Fracture mechanics approach to the study of failure in rock Claudio Scavia, Marta Castelli Politecnico di Torino Dipartimento di Ingegneria Strutturale

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



Propagation criteriaPropagation criteria






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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









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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









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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









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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









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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)

46

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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


0

300

600

900

axia

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[kP

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1

2

3

4

5 6

couple 2-couple 2-33


shear shear deformationsdeformations

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0 1 2 3 4 5


0

300

600

900

axia

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[kP

a]


1

2

3

4

5 6


shear shear deformationsdeformations

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0 1 2 3 4 5


0

300

600

900

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[kP

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1

2

3

4

5 6


Displacement Displacement vectorsvectors


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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


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


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


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


0

300

600

900

axia

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[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


0

300

600

900

axia

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[kP

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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









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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

76

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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

80

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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

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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

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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

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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

85

Cors

o d

i “Leg

gi costi

tuti

ve d

ei

geom

ate

riali” –

Novem

bre

20

05

Dott

ora

to d

i ri

cerc

a in

In

geg

neri

a

Geote

cn

ica

rock bridge failure due to

induced tensile crack

propagation

rock cliff

toppling block

Block 2: failure mechanism

86

Cors

o d

i “Leg

gi costi

tuti

ve d

ei

geom

ate

riali” –

Novem

bre

20

05

Dott

ora

to d

i ri

cerc

a in

In

geg

neri

a

Geote

cn

ica

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

Documents

Fracture mechanics approach to the study of failure in rock Claudio Scavia, Marta Castelli Politecnico di Torino Dipartimento di Ingegneria Strutturale