

The G-value, by definition, is the change of the strain energy in a linear elastic body when the crack has grown one unit of length. Therefore, to obtain the G-value the strain energy must first be estimated.

By definition, the strain energy, W, in a linear elastic body is

11


W = òòòv12 ij ij dV . 2-19

where ij and ij are the stress and strain tensors, and V is the volume of the body. The strain energy can also be calculated from the stresses and displacements along its boundary

W = 12òs

dsuu nnss )( 2-20

where s,n, us, un are the stresses and displacements in tangential and normal direction along the boundary of the elastic body. Applying Equation 2-20 to the crack system in an infinite body with far-field stresses in the shear and normal direction of the crack, (s)0 and (n)0, the strain energy, W, in the infinite elastic body is

W = 12ò

a

0( ( ) ) ( ( ) ) s s s n n nD D da 0 0 2-21

where a is the crack length, Ds is the shear displacement discontinuity and Dn is the normal displacement discontinuity of the crack. When DDM is

used to calculate the stresses and displacement discontinuities of the crack,

the strain energy can also be written in terms of the element length (ai) and the stresses and displacement discontinuities of the ith element of the crack.

)( )()( 00 )()(21 i

n

i

n

i

n

ii

si

i

s

i

s

iDaDaW 2-22

The G-value can be estimated by

G Wa

W a a W aa

( )( ) ( )

2-23

where W(a) is the strain energy governed by the original crack while W(a+a) is the strain energy governed by both the original crack, a, and its small extension, a (Figure 2-4). In Figure 2-4, a 'fictitious' element is introduced to the tip of the original crack with the length a in the direction . Both W(a) and W(a+a) can be determined easily by directly using DDM and Equation 2-23.

12


a

aCrack

Figure 2-4. Fictitious crack increment a in direction with respect to the initial crack orientation.

In the above calculation, if we restrict numerically the shear displacement of the “fictitious” element to zero, the result obtained using Equation 2-23 will be GI(). Similarly, if we restrict the normal displacement of the “fictitious” element to zero, the result obtained will be GII(). After obtaining both GI() and GII(), the F-value in Equation 2-16 can be calculated using the given fracture toughness values GIc and GIIc of a given rock type.

2.5 FRACTURE INITIATION CRITERION

In addition to the propagation of existing fractures, new fractures (cracks) may initiate at the boundaries or in the intact rock. This section describes the criteria used to detect fracture initiation.

Fracture initiation in intact rock

Fracture initiation is a complicated process. It often starts from microcrack formation. The microcracks coalesce and finally form macro-fractures. Because the FRACOD code is designed to simulate the fracturing process in macro-scale only, we ignore the process of microcrack formation. Rather, we will only focus on when and whether a macro-fracture will form at a given location with a given stress state.

The FRACOD code considers the intact rock as a flawless and homogeneous medium. Therefore, any fracture initiation from such a medium represents a localised failure of the intact rock. The localised failure can be predicted by an existing failure criterion, e.g. Mohr-Coulomb criterion. Other criteria widely used in rock mechanics and rock engineering can also be used, such as Hoek-Brown criterion etc.

A rock failure can be caused by tension or shear. Hence, a fracture initiation can be formed due to tension or shear. For tensile fracture initiation, the

13


tensile failure criterion is used in FRACOD, i.e. when the tensile stress at a given point of the intact rock exceeds the tensile strength of the intact rock, a new rock fracture will be generated in the direction perpendicular to the tensile stress (Figure 2-5)

Critical stress of fracture initiation in tension:

tensile ³ t

Direction of fracture initiation in tension:

it = (tensile)+/2

where tensile is the principal tensile stress at a given point, t is the tensile strength of the intact rock, it is the direction of the fracture initiation in tension, and (tensile) is the direction of the tensile stress.

The length of the newly generated fracture is determined by the spacing of the grid points used in the intact rock. In the current FRACOD version, it is equal to the grid point spacing in the initiation direction. The less the grid point spacing, the shorter the new fracture. However, the closer the grid points, the less different the stresses at the adjacent grid points, and hence the more likely a fracture initiation occurs in the adjacent grid points simultaneously. The newly formed short fractures link with each other to form a longer fracture. This mechanism reduces the sensitivity of the modelling results to the grid point spacing.

Figure 2-5. Fracture initiation in tension or shear in intact rock.

For a shear fracture initiation, the Mohr-Coulomb failure criterion is used in FRACOD, i.e. when the shear stress at a given point of the intact rock

14

Grid point

Tensile stressNew fractureShear stress

New fracture


exceeds the shear strength of the intact rock, a new rock fracture will be generated (Figure 2-5)

Critical stress of fracture initiation in shear:

shear ³ ntan()+c

Direction of fracture initiation in shear:

is = /2+/4

where tensile is the shear stress in the direction of is, n is the normal stress to the shear failure plane, is the internal friction angle of intact rock, c is the cohesion, and is is the direction of potential shear failure, which is measured from the direction of the minimum principal stress.

Because there are always two symmetric shear failure planes at any given point, two fractures are added in the model whenever a shear failure is detected. Often one of the two fractures will propagate predominately in later simulation of fracture propagation.

The length of the shear fracture initiation depends upon the spacing of the grid points, as discussed above for the tensile fracture initiation.

Fracture initiation at boundaries

Fracture initiation at a boundary is not as a straight forward task as that in intact rock. Because the boundary may be a straight boundary, a curved boundary, or a boundary with sharp corners, significantly stress concentration may occur at the boundary. Recent study by Shen and Rinne (2001) has highlighted the complexity of the fracture initiation at boundaries. The initiation criteria suggested by Shen and Rinne (2001) may be suitable for the cases studied but not universally for all cases. There is no simple and yet theoretically sound methods for the prediction of fracture initiation from boundaries.

To enable the simulation of fracture propagation at boundary using FRACOD, an alternative approach is taken. Instead of directly predicting the fracture initiation from a boundary, we examine the fracture initiation from the intact rock very close to the boundary, using the intact rock failure criteria as discussed before. Once an intact rock failure is detected, a fracture initiation is predicted to occur in the intact rock close to the

15


boundary. FRACOD then detects whether the newly formed fracture will link to the boundary by using the fracture propagation functions. This treatment fully utilises the advantage of the fracture propagation functions built in the code and overcomes the lack of effective methods in handling fracture initiation from the boundary.

New grid points are arranged in the intact rock along the boundary (Figure 2-6). They are set to be at a distance of one element away from the boundary since the constant DDM method does not give accurate results very close to the element. The grid points are effectively treated to be the same as other grid points in the intact rock, and the same procedure is used to detect any possible fracture initiation from these grid points. If a fracture initiation is predicted from any of the grid points close to the boundaries, a new fracture is created at the grid point in the direction of failure. The length of the fracture is a half of the length of the nearest boundary element. The code then detects whether the fracture will propagate to the boundary. If yes, the fracture will link to the boundary and effectively form a fracture initiation from the boundary.

Figure 2.6. Modelling process of a fracture initiation from boundary.

An existing fracture is treated to be the same as a boundary. The same procedure is used to detect if any fracture initiation will occur close to the surface of a fracture. In case of a fracture, grid points will be added to both sides of the fracture surface since both sides are solid rock. The fracture initiation process does not apply to the tips of an existing fracture. At a fracture tip, stress singularity occurs and any intact rock failure criterion is no longer valid. The fracture propagation modelling procedure as described in Sections 2.1-2.4 is then used.

16

Boundary

Grid point Fracture initiation

Fracture propagation


17


3 MULTIPLE REGION PROBLEMS

3.1 INTRODUCTION

Rock mass may have different properties in different regions. A multi-region problem is shown in Figure 3-1, where three different regions (concrete lining, Excavation Disturbed Zone, and in situ rock mass) need to be considered.

Figure 3-1. A shaft with concrete lining and EDZ in a fractured rock mass.

To be applicable to the above case, FRACOD needs to be further developed to simulate the multiple regions with different material properties. Because FRACOD is a boundary element code based on the mathematical solutions in an elastic, homogeneous, isotropic medium, it is not a trivial task to extend FRACOD to handle the multi-region problems. New approaches have to be taken. This chapter presents the mathematical formulations and their implementation in FRACOD for multi-region problems.

18

Rock mass

EDZConcrete

Shaft


3.1 THEORETICAL FORMULATION FOR MULTI-REGION FUNCTION

The DD method discussed in Section 2.1 is for a homogenous rock. The basic solutions in Equations (1)-(4) are based on this assumption. If the rock mass is inhomogeneous such as the case shown in Figure 3-1 where different regions with different rock properties exist, the basic solutions are no longer valid. Naturally, one may think that the basic solutions can be extended for the multi-region problem. The difficulties faced with this approach is that the mathematical solutions for all geometric cases of an unspecified number of regions is very difficult, if not impossible, to find. Other more realistic approaches are then needed.

A simple way to model a multi-region problem is to separate the problem into several individual regions, each being a homogeneous region with the same rock properties (Figure 3-2). For each homogeneous region, the basic solutions discussed in Section 2.1 apply, and systematic equations can be set up for each region to solve for stress and displacement at the internal point and on boundary. The interfaces between two regions now become boundaries in both regions. The boundary stress/displacement values at the interface boundaries, however, need to meet certain conditions to ensure the continuity of two regions at the interface. This approach is being adapted in FRACOD for the multi-region problems, and details are discussed below.

Figure 3-2. Treatment of multi-regions in FRACOD by modelling the two regions separately.

19

(E, , c, KIc etc.) 1

= +

Interface

(E, , c, KIc etc.)2

(E, , c, KIc etc.)1

(E, , c, KIc etc.)2


To make this approach and formulation easy to understand for readers, we consider a very simple problem as shown in Figure 3-3. The problem has two triangular regions with two different properties. The two regions are joined at a straight interface.

(a) (b)

Figure 3-3. A simple problem with two different regions.

To model this problem, we now separate the two regions and describe each region using three DD elements. The elements used here are:

Region 1: DD elements No. 1, 2, 5Region 2: DD elements No. 3, 4, 6 Note that element No. 5 and 6 are both representing the interface but in different region. We call them the “twin” interface elements.

For the problem with 6 DD elements, the systematic equations described by Equation (7) can be written below in full.

1616616515515414414313313212212111111snsnsssnsnsssnsnsssnsnsssnsnsssnsnsss bDADADADADADADADADADADADA 1616616515515414414313313212212111111nnnnsnsnnnsnsnnnsnsnnnsnsnnnsnsnnnsns bDADADADADADADADADADADADA 2626626525525424424323323222222121121snsnsssnsnsssnsnsssnsnsssnsnsssnsnsss bDADADADADADADADADADADADA 2626626525525424424323323222222121121nnnnsnsnnnsnsnnnsnsnnnsnsnnnsnsnnnsns bDADADADADADADADADADADADA 3636636535535434434333333232232131131snsnsssnsnsssnsnsssnsnsssnsnsssnsnsss bDADADADADADADADADADADADA 3636636535535434434333333232232131131nnnnsnsnnnsnsnnnsnsnnnsnsnnnsnsnnnsns bDADADADADADADADADADADADA 4646646545545444444343343242242141141snsnsssnsnsssnsnsssnsnsssnsnsssnsnsss bDADADADADADADADADADADADA 4646646545545444444343343242242141141nnnnsnsnnnsnsnnnsnsnnnsnsnnnsnsnnnsns bDADADADADADADADADADADADA 5656656555555454454353353252252151151snsnsssnsnsssnsnsssnsnsssnsnsssnsnsss bDADADADADADADADADADADADA 5656656555555454454353353252252151151nnnnsnsnnnsnsnnnsnsnnnsnsnnnsnsnnnsns bDADADADADADADADADADADADA

20

12

3 4

56

Region 2

Region 1


6666666565565464464363363262262161161snsnsssnsnsssnsnsssnsnsssnsnsssnsnsss bDADADADADADADADADADADADA 6666666565565464464363363262262161161nnnnsnsnnnsnsnnnsnsnnnsnsnnnsnsnnnsns bDADADADADADADADADADADADA

…………….(10)where 12

snA is the influence coefficient, representing the resultant shear stress at the center point of element 1 due to a unit normal displacement discontinuity of element 2. 1

sb is the boundary value (stress or displacement) at element 1. Because elements (1, 2, 5) and elements (3, 4, 6) are in separated regions, there will be no cross influence between them except the “twin” interface elements. Hence, the influence coefficients e.g. 14

ssA , 62ssA etc are zero.

Equation (10) is then simplified as below:1515515212212111111snsnsssnsnsssnsnsss bDADADADADADA

1515515212212111111nnnnsnsnnnsnsnnnsns bDADADADADADA 2525525222222121121snsnsssnsnsssnsnsss bDADADADADADA 2525525222222121121nnnnsnsnnnsnsnnnsns bDADADADADADA 3636636434434333333snsnsssnsnsssnsnsss bDADADADADADA 3636636434434333333nnnnsnsnnnsnsnnnsns bDADADADADADA (11)4646646444444343343snsnsssnsnsssnsnsss bDADADADADADA 4646646444444343343nnnnsnsnnnsnsnnnsns bDADADADADADA 5555555252252151151snsnsssnsnsssnsnsss bDADADADADADA 5555555252252151151nnnnsnsnnnsnsnnnsns bDADADADADADA

6666666464464363363snsnsssnsnsssnsnsss bDADADADADADA 6666666464464363363nnnnsnsnnnsnsnnnsns bDADADADADADA

The boundary values 41,..., ns bb of elements 1-4 are known since they are the

real boundaries. The boundary values 65 ,..., ns bb of the interface elements 5 and 6 are unknown. Hence in Equation (11) there are 16 unknowns (12 for element DD values and 4 for interface values), and it cannot be solved by the available 12 equations. We need to construct more equations using the interface continuity conditions.

Let’s consider the stress condition at the interface elements 5 and 6. If we assume the stresses at the interface elements 5 and 6 are 5

s , 5n , 6

s and 6n

, the last four equations in Equation (11) can be rewritten as follows:

5555555252252151151snsnsssnsnsssnsnsss DADADADADADA 5555555252252151151nnnnsnsnnnsnsnnnsns DADADADADADA (12)

21


6666666464464363363snsnsssnsnsssnsnsss DADADADADADA 6666666464464363363nnnnsnsnnnsnsnnnsns DADADADADADA

If the interface is bonded, the shear and the normal stresses at the two sides of the interface should be the same. Hence we have the following stress relations:

65

65

nn

ss

(13)

Using Equation (13) to simplify Equation (12) and after simple re-arrangement we got the following equation:

0666666555555464464363363252252151151 nsnsssnsnsssnsnsssnsnsssnsnsssnsnsss DADADADADADADADADADADADA

0666666555555464464363363252252151151 nnnsnsnnnsnsnnnsnsnnnsnsnnnsnsnnnsns DADADADADADADADADADADADA

(14)

Similar to the above process, if we consider the displacements of the interface elements 5 and 6 ( 5

sd , 5nd , 6

sd and 6nd ), we can get the following

equations for the displacement boundary conditions.

5555555252252151151snsnsssnsnsssnsnsss dDBDBDBDBDBDB 5555555252252151151nnnnsnsnnnsnsnnnsns dDBDBDBDBDBDB (15)

6666666464464363363snsnsssnsnsssnsnsss dDBDBDBDBDBDB 6666666464464363363nnnnsnsnnnsnsnnnsns dDBDBDBDBDBDB

where 51ssB etc. are the influence coefficient for displacement and 6

nd etc. are the displacements of the interface elements.

If the interface is perfectly bonded, the shear and the normal displacement displacement at the two sides of the interface should be the same, i.e.

65

65

nn

ss

dd

dd

(16)

Using the above displacement relations in Equation (15), we obtain the following equations:

22


0666666555555464464363363252252151151 nsnsssnsnsssnsnsssnsnsssnsnsssnsnsss DBDBDBDBDBDBDBDBDBDBDBDB

0666666555555464464363363252252151151 nnnsnsnnnsnsnnnsnsnnnsnsnnnsnsnnnsns DBDBDBDBDBDBDBDBDBDBDBDB

(17)

Using Equations (15) and (17) to replace the last four equations in Equation (11), we got the following complete systematic equations for the multi-region problem shown in Figure 3-3.

1515515212212111111snsnsssnsnsssnsnsss bDADADADADADA

1515515212212111111nnnnsnsnnnsnsnnnsns bDADADADADADA 2525525222222121121snsnsssnsnsssnsnsss bDADADADADADA 2525525222222121121nnnnsnsnnnsnsnnnsns bDADADADADADA 3636636434434333333snsnsssnsnsssnsnsss bDADADADADADA (18)3636636434434333333nnnnsnsnnnsnsnnnsns bDADADADADADA 4646646444444343343snsnsssnsnsssnsnsss bDADADADADADA 4646646444444343343nnnnsnsnnnsnsnnnsns bDADADADADADA

0666666555555464464363363252252151151 nsnsssnsnsssnsnsssnsnsssnsnsssnsnsss DADADADADADADADADADADADA

0666666555555464464363363252252151151 nnnsnsnnnsnsnnnsnsnnnsnsnnnsnsnnnsns DADADADADADADADADADADADA

0666666555555464464363363252252151151 nsnsssnsnsssnsnsssnsnsssnsnsssnsnsss DBDBDBDBDBDBDBDBDBDBDBDB

0666666555555464464363363252252151151 nnnsnsnnnsnsnnnsnsnnnsnsnnnsnsnnnsns DBDBDBDBDBDBDBDBDBDBDBDB

In Equation (18), there are 12 unknowns and 12 equations. So the problem becomes deterministic and solvable. Equation (18) can be rewritten in the form of matrix below:

0000

000000000000000000000000

000000000000000000000000

4

4

3

3

2

2

12

1

6

6

5

5

4

4

3

3

2

2

12

1

666655556464636352525151

666655556464636352525151

666655556464636352525151

666655556464636352525151

464644444343

464644444343

363634343333

363634343333

252522222121

252522222121

151512121111

151512121111

n

s

n

s

n

s

s

n

s

n

s

n

s

n

s

n

s

s

nnnsnnnsnnnsBnnnsnnnsnnns

snsssnsssnsssnsssnsssnss

nnnsnnnsnnnsnnnsnnnsnnns


nnnsnnnsnnns

snsssnsssnss

nnnsnnnsnnns

snsssnsssnss

nnnsnnnsnnns

snsssnsssnss

nnnsnnnsnnns

snsssnsssnss

bbbbbbbb

DDDDDDDDDDDD

BBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAA


(19)

23


Equation (19) is the final matrix for the multi-region problem shown in Figure 3-3. All components in the influence coefficient matrix at the left hand side are non-zero components. The matrix Equation (19) therefore can be solved using simple Gauss Elimination method.

After knowing the displacement discontinuities of all boundary and interface elements, the stress and displacement at any internal point can be calculated. Note that because the two regions are considered to be separate, for an internal point in, say, region 1, only the contributions from elements 1, 2 and 5 is used. Elements 3, 4 and 6 of region 6 will not have contribution to the stress and displacement of the internal point in Region 1.

When using the above formulations to consider concrete lining in an infinite rock mass, caution is needed in considering the in situ stresses in the rock mass. Because the concrete lining is normally not pre-stressed, there will be no in situ stress components in the boundary element of concrete lining. For the problem shown in Figure 3-3, if we assume Region 1 is the concrete lining and Region 2 is the rock mass with in situ stresses, the final matrix equation for this case will be the following:

00

)()(

)()()()(

000000000000000000000000

000000000000000000000000

06

06

044

044

033

033

2

2

12

1

6

6

5

5

4

4

3

3

2

2

12

1

666655556464636352525151

666655556464636352525151

666655556464636352525151

666655556464636352525151

464644444343

464644444343

363634343333

363634343333

252522222121

252522222121

151512121111

151512121111

n

s

nn

ss

nn

ss

n

s

s

n

s

n

s

n

s

n

s

n

s

s

nnnsnnnsnnnsBnnnsnnnsnnns


nnnsnnnsnnnsnnnsnnnsnnns


nnnsnnnsnnns

snsssnsssnss

nnnsnnnsnnns

snsssnsssnss

nnnsnnnsnnns

snsssnsssnss

nnnsnnnsnnns

snsssnsssnss

bb

bbbbbbbb

bbbb

DDDDDDDDDDDD

BBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAA



(20)where 03

s etc. is the in situ stress components in Region 2.

The above discussion is based on a simple problem with 6 elements and two regions. The same principles and formulations apply to a more complicated problem with many regions and elements. They are implemented in FRACOD in the next Section for any generalised cases related to the multi-region problems.

24


3.2 CODE IMPLEMENTATION

Section 3.2 provides the basic formulations for the multi region problems using the DD method. When these formulations are implemented in FRACOD, the code structure needs to be radically changed to accommodate mainly the interface elements. The detailed work for coding the multi-region FRACOD is not the main focus of this report, although it has been the most timing consuming part of the project. This chapter mainly outlines the new functions in FRACOD and provides guidelines for users to construct the input data file to solve any specific multi-region problem.

The following new functions are added in FRACOD during the code implementation stage:

Interface element option (previously only boundary element and fracture element are available);

Different mechanical properties for different regions; Concrete lining option (no in situ stresses); New graphic option for multi-region code.

With the new functions, FRACOD can now practically model up to 10 regions with different mechanical properties. Although the number of regions is theoretically unlimited in the code, we noticed that the calculation speed reduces significantly with the number of regions modelled. This is because each interface between regions requires at least two DD elements, one at each side of the interface. This results in twice as many element as for a normal boundary or fracture.

25


4 ITERATION PROCESS

Boundary element method (including DDM) is an implicit numerical method. This means that the numerical calculation will only provide the final solution at the given stress or displacement boundary conditions, ignoring the process that reaches the final solution. For elastic problems, the implicit method is the most efficient and straight forward way to get the final solution because of the linear stress-strain relation. For plasticity problem caused by joint sliding and fracture propagation, however, the implicit method could give false results because the process to reach the final solution may not be linear and the final solution will depend on the path of loading.

Iteration process is an effective method to consider the path dependent problem.

4.1 ITERATION FOR JOINT SLIDING

Figure 4-4. Iteration process to simulate complex loading path.

Let’s consider a joint element simulated by FRACOD (). The joint element is initially loaded in shear up to the maximum shear strength (s)max, then slides at the same shear stress to a specified maximum displacement (Ds)max, then is unloaded. Let’s also assume that the loading process is displacement controlled.

To model this complex process in FRACOD, we can subdivide the total maximum shear displacement into many small increment such as (Ds)i and

26

Ds

s

(Ds)i

(s)i

(Ds)j(s)j=0

(Ds)max

(s)max


(Ds)j. The correspondent increment in shear stress (s)i and (s)j can be calculated using the different equations depending upon the state of the joint element. If the joint element is still elastic such as at increment i, the shear stress increment is:

iss

is DK )()(

If the joint element is sliding such as at increment j, the shear stress increment is:

0)( is

The state of the joint element is determined by the total shear stress which is a sum of the individual stress increment during the previous loading path. For instance

in

ns

is

,1

)()(

jn

ns

js

,1

)()(

At the ith increment, (s)i < (s)max, therefore the joint element is elastic. At jth increment, (s)j = (s)max, hence the joint element is sliding.

In actual modelling, the joint element is assumed to be elastic initially in the first increment. When the resultant total shear stress is higher than the shear strength at any given increment cycle, the joint element is identified to be sliding. In the next increment cycle, the incremental joint shear stress will be recalculated using the sliding joint conditions.

For a complex joint system modelled by FRACOD, the following steps are used:

(1) Divided the final boundary stresses and/or displacement into n small equal increments. Only the incremental boundary values are used in the subsequent calculations.

(2) Calculate the incremental shear and normal stresses for all joint elements using the incremental boundary values. If this is the first increment, all joint elements are assumed to be elastic. Otherwise, the joint states are those determined from the previous increment. In this step, the normal numerical process such as setting up and solving system of matrix as described in Chapter 2 is used.

(3) Calculate the total element shear and normal stresses at each joint element by accumulating their incremental values from the previous increments.

(4) Determine if the resultant total shear stress exceeds the shear strength for each joint element. If so, the joint element will be considered sliding,

27


and sliding condition will be used for this joint element in the next increment.

(5) Go to the next increment and repeat steps (2)-(4) using the last determined joint states.

The steps (2)-(5) are repeated until the designed boundary values are reached. The incremental shear and normal displacement discontinuities of each joint element and boundary element are recorded and accumulated in each increment cycle. Their final values will be the solution of problem using the iteration method. After knowing the displacement discontinuities, the stresses and displacement at any internal point of a rock mass can be calculated.

4.2 ITERATION FOR FRACTURE PROPAGATION

The above iteration process cannot be directly applied to the cases with fracture propagation. During the process of detecting the possibility and the direction of the potential propagation using the F-criterion, a fictitious crack element is added to the candidate crack tip in different directions to simulate the possible crack growth. For each possible fracture propagation direction, a complete iteration process from the beginning of loading is required to obtain the necessary stress/displacement values of the fracture elements and boundary elements to determine the F-value. This will be extremely time consuming and practically impossible. In addition, the above treatment implies that the fictitious element existed at the beginning of the loading which is theoretically incorrect.

An alternative approach is developed to simulate the fracture propagation using iteration process and is described below.

Let’s consider a single crack tip in a finite body under external stress . The crack has grown by one element length in a given direction see Figure 4-5. The problem can be decomposed into two stages as shown in Figure 4-5: Stage 1: The existing crack and its growth element are subject to external stress . The growth element is applied with a high stress -1 so that the displacement discontinuities at the element are zero. Here 1 should be equal to the stress at the element centre calculated by considering the pre-existing crack only. This stage is equivalent to the case that the growth element does not exist.

28


Stage 2: The existing crack and its growth element are free of external stress. Only growth element is subject to internal stress 1.

In this treatment, the total resultant stress at the growth element is the sum of -1 (Stage 1) and 1 (Stage 2), i.e. zero. This is expected for mode I fracture growth.

For mode II fracture growth Figure 4-6, the surfaces of the growth element are in contact, therefore no “bonding” stress is required at Stage 1. At Stage 2, additional shear stress is applied to the growth element to composite the difference between the total resultant shear stress at Stage 1 and the shear strength.

In the two cases shown in Figure 4-5 and Figure 4-6, the crack geometry of the real problem is kept the same in the decomposed stages, and only the stresses are decomposed. This is essential to use the decomposition theory.

In both cases, Stage 1 is equivalent to the case without crack growth. It hence can be modelled by the normal iteration method described in the previous section. When a crack growth occurs, only one additional iteration step is needed to model Stage 2. This can be done by adding the growth element to the existing fracture system and applying the specified stresses to this element.

The detailed process of modelling in FRACOD is outlined blow:

Step 1: Use the iteration process to solve for the existing fracture system without fracture growth. Record the stresses and displacement discontinuities of the joint elements. Calculate the stresses at the centre of the potential growth element near the crack tip for use in the next step;Step 2: Add a growth element to the crack tip at a given direction. Apply the stresses determined from Step 1 to the growth element and solve for the new fracture system with growth element. Record the resultant stresses and displacement discontinuities of the joint elements.Step 3: Obtain the total stresses and displacement discontinuities of the joint elements by adding those from Steps (1) and (2). Calculate the F-value using the final stresses and displacement discontinuities.Step 4: Repeat Steps (2) and (3) using the growth element at a different direction. After all the desired directions are calculated, find the maximum F-value and its direction. If F-value is greater than 1.0, a real fracture growth is determined. Otherwise, the growth element is disregarded.

29


Figure 4-5. Decomposition of crack growth problem for modelling using iteration (mode I crack growth).

30

crackCrack growth

||

+

crack

crack

D

D0

D

-1

1

Stage 1

Stage 2


Figure 4-6. Decomposition of crack growth problem for modelling using iteration (mode II crack growth).

31

crack

||

+

crack

D

D

Stage 2

crack

D0

1

Stage 1

1

-(1-1tan)

1=1tan


5 TIME-DEPENDENT MODELLING

5.1 THEORETICAL BACKGROUND

Classical fracture mechanics postulates that a fracture tip which has a stress intensity equal to the material’s critical fracture toughness, IcK , will accelerate to speeds approaching the elastic wave speed in a medium (Irwin, 1958). In cases of long term loading, however, fractures can grow at stress intensities significantly below the critical values. This process is termed subcritical fracture growth and propagation velocities can vary over many orders of magnitude as a function of stress intensity.

Here the subcritical crack growth is suggested to be modelled by considering the crack length as a function of time.

5.1.1 Subcritical fracture model for a mode I fracture under pure tension

We start by considering a crack under tensional loading. When the fracture in elastic and isotropic medium is under a uniaxial far-field tension ( ) as shown in Figure 1 (at right), the stress y in front of the crack tip ( 0 ) is given by:

r

K Iy

2 (5-1)

Stress y varies with the distance r from the crack tip and it becomes infinity at the fracture tip.

Figure 5-7. An infinite plate containing a crack under biaxial loading.

32


The stress intensity factor IK determines the stress singularity at the fracture tip and its magnitude depends on the far-field stress ( ) and the crack length (crack half length = a ).

aK I (5-2)

In classical fracture mechanics the fracture initiation from the fracture tip takes place when

,IcI KK (5-3)

IcK is the mode I fracture toughness, a material constant, that can be defined by laboratory testing.

In sub-critical crack growth theory the slow crack extending takes place when

IcI KK , (5-4)

The approach presented here to model the subcritical crack growth consists of a mathematical relation between crack growth rate and the stress intensity. A variety of mathematical functions will be fitted to the laboratory data. We start with a power law relation. Charles (1958), have stated that most experimental data can be fit with an expression for subcritical velocity of the form:

11

nAKv (5-5)

where 1v is the crack velocity, A is a constant, K is the stress intensity factor and n1, the stress corrosion or crack propagation factor. The subscript “1” indicates mode I subcritical growth.

Assuming that the maximum propagation velocity (vmax)1, occurs when

IcI KK , the constant of proportionality, A, can be expressed as

nIcK

vA 1max )( (5-6)

and the expression for propagation velocity becomes

33


1

1max1 )(n

Ic

I

KK

vv

(5-7)

(Expression used by Olson (1993) and Kemeny (2002)).

Now we replace the stress intensity factor with Eq. (5-2) and we get:

1

1max1 )(n

IcKavv

(5-8)

If the stress intensity factor IK is kept constant at the fracture tip, the velocity is constant and the time dependent crack length can be easily calculated by multiplying the velocity with time.

However, in static loading conditions, as the crack extends the stress intensity will increase and it leads to an accelerating crack velocity. The constant crack length in Eq. (5-8) must be replaced by an effective crack length:

òt

eff dttvata0

10 )()( (5-9)

where 0a is the initial crack length.

Adding this into Eq. (5-8 ), we get a momentary crack velocity at time (t):

1

))(()()( 0

0

1max1

n

Ic

t

K

dttvavtv

ò

(5-10)

Because the effective crack length depends on the velocity and the velocity depends on the crack length, an iterative process is needed to calculate the crack length:

(5-11)

34


5.1.2 Subcritical fracture model for shear and compression

In the previous section the simplest case was presented; crack under pure mode I loading and the fracture extension is in the direction of the crack tip in mode I. In practise, shear or compressive or mixed mode loading is more common.

Figure 5-8. An infinite plate containing a crack under in-plane shear. i=remote in-plane shear stress.

For pure mode II loading the fracture stress intensity factor is

aK II (5-12)

It can be noted that it have a similar shape as mode I, except on the subject of shear stress (τ) instead of normal stress (σ). However, the stress conditions are much more complicated in compression and shear than under tension due to friction effects.

The classical stress criterion does not take into account the friction effect in front of the fracture tip. In FRACOD the friction on the existing fracture surface is considered by DDM. The F-criterion, based on stress energy release uses a fictitious element to model the tip part of a growing fracture and its friction is also included in the energy change.

Even though the formulation (5-5) for fracture velocity is mainly used in mode I problems, its use for mode II problems is discussed for example in Kemeny (1993). Most likely the constants A and n differs strongly for mode II loading conditions. Laboratory results in compression and shear may suggest completely different mathematical relation for the crack velocity. It is also argued that because cracking is not restricted to a single major crack

35


in compression, the term crack velocity is not appropriate (Lajtai, 1986). Anyway, as a first attempt, the subcritical crack extension for a mode II fracture will be handled here in the same manner as presented for a tensile fracture.

2

2max2 )(n

IIc

II

KK

vv

(5-13)

where 2v is the crack velocity for mode II creep, 2max )(v is the maximum mode II crack velocity, n2 is a constant, IIK is the mode II stress intensity factor and IIcK is the mode fracture toughness.


The subcritical crack growth discussed above is implemented in FRACOD using the iteration process. A time step t is used in the iteration. The following calculation steps are performed in FRACOD:

Step 1: Calculate KI and KII at any given crack tip for the given loading condition and fracture configuration. Determine the subcritical crack velocity v1 and v2 for the moment of t0.

Step 2: Calculate the length of subcritical crack growth for a time step ttvl 1 ; or tvl 2

Step 3: If the length of growth is equal to or greater than an element length, a new tip element is added to the pre-existing tip. Otherwise, the length is temporarily is stored in memory and accumulated in the next time step until it reaches one element length.

Step 4: Repeat Steps 1-4 using the new time moment t0+Nt until the specified time is reached. N is the cycle number.

In the current version of FRACOD, the time step is automatically updated based on the KI/KIC or KII/KIIC value to minimise the iteration cycles needed to reach the specified time. When the above ratio is low (K I/KIC or KII/KIIC

<<1.0), the speed of subcritical crack growth is low, hence the time step is set to be greater. When the above ratio is high close to 1.0, the speed of subcritical crack growth is high, hence the time step is set to be smaller.

36


6 GRAVITATIONAL PROBLEMS

6.1 THEORETICAL BACKGROUND

Many practical rock engineering problems involve the gravitational stresses. Rock slope stability and shallow tunnel stability are two examples where the gravity stresses cannot simply be ignored or simplified as the far-field in-situ stresses. In such cases, the uneven gravitational stresses at different depths of the rock mass have to be considered explicitly.

Modelling gravitational stresses in boundary element (BE) methods is not as straight forward as in the finite element (FE) method where the mass and weight of the rock are distributed into each element. Because in BE method the elements are only located at the boundaries, they are not able to directly represent the gravity force inside the rock body. To effectively represent the gravity force in the rock, we will need to: (1) take into account of the uneven gravity stresses at the centre of all boundary elements; (2) superposition the gravity stresses at any point inside the rock.

Discussed below is the procedure that the gravity stresses are modelled in FRACOD.

Let’s consider a simple case where an underground cavern is located in a shallow ground, see Figure 6-9. The boundary of the cavern is discretised into n elements. The centre of each element is at different depth, say, di for the ith element.

For this problem, the system of governing equations can be written as

NtoiDADA

DADA

g

i

nn

j

nn

ijN

js

j

ns

ijN

jn

i

g

i

sn

j

sn

ijN

js

j

ss

ijN

js

i

1

)(

)(

11

11

6-1

where g

i

s )( and g

i

n )( represent the initial gravitational stresses in shear

and normal directions of the ith element before the excavation was made.

37


Figure 6-9. A shallow underground excavation to demonstrate the gravity effect.

g

i

s )( and g

i

n )( can be calculated from the stresses in x- and y-direction

g

i

xx )( and g

i

yy )( and the element orientation angle.

The gravitational stresses in the x-y coordinates are

ig

i

xx

ig

i

yy

gdk

gd

)(

)(

6-2

where - rock density;g – gravity acceleration;k – ratio of horizontal stress to vertical stress.

38

i

ji

1i

Ground surface

di

d1

dj

p (xp,yp)

x

y


After solving the system of equations 6-1, the displacement discontinuities Ds and Dn of each element are known. To calculate the stresses at a given point p inside the rock, the following equations can be applied:

Ntoi

DADA

DADA

DADA

g

p

yyn

j

yyn

pjN

js

j

yys

pjN

jyy

p

n

j

xyn

pjN

js

j

xys

pjN

jxy

p

g

p

xxn

j

xxn

pjN

js

j

xxs

pjN

jxx

p

1

)(

)(

11

11

11

6-3

where xxs

pj

A etc. are the influence coefficients of a unit discontinuity on the

stresses at point p.


Theoretically speaking, implementing the gravity stresses in FRACOD should be a simple process. We only need to replace the far-field stress

00 )(,)(i

n

i

s with the gravitational stresses g

i

s )( and g

i

n )( in the code, and

ensure that the gravitational stresses are depth dependent.

In practice, however, implementing the gravitational stresses in FRACOD is not as straight forward as first thought. The real issue come from the exterior problem such as an excavation in a rock mass, as shown in Figure6-9 where the unbalanced gravitational force on the excavation boundaries will cause numerical problems.

Figure 6-10 demonstrates such a case where erroneous results are resulted from the numerical problems. During calculation, the code not only calculate the exterior1 region (the rock mass), but also the interior region (the fictitious rock block inside the tunnel) although the latter has never been used in our analysis. This is an inherent feature in the DD method and cannot be avoided. Both the exterior region and interior region are applied to the same stresses at their common boundary. Because the unbalanced force on the interior block will lead to a large rigid displacement and

1 Exterior problem means that the primary concern is the rock mass in the external region of an enclosed boundary region, for instance, an excavation in an infinite body. Interior problem is the opposite, i.e. a rock disk with finite size and volume

39


consequently very large displacement discontinuities Ds and Dn at the boundary elements, significant numerical errors are resulted in.

-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0X Axis (m)

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Y A

xis

(m)

-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0X Axis (m)

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Y A

xis

(m)

Test

-3.4

-3.2

-3.0

-2.8

-2.6

-2.4

-2.2

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0

Prin

cipa

l Maj

or S

tres

s (

Pa)

xE

5

Pxx (Pa): 0E+0 Pyy (Pa): 0E+0

Pxy (Pa): 0E+0

Max. Compres. Stress (Pa): 3.63214E+5

Max. Tensile Stress (Pa): 5.49543E+4

Elastic fracture

Open fracture

Slipping fracture

Fracture with W ater

Compressive stress

Tensile stress

Fracom Ltd

Date: 05/06/2007 09:17:56

Figure 6-10. Erroneous modelling results due to unbalanced stresses at the boundaries when gravity is considered.

To overcome the rigid movement problem, Crouch and Starfield (1983) suggested place minimum two boundary elements in the interior region with zero displacement in two different directions. This method has been trialled by the authors. However, the effects were not as good as expected.

The above issue was finally solved by using the following method. The elements at the boundaries of an internal excavation are considered to be “constrained”2 elements where the displacement discontinuities Ds and Dn of these elements will cause shear and normal stresses, i.e.

i

nn

i

n

i

ss

i

s

DK

DK

6-4

2 “Constrained” element means that the element does not have free shear and normal movement even if the stresses on the element are zero.

40


where Ks and Kn are the shear and normal stiffness of the “constrained” elements. If there is a small rigid movement of the interior block, stresses will be developed at the boundaries which resist the rigid movement.

It needs to be emphasized, however, that the joint stiffness used has to be relatively small compared with the stiffness of the rock mass. Otherwise, high additional boundary stresses may be resulted in, and the modelling results will not be accurate.

Based on many trials, it was found that the optimal stiffness values of the “constrained” elements are:

Ks = Kn = E/1×104

where E is the Young’s modulus of the host rock.

When the above method is used for the same problem as shown in Figure 6-9, the modelling results are much more stable and accurate, see Figure 6-11.

-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0X Axis (m)

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Y A

xis

(m)

-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0X Axis (m)

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Y A

xis

(m)

Test

-2.2

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0

Prin

cipa

l Maj

or S

tres

s (

Pa)

xE

5


Pxy (Pa): 0E+0



Elastic fracture

Open fracture

Slipping fracture


Compressive stress

Tensile stress

Fracom Ltd

Date: 05/06/2007 10:04:58

Figure 6-11. Correct modelling results after using “constrained” elements at excavation boundaries for gravity problems.

41


A verification test again the analytical solution is given in Example #1 Section 9.5. It was found the use of “constrained” element is effective and produces accurate results.

Note that the “constrained” elements are only needed for exterior problems (i.e. excavation in a rock mass) when the gravity is considered. For interior problems (i.e. finite body), the rigid movement are controlled by using displacement boundaries, and hence no additional measures needs to taken.

The boundaries representing ground surface (or slope surface) may also be simulated using “constrained” elements. The advantage of using “constrained” element in this case is that the numerical solution will be more stable and no “flying” blocks will be generated. It should be noted however that the “constrained” elements often produce a small normal and shear stresses on the free surface. To define “constrained” elements, one needs to use “KODE=11” in the EDGE, ARCH or ELLI command, see Section 7.

It is user’s choice whether to use “constrained” elements or the normal element. Based on our experience, we recommend that users always use “constrained” elements on the excavation boundaries if gravity is considered. In other situation, user may use normal elements.

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7 FRACOD COMMAND LIST

The FRACOD code reads the input data from a data file previously prepared with specified formats. Therefore, the user needs to construct the input file (e.g. input.dat) before running the code. This section gives a detailed instruction on how to prepare the input file for the FRACOD code. The following is an example data file which defines a borehole with cracks in the borehole wall under uniaxial compression.

------ example data file ------------------------------------------TITLEA borehole with four cracks loaded in uniaxial compression SYMMETRY -- Model symmetry 4 0.00 0.00MODULUS -- Poisson Ratio and Youngs modulus, material no. 0.25 0.40E+11 1TOUGHNESS -- Gic and Giic, material no. 50. 1000. 1PROPERIES -- mat,kn,ks,phi,coh,dilation,aperture0,aperture_r 1 0.10E+13 0.10E+13 30.0 10.0e6 0.00 10e-6 1e-6SWINDOW -- xll,xur,yll,yur,numx,numy -3.00 3.00 -3.00 3.00 30 30STRESSES -- sxx,syy,sxy 0.00E+07 -0.15E+08 0.00E+00FRACTURE -- nume,xbeg,ybeg,xend,yend,kode,jmat, mat 5 0.700 0.700 1.000 1.000 2 1 1ARCH -- nume,xcen,ycen,diam,ang1,ang2,kode,ss,sn, mat10 0.0 0.0 2.0 0.0 90.0 1 0.00 0.00CYCL 1000ENDFILESTOP

---------- end of the example data file -------------------------------------

The input data are defined by a command line, such as TITLE. The command line will, if needed, be followed by a line which defines the values. Only the first four characters of a command (e.g. TITL) are to be read by the code and hence meaningful. However, it is always desirable to write the whole word (e.g. TITLE) to help in understanding the function of this command.

All commands can be written in pure capital characters or pure small characters, or their mixture, such as “STOP”, “stop”, or “Stop”. Unacceptable commands cause no action in the code (no warning or error messages will be given).

43


The commands used by the FRACOD code are listed below. Note that the units used for the input are given in brackets.

AINT defines an arch which is the interface between two material regions

num,xc,yc,diameter,ang1,ang2, mat_neg, mat_pos

num -- number of elements on arch boundary(xc,yc) -- coordinates of arch centre (m)diam -- arch diameter (m)ang1 -- beginning angle of the arch (clockwise) (degree)ang2 – end angle of the arch (clockwise) (degree)mat_neg – material no. of the negative side of the interfacemat_pos – material no. of the positive side of the interface

Example:AINTERFACE-num,xc,yc,diameter,ang1,ang2,mat_neg,mat_pos

20 0.0 0.0 1.2 0 90 2 1

ARCH defines an arch or a tunnel/borehole

num, xcen, ycen, diam, ang1, ang2, kode, bvs, bvn, mat, gradsy, gradny

num -- number of elements on arch boundary(xcen,ycen) -- coordinates of arch centre (m)diam -- arch diameter (m)ang1 -- beginning angle of the arch (clockwise) (degree)ang2 – end angle of the arch (clockwise) (degree)kode -- type of boundary condition

= 1, shear and normal stress boundary = 2, shear and normal displacement boundary = 3, shear displacement and normal stress boundary

= 4, shear stress and normal displacement boundary= 11, shear and normal stress boundary with “constrained” elements for

exterior gravitational problems

bvs -- boundary value in shear direction (stress or displacement) (Pa or m)bvn -- boundary value in normal direction (stress or displacement) (Pa or m)mat – material no. (or region) that the arch belongs togradsy – gradient for bvs in y-directiongradny – gradient for bvn in y-direction

Warning: For an excavation opening, the arch angle starts from low to high (e.g. 0–180). For a solid disc the arch angle starts from high to low (e.g. 180– 0).

Example: arch - num,xcen,ycen,diam,ang1,ang2,kode,bvs,bvn,mat 20 0 0 1.0 0 90 1 0 -0e6 1

BOUN defines that fracture initiation at boundaries is allowed.

44


Example: Boundary fracture initiation

CREE defines creep parameters

vmax1,n1,vmax2,n2,totalT,deltaT_min,deltaT_max

vmax1 – maximum mode I creep velocity (m/s)n1 – exponential factor of creep velocity for mode Ivmax2 – maximum mode II creep velocity (m/s)n2 – exponential factor of creep velocity for mode IItotalT – total creep time to be simulated (s)deltaT_min – minimum time step deltaT_max – maximum time step The time step is automatically calculated and will be in the above range

Subcritical crack velocity: n

Ic

I

KK

vv

max

Example: CREEP - vmax1,n1,vmax2,n2,totalT,deltaT_min,deltaT_max 500, 30, 500, 30, 1000000, 1, 1000

Creep results are stored in “creep_results.dat”.

CYCL cnum starts calculation

cnum – number of cycle to be performed (one cycle often produces one step of crack growth for each unstable crack tip). If cnum is not given, the default cycle number is 1000.

Example: cycl 1000

DARC gives an increment of boundary stresses along an arch boundary

xcen, ycen, diam1, diam2, ang1, ang2, dss, dnn

xcen, ycen – coordinates of arch centre (m)diam1, diam2 – lower and higher range of the diameter (m)ang1, ang2 – beginning and finishing angle of the arch (degrees) dss – increment in boundary shear stress (Pa)dnn – increment in boundary normal stress (Pa)

Note: 1. ang1 and ang2 have to be in (-180, 180) 2. dss and dnn are displacements or stresses as defined originally by EDGE or ARCH

Example: DARCh xcen,ycen,diam1,diam2,ang1,ang2,dss,dnn 0 0 1.9 2.1 0 45 0 -20e6

45


DBOU gives an increment of boundary stresses along a straight boundary

x1, x2, y1, y2, dss, dnn

x1, x2 – range in x-direction (m)y1, y2 – range in y-direction (m)dss – increment in boundary shear stress (Pa)dnn – increment in boundary normal stress (Pa)

Note: dss and dnn are displacements or stresses as defined originally by EDGE or ARCH

Example: DBOUndary x1,x2,y1,y2,dss,dnn 0 0.10 -0.001 0.001 0 -3.21E+07

DSTR gives an increment of far-field stresses in the rock mass

dsxx, dsyy, dsxy

dsxx – increment of far-field horizontal stress (Pa)dsyy – increment of far-field vertical stress (Pa)dsxy – increment of far-field shear stress (Pa)

Example: DSTRESSES -- dsxx,dsyy,dsxy 0e6 0 0

EDGE defines a straight boundary line

num, xbeg, ybeg, xend, yend, kode, bvs, bvn, mat, gradsy, gradny

num -- number of elements along the edge(xbeg,ybeg) -- co-ordinates of the beginning point of the edge (m)(xend,yend) -- co-ordinates of the end point of the edge (m)kode -- type of boundary condition

= 1, shear and normal stress boundary = 2, shear and normal displacement boundary = 3, shear displacement and normal stress boundary= 4, shear stress and normal displacement boundary= 11, shear and normal stress boundary with “constrained” elements for

exterior gravitational problems

bvs -- boundary value in shear direction (stress or displacement) (Pa or m)bvn -- boundary value in normal direction (stress or displacement) (Pa or m)mat – material no. (or region) that the edge belongs togradsy – gradient for bvs in y-directiongradny – gradient for bvn in y-direction

46


Warning: The beginning point and the end point need to be defined in a sequence that the positive side of the edge is always the excavation. The positive side and the negative side are defined as:

Example: Edge - num,xbeg,ybeg,xend,yend,kode,bvs,bvn,mat 1 1 0 0 -1 2 0 -0e6 2

ELLI defines an elliptical opening

nume,xcen,ycen,diam1,diam2,kode,bvs, bvn, mat,gradsy,gradny

nume -- number of elements on elliptical opening(xcen,ycen) -- coordinates of ellipse centre (m)diam1 – size of the ellipse in x-direction (m)diam2 – size of the ellipse in y-direction (m)kode -- type of boundary condition

= 1, shear and normal stress boundary = 2, shear and normal displacement boundary = 3, shear displacement and normal stress boundary

= 4, shear stress and normal displacement boundary= 11, shear and normal stress boundary with “constrained” elements for

exterior gravitational problems bvs -- boundary value in shear direction (stress or displacement) (Pa or m)bvn -- boundary value in normal direction (stress or displacement) (Pa or m)mat – material no. (or region) that the arch belongs togradsy – gradient for bvs in y-directiongradny – gradient for bvn in y-direction

Example: ELLIPSE -- nume,xcen,ycen,diam1,diam2,kode,ss,sn mat 40 0.0 0.0 4.375 3.5 1 0.00E+00 0.00E+00 1

EXCA defines excavation induced random cracksd_wall, rand_e

d_wall – distance into rock where random excavation induced cracks existrand_e --percentage of internal points that have an excavation induced crack

Example: EXCAvation induced cracks (d_wall, rand_e) 0.2,0.5

ENDF defines the end of the input data file

47

xbeg,ybeg

xend,yendPositive side (opening)

Negative side (rock)


Example: endfile

FRAC defines a fracture (joint)

num, xbeg, ybeg, xend, yend, kode, jmat, mat

num -- number of elements along the fracture(xbeg,ybeg) – co-ordinates of the beginning point of the fracture (m)(xend,yend) -- co-ordinates of the end point of the fracture (m)kode – no functionjmat -- joint property ID defined before (jmat=1,2,3 … )mat – material no. (or region) that the fracture belongs to

Example FRACTURE -- nume,xbeg,ybeg,xend,yend,kode,mat 25 -1.000 -1.000 1.000 1.000 5 1 1

GRAV defines gravity parameters

density,gy,sh_sv_ratio,y_surf

dens_rock – rock density (kg/m3)gy – acceleration of gravity (m/s2)sh_sv_ratio – ratio of horizontal to vertical stressy_surf – the y-coordinate of the ground surface

Note: for non-flat ground surface, y_surf needs to be the highest point of the surface. The surface geometry needs to be represented by using additional boundary elements defined by using EDGE or ARCH.

Note 2: to model underground excavation in shallow surface, the excavation boundary needs to be defined using KODE=11 in EDGE, ARCH, or ELLI command as “constrained” elements.

Example: gravity – dens_rock,gy,sh_sv_ratio,y_surf 2500,-10,1,3

INTE defines that fracture initiation in the internal rock mass is allowed.

Example: Internal fracture initiation

ISIZ defines the fracture initiation element size.

a_ini

a_ini - fracture initiation element size

Example: Isize 0.0018

ITER defines the number of numerical iterations to reach the given stress levels (default = 20).

48


n_it

n_it – numerical iteration number (default=20)

Example: Iteration – numerical iteration number 20

IWIN defines an area window for detecting fracture initiation (used only when once particular problem area is of interests)

xmin,xmax,ymin,ymax

xmin -- left border of the window (m)xmax -- right border of the window (m)ymin -- bottom border of the window (m)ymax -- top border of the window (m)

Example: IWINDOW – xmin, xmax, ymin, ymax -2 2 -2 2

LINT defines a straight line which is the interface between two material regions

num, xbeg, ybeg, xend, yend, mat_neg, mat_pos

num -- number of elements along the line(xbeg,ybeg) -- co-ordinates of the beginning point of the line (m)(xend,yend) -- co-ordinates of the end point of the line (m)mat_neg – material no. of the negative side of the interfacemat_pos – material no. of the positive side of the interface

Example:LINTERFACE-num,xbeg,ybeg,xend,yend,mat_neg,mat_pos

10 -1.0 0.0 1.0 0.0 2 1

MLIN defines the material no. that is a concrete lining. No insitu stresses will be given to this region

mat

mat – material no. that is the concrete lining

Example: MLINING - define concrete lining material no

1

MODU gives elastic properties (modulus) of the rock medium

pr, e, mat

pr - Poisson’s ratioe - Young’s modulus (Pa)

49


mat - Material No.

Example: MODULUS 0.40E+11 1

MONI defines one monitoring point where stresses and displacement are monitored during calculation (maximum 9 points can be defined)

xmon,ymon

xmon,ymon – coordinate of the monitoring point (m)

Example: MONITORING POINT – xmon,ymon 0 0

The stresses and displacements at the monitoring point will be stored in files “hist1.dat” to “hist9.dat”

MONL defines one monitoring line where stresses and displacement are monitored during calculation (maximum 9 lines can be defined)

x1, y1 ,x2, y2, npoint

x1, y1 – coordinates of the beginning point of the monitoring line (m)x2, y2 – coordinates of the end point of the monitoring line (m)npoint – number of inserted points along the monitoring line

Example: Monline - x1,y1,x2,y2,npoint0 -0.2 0 0.2 10

The stresses and displacements at the monitoring points will be stored in files “hist_line1.dat” to “hist_line9.dat”

Note: Monitoring points along a line may be very close to the existing elements. Incorrect results could be resulted at these points. User should check the position of the monitoring points.

PERM gives permeability parameters of rock and fractures

viscosity,density,perm0

viscosity – fluid dynamic viscosity (default=1.0e-3 Pa s)density – fluid density (default=1000 kg/m3)perm0 – hydraulic conductivity of rock (default=1.0e-9 m/s)

Example: permeability -- viscosity,density,perm01e-3,1000,1e-10

PROP gives fracture surface contact properties

jmat, ks, kn, phi, coh, phid, aperture0, aperture_r

50

0m

if a

21 ;2(1 ) m

p aa

1.0m

if a

;0.1p 1.0m

if


jmat -- joint property ID (1,2,3,…)ks -- fracture shear stiffness (Pa/m)kn -- fracture normal stiffness (Pa/m)phi -- fracture friction angle (degree)coh -- fracture cohesion (Pa)phid – fracture dilation angle (degree)aperture0 – initial aperture at zero normal stress (m)aperture_r – residual aperture (m)

In FRACOD, joint aperture is calculated by

rn edee ),(max 0

where e is the joint aperture; e0 is the joint initial aperture at zero normal stress; dn is the joint normal displacement (positive values indicate closure); er is the residual joint aperture.

Example: PROPERIES -- mat, ks, kn, phi, coh, dilation, aperture0, aperture_r 1 0.10E+14 0.10E+14 30.0 0.00E+00 0 10e-6, 1e-6

QUIT stops the calculation

Example: quit

RAND defines random fracture initiation in an intact rock or at boundary.

f_ini0, l_rand

f_ini0 – stress/strength level of initiation in range of (0, 1). f_ini0=0, fracture initiates at zero stress; f_ini0=1.0, fracture initiates only when the stresses reaches the rock strength (default = 0.0)l_rand - define if the fracture initiation is random or definite. l_rand = 0, fracture initiation will not be random. It occurs whenever the stress/strength ratio reaches f_ini0; l_rand = 1, fracture initiation will be random. The probability of fracture initiation is calculated by:

;0p

where p =probability of fracture initiation; /m =ratio of the stress to strength and a= fracture initiation level. a= f_ini0

Example: RANDOM – f_ini0,l_rand

51


0.5,1

ROCK gives intact rock strength parameters for fracture initiation

rphi, rcoh, sigt, mat

rphi – Intact rock internal friction angle (degrees)rcoh – Intact rock cohesion (Pa)sigt – Intact rock tensile strength (Pa)mat – Material no. (1, 2, 3 …)

Example: ROCK STRENGTH -- rphi, rcoh, sigt, mat30 4e+06 2.492e+06 1

RWIN defines a circular window for plotting the geometry, stresses and displacements

xc0, yc0, radium, numr, numt

xc0 ,yc0—coordinates of the window’s centre point (m)radium – radium distance of the window (m)numx -- number of grid points along radial directionnumy -- number of grid points along tangential direction

Example: RWINDOW -- xc0,yc0,radium,numr,numt 0.00 0.00 8 80 90

SYMM gives symmetry conditionsksym, xsym, ysym

ksym = 0 -- no symmetryksym = 1 -- problem symmetrical against vertical line x=xsym (m)ksym = 2 -- problem symmetrical against horizontal line y=ysym (m)ksym = 3 -- problem symmetrical against point x=xsym and y=ysym (m)ksym = 4 -- problem symmetrical against line x=xsym and line y=ysym (m)

Example: SYMMETRY0 0.0 0.0

SAVE filename saves the current state of calculation into a file

Note: the saved state of modelling can be restarted later using Window commands.

Example: save run1.sav

SETF Set the cut-off level of simultaneous multiple fracture propagations

factor_f

52


factor_f – Cut-off level of instant multiple fracture propagation. If factor_f=0, all fracture tips which meet the fracture criterion will propagate at the same time; if factor_f=1.0, only the fracture tips that have the least factor of safety (i.e. highest K/Kc ratio) and meet the fracture criterion will propagate at this iteration step.

(default factor_f=0.9)

Example: SETF -- fracture initiation cut-off level (of max FoS)0.90

SETE Set the check-up level for elastic fracture growth

factor_e

factor_e – cut-off ratio of a roughly estimated K/Kc of the elastically contacting fracture tips. If K/Kc < factor_e, the elastic tip is not considered to propagate and no further checking will be conducted on this tip; if K/Kc > factor_e, the elastic tip is considered to have potential to propagate and a normal process of determining fracture propagation will be conducted on this tip.

If factor_e=0.0, all elastic fracture tips will be checked;

(default factor_e=0.5)

Example: SETE -- elastic fracture growth check up level 0.5

SETT Set the tolerance factor

tolerance

tolerance – tolerance factor that define the tolerance distance which is equals to (tolerance average element size). The tolerance distance is used in the code to judge if a point or a fracture tip is too close to the existing element. If so, they are either ignored, or merged to the existing elements.

(default tolerance=1.0)

Example: SETT – set the tolerance factor 1.0

STOP stops the calculation

Example: stop

STRE gives far-field stresses in the rock mass

Pxx, Pyy, Pxy

Pxx – Far-field horizontal stress (Pa)

53


Pyy – Far-field vertical stress (Pa)Pxy – Far-field shear stress (Pa)

Warning: if Pxy is not 0, only ksym=0 or ksym = 3 may be used.

Example: STRESSES -- Pxx,Pyy,Pxy 0e6 0 0

SWIN defines a window for plotting the geometry, stresses and displacements

xll, xur, yll, yur, numx, numy

xll -- left border of the window (m)xur -- right border of the window (m)yll -- bottom border of the window (m)yur -- top border of the window (m)numx -- number of grid points along x-directionnumy -- number of grid points along y-direction

Example: SWINDOW -- xll,xur,yll,yur,numx,numy-2 2 -2 2 80 80

SYMM gives symmetry conditionsksym, xsym, ysym

ksym = 0 -- no symmetryksym = 1 -- problem symmetrical against vertical line x=xsym (m)ksym = 2 -- problem symmetrical against horizontal line y=ysym (m)ksym = 3 -- problem symmetrical against point x=xsym and y=ysym (m)ksym = 4 -- problem symmetrical against line x=xsym and line y=ysym (m)

Example: SYMMETRY 0 0.0 0.0

TITL gives a title to the problem(words within 80 letters)

Example: TITLE A single inclined fracture under uniaxial compression

TOUG gives critical energy release rates GIc and GIIc

GIc, GIIc, mat.

GIc -- mode I fracture critical strain energy release rate (J m-2)

GIc=(1-2)KIc2/E (KIc - Fracture toughness mode I)

GIIc -- mode II fracture critical strain energy release rate (J m-2)

54


GIIc=(1-2)KIIc2/E (KIIc - Fracture toughness mode II)

mat – material no.

Example: TOUG 20 100 1

TOUK gives fracture toughness KIc and KIIc

KIc, KIIc, mat.

KIc -- mode I fracture toughness (Pa m1/2)KIIc -- mode II fracture toughness (Pa m1/2)mat – material no.

Example: TOUK 2.39e6 3.10e6 1

TUNN defines a tunnel inside which the stress/displacement will not be plotted

xcen,ycen,diam

(xcen,ycen) -- coordinates of the tunnel centre (m)diam -- tunnel diameter (m)

Example: Tunnel - xcen,ycen,diam 25 -1.000 -1.000 1.000 1.000 5 1 1

Multiple tunnels can be defined using this command

WATE defines static water pressure in fractures. Two input formats can be used:

HOLE, w_xc, w_yc, w_d, wp

‘HOLE’ – text to specify a circular range within which all fracture elements will be given a water pressurew_xc, w_yc – coordinates of the HOLE centre (m)w_d – HOLE diameter (m)wp – water pressure (Pa)

Example: WATER HOLE 0.0 0.0 6 110e6

or

RECT, w_xmin,w_xmax,w_ymin,w_ymax, wp

‘RECT’ – text to specify a rectangular range within which all fracture elements will be given a water pressurew_xmin – minimum value of the range in x-direction w_xmax– maximum value of the range in x-directionw_ymi n– minimum value of the range in y-direction

55


w_ymax– maximum value of the range in x-directionwp – water pressure (Pa)

Example: WATER RECT -5 5 -5 5,60e6

To help preparing the input file, an input pre-processor (Model Design) has been developed for the user’s convenience. The interface is fully Window based and is coupled with graphics to help in defining a model easily. An instruction of how to use Model Design is described in Appendix I.

56


8 CONDUCT AND MONITOR THE CALCU-LATION

The FRACOD code is created as an executable file (Fracod2D.exe).

To start the code FRACOD, simply double-click the executable file in Windows, a dialog menu will appear on your screen. You then need to open an input data file or a FRACOD save file, by using the open file options.

If you are starting a new problem, you need to load an input data file which has already been prepared either by using a text editor or by using the model design function provided in the code (see Appendix I).

If you are restarting a problem which has previously been run and saved, you then need to load the saved file (*.sav). FRACOD will automatically detect whether the file you are loading is an input data file or a save file.

Once a calculation has started, it will continue until it is interrupted manually, or the defined cycles finished, or no more fracture propagation is found. During the calculation, the instant geometry of the modelled fracture network is always shown on the screen so that any growth of the fractures can be monitored. The fractures are plotted in different colours to distinguish whether the fracture surfaces are in elastic contact, open or sliding (the colours can be specified or changed by users).

When a calculation is completed or is interrupted, a number of screen commands are available to plot the stress/displacement or to change the parameters etc. These commands are provided as icons on the program window and can be easily activated by clicking the mouse.

The key functions of the available screen commands are shown below.

57


File (Load, Save, Print, Exit)

Load

Load an input data file (*.dat), a saved file (*.sav) or a movie file (*.mvi)

Input File (*.dat)

Load an input data file and start a new problem which is defined in the input data file. An input data file is a text file that contains commands and values to define a problem. This is a file being prepared in advance by the user using any text editor following the format that FRACOD can recognise, or using the pre-processing functions (Appendix I) provided with the FRACOD code.

Saved File (*.sav)

Load a saved file and continue the simulation which was previously interrupted. A saved file is a file containing data of a problem run previously by using FRACOD. The data in the file is computer coded and can only be read by FRACOD itself. An ongoing fracture propagation modelling can be interrupted (see Pause) and saved (see Save) into a saved file. The saved file can then be reloaded into FRACOD to continue the previously interrupted modelling.

Movie File (*.mvi)

Load a movie file to replay the progress of fracture propagation from previous calculations. A movie file is a file containing plot data of a problem run previously by using FRACOD. It is saved automatically during FRACOD calculation using the same name as the input data file but with the extension of “.mvi”. This function provides the possibility of replaying/printing the fracture propagation process without re-runing the model.

Save (Run, Plot)

Run

Save the current status of calculation into a saved file. The saved file can later be reloaded (see Load) into FRACOD to continue the modelling.

58


Plot

Save the current plot into a file (*.emf, or *.wmf). The file has a emf (Window Meta Files) or wmf (Window Enhanced Meta Files) format. It can be copied to other Window applications (e.g. MS Word).

Print

Print the current screen plot on a default output device (printer).

Exit

Terminate the current calculation and close the FRACOD Window.

Default screen output

During simulation, the geometry of fractures and tunnels etc. will be automatically shown on the screen. The picture will be updated after each calculation cycle to trace any fracture propagation. In this way the whole process of fracture propagation can be monitored. In the screen plot, fractures are plotted with different colours to show the state of the fractures, i.e. elastic contact, sliding or open. The default fracture colour is:

Blue – elastic fractureGreen – sliding fractureRed – open fracture.

The colour can be changed by users in view/plots setup.

The magnitudes of far-field stresses are shown in the Legend plot window. Here:

Pxx – normal stress in the horizontal direction of the modelPyy – normal stress in the vertical direction of the modelPxy – shear stress

The magnitudes of stresses are in Pa unless specified otherwise.

59


Edit (Copy to Clipboard (BMP), Copy to ClipBoard (EMF))

Copy the screen plot to the clipboard in the BitMap Format (BMP) or in Window Enhance Meta Files format (EMF). The plot can be pasted to other Window applications (e.g. MS Word).

View (Model Boundary, Fractures, Acoustic Emission, Principal Stress, Max. Shear Stress, Displacement, Shear Displacement, Normal Displacement, Aperture)

Plot the stresses or displacements from a paused simulation on screen. An ongoing simulation can be paused (see Pause) and the geometry, stresses and displacements can be plotted on screen.

Model BoundaryPlot on screen the geometry of the external and internal boundaries such as tunnels in the model. The geometry plot is set as default and is automatically shown on the screen during calculation.

FracturesPlot on screen the geometry of the fractures in the model.

Principal StressPlot on screen the principal stresses in the rock mass within a defined window. Two orthogonal principal stresses, the major principal stress and the minor principal stress, will be plotted on screen as two orthogonally lines. The directions of the lines are the directions of the two principal stresses, and the length of each line is proportional to the stress magnitude. Colours are used to distinguish the compressive stress with the tensile stress:

Blue – compressive stress (default)Red – tensile stress (default)

The colour can be changed by user in View/Plot setup.

The maximum magnitude of the principal stresses in the plot is given in the Legend window (in Pa)

Max. Shear StressPlot on screen the maximum shear stress in the rock mass within a defined window. The maximum shear stress in two orthogonal directions will be plotted on the screen as two orthogonal lines. The directions of the lines are

60


the directions of the maximum shear stress, and the length of each line is proportional to the stress magnitude.

The maximum magnitude of the maximum shear stresses in the plot is given in the Legend window (in Pa)

DisplacementPlot on screen the displacement vector at specified grid points in the model. Rock displacement at a grid point will be plotted on the screen as a vector with an arrow indicating the direction of the displacement, and the length of the vector is proportional to the values of displacement.

The value of maximum displacement in the plot is given on the top of the plot window (in metres).

Shear DisplacementPlot on screen the joint shear displacements in the model. Shear displacement will be plotted on the screen as a vector with an arrow at both sides of the joint plane. The length of the vector is proportional to the values of displacement.

The value of maximum shear displacement in the plot is given on the top of the plot window (in metres). Note that the maximum shear displacement is the differential movement of the joint at both sides. Hence it is twice of individual vectors.

Normal DisplacementPlot on screen the joint normal displacements in the model. Normal displacement will be plotted on the screen as a vector with an arrow at both sides of the joint plane. The length of the vector is proportional to the values of displacement. The direction of the vector indicates if the joint is closing or open. It can also be judged from the colour of the joint (red – open joint; blue or green – closed joint).

The value of maximum normal displacement in the plot is given on the top of the plot window (in metres). Note that the maximum normal displacement is the differential movement of the joint at both sides. Hence it is twice of individual vectors.

AperturePlot on screen the joint aperture. The joint aperture is calculated by:

61


rn edee ),(max 0

where e is the joint aperture; e0 is the joint initial aperture at zero normal stress; dn is the joint normal displacement (positive values indicate closure); er is the residual joint aperture.

The aperture is plotted in the same way as the normal displacement except that the aperture is already open.

View (Displacement/stress symbol, Image, Contours, Fill material background, Clear image/ contours)

Displacement/stress symbol Show the lines and arrows that represent the direction and magnitude of stresses and displacement.

Image Plot on screen the filled contours of stresses or displacement.

Contour Plot on screen the line contours of stresses or displacement.

Fill material backgroundThis option is used together with the Image function to fill the contours.

Clear image/ contoursClean the image/ contours plots on screen

View (Rotate plot)

Rotate plotRotate the plot on screen against the original point by an angle. The angle needed to be provided on another screen window.

View (Color, Color bar, Legend, Smart Cursor, Progress bar) ColorSet or change colors for filled contour plots

Color barShow a legend vertical color bar beside the filled contour plots

62


LegendShow the legend on the plot window. Included in the Legend are: Far-field stress (Sxx,Syy,Sxy) Maximum values of the stresses or displacement appeared on the screen

plot. Colour conventions

Progress BarShow a separate window which indicate the progress of the status of the mechanical calculation and the plotting interfaces

View (Zoom in, Zoom out, Full plot)

Zoom inEnlarge the plot in a specified window (defined by dragging the Mouse).

Zoom outReduce the plot the plot in a specified window (defined by dragging the Mouse).

Full screenReturn the plot size to the originally specified window (full screen).

View (Magnifier, Plot setup, Contour setup)

MagnifierMagnify an area of the screen. To do so, locate the mouse cursor to the desired position and press down the mouse right button.

Plot setupSpecify or change the plot setup, including: Plot range Axis setting Plot attributes (line color and thickness, scale etc.) Magnifier (shape and size) Movie/Legend setting Acoustic Emission plot setting Others (grid setting etc.)

Contour setupSpecify or change the contour plot setting

63


Run (Run, Pause, Stop)

Run

Start or continue a calculation. A cycle number is requested. One cycle often produces a fracture propagation of one element length.

Pause

Pause the current calculation. A paused calculation can be reactivated and continued by using Run.

Stop

Stop the calculation. This command triggers the termination of the current calculation A stopped calculation cannot be restarted. Some calculation results (stresses and displacement at the previously specified grid points) can, however, still be shown.

Option (Far-field stress, Boundary stress, Set Stress Change from a file, Set Parameter for Factor of Safety, Set creep parameters)

Change the magnitude of the far-field stresses or boundary stresses

Far-field stress

Increase or decrease the magnitude of far-field stresses. The value of increment or reduction is requested. Note that the compressive stress is negative, so that an increment in compressive stress should be given as negative values.

This command is particularly useful in studying the change of the fracture growth path when the far field stresses are changed.

Boundary stress

Increase or decrease the magnitude of boundary stresses (or displacement if the boundary condition is specified by displacement). The value of increment or reduction of shear or normal stress is requested. Note that the compressive stress is negative, so that an increment in compressive stress should be given as a negative value.

64


This command is particularly useful in studying the change of the fracture growth path when the boundary stresses (e.g. hydraulic pressure in a borehole) are changed.

Set Stress Change from a file

Change the boundary stresses (or displacement if the boundary condition is specified by displacement) as specified in a file. This is a way to change unevenly the boundary values at different parts of the boundary. The file should contain the following information:

x1 x2 y1 y2 dss dsn

-1.0 0.0 0.059 2.01 0 -5e6 0.0 1.0 0.059 2.01 0 -5e6

x1,x2,y1,y2 define a range within which all boundary elements will have their boundary values changed;dss and dsn define the increment of the boundary values (stress or displacement as defined originally).

Set Parameters for Factor of Safety

Set the rock strength parameters for plotting the Factor of Safety contours.

Set creep parameters

Change the creep parameters during modelling.

Tools (Model design, New Password)

Model design

A pre-processor which helps the user to set up the numerical model. Details of the pre-processor are given in Appendix I.

New Password

Change password for the legal copy of this code version

Windows (…)

65


Standard Window management functions which enable users to arrange the multiple calculation Windows.

Help (…)

On-line user’s manual and helping functions.

66


9 FRACOD VERIFICATION TESTS

Three verification tests cases are listed below. The data files are provided in the program package.

9.1 SINGLE FRACTURE SUBJECTED TO NORMAL TENSILE STRESS

(1) Problem definition

A 2m fracture in an infinite rock mass is under uniaxial tensile stress of 50MPa in the direction perpendicular to the fracture plane. The elastic properties of the rock mass are:

E = 40GPa = 0.25.

The strain energy release rate in mode I for this problem is calculated by using the FRACOD code with 30 elements along the fracture.

(GI)FRACOD = 190103 J/m2

The theoretical solution of this problem gives the stress intensity factor (KI) as

mMPa

aK I

6.8811416.350

where a = half length of the fracture.

The theoretical strain energy release rate is then calculated as:

23269

2

22

/10184)106.88(104025.01

)(1)(

mJ

KE

G ItheoryI

The difference between the numerical result and the theoretical result is approximately 3%.

67


In this example, the critical strain energy release rates of fracture propagation are:

GIc =50 J/m2 GIIc =1000 J/m2 .

As the fracture propagation is pure mode I along the fracture’s original plane, only the critical strain energy release rate in mode I (GIc) is useful. The F-value obtained from the FRACOD modelling is:

38001000

050

10190

)0()0()0(

3

IIc

II

Ic

I

GG

GG

F

The F-value is by far greater than the critical value 1.0. Hence fracture propagation is detected.

(2) Input data-----------------------------------------------------------------------------TITLEA single fracture under tension SYMMETRY -- Model symmetry 0 0.00 0.00MODULUS -- Poisson's Ratio and Young's modulus, mat 0.25 0.40E+11 1TOUGHNESS -- Gic and Giic, mat 50. 1000. 1PROPERIES -- jmat, ks. kn,phi,coh, dila 1 0.10E+14 0.10E+14 30.0 0.00E+00SWINDOW -- xll,xur,yll,yur,numx,numy -5.00 5.00 -5.00 5.00 30 30STRESSES -- sxx,syy,sxy -0.0E+07 0.50E+08 0.00E+00FRACTURE -- nume,xbeg,ybeg,xend,yend,kode,jmat, mat 25 -1.000 -0.000 1.000 0.000 2 1, 1CYCL 10ENDFILE--------------------------------------------------------------------------------

68


(3) FRACOD model

-5 -4 -3 -2 -1 0 1 2 3 4 5X Axis (m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

Y A

xis

(m)

-5 -4 -3 -2 -1 0 1 2 3 4 5X Axis (m)

-5

-4

-3

-2

-1

0

1

2

3

4

5

Y A

xis

(m)

A single fracture under tensionMaximum Displacement (m): 4.1006E-3

Creep Time: 0:0:0

Creep Time Step: 0:0:0

Max. Crack velocity (m/s): 0E+0

Cycle: 10 of 10

Elastic fracture

Open fracture

Slipping fracture

Fracture with Water

Fracom Ltd

Date: 12/06/2005 09:51:26

69


9.2 SINGLE FRACTURE SUBJECTED TO PURE SHEAR STRESS

(1) Problem definition

A 2m fracture in an infinite rock mass is under pure shear stress of 50MPa. The elastic properties of the rock mass are the same as in Example 1.

According to Rao (1999), a fracture in pure shear may propagate in mode I or mode II depending on the ratio of the fracture toughness of mode I and mode II (KIc/KIIc). Only when KIc/KIIc>1.15, a mode II fracture propagation can occur. The FRACOD code is used in this example to compare with the theoretical results.

30 elements are used along the fracture. The critical mode II strain energy release rate (GIIc) is taken as 1000 J/m2. The critical mode I strain energy release rate (GIc) is varied to obtain the critical ratio (GIc/GIIc) at which the fracture propagation starts to change mode. It was found that when GIc < 1279 J/m2, fracture propagates in pure mode I in the direction of about 70 from the original fracture plane in its own plane. When GIc > 1471 J/m2 the fracture propagates in pure mode II. When 1279J/m2<GIc<1471J/m2 the fracture initiatally propagates in mode II then in mode I. When GIc > 1279 J/m2 the critical value for the fracture propagation to change mode.

If we take the average of 1375 J/m2 as the estimated critical value, and use the relation between the critical stress energy release rate and the fracture toughness

17.110001375

IIc

Ic

numericalIIc

Ic

GG

KK

The critical toughness ratio for mode II fracture propagation obtained numerically by using the FRACOD code is 1.17, close to the analytical solution of 1.15 reported by Rao (1999).

70


(2) Input data

-----------------------------------------------------------------------------TITLESingle fracture subjected to pure shear stress SYMMETRY -- Model symmetry 0 0.00 0.00MODULUS – Poisson’s Ratio and Young’s modulus 0.25 0.40E+11TOUGHNESS -- Gic and Giic 1289.0 1000. PROPERIES -- mat, kn. ks,phi,coh 1 0.0E+0 0.0E+0 0.0 0.00E+00SWINDOW -- xll,xur,yll,yur,numx,numy -2.00 2.00 -2.00 2.00 30 30STRESSES -- sxx,syy,sxy -0.0E+06 -0.00E+0 50.00E+06FRACTURE -- nume,xbeg,ybeg,xend,yend,kode,mat30 -1. 0. 1. 0. 2 1CYCL 1000ENDFILE--------------------------------------------------------------------------------

71


(3) FRACOD model

-2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8X Axis (m)

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Y A

xis

(m)

-2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8X Axis (m)

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Y A

xis

(m)

Single fracture in pure shearMaximum Displacement (m): 5.70476E-3

Creep Time: 0:0:0



Cycle: 17 of 1000

Elastic fracture

Open fracture

Slipping fracture

Fracture with Water

Fracom Ltd

Date: 12/06/2005 09:54:17

-2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8X Axis (m)

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Y A

xis

(m)

-2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8X Axis (m)

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Y A

xis

(m)

Single fracture in pure shearMaximum Displacement (m): 2.67123E-3

Creep Time: 0:0:0



Cycle: 6 of 1000

Elastic fracture

Open fracture

Slipping fracture

Fracture with Water

Fracom Ltd

Date: 12/06/2005 09:59:48

72

GIC=1270J/m2

GIIC=1000J/m2

GIC=1480J/m2

GIIC=1000J/m2


9.3 MULTIPLE REGION MODEL

A simple case of a boundary value problem in an inhomogeneous elastic body is shown in Figure 9-12. The region of interest consists of an annulus

bra with elastic constants E1 and 1 inside a circular hole of radius br in a large plate with elastic constants E2 and 2. The inside wall of the

annulus is subjected to a normal stress prr , and the plate is unstressed at infinity. The solution to this problem, satisfying continuity of radial stress and displacement at the interface br , can be constructed from standard formulas from thick-walled cylinders. The radial and tangential stresses are:

222 )/)('(')/(

)/(11 rapppbap

barr

bra

222 )/)('(')/(

)/(11 rapppbap

ba

bra (21)

2)/(' rbprr br ³2)/(' rbp br ³

in which

2

121

2

2

1

2

)/(11

111

211

)/('ba

EE

bapp

(22)

Figure 9-12. Annulus inside a circular hole in a plate.

73

a

b

r

p

E2

;2E

1

;1


The new FRACOD is applied to above problem to compare with the analytical results. In this example, the following geometrical and mechanical parameters are used:a = 0.5mb = 1.0m1 = 1 = 0.2E1 = 50GPa; E2 = 25GPap = 10MPa

60 elements are used for the internal circular boundary and 60 elements for each side of the interface. The input file for this problem is listed below:

TITLEMulti-region code verification testSYMMETRY -- Model symmetry0 0 0MODULUS -- Poisson's Ratio and Young's modulus,mat0.2 50e+9, 1MODULUS -- Poisson's Ratio and Young's modulus,may0.2 25e+9, 2TOUK -- Kic and Kiic, mat3.0e6 0.75e6 1PROPERTIES (old joints) -- mat, ks kn,phi,coh,phid1 35.5e0 65.5E0 30 0e6 0PROPERIES -- mat, ks. kn,phi,coh phid --- Tensile fractures11 10e14 10e14 30.0 4E+06 5PROPERIES -- mat, ks. kn,phi,coh phid --- Shear fractures

74

a

b

r

p

E2

;2E

1

;1


12 10e14 10e14 30.0 4E+06 5STRESSES -- sxx,syy,sxy0e6 0 0ROCK STRENGTH -- rphi, rcoh, sigt,mat30 4e+06 2.492e+06,1SWINDOW -- xll,xur,yll,yur,numx,numy-2 2 -2 2 80 80 ARCH60 0 0 0.5 0 360 1 0 -10e6 1AINTERFACE60 0 0 1.0 360 0 1 2 CYCL 1000

The modelled stress distribution using the new FRACOD is shown in Figure9-13. A comparison of the modelled radial and tangential stresses with the analytical results is shown in . A good agreement is obtained, suggesting that the new FRACOD accurately simulates the multi-region problem. Figure 9-13. Modelled stress distribution for the test problem.

75

-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7X Axis (m)

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Y A

xis

(m)

-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7X Axis (m)

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Y A

xis

(m)

Multi-region code testing, using different properties in different regionsPxx: 0E+0 Pyy: 0E+0

Pxy: 0E+0

Max. Compres. Stress: 1E+7

Max. Tensile Stress: 1.27281E+7

Elastic fracture

Open fracture

Slipping fracture

Fracture with Water

Compressive stress

Tensile stress

Fracom Ltd

Date: 27/12/2003 09:53:11


Figure 9-14. Comparison between the FRACOD results and analytical results.

9.4 SUBCRITICAL CRACK GROWTH - CREEP

Problem definition

Figure 9-15. An infinite plate containing a crack under tensional loading.

Analytical solution

Crack length when t=t

12/

1

12/0 12/

nn

c

n tK

Anaa

76

-15.0

-10.0

-5.0

0.0

5.0

10.0

15.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Distance to tunnel centre (m)

Stre

ss (M

Pa)

Srr (numerical)S00 (numerical)S1 (theory)S2 (theory)

rr

rr

1-1

σ=remote tensile stress: (10 MPa)2ao=crack length in the start (0.01m)KIC=3.80 MPa*m1/2A=material constant (500)n=stress corrosion factor (30). t=time.Note! Subcritical parameters A and n are only for demonstration purpose. Realistic factors will be defined by laboratory tests.


Time for a crack to extended from a0 to a.

n

c

nn

KAn

aat

12/

12/0

12/

From the analytical solution we get a stress intensity level of KI/KIC=0.3298 in the start (t=0).

The crack has extended to a length of 50mm after 2.021x108 seconds (~6.4 years), and the stress intensity has increased to KI/KIC = 0.7375. (After 0.04 seconds it will be KI/KIC≥1.0).

According to the analytical solution a far-field stress of 30.3 MPa would yield to KI/KIC = 1, and instant failure would take place.

To next the above example with 10MPa remote tensile stress is analysed by the FRACOD iteration code and after that by the new FRACOD CREEP code.

77

1-2


FRACOD results

Table 9-1. Material parameters.

Parameter Value and unit

INTACT ROCK Poisson’s ratio for intact rock 0.24Young’s modulus, intact rock 68.0 GPaFriction angle (no fracture initiation considered)

49 °

FRACTURE Fracture toughness in mode I (in D-THMC 3,21)

3.80 MPa*m1/2

Fracture toughness in mode II 4.40 MPa*m1/2Normal stiffness, Kn 61 GPa/m Shear stiffness, Ks 35.5 GPa/mFracture friction angle 31

Cohesion 0 MPaDilation angle 5

Figure 9-16. FRACOD calculation. Infinite plate containing a crack under 10MPa tensile load results in a stable model.

78

σ=remote tensile stress 10 MPa σ=remote tensile stress 10 MPa


Table 9-2. Creep parameters for tensile fractures etc.

Parameter Value and unit

Constant A (=max propagation velocity)

500 m/s

Stress corrosion factor n 30Time range for calculations 1x109 secTime step is automatically determined within a specified range:Delta t min 1 sDelta t max 1000sModel symmetry No symmetryNumber of elements defining the fracture (standard case)

25

Stress window and gridpoint densityNum x num y

-0.07 .07 -0.07 .07 18 x 18

Figure 9-17. Subcritical crack growth in tension. 1) Initial stage (t=0) with a crack length of 0.010m. 2) after 1.92x108 seconds, crack length is 0.012m 3) after 2.04 x108 s, the crack length is > 0.050m.

79

σ=remote tensile stress;10 MPa


Figure 9-18. Time until the initial crack (l=0.01m) has extended to a length of 0.05m is 2.04x108 seconds.

Figure 9-19. KI/KIC vs time.

80


Figure 9-20. KI/KIC vs crack velocity.

Figure 9-21. KI/KIC vs crack velocity in a logarithmic scale.

The CREEP code results very close the analytical results. Both methods suggest subcritical crack growth when the crack has grown to a length of 0.05m. In next we study the accuracy more detailed.

81


FRACOD CREEP model accuracy

Figure 9-22. Number of elements defining the pre-existing fracture vs. accuracy of the stress intensity at the fracture tip in the start. Analytical solution leads to KI/KIC=0.3298.

Figure 9-23. Number of elements defining the pre-existing fracture vs. accuracy of the approximation for Time-to-failure. Failure is defined here as time when the crack has extended to a length of 0.05m.

82


9.5 GRAVITY PROBLEMS

Example problem #1: tunnel stress analysis

A circular tunnel with diameter of 2m is at a depth of 100m. The key input parameters used are:

Rock density = 2500kg/m3

Stress ratio x/y =0.33

This problem was modelling using the gravity function with “restrained” elements at the tunnel boundary. The predicted displacement distribution is shown in Figure 9-24.

Because the tunnel is relatively deep, we can also obtain the analytical solution of this problem if we assume the tunnel is under constant far-field stress at this depth (x=0.83MPa; y=2.5MPa), see Brady and Brown, “Rock Mechanics for Underground Mining” (1985. page 162-163).

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5X Axis (m)

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

2.0

2.5

Y A

xis

(m)

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5X Axis (m)

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

2.0

2.5

Y A

xis

(m)

Test

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

Tota

l Dis

plac

emen

t (m

) xE

-5

Maximum Displacement (m): 7.00126E-5

Elastic fracture

Open fracture

Slipping fracture


Fracom Ltd

Date: 07/06/2007 13:13:50

Figure 9-24. Predicted displacement field around a circular tunnel.

83

y=0

x=0


Figure 9-25. Comparison between FRACOD results and analytical results for the circular tunnel.

The numerical results and the analytical results are compared along two monitoring lines shown in Figure 9-24. The comparison results are shown in Figure 9-25. The numerical results are nearly identical to the analytical results for most of stress and displacement component along the two lines. The maximum numerical error is less than 7.8%.

This example demonstrates that the new gravity function using the “restrained” elements is effective and accurate enough for engineering problem.

dx (y=0 line)

0.00E+00

1.00E-06

2.00E-06

3.00E-06

4.00E-06

5.00E-06

6.00E-06

7.00E-06

8.00E-06

1 1.5 2 2.5 3 3.5 4 4.5 5

X (m)

Dis

plac

emen

t (m

)FRACODAnalytical

dy (x=0 line)

-8.00E-05

-7.00E-05

-6.00E-05

-5.00E-05

-4.00E-05

-3.00E-05

-2.00E-05

-1.00E-05

0.00E+001 1.5 2 2.5 3 3.5 4 4.5 5

Y (m)

Dis

plac

emen

t (m

)

FRACOD

Analytical

Stresses (y=0 line)

-8.00E+06

-7.00E+06

-6.00E+06

-5.00E+06

-4.00E+06

-3.00E+06

-2.00E+06

-1.00E+06

0.00E+001 1.5 2 2.5 3 3.5 4

X (m)

Str

ess

(Pa)

sigxx (FRACOD)

sigyy (FRACOD)sigxx (Analytical)

sigyy (Analytical)

Stresses (x=0 line)

-2.50E+06

-2.00E+06

-1.50E+06

-1.00E+06

-5.00E+05

0.00E+00

5.00E+05

1 1.5 2 2.5 3 3.5 4

Y (m)

Str

ess

(Pa)

sigxx (FRACOD)sigyy (FRACOD)

sigxx (Analytical)sigyy (Analytical)

84


Example problem #2: rock slope stress analysis

A small rock slope is under gravity stresses only. The key input parameters used are:



Both the side boundaries and the bottom boundary are fixed with zero displacement in both normal and shear directions.

The slope geometry and calculated stress and displacement distribution in the rock mass are shown in Figure 9-26 and Figure 9-27.

-1 0 1 2 3 4 5 6 7 8 9 10 11X Axis (m)

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

Y A

xis

(m)

-1 0 1 2 3 4 5 6 7 8 9 10 11X Axis (m)

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

Y A

xis

(m)

-3.4

-3.2

-3.0

-2.8

-2.6

-2.4

-2.2

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0

Prin

cipa

l Maj

or S

tress

(P

a) x

E5


Pxy (Pa): 0E+0



Elastic fracture

Open fracture

Slipping fracture

Fracture with Water

Compressive stress

Tensile stress

Fracom Ltd

Date: 05/06/2007 12:08:59

Figure 9-26. Stress distribution around a small slope – FRACOD results.

85


-1 0 1 2 3 4 5 6 7 8 9 10 11X Axis (m)

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

Y A

xis

(m)

-1 0 1 2 3 4 5 6 7 8 9 10 11X Axis (m)

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

Y A

xis

(m)

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

Y -

Dis

plac

emen

t (m

) xE

-6


Elastic fracture

Open fracture

Slipping fracture

Fracture with Water

Fracom Ltd

Date: 05/06/2007 12:08:59

Figure 9-27. Displacement distribution around a small slope – FRACOD results.

To check the accuracy of the FRACOD modelling results, the same problem was modelled using FLAC. The detailed results obtained from FRACOD and FLAC along a monitoring line as shown in the figures are compared. The comparison results are shown in Figure 9-28. The FRACOD results agree reasonably well with the FLAC results.

In details, the difference between FRACOD and FLAC results along the monitoring line is 4.9% in the maximum vertical displacement; 1.3% in the maximum horizontal displacement; 3.5% in maximum vertical stress; and 10% in maximum horizontal stress.

86


Figure 9-28. Comparison of FRACOD results with FLAC results.

-1.E-06

0.E+00

1.E-06

2.E-06

3.E-06

4.E-06

5.E-06

6.E-06

0 1 2 3 4 5 6 7 8 9 10

X (m)

Dis

plac

emen

t (m

)

FRACOD dy

FLAC dy

FRACOD dx

FLAC dx

-2.E+05

-1.E+05

-1.E+05

-1.E+05

-8.E+04

-6.E+04

-4.E+04

-2.E+04

0.E+000 1 2 3 4 5 6 7 8 9 10

X (m)

Str

ess

(Pa)

FRACOD sigyy

FLAC syy

FRACOD sigxx

FLAC sxx

87


INPUT DATA FILE (FRACOD)

TITLERock slope stress analysisSYMMETRY -- Model symmetry0 0 0MODULUS -- Poisson's Ratio and Young's modulus0.25 60e+9 1TOUK -- Kic and Kiic, mat 1.73e6 3.07e6 1PROPERIES -- mat, ks. kn,phi,coh phid1 1200e+6 3920e+6 25.5 0,0 10e-6, 10e-6PROPERIES -- mat, ks. kn,phi,coh phid --- Tensile fractures11 3099E+9 13800E+9 33.0 0.3300E+08 2 10e-6, 10e-6PROPERIES -- mat, ks. kn,phi,coh phid --- Shear fractures12 3099E+9 13800E+9 33.0 0.3300E+08 2 10e-6, 10e-6STRESSES -- sxx,syy,sxy Depth=500m: sx=18.82Mpa; sy=20.47Mpa; sxy=-5.30Mpa; sz=12.75Mpa-0e+03 -0e+03 0.0e6gravity -- density,gy,sh_sv_ratio,y_surf 2500,-10,0.33,5ROCK STRENGTH -- rphi, rcoh, sigt/peak strength c=33, st=12.2Mpa33 33e+6 12.2e+6 1RANDOM fracture initiation - f_ini0,l_rand (initiation level, random or not)0.4 0setf -- factor for fracture initiation cut-off level (??% of max FoS)0.95sete -- elastic fracture growth check up level0.0sett -- fracture tip merging tolerance distance (*averge half length of lements) 1.0boundary fracture initiationinternal fracture initiationisize0.30SWINDOW -- xll,xur,yll,yur,numx,numy-1 11 -6 6 50 50permeability -- viscosity,density,perm01e-3,1000,1e-9monl0 -0.8 10 -0.8 49edge20 0 5 5 5 1 0 -0e3 1edge20 5 5 5 0 1 0 -0e3 1edge20 5 0 10 0 1 0 -0e3 1edge20 10 0 10 -5 2 0 -0e3 1edge40 10 -5 0 -5 2 0 0 1edge40 0 -5 0 5 2 0 0 1

cycl 1endf

88


INPUT DATA FILE (FLAC)

new

gr 50,50m elas

gen 0.0,-5.0 0.0,5.0 10.0,5.0 10.0,-5.0 rat 1.00 1

prop den=2500 bulk=40.0e9 shear=24.0e9; equivalent to E=60GPa nu=0.25

set grav=10

; boundary conditionsfix x i=1*fix y i=1fix x i=51*fix y i=51*fix x j=1fix y j=1

cyc 5000sav ini.savini xdis=0 ydis=0

hist ydisp i=1,j=28hist ydisp i=1,j=29hist ydisp i=1,j=30hist ydisp i=1,j=31hist ydisp i=1,j=32hist ydisp i=1,j=33hist ydisp i=1,j=34hist ydisp i=1,j=35

hist xdis i=28,j=26hist xdis i=29,j=26hist xdis i=30,j=26hist xdis i=31,j=26hist xdis i=32,j=26hist xdis i=33,j=26hist xdis i=34,j=26hist xdis i=35,j=26

m n i=26,50 j=26,50

fix x i=1fix y i=1fix x i=51fix y i=51fix x j=1fix y j=1

cyc 5000sav run1.savcyc 5000sav run2.savcyc 5000sav run3.sav

return

89


Example problem #3: large rock slope stress analysis

A large inclined rock slope is under gravity stresses only. The key input parameters used are:



Both the side boundaries and the bottom boundary are fixed with zero displacement in the normal direction.

The slope geometry and calculated stress distribution in the rock mass are shown in Figure 9-26.

-600 -550 -500 -450 -400 -350 -300 -250 -200 -150 -100 -50 0 50 100 150 200 250 300 350 400 450X Axis (m)

-600

-550

-500

-450

-400

-350

-300

-250

-200

-150

-100

-50

0

50

100

150

200

250

300

350

400

450

Y A

xis

(m)

-600 -550 -500 -450 -400 -350 -300 -250 -200 -150 -100 -50 0 50 100 150 200 250 300 350 400 450X Axis (m)

-600

-550

-500

-450

-400

-350

-300

-250

-200

-150

-100

-50

0

50

100

150

200

250

300

350

400

450

Y A

xis

(m)

Large rock slop stress analysis

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

Prin

cipa

l Maj

or S

tres

s (

Pa)

xE

7


Pxy (Pa): 0E+0



Cycle: 1 of 1

Elastic fracture

Open fracture

Slipping fracture

Fracture with Water

Compressive stress

Tensile stress

Fracom LtdDate: 12/06/2007 12:47:19

Figure 9-29. Stress distribution around a small slope – FRACOD results.

To check the accuracy of the FRACOD modelling results, the same problem was modelled using FLAC. The detailed results obtained from FRACOD and FLAC along a monitoring line as shown in the figures are compared. The comparison results are shown in Figure 9-30. The FRACOD results

90

Monitoring line


agree very well with the FLAC results (the overall difference is less than 5%)

-8.00E+06

-7.00E+06

-6.00E+06

-5.00E+06

-4.00E+06

-3.00E+06

-2.00E+06

-1.00E+06

0.00E+00

1.00E+06

2.00E+06

3.00E+06

-600 -500 -400 -300 -200 -100 0 100 200

x (m)

Stre

ss (P

a)

sxx (FLAC)syy (FLAC)sxy (FLAC)sxx (Fracod)syy (Fracod)sxy (Fracod)

Figure 9-30. Comparison of FRACOD results with FLAC results.

91


INPUT DATA FILE (FRACOD)

TITLELarge rock slop stress analysis SYMMETRY -- Model symmetry0 0 0MODULUS -- Poisson's Ratio and Young's modulus0.25 60e+9 1TOUK -- Kic and Kiic, mat 1.73e5 3.07e5 1PROPERIES -- mat, ks. kn,phi,coh phid1 12.00e+9 39.20e+9 25.5 0,0 10e-6, 10e-6PROPERIES -- mat, ks. kn,phi,coh phid --- Tensile fractures11 30.99E+9 138.00E+9 33.0 0.00E+08 0 10e-6, 10e-6PROPERIES -- mat, ks. kn,phi,coh phid --- Shear fractures12 30.99E+9 138.00E+9 33.0 0.00E+08 0 10e-6, 10e-6STRESSES -- sxx,syy,sxy -0e+03 -0e+03 0.0e6gravity -- density,gy,sh_sv_ratio,y_surf 2500,-10,0.33,400ROCK STRENGTH -- rphi, rcoh, sigt30 1.0e+9 1.22e+9 1RANDOM fracture initiation - f_ini0,l_rand (initiation level, random or not)0.5 1setf -- factor for fracture initiation cut-off level (??% of max FoS)0.00sete -- elastic fracture growth check up level0.0sett -- fracture tip merging tolerance distance (*averge half length of lements) 1.0boundary fracture initiationinternal fracture initiationisize20SWINDOW -- xll,xur,yll,yur,numx,numy-600 500 -600 500 60 60permeability -- viscosity,density,perm01e-3,1000,1e-9IWINDOW -- xll,xur,yll,yur-300 300 -300 300monline-600 129 71 129 50

edge25 -600 400 -200 400 1 0 -0e3 1edge30 -200 400 200 0 1 0 -0e3 1edge25 200 0 600 0 1 0 -0e3 1edge20 600 0 600 -400 4 0 -0e3 1edge40 600 -400 -600 -400 4 0 0 1edge40 -600 -400 -600 400 4 0 0 1

cycl 1endf

92


INPUT DATA FILE (FLAC)

new

grid 60 60mod elas

gen -600,-400 -600,0 600,0 600,-400 j=1,30

gen same -600 400 -200,400 same i=1,41 j=30,61

model null i=41,60 j=30,60

prop den=2500 bulk=40.0e9 shear=24.0e9; equivalent to E=60GPa nu=0.25

set grav=10

; boundary conditionsfix x i=1fix x i=61fix y j=1

cyc 5000sav ini1.sav*ini xdis=0 ydis=0

m n i=41,60 j=30,60

fix x i=1*fix y i=1fix x i=61*fix y i=61*fix x j=1fix y j=1

cyc 5000sav run1a.savcyc 5000sav run2a.savcyc 5000sav run3a.sav

return

pause

93


Example problem #4: Large rock slope failure analysis

A high (400m) rock slope is under gravity stresses. The key mechanical input parameters are listed below:


Stress ratio x/y =1.0Rock mass friction angle = 30 degRock mass cohesion = 1.0MPaRock mass tensile strength = 1.22MPaMode I fracture toughness = 0.17MPam1/2

Mode II fracture toughness = 0.31MPam1/2

Both the side boundaries and the bottom boundary are fixed with zero displacement in normal and shear directions.

Fracture initiation and propagation are allowed in the numerical model, so that the failure initiation of the rock slope can be investigated.

An interim stage of the progressive failure process is shown in Figure 9-31. Fracturing starts near the toe of the rock slope and propagate upward.

-500 -450 -400 -350 -300 -250 -200 -150 -100 -50 0 50 100 150 200 250 300 350 400 450 500X Axis (m)

-500

-450

-400

-350

-300

-250

-200

-150

-100

-50

0

50

100

150

200

250

300

350

400

450

500

Y A

xis

(m)

-500 -450 -400 -350 -300 -250 -200 -150 -100 -50 0 50 100 150 200 250 300 350 400 450 500X Axis (m)

-500

-450

-400

-350

-300

-250

-200

-150

-100

-50

0

50

100

150

200

250

300

350

400

450

500

Y A

xis

(m)

Rock slop failure process


Pxy (Pa): 0E+0



Elastic fracture

Open fracture

Slipping fracture


Compressive stress

Tensile stress

Fracom Ltd

Date: 07/06/2007 15:44:43

Figure 9-31. Simulated fracturing process of a high rock slope.

94


INPUT DATA FILE

TITLERock slop failure process SYMMETRY -- Model symmetry0 0 0MODULUS -- Poisson's Ratio and Young's modulus0.36 49.4e+9 1TOUK -- Kic and Kiic, mat 1.73e5 3.07e5 1PROPERIES -- mat, ks. kn,phi,coh phid1 12.00e+9 39.20e+9 25.5 0,0 10e-6, 10e-6PROPERIES -- mat, ks. kn,phi,coh phid --- Tensile fractures11 30.99E+9 138.00E+9 33.0 0.00E+08 0 10e-6, 10e-6PROPERIES -- mat, ks. kn,phi,coh phid --- Shear fractures12 30.99E+9 138.00E+9 33.0 0.00E+08 0 10e-6, 10e-6STRESSES -- sxx,syy,sxy -0e+03 -0e+03 0.0e6gravity -- density,gy,sh_sv_ratio,y_surf 2600,-9.81,1.0,400ROCK STRENGTH -- rphi, rcoh, sigt30 1.0e+6 1.22e+6 1RANDOM fracture initiation - f_ini0,l_rand 0.5 1setf -- factor for fracture initiation cut-off level0.00sete -- elastic fracture growth check up level0.0sett -- fracture tip merging tolerance distance 1.0boundary fracture initiationinternal fracture initiationisize20SWINDOW -- xll,xur,yll,yur,numx,numy-500 500 -500 500 60 60permeability -- viscosity,density,perm01e-3,1000,1e-9IWINDOW -- xll,xur,yll,yur-300 300 -300 300edge25 -600 400 -200 400 11 0 -0e3 1edge30 -200 400 200 0 11 0 -0e3 1edge25 200 0 600 0 11 0 -0e3 1

cycl 20endf

95


Example problem #5: shallow tunnel stress analysis

A shallow tunnel is located 3m below ground surface. The key mechanical input parameters are listed below:



The tunnel has a diameter of 4m. The modelled stress and displacement distribution in the rock mass surrounding the tunnel are shown in Figure 9-32 and Figure 9-33

-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0X Axis (m)

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Y A

xis

(m)

-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0X Axis (m)

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Y A

xis

(m)

Test

-2.2

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0

Prin

cipa

l Maj

or S

tres

s (

Pa)

xE

5


Pxy (Pa): 0E+0



Elastic fracture

Open fracture

Slipping fracture


Compressive stress

Tensile stress

Fracom Ltd

Date: 05/06/2007 13:01:00

Figure 9-32. Modelled stress distribution in the vicinity of a shallow tunnel.

96


-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0X Axis (m)

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Y A

xis

(m)

-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0X Axis (m)

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Y A

xis

(m)

Test

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Tot

al D

ispl

acem

ent (

m)

xE-5


Elastic fracture

Open fracture

Slipping fracture


Fracom Ltd

Date: 05/06/2007 13:01:00

Figure 9-33. Modelled displacement distribution in the vicinity of a shallow tunnel.

INPUT DATA FILE

TITLETestSYMMETRY -- Model symmetry0 0 0MODULUS -- Poisson's Ratio and Young's modulus0.25 60e+9 1TOUK -- Kic and Kiic, mat 1.73e6 3.07e6 1PROPERIES -- mat, ks. kn,phi,coh phid1 1000e+9 10000e+9 0.0 0,0 10e-6, 10e-6PROPERIES -- mat, ks. kn,phi,coh phid --- Tensile fractures11 3099E+11 13800E+11 0.0 0.3300E+00 0 10e-6, 10e-6PROPERIES -- mat, ks. kn,phi,coh phid --- Shear fractures12 3099E+11 13800E+11 0.0 0.3300E+00 0 10e-6, 10e-6

STRESSES -- sxx,syy,sxy -0e+06 -0e+6 0.0e6

gravity -- density,gy,sh_sv_ratio,y_surf 2500,-10,1,3.0

ROCK STRENGTH -- rphi, rcoh, sigt33 33e+6 12.2e+6 1

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RANDOM fracture initiation - f_ini0,l_rand 0.4 0setf -- factor for fracture initiation cut-off level 0.95sete -- elastic fracture growth check up level0.0sett -- fracture tip merging tolerance distance 1.0

*boundary fracture initiation*internal fracture initiation

isize0.30

SWINDOW -- xll,xur,yll,yur,numx,numy-4 4 -4 4 60 60

permeability -- viscosity,density,perm01e-3,1000,1e-9

*IWINDOW -- xll,xur,yll,yur-8 8 8 8

monline2 0 10 0 20monline0 2 0 10 20monline2 2 10 10 20

ARCH -- nume,xcen,ycen,diam,ang1,ang2,kode,ss,sn mat 64 0.0 0.0 4 -180.0 180.0 11 0.00E+00 0.00E+00 1

edge 80 -10 3 10 3 1 0 0 1

cycl 1endfile

Note that, in the ARCH command, kode=11 implies a stress boundary with “constrained” elements. Refer to Section 6.2 for explanations.

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Crouch S.L., 1976. Solution of plane elasticity problems by the displacement discontinuity method. Int. J. Num. Methods Engng. 10, 301-343.

Crouch S.L. and Starfield A.M., 1983. Boundary element methods in solid mechanics. George Allen & Unwin (publisher).

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Griffith, A., 1921. The phenomena and rupture flow in solids. Phil. Trans. R. Soc. London. A221, 163-198.

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Hellan K., 1985. Introduction to fracture mechanics. McGraw-Hill Book Company (publisher).

Hoori H. and Nemat-Nasser S., 1985. Compression-induced microcrack growth in brittle solid: axial splitting and shear failure. J. Geophy. Res. 90(B4), 3105-3125.

Horri H. and Nemat-Nasser S., 1986. Brittle failure in compression: splitting, faulting and brittle-ductile transition. Phil. Trans. Roy. Soc., 319 (1549), 337-374.

Hussain M.A., Pu S.L. and Underwood J., 1974. Strain energy release rate for a crack under combined mode I and mode II. Fracture Analysis, ASTM-STP. 560, 2-28. Am. Soc. Testing Materials, Philadelphia.

Ingrafea A., 1987. Finite element models for rock fracture mechanics. Int. J. Num. Ana. Meth. Geomech. 4, 24-43.

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Kemeny, J.M., 1991. A model for non-linear rock deformation under compression due to subcritical crack growth. Int. J. Rock Mech. Min. Sci. 28, 459-467.

Kemeny J M, 2002. The Time-Dependent Reduction of Sliding Cohesion due to Rock Bridges Along Discontinuities: A Fracture Mechanics Approach. Department of Mining and Geological Engineering, University of Arizona, Tucson, U.S.A. Rock Mech. Rock Engng. (2003) 36 (1), 27-38.

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Lee H-S., Jing L., Shen B. Rinne M., Stephansson O. 2003 Modelling brittle fracture and damage between deposition holes by excavation and thermal loading with a stress reconstruction technique. In: Impact of the Excavation Disturbed or Damaged Zone (EDZ) on the performance of Radiaoactive Waste Geological Repositorteory. Proceedings of a European Commission CLUSTER Conference, Luxembourg, 3-5 November 2003. pp.150-154. Li V.C., 1991. Mechanics of shear rupture applied to earthquake zones. In: Fracture mechanics of rock, Atkinson K.B. (ed). Academic Press, London, 351-428.

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Petit J.-P. and Barquins M., 1988. Can natural faults propagate under mode II conditions? Tectonics, 7(6), 1243-1256. Reyes O. and Einstein H.H., 1991. Failure mechanism of fractured rock A fracture coalescence model. Proc. 7th Int. Con. on Rock Mechanics, 1, 333-340.

Rao Q., 1999. Pure shear fracture of brittle rock – A theoretical and laboratory study. PhD Thesis 1999:08, Lulea University of Technology.

Rinne M., Shen B, Lee H-S, Jing L., 2003 Thermo-mechanical simulations of pillar spalling in SKB APSE test by FRACOD. GeoProc International Symposium, Stockholm, Sweden, August 2003.

Savilahti T., Nordlund E. and Stephansson O., 1990. Shear box testing and modelling of joint bridges. In: Rock Joints, Barton & Stephansson (eds). Proc. Int. Symp. Rock Joints (Norway). 295-300.

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APPENDIX I – HOW TO USE THE PREPROCESSOR TO SET UP MODELS

FRACOD provides a pre-processor to help users in setting up the numerical model. The pre-processor is a Window based interface which enables users to see instantly the geometry of the fractures and boundaries they have defined. It also provides pop-up windows to guide the input whenever values are needed. After all the fractures and parameters are defined for a problem, a FRACOD input data file can then be created in a format that FRACOD can read and it is equivalent to the data file created manually by using a text editor.

The pre-processor can be activated by clicking Model Design in the display Window (i.e. the default Window when FRACOD is activated). A second Window, i.e. the Model Design Window will then pop up. The following key functions are included in the Model Design Window.

File (Load, SaveAs, Print)

Load

Open an existing FRACOD input data file. The model geometry and mechanical properties defined by the file will be loaded into the memory and can be shown on the screen. They can also be modified by the user.

SaveAs

Save the defined model into a FRACOD input data file.

Print

Print the current model geometry.

Edit (Copy to Clipborad (BMP); Copy to Clipborad (EMF))

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Copy to Clipborad (BMP)

Copy the current model geometry to Clipborad in BitMap format. It can later be pasted to other Window applications (such as MS Word).

Copy to Clipborad (EMF)

Copy the current model geometry to Clipborad in Enhanced Window Meta Format. It can later be pasted to other Window applications (such as MS Word).

View (Model Properties)

Model Properties

View the properties of a selected object (fracture, edge, arc etc.). The geometrical properties will be shown immediately after this function is selected.

If the selected object is a fracture, the mechanical properties (shear and normal stiffness, friction angle and cohesion) can be viewed and modified by clicking icon “Define Fracture Properties” in the current Window.

If the selected object is a boundary (edge, hole etc.), the boundary conditions of the object can be viewed and modified by clicking icon “Define Boundary Conditions”.

SetUp (Set Parameters)

Set Parameters

Set up the model geometrical and mechanical parameters. Options include:

Caption: Give a title to the current model.

Symmetry: Define the symmetry condition of the model.

XY Range: Define the range of display for both Model Design and the fracture propagation modelling

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Properties: Define the mechanical properties of intact rock (Young’s modulus, Poisson’s ratio, critical strain energy release rates GIc and GIIc) and fractures (shear and normal stiffness, friction angle, cohesion). Up to 10 different fracture properties can be given, each with a material index number (=1-10). Different fractures can be assigned with different fracture properties. Also should be given here are the far-field stresses applied in the model. Intact rock properties for fracture initiation.

Shapes This is an interactive function allowing users to define the model geometry such as boundaries and fractures. It includes the following options:

Arc-Disc (in: Shapes/Arc/Arc-Disc)

Define a disc or part of a disc. The geometrical properties as well as the boundary conditions can be defined and altered in View (Object Properties). The disc can also be repositioned and resized by selecting and dragging the object using mouse.

A disc is defined by giving the coordinate of the centre point, the diameter and the start and end angles (default 180 to -180). The start and end angles have to be defined in clockwise.

Arc-Hole (in: Shapes/Arc/Arc-Hole)

Define a hole (tunnel) in a rock mass. The geometrical properties as well as the boundary conditions can be defined and altered in View (Object Properties). The hole can also be repositioned and resized by selecting and dragging the object using mouse.

A hole is defined by giving the coordinate of the centre point, the diameter and the start and end angles (default -180 to 180). The start and end angles have to be defined in anti-clockwise.

Edge (in: Shapes/Line/Edge)

Define an edge (i.e. a straight boundary) in a rock mass. The geometrical properties as well as the boundary conditions can be defined and altered in View (Object Properties). The edge can also be repositioned and resized by selecting and dragging the object using mouse.

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An edge is defined by giving the coordinates of the start and end points. The start and end points have to be arranged in such a way that the negative side of the edge is the rock mass, as shown below.

Fracture (in: Shapes/Line/Fracture)

Define a fracture in a rock mass. The geometrical properties as well as the mechanical properties of the fracture can be defined and altered in View (Object Properties). The fracture can also be repositioned and resized by selecting and dragging the object using mouse.

A fracture is defined by giving the coordinates of the start and end points. The definition of a fracture is not sensitive to the sequence of the start and end points.

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Start point

End point

Positive side(opening)

Negative side(rock)

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