Modified 12/06/2007
2D
FR
ACO
D

AFR
A
C
O
MP
roduct
TWO DIMENSIONAL FRACTURE
PROPAGATION CODE
(VERSION 2.3)
USER’S MANUAL
FRACOM Ltd.
info@fracom.com.fi
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 nonsymmetrical, and randomly distributed fractures.
FRACOD2D Version 2.3 can simulate the following problems:
(1)
(2)
(3)
(4)
Complex fracture propagations in jointed rock mass;
Multiple region problems;
Time-dependent problems;
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.
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FRACOD V2.3 User’s Manual – Modified 12/06/2007
TABLE OF CONTENTS
ABSTRACT
i
1
INTRODUCTION
1
2
THEORETICAL BACKGROUND
2
2.1
DISPLACEMENT DISCONTINUITY METHOD (DDM)
2
2.1.1
2.1.2
2.2
2.3
2.4
DISPLACEMENT DISCONTINUITY METHOD IN AN INFINITE SOLID
NUMERICAL PROCEDURE
2
4
2.5
SIMULATION OF ROCK DISCONTINUITIES
7
FRACTURE PROPAGATION CRITERION
9
DETERMINATION OF FRACTURE PROPAGATION USING
DDM
11
FRACTURE INITIATION CRITERION
13
3
MULTIPLE REGION PROBLEMS
18
3.1
3.2
18
3.3
INTRODUCTION
THEORETICAL
FORMULATION
FUNCTION
CODE IMPLEMENTATION
4
ITERATION PROCESS
26
4.1
4.2
ITERATION FOR JOINT SLIDING
ITERATION FOR FRACTURE PROPAGATION
26
28
5
TIME-DEPENDENT MODELLING
32
5.1
THEORETICAL BACKGROUND
32
5.1.1
TENSION
5.1.2
FOR
MULTI-REGION
19
25
SUBCRITICAL FRACTURE MODEL FOR A MODE I FRACTURE UNDER PURE
32
SUBCRITICAL FRACTURE MODEL FOR SHEAR AND COMPRESSION
35
5.2
CODE IMPLEMENTATION
36
6
GRAVITATIONAL PROBLEMS
38
6.1
6.2
THEORETICAL BACKGROUND
CODE IMPLEMENTATION
38
40
7
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
SINGLE FRACTURE SUBJECTED TO PURE SHEAR STRESS
MULTIPLE REGION MODEL
SUBCRITICAL CRACK GROWTH - CREEP
GRAVITY PROBLEMS
68
71
74
77
83
9.2
9.3
9.4
9.5
REFERENCES
99
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FRACOD V2.3 User’s Manual – Modified 12/06/2007
APPENDIX I – HOW TO USE THE PREPROCESSOR TO SET UP MODELS
104
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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 nonsymmetrical 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.
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FRACOD User’s Manual
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:
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FRACOD User’s Manual
Dx  u x ( x,0 )  u x ( x,0 )
D y  u y ( x,0 )  u y ( x,0 )
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.
y
+Dy
+Dx
x
2a
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:






u x  Dx 2(1  ) f , y  yf , xx  D y  (1  2 ) f , x  yf , xy
u y  Dx (1  2 ) f , x  yf , xy  D y 2(1  ) f , y  yf , yy


2-2
and



 yy  2GDx  yf, xyy  2GDy  f, yy  yf, yyy 
 xy  2GDx  f, yy  yf, yyy  2GDy  yf, xyy 
 xx  2GDx  2 f, xy  yf, xyy  2GDy f, yy  yf, yyy

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:
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FRACOD User’s Manual
f ( x, y ) 
1 
y
y
y
(arctan

arctan
)
4 (1   ) 
xa
xa
( x  a)
 ( x  a) ln
2

( x  a)
 y 2  ( x  a) ln
2

 y2 

2-4
2.1.2
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.
(a)
(b)
s
s
y
n

n
n=0+
N
(x,y)
j
j
2a
N
n=0-
Ds
j
j
x
Dn
i
2a
i
i
i
3
1 2
3
1 2
s
n
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
j
j
donated as D s and D n . These quantities are defined as follows:
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FRACOD User’s Manual
j
j
j
j
j

n
j

n
Ds  u s  u s
Dn  u  u
2-5
j
j
In these definitions, u s and u n 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.
j
j
The local displacements u s and u n 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
j
displacement discontinuity D n is positive if the two surfaces of the crack
j
displace toward one another. Similarly, the shear component D s 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:

i=1 to N
ij
j
ij
j 
i

 n  Ans Ds  Ann Dn 
i
ij
j
ij
j
 s  Ass Ds  Asn Dn 
2-6
ij
where A ss ,etc., are the boundary influence coefficients for the stresses. The
ij
coefficient A ns , for example, gives the normal stress at the midpoint of the
j
ith element (i.e.  n ) due to a constant unit shear displacement discontinuity
j
over the jth element (i.e. D s =1).
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FRACOD User’s Manual
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,
i
N
ij
N
j
ij
j

 s   Ass Ds   Asn Dn 

 i=1 to N
 n   Ans Ds   Ann Dn 

j 1
j 1
j 1
i
j 1
N
ij
N
j
ij
2-7
j
j
j
If we specify the values of the stress  s and  n 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
j
j
D s and D n . 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
j 
i
N ij j
N ij
u s   Bss Ds   Bsn Dn 
j 1
j 1

 i=1 to N
j
j
i
N ij
N ij

u n   Bns Ds   Bnn Dn 
j 1
j 1

2-8
ij
where B ss , 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
ij
ij
Bsn  Bns  0
ij
ij
1
1
Bss  Bnn   (n  0  ); (n  0  );
2
2
2-9
The remaining coefficients (i.e. the ones for which ij) are continuous and
they can be obtained by using Equations 2-1, 2-2 and 2-3 in Section 2.1.1.
i
i
Displacements u s and u n in Equation 2-8 will exhibit constant
i
i
discontinuities D s and D n , as required.
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FRACOD User’s Manual
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
i
ij
N
j
ij
j
i

 s   Ass Ds   Asn Dn  ( s ) 0 

 i=1 to N
i
i
 n   Ans Ds   Ann Dn  ( n ) 0 

j 1
j 1
j 1
N
j 1
ij
N
j
i
ij
2-10
j
i
where  s and  n represent the shear and normal stresses of the ith element
i
i
respectively; ( s ) 0 , ( n ) 0 are the far-field stresses transformed in the crack
ij
ij
shear and normal directions. Ass , ... , Ann are the influence coefficients, and
j
j
Ds , Dn 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
i
i
following ways, depending on the shear and normal stresses (  s and  n )
of the crack.

i
i
For an open crack  s =  n = 0, therefore the system of governing
equations 2-10 can be rewritten as:
N
i
ij
N
j
ij
j
i

 s  0   Ass Ds   Asn Dn  ( s ) 0 

 i=1 to N
i
i
 n  0   Ans Ds   Ann Dn  ( n ) 0 

j 1
j 1
j 1
N

j 1
ij
N
j
ij
j
2-11
When the two crack surfaces are in elastic contact, the magnitude of
i
i
 s and  n will depend on the crack stiffness (Ks, Kn) and the displacement
j
j
discontinuities ( Ds , Dn )
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FRACOD User’s Manual
i
i
i
i
 s  K s Ds
2-12
 n  K n Dn
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
N
ij
j
ij
j
i
i 
0   Ass Ds   Asn Dn  ( s ) 0  K s Ds 
j 1
j 1

 i=1 to N
N
N
ij
j
ij
j
i
i
0   Ans Ds   Ann Dn  ( n ) 0  K n Dn 

j 1
j 1

2-13
For a crack with its surfaces sliding
i
i
 n  K n Dn
i
i
i
 s    n tan    K n Dn tan 
2-14
i
where  is the friction angle of the crack surfaces. The sign of  s depends
on the sliding direction. Consequently, the system of equations 2-10 can be
presented as:
N ij
N
j
ij
j
i
i

0   Ass Ds   Asn Dn  ( s )0  K n Dn tan  
j 1
j 1

 i=1 to N
N
N
ij
j
ij
j
i
i

0   Ans Ds   Ann Dn  ( n )0  K n Dn

j 1
j 1
j
2-15
j
The displacement discontinuities ( Ds , Dn ) 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
i
(  s ,  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 MohrCoulomb failure criterion:
(1) open joint:
n > 0
(2) elastic joint:
n < 0, |s| < c + |n|tan
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FRACOD User’s Manual
n < 0, |s|  c + |n|tan
(3) sliding joint:
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
i
i
 n  K n Ds tan  d
For dilasive joint, Equation (2-15) becomes
N
ij
j
ij
j
N



  i  1  to  N
ij
j
i
i
i
Ann Dn  ( n ) 0  K n Dn  K n Ds tan  d 

ij
j
i
i
0   Ass Ds   Asn Dn  ( s ) 0  K n Dn tan 
j 1
N
j 1
N
0   Ans Ds  
j 1
j 1
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
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FRACOD User’s Manual
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
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
Original
surface
New
surface
(a)
=
GI
+
GII
Growth
(b)
(c)
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FRACOD User’s Manual
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.
(1). In an arbitrary direction () at a fracture tip there exists a F-value, which
is calculated by
F ( ) 
GI () GII ()

GIc
GIIc
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 ( )
 0
 max .
2-17
(3). When the maximum F-value reaches 1.0, the fracture tip will propagate,
i.e.
F ( )   0  1.0
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
USING DDM
OF
FRACTURE
PROPAGATION
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.
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FRACOD User’s Manual
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
W = 
1
  dV .
v 2 ij ij
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=
1
( s u s   n u n )ds
2 s
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=
1 a
 (  s  (  s ) 0 ) Ds  (  n  (  n ) 0 ) Dn da
2 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.
W
i
i
i
i
i
i
i
i
1
(
a
(


(

)
)
D

a
(


(

)
)
D
 s s0 s
n
n 0
n)
2 i
2-22
The G-value can be estimated by
G () 
W ( a  a )  W ( a )
W

a
a
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
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FRACOD User’s Manual
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.
Crack
a 
a
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
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FRACOD User’s Manual
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
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  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.
New fracture
Tensile stress
Grid point
Shear stress
New fracture
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FRACOD User’s Manual
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
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  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
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FRACOD User’s Manual
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
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.
Grid point
Fracture
initiation
Fracture
propagation
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
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FRACOD User’s Manual
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.
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FRACOD User’s Manual
3
MULTIPLE REGION PROBLEMS
3.1
INTRODUCTION
Rock mass may have different properties in different regions. A multiregion 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.
Shaft
Concrete
EDZ
Rock mass
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.
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FRACOD User’s Manual
3.2
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.
(E, , c,
KIc etc.)2
(E, , c,
KIc etc.) 1
(E, , c,
KIc etc.)2
=

+
(E, , c,
KIc etc.)1
I
n
t
e
r
Figure 3-2. Treatment of multi-regions in FRACOD by modelling
f the two
regions separately.
a
c
e
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
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FRACOD User’s Manual
two triangular regions with two different properties. The two regions are
joined at a straight interface.
Region 2
3
4
6
5
Region 1
1
2
(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, 5
Region 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.
11 1
12
13 3
14
15 5
16
Ass11 Ds1  Asn
Dn  Ass12 Ds2  Asn
Dn2  Ass13 Ds3  Asn
Dn  Ass14 Ds4  Asn
Dn4  Ass15 Ds5  Asn
Dn  Ass16 Ds6  Asn
Dn6  bs1
11 1
11 1
12
12
13 3
13 3
14
14
15 5
15 5
16
16 6
Ans
Ds  Ann
Dn  Ans
Ds2  Ann
Dn2  Ans
Ds  Ann
Dn  Ans
Ds4  Ann
Dn4  Ans
Ds  Ann
Dn  Ans
Ds6  Ann
Dn  bn1
Ass21Ds1  Asn21Dn1  Ass22 Ds2  Asn22 Dn2  Ass23 Ds3  Asn23 Dn3  Ass24 Ds4  Asn24 Dn4  Ass25 Ds5  Asn25 Dn5  Ass26 Ds6  Asn26 Dn6  bs2
21 1
22
23 3
24
25 5
26
Ans21Ds1  Ann
Dn  Ans22 Ds2  Ann
Dn2  Ans23 Ds3  Ann
Dn  Ans24 Ds4  Ann
Dn4  Ans25 Ds5  Ann
Dn  Ans26 Ds6  Ann
Dn6  bn2
Ass31Ds1  Asn31Dn1  Ass32 Ds2  Asn32 Dn2  Ass33 Ds3  Asn33 Dn3  Ass34 Ds4  Asn34 Dn4  Ass35 Ds5  Asn35 Dn5  Ass36 Ds6  Asn36 Dn6  bs3
31 1
32 2
33 3
34 4
35 5
36 6
Ans31Ds1  Ann
Dn  Ans32 Ds2  Ann
Dn  Ans33 Ds3  Ann
Dn  Ans34 Ds4  Ann
Dn  Ans35 Ds5  Ann
Dn  Ans36 Ds6  Ann
Dn  bn3
Ass41Ds1  Asn41Dn1  Ass42 Ds2  Asn42 Dn2  Ass43 Ds3  Asn43 Dn3  Ass44 Ds4  Asn44 Dn4  Ass45 Ds5  Asn45 Dn5  Ass46 Ds6  Asn46 Dn6  bs4
41 1
42 2
43 3
44 4
45 5
46 6
Ans41Ds1  Ann
Dn  Ans42 Ds2  Ann
Dn  Ans43 Ds3  Ann
Dn  Ans44 Ds4  Ann
Dn  Ans45 Ds5  Ann
Dn  Ans46 Ds6  Ann
Dn  bn4
Ass51Ds1  Asn51Dn1  Ass52 Ds2  Asn52 Dn2  Ass53 Ds3  Asn53 Dn3  Ass54 Ds4  Asn54 Dn4  Ass55 Ds5  Asn55 Dn5  Ass56 Ds6  Asn56 Dn6  bs5
51 1
52
53 3
54
55 5
56 6
Ans51Ds1  Ann
Dn  Ans52 Ds2  Ann
Dn2  Ans53 Ds3  Ann
Dn  Ans54 Ds4  Ann
Dn4  Ans55 Ds5  Ann
Dn  Ans56 Ds6  Ann
Dn  bn5
61 1
62
63 3
64
65 5
66
Ass61Ds1  Asn
Dn  Ass62 Ds2  Asn
Dn2  Ass63 Ds3  Asn
Dn  Ass64 Ds4  Asn
Dn4  Ass65 Ds5  Asn
Dn  Ass66 Ds6  Asn
Dn6  bs6
61 1
62
63 3
64
65 5
66 6
Ans61Ds1  Ann
Dn  Ans62 Ds2  Ann
Dn2  Ans63 Ds3  Ann
Dn  Ans64 Ds4  Ann
Dn4  Ans65 Ds5  Ann
Dn  Ans66 Ds6  Ann
Dn  bn6
…………….(10)
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FRACOD User’s Manual
12
where Asn
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. bs1 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. Ass14 , Ass62 etc are zero.
Equation (10) is then simplified as below:
11 1
12
15 5
Ass11Ds1  Asn
Dn  Ass12 Ds2  Asn
Dn2  Ass15 Ds5  Asn
Dn  bs1
11 1
11 1
12
12
15 5
15
Ans
Ds  Ann
Dn  Ans
Ds2  Ann
Dn2  Ans
Ds  Ann
Dn5  bn1
Ass21 Ds1  Asn21 Dn1  Ass22 Ds2  Asn22 Dn2  Ass25 Ds5  Asn25 Dn5  bs2
21 1
22
25 5
Ans21 Ds1  Ann
Dn  Ans22 Ds2  Ann
Dn2  Ans25 Ds5  Ann
Dn  bn2
Ass33 Ds3  Asn33 Dn3  Ass34 Ds4  Asn34 Dn4  Ass36 Ds6  Asn36 Dn6  bs3
33 3
34
36
Ans33 Ds3  Ann
Dn  Ans34 Ds4  Ann
Dn4  Ans36 Ds6  Ann
Dn6  bn3
(11)
Ass43 Ds3  Asn43 Dn3  Ass44 Ds4  Asn44 Dn4  Ass46 Ds6  Asn46 Dn6  bs4
43 3
44
46
Ans43 Ds3  Ann
Dn  Ans44 Ds4  Ann
Dn4  Ans46 Ds6  Ann
Dn6  bn4
Ass51 Ds1  Asn51 Dn1  Ass52 Ds2  Asn52 Dn2  Ass55 Ds5  Asn55 Dn5  bs5
51 1
52
55 5
Ans51 Ds1  Ann
Dn  Ans52 Ds2  Ann
Dn2  Ans55 Ds5  Ann
Dn  bn5
Ass63 Ds3  Asn63 Dn3  Ass64 Ds4  Asn64 Dn4  Ass66 Ds6  Asn66 Dn6  bs6
63 3
64
66
Ans63 Ds3  Ann
Dn  Ans64 Ds4  Ann
Dn4  Ans66 Ds6  Ann
Dn6  bn6
The boundary values bs1 ,..., bn4 of elements 1-4 are known since they are the
real boundaries. The boundary values bs5 ,..., bn6 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  s5 ,  n5 ,  s6
and  n6 , the last four equations in Equation (11) can be rewritten as follows:
Ass51 Ds1  Asn51 Dn1  Ass52 Ds2  Asn52 Dn2  Ass55 Ds5  Asn55 Dn5   s5
51 1
52
55 5
Ans51 Ds1  Ann
Dn  Ans52 Ds2  Ann
Dn2  Ans55 Ds5  Ann
Dn   n5
(12)
Ass63 Ds3  Asn63 Dn3  Ass64 Ds4  Asn64 Dn4  Ass66 Ds6  Asn66 Dn6   s6
63 3
64
66
Ans63 Ds3  Ann
Dn  Ans64 Ds4  Ann
Dn4  Ans66 Ds6  Ann
Dn6   n6
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FRACOD User’s Manual
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:
 s5   s6
(13)
 n5   n6
Using Equation (13) to simplify Equation (12) and after simple rearrangement we got the following equation:
Ass51 Ds1  Asn51 Dn1  Ass52 Ds2  Asn52 Dn2  Ass63 Ds3  Asn63 Dn3  Ass64 Ds4  Asn64 Dn4  Ass55 Ds5  Asn55 Dn5  Ass66 Ds6  Asn66 Dn6  0
51 1
52 2
63 3
64 4
55 5
66 6
Ans51Ds1  Ann
Dn  Ans52 Ds2  Ann
Dn  Ans63 Ds3  Ann
Dn  Ans64 Ds4  Ann
Dn  Ans55 Ds5  Ann
Dn  Ans66 Ds6  Ann
Dn  0
(14)
Similar to the above process, if we consider the displacements of the
interface elements 5 and 6 ( d s5 , d n5 , d s6 and d n6 ), we can get the following
equations for the displacement boundary conditions.
Bss51 Ds1  Bsn51 Dn1  Bss52 Ds2  Bsn52 Dn2  Bss55 Ds5  Bsn55 Dn5  d s5
51 1
52
55 5
Bns51 Ds1  Bnn
Dn  Bns52 Ds2  Bnn
Dn2  Bns55 Ds5  Bnn
Dn  d n5
B D B D B D B D B D B D d
63
ss
3
s
63
sn
3
n
64
ss
4
s
64
sn
4
n
66
ss
6
s
66
sn
6
n
(15)
6
s
63 3
64
66
Bns63 Ds3  Bnn
Dn  Bns64 Ds4  Bnn
Dn4  Bns66 Ds6  Bnn
Dn6  d n6
where Bss51 etc. are the influence coefficient for displacement and d n6 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.
d s5  d s6
d n5  d n6
(16)
Using the above displacement relations in Equation (15), we obtain the
following equations:
 Bss51Ds1  Bsn51Dn1  Bss52 Ds2  Bsn52 Dn2  Bss63 Ds3  Bsn63 Dn3  Bss64 Ds4  Bsn64 Dn4  Bss55 Ds5  Bsn55 Dn5  Bss66 Ds6  Bsn66 Dn6  0
51 1
52 2
63 3
64 4
55 5
66 6
 Bns51Ds1  Bnn
Dn  Bns52 Ds2  Bnn
Dn  Bns63 Ds3  Bnn
Dn  Bns64 Ds4  Bnn
Dn  Bns55 Ds5  Bnn
Dn  Bns66 Ds6  Bnn
Dn  0
(17)
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FRACOD User’s Manual
Using Equations (15) and (17) to replace the last four equations in Equation
(11), we got the following complete systematic equations for the multiregion problem shown in Figure 3-3.
11 1
12
15 5
Ass11Ds1  Asn
Dn  Ass12 Ds2  Asn
Dn2  Ass15 Ds5  Asn
Dn  bs1
11 1
11 1
12
12
15 5
15
Ans
Ds  Ann
Dn  Ans
Ds2  Ann
Dn2  Ans
Ds  Ann
Dn5  bn1
Ass21 Ds1  Asn21 Dn1  Ass22 Ds2  Asn22 Dn2  Ass25 Ds5  Asn25 Dn5  bs2
21 1
22
25 5
Ans21 Ds1  Ann
Dn  Ans22 Ds2  Ann
Dn2  Ans25 Ds5  Ann
Dn  bn2
Ass33 Ds3  Asn33 Dn3  Ass34 Ds4  Asn34 Dn4  Ass36 Ds6  Asn36 Dn6  bs3
(18)
33 3
34
36
Ans33 Ds3  Ann
Dn  Ans34 Ds4  Ann
Dn4  Ans36 Ds6  Ann
Dn6  bn3
Ass43 Ds3  Asn43 Dn3  Ass44 Ds4  Asn44 Dn4  Ass46 Ds6  Asn46 Dn6  bs4
43 3
44
46
Ans43 Ds3  Ann
Dn  Ans44 Ds4  Ann
Dn4  Ans46 Ds6  Ann
Dn6  bn4
Ass51 Ds1  Asn51 Dn1  Ass52 Ds2  Asn52 Dn2  Ass63 Ds3  Asn63 Dn3  Ass64 Ds4  Asn64 Dn4  Ass55 Ds5  Asn55 Dn5  Ass66 Ds6  Asn66 Dn6  0
51 1
52 2
63 3
64 4
55 5
66 6
Ans51Ds1  Ann
Dn  Ans52 Ds2  Ann
Dn  Ans63 Ds3  Ann
Dn  Ans64 Ds4  Ann
Dn  Ans55 Ds5  Ann
Dn  Ans66 Ds6  Ann
Dn  0
 Bss51 Ds1  Bsn51 Dn1  Bss52 Ds2  Bsn52 Dn2  Bss63 Ds3  Bsn63 Dn3  Bss64 Ds4  Bsn64 Dn4  Bss55 Ds5  Bsn55 Dn5  Bss66 Ds6  Bsn66 Dn6  0
51 1
52
63 3
64
55 5
66
 Bns51 Ds1  Bnn
Dn  Bns52 Ds2  Bnn
Dn2  Bns63 Ds3  Bnn
Dn  Bns64 Ds4  Bnn
Dn4  Bns55 Ds5  Bnn
Dn  Bns66 Ds6  Bnn
Dn6  0
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:
11
 Ass11
Asn
Ass12
 11
11
12
Ann
Ans
 Ans
 Ass21
Asn21
Ass22
 21
21
Ann
Ans22
 Ans
 0
0
0

0
0
 0
 0
0
0

0
0
0

51
 A51
Asn
Ass52
 ss51
51
Ann
Ans52
 Ans
 B 51  B 51  B 52
sn
ss
 ss
51
 Bns51  Bnn
 Bns52
12
Asn
12
Ann
Asn22
22
Ann
0
0
0
0
Asn52
52
Ann
 Bsn52
52
 Bnn
0
0
0
0
Ass33
Ans33
Ass43
Ans43
 Ass63
 Ans63
Bss63
Bns63
0
0
0
0
Asn33
33
Ann
Asn43
43
Ann
 Asn63
63
 Ann
Bsn63
63
BBnn
0
0
0
0
Ass34
Ans34
Ass44
Ans44
 Ass64
 Ans64
Bss64
Bns64
0
0
0
0
Asn34
34
Ann
Asn44
44
Ann
 Asn64
64
 Ann
Bsn64
64
Bnn
Ass15
15
Ans
Ass25
Ans25
0
0
0
0
Ass55
Ans55
 Bss55
 Bns55
15
Asn
15
Ann
Asn25
25
Ann
0
0
0
0
Asn55
55
Ann
 Bsn55
55
 Bnn
0
0
0
0
Ass36
Ans36
Ass46
Ans46
 Ass66
 Ans66
Bss66
Bns66
0   Ds1   bs1 
    
0   D21   b21 
0   Ds2  bs2 
    
0   Dn2  bn2 
Asn36   Ds3  bs3 
 3
36   3 
Ann
   Dn   bn 
Asn46   Ds4  bs4 
  4  4
46
Ann
  Dn  bn 
66   5 
0
 Asn
Ds



 
66
5
 Ann
  Dn   0 
Bsn66   Ds6   0 
    
66
  Dn6   0 
Bnn
(19)
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.
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FRACOD User’s Manual
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:
 Ass11
 11
 Ans
 Ass21
 21
 Ans
 0

 0
 0

 0
 A51
 ss51
 Ans
 B 51
 ss
 Bns51
11
Asn
Ass12
11
12
Ann
Ans
21
Asn
Ass22
21
Ann
Ans22
0
0
0
0
0
0
0
0
Asn51
Ass52
51
Ann
Ans52
51
 Bsn  Bss52
51
 Bnn
 Bns52
12
Asn
12
Ann
Asn22
22
Ann
0
0
0
0
Asn52
52
Ann
 Bsn52
52
 Bnn
where 

3
s 0
0
0
0
0
Ass33
Ans33
Ass43
Ans43
 Ass63
 Ans63
Bss63
Bns63
0
0
0
0
Asn33
33
Ann
Asn43
43
Ann
 Asn63
63
 Ann
63
Bsn
63
BBnn
0
0
0
0
Ass34
Ans34
Ass44
Ans44
 Ass64
 Ans64
Bss64
Bns64
0
0
0
0
Asn34
34
Ann
Asn44
44
Ann
 Asn64
64
 Ann
64
Bsn
64
Bnn
Ass15
15
Ans
Ass25
Ans25
0
0
0
0
Ass55
Ans55
 Bss55
 Bns55
15
Asn
15
Ann
Asn25
25
Ann
0
0
0
0
Asn55
55
Ann
 Bsn55
55
 Bnn
0
0
0
0
Ass36
Ans36
Ass46
Ans46
 Ass66
 Ans66
Bss66
Bns66

0   Ds1  
bs1
  1 

1
0   D2  
b2


0   Ds2  
bs2
  2 

2
0   Dn  
bn

Asn36   Ds3   bs3  (bs3 ) 0 
 3

36   3 
3
Ann
   Dn    bn  (bn ) 0 
Asn46   Ds4  bs4  (bs4 ) 0 
  4  4

46
4
Ann
  Dn  bn  (bn ) 0 
 Asn66   Ds5   (bs6 ) 0 
  5 

66
 Ann
  Dn   (bn6 ) 0 

Bsn66   Ds6  
0
   

66
  Dn6  

Bnn
0
(20)
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 multiregion problems.
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FRACOD User’s Manual
3.3
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.
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FRACOD User’s Manual
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
s
(s)max
(Ds)j
(s)j=0
(s)i
(Ds)i
(Ds)max
Ds
Figure 4-1. Iteration process to simulate complex loading path.
Let’s consider a joint element simulated by FRACOD (
Figure 4-1). 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.
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FRACOD User’s Manual
To model this complex process in FRACOD, we can subdivide the total
maximum shear displacement into many small increment such as (Ds)i and
(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:
( s ) i  K s (Ds ) i
If the joint element is sliding such as at increment j, the shear stress
increment is:
(  s ) i  0
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
( s ) i 
( s ) j 
 (
n 1,i
s
 (
n 1, j
)n
s
)n
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.
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FRACOD User’s Manual
(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,
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 4-2.
The problem can be decomposed into two stages as shown in Figure 4-2:
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 preexisting crack only. This stage is equivalent to the case that the growth
element does not exist.
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FRACOD User’s Manual
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 4-3, 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 4-2 and Figure 4-3, 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.
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FRACOD User’s Manual

Crack growth
crack
D
||

Stage 1
-1
crack
D0
+
Stage 2
1
crack
D
Figure 4-2. Decomposition of crack growth problem for modelling using
iteration (mode I crack growth).
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FRACOD User’s Manual

1
=1tan
crack
D
||

Stage 1
1
1
crack
D0
+
Stage 2
crack
-(1-1tan)
D
Figure 4-3. Decomposition of crack growth problem for modelling using
iteration (mode II crack growth).
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FRACOD User’s Manual
5
5.1
TIME-DEPENDENT MODELLING
THEORETICAL BACKGROUND
Classical fracture mechanics postulates that a fracture tip which has a stress
intensity equal to the material’s critical fracture toughness, K Ic , 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:
y 
KI
2  r
(5-1)
Stress y varies with the distance r from the crack tip and it becomes infinity
at the fracture tip.
Figure 5-1. An infinite plate containing a crack under biaxial loading.
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FRACOD User’s Manual
The stress intensity factor K I 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 ).
KI     a
(5-2)
In classical fracture mechanics the fracture initiation from the fracture tip
takes place when
K I  K Ic ,
(5-3)
K Ic 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
K I  K Ic ,
(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:
v1  AK n1
(5-5)
where v1 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
K I  K Ic , the constant of proportionality, A, can be expressed as
A
(v max )1
K Icn
(5-6)
and the expression for propagation velocity becomes
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FRACOD User’s Manual
K
v1  (v max )1  I
 K Ic



n1
(5-7)
(Expression used by Olson (1993) and Kemeny (2002)).
Now we replace the stress intensity factor with Eq. (5-2) and we get:
   a 

v1  (v max )1 

 K Ic 
n1
(5-8)
If the stress intensity factor K I 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:
t
aeff (t )  a0   v1 (t )dt
(5-9)
0
where a 0 is the initial crack length.
Adding this into Eq. (5-8 ), we get a momentary crack velocity at time (t):
n1
t


    (a  v(t )dt ) 
0
0


(5-10)
v1 (t )  (v max )1 

K Ic






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:
a(t t )
     at 1
 (vmax )1 

K Ic

n1
t t
t 1

   (t )   (a0  a1  a2  ...  at 1 )
 t t 1
t 0

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FRACOD User’s Manual
(5-11)
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 5-2. 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
K II     a
(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
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FRACOD User’s Manual
is also argued that because cracking is not restricted to a single major crack
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.
K
v 2  (v max ) 2  II
 K IIc



n2
(5-13)
where v2 is the crack velocity for mode II creep, (vmax ) 2 is the maximum
mode II crack velocity, n2 is a constant, K II is the mode II stress intensity
factor and K IIc is the mode fracture toughness.
5.2
CODE IMPLEMENTATION
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 t
l  v1t ; or l  v2 t
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+Nt 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
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FRACOD User’s Manual
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.
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FRACOD User’s Manual
6
6.1
GRAVITATIONAL PROBLEMS
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 insitu 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 6-1. 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
i
N
ij
N
j
ij
j
i

 s   A ss D s   A sn D n  ( s ) g 
j 1
i
N
j 1
ij
N
j
 n   A ns D s  
j 1
i
j 1

  i  1  to  N
ij
j
i
A nn D n  ( n ) g 

6-1
i
where ( s ) g and ( n ) g represent the initial gravitational stresses in shear
and normal directions of the ith element before the excavation was made.
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FRACOD User’s Manual
y
Ground surface
dj
di
d1
j
i
p (xp,yp)
i
1
i
x
Figure 6-1. A shallow underground excavation to demonstrate the gravity
effect.
i
i
( s ) g and ( n ) g can be calculated from the stresses in x- and y-direction
i
i
( xx ) g and ( yy ) g and the element orientation angle.
The gravitational stresses in the x-y coordinates are
i
( yy ) g  gd i
i
( xx ) g  kgd i
where
6-2
 - rock density;
g – gravity acceleration;
k – ratio of horizontal stress to vertical stress.
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FRACOD User’s Manual
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:
N
p
pj
j

N
pj
j 1


A xyn D n
  i  1  to  N

pj
j
p

A yyn D n  ( yy ) g 

j
p
 xx   A xxs D s   A xxn D n  ( xx ) g 
j 1
N
p
pj
j
N
 xy   A xys D s  
j 1
N
p
j 1
pj
j
N
 yy   A yys D s  
j 1
j 1
pj
j
6-3
pj
where A xxs etc. are the influence coefficients of a unit discontinuity on the
stresses at point p.
6.2
CODE IMPLEMENTATION
Theoretically speaking, implementing the gravity stresses in FRACOD
should be a simple process. We only need to replace the far-field stress
i
i
i
i
( s ) 0 , ( n ) 0 with the gravitational stresses ( s ) g and ( n ) g 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 Figure
6-1 where the unbalanced gravitational force on the excavation boundaries
will cause numerical problems.
Figure 6-2 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
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FRACOD User’s Manual
consequently very large displacement discontinuities Ds and Dn at the
boundary elements, significant numerical errors are resulted in.
Test
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
X Axis (m)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
3.5
3.5
3.0
3.0
2.5
2.5
2.0
2.0
1.5
1.5
1.0
1.0
0.5
0.5
0
-0.2
-0.4
0
-0.5
-0.5
-1.0
-1.0
-1.0
-1.2
-1.4
-1.6
-1.8
-2.0
Pxx (Pa): 0E+0
-1.5
Pyy (Pa): 0E+0
-1.5
Pxy (Pa): 0E+0
-2.2
Principal Major Stress (Pa) xE5
0
-0.8
Y Axis (m)
Y Axis (m)
-0.6
Max. Compres. Stress (Pa): 3.63214E+5
Max. Tensile Stress (Pa): 5.49543E+4
-2.0
Elastic fracture
-2.0
Open fracture
-2.5
Slipping fracture
-2.6
-2.5
-2.8
Fracture with Water
Compressive stress
-3.0
Tensile stress
-3.0
Fracom Ltd
-3.5
Date: 05/06/2007 09:17:56
-2.4
-3.5
-3.0
-3.2
-3.4
-4.0
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0
0.5
X Axis (m)
1.0
1.5
2.0
2.5
3.0
3.5
-4.0
4.0
Figure 6-2. 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
i
i
i
 s  K s Ds
6-4
 n  K n Dn
“Constrained” element means that the element does not have free shear and normal
movement even if the stresses on the element are zero.
2
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FRACOD User’s Manual
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
6-1, the modelling results are much more stable and accurate, see Figure
6-3.
Test
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
X Axis (m)
-0.5
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
3.5
3.5
3.0
3.0
2.5
2.5
2.0
2.0
0
-0.2
1.5
1.5
1.0
1.0
0.5
0.5
-0.4
0
-0.5
-1.0
Pxx (Pa): 0E+0
-1.5
-0.5
-1.0
-1.0
-1.2
Pyy (Pa): 0E+0
-1.5
Pxy (Pa): 0E+0
-1.4
Max. Compres. Stress (Pa): 2.22553E+5
Max. Tensile Stress (Pa): 2.60582E+4
-2.0
Elastic fracture
-2.0
-1.6
Open fracture
-2.5
Slipping fracture
-2.5
Fracture with Water
Compressive stress
-3.0
Tensile stress
-1.8
-3.0
-2.0
Fracom Ltd
-3.5
-4.0
-4.0
Date: 05/06/2007 10:04:58
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0
0.5
X Axis (m)
1.0
1.5
2.0
2.5
-0.8
3.0
3.5
-3.5
-4.0
4.0
-2.2
Figure 6-3. Correct modelling results after using “constrained” elements at
excavation boundaries for gravity problems.
42
Principal Major Stress (Pa) xE5
0
Y Axis (m)
Y Axis (m)
-0.6
FRACOD User’s Manual
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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FRACOD User’s Manual
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 -----------------------------------------TITLE
A borehole with four cracks loaded in uniaxial compression
SYMMETRY -- Model symmetry
4 0.00 0.00
MODULUS -- Poisson Ratio and Youngs modulus, material no.
0.25 0.40E+11 1
TOUGHNESS -- Gic and Giic, material no.
50.
1000.
1
PROPERIES -- 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-6
SWINDOW -- xll,xur,yll,yur,numx,numy
-3.00
3.00
-3.00
3.00
30
30
STRESSES -- sxx,syy,sxy
0.00E+07 -0.15E+08
0.00E+00
FRACTURE -- nume,xbeg,ybeg,xend,yend,kode,jmat, mat
5
0.700
0.700
1.000
1.000 2 1 1
ARCH -- nume,xcen,ycen,diam,ang1,ang2,kode,ss,sn, mat
10
0.0
0.0
2.0
0.0
90.0 1 0.00
0.00
CYCL 1000
ENDFILE
STOP
---------- 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).
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FRACOD User’s Manual
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 interface
mat_pos – material no. of the positive side of the interface
Example:AINTERFACE-num,xc,yc,diameter,ang1,ang2,mat_neg,mat_pos
20
ARCH
0.0 0.0
1.2
0
90
2
1
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 to
gradsy – gradient for bvs in y-direction
gradny – 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.
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FRACOD User’s Manual
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 I
vmax2 – maximum mode II creep velocity (m/s)
n2 – exponential factor of creep velocity for mode II
totalT – 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
K
Subcritical crack velocity: v  v max  I
 K Ic



n
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
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FRACOD User’s Manual
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 to
gradsy – gradient for bvs in y-direction
gradny – gradient for bvn in y-direction
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FRACOD User’s Manual
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:
xend,yend
Positive side
(opening)
Negative
side (rock)
xbeg,ybeg
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 to
gradsy – gradient for bvs in y-direction
gradny – 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 cracks
d_wall, rand_e
d_wall – distance into rock where random excavation induced cracks exist
rand_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
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FRACOD User’s Manual
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 function
jmat -- 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 stress
y_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).
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FRACOD User’s Manual
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 interface
mat_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 ratio
50
FRACOD User’s Manual
e - Young’s modulus (Pa)
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,npoint
0 -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,perm0
1e-3,1000,1e-10
PROP
gives fracture surface contact properties
jmat, ks, kn, phi, coh, phid, aperture0, aperture_r
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FRACOD User’s Manual
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
e  max (e0  d n ), er 
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:



if  0 
 a
m


p  0;
p
2



a

 ;

(1  a )2   m
1
p  1.0;



if  a 
 1.0 
m




if 
 1.0 
 m

where p =probability of fracture initiation;
/m =ratio of the stress to strength and
a= fracture initiation level. a= f_ini0
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FRACOD User’s Manual
Example: RANDOM – f_ini0,l_rand
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, mat
30 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 direction
numy -- number of grid points along tangential direction
Example: RWINDOW -- xc0,yc0,radium,numr,numt
0.00
0.00
8
80
90
SYMM
gives symmetry conditions
ksym, xsym, ysym
ksym = 0
ksym = 1
ksym = 2
ksym = 3
ksym = 4
-- no symmetry
-- problem symmetrical against vertical line x=xsym (m)
-- problem symmetrical against horizontal line y=ysym (m)
-- problem symmetrical against point x=xsym and y=ysym (m)
-- problem symmetrical against line x=xsym and line y=ysym (m)
Example: SYMMETRY
0 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
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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
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FRACOD User’s Manual
Pxx – Far-field horizontal stress (Pa)
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-direction
numy -- number of grid points along y-direction
Example: SWINDOW -- xll,xur,yll,yur,numx,numy
-2 2 -2 2 80 80
SYMM gives symmetry conditions
ksym, xsym, ysym
ksym = 0
ksym = 1
ksym = 2
ksym = 3
ksym = 4
-- no symmetry
-- problem symmetrical against vertical line x=xsym (m)
-- problem symmetrical against horizontal line y=ysym (m)
-- problem symmetrical against point x=xsym and y=ysym (m)
-- 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)
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FRACOD User’s Manual
GIIc=(1-2)KIIc2/E
(KIIc - Fracture toughness mode II)
mat – material no.
Example: TOUG
20
TOUK
100
1
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 pressure
w_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 pressure
w_xmin – minimum value of the range in x-direction
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FRACOD User’s Manual
w_xmax– maximum value of the range in x-direction
w_ymi n– minimum value of the range in y-direction
w_ymax– maximum value of the range in x-direction
wp – 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.
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FRACOD User’s Manual
8
CONDUCT AND MONITOR THE CALCULATION
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.
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FRACOD User’s Manual
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.
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FRACOD User’s Manual
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 fracture
Green – sliding fracture
Red – 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 model
Pyy – normal stress in the vertical direction of the model
Pxy – shear stress
The magnitudes of stresses are in Pa unless specified otherwise.
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FRACOD User’s Manual
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 Boundary
Plot 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.
Fractures
Plot on screen the geometry of the fractures in the model.
Principal Stress
Plot 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 Stress
Plot 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
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FRACOD User’s Manual
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)
Displacement
Plot 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 Displacement
Plot 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 Displacement
Plot 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.
Aperture
Plot on screen the joint aperture. The joint aperture is calculated by:
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FRACOD User’s Manual
e  max (e0  d n ), er 
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 background
This option is used together with the Image function to fill the contours.
Clear image/ contours
Clean the image/ contours plots on screen
View (Rotate plot)
Rotate plot
Rotate 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)
Color
Set or change colors for filled contour plots
Color bar
Show a legend vertical color bar beside the filled contour plots
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FRACOD User’s Manual
Legend
Show 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 Bar
Show 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 in
Enlarge the plot in a specified window (defined by dragging the Mouse).
Zoom out
Reduce the plot the plot in a specified window (defined by dragging the
Mouse).
Full screen
Return the plot size to the originally specified window (full screen).
View (Magnifier, Plot setup, Contour setup)
Magnifier
Magnify an area of the screen. To do so, locate the mouse cursor to the
desired position and press down the mouse right button.
Plot setup
Specify 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 setup
Specify or change the contour plot setting
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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.
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FRACOD User’s Manual
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
0.0
1.0
0.059
0.059
2.01
2.01
0
0
-5e6
-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 (…)
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FRACOD User’s Manual
Standard Window management functions which enable users to arrange the
multiple calculation Windows.
Help (…)
On-line user’s manual and helping functions.
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9
FRACOD VERIFICATION TESTS
Three verification tests cases are listed below. The data files are provided in
the program package.
9.1
SINGLE FRACTURE
TENSILE STRESS
(1)
SUBJECTED
TO
NORMAL
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 = 190103 J/m2
The theoretical solution of this problem gives the stress intensity factor (KI)
as
K I   a
 50  3.1416  1  88.6MPa m
where a = half length of the fracture.
The theoretical strain energy release rate is then calculated as:
1  2
(K I ) 2
E
1  0.25 2

 (88.6  10 6 ) 2  184  10 3 J / m 2
9
40  10
(G I ) theory 
The difference between the numerical result and the theoretical result is
approximately 3%.
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FRACOD User’s Manual
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:
F (0) 
G I (0) G II (0)

G Ic
G IIc
190  10 3
0


 3800
50
1000
The F-value is by far greater than the critical value 1.0. Hence fracture
propagation is detected.
(2)
Input data
----------------------------------------------------------------------------TITLE
A single fracture under tension
SYMMETRY -- Model symmetry
0 0.00 0.00
MODULUS -- Poisson's Ratio and Young's modulus, mat
0.25 0.40E+11 1
TOUGHNESS -- Gic and Giic, mat
50. 1000. 1
PROPERIES -- jmat, ks. kn,phi,coh, dila
1 0.10E+14 0.10E+14 30.0 0.00E+00
SWINDOW -- xll,xur,yll,yur,numx,numy
-5.00
5.00 -5.00
5.00 30 30
STRESSES -- sxx,syy,sxy
-0.0E+07 0.50E+08 0.00E+00
FRACTURE -- nume,xbeg,ybeg,xend,yend,kode,jmat, mat
25 -1.000 -0.000 1.000 0.000 2 1, 1
CYCL 10
ENDFILE
--------------------------------------------------------------------------------
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(3)
FRACOD model
A single fracture under tension
Maximum Displacement (m): 4.1006E-3
Creep Time: 0:0:0
Creep Time Step: 0:0:0
5
Max. Crack velocity (m/s): 0E+0
-5
-4
-3
-2
-1
X Axis (m)
0
1
2
3
4
5
5
Cycle: 10of 10
Elastic fracture
Open fracture
Slipping fracture
4
4
3
3
2
2
1
1
0
0
-1
-1
-2
-2
-3
-3
-4
-4
Fracture with Water
-5
Y Axis (m)
Y Axis (m)
Fracom Ltd
Date: 12/06/2005 09:51:26
-5
-5
-4
-3
-2
-1
0
X Axis (m)
1
2
3
4
5
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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
 K Ic

 K IIc

G Ic
1375



 1.17
G IIc
1000
 numerical
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).
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(2)
Input data
----------------------------------------------------------------------------TITLE
Single fracture subjected to pure shear stress
SYMMETRY -- Model symmetry
0 0.00 0.00
MODULUS – Poisson’s Ratio and Young’s modulus
0.25 0.40E+11
TOUGHNESS -- Gic and Giic
1289.0 1000.
PROPERIES -- mat, kn. ks,phi,coh
1 0.0E+0 0.0E+0 0.0 0.00E+00
SWINDOW -- xll,xur,yll,yur,numx,numy
-2.00
2.00 -2.00
2.00 30 30
STRESSES -- sxx,syy,sxy
-0.0E+06 -0.00E+0 50.00E+06
FRACTURE -- nume,xbeg,ybeg,xend,yend,kode,mat
30 -1. 0. 1. 0. 2 1
CYCL 1000
ENDFILE
--------------------------------------------------------------------------------
72
FRACOD User’s Manual
(3)
FRACOD model
Single fracture in pure shear
Maximum Displacement (m): 5.70476E-3
Creep Time: 0:0:0
-2.0
Creep Time Step: 0:0:0
-1.8
-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
X Axis (m)
-0.2
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Max. Crack velocity (m/s): 0E+0
Cycle: 17of 1000
1.8
Elastic fracture
Open fracture
Slipping fracture
Fracture with Water
1.8
GIC=1270J/m2
GIIC=1000J/m2
1.6
1.4
1.6
1.4
Y Axis (m)
1.2
1.2
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1.0
-1.0
-1.2
-1.2
-1.4
-1.4
-1.6
-1.6
-1.8
-1.8
-2.0
Y Axis (m)
Fracom Ltd
Date: 12/06/2005 09:54:17
-2.0
-2.0
-1.8
-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
-2.0
-1.8
-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0
0.2
X Axis (m)
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Single fracture in pure shear
Maximum Displacement (m): 2.67123E-3
Creep Time: 0:0:0
Creep Time Step: 0:0:0
-0.4
X Axis (m)
-0.2
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Max. Crack velocity (m/s): 0E+0
1.8
Elastic fracture
Open fracture
1.8
2
GIC=1480J/m
GIIC=1000J/m2
1.6
Slipping fracture
Fracture with Water
1.4
Fracom Ltd
Y Axis (m)
Date: 12/06/2005 09:59:48
1.6
1.4
1.2
1.2
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1.0
-1.0
-1.2
-1.2
-1.4
-1.4
-1.6
-1.6
-1.8
-1.8
-2.0
-2.0
-2.0
-1.8
-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0
0.2
X Axis (m)
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
73
Y Axis (m)
Cycle: 6 of 1000
FRACOD User’s Manual
9.3
MULTIPLE REGION MODEL
A simple case of a boundary value problem in an inhomogeneous elastic
body is shown in Figure 9-1. The region of interest consists of an annulus
a  r  b with elastic constants E1 and 1 inside a circular hole of radius
r  b in a large plate with elastic constants E2 and 2. The inside wall of the
annulus is subjected to a normal stress  rr   p , and the plate is unstressed
at infinity. The solution to this problem, satisfying continuity of radial stress
and displacement at the interface r  b , can be constructed from standard
formulas from thick-walled cylinders. The radial and tangential stresses are:
1

 rr 
p(a / b)2  p'  ( p  p' )( a / r )2
arb
2
1  ( a / b)
  




1

p(a / b)2  p'  ( p  p' )( a / r )2
2
1  ( a / b)
arb
 rr   p' (b / r )2
rb
    p' (b / r )2
rb
(21)
in which
p' 
p ( a / b) 2
1  E 1  2
1
1   1

2
2  E2 1  1 1  1
(22)

1  (a / b)2 

E
2
;

E
2
r
1
p
a ;

1
b
Figure 9-1. Annulus inside a circular hole in a plate.
74
FRACOD User’s Manual
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 = 0.5m
b = 1.0m
1 = 1 = 0.2
E1 = 50GPa; E2 = 25GPa
p = 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:
TITLE
Multi-region code verification test
SYMMETRY -- Model symmetry
000
MODULUS -- Poisson's Ratio and Young's modulus,mat
0.2 50e+9, 1
MODULUS -- Poisson's Ratio and Young's modulus,may
0.2 25e+9, 2
TOUK -- Kic and Kiic, mat
3.0e6 0.75e6 1
PROPERTIES (old joints) -- mat, ks kn,phi,coh,phid
1 35.5e0 65.5E0 30 0e6 0
PROPERIES -- mat, ks. kn,phi,coh phid --- Tensile fractures
11 10e14 10e14 30.0 4E+06 5
PROPERIES -- mat, ks. kn,phi,coh phid --- Shear fractures
12 10e14 10e14 30.0 4E+06 5
STRESSES -- sxx,syy,sxy
0e6 0 0
ROCK STRENGTH -- rphi, rcoh, sigt,mat
30 4e+06 2.492e+06,1
SWINDOW -- xll,xur,yll,yur,numx,numy
-2 2 -2 2 80 80
ARCH
60 0 0 0.5 0 360 1 0 -10e6 1
AINTERFACE
60 0 0 1.0 360 0 1 2
CYCL 1000
75
FRACOD User’s Manual
The modelled stress distribution using the new FRACOD is shown in Figure
9-2. A comparison of the modelled radial and tangential stresses with the
analytical results is shown in
Figure 9-3. A good agreement is obtained, suggesting that the new
FRACOD accurately simulates the multi-region problem.
Pxx: 0E+0
Multi-region code testing, using different properties in different regions
Pyy: 0E+0
Pxy: 0E+0
-0.8
Max. Compres. Stress: 1E+7
Max. Tensile Stress: 1.27281E+7
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
X Axis (m)
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
Elastic fracture
Open fracture
Slipping fracture
Fracture with Water
Compressive stress
Tensile stress
Y Axis (m)
Date: 27/12/2003 09:53:11
0
0
-0.1
-0.1
-0.2
-0.2
-0.3
-0.3
-0.4
-0.4
-0.5
-0.5
-0.6
-0.6
-0.7
Y Axis (m)
Fracom Ltd
-0.7
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
X Axis (m)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Figure 9-2. Modelled stress distribution for the test problem.
15.0
Srr
 (numerical)
S00
rr (numerical)
S1
 (theory)
S2
rr (theory)
10.0
Stress (MPa)
5.0
0.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-5.0
-10.0
-15.0
Distance to tunnel centre (m)
Figure 9-3. Comparison between the FRACOD results and analytical
results.
76
FRACOD User’s Manual
9.4
SUBCRITICAL CRACK GROWTH - CREEP
Problem definition
σ=remote tensile stress: (10 MPa)
2ao=crack length in the start (0.01m)
KIC=3.80 MPa*m1/2
A=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.
Figure 9-4. An infinite plate containing a crack under tensional loading.
Analytical solution
Crack length when t=t
1

    n   n / 21
1-1
 n / 21
  t 
a   a0
  n / 2  1 A


K 

  c 
 
Time for a crack to extended from a0 to a.
 n / 2 1
a  n / 21  a0
t
n
  

 n / 2  1A
 Kc 
From the analytical solution we get a stress intensity level of K I/KIC=0.3298
1-2
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
FRACOD User’s Manual
FRACOD results
Table 9-1. Material parameters.
Parameter
Value
unit
and
INTACT ROCK
Poisson’s ratio for intact rock
0.24
Young’s modulus, intact rock
Friction angle (no fracture initiation
considered)
FRACTURE
Fracture toughness in mode I (in DTHMC 3,21)
Fracture toughness in mode II
68.0 GPa
49 °
Normal stiffness, Kn
61 GPa/m
Shear stiffness, Ks
35.5 GPa/m
Fracture friction angle
31
Cohesion
0 MPa
Dilation angle
5
σ=remote tensile stress 10 MPa
3.80 MPa*m1/2
4.40 MPa*m1/2
σ=remote tensile stress 10 MPa
Figure 9-5. FRACOD calculation. Infinite plate containing a crack under
10MPa tensile load results in a stable model.
78
FRACOD User’s Manual
Table 9-2. Creep parameters for tensile fractures etc.
Parameter
Constant A (=max propagation
velocity)
Stress corrosion factor n
Time range for calculations
Time step is automatically determined within
a specified range:
Delta t min
Delta t max
Model symmetry
Number of elements defining the
fracture (standard case)
Stress window and gridpoint density
Num x num y
Value
and unit
500 m/s
30
1x109 sec
1s
1000s
No symmetry
25
-0.07 .07 -0.07 .07
18 x 18
σ=remote tensile stress;10 MPa
Figure 9-6. 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
FRACOD User’s Manual
Figure 9-7. Time until the initial crack (l=0.01m) has extended to a length
of 0.05m is 2.04x108 seconds.
Figure 9-8. KI/KIC vs time.
80
FRACOD User’s Manual
Figure 9-9. KI/KIC vs crack velocity.
Figure 9-10. 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 User’s Manual
FRACOD CREEP model accuracy
Figure 9-11. 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 9-12. 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
FRACOD User’s Manual
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 9-13.
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).
Test
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
X Axis (m)
0
0.5
1.0
1.5
2.0
2.5
Maximum Displacement (m): 7.00126E-5
Elastic fracture
Open fracture
2.5
2.5
Slipping fracture
Fracture with Water
Fracom Ltd
2.0
2.0
x=0
Date: 07/06/2007 13:13:50
1.5
1.5
1.0
1.0
6.5
6.0
5.5
0.5
0
y=0
-0.5
4.5
4.0
-0.5
3.5
-1.0
-1.0
-1.5
-1.5
3.0
Total Displacement (m) xE-5
0
5.0
Y Axis (m)
Y Axis (m)
0.5
2.5
2.0
-2.0
-2.0
-2.5
-2.5
1.5
1.0
0.5
-3.0
-3.0
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0
X Axis (m)
0.5
1.0
1.5
2.0
2.5
Figure 9-13. Predicted displacement field around a circular tunnel.
83
FRACOD User’s Manual
dx (y=0 line)
Stresses (y=0 line)
8.00E-06
0.00E+00
1.5
2
2.5
3
3.5
-2.00E+06
Stress (Pa)
5.00E-06
4.00E-06
3.00E-06
-3.00E+06
-4.00E+06
-5.00E+06
2.00E-06
sigxx (FRACOD)
-6.00E+06
sigyy (FRACOD)
1.00E-06
-7.00E+06
0.00E+00
1
1.5
2
2.5
3
3.5
4
4.5
5
sigxx (Analytical)
sigyy (Analytical)
-8.00E+06
X (m)
X (m)
dy (x=0 line)
Stresses (x=0 line)
0.00E+00
5.00E+05
1
1.5
2
2.5
3
-1.00E-05
3.5
4
FRACOD
4.5
5
0.00E+00
Analytical
-2.00E-05
1
1.5
2
2.5
3
3.5
4
-5.00E+05
-3.00E-05
Stress (Pa)
Displacement (m)
4
-1.00E+06
Analytical
6.00E-06
Displacement (m)
1
FRACOD
7.00E-06
-4.00E-05
-5.00E-05
-1.00E+06
-1.50E+06
-6.00E-05
-2.00E+06
-7.00E-05
-2.50E+06
-8.00E-05
Y (m)
sigxx (FRACOD)
sigyy (FRACOD)
sigxx (Analytical)
sigyy (Analytical)
Y (m)
Figure 9-14. 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 9-13. The comparison results are shown in
Figure 9-14. 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.
84
FRACOD User’s Manual
Example problem #2: rock slope stress analysis
A small rock slope is under gravity stresses only. The key input parameters
used are:
Rock density = 2500kg/m3
Stress ratio x/y =0.33
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 9-15 and Figure 9-16.
6
-1
0
1
2
3
4
X Axis (m)
5
6
7
8
9
Pxx (Pa): 0E+0
5
10
11
6
Pyy (Pa): 0E+0
5
Pxy (Pa): 0E+0
Max. Compres. Stress (Pa): 3.42044E+5
Max. Tensile Stress (Pa): 3.46191E+4
Elastic fracture
4
4
Open fracture
0
Slipping fracture
Fracture with Water
3
-0.2
3
Compressive stress
Tensile stress
-0.4
Fracom Ltd
2
2
Date: 05/06/2007 12:08:59
-0.6
-0.8
1
1
0
-1
-1
-1.2
-1.4
-1.6
-1.8
-2
-2
-3
-3
-4
-4
-2.0
Principal Major Stress (Pa) xE5
0
Y Axis (m)
Y Axis (m)
-1.0
-2.2
-2.4
-2.6
-2.8
-3.0
-5
-5
-3.2
-6
-6
-1
0
1
2
3
4
5
X Axis (m)
6
7
8
9
10
-3.4
11
Figure 9-15. Stress distribution around a small slope – FRACOD results.
85
FRACOD User’s Manual
6
-1
0
1
2
3
4
X Axis (m)
5
6
7
8
9
10
11
6
Maximum Displacement (m): 6.88985E-6
5
5
Elastic fracture
Open fracture
Slipping fracture
Fracture with Water
4
4
Fracom Ltd
Date: 05/06/2007 12:08:59
6.5
3
3
2
2
6.0
1
0
0
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
5.0
4.5
4.0
3.5
3.0
Y - Displacement (m) xE-6
1
Y Axis (m)
Y Axis (m)
5.5
2.5
2.0
1.5
1.0
0.5
-6
-6
-1
0
1
2
3
4
5
X Axis (m)
6
7
8
9
10
0
11
Figure 9-16. 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 9-17. 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
FRACOD User’s Manual
6.E-06
FRACOD dy
5.E-06
FLAC dy
FRACOD dx
Displacement (m)
4.E-06
FLAC dx
3.E-06
2.E-06
1.E-06
0.E+00
0
1
2
3
4
5
6
7
8
9
10
-1.E-06
X (m)
FRACOD sigyy
-2.E+05
FLAC syy
-1.E+05
FRACOD sigxx
FLAC sxx
Stress (Pa)
-1.E+05
-1.E+05
-8.E+04
-6.E+04
-4.E+04
-2.E+04
0.E+00
0
1
2
3
4
5
X (m)
6
7
8
9
10
Figure 9-17. Comparison of FRACOD results with FLAC results.
87
FRACOD User’s Manual
INPUT DATA FILE (FRACOD)
TITLE
Rock slope stress analysis
SYMMETRY -- Model symmetry
0 0 0
MODULUS -- Poisson's Ratio and Young's modulus
0.25 60e+9 1
TOUK -- Kic and Kiic, mat
1.73e6
3.07e6
1
PROPERIES -- mat, ks. kn,phi,coh phid
1 1200e+6 3920e+6 25.5 0,0 10e-6, 10e-6
PROPERIES -- mat, ks. kn,phi,coh phid --- Tensile fractures
11 3099E+9 13800E+9 33.0 0.3300E+08 2 10e-6, 10e-6
PROPERIES -- mat, ks. kn,phi,coh phid --- Shear fractures
12 3099E+9 13800E+9 33.0 0.3300E+08 2 10e-6, 10e-6
STRESSES -- sxx,syy,sxy Depth=500m:
sx=18.82Mpa;
sy=20.47Mpa; sxy=-5.30Mpa; sz=12.75Mpa
-0e+03 -0e+03 0.0e6
gravity -- density,gy,sh_sv_ratio,y_surf
2500,-10,0.33,5
ROCK STRENGTH -- rphi, rcoh, sigt/peak strength c=33,
st=12.2Mpa
33 33e+6 12.2e+6 1
RANDOM fracture initiation - f_ini0,l_rand (initiation level,
random or not)
0.4 0
setf -- factor for fracture initiation cut-off level (??% of
max FoS)
0.95
sete -- elastic fracture growth check up level
0.0
sett -- fracture tip merging tolerance distance (*averge half
length of lements)
1.0
boundary fracture initiation
internal fracture initiation
isize
0.30
SWINDOW -- xll,xur,yll,yur,numx,numy
-1 11 -6 6 50 50
permeability -- viscosity,density,perm0
1e-3,1000,1e-9
monl
0 -0.8 10 -0.8 49
edge
20 0 5 5 5
1 0 -0e3 1
edge
20 5 5 5 0
1 0 -0e3 1
edge
20 5 0 10 0
1 0 -0e3 1
edge
20 10 0 10 -5 2 0 -0e3 1
edge
40 10 -5 0 -5 2 0 0 1
edge
40 0 -5 0 5 2 0 0 1
cycl 1
endf
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INPUT DATA FILE (FLAC)
new
gr 50,50
m 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 conditions
fix x i=1
*fix y i=1
fix x i=51
*fix y i=51
*fix x j=1
fix y j=1
cyc 5000
sav ini.sav
ini xdis=0 ydis=0
hist
hist
hist
hist
hist
hist
hist
hist
ydisp
ydisp
ydisp
ydisp
ydisp
ydisp
ydisp
ydisp
hist
hist
hist
hist
hist
hist
hist
hist
xdis
xdis
xdis
xdis
xdis
xdis
xdis
xdis
i=1,j=28
i=1,j=29
i=1,j=30
i=1,j=31
i=1,j=32
i=1,j=33
i=1,j=34
i=1,j=35
i=28,j=26
i=29,j=26
i=30,j=26
i=31,j=26
i=32,j=26
i=33,j=26
i=34,j=26
i=35,j=26
m n i=26,50 j=26,50
fix
fix
fix
fix
fix
fix
x
y
x
y
x
y
i=1
i=1
i=51
i=51
j=1
j=1
cyc
sav
cyc
sav
cyc
sav
5000
run1.sav
5000
run2.sav
5000
run3.sav
return
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Example problem #3: large rock slope stress analysis
A large inclined rock slope is under gravity stresses only. The key input
parameters used are:
Rock density = 2500kg/m3
Stress ratio x/y =0.33
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 9-15.
Large rock slop stress analysis
X Axis (m)
-600 -550 -500 -450 -400 -350 -300 -250 -200 -150 -100 -50
0
50
Pxx (Pa): 0E+0
450
Pyy (Pa): 0E+0
Pxy (Pa): 0E+0
Max. Compres. Stress (Pa): 2.01047E+7
400
350
100 150 200 250 300 350 400 450
Max. Tensile Stress (Pa): 7.52593E+5
Cycle: 1 of 1
Monitoring line
400
350
Elastic fracture
Open fracture
300
450
300
Slipping fracture
Fracture with Water
250
250
Compressive stress
200
Tensile stress
Fracom Ltd
Date: 12/06/2007 12:47:19
150
200
-0.2
150
100
100
50
50
0
0
-0.4
-50
-100
-150
-150
-200
-200
-250
-250
-300
-300
-350
-350
-400
-400
-450
-450
-500
-500
-550
-550
-0.8
-1.0
-1.2
Principal Major Stress (Pa) xE7
-50
-100
Y Axis (m)
Y Axis (m)
-0.6
-1.4
-1.6
-600
-600 -550 -500 -450 -400 -350 -300 -250 -200 -150 -100 -50
0
X Axis (m)
-600
50
-1.8
-2.0
100 150 200 250 300 350 400 450
Figure 9-18. 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 9-19. The FRACOD results
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FRACOD User’s Manual
agree very well with the FLAC results (the overall difference is less than
5%)
3.00E+06
2.00E+06
1.00E+06
Stress (Pa)
0.00E+00
-600
-1.00E+06
-500
-400
-300
-200
-100
0
100
200
-2.00E+06
-3.00E+06
-4.00E+06
sxx (FLAC)
-5.00E+06
syy (FLAC)
sxy (FLAC)
-6.00E+06
sxx (Fracod)
-7.00E+06
syy (Fracod)
sxy (Fracod)
-8.00E+06
x (m)
Figure 9-19. Comparison of FRACOD results with FLAC results.
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FRACOD User’s Manual
INPUT DATA FILE (FRACOD)
TITLE
Large rock slop stress analysis
SYMMETRY -- Model symmetry
0 0 0
MODULUS -- Poisson's Ratio and Young's modulus
0.25 60e+9 1
TOUK -- Kic and Kiic, mat
1.73e5
3.07e5
1
PROPERIES -- mat, ks. kn,phi,coh phid
1 12.00e+9 39.20e+9 25.5 0,0 10e-6, 10e-6
PROPERIES -- mat, ks. kn,phi,coh phid --- Tensile fractures
11 30.99E+9 138.00E+9 33.0 0.00E+08 0 10e-6, 10e-6
PROPERIES -- mat, ks. kn,phi,coh phid --- Shear fractures
12 30.99E+9 138.00E+9 33.0 0.00E+08 0 10e-6, 10e-6
STRESSES -- sxx,syy,sxy
-0e+03 -0e+03 0.0e6
gravity -- density,gy,sh_sv_ratio,y_surf
2500,-10,0.33,400
ROCK STRENGTH -- rphi, rcoh, sigt
30 1.0e+9 1.22e+9 1
RANDOM fracture initiation - f_ini0,l_rand (initiation level,
random or not)
0.5 1
setf -- factor for fracture initiation cut-off level (??% of
max FoS)
0.00
sete -- elastic fracture growth check up level
0.0
sett -- fracture tip merging tolerance distance (*averge half
length of lements)
1.0
boundary fracture initiation
internal fracture initiation
isize
20
SWINDOW -- xll,xur,yll,yur,numx,numy
-600 500 -600 500 60 60
permeability -- viscosity,density,perm0
1e-3,1000,1e-9
IWINDOW -- xll,xur,yll,yur
-300 300 -300 300
monline
-600 129 71 129 50
edge
25 -600 400 -200 400 1 0 -0e3 1
edge
30 -200 400 200 0 1 0 -0e3 1
edge
25 200 0 600 0 1 0 -0e3 1
edge
20 600 0 600 -400 4 0 -0e3 1
edge
40 600 -400 -600 -400 4 0 0 1
edge
40 -600 -400 -600 400 4 0 0 1
cycl 1
endf
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INPUT DATA FILE (FLAC)
new
grid 60 60
mod 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 conditions
fix x i=1
fix x i=61
fix y j=1
cyc 5000
sav ini1.sav
*ini xdis=0 ydis=0
m n i=41,60 j=30,60
fix x i=1
*fix y i=1
fix x i=61
*fix y i=61
*fix x j=1
fix y j=1
cyc
sav
cyc
sav
cyc
sav
5000
run1a.sav
5000
run2a.sav
5000
run3a.sav
return
pause
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FRACOD User’s Manual
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:
Rock density = 2500kg/m3
Stress ratio x/y =1.0
Rock mass friction angle = 30 deg
Rock mass cohesion = 1.0MPa
Rock mass tensile strength = 1.22MPa
Mode 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 9-20.
Fracturing starts near the toe of the rock slope and propagate upward.
Rock slop failure process
-500 -450 -400 -350 -300 -250 -200 -150 -100
500
X Axis (m)
-50
0
50
100
150
200
250
300
350
Pxx (Pa): 0E+0
450
400
450
Pyy (Pa): 0E+0
500
500
450
Pxy (Pa): 0E+0
Max. Compres. Stress (Pa): 2.45205E+7
400
Max. Tensile Stress (Pa): 1.39879E+6
Elastic fracture
350
350
Open fracture
Slipping fracture
300
Fracture with Water
Compressive stress
250
Tensile stress
Fracom Ltd
200
400
300
250
200
150
150
100
100
50
50
0
0
-50
-50
-100
-100
-150
-150
-200
-200
-250
-250
-300
-300
-350
-350
-400
-400
-450
-450
-500
-500 -450 -400 -350 -300 -250 -200 -150 -100
-50
0
50
X Axis (m)
100
150
200
250
300
350
400
450
Y Axis (m)
Y Axis (m)
Date: 07/06/2007 15:44:43
-500
500
Figure 9-20. Simulated fracturing process of a high rock slope.
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FRACOD User’s Manual
INPUT DATA FILE
TITLE
Rock slop failure process
SYMMETRY -- Model symmetry
0 0 0
MODULUS -- Poisson's Ratio and Young's modulus
0.36 49.4e+9 1
TOUK -- Kic and Kiic, mat
1.73e5
3.07e5
1
PROPERIES -- mat, ks. kn,phi,coh phid
1 12.00e+9 39.20e+9 25.5 0,0 10e-6, 10e-6
PROPERIES -- mat, ks. kn,phi,coh phid --- Tensile fractures
11 30.99E+9 138.00E+9 33.0 0.00E+08 0 10e-6, 10e-6
PROPERIES -- mat, ks. kn,phi,coh phid --- Shear fractures
12 30.99E+9 138.00E+9 33.0 0.00E+08 0 10e-6, 10e-6
STRESSES -- sxx,syy,sxy
-0e+03 -0e+03 0.0e6
gravity -- density,gy,sh_sv_ratio,y_surf
2600,-9.81,1.0,400
ROCK STRENGTH -- rphi, rcoh, sigt
30 1.0e+6 1.22e+6 1
RANDOM fracture initiation - f_ini0,l_rand
0.5 1
setf -- factor for fracture initiation cut-off level
0.00
sete -- elastic fracture growth check up level
0.0
sett -- fracture tip merging tolerance distance
1.0
boundary fracture initiation
internal fracture initiation
isize
20
SWINDOW -- xll,xur,yll,yur,numx,numy
-500 500 -500 500 60 60
permeability -- viscosity,density,perm0
1e-3,1000,1e-9
IWINDOW -- xll,xur,yll,yur
-300 300 -300 300
edge
25 -600 400 -200 400
11 0 -0e3 1
edge
30 -200 400 200 0
11 0 -0e3 1
edge
25 200 0 600 0
11 0 -0e3 1
cycl 20
endf
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FRACOD User’s Manual
Example problem #5: shallow tunnel stress analysis
A shallow tunnel is located 3m below ground surface. The key mechanical
input parameters are listed below:
Rock density = 2500kg/m3
Stress ratio x/y =1.0
The tunnel has a diameter of 4m. The modelled stress and displacement
distribution in the rock mass surrounding the tunnel are shown in Figure
9-21 and Figure 9-22
Test
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
X Axis (m)
-0.5
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
3.5
3.5
3.0
3.0
2.5
2.5
2.0
2.0
1.5
1.5
1.0
1.0
0.5
0.5
0
-0.2
-0.4
0
-0.8
-0.5
-0.5
-1.0
-1.0
-1.0
-1.2
Pxx (Pa): 0E+0
-1.5
Pyy (Pa): 0E+0
-1.5
Pxy (Pa): 0E+0
-1.4
Principal Major Stress (Pa) xE5
0
Y Axis (m)
Y Axis (m)
-0.6
Max. Compres. Stress (Pa): 2.22185E+5
Max. Tensile Stress (Pa): 2.64097E+4
-2.0
-2.0
Elastic fracture
-1.6
Open fracture
-2.5
Slipping fracture
-2.5
Fracture with Water
Compressive stress
-3.0
-1.8
-3.0
Tensile stress
-2.0
Fracom Ltd
-3.5
-3.5
Date: 05/06/2007 13:01:00
-4.0
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0
0.5
X Axis (m)
1.0
1.5
2.0
2.5
3.0
3.5
-4.0
4.0
-2.2
Figure 9-21. Modelled stress distribution in the vicinity of a shallow tunnel.
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FRACOD User’s Manual
Test
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
X Axis (m)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
3.5
3.5
3.0
3.0
2.5
2.5
2.0
2.0
1.0
1.5
1.5
0.9
1.0
1.0
0.5
0.5
0
-0.5
-0.5
-1.0
-1.0
-1.5
-1.5
-2.0
0.7
0.6
0.5
Total Displacement (m) xE-5
0
Y Axis (m)
Y Axis (m)
0.8
0.4
-2.0
Maximum Displacement (m): 1.06232E-5
0.3
Elastic fracture
-2.5
Open fracture
-2.5
Slipping fracture
Fracture with Water
-3.0
0.2
-3.0
Fracom Ltd
Date: 05/06/2007 13:01:00
-3.5
-4.0
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0
0.5
X Axis (m)
1.0
1.5
2.0
2.5
3.0
3.5
-3.5
0.1
-4.0
4.0
Figure 9-22. Modelled displacement distribution in the vicinity of a shallow
tunnel.
INPUT DATA FILE
TITLE
Test
SYMMETRY -- Model symmetry
0 0 0
MODULUS -- Poisson's Ratio and Young's modulus
0.25 60e+9 1
TOUK -- Kic and Kiic, mat
1.73e6
3.07e6
1
PROPERIES -- mat, ks. kn,phi,coh phid
1 1000e+9 10000e+9 0.0 0,0 10e-6, 10e-6
PROPERIES -- mat, ks. kn,phi,coh phid --- Tensile fractures
11 3099E+11 13800E+11 0.0 0.3300E+00 0 10e-6, 10e-6
PROPERIES -- mat, ks. kn,phi,coh phid --- Shear fractures
12 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, sigt
33 33e+6 12.2e+6 1
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FRACOD User’s Manual
RANDOM fracture initiation - f_ini0,l_rand 0.4 0
setf -- factor for fracture initiation cut-off level 0.95
sete -- elastic fracture growth check up level
0.0
sett -- fracture tip merging tolerance distance
1.0
*boundary fracture initiation
*internal fracture initiation
isize
0.30
SWINDOW -- xll,xur,yll,yur,numx,numy
-4 4 -4 4 60 60
permeability -- viscosity,density,perm0
1e-3,1000,1e-9
*IWINDOW -- xll,xur,yll,yur
-8 8 8 8
monline
2 0 10 0 20
monline
0 2 0 10 20
monline
2 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 1
endfile
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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FRACOD User’s Manual
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FRACOD User’s Manual
Sih G.C., 1974. Strain-energy-density factor applied to mixed mode crack
problems. Int. J. Fracture. 10(3), 305-321.
Wang R., Zhao Y., Chen Y., Yan H., Yin Y., Yao C. and Zhang H.,
1987. Experimental and finite element simulation of X-type shear fractures
from a crack in marble. Tectonophysics. 144, 141-150.
Wong, T-F, 1982. Micromechanics of faulting in Westerly granite. Int. J.
Rock Mech. Min. Sci. 19, 49-64.
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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.
End point
Positive side
(opening)
Negative side
(rock)
Start point
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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