Μεραληθή Πεηξσκάησλ (Rock Mechanics) Μέξνο 1 νλ «Εηζαγσγηθά»

Μεραληθή Πεηξσκάησλ (Rock Mechanics) Μέξνο 1 νλ «Εηζαγσγηθά» Exadaktylos G. Exadaktylos notes Rock Mechanics part I Slide 1 of 31

Η Μετανική ηων Πεηρωμάηων (Rock Mechanics) αζτολείηαι με ηεν μελέηε ηες ζηαηικής και δσναμικής ζσμπεριθοράς ηοσ αρρήκηοσ πεηρώμαηος και ηες βρατόμαδας (ρεγμαηωμένοσ πεηρώμαηος). Η Μετανοηετνία ηων Πεηρωμάηων (Rock engineering) εθαρμόδεηαι για ηον ζτεδιαζμό και ηε δεμιοσργία καηαζκεσών πάνω και μέζα ζηα πεηρώμαηα. Exadaktylos notes Rock Mechanics part I Slide 2 of 31

Παξαδείγκαηα θαηαζθεπώλ ζε πεηξώκαηα - Πξαλή (ππαίζξηεο εθκεηαιιεύζεηο) - Υπόγεηα έξγα (ζήξαγγεο, ζάιακνη, θξέαηα, θεθιηκέλα) - Βαζηέο γεσηξήζεηο - Θεκειηώζεηο (θξάγκαηα, γέθπξεο, πάζζαινη, ) -Τνίρνη αληηζηήξημεο Exadaktylos notes Rock Mechanics part I Slide 3 of 31

Καηαζθεπέο εληόο ησλ πεηξσκάησλ θαη επί απηώλ. Exadaktylos notes Rock Mechanics part I Slide 4 of 31

Η Μεραληθή ησλ Πεηξσκάησλ είλαη ην «εξγαιείν» γηα ηελ πξόβιεςε ηεο κειινληηθήο ζπκπεξηθνξάο ησλ βξαρνκαδώλ. Λ.ρ. ηη ζα ζπκβεί αλ κία ζήξαγγα ή έλαο ζάιακνο νξπρζεί κε έλα ζπγθεθξηκέλν πξνζαλαηνιηζκό θαη ζε έλα νξηζκέλν βάζνο ζε κηα βξαρόκαδα? Τη ζα γίλεη αλ αιιάμεη ν πξνζαλαηνιηζκόο θαη ην βάζνο? Π.ρ. κεηά ηελ θαηάξξεπζε ηνπ ζαιάκνπ ζην Sao Paulo to 2007 κεξηθνί πξόηεηλαλ όηη ππόγεηνη ζηαζκνί ηνπ κεηξό ζε θαηνηθεκέλεο πεξηνρέο πξέπεη λα γίλνληαη ζε βάζε κεγαιύηεξα από 20 m. Oη απαληήζεηο ζε ηέηνηνπ είδνπο εξσηήκαηα απαηηνύληαη ζην rock engineering design. Ο Βξαρνκεραληθόο πξέπεη λα έρεη δπλαηόηεηεο πξόβιεςεο: ρσξίο απηέο δελ ππάξρεη ε βάζε γηα ζπλεθηηθό ζρεδηαζκό! Ο ηξόπνο (κέζνδνο) απάληεζεο ζε ηέηνηνπ είδνπο εξσηήκαηα είλαη θαη ζθνπόο ηνπ καζήκαηνο. Exadaktylos notes Rock Mechanics part I Slide 5 of 31

Γηα λα κελ αληηκεησπίζνπκε ιόγσ θαθνύ ζρεδηαζκνύ ηέηνηα πξνβιήκαηα αιιά θαη άιια κεξηθά από ηα νπνία ζα δνύκε θαη ζε άιιεο δηαθάλεηεο. Καηάξξεπζε κεηώπνπ ζηε Σήξαγγα Galgenberg, Aπζηξία 6 Exadaktylos notes Rock Mechanics part I Slide 6 of 31

Exadaktylos notes Rock Mechanics part I Slide 7 of 31 Ρσγκέο ζηελ ζθπξνδεηεκέλε επέλδπζε

F 1 F 2 Intact rock F 3 Excavation Boundary conditions F n Κύξηνη παξάγνληεο ζρεδηαζκνύ Water flow Fractures Γεσινγία (Geology) Επί ηόπνπ ηάζεηο (Rock stress) Αξξεθην πέηξσκα (Intact rock) Αζπλέρεηεο (Discontinuities, Fractures) Παξάκεηξνη βξαρνκάδαο (Rock mass properties) Τπόγεηα ύδαηα (Water flow) Οξπμε, ππνζηήξημε (Engineering activities) Μνληεινπνίεζε (Modelling) Exadaktylos notes Rock Mechanics part I Slide 8 of 31

It is essential for rock engineering to understand the geology at a project site, for example at the JinPing II site Exadaktylos notes Rock Mechanics part I Slide 9 of 31 9

Aπνηύπσζε ξεγκάησλ ζηε ζήξαγγα Galgenberg, Aπζηξία Exadaktylos notes Rock Mechanics part I Slide 10 of 31

DTM with superimposed faults of the corridor area (corridor=possible path of the tunnel) DTM = Digital Terrain Model Exadaktylos notes Rock Mechanics part I Slide 11 of 31

Moξθνινγία θαλνληθώλ ξεγκάησλ (normal faults), La Linea Tunnel, Colombia Exadaktylos notes Rock Mechanics part I Slide 12 of 31

Exadaktylos notes Rock Mechanics part I Slide 13 of 31 3-δηάζηαην γεσινγηθν-ηεθηνληθό κνληέιν

Input: Γεσινγηθό κνληέιν απαξηηδόκελν από ηα ζηεξεά κνληέια θάζε ιηζνινγηθήο θάζεο Exadaktylos notes Rock Mechanics part I Slide 14 of 31

Exadaktylos notes Rock Mechanics part I Slide 15 of 31

Xαξαθηεξηζκόο βξαρόκαδαο από ππξήλεο γεσηξήζεσλ (rock cores) 1 2 m Exadaktylos notes Rock Mechanics part I Slide 16 of 31 16

Borehole core information 1 m Exadaktylos notes Rock Mechanics part I Slide 17 of 31

RQD Rock Quality Designation (RQD) is defined as the percentage of rock cores that have length equal or greater than 10 cm over the total drill length. Deere D. U. Technical description of rock cores for engineering purposes. Rock Mech. Eng. Geol. 1, 17-22 (1964). Exadaktylos notes Rock Mechanics part I Slide 18 of 31

Πεξηνξηζκνί ζηε ρξήζε ηνπ RQD γηα ηελ πνζνηηθή πεξηγξαθή ηνπ βαζκνύ ξεγκάησζεο Exadaktylos notes Rock Mechanics part I Slide 20 of 31

Εθηηκήζαηε ην RQD ησλ ππξήλσλ καξκάξνπ. Exadaktylos notes Rock Mechanics part I Slide 21 of 31

Exadaktylos notes Rock Mechanics part I Slide 22 of 31 Εθηηκήζαηε ην RQD ησλ ππξήλσλ καξκάξνπ.

Εθηηκήζαηε ην RQD αλά 2 m ησλ ππξήλσλ καξκάξνπ. Exadaktylos notes Rock Mechanics part I Slide 23 of 31

Sampling issues Location of borehole (1. Πξνζαλαηνιηζκόο γεώηξεζεο ζε ζρέζε κε ηνλ πξνζαλαηνιηζκό ησλ ξεγκάησλ θαη ησλ δηαθιάζεσλ) 2. Θέζε ηεο γεώηξεζεο ε ζρέζε κε ηελ ζέζε ησλ ξεγκαηώζεσλ Orientation of borehole 3 2 4 1 24 Exadaktylos notes Rock Mechanics part I Slide 24 of 31

Κύξηνη παξάγνληεο ζρεδηαζκνύ Γεσινγία (Geology) Επί ηόπνπ ηάζεηο (Rock stress) Αξξεθην πέηξσκα (Intact rock) Αζπλέρεηεο (Discontinuities, Fractures) Παξάκεηξνη βξαρνκάδαο (Rock mass properties) Τπόγεηα ύδαηα (Water flow) Οξπμε, ππνζηήξημε/ελίζρπζε (Engineering activities) Μνληεινπνίεζε (Modelling) Exadaktylos notes Rock Mechanics part I Slide 27 of 31

Stress is a tensor quantity A SCALAR is a quantity with magnitude only. Examples of scalars are temperature, time, mass and pure colour - they are described completely by one value, e.g. degrees, seconds, kilograms and frequency. A VECTOR is a quantity with magnitude and direction. Examples of vectors are force, velocity, and the frequency of fractures encountered along a line in a rock mass - they are described completely by three values, for example, x, y, z components which together specify both direction and magnitude. A TENSOR is a quantity with magnitude, direction and the plane under consideration. Examples of tensors are stress, strain, permeability and moment of inertia-they are described completely by six values, Exadaktylos notes Rock Mechanics part I Slide 28 of 31

A solid can sustain a shear force, whereas a liquid or gas cannot. A liquid or gas contains a pressure, i.e. a force per unit area, which acts equally in all directions and hence is a scalar quantity. Rock can sustain a shear force and hence contains stress, which is a tensor quantity. Exadaktylos notes Rock Mechanics part I Slide 29 of 31

Οη 3 θύξηεο ηάζεηο Exadaktylos notes Rock Mechanics part I Slide 30 of 31

ύκβαζε 1: Θιηπηηθέο ηάζεηο ζεσξνύληαη ζεηηθέο πνζόηεηεο ύκβαζε 2: ζ 1 >ζ 2 >ζ 3 Exadaktylos notes Rock Mechanics part I Slide 31 of 31

(c) Previous tectonic forces. There may also be residual stresses from earlier tectonic events: when a fractured rock mass is compressed and then unloaded, stresses can be left locked in the rock mass. An analogue example of this effect is a pre-stressed car windscreen. Παξακέλνπζεο ηάζεηο ιόγσ πξνεγνύκελεο ηεθηνληθήο δξαζηεξηόηεηαο θαη απνθόξηηζεο Exadaktylos notes Rock Mechanics part I Slide 32 of 31

A pop-up observed at a quarry site in granite in Southeastern Manitoba and the horizontal stress determined using the USBM Borehole Deformation Gauge in a vertical borehole. (Quarry popup. (b) Measured horizontal stress) Exadaktylos notes Rock Mechanics part I Slide 33 of 31

Tεθηνληθέο ηάζεηο θνληά ζε ξήγκα ή αζπλέρεηα Exadaktylos notes Rock Mechanics part I Slide 34 of 31

Geological effect Engineering effect A B C F rac tu re 1 0 0 0 0 2 0 0 0 2, 1 3 = 0 R ock m as s Principal stresses are parallel and perpendicular to open fracture surfaces and excavation surfaces An open fracture will perturb the stress field and cause the principal stresses to be locally parallel and perpendicular to the fracture surface. Free surfaces in a rock mass cause the local principal stresses to be parallel and perpendicular to the free surface. Exadaktylos notes Rock Mechanics part I Slide 35 of 31

Principal stresses occur on planes with no shear stress acting on them. Hence... 1 0 0 0 2 0. 1 0 0 0 0 0 0 2 0. 0 0 0 Principal stresses parallel to excavation surface 1 Principal stresses parallel perpendicular to excavation to surface excavation surface 1 Principal stress perpendicular to This is for the case where there are excavation no tensile surface stresses in the rock, i.e. the 3 value of zero is the lowest of the three principal stresses. This is for the case where there are no tensile stresses in the rock, i.e. the 3 value of zero is the lowest of the three principal stresses. 2 2 3 = 0 3 = 0 Exadaktylos notes Rock Mechanics part I Slide 36 of 31

In situ stress In the majority of rock mechanics models, the in situ rock stress is required as a boundary condition input to the model. However, it is difficult to validate the in situ stress assumption locally, both in terms of the principal stress values/directions and their consistency through the fractured rock mass. In situ stress picture Cornwall hydrofrac? World stress map? Mueller, B., Reinecker, J., Heidbach, O. and Fuchs, K. (2000): The 2000 release of the World Stress Map (available online at www.world-stress-map.org) 37 Exadaktylos notes Rock Mechanics part I Slide 37 of 31

Predicted European tectonic stresses based on mid-atlantic ridge push and one type of model (from Gölke and Coblentz (1996). Επί ηόπνπ ηάζεηο ιόγσ θίλεζεο ιηζνζθαηξηθώλ ή ηεθηνληθώλ πιαθώλ (θειπθώλ). H θίλεζε δείρλεηαη κε ηα βέιε ελώ ηα ηξίγσλα ζηα ζύλνξα ηεο πιάθαο (θάησ θαη δεμηά) ζπκβνιίδνπλ αθίλεηα ζύλνξα. Exadaktylos notes Rock Mechanics part I Slide 38 of 31 38

Επί ηόπνπ ηάζεηο ιόγσ θίλεζεο ιηζνζθαηξηθώλ πιαθώλ (θειπθώλ) Predicted European tectonic stresses based on mid-atlantic ridge push and one type of model (from Gölke and Coblentz, 1996). 52 60 43 52 35 43 27 35 18 27 10 18 MPa Σξη-δηάζηαην κνληέιν γηα ηνλ ππνινγηζκό ησλ ηάζεσλ Figure 2. Example of 3DEC modelling of the influence of brittle deformation zones on rock stresses, Laxemar site, Sweden, horizontal section, ~ 7 km across (prepared for SKB, by Eva Hakami, Itasca, Sweden). Exadaktylos notes Rock Mechanics part I Slide 39 of 31 39

Ρεγκαησκέλε δώλε πιεξσκέλε κε άξγηιν Επίδξαζε ηεο ηνπνγξαθίαο θαη πξνϋπαξρόλησλ ξεγκάησλ ζηηο επί ηόπνπ ηάζεηο. Exadaktylos notes Rock Mechanics part I Slide 40 of 31

z z z Σάζεηο ιόγσ βαξύηεηαο ζε πξαλέο ζ 1 = γz, γ=0.027 ΜPa/m Exadaktylos notes Rock Mechanics part I Slide 41 of 31

Kαηαλνκή θύξηαο επί ηόπνπ ηάζεο ζ 1 ιόγσ βαξύηεηαο ζε πξαλέο Kύξηεο ηάζεηο Exadaktylos notes Rock Mechanics part I Slide 42 of 31

Kαηαλνκή ειάρηζηεο θύξηαο επί ηόπνπ ηάζεο ζ 3 ιόγσ βαξύηεηαο ζε πξαλέο ζ 3 = Κ x ζ 1, Κ = ζπληειεζηήο πιεπξηθήο ηάζεο Exadaktylos notes Rock Mechanics part I Slide 43 of 31

Slide from: The Recent Tectonic Stress Field and Strong Earthquakes in China Xie Furen Institute of Crustal Dynamics, CEA Dynamic regions of China and adjacent areas 欧亚板块北美板块太平洋板块印度板块菲律宾板块 Σεθηνληθέο ηάζεηο Exadaktylos notes Rock Mechanics part I Slide 44 of 31

Exadaktylos notes Rock Mechanics part I Slide 45 of 31 Σεθηνληθό κνληέιν ηεο Βξεηαλίαο κε ηελ ζρεηηθή θίλεζε ησλ ηεκαρώλ κεηαμύ ησλ ξεγκάησλ (θαίλεηαη κε βέιε) γηα ηελ πξόβιεςε ησλ πεξηνρώλ πνπ κπνξνύλ λα «δώζνπλ» ζεηζκό (κε κπιε ρξώκα).

Characterizing rock stress at a site F 1 F 2 F 3 Intact rock Water flow Excavation Boundary conditions F n N Trend 16 MPa Plunge 10 MPa 7 MPa Fractures Natural stress: the in situ stress which exists prior to engineering. Gravitational stress: the stress state caused by the weight of the rock above. Induced stress: the natural stress state as perturbed by engineering. Tectonic stress: the stress state caused by tectonic plate movement. Residual stress: the stress state caused by previous tectonic activity. Thermal stress: the stress state caused by temperature change. Paleostress: a previous natural stress that is no longer acting. Near-field stress: the stress state in the region of an engineering perturbation. Far-field stress: the stress state beyond the near-field. Local stress: the stress state in a region of interest. Exadaktylos notes Rock Mechanics part I Slide 46 of 31 46

Exadaktylos notes Rock Mechanics part I Slide 49 of 31 P b = πίεζε ζξαύζεο (breakdown or opening pressure) P s = shut-in pressure < P b γηαηί δελ ρξεηάδεηαη λα ππεξληθεζεί ε εθειθπζηηθή αληνρή ηνπ πεηξώκαηνο

Exadaktylos notes Rock Mechanics part I Slide 52 of 31 ύκβαζε πξνζήκνπ ηάζεσλ: Θιηπηηθέο ηάζεηο > 0, Εθειθπζηηθέο ηάζεηο <0

ζζ/ζh 2 1 R 1 ( H h) 1 ( H h) 1 2 2 r 2 ( ) 2( )cos2 P H h H h 3R 4 r 4 cos2 P R r 2 2 rr 2.5 2 1.5 1 0.5 0-0.5-1 compression 0 45 90 135 180 225 270 315 360 tension Circular hole (ζh/ζh=2/3) ζ [degrees] P/sigmaH=0 P/sigmaH=0.4 P/sigmaH=0.8 P/sigmaH=1.2 P/sigmaH=1.4 P/sigmaH=1.6 the highest stress concentration occurs always at the points of the contour corresponding to ζ=0 o, 180 o, i.e. along the direction of the minimum principal stress. Ιncreasing the borehole pressure P results in a decrease of the tangential stress at ζ=0 o, 180 o and increase of tension at ζ=90 ν, 270 ν Exadaktylos notes Rock Mechanics part I Slide 53 of 31

Exadaktylos notes Rock Mechanics part I Slide 54 of 31 sin 2 3 2 1 ) ( 2 1 cos2 3 1 ) ( 2 1 1 ) ( 2 1 cos2 3 4 1 ) ( 2 1 1 ) ( 2 1 4 4 2 2 ) ( 4 4 2 2 ) ( 4 4 2 2 2 2 ) ( r R r R r R r R r R r R r R h H o r h H h H o h H h H o r 0,, 2 2 ) ( 2 2 ) ( r i i r r R P r R P Kirsch s (1898) stress solution assuming isotropic linear elasticity Stability of deep boreholes b h H b H h T h H h H h H h H R r i o P T P T P P 3 3 ) 2( ) ( )cos180 2( ) ( 0 ) ( Τπέξζεζε (ιόγσ γξακκηθόηεηαο) -Σ = εθειθπζηηθή αληνρή ηνπ πεηξώκαηνο (<0) Πίεζε πόξσλ = 0, Αιιηώο έλλνηα «ελεξγήο ηάζεο» θαηά Terzaghi

Exadaktylos notes Rock Mechanics part I Slide 55 of 31 p b h H p b p h p H P P T P P T P P 3 3 Πίεζε πόξσλ = Pp, έλλνηα «ελεξγήο ηάζεο» θαηά Terzaghi

Exadaktylos notes Rock Mechanics part I Slide 57 of 31 Issue of this journal devoted to rock stress estimation which contains the four International Society for Rock Mechanics Suggested Methods and 17 papers on experiences of stress measurement

εκαζία ηνπ κέηξνπ ειαζηηθόηεηαο θαη ηνπ ιόγνπ ηνπ Poisson ζην ζρεδηαζκό πδξαπιηθήο ζξαύζεο. Mεηά ην πξόβιεκα ηνπ Kirsch, ην πξόβιεκα ηνπ ειιεηπηηθνύ αλνίγκαηνο είλαη ην πην ζεκαληηθό θαη ιύζεθε από ηνλ Inglis (1913). Exadaktylos notes Rock Mechanics part I Slide 60 of 31

εκαζία ηνπ κέηξνπ ειαζηηθόηεηαο θαη ηνπ ιόγνπ ηνπ Poisson ζηελ ζπκπεξηθνξά ηεο ζεκειίσζεο Exadaktylos notes Rock Mechanics part I Slide 61 of 31

(plane Ox 2 x 3 parallel to the Earth s surface) Ιζόηξνπν πέηξσκα Νόκνο ηνπ Hooke ij 1 E 1, (i, j 1,2,3 ) ij kk ij Exadaktylos notes Rock Mechanics part I Slide 63 of 31

11 22 33 1 a 1 a 1 a 11 22 33 22 33, a 33 11, a 11 22. a a E Θέηνπκε 11 v z 22 33 0, 22 33 h Καη βξίζθνπκε h K, K v 1 Nα απνδεηρζεί από εζάο! Exadaktylos notes Rock Mechanics part I Slide 64 of 31

Sheorey (1994) based on an elasto-static thermal model of the earth (curvature of the crust, variation of elastic constants, density and thermal expansion coefficients through the crust and mantle), gave the eqn K 0.25 7 E h 0.001 Πσο ζα ζρνιηάδαηε απηό ην ζρήκα? 1 z Exadaktylos notes Rock Mechanics part I Slide 65 of 31

Hoek and Brown collated stress data Exadaktylos notes Rock Mechanics part I Slide 66 of 31

Due to the action of significant geostatic and tectonic forces that are encountered at great depths, deep boreholes suffer from severe instabilities, breakouts and exfoliations that can become critical for the progress of the drilling process and may interrupt or even stop energy production. Breakouts lead in general to progressive deterioration of the borehole. Θιηπηηθή ζξαύζε γεσηξήζεσλ (borehole breakouts) Exadaktylos notes Rock Mechanics part I Slide 67 of 31

(the main cause of failure: stress concentration around holes) http://astro.temple.edu/~davatzes/methods/methods/borehole_analysis.html Exadaktylos notes Rock Mechanics part I Slide 68 of 31

Introductory remarks Theoretical approach: The ability to predict fracture and stress orientations is becoming increasingly important in development of production wells. Experimental approach: Wireline borehole scanner data to identify the borehole breakouts and its orientation to maximum stress zones. http://csgexploration.com/structural%20geology%20and%20tectonic%20analysis.html Exadaktylos notes Rock Mechanics part I Slide 69 of 31

(World stress map (Mediterranean region) Approximately 19% of the stress orientation indicators in the World Stress Map (WSM) database have been determined from borehole breakouts. http://dc-app3-14.gfz-potsdam.de/pub/stress_data/stress_data_frame.html Exadaktylos notes Rock Mechanics part I Slide 70 of 31

The danger of damage to the tunnel can be reduced by, if possible, orientating the tunnel so that it is parallel to the major principal stress 1 perpendicular to tunnel axis 1 parallel to tunnel axis 1 is vertical Exadaktylos notes Rock Mechanics part I Slide 71 of 31

The problem of the possible proximity of faults: causing major alterations to the stress field outside the overall variation Horizontal stress Without faults Exadaktylos notes Rock Mechanics part I Slide 72 of 31

Horizontal stress With fault Exadaktylos notes Rock Mechanics part I Slide 73 of 31

Conclusion: Cannot predict these local macroperturbations but they will be relatively rare and so will have to be treated as extreme values Exadaktylos notes Rock Mechanics part I Slide 74 of 31

Αλ ζέιαηε λα θαηαζθεπάζεηε έλα ππόγεην ζάιακν πσο ζα ζρεδηάδαηε ηε δηαηνκή ηνπ (H/W??) ιακβάλνληαο ππόςηλ ην επί ηόπνπ πεδίν ηάζεσλ? Η Αλαπηύμαηε ηε κέζνδν ζαο. W Exadaktylos notes Rock Mechanics part I Slide 75 of 31

Will rock spalling occur? Pre-existing natural rock stress The strength of the rock is compared to the local stress values Spalling can start to occur at about half the uniaxial compressive strength Tunnels parallel to the maximum in situ principal stress are subject to less stress concentration Excavationinduced stresses around the excavation Stress diagram from work by Dr Erik Johansson, Saanio and Riekkola, Finland Exadaktylos notes Rock Mechanics part I Slide 76 of 31 76

Exadaktylos notes Rock Mechanics part I Slide 77 of 31 Spalling in a shaft

Exadaktylos notes Rock Mechanics part I Slide 78 of 31 Worst case scenario spalling in the South African gold mines

Επίδξαζε ηεο αληνρήο ηεο βξαρνκάδαο ζηελ έθηαζε ηεο δώλεο κηθξνξσγκάησζεο γύξσ από ππόγεηνπο ζαιάκνπο 1. Πσο κπνξεί λα εθηηκεζεί ε έθηαζε ηεο δώλεο κηθξνξσγκάησζεο γύξσ από ππόγεην άλνηγκα? 2. Πνηεο είλαη νη παξάκεηξνη πνπ επεξεάδνπλ ηελ έθηαζε ηεο? Exadaktylos notes Rock Mechanics part I Slide 79 of 31

Exadaktylos notes Rock Mechanics part I Slide 80 of 31 Επί ηόπνπ ηάζεηο? (κάιινλ ην μέραζε ν πνηεηήο!)

Gravity-driven failure Stress-driven failure Exadaktylos notes Rock Mechanics part I Slide 81 of 31