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Modelización de la anisotropía de los macizos rocosos
Dr. Alejo O. SfrisoUniversidad de Buenos Aires materias.fi.uba.ar/6408 [email protected] Consulting (Argentina) latam.srk.com [email protected] www.aosa.com.ar [email protected]
Isotropía, anisotropía, ortotropía
• Isotropía: mismas propiedades en todas las direcciones (rocas ígneas intactas)
• Ortotropía: dos o tres ejes ortogonales de simetría (algunas rocas sedimentarias)
• Anisotropía: propiedades diferentes en diferentes direcciones
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AxialRadialCircunfe-rencial
es.wikipedia.org/wiki/Material_ortótropo
Aliviadero Caracoles
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tests if both the stress direction and the fabric orientation are set toalign in the same fixed coordinate, such as the cases shown inFig. 1a, the yield surface can be plotted as shown in Fig. 1b (the yieldsurfaces do not cross the origin of the coordinate system due to theexistence of bonding) and Fig. 1c (yield loci in the deviatoric planewith different values of hardening parameter). The isotropic failuresurface is shown in the deviatoric plane in comparison with theanisotropic one. Note that in Fig. 1a we denote the angle betweenthe current stress state with the vertical stress axes in the deviatoricplane by h, and the deviatoric plane is partitioned into three zonesas shown in Fig. 1a. The same convention will be followed in thesubsequent sections.
3.2. Hardening law
The following evolution law for H is proposed:
dH ¼ hdLirH ¼ hdLiGchfHprðMf # HÞ ð7Þ
where rH denotes the evolution direction of H and is always greaterthan or equal to zero; dL is a loading index and hxi denotes theMacauley bracket with hxi = 0 when x 6 0 and hxi = x when x > 0;ch is a positive constant. Following Li and Dafalias [37] and Dafaliaset al. [13], we introduce the following f as a scaling factor to accountfor the effect of fabric anisotropy on the soil stiffness
f ¼ exp½#kðAþ 1Þ' ð8Þ
where k is a positive model parameter. Evidently, f in form of Eq. (8)is a decreasing function of A. This is supported by experimentalobservations that, under otherwise identical conditions, the re-sponse of a soil becomes softer as the major principal stress direc-tion deviates away from the direction of deposition (A decreaseswith this change) [42,59]. Note that in case of conventional triaxialcompression with the axis of deposition co-axial with the axialcompression direction, A = #1, such that f ( 1. The special featureof this shear mode makes it suitable to be used as a reference formodel calibration, which will be discussed in a later section.
Experimental observations [1,58] show that the bonding of soilsis gradually weakened due to the development of plastic deforma-tion, which leads to significant degradation in shear modulus dur-ing the post peak stage. In the present model, a simple linearrelation between the rate of de-bonding and the plastic deviatoricstrain increment is assumed,
dr0 ¼ hdLir0 ð9Þ
where
r0 ¼#mðH=Mf Þ2000r0 for r0 > 00 for r0 6 0
(
ð10Þ
where r0 denotes the current triaxial tensile strength of the mate-rial and m is a non-negative model parameter. The above evolutionlaw ensures that r0 is always less than or equal to zero and the pro-cess of de-bonding proceeds steadily with plastic straining until r0
reaches zero. It is assumed that elastic deformation does not causede-bonding in this evolution law. Since the initial value of the triax-ial tensile strength r0i is determined based on the peak strengthstate of cemented sand (see the case for cemented Ottawa sandshown in Fig. 2), the term (H/Mf)2000 is used to force the de-bondingrate to become very small before the peak strength state.
3.3. Dilatancy and flow rule
Dilatancy relation is the cornerstone of a constitutive model forsand. To incorporate the effect of bonding and fabric anisotropyinto the dilatancy of sand, we propose the following dilatancy rela-tion based on the work by Li and Dafalias [37],
D ¼ depvffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2=3depijdep
ij
q ¼ d1
expðRhdLiÞ
ðMpdCdF # HÞ ð11Þ
where depv is the plastic volumetric strain increment; dep
ij
(¼ depij #dep
vdij=3) is the plastic deviatoric strain increment; d1 is apositive model parameter; Mp is the phase transformation stress ra-tio measured in conventional triaxial compression tests on remol-ded samples. The role of the denominator in Eq. (11) is to controlthe volume change, especially when the strain level is high. Asthe sample is sheared to the critical state, the increment of plasticdeviatoric strain will not be limited. As such the denominator term
(a)
(b)
(c)
zσ
xσ240θ =
I
II
III
I
II
III
yσ
0θ =
60θ =
120θ =180θ =
300θ =
3σ
2σ
1σ
x
y
z
3σ
1σ
2σ
2σ
1σ
3σ
θ
Fig. 1. (a) Definition of the angle h and partition of the deviatoric plane under thetrue triaxial test condition (after [46]); (b) the yield surface in the three-dimensional space and (c) the yield loci in the deviatoric plane.
60 Z. Gao, J. Zhao / Computers and Geotechnics 41 (2012) 57–69
Estrategias de modelizaciónde anisotropía
Medio continuo• Anisótropía elástica / elastoplástica• Juntas difusas• Interfases distribuidas• Interfases explícitas• Mesomecánica (SRM)Modelos de contacto/bloques• UDEC/3DEC• PFC• Slope model• SRK: Frack_Rock (Gibson)3
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(Gao & Zhao 2012)
Anisótropo
Isótropo
(Gibson 2016)
Fluencia anisotrópicadentro de la mecánica del continuo
Los modelos de plasticidad simples son isotrópicos(p.ej. Mohr-Coulomb o Hoek-Brown)
Las discontinuidades agregan mecanismos adicionales de deformación anisotrópica (modelos de juntas difusas)
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T
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Fluencia anisotrópicadentro de la mecánica del continuo
• No hay “distancia” entre discontinuidades: siempre “existe” una discontinuidad en la posición desfavorable
• Limitación en 2D: sólo es válido si las discontinuidades son normales al modelo
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Modelling Rock in Plaxis
CG2 - Buenos Aires, Argentina - Octobre 2010 5
Jointed Rock model, 2D Example
α1= 0°
Incrementaldisplacements
IncrementalShear strains
Plastic points
γ= 0, c=1, φ=0E1 = E2
Jointed Rock model, 2D Example
α1= 30°
Incrementaldisplacements
IncrementalShear strains
Plastic points
γ= 0, c=1, φ=0E1 = E2
Modelling Rock in Plaxis
CG2 - Buenos Aires, Argentina - Octobre 2010 5
Jointed Rock model, 2D Example
α1= 0°
Incrementaldisplacements
IncrementalShear strains
Plastic points
γ= 0, c=1, φ=0E1 = E2
Jointed Rock model, 2D Example
α1= 30°
Incrementaldisplacements
IncrementalShear strains
Plastic points
γ= 0, c=1, φ=0E1 = E2
Modelling Rock in Plaxis
CG2 - Buenos Aires, Argentina - Octobre 2010 5
Jointed Rock model, 2D Example
α1= 0°
Incrementaldisplacements
IncrementalShear strains
Plastic points
γ= 0, c=1, φ=0E1 = E2
Jointed Rock model, 2D Example
α1= 30°
Incrementaldisplacements
IncrementalShear strains
Plastic points
γ= 0, c=1, φ=0E1 = E2
Modelling Rock in Plaxis
CG2 - Buenos Aires, Argentina - Octobre 2010 5
Jointed Rock model, 2D Example
α1= 0°
Incrementaldisplacements
IncrementalShear strains
Plastic points
γ= 0, c=1, φ=0E1 = E2
Jointed Rock model, 2D Example
α1= 30°
Incrementaldisplacements
IncrementalShear strains
Plastic points
γ= 0, c=1, φ=0E1 = E2
Juntas horizontales Juntas inclinadas
(Waterman 2010)(Waterman 2010)
Discontinuidades explícitas
Superficies pre-definidas en el modelocon propiedades resistentes propiasVentajas• Fallas y otras estructuras bien
caracterizadas, no “promediadas”• Ablandamiento (de discontunuidad)
no induce dependencia de la mallaDesventajas• Requiere caracterización mecánica• Modelización difícil de superficies
curvas y/o con puentes de roca6
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2 DEVELOPMENT OF A UBIQUITOUS JOINT ROCK MASS (UJRM) MODEL 2.1 The Subiquitous (Strain-Softening Ubiquitous Joint) constitutive model
The Subiquitous constitutive model in FLAC3D is routinely used to represent laminated materials that exhibit nonlinear material hardening or softening. Clark (2006) used FLAC (Itasca 2005) to demonstrate that the assignment of ubiquitous joint orientations at the zone level (from a known joint-orientation distribution) results in realistic rock mass behavior and can yield properties that are consistent with empirical techniques. The methodology detailed by Clark (2006) has been extended to FLAC3D to allow for the characterization of strength anisotropy and sample scale effects.
Within the Subiquitous constitutive model, both matrix and joint properties are specified (see Fig. 1). In order for the UJRM testing methodology to be practical and honor existing rock mechanics relations, it has been assumed that the matrix and joint properties can be derived directly from the intact or SRM testing results. By modifying these input strength parameters, the calibration of Young’s Modulus, unconfined compressive strength (UCS), tensile strength and the softening behavior of different sample sizes, in different loading directions have been completed. In addition, SRM failure mechanisms within the UJRM samples also have been honored through the monitoring of progressive matrix degradation, joint slip and joint dislocation. An example of the damage propagation behaviors within a UJRM sample can be seen through the progressive degradation of matrix cohesion and ubiquitous joint-failure plots at various stages of UJRM UCS sample loading – illustrated in Figure 2.
Figure 1. UJRM model: matrix and joint Figure 2. Stages of damage within a UJRM specimen. properties.
2.2 Establishment of a standard UJRM laboratory testing environment
To date, SRM testing has been performed on one sample size that has been subjected to one stress-path loading condition that simulates the expected stress path in situ. This has made the material properties derived from this technique specific to one application. As discussed in Mas Ivars et al. (2008), the SRM methodology has been developed further to achieve calibration of the rock mass (a) in three opposing loading directions, and (b) at a number of different scales. This ensures that the material properties derived from the technique are not specific to one particular stress path and may
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Discrete Fracture Network (introducción)
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(Sainsbury 2008)
Modelos discontinuos: bloques en contacto
Mecánica del continuo dentro de cada bloqueTeorías de contacto entre bloquesVentajas• Puede propagar fracturas
(en contactos pre-definidos)• Permite modelar localización
de deformacionesDesventajas• Bloques elásticos: puede bloquear• Bloques elastoplásticos: alto costo
computacional8
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(SRK Consulting, Severin 2012)
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PFC2D model showing toppling on majorstructures (Cundall, 2007)
UDEC model showing lithology, discontinuities and anual pit geometries. (Lorig and Calderón, 2002)
SLOPE MODEL model showing toppling on major structures (LOP, 2009)
Modelos discontinuos en gran escala: Chuquicamata Pared Oeste
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(Silva et al 2015)
El problema de la interpretaciónde los resultados
Macizo rocosoFS = 1.65
Juntas difusasFS = 1.17
DFN en FLAC3DFS = 0.97
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(SRK Consulting, Severin 2014)10
