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

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Presentation on theme: "Fracture mechanics approach to the study of failure in rock Claudio Scavia, Marta Castelli Politecnico di Torino Dipartimento di Ingegneria Strutturale."— Presentation transcript:

1 Fracture mechanics approach to the study of failure in rock Claudio Scavia, Marta Castelli Politecnico di Torino Dipartimento di Ingegneria Strutturale e Geotecnica Corso di Leggi costitutive dei geomateriali Dottorato di Ricerca in Ingegneria Geotecnica

2 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 2 Index Introduction Introduction Basic concepts of Linear Elastic Fracture Mechanics Basic concepts of Linear Elastic Fracture Mechanics Propagation criteria Propagation criteria Non linear Fracture Mechanics Non linear Fracture Mechanics Numerical modelling of cracked rock structures Numerical modelling of cracked rock structures Ø The Displacement Discontinuity Method Ø Numerical simulation of experimental results Ø Application to slope stability

3 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 3 Introduction 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 Tay bridge (Scotland, 1898)Schenectady ship (1943)

4 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 4 Introduction Large natural defects (faults, joints…) exist in rock masses Example: 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 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

5 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 5 Introduction Main steps in Fracture Mechanics Evaluation of stress concentration Choice of a propagation criterion Definition of a methodology for the simulation of crack propagation Ø stable propagation Ø unstable propagation Analysis of the state of stress

6 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 6 Modes of failure in rocks 1 3Shearband 1 Direct tension Axialsplitting 1 Indirect tension 1 At the scale of the laboratory

7 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 7 Modes of failure in rocks At the scale of the rock mass Indirecttension Shear Directtension

8 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 8 Index Introduction Introduction Basic concepts of Linear Elastic Fracture Mechanics Basic concepts of Linear Elastic Fracture Mechanics Propagation criteria Propagation criteria Non linear Fracture Mechanics Non linear Fracture Mechanics Numerical modelling of cracked rock structures Numerical modelling of cracked rock structures Ø The Displacement Discontinuity Method Ø Numerical simulation of experimental results Ø Application to slope stability

9 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 9 Linear Elastic Fracture Mechanics Elastic behaviour of the material Inelastic behaviour of crack surfaces Determination of stress concentration at the crack tip Ø fracture energy Ø stress intensity factor Definition of the conditions for crack to propagate, through energetic or stress intensity balances

10 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 10 Stress concentration circular hole elliptical hole 2b a crack a 2b 0 max = 3 r max max = f(a, b) r max r

11 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 11 Energetic approach (Griffith, 1921) 2 = fracture energy G c 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 Condition for crack propagation elastic energy release rate surface energy

12 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 12 Tensional approach (Irwin, 1957) Crack propagation can be studied through the superposition of the effects of three independent load application modes opening mode I opening - loads are orthogonal to the fracture plane slip mode II slip - loads are tangent to the fracture plane in the direction of maximum dimension tear mode III tear - loads are contained in the fracture plane and act perpendicularly to mode II

13 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 13 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:

14 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 14 Tensional approach For =0 i.e. for a point at a distance r along the line of the crack: For relative displacements û between the crack faces at a small distance x from the crack tip: r y x ûyûy

15 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 15 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

16 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 16 Meaning of the Stress Intensity Factors The vertical stress, y, around the crack tip is given by the theory of elasticity: a 2b 0 y r The specific boundary conditions of the problem affect the value of y through a constant term K I which is given by: Example: crack of length 2a, located in a plate subjected to a uniform vertical tensile stress

17 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 17 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 K C 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

18 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 18 Index Introduction Introduction Basic concepts of Linear Elastic Fracture Mechanics Basic concepts of Linear Elastic Fracture Mechanics Propagation criteria Propagation criteria Non linear Fracture Mechanics Non linear Fracture Mechanics Numerical modelling of cracked rock structures Numerical modelling of cracked rock structures Ø The Displacement Discontinuity Method Ø Numerical simulation of experimental results Ø Application to slope stability

19 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 19 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 K IC (or fracture energy G Ic ) 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

20 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 20 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 K IC is the material toughness

21 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 21 Open cracks (Erdogan & Sih, 1963) For pure mode I: For pure mode II:

22 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 22 Open cracks K I > 0 K II = 0 K I > 0 K II 0 K I < 0 K II 0 K I = 0 K II 0 K I < 0 K II = 0

23 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 23 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 K IC (Erdogan & Sih, 1963) K IC Shear propagation: (mode II) The original crack is compressed, and it propagates in compressive stress fields K IIC ?

24 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 24 Shear propagation criteria A 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 (K IIC ) is still under discussion

25 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 25 Fracture toughness: mode I Experimental determination Suggested methods (ISRM, 1988) Short rod (SR) Chevron bend (CB)

26 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 26 Short rod uncut rock or ligament a0a0 a W a1a1 notch P D Pload on specimen Ddiameter of short rod specimen Wlength of specimen hdepth of crack in notch flank chevron angle tnotch width a 0 chevron tip distance a crack length a 1 maximum depth of chevron flanks D/2 t

27 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 27 Chevron bend Pload on specimen Aprojected ligament area L specimen length Sdistance between support points Ddiameter of chevron bend specimen CMODrelative opening of knife edges hdepth of crack in notch flank chevron angle = 90° a 0 chevron tip distance a crack length a CMOD uncut rock or ligament notch knife a0a0 a h P S L loading roller Support roller D A

28 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 28 Chevron bend

29 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 29 When is a LEFM approach applicable? process zone 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 Extremely high stress values involved in the phenomenon of crack propagation:

30 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 30 Index Introduction Introduction Basic concepts of Linear Elastic Fracture Mechanics Basic concepts of Linear Elastic Fracture Mechanics Propagation criteria Propagation criteria Non linear Fracture Mechanics Non linear Fracture Mechanics Numerical modelling of cracked rock structures Numerical modelling of cracked rock structures Ø The Displacement Discontinuity Method Ø Numerical simulation of experimental results Ø Application to slope stability

31 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 31 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 Ø 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

32 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 32 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

33 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 33 Non Linear Fracture Mechanics t inelastic stress distribution Visible crack true crackprocess zone stress freeelastic stress distribution c Open cracks (tension): the Cohesive Crack Model (Dugdale, 1960; Barenblatt, 1962)

34 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 34 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 tipreal tip process zone real crack p r r G Closed cracks (compression and shear): the Slip-Weakening Model the Slip-Weakening Model (Palmer & Rice, 1973)

35 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 35 Index Introduction Introduction Basic concepts of Linear Elastic Fracture Mechanics Basic concepts of Linear Elastic Fracture Mechanics Propagation criteria Propagation criteria Non linear Fracture Mechanics Non linear Fracture Mechanics Numerical modelling of cracked rock structures Numerical modelling of cracked rock structures Ø The Displacement Discontinuity Method Ø Numerical simulation of experimental results Ø Application to slope stability

36 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 36 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 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

37 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 37 Numerical modelling of cracked rock structures Displacement Discontinuity Method (Crouch & Starfield, 1983) allows to simulate the crack as Displacement Discontinuity elements D s = u s (s, 0 - ) - u s (s, 0 + ) D n = u n (s, 0 - ) - u n (s, 0 + ) n +D s s 2a +D n

38 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 38 influence coefficients of D s (j) and D n (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

39 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 39 Open elements Tensile stress fields D n < 0 (opening) s (i), n (i) = 0 Compressive stress fields D n > 0 (closure) s (i), n (i) = 0

40 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 40 Closed elements Compressive stress fields D n = 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 DsDs KsKs sr = n ·tan

41 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 41 Simulation of crack propagation open cracks open cracks Erdogan & Sihs propagation criterion, based on the Stress Intensity factors calculation at the tip of the crack closed cracks closed cracks Ø induced-tensile propagation: Erdogan & Sihs criterion Ø 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: Ø stable propagation may develop only if the load is increased Ø unstable propagation: develops without any load increment (Scavia, 1995; Scavia et al., 1997)

42 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 42 Numerical implementation of the SWM Computer code BEMCOM (Allodi et al., 2002)

43 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 43 Adopted slip-weakening laws intact material (tip element): c p, p real crack: c = 0, = r process zone: linear variation of c and as a function of c * and * Cohesion (c)Friction angle ( ) 0 cpcp c * c 0 * p r

44 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 44 Index Introduction Introduction Basic concepts of Linear Elastic Fracture Mechanics Basic concepts of Linear Elastic Fracture Mechanics Propagation criteria Propagation criteria Non linear Fracture Mechanics Non linear Fracture Mechanics Numerical modelling of cracked rock structures Numerical modelling of cracked rock structures Ø The Displacement Discontinuity Method Ø Numerical simulation of experimental results Ø Application to slope stability

45 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 45 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 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 46 Induced-tensile propagation (Castelli, 1998) Experimental work on concrete samples containing two open slits subjected to uni-axial compression Characteristic of the materialGeometry and load configuration

47 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 47 Experimental results Stress-strain diagramStrain directions horizontal oblique longitudinal

48 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 48 Propagation trajectories NumericalExperimental

49 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 49 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 diagram measured displacements (stereo-photogrammetry) c5-c7 c7-c8

50 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 50 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

51 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 51 Propagation trajectories NumericalExperimental c5c7c7c8 c5c7c7c8

52 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 52 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) numericalexperimental

53 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 53 NLFM approach to shear propagation stereo-photogrammetric analysis Photographs of the specimens during the tests in order to carry out a stereo-photogrammetric analysis (Desrues, 1995) LB-01 LB-02LB-04 MB-09MB-10MB-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 mm 3 85 x 40 x 35 mm 3 (LB-02, LB-04) Experimental results (Marello, 2004)

54 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 54 couple 1-2 shear deformations Experimental results: test MB-11

55 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 55 couple 2-3 Experimental results: test MB-11 shear deformations

56 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 56 Experimental results: test MB-11 couple 3-4 shear deformations

57 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 57 Experimental results: test MB-11 Displacement vectors couple 5-6

58 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 58 Experimental results: test MB-11 The specimen at the end of the test

59 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 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

60 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 60 Numerical simulation: results Stress-strain global behaviour

61 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 61 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

62 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 62 Incremental displacements: points 2 and II Numerical (II) u y u y Experimental(2)

63 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 63 Incremental displacements: points 3 and III Experimental(3) u y u y Numerical (III) u y

64 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 64 Incremental displacements: points 4 e IV Experimental (4) u y u y Numerical (IV) u y u y

65 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 65 Index Introduction Introduction Basic concepts of Linear Elastic Fracture Mechanics Basic concepts of Linear Elastic Fracture Mechanics Propagation criteria Propagation criteria Non linear Fracture Mechanics Non linear Fracture Mechanics Numerical modelling of cracked rock structures Numerical modelling of cracked rock structures Ø The Displacement Discontinuity Method Ø Numerical simulation of experimental results Ø Application to slope stability

66 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 66 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 rocks soft rocks, hard soils

67 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 67 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

68 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 68 Cross-section of the slope observed portion of the actual slip surface (Skempton, 1964)

69 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 69 Shear strength parameters Laboratory tests Laboratory tests (Skempton, 1964) c p ' = 15.3 kPa p ' = 20°peak c r ' = 0 r ' = 16°residual Back Analyses Back Analyses according to the Limit Equilibrium Method with circular sliding surface (Skempton, 1964) c' = 6.72 kPa ' = 18° Back Analyses 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

70 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 70 The numerical model Assumptions peak shear strength values for intact material residual shear strength values for the surface of the crack Failure process starting at the foot of the slope Failure taking place at the end of the excavation works in drained conditions LEFM approach

71 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 71 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

72 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 72 Toe of the slope Top of the slope Numerical failure surface before propagation after propagation sliding surface

73 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 73 Mobilisation ratio At the end of the excavation process The propagation will take place in the direction where R is maximum

74 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 74 Computed relative displacements At the end of the excavation process Maximum relative displacement = 19.3 cm

75 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 75 Example of application to hard rock slopes Mattsand (CH) (Amatruda et al., 2004) : Back analysis of the rockfall occurred in October 1998 in Mattsand (CH) (Amatruda et al., 2004) : steep gneiss slope a volume of about 300 m 3, triggered from a steep gneiss slope, fell into a water reservoir and damaged a road MATTSAND

76 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 76 Detaching zone Road Water reservoir

77 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 77 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 J2 J1

78 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 78 Geometry and structural configuration J1 S

79 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 79 Localisation and extension of rock bridges

80 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 80 Proposed failure mechanisms Consecutive toppling of three blocks, due to the tensile failure of rock bridges

81 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 81 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)

82 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 82 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

83 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 83 Geometrical configurations DD open elements (edges) DD open elements DD closed elements Block 2 Block 3

84 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 84 Numerical results: block 2 Open crack propagation in mixed mode conditions (K I and K II 0) Propagation takes place for: K IC = 0.34 MPa m A B

85 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 85 rock bridge failure due to induced tensile crack propagation rock cliff toppling block Block 2: failure mechanism

86 Corso di Leggi costitutive dei geomateriali – Novembre 2005 Dottorato di ricerca in Ingegneria Geotecnica 86 Numerical results: block 2 Open crack Closed crack tangential stress normal stress n


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