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Fracture mechanics Fracture mechanics approach to the approach to the
study of failure in study of failure in rockrock
Claudio Scavia, Marta Castelli Claudio Scavia, Marta Castelli
Politecnico di Torino Dipartimento di Ingegneria Strutturale e Geotecnica
Corso di “Leggi costitutive dei geomateriali”
Dottorato di Ricerca in Ingegneria Geotecnica
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Index
IntroductionIntroduction
Basic concepts of Linear Elastic Fracture Basic concepts of Linear Elastic Fracture MechanicsMechanics
Propagation criteria Propagation criteria
Non linear Fracture MechanicsNon linear Fracture Mechanics
Numerical modelling of cracked rock structuresNumerical modelling of cracked rock structures
The Displacement Discontinuity MethodThe Displacement Discontinuity Method
Numerical simulation of experimental resultsNumerical simulation of experimental results
Application to slope stabilityApplication to slope stability
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IntroductionSince Coulomb (1776) the problem of failure in natural and
man-made material have been approached on the basis of the traditional concept of
Material strengthMaterial strength
This approach cannot explain some disastrous brittle failures and can be (depending on the scale) a great
oversimplification of the crack initiation process
Tay bridge (Scotland, 1898)
Schenectady ship (1943)
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Introduction
Large natural defects (faults, joints…) exist in rock masses
Example: progressive failure in slopesExample: progressive failure in slopes
The main cause of fracture initiation is the presence of defects in the material, which concentrate the stress at their tips
Fracture MechanicsFracture 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
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IntroductionMain steps in Fracture MechanicsMain steps in Fracture Mechanics
Evaluation of Evaluation of stress stress
concentrationconcentration
Choice of a propagation criterion
Definition of a methodology for the simulation of crack propagation
stable propagation unstable propagation
Analysis of the state of stress
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Modes of failure in rocks
1
3
Shear Shear bandband
1
Direct Direct tensiontension
Axial Axial splittingsplitting
1
Indirect Indirect tensiontension
1
At the scale of the At the scale of the laboratorylaboratory
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Modes of failure in rocksAt the scale of the rock massAt the scale of the rock mass
Indirect Indirect tensiontension
Shear Shear
Direct Direct tensiontension
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Index
IntroductionIntroduction
Basic concepts of Linear Elastic Fracture Basic concepts of Linear Elastic Fracture MechanicsMechanics
Propagation criteriaPropagation criteria
Non linear Fracture MechanicsNon linear Fracture Mechanics
Numerical modelling of cracked rock structuresNumerical modelling of cracked rock structures
The Displacement Discontinuity MethodThe Displacement Discontinuity Method
Numerical simulation of experimental resultsNumerical simulation of experimental results
Application to slope stabilityApplication to slope stability
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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
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Stress concentration
circular hole
elliptical hole
2b
a
crack
a
2b0
ba
21σσmax
max = 3
r
max
max = f(a, b)
r
max
r
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Energetic approach (Griffith, 1921)
22 = fracture energy = fracture energy GGcc
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
dadW
dadW se
Condition for crack propagation
aE
2
Ea
W22e
elastic energy release rate
surface energy
aWs 4
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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 openingopening - loads are orthogonal to the fracture plane
mode II slip slip - loads are tangent to the fracture plane in the direction of maximum dimension
mode III teartear - loads are contained in the fracture plane and act perpendicularly to mode II
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1cos3KsinK2
cosr2
1
sinK23
2cosK
2cos
r21
2tanK2sinK
23
2sin1K
2cos
r21
IIIr
II2
I
IIII2
Ir
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:
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Tensional approach
For =0 i.e. for a point at a distance r along the line of the crack:
r21
K
r21
K
r21
K
IIxy
Iy
Ix
21Gx4
Kû
21Gx4
Kû
IIy
Ix For relative displacements û between the crack faces at a small distance x from the crack tip:
ry
x ûy
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Tensional approach
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
r
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Meaning of the Stress Intensity Factors
The vertical stress, y, around the crack tip is given by the theory of elasticity:
a
2b0
y
r
ry 2
a
The specific boundary conditions of the problem affect the value of y through a constant term KI which is given by:
aKI
Example: crack of length 2a, located in a plate subjected to a uniform vertical tensile stress
rKI
21
y
x
Gy 4
21 ûKI
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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 criterionpropagation criterion
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Index
IntroductionIntroduction
Basic concepts of Linear Elastic Fracture Basic concepts of Linear Elastic Fracture MechanicsMechanics
Propagation criteriaPropagation criteria
Non linear Fracture MechanicsNon linear Fracture Mechanics
Numerical modelling of cracked rock structuresNumerical modelling of cracked rock structures
The Displacement Discontinuity MethodThe Displacement Discontinuity Method
Numerical simulation of experimental resultsNumerical simulation of experimental results
Application to slope stabilityApplication to slope stability
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Propagation criteria
open cracks: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: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
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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:
where KIC is the material toughness
0II
02I
0eqIC sinK
2
3
2cosK
2cosKKr2
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Open cracks (Erdogan & Sih, 1963)
13cosKsK2
cos0 0II0I0 inr
01cos3sin 00 III KK
For pure mode I:
0IIK
00sin 00 IK
For pure mode II:
0IK
5.7001cos3 00 IIK
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Open cracks
KI > 0
KII = 0
KI > 0KII 0
KI < 0KII 0
KI = 0
KII 0
KI < 0
KII = 0
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Closed cracks
Failure by shear
Propagated crack
Induced-tensile propagation: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) KKICIC
Shear propagation:Shear propagation:
(mode II)
The original crack is compressed, and it propagates in compressive stress fields
KKIICIIC??
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Shear propagation criteriaA controversial issue is whether or not it is possible to apply LEFM concepts to the analysis of shear failure
1
3
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
The propagation is accompanied by considerable energy dissipation due to friction
The meaning of fracture toughness in mode II (KThe meaning of fracture toughness in mode II (KIICIIC) is ) is still under discussionstill under discussion
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Fracture toughness: mode I
Experimental determinationExperimental determination
Suggested methods (ISRM, 1988)
Short rod (SR)Short rod (SR) Chevron bend (CB)Chevron bend (CB)
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Short rod
uncut rock or ligament
a0
a
Wa1
notch
P
D
P load on specimenD diameter of short rod specimenW length of specimenh depth of crack in notch flank chevron angle t notch widtha0 chevron tip distance
a crack lengtha1 maximum depth of chevron flanks
D/2
t
1.5
maxIC D
24PK
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Chevron bend
P load on specimenA projected ligament areaL specimen lengthS distance between support pointsD diameter of chevron bend specimenCMOD relative opening of knife edgesh depth of crack in notch flank chevron angle = 90°a0 chevron tip distance
a crack length
a
CMOD
uncut rock or ligamentnotc
h
knife
a0a
hP
SL
loading roller
Support roller
1.5
maxIC D
PAK
D
A
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Chevron bend
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When is a LEFM approach applicable?
a zone of material exhibiting a non linear behaviour (process zoneprocess 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
Extremely high stress values involved in the Extremely high stress values involved in the phenomenon of crack propagation:phenomenon of crack propagation:
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Index
IntroductionIntroduction
Basic concepts of Linear Elastic Fracture Basic concepts of Linear Elastic Fracture MechanicsMechanics
Propagation criteria Propagation criteria
Non linear Fracture MechanicsNon linear Fracture Mechanics
Numerical modelling of cracked rock structuresNumerical modelling of cracked rock structures
The Displacement Discontinuity MethodThe Displacement Discontinuity Method
Numerical simulation of experimental resultsNumerical simulation of experimental results
Application to slope stabilityApplication to slope stability
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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
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Non linear Fracture Mechanics
Process zone at the crack tipProcess 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
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Non Linear Fracture Mechanics
t
inelastic
stress distribution
Visible crack
true crack process zone
stress free elastic
stress distribution
c
Open cracks (tension)Open cracks (tension): t: the he Cohesive Crack ModelCohesive Crack Model
(Dugdale, 1960; Barenblatt, 1962)
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Non Linear Fracture Mechanics
A process zone is introduced at the crack tip, where the damage is concentrated
Here, a relation is assumed between relative displacement and shear stress
A residual shear strength r occurs when reaches a critical value *
= process zone extension
G = energy amount stored inside the process zone
*
n
n
r
fictitious tip
real tip
process zone
real crack
p
r
r
G
Closed cracks (compression and shear)Closed cracks (compression and shear): :
tthe he Slip-Weakening ModelSlip-Weakening Model (Palmer & Rice, 1973)
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Index
IntroductionIntroduction
Basic concepts of Linear Elastic Fracture Basic concepts of Linear Elastic Fracture MechanicsMechanics
Propagation criteriaPropagation criteria
Non linear Fracture MechanicsNon linear Fracture Mechanics
Numerical modelling of cracked rock structuresNumerical modelling of cracked rock structures
The Displacement Discontinuity MethodThe Displacement Discontinuity Method
Numerical simulation of experimental resultsNumerical simulation of experimental results
Application to slope stabilityApplication to slope stability
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Numerical modelling of cracked rock structures
Resort to numerical techniques for the analysis of cracked rock structures proves necessary because of the
geometrical complexity of most application problems
Analysis of the state of stress and Analysis of the state of stress and
simulation of the propagation simulation of the propagation
Boundary Element Method (BEM)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
Finite Element Method (FEM)Finite Element Method (FEM)
Needs a re-meshing at each crack propagation step
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Numerical modelling of cracked rock structuresDisplacement Discontinuity MethodDisplacement Discontinuity Method ((Crouch & Starfield, 1983Crouch & Starfield, 1983))
allows to simulate the crack as Displacement Discontinuity elements
Ds = us(s, 0-) - us (s, 0+)
Dn = un(s, 0-) - un (s, 0+)
n
+Ds
s
2a
+Dn
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jDj,iAjDj,iAi
jDj,iAjDj,iAi
n
N
1jnns
N
1jnsn
n
N
1jsns
N
1jsss
influence coefficients of Ds(j) and Dn(j) on stresses or displacements
over the i-th element
known tangential and normal stresses or
displacements acting on the i-th
element
The Displacement Discontinuity Method
(1) i
s
n
s
n (N)
j
(i)
(j)
unknown displacement discontinuities in the tangential and normal directions, in the centre of
the j-th element
computer code
BEMCOM
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Open elements
Tensile stress fieldsTensile stress fields
Dn < 0 (opening)
s(i), n(i) = 0
Compressive stress fieldsCompressive stress fields
Dn> 0 (closure)
s(i), n(i) = 0
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Closed elements
Compressive stress fieldsCompressive stress fields
Dn = 0
s(i), n(i) 0
No Displacement Discontinuities in the normal direction
A tangential Displacement Discontinuity occurs if and when the available frictional shear strength is mobilised
s
Ds
Ks
sr = n·tan
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Simulation of crack propagation open cracksopen cracks
Erdogan & Sih’s propagation criterion, based on the Stress Intensity factors calculation at the tip of the crack
closed cracksclosed 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
(Scavia, 1995; Scavia et al., 1997)
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Numerical implementation of the SWM
real cracknon-
cohesive process
zone
(Ds)
*
r
p
tip element
n
c*
cohesive process
zone
Computer code Computer code BEMCOMBEMCOM
(Allodi et al., 2002)
Computer code Computer code BEMCOMBEMCOM
(Allodi et al., 2002)
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Adopted slip-weakening laws
intact material (tip element): cp, p
real crack: c = 0, = r
process zone: linear variation of c and as a function of c* and *
Cohesion (c)
Friction angle ()
0
cp
c*
c
0 *
p
r
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Index
IntroductionIntroduction
Basic concepts of Linear Elastic Fracture Basic concepts of Linear Elastic Fracture MechanicsMechanics
Propagation criteria Propagation criteria
Non linear Fracture MechanicsNon linear Fracture Mechanics
Numerical modelling of cracked rock structuresNumerical modelling of cracked rock structures
The Displacement Discontinuity MethodThe Displacement Discontinuity Method
Numerical simulation of experimental resultsNumerical simulation of experimental results
Application to slope stabilityApplication to slope stability
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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)
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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) 20800 Es50 (MPa) 17600 t50 (-) 0.21 s50 (-) 0.11 C0 (MPa) 74 T0 (MPa) 3.53 K1C (MPa*m) 0.94 c (MPa) 23.7 p (°) 35.5
Characteristic of the material
Geometry and load configuration
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Experimental results
Stress-strain diagram Strain directions
horizontal
oblique
longitudinal
0 5000 10000 15000
stra ins (m icrostra in)
0
2
4
6
8
10
12
axia
l str
ess
(MP
a)
obliquelongitudinal
horizontal
onset o f p ropagation(num er ica l s imula tion)
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Propagation trajectories
NumericalExperimental
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Shear propagation (Scavia et al., 1997)Experimental work on Beaucaire marl samples subjected to uni-axial compression in plane-strain conditions (Tillard, 1992)
Axial load-axial strain diagrammeasured displacements(stereo-photogrammetry)
c5 c7 c7 c8c5-c7 c7-c8
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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) 81 (-) 0.35 c (MPa) 0.33 p (°), intact material 28 r (°), crack surfaces 20 C0 (MPa) 1.10 Sample height (mm) 120 Sample width (mm) 60 Sample thickness (mm) 35
= 28°l = 2mm
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Propagation trajectories
NumericalExperimental
c5c7 c7c8 c5c7 c7c8
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Limit of a LEFM approach
The numerical model is unable to simulate the global response of a specimen under load (no energy dissipation in the elastic material)
0 0.4 0.8 1.2 1.6 2.0
Axial strain (%)
0.00.20.40.6
0.81.01.21.41.6
1.82.0
Ax
ial
load
(kN
)
numericalexperimental
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NLFM approach to shear propagation
0 1 2 3 4 5 6global axia l stra in [% ]
0
200
400
600
800
1000
1200
axia
l str
ess
[kP
a]
LB-01LB-02LB-04MB-09MB-10MB-11
Photographs of the specimens during the tests in order to carry out a stereo-photogrammetric analysisstereo-photogrammetric analysis (Desrues,
1995)
LB-01
LB-02
LB-04
MB-09 MB-10 MB-11
Biaxial compression tests in plane strain conditions
Axial load under displacement control
No lateral confinement
Prismatic specimens of Beaucaire marl (two different samples)
Specimen dimensions:
170 x 80 x 35 mm3
85 x 40 x 35 mm3
(LB-02, LB-04)
Experimental results Experimental results
(Marello, 2004)(Marello, 2004)
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0 1 2 3 4 5
global axial strain [% ]
0
300
600
900
axia
l str
ess
[kP
a]
"experim enta l" photographs
1
2
3
4
5 6
couple 1-2couple 1-2 shear shear deformationsdeformations
Experimental results: test MB-11
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0 1 2 3 4 5
global axial strain [% ]
0
300
600
900
axia
l str
ess
[kP
a]
"experim enta l" photographs
1
2
3
4
5 6
couple 2-couple 2-33
Experimental results: test MB-11
shear shear deformationsdeformations
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Experimental results: test MB-11
0 1 2 3 4 5
global axial strain [% ]
0
300
600
900
axia
l str
ess
[kP
a]
"experim enta l" photographs
1
2
3
4
5 6
couple 3-couple 3-44
shear shear deformationsdeformations
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0 1 2 3 4 5
global axial strain [% ]
0
300
600
900
axia
l str
ess
[kP
a]
"experim enta l" photographs
1
2
3
4
5 6
Experimental results: test MB-11
Displacement Displacement vectorsvectors
couple 5-couple 5-66
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Experimental results: test MB-11
The specimen at the end of the testThe specimen at the end of the test
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170 mm
80 mm
x
y
Numerical simulation (Allodi
et al., 2002) l Uniform axial displacement to
the upper surface of the specimen
Mechanical parameters:
E = 45 MPa
= 0.35
c = 0.27 MPa
p = 28°
r = 24°
* = 2 mm
c* = 1 mm
from the literature(Skempton 1964, Li 1987)
Initial notch with orientation =/4 + p/2, approximately equal to the initial orientation of the experimentally observed crack
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0 1 2 3 4 5
global axial strain [% ]
0
300
600
900ax
ial s
tres
s [k
Pa]
II
I
III
IV
1
2
3
4
5 6
Numerical simulation: results
experim ental resultsnum erical sim ulation
"experim enta l" photographs"num erical" photographs
Stress-strain global behaviourStress-strain global behaviourStress-strain global behaviourStress-strain global behaviour
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0 1 2 3 4 5
global axial strain [% ]
0
300
600
900ax
ial s
tres
s [k
Pa]
II
I
III
IV
1
2
3
4
5 6
Numerical simulation: results
pre-failure phase: displacements are
homogeneous all over the sample
surface
peak load: a shear propagation
evolves inside the specimen with the same orientation
of the initial notch
end of the analysis: the band reaches the opposite side
of the specimen and all the elements reach their residual strength
post-failure phase:the formation of a
second band cannot be numerically simulated
the different stress level observed in points 4 and IV can be due to the values of
* and c
* chosen for the numerical simulation
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0 1 2 3 4 5
global axial strain [% ]
0
300
600
900
axia
l str
ess
[kP
a]
II
I
III
IV
1
2
3
4
5 6
Incremental displacements: points 2 and II
NumericaNumerical l (II)(II)
uu
yy
ExperimentaExperimentall
(2)(2)uu
yy
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0 1 2 3 4 5
global axial strain [% ]
0
300
600
900
axia
l str
ess
[kP
a]
II
I
III
IV
1
2
3
4
5 6
Incremental displacements: points 3 and III
ExperimentaExperimentall
(3)(3)uu
yy
NumericaNumerical l (III)(III)
uy
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0 1 2 3 4 5
global axial strain [% ]
0
300
600
900
axia
l str
ess
[kP
a]
II
I
III
IV
1
2
3
4
5 6
Incremental displacements: points 4 e IV
ExperimentaExperimental l (4)(4)
uu
yy
NumericaNumericall (IV) (IV)
uu
yy
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Index
IntroductionIntroduction
Basic concepts of Linear Elastic Fracture Basic concepts of Linear Elastic Fracture MechanicsMechanics
Propagation criteria Propagation criteria
Non linear Fracture MechanicsNon linear Fracture Mechanics
Numerical modelling of cracked rock structuresNumerical modelling of cracked rock structures
The Displacement Discontinuity MethodThe Displacement Discontinuity Method
Numerical simulation of experimental resultsNumerical simulation of experimental results
Application to slope stabilityApplication to slope stability
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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
stepped failure surface
pre-existing discontinuity
failure surface
hard hard rocksrocks
soft rocks, hard soilssoft rocks, hard soils
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Example of application to soft rocks
Back Analysis of the Northold instability (Great Back Analysis of the Northold instability (Great Britain)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
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Cross-section of the slope
observed portion of the actual slip surface
(Skempton, 1964)
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Shear strength parameters
Laboratory testsLaboratory tests (Skempton, 1964)
cp' = 15.3 kPa p' = 20° peak
cr' = 0 r' = 16° residual
Back AnalysesBack Analyses according to the Limit Equilibrium Method with circular sliding surface (Skempton, 1964)
c' = 6.72 kPa ' = 18°
Back AnalysesBack Analyses according to the try and error procedure, based on the Limit Equilibrium Method (Duncan & Stark, 1986)
c' = 0.95 kPa ' = 24° circular surface
c' = 0.72 kPa ' = 25° non-circular surface
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The numerical model
AssumptionsAssumptions
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
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The Numerical model
Geometrical and mechanical configurationGeometrical 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 ’ = 20° intact material
c’ = 0 ’ = 16° surface of the crack
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Toe of the slope
Top of the slope
Numerical failure surface
before propagation
after propagation
sliding surface
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Mobilisation ratio
10
m
(1/1R)max
At the end of the excavation process
The propagation will take place in the direction where R is maximum
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Computed relative displacements
10 m
25 m
At the end of the excavation process
Maximum relative displacement = 19.3 cm
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Example of application to hard rock slopes
Back analysis of the rockfall occurred in October 1998 in
Mattsand (CH) Mattsand (CH) (Amatruda et al., (Amatruda et al., 2004)2004)::
a volume of about 300 m3, triggered
from a steep gneiss slopesteep gneiss slope, fell into a water reservoir and damaged a road
MATTSAND
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Detaching zone
Road
Water reservoir
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35°
35°
75°
30°4.1 m
7.4
m
2 m
5.5 m
3 m
T
E
D
C
B
A
S
J1
Geometry and structural configuration
Discontinuity systems:
J1: (65°, 75°)S: (245°, 35°)
making up the failure surface
J2: (130°, 85°)laterally delimiting the
falling mass
J2J1
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Geometry and structural configuration
J1
S
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Localisation and extension of rock bridges
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Proposed failure mechanisms
Consecutive toppling of three blocks, due to the tensile failure of
rock bridges
3 5 °
3 5 °
7 5 °
3 0 °
3
2
1
W 1
W 2
W 3
d i s c o n t i n u i t y J 1
r o c k t o o t h
81
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geg
neri
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ica Indirect tensile strength T0 (MPa) 9.2
Toughness (MPam) 0.56 Basic friction angle b (°) 33° JRC (-) 4.5 JCS (MPa) 32
Geomechanical Parameters
Through laboratory and in-situ tests, the following geomechanical parameters (mean values) have been
obtained for intact rock and discontinuities:
Peak friction angle on the scistosity surface (Barton, Peak friction angle on the scistosity surface (Barton, 1976)1976)
43JCS
logJRC bn
10p
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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 parametersAssumed mechanical and geometrical parameters
Young modulus E (MPa) 25000 Poisson ratio 0.2 peak friction angle p (°) 43° Toughness (MPam) 0.34
Block 2 1 Length of rock bridges (m)
Block 3 0.6
83
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Geometrical configurations
DD open elements (edges)DD open elementsDD closed elements
Elementi DD di contorno (aperti)
Elementi DD aperti
Elementi DD chiusi
AB s n
13
Misure in m
Elementi DD di contorno (aperti)
Elementi DD aperti
Elementi DD chiusis n
A
B
1
3
Misure in m
Block 2 Block 3
84
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Numerical results: block 2
Open crack propagation in mixed mode conditions
(KI and KII 0)
Propagation takes place for:
KIC = 0.34 MPam
1 m m
S c a l a d e g l i s p o s t a m e n t i
C o n fi g u r a z i o n e i n d e f o r m a t a
P r o p a g a z i o n e d e g l i a p i c i
C o n fi g u r a z i o n e d e f o r m a t a
5 m m
T i p p r o p a g a t i o n
I n i t i a l c o n fi g u r a t i o n
F i n a l c o n fi g u r a t i o n
A
B
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rock bridge failure due to
induced tensile crack
propagation
rock cliff
toppling block
Block 2: failure mechanism
86
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Numerical results: block 2
-1,5
-1
-0,5
0
0,5
1
0,1
0
0,3
0
0,5
0
0,7
0
0,9
0
1,1
0
1,3
0
1,5
0
1,7
0
1,9
0
2,1
0
2,3
0
2,5
0
2,7
0
2,9
0
Local coordinate [m]
Str
ess [
MP
a]
Open crack Closed crack
tangential stress
normal stress n