Modelización de la anisotropía de los macizos rocosos
Dr. Alejo O. Sfriso Universidad de Buenos Aires SRK Consulting (Argentina) AOSA
materias.fi.uba.ar/6408 latam.srk.com www.aosa.com.ar
[email protected] [email protected] [email protected]
Modelización de anisotropía en macizos rocosos
Isotropía, anisotropía, ortotropía
2
• 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
Axial Radial Circunferencial es.wikipedia.org/wiki/ Material_ortótropo
Aliviadero Caracoles
1
60
Z. Gao, J. Zhao / Computers and Geotechnics 41 (2012) 57–69
z
(a)
The following evolution law for H i σ2
θ
Estrategias de modelización de anisotropía θ = 240 σx
dH ¼ hdLir H ¼ hdLi
σ3
θ = 60
θ = 300
Modelización de anisotropía en macizos rocosos
σ1
θ =0
y x
3
3.2. Hardening law
σz
I
σ2
I
II
σ1
II σ3
III
III σ3
θ = 180
θ = 120 σy
σ1 σ2
Medio continuo (b) • Anisótropía elástica / elastoplástica • Juntas difusas • Interfases distribuidas • Interfases explícitas • Mesomecánica (SRM) Modelos de contacto/bloques • UDEC/3DEC (c) • PFC • Slope model • SRK: Frack_Rock (Gibson)
Anisótropo
Isótropo (Gao & Zhao 2012)
where rH denotes the evolution directio than or equal to zero; dL is a loading Macauley bracket with hxi = 0 when x ch is a positive constant. Following Li an et al. [13], we introduce the following f for the effect of fabric anisotropy on th
f ¼ exp½#kðA þ 1Þ'
where k is a positive model parameter. E is a decreasing function of A. This is observations that, under otherwise id sponse of a soil becomes softer as the m tion deviates away from the direction with this change) [42,59]. Note that in compression with the axis of deposit compression direction, A = #1, such th of this shear mode makes it suitable to model calibration, which will be discus Experimental observations [1,58] sh is gradually weakened due to the deve tion, which leads to significant degrada ing the post peak stage. In the prese relation between the rate of de-bondin strain increment is assumed,
dr0 ¼ hdLir 0 where
r0 ¼
(Gibson 2016)
Fig. 1. (a) Definition of the angle h and partition of the deviatoric plane under the true triaxial test condition (after [46]); (b) the yield surface in the threedimensional space and (c) the yield loci in the deviatoric plane.
Modelización de anisotropía en macizos rocosos
Gch f ðM f # HÞ Hpr
tests if both the stress direction and the fabric orientation are set to align in the same fixed coordinate, such as the cases shown in Fig. 1a, the yield surface can be plotted as shown in Fig. 1b (the yield surfaces do not cross the origin of the coordinate system due to the existence of bonding) and Fig. 1c (yield loci in the deviatoric plane with different values of hardening parameter). The isotropic failure surface is shown in the deviatoric plane in comparison with the anisotropic one. Note that in Fig. 1a we denote the angle between the current stress state with the vertical stress axes in the deviatoric plane by h, and the deviatoric plane is partitioned into three zones as shown in Fig. 1a. The same convention will be followed in the subsequent sections.
Fluencia anisotrópica dentro de la mecánica del continuo
Los modelos de plasticidad simples son isotrópicos (p.ej. Mohr-Coulomb o Hoek-Brown)
(
#mðH=M f Þ2000 r0 0
for for
r0 > 0 r0 6 0
where r0 denotes the current triaxial t rial and m is a non-negative model para law ensures that r0 is always less than o cess of de-bonding proceeds steadily w reaches zero. It is assumed that elastic de-bonding in this evolution law. Since ial tensile strength r0i is determined state of cemented sand (see the case shown in Fig. 2), the term (H/Mf)2000 is u rate to become very small before the p 3.3. Dilatancy and flow rule
Dilatancy relation is the cornerstone sand. To incorporate the effect of bon into the dilatancy of sand, we propose t tion based on the work by Li and Dafa
depv d1 R ðM p dC dF D ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ p p 2=3deij deij expð hdLiÞ
where depv is the plastic volumet (¼ depij #depv dij =3) is the plastic deviato positive model parameter; Mp is the pha tio measured in conventional triaxial c ded samples. The role of the denomina the volume change, especially when the sample is sheared to the critical sta deviatoric strain will not be limited. As
N T
Las discontinuidades agregan mecanismos adicionales de deformación anisotrópica (modelos de juntas difusas) N T
4
2
Incremental displacements γ= 0, c=1, φ=0 E1 = E2
γ= 0, c=1, φ=0 E1 = E2
Incremental Shear strains
Modelización de anisotropía en macizos rocosos
Modelling Rock in Plaxis
Fluencia anisotrópica dentro de la mecánica del continuo • No hay “distancia” entre discontinuidades: siempre “existe” una discontinuidad en la posición desfavorable Jointed Rock Example Jointed Rock model, 2D Example • Limitación en model, 2D: sólo 2D es válido si las discontinuidades son normales al modelo α = 0° α = 30°
1 Jointed Rock model, 2D Example 1 nted Rock model, 2D Example
α1= 30°
(Waterman 2010)
(Waterman 2010)
Plastic points Plastic points
Incremental 5 displacements
= 0, c=1, φ=0 1 = E2
Incremental displacements
Incremental Incremental γ= 0,Juntas c=1, φ=0horizontales Shear strains displacements E1 = E2
Plastic points
Incremental Incremental Shear strains displacements
Plastic points
Juntas γ= 0, c=1, φ=0 E1 = E2
Inc She
Incremental
inclinadas Shear strains
γ= 0, c=1, φ=0 E1 = E2
Modelización de anisotropía en macizos rocosos
Discontinuidades explícitas CG2 - Buenos Aires, Argentina - Octobre 2010
CG2 - Buenos Aires,Superficies Argentina -pre-definidas Octobre 2010en el modelo
5
con propiedades resistentes propias
Jointed Ventajas Rock model, 2D Example
y otras estructuras bien α•1= Fallas 30° ted Rock model, 2D Example
°
caracterizadas, no “promediadas” • Ablandamiento (de discontunuidad) no induce dependencia de la malla Desventajas Plastic • Requiere caracterización mecánica points Incremental Plastic • Modelización difícil de superficies displacements points curvas y/o con puentes de roca Incremental
Incremental 6 displacements
γ= 0, c=1, φ=0 E1 = E2
Shear strains
Incremental Shear strains (SRK Consulting, Severin 2012)
= 0, c=1, φ=0 1 = E2
3
Modelización de anisotropía en macizos rocosos
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.
Discrete Fracture Network (introducción)
7
(Sainsbury 2008)
Figure 1. UJRM model: matrix and joint properties.
Figure 2. Stages of damage within a UJRM specimen.
2.2 Establishment of a standard UJRM laboratory testing environment
Modelización de anisotropía en macizos rocosos
To date, SRM testing has been performed on one sample size that has been subjected to one stresspath 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
8
Modelos discontinuos: bloques en contacto
2
Mecánica del continuo dentro de cada bloque Teorías de contacto entre bloques Ventajas • Puede propagar fracturas (en contactos pre-definidos) • Permite modelar localización de deformaciones Desventajas • Bloques elásticos: puede bloquear • Bloques elastoplásticos: alto costo computacional (SRK Consulting, Severin 2012)
4
Modelización de anisotropía en macizos rocosos
Modelos discontinuos en gran escala: Chuquicamata Pared Oeste
UDEC model showing lithology, discontinuities and anual pit geometries. (Lorig and Calderón, 2002)
PFC2D model showing toppling on major structures (Cundall, 2007)
SLOPE MODEL model showing toppling on major structures (LOP, 2009)
Modelización de anisotropía en macizos rocosos
9
10
(Silva et al 2015)
El problema de la interpretación de los resultados
Macizo rocoso FS = 1.65
Juntas difusas FS = 1.17
DFN en FLAC3D FS = 0.97 (SRK Consulting, Severin 2014)
5
