Corso di “Leggi costitutive dei geomateriali”
Dottorato di Ricerca in Ingegneria Geotecnica
Fracture mechanics approach to
the study of failure in rock
Claudio Scavia, Marta Castelli
Politecnico di Torino
Dipartimento di Ingegneria Strutturale e Geotecnica
Corso di “Leggi costitutive dei geomateriali” – Novembre 2005
Dottorato di ricerca in Ingegneria Geotecnica
Index

Introduction

Basic concepts of Linear Elastic Fracture Mechanics

Propagation criteria

Non linear Fracture Mechanics

Numerical modelling of cracked rock structures

The Displacement Discontinuity Method

Numerical simulation of experimental results

Application to slope stability
2
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Introduction
Since Coulomb (1776) the problem of failure in natural and manmade material have been approached on the basis of the
traditional concept of
Material strength
This approach cannot explain some disastrous brittle failures and
can be (depending on the scale) a great oversimplification of the
crack initiation process
Schenectady ship (1943)
Tay bridge (Scotland, 1898)
3
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Introduction
The main cause of fracture initiation is the presence of defects
in the material, which concentrate the stress at their tips
Large natural defects (faults, joints…) exist in rock masses
Example: progressive failure in slopes
Fracture Mechanics
makes it possible to take such phenomenon into account
through a study of the triggering and propagation of cracks
starting from natural defects or discontinuities
4
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Introduction
Main steps in Fracture Mechanics
 Analysis of the state of stress
Evaluation of stress
concentration
 Choice of a propagation criterion
 Definition of a methodology for the simulation of crack
propagation


stable propagation
unstable propagation
5
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Modes of failure in rocks
At the scale of the laboratory
Direct
tension
Indirect
tension
1
1
Axial
splitting
1
Shear
band
1
3
6
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Modes of failure in rocks
At the scale of the rock mass
Direct
tension
Indirect
tension
Shear
7
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Index

Introduction

Basic concepts of Linear Elastic Fracture Mechanics

Propagation criteria

Non linear Fracture Mechanics

Numerical modelling of cracked rock structures

The Displacement Discontinuity Method

Numerical simulation of experimental results

Application to slope stability
8
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Linear Elastic Fracture Mechanics

Elastic behaviour of the material

Inelastic behaviour of crack surfaces

Determination of stress concentration at the crack tip
 fracture energy
 stress intensity factor

Definition of the conditions for crack to propagate,
through energetic or stress intensity balances
9
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Stress concentration



circular hole
elliptical hole
crack
2b
2b0
a
a



a
σ max  σ 1  2  
b


max
3
max = 3
max = f(a, b)



r
max  
r
r10
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Energetic approach (Griffith, 1921)
Condition for crack propagation
dWe dWs

da
da
a22
We 
E
Ws  4ag
elastic energy
release rate
surface energy

2gE
a
2g = fracture energy Gc
fracture energy is a material characteristic which accounts for the
energy required to create the new surface area, and for any additional
energy absorbed by the fracturing process, such as plastic work
11
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Tensional approach (Irwin, 1957)
Crack propagation can be studied through the superposition of the
effects of three independent load application modes


(I)


(II)


(III)
mode I  opening - loads are orthogonal to the fracture plane
mode II  slip
- loads are tangent to the fracture plane in the
direction of maximum dimension
mode III  tear
- loads are contained in the fracture plane and
act perpendicularly to mode II
12
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Tensional approach

The state of stress in plane conditions (modes I and II) at
a point P close to the crack tip is given as:
1
 
3

2 
r 
cos K I 1  sin   K IIsin  2K II tan 
2r
2 
2 2
2
1

 3

cos K Icos 2  K IIsin
2r
2
2 2

1

r 
cos K Isin  K II 3cos   1
2r
2
 
13
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Tensional approach

For =0
i.e. for a point at a distance r along
the line of the crack:
x  K I
y  K I
 xy  K II
y
x

ûy
r

For relative displacements û
between the crack faces at a small
distance x from the crack tip:
1
2r
1
2r
1
2r
ûx  K I
4 x
G1    2
ûy  K II
4 x
G1    2
14
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Tensional approach

r


stresses tend to infinity when r  0
the Stress Intensity Factors K quantify the effect of
geometry, loads, and restraints on the magnitude of the
stress field near the tip
15
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Meaning of the Stress Intensity Factors
Example: crack of length 2a, located in a plate subjected to a
uniform vertical tensile stress

The vertical stress, y, around the crack tip is given
by the theory of elasticity:
y  
a
2r
2b0
a
The specific boundary conditions of the problem
affect the value of y through a constant term KI
which is given by:
K I   a
1
KI   y
2r
G1    2
KI  û y
4 x

y


r
16
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Meaning of the Stress Intensity Factors

The value of K is representative of the stress field around
the crack tip

for known geometrical characteristics of the specimens, it is
possible to determine the critical value of K (toughness of
the material) that will trigger propagation

A comparison between the experimental values of KC and
the values computed at the tips of cracks makes it possible
to establish whether or not they can propagate, provided
that the behaviour of the rock material is assumed to be
linear-elastic
propagation criterion
17
Corso di “Leggi costitutive dei geomateriali” – Novembre 2005
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Index

Introduction

Basic concepts of Linear Elastic Fracture Mechanics

Propagation criteria

Non linear Fracture Mechanics

Numerical modelling of cracked rock structures

The Displacement Discontinuity Method

Numerical simulation of experimental results

Application to slope stability
18
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Propagation criteria
open cracks:
mode I propagation takes place in most brittle materials, and
a Linear Elastic Fracture Mechanics approach is suitable for the
simulation of the phenomenon, on the basis of the fracture
toughness KIC (or fracture energy GIc)
closed and compressed cracks:
several mechanisms must be taken into account, and different
criteria are to be chosen for the study of induced-tensile and
shear propagation
In some case it is necessary to resort to a non linear
approach, depending on the extension of the zone of localized
deformation
19
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Open cracks (Erdogan & Sih, 1963)

cracks spread radially starting from their tips;

the direction of propagation, defined by an angle 0, is
perpendicular to the direction along which the maximum
tensile stress, (0), is found;

crack begins to spread when (0) reaches a critical value
(0)C;

By expressing (0) and (0)C as a function of the stress
intensity factors, the propagation criterion can be written
in this form:

0 
0
3

2r  K IC  K eq  cos K I cos
 K IIsin  0 
2 
2 2

2
where KIC is the material toughness
20
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Open cracks (Erdogan & Sih, 1963)
 r  0  cos
0
2
K Isin 0  K II 3cos 0  1
K I sin  0  K II 3 cos  0  1  0

For pure mode I:

For pure mode II:
K II  0
KI  0
K I sin  0  0   0  0
K II 3 cos 0  1  0  0  70.5
21
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Open cracks


KI > 0
KII = 0

KI < 0
KII  0

KI < 0
KII = 0
KI = 0
KII  0

KI > 0
KII  0
22
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Closed cracks
Induced-tensile propagation:
Brittle phenomenon (mixed mode)
The original crack is compressed, while the
part that propagates is open and in a tensile
stress field
(Erdogan & Sih, 1963)  KIC
Shear propagation:
(mode II)
The original crack is compressed, and it
propagates in compressive stress fields
KIIC?
23
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Shear propagation criteria
A controversial issue is whether or not it is possible to apply LEFM
concepts to the analysis of shear failure
Experimental evidence show that
compressed cracks in brittle materials
evolve along shear fracture planes
only after a long process involving
the formation of microcracks under
tensile stresses, their propagation
and coalescence in large-scale shear
progressive failure
1
3
The propagation is accompanied by
considerable energy dissipation due
to friction
The meaning of fracture toughness in mode II (KIIC) is still
under discussion
24
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Fracture toughness: mode I
Experimental determination
Suggested methods (ISRM, 1988)
Short rod (SR)
Chevron bend (CB)
25
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Short rod
P
D
W
h

t
a0
a
a1
D
P
load on specimen
diameter of short rod specimen
length of specimen
depth of crack in notch flank
chevron angle
notch width
chevron tip distance
crack length
maximum depth of chevron flanks
t
notch
a0
a
a1
24Pmax
K IC  1.5
D
W

uncut rock or ligament
D/2
26
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Chevron bend
h
P
Support roller
loading roller

a
a0
A
notch
S
L
a
CMOD
knife
A  Pmax
K IC  1.5
D
uncut rock
or ligament
D
P
A
L
S
D
CMOD
h

a0
a
load on specimen
projected ligament area
specimen length
distance between support points
diameter of chevron bend specimen
relative opening of knife edges
depth of crack in notch flank
chevron angle = 90°
chevron tip distance
crack length
27
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Chevron bend
28
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When is a LEFM approach applicable?
Extremely high stress values involved in the phenomenon of
crack propagation:
a zone of material exhibiting a non linear behaviour (process
zone) always forms at the crack tips, where the actual
evolution of stresses is bound to deviate from the theoretical
elastic values
only when this zone is small compared to the size of the
structure, the actual evolution of stresses will still be
governed by K and the Linear Elastic Fracture Mechanics
procedure can be applied
29
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Index

Introduction

Basic concepts of Linear Elastic Fracture Mechanics

Propagation criteria

Non linear Fracture Mechanics

Numerical modelling of cracked rock structures

The Displacement Discontinuity Method

Numerical simulation of experimental results

Application to slope stability
30
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Non Linear Fracture Mechanics

Elastic behaviour of the material

Inelastic behaviour inside the process zone and on
crack surfaces

Stress distribution does not present any singularity at
the crack tip


stresses must be computed taking into account
different constitutive models for intact material and
the process zone
Definition of the conditions for the propagation of
the crack and the process zone on the basis of
material strength
31
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Non linear Fracture Mechanics
Process zone at the crack tip
zone accompanying crack initiation and propagation in which
inelastic material response is occurring
The micro-structural process of breakdown near the crack tip
can be interpreted by assuming that it gives rise to cohesive
stresses, which oppose the action of applied loads
32
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Non Linear Fracture Mechanics
Open cracks (tension): the Cohesive Crack Model
(Dugdale, 1960; Barenblatt, 1962)
stress free
inelastic
stress distribution
elastic
stress distribution
t
dc
Visible crack
true crack
process zone
33
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Non Linear Fracture Mechanics
Closed cracks (compression and shear):
the Slip-Weakening Model (Palmer & Rice, 1973)
real crack
process
zone
G
r
n
d
real tip

p
r
fictitious tip
r
 Here, a relation is assumed
between relative
displacement d and shear
stress 
 A residual shear strength r
occurs when d reaches a
critical value d*
d
n
 A process zone is introduced
at the crack tip, where the
damage is concentrated

  = process zone extension
d*
d
 G = energy amount stored
inside the process zone
34
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Index

Introduction

Basic concepts of Linear Elastic Fracture Mechanics

Propagation criteria

Non linear Fracture Mechanics

Numerical modelling of cracked rock structures

The Displacement Discontinuity Method

Numerical simulation of experimental results

Application to slope stability
35
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Numerical modelling of cracked rock structures
Analysis of the state of stress and
simulation of the propagation
Resort to numerical techniques for the analysis of cracked rock
structures proves necessary because of the geometrical complexity
of most application problems

Finite Element Method (FEM)
Needs a re-meshing at each crack propagation step

Boundary Element Method (BEM)
requires only the discretisation of the structure boundaries
and hence it is suited to deal with problems characterised by
evolving geometries
36
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Numerical modelling of cracked rock structures
Displacement Discontinuity Method
(Crouch & Starfield, 1983)
allows to simulate the crack as Displacement Discontinuity elements
n
Ds = us(s, 0-) - us (s, 0+)
+Dn
Dn = un(s, 0-) - un (s, 0+)
+Ds
s
2a
37
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The Displacement Discontinuity Method
computer code
(N)
BEMCOM
bj
(1)
bi
known tangential and
normal stresses or
displacements acting
on the i-th element
influence coefficients of Ds(j) and Dn(j)
on stresses or displacements over
the i-th element
N
N
j1
j1
N
N
j1
j1
s i   A ss i, j  Ds  j   A sn i, j  Dn  j
n i   A ns i, j  Ds  j   A nn i, j  Dn  j
unknown displacement discontinuities in
the tangential and normal directions, in
the centre of the j-th element
38
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Open elements
Tensile stress fields

Dn < 0 (opening)

s(i), n(i) = 0
Compressive stress fields

Dn> 0 (closure)

s(i), n(i) = 0
39
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Closed elements
Compressive stress fields


Dn = 0
s(i), n(i)  0
s
sr = n·tan
Ks
Ds
No Displacement Discontinuities in the normal direction
A tangential Displacement Discontinuity occurs if and when the
available frictional shear strength is mobilised
40
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Simulation of crack propagation

open cracks
(Scavia, 1995; Scavia et al., 1997)
Erdogan & Sih’s propagation criterion, based on the Stress
Intensity factors calculation at the tip of the crack

closed cracks


induced-tensile propagation: Erdogan & Sih’s criterion
shear propagation: calculation of the stress field near the tip
and its comparison with the Mohr Coulomb strength criterion

The load is applied in step, and the possibility of crack
propagation is evaluated at each step. If such possibility is
verified, a new element is added at the crack tip

Two kind of propagation may occur:


stable propagation may develop only if the load is increased
unstable propagation: develops without any load increment
41
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Numerical implementation of the SWM
Computer code BEMCOM
(Allodi et al., 2002)
tip element
real crack
non-cohesive
process zone
cohesive
process zone
42
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Adopted slip-weakening laws
c

cp
p
r
0
dc*
d
Cohesion (c)
d*
0
Friction angle ()

intact material (tip element): cp, p

real crack: c = 0,  = r

d
process zone: linear variation of c and  as a function of
dc* and d*
43
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Index

Introduction

Basic concepts of Linear Elastic Fracture Mechanics

Propagation criteria

Non linear Fracture Mechanics

Numerical modelling of cracked rock structures

The Displacement Discontinuity Method

Numerical simulation of experimental results

Application to slope stability
44
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Numerical simulation of experimental results
The computer code BEMCOM has been used to simulate
some experimental results through a
LEFM approach:

Induced-tensile propagation in hard rock bridges (Castelli, 1998)
Experimental work on concrete samples containing two open slits
subjected to uni-axial compression

Shear propagation in soft rocks (Scavia et al., 1997)
Experimental work on Beaucaire marl samples subjected to uni-axial
compression in plane-strain conditions (Tillard, 1992)
45
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Induced-tensile propagation (Castelli, 1998)
Experimental work on concrete samples containing two open
slits subjected to uni-axial compression
Et50 (MPa)
Es50 (MPa)
t50 (-)
s50 (-)
C0 (MPa)
T0 (MPa)
K1C (MPa*m)
c (MPa)
p (°)
20800
17600
0.21
0.11
74
3.53
0.94
23.7
35.5
Characteristic of the material
Geometry and load configuration
46
12
axial stress (MPa)
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Experimental results
horizontal
10
longitudinal
oblique
longitudinal
8
6
onset of propagation
(numerical simulation)
4
oblique
2
0
0
5000
10000
15000
horizontal
strains (microstrain)
Stress-strain diagram
Strain directions
47
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Propagation trajectories
Experimental
Numerical
48
Experimental work on Beaucaire marl samples subjected to uniaxial compression in plane-strain conditions (Tillard, 1992)
Axial load (KN)
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Shear propagation (Scavia et al., 1997)
2.5
2.0
3
1.5
4
5 6 7
8
9 10 11 12
2
1.0
0.5
1
0
0
1
2
3
4
axial strain (%)
Axial load-axial strain diagram
5
c5c7
c5-c7
c7c8
c7-c8
measured displacements
(stereo-photogrammetry)
49
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Numerical simulation


two initial notches, 2 mm long and inclined 28° to the
vertical, are inserted at the upper corners of the specimen
onset of propagation occurs at an axial applied stress
equal to 0.9 MPa
E (MPa)
 (-)
c (MPa)
p (°), intact material
r (°), crack surfaces
C0 (MPa)
Sample height (mm)
Sample width (mm)
Sample thickness (mm)
81
0.35
0.33
28
20
1.10
120
60
35
b = 28°
l = 2mm
50
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Propagation trajectories
c5c7
c7c8
Experimental
c5c7
c7c8
Numerical
51
The numerical model is unable to simulate the global response
of a specimen under load (no energy dissipation in the elastic
material)
2.5
2.0
3
1.5
4
5 6 7
Axial load (kN)
Axial load (KN)
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Limit of a LEFM approach
8
9 10 11 12
2
1.0
0.5
1
0
0
1
2
3
axial strain (%)
4
5
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
0.4
0.8
1.2
1.6
2.0
Axial strain (%)
experimental
numerical
52
NLFM approach to shear propagation
 Biaxial compression tests in
plane strain conditions
(Marello, 2004)
1200
 Axial load under
displacement control
 No lateral
confinement
LB-01
LB-02
LB-04
 Prismatic specimens of
Beaucaire marl (two
different samples)
 Specimen dimensions:
170 x 80 x 35 mm3
MB-09
MB-10
mm3
85 x 40 x 35
(LB-02, LB-04)
MB-11
LB-01
LB-02
LB-04
MB-09
MB-10
MB-11
1000
axial stress [kPa]
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Experimental results
800
600
400
200
0
0
1
4
3
2
global axial strain [%]
5
6
Photographs of the specimens during the tests in order to carry out a
stereo-photogrammetric analysis (Desrues, 1995)
53
couple 1-2
axial stress [kPa]
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Experimental results: test MB-11
900
2
600
3
shear deformations
4
300
5
6
1
0
0
1
2
3
4
5
global axial strain [%]
"experimental" photographs
54
shear deformations
900
axial stress [kPa]
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Experimental results: test MB-11
2
couple 2-3
3
600
4
300
5
6
1
0
0
1
2
3
4
5
global axial strain [%]
"experimental" photographs
55
axial stress [kPa]
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Experimental results: test MB-11
shear deformations
900
2
600
3
4
300
5
couple 3-4
6
1
0
0
1
2
3
4
5
global axial strain [%]
"experimental" photographs
56
axial stress [kPa]
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Experimental results: test MB-11
Displacement vectors
900
2
600
3
4
300
5
couple 5-6
6
1
0
0
1
2
3
4
5
global axial strain [%]
"experimental" photographs
57
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Experimental results: test MB-11
The specimen at the end of the test
58
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Numerical simulation (Allodi et al., 2002)
l
 Uniform axial displacement to the
upper surface of the specimen
 Initial notch with orientation
 =/4 + p/2, approximately equal
to the initial orientation of the
experimentally observed crack
170 mm

 Mechanical
parameters:
E = 45 MPa
 = 0.35
c = 0.27 MPa
y
80 mm
x
d* = 2 mm
dc* = 1 mm
p = 28°
r = 24°
from the literature
(Skempton 1964, Li 1987)
59
Stress-strain global behaviour
axial stress [kPa]
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Numerical simulation: results
900
II
600
III
2
3
4
300
IV
I
5
6
1
0
0
1
2
3
4
5
global axial strain [%]
experimental results
numerical simulation
"experimental" photographs
"numerical" photographs
60
a shear propagation
evolves inside the specimen
with the same orientation
of the initial notch
900
pre-failure phase:
displacements are
600
homogeneous
all over the sample
surface
axial stress [kPa]
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Numerical
simulation: results
peak load:
II
III
2
the different stress level
observed in points 4 and IV
can be due to the values of d*
and dc* chosen for the
numerical simulation
3
4
300
IV
I
post-failure phase:
the formation of a
second band cannot be
numerically simulated
5
6
1
0
1 of the
2 analysis:
3
4
end
axial
[%]
the bandglobal
reaches
thestrain
opposite
side
of the specimen and all the elements
reach their residual strength
0
5
61
Numerical
(II)
uy
axial stress [kPa]
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Incremental displacements: points 2 and II
900
II
600
III
2
3
Experimental
(2)
4
300
IV
I
5
6
1
0
0
1
2
3
4
5
uy
global axial strain [%]
62
Numerical
(III)
uy
axial stress [kPa]
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Incremental displacements: points 3 and III
900
II
600
III
2
3
4
300
IV
I
5
Experimental
(3)
6
1
0
0
1
2
3
global axial strain [%]
4
5
uy
63
Numerical
(IV)
axial stress [kPa]
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Incremental displacements: points 4 e IV
900
II
600
III
3
4
300
IV
I
uy
Experimental
(4)
2
5
6
1
0
0
1
2
3
4
5
uy
global axial strain [%]
64
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Index

Introduction

Basic concepts of Linear Elastic Fracture Mechanics

Propagation criteria

Non linear Fracture Mechanics

Numerical modelling of cracked rock structures

The Displacement Discontinuity Method

Numerical simulation of experimental results

Application to slope stability
65
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Application of the method to slope stability


The BEMCOM numerical code has been applied to the
study of the stability of rock slopes with non persistent
natural discontinuities (Scavia,1995; Castelli, 1998).
crack propagation inside the rock mass is simulated
hard rocks
soft rocks, hard soils
failure surface
stepped failure surface
pre-existing discontinuity
66
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Example of application to soft rocks
Back Analysis of the Northold instability (Great Britain)
(Skempton, 1964; Duncan & Stark, 1986)

10 m high slope, with an inclination of 22°, excavated in
London clay in 1903, reshaped in 1936 and collapsed in
1955;

strength parameters determined through extensive
laboratory tests and back analyses

the position of the phreatic surface and portions of the
sliding surface are known
67
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Cross-section of the slope
observed portion of the actual slip surface
(Skempton, 1964)
68
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Shear strength parameters

Laboratory tests (Skempton, 1964)
cp' = 15.3 kPa
cr ' = 0

peak
residual
Back Analyses according to the Limit Equilibrium Method
with circular sliding surface (Skempton, 1964)
c' = 6.72 kPa

p' = 20°
r' = 16°
' = 18°
Back Analyses according to the try and error procedure,
based on the Limit Equilibrium Method (Duncan & Stark, 1986)
c' = 0.95 kPa
c' = 0.72 kPa
' = 24°
' = 25°
circular surface
non-circular surface
69
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The numerical model
Assumptions

peak shear strength values for intact material

residual shear strength values for the surface of the crack

Failure process starting at the foot of the slope

Failure taking place at the end of the excavation works in
drained conditions
LEFM approach
70
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The Numerical model
Geometrical and mechanical configuration

The propagation process was triggered by a crack located
at the foot of the slope, with length l=5m and inclination
=5°

excavation works were simulated through10 steps

the strength parameters were taken to be same as the
effective parameters determined experimentally by
Skempton (1964):
c’ = 15.30 kPa
c’ = 0
’ = 20°
’ = 16°
intact material
surface of the crack
71
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Numerical failure surface
before propagation
after propagation
Top of the slope
Toe of the slope
sliding surface
72
At the end of the excavation process
The propagation will take place in the direction where R is
maximum
10 m
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Mobilisation ratio
(1/1R) max
73
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Computed relative displacements
At the end of the excavation process
Maximum relative displacement = 19.3 cm
10 m
25 m
74
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Example of application to hard rock slopes
MATTSAND
Back analysis of the rockfall
occurred in October 1998 in
Mattsand (CH) (Amatruda et al., 2004):
a volume of about 300 m3,
triggered from a steep gneiss
slope, fell into a water reservoir
and damaged a road
75
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Detaching zone
Water reservoir
Road
76
Discontinuity systems:
T
J1: (65°, 75°)
S: (245°, 35°)
J2 surface
making up the failure
30°
35°
m
E
7.4 m
J1
J2: (130°, 85°)
laterally delimiting the falling
mass
4.1
S
D
2m
J1
C
3m
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Geometry and structural configuration
75°
5.5
B
m
35°
A
77
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Geometry and structural configuration
J1
S
78
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Localisation and extension of rock bridges
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Proposed failure mechanisms
30°
3
2
Consecutive toppling
of three blocks, due to
the tensile failure of
rock bridges
1
W3
W2
35°
W1
75°
rock tooth
35°
discontinuity J1
80
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Geomechanical Parameters
Through laboratory and in-situ tests, the following
geomechanical parameters (mean values) have been
obtained for intact rock and discontinuities:
Indirect tensile strength T0 (MPa)
Toughness (MPam)
Basic friction angle b (°)
JRC (-)
JCS (MPa)
9.2
0.56
33°
4.5
32
Peak friction angle on the scistosity surface (Barton, 1976)
p  JRC  log10
JCS
 b  43
n
81
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Numerical back analysis


The toppling failure of blocks 2 and 3 is analysed using
the numerical method, through the simulation of a tensile
crack propagation into the rock bridges
Block 1 is considered as failed, since it was not possible to
survey any rock bridge on its surfaces
Assumed mechanical and geometrical parameters
Young modulus E (MPa)
Poisson ratio 
peak friction angle p (°)
Toughness (MPam)
Length of rock bridges (m)
Block 2
Block 3
25000
0.2
43°
0.34
1
0.6
82
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Geometrical configurations
Elem
3
1
Elem
Elem
B s n
A
Block 2
Misure in m
Block 3
DD open elements (edges)
DD open elements
DD closed elements
83
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Numerical results: block 2

Initial configuration
Final configuration
(KI and KII  0)
Configurazione indeformata
Tip propagation

1 mm5
Propagazione degli apici
Propagation takes
Configurazione
deformata
place
for:
KIC = 0.34 MPam
B
Scala degli spostamenti
Open crack
propagation in mixed
mode conditions
A
mm
84
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Block 2: failure mechanism
rock cliff
toppling block
rock bridge failure
due to induced
tensile crack
propagation
85
tangential stress 
0,10
0,30
0,50
0,70
Open crack
0,90
1,10
1,30
1,50
1,70
1,90
2,10
2,30
2,50
2,70
2,90
Stress [MPa]
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Numerical results: block 2
1
0,5
0
-0,5
Closed crack
-1
-1,5
Local coordinate [m]
normal stress n
86