Integration of Elastic Stiffness Anisotropy into ...

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ScienceDirect Procedia Engineering 00 (2017) 000–000 www.elsevier.com/locate/procedia

Symposium of the International Society for Rock mechanics

Integration of Elastic Stiffness Anisotropy into Ubiquitous Joint Model Mohamed Ismaela,* , Heinz Konietzkya a

Geotechnical Institute, Technische Universität Bergakademie Freiberg, Freiberg 09599, Germany

Abstract This paper applies a modified ubiquitous joint plane model (Modified Ubi) to describe the mechanical behavior of layered rock masses. The constitutive model is an implicit-continuum based one where the joints are smeared across the rock mass. This modified model concerns not only the strength anisotropy but also it integrates the elastic stiffness anisotropy. Thus, the elastic stress increments and the plastic corrections from the original ubiquitous joint model have been altered. Now, it is possible to simulate the elastic and plastic behavior of transversely isotropic rocks. Modified Ubi is applied to simulate the behavior of the transverse isotropic rock samples in uniaxial compressive loading and triaxial loading tests. The results out of the modified model were compared to the analytical solution from Jaeger and the CANISO model from FLAC 8.0. © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EUROCK 2017. Keywords: Stiffness Anisotropy; Transverse Isotropy; Ubiquitous Joint Plane; FLAC

1. Introduction Determining the rock mass properties and expecting its behavior are matter of the modelling of the discontinuities in the rock continua. Because of the complex mechanism of the rock mass which may not be directly predicted using the conventional ways of modeling (such as: closed form solutions or physical modelling); numerical modelling is one of the most trending means to model behavior of rock mass [1]. The condition of the rock mass (such as: excessively fractured or reasonably fractured or intact massive) and the scale of the engineering application (i.e.: layering is

* Corresponding author. Tel.: +49-3731-392515; fax: +49-3731-393638. E-mail address: [email protected] 1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EUROCK 2017.

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Author name / Procedia Engineering 00 (2017) 000–000

significantly smaller or bigger than the scale of the application) affect strongly the optimum approach selection of the applied numerical model, either continuum-based approach or discontinuum-based one [2]. Modelling of layered rock mass is still a point of interest between different computational tools which either consider the joint implicitly (i.e.: FLAC – Ubiquitous joint model) or use the explicit representation of the discontinuities (i.e.: UDEC – Discreet element model) [3]. However, both the computational power and time consumed in calculation limit the usage of such discrete approaches. The accuracy of different implicit joint models has been investigated between both the ubiquitous joint model and Cosserat model [4]. This paper adopts the continuum-based approach to implicitly model an isotropic rock mass (i.e.: smeared joints). The elasto perfect-plastic ubiquitous joint plane model is one of the numerical models used to analyze the jointed rock masses [5]. It is found in the FLAC package as “an anisotropic plasticity model that includes weak planes of specific orientation embedded in a Mohr-Coulomb (M-C) solid” [6]. This model can predict the strength anisotropy for a rock mass containing a weak plane reasonably accurate. However, the presence of the joint is accounted into the plastic corrections but has no effect on the elastic behavior and the model is restricted to one set of joints [4]. A modified ubiquitous joint model (Comba model) has been presented to simulate the behavior of columnar basalt with the presence of up to four arbitrary orientation of weakness planes. Although, the elastic stiffness matrix in Comba model is non-elastic because each joint set’s orientation, spacing and stiffness are considered; it is difficult to set the joints’ orientation and stiffness values that equalize the elastic stiffness matrix of the transverse isotropy for a specific orientation [7]. On the other hand, a new constitutive model has been released and embedded into FLAC 8.0 named CANISO model. However, this model considers the rock matrix as elastic continua without plastic corrections [6]. The paper introduces the modification of the ubiquitous joint model to simulate, for both the rock matrix and the joint plane, the elastic and plastic behavior of transverse isotropic rock mass. 2. Model methodology In principle, the modified model is using the same methodology as the original ubiquitous joint model in which only one joint is considered in a zone. Symmetry characteristics and corresponding properties are illustrated in figure 1.

Fig. 1. Illustration of modified ubiquitous joint model

Thus, the zone represents both, a transverse isotropic elasto-plastic rock matrix and a joint, which can fail in tension or shear. Mohr-Coulomb parameters are given for the rock matrix and the joint plane as well in addition to the transverse isotropic elastic matrix parameters. 2.1. Input parameters The rock matrix is defined as a M-C solid with cohesion (c), friction angle (ϕf), dilation angle (ѱ) and tensile strength (σt). There is a specific input for the joint orientation and therefore the plane of isotropy. The orientation is given either by the dip (Dip) and the dip direction (dd) or by specifying the three components (nx, ny, nz) of the unit vector normal to the joint plane. Five independent elastic stiffness parameters have to be given parallel or normal to the joint plane, respectively.


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2.2. Elastic increments The elastic stiffness matrix {S} is described in local coordinates of the joint plane (x1, x2, x3). It is simpler to provide the compliance matrix {C} which is the inverse of the stiffness matrix. The independent elastic properties in the plane parallel to the joint plane are Young’s Modulus (E1) and Poisson’s ratio (υ1), while the other elastic properties are in the plane normal to the joint plane (E3, υ3 and G3), as indicated previously in figure 1.

 1  E    E    3 E [C ]   3   0   0    0 





E 1 E





3



3

E3

0

0

0

0

0

0

E3

E3 1 E3

0

0

E1 2(1   1 )

0

0

0

0

1 G3

0

0

0

0

3

 0   0    0    0   0   1  G3 

(1)

Based on the orientation of the joint plane, a transformation of the elastic stiffness matrix from local coordinates the global coordinates (X, Y, Z) is implemented. The calculation scheme of the elastic increments is always performed in global coordinates. 2.3. Plastic corrections for rock matrix and joint plane During the elastic stress incrimination, the yield condition is always monitored for both, the rock matrix and the joint plane. For checking the failure condition in the rock matrix, the global stress state [σ] is transformed into the principal stress state [σP] as shown in equation 2. Also, the stiffness matrix {SG} is transformed into the principal coordinates (xP, yP, zP) leading to {SP}.

 1P  S11P 1P  S12P  2P  S13P  3P  2P  S 21P  1P  S 22P  2P  S 23P  3P

(2)

 1P  S31P  1P  S32P  2P  S33P  3P Once the stress state violates the yield surface, plastic corrections are performed. The failure criterion is a MohrCoulomb criterion with tension cut-off as defined in equations 3 and 4.

f s   3P   1P  ( 3P   1P ) sin  f  2c cos  f  0

(3)

ft   3P   t  0

(4)

The plastic shear potential corresponds to a non-associated flow rule by using dilation angle ѱ instead of the friction angle (equation 5), while the tensile plastic potential follows an associated flow rule (equation 6).

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g s   3P   1P  ( 3P   1P ) sin  const.

(5)

gt   3P

(6)

Once the stress state exceeds the yield conditions (fs > 0 or ft > 0), a series of plastic corrections are performed to return the stress state to the defined yield surface. 2.3.1. Plastic corrections for yielding in shear In the case of shear failure the obtained corrections (  plastic multiplier which is calculated by equation 8.

PC

) are given by equation 7, where

s represents the shear

 1P  s ( S11P  S13P N ) C

P  2P  s ( S 21  S 23P N ) C

(7)

 3P  s ( S31P  S33P N ) C

f s ( 1P , 3P ) s  P P ( S11  S13 N )  ( S 31P  S 33P N ) N f O

Where,

N 

1  sin 1  sin

O

and

N f 

(8)

1  sin  f 1  sin  f

The notations used for principal stresses (  P ) differ between the old corrected values and the new demanded values. O

2.3.2. Plastic corrections for yielding in tension Similar to the above mentioned procedure used for yielding in shear, the corrections for yielding in tension are performed as shown by equation 9.

 1P  t S13P C

 2P  t S 23P C

(9)

 3P  t S 33P C

The tensile plastic multiplier is defined by equation 10.

f ( P ) t  t P3 S33 O

(10)

Thus, the new principal stress state is given by equation (11), whether it is corrected due to the shear or tensile yielding.


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 1P   1P   C    2P   2P   P   PC  3   3 C

  PN

(11)

The procedure of the plastic corrections for the joint planes are the same as those implemented in the model CANISO [6]. 3. Verification: single joint plane in isotropic matrix For verification, the modified ubiquitous joint model is used to predict the compressive strength of a rock sample with inherent anisotropy planes. The elastic stiffness matrix is adjusted to be isotropic which results in a bulk modulus K of 100 MPa and a shear modulus G of 70 MPa (table 1). The results are compared with the analytical solution from Jaeger and Cook [8], where the uniaxial compressive strength is calculated by equation (12) [9].

𝜎𝑐 = {

𝑚𝑖𝑛⁡{2𝑐 ⁄ N f , 2𝑐𝑗 ⁄(1 − 𝑡𝑎𝑛𝜙𝑓𝑗 𝑡𝑎𝑛𝛽)𝑠𝑖𝑛2𝛽⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑖𝑓(1 − 𝑡𝑎𝑛𝜙𝑓𝑗 𝑡𝑎𝑛𝛽) > 0

(12)

2𝑐 ⁄ N f ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑖𝑓(1 − 𝑡𝑎𝑛𝜙𝑓𝑗 𝑡𝑎𝑛𝛽) > 0

Modified ubiquitous joint model and analytical solution show nearly identical results with relative error less than 1% for all values of β as documented in figure 2. In the case of rock matrix failure, the uniaxial compressive strength σc is limited to 8.58 KPa. For this constellation, modified and original model give identical results. Table 1. Properties of the tested rock sample [9].

Elastic stiffness matrix components E1=E3 υ1= υ3 G1= G3 K 170 MPa 0.22 70 MPa 100 MPa

G 70 MPa

Modified Ubi σc [KPa] 10 9 8 7 6 5 4 3 2 0

15

30

Rock matrix M-C input c ϕf σt o 2 KPa 40 2.4 KPa

Joint M-C input cj ϕfj 1 KPa 30o

Analytical Solution

45

60

75

90

β [Degree] Fig.2. Comparison between the analytical and numerical solution using modified ubiquitous joint model [9].

σtj 2 KPa

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4. Application: single joint plane in transverse isotropic matrix Simulations have been conducted to check the applicability of the modified ubiquitous joint model under uniaxial and triaxial loading conditions. Results are compared with the CANISO model and the analytical solution from Jaeger [8]. 4.1. Uniaxial compression test: Bossier shale Used input data are based on experimental results of Bossier shale presented in table 2 [10]. This shale is a reservoir rock, highly laminated with organic-filled weak planes. The tensile strength values for both, rock matrix and the joints are estimated based on equation (13) which is considered as maximum value. t  max  c tan  f

(13)

The value for G3 is estimated as suggested by Lekhnitskii [11] and also based on laboratory results [6].

G3 

E1E3 E1 (1  23 )  E3

(14)

Table 2. Parameters for Bossier shale [10]

Elastic stiffness matrix components E1 [GPa] E3 [GPa] υ1 υ3 15.2 29.65 0.2 0.22

G3 [GPa] 5.86

Rock matrix M-C input c [MPa] ϕf σt [MPa] o 26 29 46.9

Joint M-C input cj [MPa] ϕfj σtj [MPa] o 14 24 31.4

The results of Bossier shale are presented in figure 3. The modified ubiquitous joint model shows the same behaviour as the modified analytical solution. However, the CANISO model can only detect joint failure and cannot predict the yielding in the rock matrix. Modified Ubi

CANISO

Analytical

Real

12

σc [x10 MPa]

10

8 6 4

2 0 0

15

30

45 β [Degree]

60

75

90

Fig. 3. Uniaxial compression tests: Numerical and analytical solutions for Bossier shale.


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4.2. Triaxial compression test: Martinsburg slate Both, the modified ubiquitous and the CANISO model are used to simulate triaxial tests (confining pressure σ3=3.5 and 10.5 MPa, respectively) on Martinsburg slate [12]. This rock behaves like a transverse isotropic rock. E1, E3, σt and σtj are estimated under triaxial stress conditions at β = 0o and 90o using RocLab [13], assuming GSI=95 and MR=450. Shear modulus G3 is computed by equation 14, while υ1 and υ3 are assumed as 0.22. M-C parameters for rock matrix and joints are given in [10] and [14], respectively. Table 3 shows the data applied to the numerical simulations and to Jaeger’s analytical solution. Results obtained from numerical simulations and the analytical solution are shown in figure 4. Table 3. Parameters for Martinsburg slate.

Elastic stiffness matrix components E1 [GPa] E3 [GPa] υ1 υ3 43.6 69.7 0.22 0.22

G3 [GPa] 23

Rock matrix M-C input c [MPa] ϕf σt [MPa] o 25 45 23.8

Joint M-C input cj [MPa] ϕfj σtj [MPa] o 14 24 7.5

Fig. 4.Mod. The numerical simulations for triaxial testing of Martinsburg slate for both σ3=3.5 and 10.5 MPa. Ubi (σ3=3.5 Mpa) and analytical solution CANISO (σ3=3.5 Mpa) Analy. (σ3=3.5 Mpa)

σ3=3.5 Mpa Analy. (σ3=10.5 Mpa)

Mod. Ubi (σ3=10.5 Mpa) σ3=10.5 Mpa

CANISO (σ3=10.5 Mpa)

27 24 21

σ1 x10[MPa]

18 15 12 9 6 3 0 0

15

30

45 β [Degree]

60

75

Fig. 4. Triaxial compression tests: Numerical and analytical solutions for Martinsburg slate.

90

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As Figure 4 shows, the CANISO model behaves elastic considering the rock matrix. Thus, a maximum driving strain (εmax) must be assigned. However, the failure in the rock matrix is limited to the Mohr-Coulomb matrix failure surface and corresponds therefore to the modified analytical solution of Jaeger. Generally, both numerical models and the analytical solution show the same behavior in case of joint yielding. 5. Conclusion The stiffness anisotropy of transverse isotropic rock matrix has been integrated into the ubiquitous joint model. The mathematical formulation for the plastic corrections for the rock matrix was introduced for both, shear and tension yielding. Numerical simulations on the basis of uniaxial compression tests were carried out to verify this model and results were compared to the modified analytical solution of Jaeger. Furthermore, the modified ubiquitous joint model and the CANISO model were tested under uniaxial and triaxial compression using lab data of two different rocks. Results indicate that the modified ubiquitous joint model is able to reproduce elastic as well as plastic behavior for both, joint and rock matrix. The stiffness anisotropy has a major influence in the elastic range. Finally, the new proposed model is an interesting alternative to describe anisotropic rock behavior and is advantageous in case of potential matrix failure. References [1] L. Jing, J.A. Hudson, Numerical models in rock mechanics, Int. J. Rock Mech. Min. Sci. 39 (2002) 409–427. [2] C. Edelbro, Rock mass strength—a review, Technical report (2003) 16, Lulea University of Technology, ISSN 1402–1536 [3] J.M. Davila, W. Schubert, Rock layering influence on rock mass displacements in tunnelling, in: Shimizu, Kaneko, Kodama (Eds.), Rock Mechanics for Global Issues, ©2014 by Japanese Committee for Rock Mechanics, ISBN 978-4-907430-03-0. [4] A.R. Dehkordi, 3D finite element cosserat continuum simulation of layered geomaterials, PhD Thesis (2008) University of Toronto: Department of Civil Engineering. [5] T.T. Wang, T.H. Huang, A constitutive model for the deformation of a rock mass containing sets of ubiquitous joints, Int. J. Rock Mech. Min. Sci. 46 (2009) 521–530. [6] Itasca Consulting Group, Inc. FLAC – Fast Lagranian Analysis of Continua, constitutive models manual, Ver. 8.0 (2016) Minneapolis: Itasca. [7] C. Detournay, G. Meng, P, Cundall, Development of a constitutive model for columnar basalt, in: Gomez, Detournay, Hart, Nelson (Eds.), Applied Numerical Modelling in Geomechanics, ©2016 Itasca International Inc., Minneapolis, ISBN 978-0-9767577-4-0. [8] J.C. Jaeger, N.G.W. Cook, Fundamentals of Rock Mechanics, third ed., Chapman & Hall, London, 1976. [9] Itasca Consulting Group, Inc. FLAC – Fast Lagranian Analysis of Continua, Verification problems manual, Ver. 8.0 (2016) Minneapolis: Itasca. [10] J. Ambrose, Failure of anisotropic shales under triaxial stress conditions, PhD Thesis (2014) Imperial College London: Department of Earth Science and Engineering. [11] S. G. Lekhnitskii, Theory of elasticity of an anisotropic body, Mir Publishers, Moscow, 1981. [12] F.A. Donath, Experimental study of shear failure in anisotropic rocks, Geol. Soc. America Bulletin, 72 (1961) 985–990. [13] Rocscience Inc., RocLab, Ver. 1.0 (2007) Toronto, Ontario, Canada. [14] Y.M. Tien, M.C. Kuo, A failure criterion for transversely isotropic rocks, Int. J. Rock Mech. Min. Sci., 38 (2001) 399–412.