Continuum modeling of discontinuous rock structures

Continuum modeling of discontinuous rock structures Homogenization of discrete particle assemblies I. Stefanou* and J. Sulem** * Department of Applied Mathematics and Physics National Technical University of Athens Zografou Campus, Greece [email protected] http://geolab.mechan.ntua.gr/ ** UR Navier, CERMES, Ecole des Ponts ParisTech, Université Paris-Est Marne-la-Vallée, France [email protected] http://navier.enpc.fr/~sulem/index.html RÉSUMÉ.

On présente dans cet article un modèle continu pour les massifs rocheux discontinus.. Le système discret est formé par trois groupes de joints avec des propriétés géométriques et mécaniques arbitraires. Dans le cas général, les blocs sont des parallélépipèdes qui interagissent en transférant des forces et des moments par leurs interfaces. A cause de la géométrie du système, le tenseur d'inertie des blocs est anisotrope et la flexion sur les interfaces est non-symétrique. Le milieu continu que l’on obtient est un milieu Cosserat tri-dimensionnel, qui approxime le système discret pour des longueurs d'onde supérieure à cinq fois la longueur caractéristique du bloc rocheux typique. ABSTRACT.

A continuum model for discontinuous rock masses is presented here. The discrete system is formed by three joint sets of arbitrary geometrical and mechanical properties. In the general case the joint blocks are parallelepipeds that interact through their interfaces transferring both forces and torques. For geometrical reasons the inertia tensor of the blocks is not isotropic and the bending at the interfaces is non-symmetric. The derived continuum is an anisotropic three dimensional Cosserat continuum, which approximates the discrete system for wavelengths five times bigger the characteristic length of the typical rock block. MOTS-CLÉS

: Milieu continu de Cosserat, milieu micromorphique, homogénéisation joints,

rocheux,. KEYWORDS:

Cosserat continuum, Micromorphic continuum, homogenization, rock joints

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1. Introduction Modeling a discontinuous medium, such as a rock mass or a masonry wall , sounds oxymoron in the context of Continuum Mechanics. Nevertheless, Continuum Mechanics is frequently used for the description of discontinuous systems after applying proper homogenization techniques. From the mathematical point of view, the objective of homogenization is to replace the various differential equations that describe each discrete particle of the medium by a system of partial differential equations (PDE’s) that reflect the behavior of the system at the macro-scale. As a result, homogenization could be seen as a way of bridging micro- and macro-scale, revealing the fundamental properties of the medium. According to Pasternak & Mühlhaus (2005), the existing homogenization techniques can be grouped in four families. These are a) the asymptotic averaging techniques, b) the direct averaging techniques, c) the homogenization by integral transformation techniques and d) the differential expansion techniques. Except the latter, all of the aforementioned techniques premise an averaging of the various fields of the medium (displacements, strains, stresses etc.). On the contrary, in the last mentioned technique, averaging is avoided and the equivalent continuum is gradually built starting from the microstructure (cf. also Stefanou et al., 2010; generalized differential expansion technique). The necessary theoretical background is found in the work of Germain (1973). The formulation of Germain differs from others (e.g. Mindlin, 1964; Eringen, 1999) in the sense that it is based on the principle of virtual power, which offers the flexibility to generalize the theory to nth order micromorphic continua (cf. also Maugin, 1980). Homogenization has been applied to the modeling of discontinuous rock structures with applications in rock mechanics (see Salamon, 1968; Gerrard, 1982; Besdo, 1985; Papamichos et al., 1990; Dai et al., 1996; Adhikary & Dyskin, 1997; Adhikary et al., 2001; Sulem et al., 2002; Yang & Wang, 2009, among others) and in masonry structures (see Masiani et al., 1995; Sulem & Mühlhaus, 1997; Cerrolaza et al., 1999; Cecchi & Sab, 2002; Cecchi & Sab, 2004; Massart et al., 2005; Massart et al., 2007; Cecchi & Milani, 2008, just to mention few). Here, we focus on the modeling of fractured rock masses by generalizing existing continuum models in three dimensions and by introducing additional parameters describing the geometry of the fractures. 2. Simplified model for jointed rock masses in three dimensions Even though two dimensional modeling may be adequate in some practical applications, there are many other cases where three dimensional analyses are required. Take for instance the SE/E fractured rock corner slope of Acropolis Hill in Athens (Stefanou & Vardoulakis, 2005) or the protruding corner of the excavation (Karantzoulis, 2009) in Figure 1, whose three dimensional geometrical configuration defies common two dimensional geotechnical approaches.

Continuum modeling of discontinuous rock structures

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It is well known that one of the fundamental characteristics of rock masses is their discontinuous nature because of the various geological mechanisms that the rock is subjected to (tectonic movements and deformations, temperature changes, water, erosion etc.). The discontinuities or the joints, as they will be called here, do not occur at completely random orientations, but they appear with some degree of ‘clustering’ around preferred orientations associated with the formation process (Hudson & Harisson, 2000). It is reasonable, therefore, to group them into joint sets. The main geological data required to characterize the joint sets are the spacing, the orientation and the inclination of them (Goodman & Shi, 1985). In Figure 2 we present a schematic of the main geometrical properties of the discontinuities in a rock mass (Hudson & Harisson, 2000) and in Figure 3 we present the terms used to describe the geometric parameters of a discontinuity (Goodman & Shi, 1985), i.e. the dip angle, α, which is a positive angle measured downward from the horizontal (xy) plane and the dip direction, β, which is a positive angle measured clockwise from north (y).

Figure 1. Left: Photo of the SE/E part of the fractured rock corner slope of Acropolis Hill in Athens in 1880 (Stefanou & Vardoulakis, 2005); Right: Protruding corner of an excavation in Offenbach, Germany (Karantzoulis, 2009).

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Figure 2. Schematic of the main geometrical properties of discontinuities in a rock mass (from Hudson & Harisson, 2000).

Up (z) North (y)

z

β = Dip direction cos 

α = Dip

n

α

sin  cos 

East (x) 

Joint plane

si

y

β sin

 sin

n

x

Figure 3. Terms describing the geometric parameters of a discontinuity: dip angle, α, is positive measured downward from the horizontal (xy) plane; dip direction, β, is positive measured clockwise from north (y) (Goodman & Shi, 1985; Itasca, 2007). Being derived from the Latin word ‘modus’ (a measure), the word model implies a change of scale in the sense of an abstraction of the natural object, which in the present case is the rock mass (Rutherford, 1994). Let’s assume that the discontinuous rock mass is characterized by three joint sets forming parallelepiped

Continuum modeling of discontinuous rock structures

5

joint blocks in the three dimensional space. The assumption of parallel joint block faces is reasonable for many applications (Goodman & Shi, 1985). It should be noticed that the values of the various geometrical and mechanical parameters that determine the behavior of a rock mass are not deterministic. They rather obey statistical distributions (see Hudson & Harisson, 2000; Goodman & Shi, 1985; Priest & Hudson, 1981; Goodman, 1989). However, in the present paper we focus on the development of a continuous model through homogenization and we assign certain values for the required parameters of the discontinuous rock mass. A geometrical representation of the rock mass considered here is presented in Figure 4. Evidently, the structure is periodic in 3D space following a certain pattern. In solid state physics terminology (Kittel, 1996) this pattern is called “lattice”, while the repeated cell is called “basis”:

structure  lattice  basis

[1]

The basis or the “elementary cell”, as it is called here, must contain all the necessary information for the constitutive description of the periodic structure. It has to be mentioned though, that generally the elementary cell is not unique and that its choice affects the obtained homogenized continuum. For this rather well known point the reader can refer to the book of Novozhilov (1961). The chosen elementary cell of the lattice (basis), depicted in Figure 4, is identified with the primitive cell defined by the primitive axes α i . Note that a primitive cell is a minimum volume cell that fills all space by suitable repetition and translation. The rock structure is generated by repeating and translating the chosen basis over the lattice points (Figure 5). Translation is the simplest among 17 possible ways for generating a two dimensional pattern (Ernst, 1983). The lattice translational vector is (Kittel, 1996): T  l1 , l2 , l3   l1α1  l2α 2  l3α3

[2]

where l1 , l 2 , l3 are arbitrary integers and: α1  d1  n 2  n3  , α 2  d 2  n3  n1  , α 3  d3  n1  n 2 

[3]

ei are the unit vectors of the Cartesian global system and di   1 Di , with Di the

spacing of the discontinuity set ‘i’ in the direction of n i and   n1   n 2  n3  .

Each node of the lattice is given three indices representing its position in space (Figure 5). Hence, the coordinates of node  I , J , K  are: X i( I , J , K )   I α1  J α 2  K α 3  ei , i  1, 2,3

[4]

The volume of the elementary cell is: V  α3   α1  α 2 

[5]

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Joint set No.1 Joint set No.2

Joint set No.3

d1 α3

C α2

α1

d3 d2

Typical joint block

Figure 4. Geometrical model of the rock mass considered herein. The joint blocks are parallelepipeds.

 K  1 α3

K  α1

e3

e1

I 

 I  1

Figure 5. Periodic arrangement of the elementary cells and nodes of the lattice (2D section). Each node of the lattice is given three indices  I , J , K  representing its position in space. The J index is not shown as it is perpendicular to the section plane of the figure.

For the mechanical description of the structure we assume that the joint blocks are rigid with deformable interfaces (soft-contacts). This assumption implies that the deformation is concentrated in the interfaces of the blocks and that it is small

Continuum modeling of discontinuous rock structures

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compared to their dimensions. Similar assumptions are made in masonry structures, for which the rigidity of the interface (block-mortar-block) is smaller that the one of the mortar itself (Raffard, 2000). Generally, the assumption of rigid building blocks with deformable interfaces is adopted by many researchers in similar homogenization approaches of discontinues rock systems (cf. Besdo, 1985; Sulem & Mühlhaus, 1997; Cerrolaza et al., 1999; Cecchi & Sab, 2004; Cecchi & Milani, 2008). Moreover, other methods in rock mechanics, like for instance the Key-Block theory (Goodman & Shi, 1985), are based on the assumption of rigid rock blocks. The deformation of the rock mass involves translations and rotations of the joint blocks in three dimensions. Therefore, the interfaces between the blocks transfer both forces and torques. Each joint block has six degrees of freedom (dof’s). With U i(b ) we denote the translation of the center mass of block b parallel to ei axis and with i(b ) the rotation of block b around ei axis. For infinitesimal rotations (small strains) the displacement of a point of a block is: Pi (b )  R(j b )   U i(b )   ijk (kb )  R(j b)  C (j b) 

[6]

where  ijk is the Levi-Civita tensor, C the center of mass and R the position vector of the considered point of block b. With capital letters we refer to quantities expressed in the global coordinate system. In Figure 6 the numbering of the interfaces of the elementary cell is shown, which in the present case is identical to  the typical joint block. Let bA, bB be two blocks interacting through interface  and ( b A , bB ),  ( bB , b A ),  (b A , bB ),  (bB , b A ),  (resp. Fi ) and M i (resp. M i ) the total force and the Fi total moment exerted by block bB over bA (resp. bA over bB) expressed at the geometric center of the inteface. Then a set of self-balanced forces and moments is developed in the lattice. Notice that the elementary cell interacts with the adjacent 1 6 cells along six interfaces  -  . In more complicated patterns, as the one of the diatomic masonry pattern presented in Figure 7 (Stefanou et al., 2010; Stefanou & Sulem, 2010), the elementary cell may also have internal interfaces, i.e. interface 0 . (b ) i

(b ) i

Σ3

α3

2

4

Σ 6

5

α2

Σ1

α1

Figure 6. Numbering of interfaces Σβ of the elementary cell with its adjacent cells. With gray color we denote the interface numbers that correspond to the hidden faces of the joint block. In the present case the elementary cell is identical to the joint block.

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6 

7

5

4 3

0

8 9

 10

2 1 

Figure 7. Diatomic pattern of a masonry wall structure (Stefanou et al., 2010). The chosen elementary cell is denoted by the dark region and contains two different blocks. Except of the interfaces of the elementary cell with its adjacent cells there is also the interface Σ0 inside the elementary cell.

3. Homogenization The enriched kinematics of generalized micromorphic continua makes the theory suitable for describing materials with microstructure. As opposed to the static regime (Salerno & de Felice, 2009), in the dynamic regime the richer the structure of the continuum model is, the more refined the homogenization (identification) scheme should be. Otherwise, the dispersion functions of the continuous approximation would not converge to the discrete ones, which would contradict the equivalence between the continuum and the discrete system. The homogenization procedure followed here is based on the construction of a continuum, which satisfies the two following criteria: – The kinematics of the discrete system is identical to the kinematics of the continuum. – The power of the internal forces and the kinetic energy of the continuum are equal to the power of the internal forces and the kinetic energy of the discrete system for any virtual kinematic field. The formulation presented here follows Germain (1973), by identification of the elementary cell to the particle of the corresponding micromorphic continuum: particle  elementary cell

[7]

Therefore, in the specific case of the idealized jointed rock mass, the particle P ( M ) should have six degrees of freedom (3 translational and 3 rotational in 3D). Assuming a system of particles and following Germain’s notation (1973), M is the center of mass of the particle P ( M ) , M  a point of P ( M ) , Vi C the velocity of M ,

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9

xi the coordinates of M  in a Cartesian frame parallel to the given frame X i with M its origin, Vi the velocity of M  with respect to the given frame and X i the coordinates of M in the given frame (Figure 8). D denotes the control volume. For a given particle, it is natural to look at the Taylor expansion of Vi with respect to xj :

Vi  Vi C   ij x j   ijk x j xk    ijk x j xk  x  

[8]

 ij , ijk , ijk are called micro-deformation rate tensors. Assuming that Vi is continuous in xi , the tensors ijk , ijk are fully symmetric with respect to the indices j , k , . The identification of the particle with the elementary cell (Eq.[7]) results in the following definitions that relate the velocities of the continuous with the velocities of the discrete system:

Vi

ViC

x i

M

M

P (M )

Xi

D

Figure 8. Continuum with microstructure.



Vi  r b

 (kI , J , K )

1   ijk 2

where .,i 

  .  xi

 V  r 

 I ,J ,K 

U i( I , J , K )

b

[9]

I ,J ,K 

i, j

(i  1, 2,3) and r b

 I , J ,K 

the coordinates of the center mass of the

joint block b I , J , K  corresponding to the elementary cell

I, J, K 

expressed in a

Cartesian frame parallel to the given frame X i with M its origin. As in the present case the center of mass of the joint block coincides with the center of mass of the  I , J ,K  elementary cell it is rb  0 . It should be mentioned that the rotations of the blocks in Eqs.[9] are defined in the continuum through the Curl operator on the

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vector field Vi  ,   Vi  . Additionally we define the quantity  ij( b ) , which may be interpreted as a homogeneous deformation rate tensor of the joint block of the elementary cell. For rigid blocks it holds: ij( I , J )









 I , J ,K   I , J ,K  1 0 Vi , j r b  V j,i r b   2

[10]

Equations [9] and [10] map exactly (one-to-one correspondence) the discrete dof’s to the continuum kinematic quantities. From the continuum point of view, the dislocations and the disclinations that appear at the interfaces of the blocks (cf. also screw dislocations Hirth & Lothe, 1982) have no effect on the derived continuum. This is because we focus only on the centers of the blocks of the structure and consequently there is no implication of field discontinuities in the formulation of the equivalent continuum. Combining the aforementioned equations we obtain: U i( I , J , K )  Vi C 1 (kI , J , K )   eijk  jk 2 11   22   33  0;  ij    ji

[11]

where Vi C  Vi C  X i( I , J , K )  and ij  ij  X i( I , J , K )  . It should be mentioned that the above set of equations was derived from the general definitions of Eqs.[9] and [10]. This inductive derivation is quite important in more complex elementary cells consisting of more than one blocks as no ad-hoc assumptions are needed for the identification of the continuum fields with the discrete dof’s. The presence of only first order terms  ij in Eq.[11] implies that a first order micromorphic continuum will be needed to describe the microstructure. Moreover the antisymmetry of the first order microdeformation rate tensor  ij indicates that a Cosserat continuum (or Micropolar continuum) is sufficient to describe the microstructure (see Cosserat & Cosserat, 1909; Eringen, 1968; Vardoulakis & Sulem, 1995). As the interaction of the elementary cell is limited to their first neighbors, a first order Taylor expansion from particle to particle of the velocities and microdeformation rate tensors is sufficient:

Vi C ( I l1 , J l2 , K  l3 )  Vi C ( I , J , K )  T j  l1 , l2 , l3  Vi ,Cj( I , J , K )

ij( I l , J l , K l )  ij( I , J , K )  Tk  l1 , l2 , l3   ijk( I , J , K ) 1

2

[12]

3

where  ijk   ijk( I , J , K )  ij , k  X i( I , J , K )  . A first order Taylor expansion of the kinematic fields (from particle to particle) seems suitable for most applications. Exceptions are the applications where the forces between the elementary cells (particles) are not limited to the first neighbor (Mindlin, 1965). In other words, when the elementary cell does not interact only with its adjacent elementary cells but further with the second, third, etc. neighbor cells, higher order partial derivatives of the velocities and of the micro-deformation

Continuum modeling of discontinuous rock structures

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rates are needed. Alternatively, the elementary cell (particle) could be enlarged to contain all the interacting neighbors, with the price of a higher order micromorphic continuum. The power of internal forces for a Micromorphic continuum of 1st order is given as follows (Germain, 1973): pcon   ijVi ,Cj  sij ij  ijk  ijk

[13]

 ij

[14]

with

 ij  sij

where  ij is the stress tensor,  ij is the intrinsic stress tensor (symmetric), sij is the intrinsic microstress tensor and  ijk is the intrinsic second microstress tensor. In the particular case of a Cosserat continuum sij   ij  , because of the antisymmetry 1 of  ij , and  ijk    ijl mlk with mlk being the couple stress tensor (Figure 9). The 2 objectivity of [13] is guaranteed by the symmetry of  ij . d 22

x2 d12

d 32

dm22

d 23

dm12

dm32

dm23

dm21

d 33

d 21 d11

d13 d 31

O

dm31

dm13

dm11

x3

x1

dm33

Figure 9. Stresses and couple stresses on Cosserat element  dx1 , dx2 , dx3  .

Having defined and linked the kinematics of the discrete and of the continuum, we set for any virtual kinematic field, Vi C ,  ij , the kinetic energy density and the power of the internal forces of the continuum equal to the kinetic energy and the power of the internal forces of the discrete system: pcon  pcell and kcon  kcell

where

[15]

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pcell 

1 V

F   3

( b A , b B ), 

i

1

U (j b

A

, b B ), 

 M i(b

1 1 1  kcell   mbU ibU ib  ib I ijb bj  V 2 2 

A

, b B ), 

(jb

A

, b B ), 

 [16]

and mb , I ijb the mass and the inertia tensor of block b. Notice, that in the general case of dip angle and orientation of the joint sets, the inertia tensor is not isotropic, the principal axes of the blocks are not parallel to the global coordinate system axes and the bending at the interface is non-symmetric. By introducing Eqs.[11] into Eq.[16] and using Eqs.[13] and [15] we derive the constitutive equations of the continuum:

 ij 

pcon p p ; sij   con ;  ijk  con ij  ijk Vi ,Cj

[17]

It is worth mentioning that until this point nothing was assumed regarding the constitutive relations of the rock mass interfaces. Moreover, it has to be mentioned that because of the general formulation of the power of the internal forces in the discrete system (Eq.[16]), the constitutive law of the derived continuum (Eqs.[17]) is expressed in function of the internal forces and moments of the elementary cell. Therefore, yield criteria can be set out in the continuum at the macrolevel by considering the internal forces and moments developed at the microlevel, i.e. at the interfaces of the microstructure. Various yielding mechanisms such as sliding, rocking, wobbling and twisting can be considered and expressed in terms of internal forces and moments at the microlevel (e.g. Sulem & Mühlhaus, 1997). Failure criteria at the microlevel depend on the mechanical properties of the rock blocks and their interfaces. These mechanical properties can be either specified according to existing interface models (Orduña & Lourenço, 2005) or determined experimentally on a per case basis. The dynamic partial differential equations of the aforementioned 3D Cosserat continuum can be derived following a variational approach (cf. also Germain, 1973; Georgiadis & Velgaki, 2003; Stefanou et al., 2008) as follows:

 ij , j  fi   Vi ekpq qp  mk

,

 k 

I kj VP ( M )

j

[18]

where ij   ijk k , Assuming small deformations, the normal time derivative in Eqs.[18] is identical to the material derivative. Therefore, the microinertia terms consist only in time derivatives and do not contain spatial derivatives as in the case of restricted continua (e.g. Georgiadis & Velgaki, 2003). The boundary conditions are given by Eqs.[11], for the part of the boundary where the displacements and rotations of the bricks of

Continuum modeling of discontinuous rock structures

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the elementary cells are prescribed and by the following equations for the complementary part of the boundary where generalized tractions are imposed: Ti   ij n j M ij   ijk nk

[19]

where ni is the unit vector of the boundary. The generalized tractions are related to the forces and moments applied to the blocks of the boundary of the structure through the relations: i  Fi ex [20] M ij   ijk M kex where Fi ex and M iex are respectively the resultant force and moment of the forces exerted at the boundary of the block b , transferred to its center mass. Obviously, a free boundary has zero generalized traction. 4. Validation For linear elastic interfaces the constitutive law can be directly determined in function of the deformation measures of the continuum. As the full form of the constitutive law is too long to be presented here we present only its structure1: S  CX

[21]

where S   ij , sij , vijk  , X  Vi ,Cj , ij ,  ijk  and C a matrix containing the constitutive relations. The domain of validity of the resulting continuum is evaluated by comparing its dynamic response with the dynamic response of the discrete-lattice model. The dynamic response of a structure is characterized by its dispersion functions that relate the wave propagation frequency to the wavelength. Hence, the dispersion functions of the homogenized continuum are compared with those of the discrete structure in order to assess the validity of the homogenization. Notice that if the homogenization procedure is inadequate, then the dispersion curves between the continuum and the discrete diverge, reflecting that the two systems have (a) different degrees of freedom, (b) different rigidities and (c) different inertial properties. The evaluation of the dispersion curves for each system is performed by Fourier transformation of the ordinary differential equations for the discrete system and of the partial differential equations for the continuum (cf. also Achenbach, 1975). The normalized wavelength and group velocity of the propagating waves are respectively: 1

All the analytical calculations in the present paper have been performed using the symbolic language mathematical package Mathematica (Wolfram Research, 2009). The Mathematica files are available to the reader upon request.

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ˆ 

2 ˆ ; cˆ  ˆ ˆ

[22]

1 2 where ˆ  L , ˆ  L1 2 g 1 2, ˆ  L1 , cˆ   gL  c, L  V 1/3 and g is the acceleration of gravity. The propagating waves are plane waves satisfying κ   sin a sin  ,sin a cos  , cos a  , with a and   the dip angle and the dip direction of the wave plane, respectively.

Table 1. Geometrical parameters of the joint sets considered for the numerical example. Joint set 1 2 3

Dip angle 80° 80° 30°

Dip direction 45° 0° 0°

Spacing 1.0 m 1.0 m 1.0 m

Let the joint sets have the geometrical characteristics presented in Table 1 and common normal and shear stiffness: kn  80 MPa m , ks  40 MPa m . Let also a 3 density value of   2000 Kg m to be assigned to the rock mass. Figure 10 and Figure 11 show that the discrete and the continuum descriptions converge asymptotically as the wavelength increases. Actually, the relative error, as defined below and for the parameters considered in the present numerical example, is less than 5% for ˆ  10 or 10% for ˆ  5 .

e%  cˆ  

1 6 cˆcon  m   cˆdsc  m   cˆ  m 6 m 1 con

[23]

The number of the dispersion curves is six because of the six independent degrees of freedom of the elementary cell. Each dispersion curve corresponds to a different oscillation mode, which activates different degrees of freedom of the blocks. For large wave lengths (   0 ), the oscillation modes 1, 2 and 3 are characterized by the translation of the blocks, while oscillation modes 4, 5 and 6 are dominated by the rotations of the joint blocks.

Continuum modeling of discontinuous rock structures

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Figure 10. Average relative error between the discrete and the continuum description of the rock mass for the six oscillation modes and for propagating waves in direction e1. The discrete and the continuum descriptions converge for large wavelengths.

153

6 5  4

 3  2 

1

Figure 11. Comparison of the discrete (dashed lines) and continuum (solid lines) dispersion functions for propagating waves in direction e1. Continuum and discrete dispersion functions are identical for large wavelengths.

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6. Conclusions A three dimensional continuum model was proposed here for discontinuous rock masses with three joint sets. The geometry of the system is general enough to account for the dip angle, the dip orientation and the spacing of the joint sets. Under the assumption of soft contacts and rigid joint blocks, the medium that describes the dynamic behavior of the system is an anisotropic Cosserat continuum. The homogenization procedure that was followed here is based on the generalized differential expansion homogenization technique and differs from the direct averaging (Aboudi, 1991) and the asymptotic averaging homogenization techniques (Bensoussan et al., 1978; Bakhvalov & Panasenko, 1984; Sanchez-Palencia & Zaoui, 1987) in the sense that the latter are based on the averaging of the discrete quantities. The present approach avoids the ad-hoc omission of the higher order derivatives of the continuous fields for displacements and rotations that usually appear in other formulations (cf. Bazant & Christensen, 1972; Eringen, 1999; Kumar & McDowell, 2004; Stefanou et al., 2008). Furthermore, it avoids the identification by direct comparison of the terms of the PDE’s of the continuum with the equations of the discrete system, after having replaced in the latter the discrete quotients by differential quotients based on Taylor expansions of some order (Eringen, 1999). The order of the Taylor expansion of the kinematic fields is not an a priori assumption of the method as it is in previous approaches. The necessary order of the Taylor expansion of the kinematic field of the continuum particle is inferred by equating the degrees of freedom of the elementary cell of the discrete system with the equivalent measures of the particle itself. The validity of the derived continuum was investigated by juxtaposing the dispersion curves of the discrete and the continuum models in the elastic regime. The results are quite satisfactory for engineering applications as the continuum description is a large wavelength approximation of the discrete system. In particular the continuous model behaves well for wavelengths five times bigger than the characteristic length of the joint blocks. This means that the discrete and the continuous approximation share the same oscillation modes, have the same degrees of freedom, the same rigidity and the same inertia properties. Of course, the present model assumes certain values for the mechanical and geometrical properties of the rock mass leading to an ideal periodic configuration. However, this idealization may not be effective in some practical cases and the geometrical and mechanical parameters of the model should rather obey to statistical distributions to better represent the randomness of the physical system. This probabilistic approach will be addressed in a future work. Finally, failure criteria should be derived in the future for the continuum on the base of the micromechanical considerations presented herein and of appropriate constitutive laws for the joint sets.

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