# Fracture mechanics approach to the study of failure in rock

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Fracture mechanics approach to the study of failure in rock
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

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

Introduction Material strength
Since Coulomb (1776) the problem of failure in natural and man-made 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)

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

Main steps in Fracture Mechanics Evaluation of stress concentration
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

Modes of failure in rocks
At the scale of the laboratory s1 Direct tension Indirect tension 1 Axial splitting 1 1 3 Shear band

Modes of failure in rocks
At the scale of the rock mass Indirect tension Direct tension Shear

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

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

Stress concentration s s s  s 3s smax = 3s r circular hole s smax
smax = f(a, b) r s elliptical hole 2b a s crack a 2b0 s smax   r

Energetic approach (Griffith, 1921)
Condition for crack propagation elastic energy release rate surface energy 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

Tensional approach (Irwin, 1957)
Crack propagation can be studied through the superposition of the effects of three independent load application modes 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

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: Commento alla figura: If teta=0, for example in a point at a distance r along the line of the crack the state of stress is a function of KI and KII (girare).

Tensional approach For =0
i.e. for a point at a distance r along the line of the crack: y r x ûy q Also relative displacements between crack faces at a distance x from the crack tip are a function of KI and KII. G and  are the shear modulus and Poisson’s ratio of the elastic material KI and KII are the Stress Intensity Factors.They are representative of the stress field around the crack tip and depends on the boundary conditions and the geometry. They can be analytically determined in simple cases and numerically computed in more complex cases, through theses equations, starting from the computation of stresses and displacements in the crack tip zone. The value of K that triggers propagation is called critical and represented as Kc The study of a large-scale behaviour of discontinuities on the basis of LEFM involves the choice of a propagation criterion for open and closed cracks For relative displacements û between the crack faces at a small distance x from the crack tip:

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

Meaning of the Stress Intensity Factors
Example: crack of length 2a, located in a plate subjected to a uniform vertical tensile stress s a 2b0 sy r The vertical stress, y, around the crack tip is given by the theory of elasticity: The specific boundary conditions of the problem affect the value of y through a constant term KI which is given by:

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

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

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

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 For open and closed cracks propagation in mixed mode conditions, the criterion used is the one developed by Erdogan and Sih, summarised here: the assumption are:

Open cracks (Erdogan & Sih, 1963)
For pure mode I: For pure mode II: For open and closed cracks propagation in mixed mode conditions, the criterion used is the one developed by Erdogan and Sih, summarised here: the assumption are:

Open cracks      KI > 0 KII = 0 KI < 0 KII = 0 KI = 0

Induced-tensile propagation:
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? If we want to study crack propagation phenomena at the scale of the specimen, we can consider both the modes of failure as crack propagation. In particular, in compressive stress fields, we will have to deal with two different types of crack propagation: - If the process zone is small compared to the size of the structure (in many problems related to rock crack propagation this is true), the approximation made by Linear Elastic Fracture Mechanics is acceptable.

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 The propagation is accompanied by considerable energy dissipation due to friction 1 3 The meaning of fracture toughness in mode II (KIIC) is still under discussion

Fracture toughness: mode I
Experimental determination Suggested methods (ISRM, 1988) Short rod (SR) Chevron bend (CB)

Short rod P t a0 a a1  D W D/2 notch uncut rock or ligament
P load on specimen D diameter of short rod specimen W length of specimen h depth of crack in notch flank  chevron angle t notch width a0 chevron tip distance a crack length a1 maximum depth of chevron flanks P D D/2 t a0 a W a1 notch uncut rock or ligament

Chevron bend  a a0 a h P A S D L CMOD Support roller loading roller
uncut rock or ligament notch a CMOD P load on specimen A projected ligament area L specimen length S distance between support points D diameter of chevron bend specimen CMOD relative opening of knife edges h depth of crack in notch flank  chevron angle = 90° a0 chevron tip distance a crack length knife

Chevron bend

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

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

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

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

Non Linear Fracture Mechanics
Open cracks (tension): the Cohesive Crack Model (Dugdale, 1960; Barenblatt, 1962) st inelastic stress distribution Visible crack true crack process zone stress free elastic dc

Non Linear Fracture Mechanics
Closed cracks (compression and shear): the Slip-Weakening Model (Palmer & Rice, 1973) * n r fictitious tip real tip process zone real crack p r G 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 The Slip-Weakening Model has been developed by Palmer & Rice in 1973 for the study of compressed cracks subjected to shear On the left we have a schematic representation of such model: A process zone is introduced … Here, a relation between… as shown in the diagram above  so at the process zone tip (fictitious tip) is associated the peak shear resistance of the material and a displacement  equal to zero (as shown in the diagram below), from this point, inside the process zone, the damage induces an increasing displacement, while the shear resistance decreases. A residual…  here the real crack is formed The process zone extension is a function of delta* The dashed area in the diagram above represents the energy amount stored inside the process zone, which is dissipated for unit surface during crack propagation. When such amount reaches a critical value, the propagation of the process zone occurs

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

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

Numerical modelling of cracked rock structures
Displacement Discontinuity Method (Crouch & 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 Numerical techniques prove necessary in the analysis of cracked rock structures because of the geometrical complexity of most problems. Among the available numerical methods, the BEM requires only the discretisation of the structure’s boundary, and hence is suited to deal with problems characterised by evolving geometry, such as crack propagation. The BEM technique of the DDM, developed by Crouch & Starfield in 1983 for the study of the behaviour of crack surfaces is used. In the DDM a crack is represented by a line discretised into elements The DDM is based on the assumption that the distribution of Displacement Discontinuity along a crack in a linear elastic body can be approximated by subdividing the crack into elements and by assuming the DDs to vary over the individual elements according to a predetermined mode (constant, linear, parabolic…) The DD are defined as the difference between the displacement of the lower and the upper surface of the crack in the normal and tangential direction.

The Displacement Discontinuity Method
(1) bi s n (N) bj (i) (j) computer code BEMCOM 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 discretisation of Displacement Discontinuities is made by subdividing the crack into a number N of DD elements. In the hypothesis of elastic behaviour of the material, knowing the analytical solution for a single DD element, it is possible to calculate at any point of the body, stresses and displacements due to the crack, by adding up the effects of the N elements the solving system, made up of 2N equation is given here, where (click mouse) The method has been implemented in the computer code BEMCOM, which allows the simulation of open and closed cracks, both in tensile and compressive stress fields. unknown displacement discontinuities in the tangential and normal directions, in the centre of the j-th element

Open elements Tensile stress fields Dn < 0 (opening)
s(i), n(i) = 0 Compressive stress fields Dn> 0 (closure) As regards open elements… Claudio???

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 If the surfaces of a crack are in contact in a compressive stress field, a deformation takes place without giving rise to Displacement Discontinuities in the normal direction. In the tangential direction, if a shear stress is also applied, frictional stresses will develop on the crack surfaces. When the shear stress equals the value of the available strength, the surfaces begin to slide apart and tangential displacement discontinuity occur an elastic-ideally plastic constitutive equation is assumed for crack surfaces

Simulation of crack propagation
(Scavia, 1995; Scavia et al., 1997) open cracks 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 The method is able to simulate the propagation of both open and closed cracks: In the case of open cracks, a tensile propagation occurs: the Erdogan & Sih propagation criterion is used, based on the stress intensity factors calculation at the tip of the crack and their comparison with the critical value K1c. In the case of closed and compressed cracks, two different kind of propagation may occur, and two different propagation criteria are considered: induced-tensile propagation: the phenomenon is very similar to the one analysed for open cracks, the Erdogan & Sih’s criterion is again used. shear propagation: the possibility for a crack to propagate is evaluated on the basis of the calculation of the stress field near the tip and its comparison with the Mohr Coulomb strength criterion. In the analysis of a typical boundary value problem, the load is applied in steps and the possibility of crack propagation is evaluated at each step. If the condition for crack propagation is met, a new element is added at the crack tip, in the direction depicted by the propagation criterion In

Numerical implementation of the SWM
Computer code BEMCOM (Allodi et al., 2002) real crack non-cohesive process zone  (Ds) j* tr tp t tip element n c* cohesive process zone The Palmer & Rice theoretical model well suits the DDM, because the process zone can be simulated with a number of elements where the shear resistance is greater than the residual one, as a function of the computed tangential displacement discontinuity in this picture the numerical implementation of the SWM in the computer code BEMCOM is shown. The problem is related to the slip-weakening law to be associated to the process zone elements. As a matter of facts, the Palmer & Rice model, has been developed for a shear test, where the normal stress on crack surface is maintained constant. But if the normal stress on crack surface is not constant, the peak and residual values of the shear strength vary as a consequence. To overcome this problem, an indirect application of the model has been numerically implemented

cp c* c j * jp jr Cohesion (c) Friction angle () intact material (tip element): cp, p real crack: c = 0,  = r process zone: linear variation of c and  as a function of c* and * Assuming the strength characteristics of intact material and crack surfaces (friction angle and cohesion), as independent from the normal stress sigman, a variation of such characteristics is given as a function of the computed relative tangential displacement on the process zone elements. In this way, the friction angle decays from a peak (characteristic of intact material) to a residual (characteristic of crack surface) value, and the cohesion from a peak to zero. Here linear laws are considered, but they can be any. As a consequence, the shear strength decreases on process zone elements from the peak to the residual value Two different values of the critical slip should be used to define the decrease of cohesion and friction angle, due to the experimental evidence that the resistance term due to cohesion disappears very quickly inside the process zone during the propagation As shown in the previous figure, the process zone is therefore made of a cohesive part, located near the tip, where cohesion is greater than zero, and a non-cohesive part where a rough crack is formed with c = 0 and p <  < r Propagation occurs when the shear stress computed at the tip element exceeds the shear strength. A new element can then be added in the critical direction with regards to shear failure, following the Mohr- Coulomb criterion. Peak values of the material characteristics are associated to this element, while for elements behind new reduced resistant values are computed.

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

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)

Induced-tensile propagation (Castelli, 1998)
Experimental work on concrete samples containing two open slits subjected to uni-axial compression Characteristic of the material Geometry and load configuration

Stress-strain diagram
Experimental results horizontal oblique longitudinal Stress-strain diagram Strain directions

Propagation trajectories
Experimental Numerical

Shear propagation (Scavia et al., 1997)
Experimental work on Beaucaire marl samples subjected to uni-axial compression in plane-strain conditions (Tillard, 1992) c5-c7 c7-c8 measured displacements (stereo-photogrammetry) Axial load-axial strain diagram

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  = 28° l = 2mm

Propagation trajectories
c5c7 c7c8 c5c7 c7c8 Experimental Numerical

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) From the numerical point of view, a snap-back phenomenon is seen, the method is not able to simulate the global behaviour of the sample. This is related to the linear elasticity, no energy dissipation is possible in the elastic material. NB il secondo limite è legato al criterio di propagazione per taglio. La singolarità nella stato tensionale agli apici, legata alla teoria dell’elasticità, fa si che sia necessario allontanarsi dall’apice per poter applicare il criterio di Coulomb. La distanza a cui vengono calcolate le tensioni è però arbitraria, e la sua influenza deve essere valutata di volta in volta. In order to overcome suck limits, a non linear approach is needed, and this is the most recent development of the method. We told that we can think at the formation of a shear band in rock, from a microscopical point of view, as a complex breakdown process, due to the propagation, interaction and coalescence of a few microcracks, which evolve by inducing microcrack growing at its front (damage). This microstructural process of breakdown (damage) near the crack tip can be interpreted by assuming that it gives rise to cohesive stresses, which opposes the action of applied loads. So, an appropriate macroscopical representation of shear crack propagation in rock seems to be a cohesive-zone model of the kind introduced by Dugdale and Barenblatt for tensile crack. experimental numerical

NLFM approach to shear propagation
Experimental results (Marello, 2004) 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) LB-01 LB-02 LB-04 MB-09 MB-10 MB-11 The experimental research considered in this work has been carried out by Stefania Marello (2002) with the collaboration of Cino Viggiani, at the Laboratoire 3S di Grenoble. Here the characteristics of the tests are listed: some biaxial compression tests have been carried out in plane strain conditions by applying axial load under displacement control no confinement… six prismatic specimens… dimensions… Here the failure pattern developed at the end of the test is shown for the six specimens. Two different main failure mechanisms can be observed: in the case of samples LB some Y-shaped cracks develop in the upper part of the specimen, as a result of the tensile propagation of two or more cracks. In the case of samples MB on the contrary, the failure is due to the formation of a single shear band and only after its full evolution secondary mechanisms are observed, leading to the final pattern. During the test some photographs are taken from the same point of view in order to follow the strain field evolution on the surface of the specimen. The photographs are analyzed using the method of False Relief Stereo-photogrammetry (FRS), based on the photogrammetric analysis of successive pairs of photographs. Photographs of the specimens during the tests in order to carry out a stereo-photogrammetric analysis (Desrues, 1995)

Experimental results: test MB-11
couple 1-2 shear deformations The results of test MB-11 have be considered for the numerical simulation. Here the the axial stress - axial strain diagram is shown. the instants when photographs have been taken are noted on the curve by numbers 1 to 6. The behaviour of the specimen is strongly brittle the stereo-photogrammetric analysis on couple 1-2 show a strain localisation on the upper surface of the specimen at the peak load After the peak, a band propagates inside the specimen towards the right side in point 4 the fissure is completely developed (couple 3-4). The strain increment observed at point 4 is probably due to the tensile propagation of a second band, where further deformation localizes. When the residual strength is reached (couple 5-6), this second band is completely developed this is the final failure pattern of the specimen

Experimental results: test MB-11
shear deformations couple 2-3 The results of test MB-11 have be considered for the numerical simulation. Here the the axial stress - axial strain diagram is shown. the instants when photographs have been taken are noted on the curve by numbers 1 to 6. The behaviour of the specimen is strongly brittle the stereo-photogrammetric analysis on couple 1-2 show a strain localisation on the upper surface of the specimen at the peak load After the peak, a band propagates inside the specimen towards the right side in point 4 the fissure is completely developed (couple 3-4). The strain increment observed at point 4 is probably due to the tensile propagation of a second band, where further deformation localizes. When the residual strength is reached (couple 5-6), this second band is completely developed this is the final failure pattern of the specimen

Experimental results: test MB-11
shear deformations couple 3-4 The results of test MB-11 have be considered for the numerical simulation. Here the the axial stress - axial strain diagram is shown. the instants when photographs have been taken are noted on the curve by numbers 1 to 6. The behaviour of the specimen is strongly brittle the stereo-photogrammetric analysis on couple 1-2 show a strain localisation on the upper surface of the specimen at the peak load After the peak, a band propagates inside the specimen towards the right side in point 4 the fissure is completely developed (couple 3-4). The strain increment observed at point 4 is probably due to the tensile propagation of a second band, where further deformation localizes. When the residual strength is reached (couple 5-6), this second band is completely developed this is the final failure pattern of the specimen

Experimental results: test MB-11
Displacement vectors couple 5-6 The results of test MB-11 have be considered for the numerical simulation. Here the the axial stress - axial strain diagram is shown. the instants when photographs have been taken are noted on the curve by numbers 1 to 6. The behaviour of the specimen is strongly brittle the stereo-photogrammetric analysis on couple 1-2 show a strain localisation on the upper surface of the specimen at the peak load After the peak, a band propagates inside the specimen towards the right side in point 4 the fissure is completely developed (couple 3-4). The strain increment observed at point 4 is probably due to the tensile propagation of a second band, where further deformation localizes. When the residual strength is reached (couple 5-6), this second band is completely developed this is the final failure pattern of the specimen

Experimental results: test MB-11
The specimen at the end of the test

Numerical simulation (Allodi et al., 2002)
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 80 mm x y 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) Numerical simulations of the test have been carried out using the computer code BEMCOM. the geometrical configuration of the specimen is shown here. The simulation has been carried out by applying a uniform axial displacement… An initial notch is located at the upper left corner of the specimen, with orientation α = π/4 + φp/2 (Mohr-Coulomb failure angle), approximately equal to the orientation of the experimentally observed crack. Since the numerical simulation is carried out inducing a uniform axial displacement to the upper surface of the specimen, the notch has been located on the lateral side to obtain a relative tangential displacement on its surfaces. the mechanical characteristics taken into account in the simulation are listed here. As far as the critical relative displacement δ* is concerned, no indication can be obtained from the experimental observations. Data obtained from the literature have then been considered, referring to soft materials with mechanical characteristics equivalent to the Beaucaire marl (Skempton 1964, Li 1987).

Numerical simulation: results
Stress-strain global behaviour The comparison between the axial stress vs. axial strain curves experimentally and numerically obtained is shown. The numerical simulation seems to fit quite well the experimental result. In the pre-failure phase displacements are homogeneous all over the sample surface A shear propagation is triggered at the peak load and evolves inside the specimen with the same orientation of the initial notch. No global axial deformation increment is seen during the propagation of the shear band and the post-peak behavior of the specimen is perfectly brittle. As soon as the band reaches the opposite side of the specimen, the analysis stops (point IV) as all the DD elements reaches their residual strength. For a comparison with the experimental results, the experimental curve should be considered only until point 4, since the formation of a second band cannot be numerically simulated because the strain field outside the shear band is homogeneous. The different stress level observed in points 4 and IV can be due to the values of * c* chosen for the numerical simulation. Hence, a deep investigation is needed on the role of the critical slip * in the phenomenon and its influence on the decay of resistance parameters cohesion and friction angle. Furthermore, a linear slip-weakening law has been adopted for the resistance parameters (figure 4) and the influence of non-linear laws on the behavior of the process zone should be analyzed in the future as well.

Numerical simulation: results
peak load: a shear propagation evolves inside the specimen with the same orientation of the initial notch post-failure phase: the formation of a second band cannot be numerically simulated pre-failure phase: displacements are homogeneous all over the sample surface the different stress level observed in points 4 and IV can be due to the values of * and c* chosen for the numerical simulation The comparison between the axial stress vs. axial strain curves experimentally and numerically obtained is shown. The numerical simulation seems to fit quite well the experimental result. In the pre-failure phase displacements are homogeneous all over the sample surface A shear propagation is triggered at the peak load and evolves inside the specimen with the same orientation of the initial notch. No global axial deformation increment is seen during the propagation of the shear band and the post-peak behavior of the specimen is perfectly brittle. As soon as the band reaches the opposite side of the specimen, the analysis stops (point IV) as all the DD elements reaches their residual strength. For a comparison with the experimental results, the experimental curve should be considered only until point 4, since the formation of a second band cannot be numerically simulated because the strain field outside the shear band is homogeneous. The different stress level observed in points 4 and IV can be due to the values of * c* chosen for the numerical simulation. Hence, a deep investigation is needed on the role of the critical slip * in the phenomenon and its influence on the decay of resistance parameters cohesion and friction angle. Furthermore, a linear slip-weakening law has been adopted for the resistance parameters (figure 4) and the influence of non-linear laws on the behavior of the process zone should be analyzed in the future as well. end of the analysis: the band reaches the opposite side of the specimen and all the elements reach their residual strength

Incremental displacements: points 2 and II
Numerical (II) uy Experimental (2) uy Now the comparison is made in terms of incremental displacement fields on the surface of the specimen. The experimental and numerical displacement increments Δuy and Δux occurred during each couple are shown. At the peak load displacements are homogeneous over all the sample surface, except in the upper right corner, where a strain localization can be observed.

Incremental displacements: points 3 and III
Numerical (III) uy Experimental (3) uy

Incremental displacements: points 4 e IV
Experimental (4) uy Numerical (IV) uy

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

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 stepped failure surface pre-existing discontinuity failure surface Two theoretical example of application are shown here. On the left we have a slope in of hard rocks, containing a system of natural discontinuities. A stepped failure surface is found, due to the coalescence of the pre-existing discontinuities and the failure of rock bridge. On the right, a slope in soft rocks containing a single defect at its toe. A shear propagation of such defect can reach the top of the slope and form a curvilinear failure surface

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 As an example of… I will present the back analysis carried out on the instability phenomenon occurred in Northold. The phenomenon has bee observed and described by…. And involve a slope 10 m high … This case has been selected because the material is described in detail in the literature and the strength parameters have been determined through extensive laboratory tests and many back analyses. Furthermore, the position of the phreatic surface and portion of the sliding surface are known.

Cross-section of the slope
observed portion of the actual slip surface In this picture, made by Skempton, a cross section of the slope is represented, where the position of the phreatic surface and the observed portion of the actual failure surface are indicated (Skempton, 1964)

Shear strength parameters
Laboratory tests (Skempton, 1964) cp' = 15.3 kPa p' = 20° peak cr' = 0 r' = 16° residual Back Analyses according to the Limit Equilibrium Method with circular sliding surface (Skempton, 1964) c' = 6.72 kPa ' = 18° Back 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 Here the shear strength parameters, obtained by shear laboratory tests, and back analyses based on the limit equilibrium method with circular sliding surface (both by Skempton) are summarised. It can be noted that these latter values fall within the range identified by lab tests. The results obtained by Duncan and Stark through back analyses based on the procedure try and error are also shown. Such procedure is based on the consideration that, for the sliding surface to be univocally determined, the safety factor must be 1 on the surface and greater than 1 on the surface immediately below and above. This principle was applied by Duncan and Stark to the circular surface previously identified by Skempton and to a non-circular surface containing the known portion of the actual failure surface. The difference between the resistance values obtained from back analyses and those determined in laboratory may arise from the fact that failure in the actual slope took place progressively over a long period of time (think to the years passed from the excavation to the collapse), with the sliding surface propagating from the toe of the slope. Such phenomenon cannot be analysed with a limit equilibrium method, which assumes that the shear strength parameters are the same along the whole imposed failure surface. On the contrary, a fracture mechanics approach lends itself well to the study of progressive failure, thanks to the possibility of simulating the propagation by shear of a crack located at the foot of the slope, by adopting different strength parameters values for intact material and the surface of the crack

The numerical model LEFM approach 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 So a series of numerical analyses has been carried out on the basis of the following assumptions: - NB Drained conditions is the assumption for Skemton back analyses LEFM approach

Geometrical and mechanical configuration
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’ = kPa ’ = 20° intact material c’ = 0 ’ = 16° surface of the crack Prima leggere A first analysis carried out in the absence of water did not give rise to propagation: the slope turned out to be stable Assuming the phreatic surface to be present, unstable propagation by shear was triggered at the end of the excavation process

Numerical failure surface
Top of the slope before propagation after propagation Toe of the slope The resulting sliding surface is shown here, together with the profile of the slope before and after propagation, in order to highlight the deformation caused by the propagation process. In particular, we can see a zoom of the top of the slope and the toe. In the proximity of the top, a propagation process was triggered by tensile stresses, which stopped before complete coalescence. The intact portion of the of the slope is small, this let us conclude that the slope is unstable sliding surface

Mobilisation ratio At the end of the excavation process
The propagation will take place in the direction where R is maximum Ratio between the the maximum acting principal stress 1, and the maximum principal resisting stress 1R inside the intact material before the onset of propagation. The propagation will take place in the direction where R is maximum

Computed relative displacements
At the end of the excavation process Maximum relative displacement = 19.3 cm Relative displacement of the slope profile The elastic component (due to the excavation) of the displacement has been deducted The maximum relative displacement at the foot is ... No comparative values are available since in situ displacement were not measured, however, the direction of the displacement seems realistic As a conclusion, we can say that the application of fracture mechanics to the study of progressive failure of a soft rock slope gave good results in terms of failure surface and strength parameters of the material, even with a linear elastic approach. This is encouraging for the application of the non-linear approach, which should allow to take into account the dissipation of energy occurring on the failure surface

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

Panoramic view of the rock mass involved in the rockfall The blocks broke off from the zone indicated with a pink circle and involved the road at the bottom. Behind the hill there’s the water reservoir where blocks fell. Water reservoir Road

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 S J1 J1 J2 The stepped failure surface is made up by a combination of planes belonging to the following discontinuity systems: J1 : (dip direction 65°, dip 75°) S: (dip direction 245°, dip 35°) the rock block failed on this surface and laterally delimited by a third system: J2: (dip direction 130°, dip 85°) The picture show the stepped failure surface, where planes E and C belong to the system J1 and planes D and B to the system S. Here we have a photo showing the same planes (foto!) The existence of fresh rock zones resulting from the failure of rock bridges was detected on planes E and C. A tensile failure of such rock bridges was assumed due to the high roughness of the fresh rock surfaces the rock mass was prevented from sliding due to the existence of intact rock teeth on the failure surface (foto!) In the picture we can also note a vertical discontinuity, belonging to system J1 All these observation lead the geologists to assume a toppling mechanisms of three separate blocks

Geometry and structural configuration
J1 S

Localisation and extension of rock bridges

Proposed failure mechanisms
Consecutive toppling of three blocks, due to the tensile failure of rock bridges The following kinematism seems to be realistic: Block 1 is assumed to be toppled due to the failure of contrasting rock located in its lower part. No failure of rock bridges can be observed between block 1 and block 2, but the presence of a discontinuity belonging to J1 can be assumed (indicare discontinutà) Block 2 is assumed to be toppled because of the collapse of block 1. Failure of rock bridges can be observed on plane C (behind block 2) Block 3 is assumed to be toppled because of the failure of blocks 1 and 2. Failure of rock bridges can be observed on plane E (behind block 1)

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, 1976)

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 On the basis of the assumed failure mechanism and the laboratory and in-situ investigations, a numerical back analysis has been carried out. The toppling failure… leggere tutto The length of rock bridges for block 2 and 3 has been determined on the basis of the location and extension of rock bridges surveyed by the geologists on planes C and D of the stepped failure surface.

Geometrical configurations
Block 2 Block 3 Here, the geometrical model assumed for blocks 2 and 3 is shown, and the length and position of rock bridges is highlighted. In black are are presented the edges of the structure, in red the open elements and in blue the closed elements DD open elements (edges) DD open elements DD closed elements

Numerical results: block 2
Open crack propagation in mixed mode conditions (KI and KII  0) Propagation takes place for: KIC = 0.34 MPam A toppling failure mechanisms is found, with failure of the rock bridge. The propagating crack inside the rock bridge is open and the curved trajectory is due to the occurrence of mixed-mode conditions (the stress intensity factors in mode I and II are not nil. This confirms the tensile character of the failure and the observed rough fresh rock surfaces. The value of toughness for which failure of the rock bridge occurs is lower than the experimental value. This can be due to scale effects (???)

Block 2: failure mechanism
rock cliff toppling block rock bridge failure due to induced tensile crack propagation

Numerical results: block 2
Open crack Closed crack The variation of the normal and tangential stress along the base surface is reported here. This shows how the toppling mechanism leads to the opening of the upper part of the basis (open elements on the left of the diagram, with n and  equal to zero), while in the lower part of the basis the stress are different from zero and increasing (compressions are negative) ELIMINARE IL PICCO???? tangential stress  normal stress n

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