Numerical simulation of damage and failure in brittle

Computers and Geotechnics 64 (2015) 48–60

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Research Paper

Numerical simulation of damage and failure in brittle rocks using a modified rigid block spring method C. Yao a,b, Q.H. Jiang a, J.F. Shao b,⇑ a b

School of Civil Engineering and Architecture, Nanchang University, Nanchang, China Laboratory of Mechanics of Lille, UMR8107 CNRS, University of Lille, Cité scientifique, 59655 Villeneuve d’Ascq, France

a r t i c l e

i n f o

Article history: Received 17 January 2014 Received in revised form 23 September 2014 Accepted 18 October 2014

Keywords: Damage Failure Discrete approach Micromechanics Rigid block spring method Brittle rocks

a b s t r a c t This paper is devoted to the numerical simulation of damage and failure in brittle rocks using a modified rigid block spring method. The brittle rocks are represented by an assembly of rigid blocks based on a Voronoi diagram. The macroscopic mechanical behavior is related to that of interfaces between blocks. The mechanical behavior of each interface is described by a uniform distribution of normal and tangential springs, which together define the deformation and failure process of the interface. The elastic stiffness of springs can explicitly be related to the macroscopic elastic properties of material. The local failure process of interface is controlled by both normal stress and shear stress. Both tensile and shear failures are considered. Numerical simulations are presented and compared with experimental data to show the performance of the proposed model. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Damage by the growth of microcracks is the main mechanism of inelastic deformation and failure in brittle rocks. The transition from diffused microcracks to localized fractures is the key challenge for the description of progressive failure process in such materials. Many phenomenological and micromechanical damage models have been developed during the last decades for brittle materials (we do not give here an exhaustive list of all these models). These models are generally able to describe the average macroscopic stress–strain behaviors until the post-peak regime, but fail to explicitly describe the onset of macroscopic fractures. On the other hand, extended finite element methods have also been proposed in order to consider the onset and propagation of fractures. These methods indeed opened ambitious perspectives for modeling of failure process but so far are generally limited to 2D conditions and fail to deal with problems with multiple fractures. As an alternative way, various discrete element methods have been intensively developed during the last decades either for granular media and cohesive materials [1–4 just to mention a few]. With this class of approaches, a cohesive continuum is replaced by an equivalent discrete assemblage of blocks or elements which are bonded together by cohesive forces, contact ⇑ Corresponding author. E-mail address: [email protected] (J.F. Shao). http://dx.doi.org/10.1016/j.compgeo.2014.10.012 0266-352X/Ó 2014 Elsevier Ltd. All rights reserved.

forces or cementation interfaces. Various discrete approaches have been successfully applied to modeling the damage and failure in heterogeneous geomaterials. For instance, Lan et al. [5] have studied the effect of heterogeneity of brittle rock on micromechanical extensile behavior during compression loading. Using discrete numerical simulations, Kazerani and Zhao [6] have investigated micromechanical parameters controlling compressive and tensile failure in crystallized rock. Among the discrete approaches for cohesive geomaterials, the bonded particle model proposed by Potyondy and Cundall [4] is based on the distinct element method and can reproduce many features of the mechanical behavior of brittle rocks, including elastic deformation, damage due to microcrack propagation and induced material anisotropy, hysteresis loops during unloading and reloading cycles, volumetric dilation, microcracks coalescence and postpeak softening, etc. This method is so far considered as one of leading methods among discrete approaches. However, it is shown that the particle size has a strong influence on numerical results. Different packing assemblies of particles can also affect macroscopic behaviors of materials [7–9]. Therefore, each randomly generated packing specimen must be calibrated by using specific algorithms, such as that proposed by Lan et al. [5]. More generally, the mesh dependency is a common shortcoming for many discrete element methods. The main objective of this study is to propose a discrete numerical model for modeling the mechanical behavior of brittle rocks by minimizing the effects of mesh size and arrangement.

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C. Yao et al. / Computers and Geotechnics 64 (2015) 48–60

(b) Contact model for the original RBSM

(a) Contact model for the modified RBSM

Fig. 1. Illustration of contact models for the modified and original RBSM.

Block 1 ( x1 , y1 )

Δus

h1

P1

( x, y )

Δun

h2

y

( x2 , y2 )

P2

Block 2 o

x Fig. 2. Local displacement for a point on the contact interface.

The rigid block spring method (RBSM), initially proposed by Kawai [10], is a very suitable method for modeling damage and failure in brittle rocks and has also been applied to concrete and concrete structures [11–14]. This method has been successively improved by Qian and Zhang [15], Zhang [16] and Chen et al. [17] and used in the limit analysis of rock slopes and other engineering structures. In view of its computational scheme, this method is very similar to the DDA model (Discontinuous Deformation Analysis) initiated by Shi [18] and thus can be viewed as an implicit discrete element method. Their main difference lies in the fact that no deformation in blocks and no contact update are considered in the RBSM method. One of key points of this method is the mesh generation. Bolander and Saito [19] used the RBSM for fracture analysis of concrete with a uniform Voronoi diagram as the basic mesh. They found that this type of mesh has the advantage of ensuring elastic uniformity and maximizing the degree of isotropy with respect to potential crack direction. Voronoi diagram was also used for dense packing simulation based on distinct element method [20]. Moreover, the widely used software UDEC (Universal Distinct Element Code) has also incorporated Voronoi tessellation [21]. Damjanac and Fairhurst [22] used this tool to study the effect of decreasing fracture toughness due to stress corrosion on the strength of a crystalline rock and Gao and Stead [23] proposed a modified Voronoi logic for brittle fracture modeling at both laboratory and field scales. Based on the previous works, a modified RBSM is proposed in this study for modeling the mechanical behaviors of brittle rocks. Since this model is in an implicit form and no contact distribution needs to be updated, it is much easier to be implemented and can provide faster convergence. There is no need to introduce an artificial damping coefficient,

compared to the explicit discrete element methods such as UDEC, PFC, and DEM. Furthermore, as it will be shown through examples in this paper, the proposed model is not less efficient in producing damage and failure process of brittle rocks under compression. The cohesive brittle rocks are represented by an assemblage of polygonal discrete elements which are based on uniformly and randomly generated Voronoi cells. A specific criterion is proposed to deal with both tensile and shear failure modes of interfaces. Macroscopic elastic and inelastic deformations, the transition from diffused damage to localized failure will be analyzed. In particular, effects of element size and mesh arrangement on the macroscopic responses will be investigated.

2. Basic principle of the modified rigid block spring method In the modified rigid block spring method (RBSM), the cohesive brittle rock is represented by an assemblage of rigid blocks which are interconnected along their boundary surfaces. Each block has three degrees of freedom defined at its centroid, two translational and one rotational. Two neighboring blocks share a common boundary surface, the interface. The interface between two neighboring blocks is mechanically represented by a series of normal and tangential springs, uniformly distributed along the interface as shown in Fig. 1a. Inspired by the contact model proposed by Goodman [24], the elastic property of springs is characterized by the normal and tangential elastic stiffness coefficients, kn and ks. The basic difference of this modified RBSM with the original RBSM is that, in the original RBSM there are only three springs positioned at the mid-point of the interface, a normal spring, a tangential

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(a) Step 1: Points generation

(b) Step 2: Voronoi diagram

Fig. 3. Illustration of mesh generation procedure.

spring and a rotational spring, as shown in Fig. 1b. Since the distributed springs have the capacity to represent the moment, the rotational spring is eliminated in the modified RBSM. Compared with the original RBSM, the modified RBSM allows the description of non-uniform distribution of stress and strain along the interface, and then the progressive damage and failure of the interface. Another advantage of the modified RBSM is that only two local parameters; say normal and shear stiffness of springs, are needed instead of three in the original one. Note that the modified RBSM can also be used in the case when the two contacting faces are not parallel as the displacement jumps on an interface are not necessarily uniform. Consider two arbitrary blocks, block 1 and block 2, next to each other, the centroids of which are (x1, y1) and (x2, y2) respectively, as shown in Fig. 2. The point P1 on the block 1 and the point P2 on the block 2 coincide in the same point (x, y) along the boundary between the two blocks. Assuming that rotations are small, the relative displacements between the points P1 and P2, {Du}, can be expressed by the displacements of centroids of the two blocks {U}12 as:

fDug ¼ fDun

Dus gT ¼ ½B½NfU g12

 fU g12 ¼ U 1x

U 1y

U 1h

U 2x

U 2y

ð1Þ U 2h

T

ð2Þ

U1x U1y U1h are respectively the translational displacement in x direction, translational displacement in y direction, rotational displacement of the centroid of block 1, while U2x U2y U2h are the displacement components of the centroid of block 2. Dun and Dus are respectively the normal and tangential relative displacements

Failure surface

Mode II

between the points P1 and P2. The matrix [B] is defined by the components of the unit normal vector of the interface as:

 ½B ¼

 ½N ¼

m

l

 ð3Þ

0

1 0

y10

1 0 y20 0 1

1 x10

 ð4Þ

x20

x10 ¼ x  x1 ; y10 ¼ y  y1 ; x20 ¼ x  x2 ; y20 ¼ y  y2

ð5Þ

The stresses induced by the relative displacements can be expressed by a suitable constitutive relation of the interface such as:

frg ¼ frn

rs gT ¼ ½DfDug; ½D ¼



kn

0

0

ks

 ð6Þ

rn and rs denote the normal and tangential stresses respectively. kn and ks are respectively the stiffness coefficients of the normal and tangential springs. Applying the virtual work theorem, for any block in the blocks system, the following equilibrium condition stands: X Z le0

e

dfDugT frgdl þ

Z ler

dfugT fFgdl þ

Z Z Se

! dfugT fPgds

¼0 ð7Þ

e e l0 , lr

e

and s are respectively the interface boundary, the force boundary and the whole domain of the block. {F} and {u} are the external force and displacement of points on the force boundary. {P} and {u} denote the body force and displacement of points inside the block. For the sake of simplicity, in this study, the body force related to the gravity of blocks is ignored. Therefore, the relation (7) is simplified to:

e

le0

XZ dfDugT frgdl ¼  dfugT fFgdl e

ler

ð8Þ

Together with the kinematic and constitutive relations presented above, the global equilibrium equation for the whole block system can be obtained and written in the general form:

Residual Strength

Mode I O

m

The matrix [N] is defined by:

XZ

C

l

T

Fig. 4. Failure criterion of contact interface with tensile mode and shear mode.

½KfUg ¼ fFg

ð9Þ

The global stiffness matrix [K] is obtained through an assemblage process similar to that used in the finite element method. More details can be found in Shi [18] and Wang et al. [25] (see Fig. 3).

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C. Yao et al. / Computers and Geotechnics 64 (2015) 48–60

than their distance to any other site Pj, denoted as d(x, Pj), which can be more formally expressed as:

3. Mesh generation The modified RBSM described above can be applied to any polygon-based mesh arrangement. For the sake of automatic mesh generation, in the present work, the specimen of brittle rock is considered as a rectangular domain (specimen), divided by uniformly and randomly distributed Voronoi cells, which are also called as the ‘‘mesh’’ here. The mesh generation procedure consists of two main steps as described in [10,26]. The first step is to randomly generate a set of uniformly distributed points in the domain. If the minimum distance between any two points is set to be dmin, in a rectangular domain a  b, the maximum number of points n can be calculated by: 2

n  0:68ab=dmin

ð10Þ

With the maximum number of points and the minimum distance between two arbitrary points, a uniformly distributed set of points can be obtained through an iterative random process using Monte Carlo method. The calculation time can be high when the number of points generation is large since each pair of points should be checked whether their distance exceed the minimum distance dmin. To make this process effective, a grid search algorithm is used as proposed in [27]. The second step is to generate a Voronoi diagram with the points obtained in the first step. The Voronoi diagram is a special kind of decomposition of a given space determined by a set of sites in the space. In its simplest case with Euclidean plane, each site Pk is simply a point, and its corresponding Voronoi cell Rk consisting of all points whose distance to Pk, denoted as d(x, Pk), is not greater

Rk ¼ fx 2 Xjdðx; Pk Þ 6 dðx; Pj Þ; for all j–kg

ð11Þ

A Voronoi diagram is uniquely determined by the set of sites specified. As a result, an appropriate Voronoi diagram can be obtained through just adjusting the number and the distribution of sites. With points get in the first step, a uniformly and randomly distributed set of Voronoi cells can be obtained. There are plenty of effective algorithms for Voronoi diagram generation in literatures. In this study, an open code originated from Fortune [28] is employed.

4. Failure criterion of interface In the Voronoi based modified RBSM, the basic element, the Voronoi cell is rigid (without deformation) and unbreakable. All the internal energy is restored in the element interface. Assuming that the interface failure is related to the energy dissipation in tension and shearing, two failure criteria are introduced at the interface level respectively for two modes of failure. For the tensile failure called as Mode I, an elastic brittle behavior is assumed and characterized by the tensile strength T for all interfaces. Once the tensile stress rn of a point on a contact interface reaches the tension strength T, the brittle failure occurs by instantaneously reducing the contact stresses rn and rs to zero. For the shear failure called as Mode II, a Mohr-Column type criterion is employed to define the shear strength of interface. The shear strength of material is defined by the friction angle /,

ν

0.5

n=1000

ν

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

n=2000

0 0

0.2

0.4

0.6

0.8

0

1

0.2

r=ks/kn

ν

n=1000

n=2000

n=5000

n=5000

ν

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0.2

0.4

0.6

0.8

0

1

0.2

ν

n=20000

0.3

0.3

0.2

0.2

0.1

0.1 0.4

0.6

ν

0.5 0.4

0.2

0.6

0.8

1

0.8

1

(d) n=10000

0.4

0

0.4

r=ks/kn

0

n=20000

1

n=10000

r=ks/kn (c) n=5000

n=10000

0.8

0

0

0.5

0.6

r=ks/kn (b) n=2000

(a) n=1000 0.5

0.4

0.8

1

n=50000

0 0

0.2

0.4

0.6

r=ks/kn (f) n=50000

r=ks/kn (e) n=20000

n=50000

Fig. 5. Typical numerical specimens with different number of elements.

Fig. 6. Relationships between specimens.

v

and r obtained respectively in six series of

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C. Yao et al. / Computers and Geotechnics 64 (2015) 48–60

ν

0.5

1.1

n=1000 n=2000 n=5000 n=10000 n=20000 n=50000

0.4

0.3

E/E

1 0.9 0.8

0.2

0.7

0.1

0.6

n=1000

n=2000

n=5000

n=10000

n=20000

n=50000

0.5 0

0 0

0.2

0.4

0.6

0.8

0.2

0.4

1

0.6

0.8

1

r=ks/kn

r=ks/kn

Fig. 9. Relationships between E=E and r obtained from average results of six series. Fig. 7. Relationships between

1.10 1.00 0.90 0.80 0.70 0.60 0.50

E/E

0

0.2

v and r obtained from average results of six series.

1.10 1.00 0.90 0.80 0.70 0.60 0.50

n=1000

0.4

0.6

0.8

E/E

0

1

0.2

E/E

0

0.2

0.6

r=ks/kn

1.10 1.00 0.90 0.80 0.70 0.60 0.50 0.8

1

E/E

0

0.2

E/E

0

0.2

0.6

1

0.8

1

0.8

1

n=10000

0.4

0.6

(d) n=10000

n=20000

0.4

0.8

r=ks/kn

(c) n=5000 1.10 1.00 0.90 0.80 0.70 0.60 0.50

0.6

(b) n=2000

n=5000

0.4

0.4

r=ks/kn

r=ks/kn (a) n=1000 1.10 1.00 0.90 0.80 0.70 0.60 0.50

n=2000

0.8

1

r=ks/k n (e) n=20000

1.10 E / E 1.00 0.90 0.80 0.70 0.60 0.50 0 0.2

n=1000

0.4

0.6

r=ks/kn (f) n=50000

Fig. 8. Relationships between E=E and r obtained respectively from six series.

cohesion coefficient C and the critical normal stress rcr, as illustrated in Fig. 4. The shear strength is then determined by:

jrsmax j ¼ rn tan / þ C; jrsmax j ¼ rcr tan / þ C;

rn P rcr rn < rcr

ð12Þ

Note that in many rocks, when the confining pressure is high enough, the mechanical strength become queasily independent on the confining pressure. This means that frictional effect progressively vanishes. In order to interpret this phenomenon in a very simply way, the critical normal stress rcr is then introduced. When

the normal compressive stress is higher than this critical value, the shear strength of interface becomes constant and independent on the normal stress. Further, for the sake of simplicity, the tensile strength T is set to be proportional to the cohesion coefficient, such as T = 0.3 C. According to the previous description, the stress state of a material point along a single interface is calculated using Eq. (6). The failure condition at this point is determined by the failure criterion defined above. The stress state can vary along the interface length if the displacement jumps between the two neighboring blocks are not uniform. Thus, some part of the interface can verify the failure criterion but not entirely. The proposed model can describe the progressive failure of the interface. In practice, we have used a four-point Gauss–Legendre integration method for the calculation of the global system of equilibrium equations. Therefore, the failure condition is checked only on Gauss points. In this way the complexity induced by the continuous distribution of stress along the interface is greatly reduced. The stress state can be different at the different Gauss points. In a given loading step, one Gauss point can meet the failure condition while the others not. In the subsequent step, some remaining Gauss points may also meet the failure condition. The interface is then progressively failed. The modified Newton–Raphson method is adopted for the nonlinear iterative calculation. A home-made code named Voronoi RBSM is developed with Visual C++ based on the proposed method. For all numerical results presented in this paper, the displacement step is chosen as 1  107 m for compression tests and 1  108 m for tension tests. The convergence is checked for each loading step against the following tolerance condition:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P3n k 2 i¼1 ðDU i Þ 6v tolerence ¼ 3n

ð13Þ

DU ki is the value of the computed ith displacement in the kth step, and n is the total number of blocks. The tolerance limit is set as v = 1  1012 m. In all calculations presented in this study, the convergence condition was easily achieved before peak stress. After the peak stress, the convergence rate becomes quite slow compared to the pre-peak stage due to the development of large amount of microcracks. To make a compromise between efficiency and accuracy, as in many codes for computational mechanics, a maximum

Table 1 Representative parameters for mesh sensitivity study on mechanical strength. E

m

tan /

C

T

rcr

30 GPa

0.18

0.8

15 MPa

4.5 MPa

20 MPa

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Confining pressure

0 MPa

5 MPa

10 MPa

Series

l (MPa)

Cv (%)

l (MPa)

Cv (%)

l (MPa)

Cv (%)

1 (n = 2000) 2 (n = 5000) 3 (n = 10,000)

40.98 40.37 40.06

1.42 0.57 0.74

58.75 58.53 58.82

1.22 0.74 0.49

65.65 65.74 65.98

0.71 0.63 0.49

l is the mean value of strength of the ten specimens in each series. Cv is the coefficient of variation, Cv = r/l, r being the standard deviation.

iteration number is usually set. This number was here fixed as 400 iterations. As can be seen in Section 5.4, main features of post peak behaviors are well captured with this maximum iteration number. 5. Results and discussions In this section, the proposed modified RBSM is applied to describe the mechanical behavior of brittle rocks. In particular, the macroscopic elastic property, failure strength as well as postpeak response will be studied. The influence of mesh size will be also investigated. 5.1. Macroscopic elastic parameters In the present work, only isotropic materials are considered. The macroscopic elastic property is defined by Young’s modulus E and Poisson’s ratio m. At the local level, the elastic property of interface is characterized by the normal and tangential stiffness coefficients kn and ks. The objective here is to discuss the relationship between the macroscopic and local elastic properties. For this purpose, a series of numerical simulations is performed. The size of the rectangular specimen is 0.05 m  0.1 m. Six groups of numerical specimens are generated through the mesh generation procedure described above. Each group has 10 different randomly generated meshes having a same number of elements, respectively being 1000, 2000, 5000, 10,000, 20,000, 50,000. Typical specimens for each group are illustrated in Fig. 5. For the sake of convenience, the local elastic parameters of interfaces are expressed by the following relations:

E kn ¼ ; ks ¼ r  kn h1 þ h2

ð14Þ

E is a predetermined modulus having the same order of magnitude as that of the macroscopic elastic modulus E. The coefficient r defines the ratio between ks and kn. h1 and h2 denote the distances from the centroids of two neighboring elements to their connecting interface, respectively, as illustrated in Fig. 2. Uniaxial tests are first simulated on each specimen under elastic conditions without considering interface failure. The loading is controlled by displacement, denoted by d. The value of coefficient E is chosen as 30 GPa and that of r between 0.1 and 0.96, with a step gap of 0.05. The macroscopic Young’s modulus E is calculated by:

E¼

P

e1

;

e1 ¼ d=b

ð15Þ

P is the average stress calculated over the upper or lower boundary of the specimen, e1 is the average macroscopic axial strain and b being the height of the specimen. The macroscopic Poisson’s ratio is calculated by:

v¼

e3 ; e ¼ ðdr  dl Þ=a e1 3

ð16Þ

e3 is the average macroscopic lateral strain. dl, dr denote the average displacement of left and right boundaries respectively and a being the width of the specimen.

In Fig. 6, we present the variations of macroscopic Poisson’s ratio as functions of the coefficient r for each group of specimens. It can be seen that when the number of elements is 1000, there is a little variance between the 10 different meshes having the same number of elements. When the number of elements increases, the variance significantly decreases and nearly vanished for the groups of specimens with 10,000, 20,000 and 50,000 elements. This indicates that when the number of elements is large enough, higher than 2000 in the present case, the mesh arrangement has negligible impact on the relationship between the macroscopic Poisson’s ratio and the local elastic coefficient r. In Fig. 7, the average curves of 6 groups are put together. It can be observed that the curves obtained from 6 groups of specimens almost coincide with each other, showing that the relationship between the macroscopic Poisson’s ratio and the local elastic coefficient r is nearly not affected by the number of elements or the element size. The relationships between the macroscopic Young’s modulus and r is also investigated. In Fig. 8, we show the variations of modulus ratio E=E as functions of r. It can be seen that as for the macroscopic Poisson’s ratio, the relationship between the macroscopic elastic modulus and the local elastic coefficient r is not affected by the mesh arrangement when the number of elements is higher than 2000. In Fig. 9, the average curves of 6 groups are put together. It can be observed that the relationship is slightly affected by the number of elements. However, the curves obtained with 10,000, 20,000 and 50,000 elements are very close each other, showing that element size can affect the relationship between E=E and r only if the number of elements is not large enough, for instance less than 10,000 in the present case.

6

Average coordinaon number

Table 2 Effects of mesh on compression strength with different confining pressures.

5.8

5.6

5.4

5.2

5 0

20000

40000

60000

Number of blocks Fig. 10. Average coordination number of each series.

Cv (%) of coordination number

0.2

0.15

0.1

0.05

0 0

10000

20000

30000

40000

50000

Number of blocks Fig. 11. Variation coefficient of coordination number of each series.

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(a) n=1000

(b) n=2000

(c) n=5000

(f) n=50000

(e) n=20000

(d) n=10000

Fig. 12. Typical interface orientation distribution of each series of numerical specimens with different block number.

12

Ci (%)

σ1

σ1

0.05m

0.05m

10 8 6

σ3

0.1m

σ3

2

0.1m

4

0 0

20000

40000

60000

Number of blocks Fig. 13. Average versus number of blocks.

Table 3 Representative parameters of Vienne granite used in biaxial compression tests. E

m

tan /

C

T

rcr

72 GPa

0.22

0.95

57 MPa

17.1 MPa

100 MPa

Based on the numerical results discussed above, it can be concluded that when the number of elements is large enough, there is very a small influence of element size and mesh arrangement

Biaxial compression test

Uniaxial tension test

Fig. 14. Boundary conditions and specimen geometry for tension and biaxial compression tests.

on the relationships between macroscopic elastic parameters and local elastic coefficients. As a consequence, two empirical relations are determined from the average results of the group with 50,000 elements:

C. Yao et al. / Computers and Geotechnics 64 (2015) 48–60

55

(b) σ3=10MPa

(a) σ3=0MPa

(c) σ3=20MPa

(d) σ3=40MPa

Fig. 15. Comparison between numerical results obtained in biaxial compression and experiment data in conventional triaxial compression tests. Numerical results are computed with parameters listed in Table 3 (Unit: stress: MPa, strain: 106).

r ¼ ks =kn ¼ 4:025m4  6:087m3 þ 6:022m2  3:966v þ 1

ð17Þ

E=E ¼ 0:6291r 4 þ 1:617r3  1:678r 2 þ 1:174r þ 0:5162

ð18Þ

Therefore, using these relations, it is easy to determine the local elastic parameters of interfaces (kn and ks) from the macroscopic elastic properties (E and m). Since h1 and h2 (in Eq. (14)) are different among different interfaces, kn and ks are not the same for different interfaces. In other words, for the same mesh, the values of kn and ks vary among interfaces. It is impossible to list all these parameters. Instead, E and v are used as input parameters. 5.2. Effects of element size and mesh arrangement on strength In order to investigate effects of element size and mesh arrangement on the mechanical strength, three comparative series of numerical simulations are conducted. Each series contain 10

randomly generated meshes while the size of specimen remains the same for all series. A representative set of mechanical parameters are chosen for all cases and listed in Table 1. But the number of element in each series is different and equal to 2000, 5000 and 10,000 respectively as shown in Table 2. Further, inside each series, 10 different mesh arrangements are randomly generated and considered. Three values of confining stress are chosen, namely uniaxial, 5 MPa, and 10 MPa. The numerical results obtained are shown in Table 2. The average values of the peak stress difference and the relative scatters are presented for three series. It is observed that for each series, the relative scatters between ten different meshes arrangements are relatively small. The maximum scatter is obtained for the case of uniaxial compression in the series No. 1 with 1000 elements. This indicates that the mesh form has a very limited influence on the mechanical strength predicted by the model. Further, the differences between three series are also very small. With this fact, it can be stated that the mechanical strength predicted by the Voronoi based modified RBSM is nearly independent on the element size and mesh arrangement. Note that due to the strong nonlinearity of failure process, it is generally not possible to determine explicit relationships between the microscopic failure parameters and macroscopic mechanical strength, as those for the elastic properties shown above. However, according to the numerical investigations presented above and below, it is clearly that the macroscopic mechanical strength is inherently dependent on the microscopic failure process of interfaces.

5.3. Geometrical effect of Voronoi based mesh

Fig. 16. Comparison of failure stress between numerical predictions and experiment data. Numerical results are computed with parameters listed in Table 3.

Macroscopic behaviors predicted by discrete element methods can significantly depend on the geometrical characteristics of generated numerical meshes. These geometrical characteristics can be related to the microstructure properties of heterogeneous materials. Therefore, the strength of discrete approaches lies in their capability to describe effects of microstructural heterogeneities on

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macroscopic behaviors by adopting suitable numerical meshes. On the other hand, in order to reproduce the macroscopic behavior of a statistically homogeneous continuum at the macroscopic scale by using a discrete method such the RBSM in this paper, it is important to study the mesh independency of numerical results obtained. This means that the proposed Voronoi based mesh is representative of main microstructural characteristics of the brittle rock despite of different element size and mesh arrangement. According to the previous work by Scholtes et al. [29], for a given porosity (which is constant and equal to zero in the present study) and a given set of micromechanical parameters, it can be shown that the macroscopic properties depend only on the coordination number, defined as the number of neighboring blocks per block. The average coordination number of each series of specimens in Section 5.1 is given in Fig. 10. We can see that all series have a similar coordination number, though with a minor increase as the block number gets larger from 5.75 when n = 1000 to 5.96 when n = 50,000. This minor increase may help explain the small variation in the relationship between E/E0 and r among different series shown in Fig. 9. The variation coefficient of coordination number Cv in each series is illustrated in Fig. 11, in which it is shown that Cv decreases as n gets larger, indicating that the geometrical variation among different specimens with same block number decreases as block number increases. In addition, the interfaces are distributed in an isotropic manner in the proposed mesh. Illustrated in Fig. 12 are typical interface orientation distributions for each series of numerical specimens. We divide all orientations into 36 sections with a step of 5° and count the number of interfaces lying in each section. It can be observed that interfaces are almost uniformly distributed in orientation for each specimen with different block number and there is a trend that as block number increases, the distribution gets more isotropic. We can demonstrate this point in a more quantitative way. Here we use the parameter Ci = ri/li to define the isotropy of interface distribution, in which ri is the standard deviation of number of interfaces in each 5° for one specimen and li is the average number of interfaces in each 5°. The lower Ci is, the more isotropic the mesh

(a) σ3=0MPa

(c) σ3=20MPa

is. The average Ci of each series of specimens with different number of blocks is shown in Fig. 13. As the number of blocks gets greater, the average Ci decreases. 5.4. Comparison with experiment data In order to check the performance of the proposed model in producing the mechanical behavior of brittle rocks, numerical results are now compared with typical experimental data. For this purpose, a numerical specimen with 10,000 elements is adopted. Uniaxial and triaxial compression tests have been performed on granite with different confining pressures [30]. The macroscopic elastic parameters (E and m) are measured from the linear parts of stress–strain curves. The local elastic parameters of interfaces are then deduced using the empirical relations presented above. The values of interface failure, tan /, C and rcr, are identified through an trial and error optimization algorithm with the stress–strain curves of conventional tri-axial compression test. In each step of calibration, simulation results are compared with experimental data at all confining pressures. Calibration progresses until simulation results match well with experimental data on the whole. The values of parameters obtained are given in Table 3. These values are used in all numerical simulations compared with experimental data presented in this paper. The boundary conditions used in numerical simulations are shown in Fig. 14 for both biaxial compression and uniaxial tension tests. Comparisons between numerical results and experimental data are presented in Fig. 15 for various biaxial tests with different lateral stress or confining pressure. The values of E and v obtained from stress strain curves shown in Fig. 15 are respectively 72.15 GPa and 0.217, with very small differences compared to the input values listed in Table 3. This shows that the relationship established in the relations 17 and 18 works well. Overall, the numerical results fit well with the experiment data. The proposed model correctly describes the main features of mechanical behavior of brittle rocks, such as the nonlinear strain due to crack

(b) σ3=10MPa

(d) σ3=40MPa

Fig. 17. Stress strain curves together with microcrack count evolution during the whole process of compression under different confining pressures. Numerical results are computed with parameters listed in Table 3 (Unit: stress: MPa, strain: 106).

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A1

B1

A2

B2

A3

B3

A4

(a) σ3= 0MPa

(b) σ3= 10MPa

(c) σ3= 20MPa

(d) σ3= 40MPa

B4

Fig. 19. Macro failure mode under different confining pressure produced by numerical simulation. Numerical results are computed with parameters listed in Table 3 (horizontal displacement is amplified by 10 for a, b, c, and by 40 for d).

C1

D1

C2

D2

C3

D3

C4

D4

Fig. 18. Progressive process of micro crack development during compression under different confining pressures. A, B, C, D respectively corresponds to confining pressure of 0 MPa, 10 MPa, 20 MPa and 40 MPa. And the number 1, 2, 3, 4 indicate their positions on corresponding stress–strain curves which are shown in Fig. 17. Numerical results are computed with parameters listed in Table 3. (Blue: tensile cracks; Pink: shear cracks). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

propagation, macroscopic failure by crack coalescence, residual mechanical strength due to crack frictional effects and interactions. As an example, the residual uniaxial compression strength is about 40 MPa from Fig. 15a. In Fig. 16, the peak stress difference is expressed as a function of lateral stress. To demonstrate again the mesh independency of strength, results from simulation on a mesh of 2000 blocks are also presented. There is a good agreement between the numerical predictions and experimental values. The pressure dependency of mechanical strength is well reproduced by the proposed model. In addition, results from 10,000 blocks almost coincide with which from 2000 blocks, indicating again that our model is mesh independent in predicting the mechanical

strength. It is important to point out that the conventional triaxial tests performed on cylindrical samples were realized in axisymmetric 3D conditions. These tests should be investigated by 3D numerical simulations. However, at this stage, our computational code is operational only for 2D conditions. As a traditional method of approximation, the conventional triaxial compression tests on cylindrical samples are here replaced by bi-axial compression tests on rectangular specimens in plane strain conditions. Accordingly, the model’s parameters are calibrated by an optimization procedure from the experimental data of some triaxial compression tests which are also simulated in 2D conditions. Therefore, the comparisons given here give only a qualitative verification of the proposed model in the description of mechanical behavior of brittle rocks. When the proposed model is extended to 3D conditions, the model’s parameters should be updated in order to account for geometrical effects. In Fig. 17, we show the stress–strain curves together with the microcrack event accumulation process during compression under different confining pressures. The volumetric deformation curves are also shown. Generally, four phases are observed during the whole process of compression. In the first phase, no microcrack occurs and the stress–strain curve is elastically linear. In the second phase, tensile cracks initiate and its number increases rapidly. At the same time, there is a minor decrease in the elastic modulus and the volumetric dilation starts to occur. In the third phase, shear cracks initiate and develop significantly. The stress strain curve becomes obviously non-linear, the peak stress occurs, and there is a clear transition from the volumetric compressibility to dilatancy in this phase. The fourth phase takes place in the residual stage. In this phase, the development of both tensile crack and shear crack becomes stable. As confining pressure increases, the

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Fig. 20. Illustration of stress–strain curve and failure mode under uniaxial tension (tensile crack in blue and shear cracks in pink). Numerical results are computed with parameters listed in Table 3 (horizontal deformation amplified by 50). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 23. Effects of rcr on stress envelope. Fig. 21. Effects of C on macroscopic strength envelope.

Fig. 22. Effects of tan / on stress envelope.

Fig. 24. Effects of T on macro stress envelope.

weight of shear cracks gets higher when failure occurs, and the initiation stress of both tensile crack and shear crack gets greater. This is the fundamental mechanism of brittle–ductile transition at the microscopic scale, as pointed out by Schopfer [31]. This can also help to explain the confining pressure dependency of strength, which is shown in Fig. 16. In order to illustrate the progressive growth of microcracks, four points on the stress strain curve of each confining pressure are chosen (for example A1 in Fig. 17a). Microcrack distributions of each case are shown in Fig. 18. It is observed that the growth of microcracks starts before the peak strength is reached. There is a propagation of microcracks with the increase of axial strain. The material failure is due to the coalescence of microcracks to the formation of oriented macroscopic

fractures. The progressive failure process from diffusion to localization is well reproduced. As confining pressure increases, shearing failure gradually becomes the dominant failure mechanism. The post-peak behavior is controlled by the residual strength of interfaces and also interactions between microcracks. In Fig. 19, we present the macroscopic failure mode at the points A4, B4, C4 and D4, which are shown in Fig. 18. It can be observed that the failure mode turns from a tensile rupture failure when r3 = 0 to a conjugate shear failure when r3 = 40 MPa. In Fig. 20, we present the numerical stress–strain curve and failure mode in the uniaxial tension test. As expected, the macroscopic behavior of material is elastic brittle. The material strength is sharply reduced to zero after the peak stress. The macroscopic failure is

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Fig. 25. Effects of T on macro tensile strength.

Fig. 26. Typical structure of Voronoi mesh.

characterized by the formation of horizontal fractures which are perpendicular to the axial stress and most microcracks are tensile ones (in blue color). 5.5. Sensitivity analysis of microscopic parameters Since an empirical relationship is established between microscopic elastic parameters (kn, ks) and macroscopic ones (E, v) so that the former can be directly derived from the latter, only four other microscopic parameters are needed to reproduce the macroscopic behavior of brittle rocks, which are C, tan /, rcr and T. In this part, their effects on the macroscopic failure stress envelope are discussed separately. As mentioned above, it is so far not possible to establish explicit relationships between the microscopic failure parameters and macroscopic mechanical strength. However, a comprehensive sensitivity analysis can provide some qualitative correlations between the microscopic and macroscopic quantities. In the following we will investigate the influences of main microscopic failure parameters on the macroscopic response. When one parameter is being discussed, others are remained the same as listed in Table 3. Here, to make calculation more efficient, a mesh with 2000 blocks is adopted. As shown in Fig. 16, with the same parameters listed in Table 3, the stress envelope produced with this mesh almost coincides with which produced with the mesh of 10,000 blocks. Effects of cohesion C on the stress envelop are shown in Fig. 21. Three levels of C are tested, i.e. 30 MPa, 57 MPa and 80 MPa. Generally, a higher value of C produces a higher strength. But when the confining pressure increases, the effect of C decreases. Effects of

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tan / is shown in Fig. 22. Generally, a higher value of tan / results in a higher strength and a higher macroscopic internal frictional angle. But its effect gets greater as confining pressure increases, which is contrary to C’s effects. Effects of rcr is shown in Fig. 23, which is very similar to tan /. A higher value of rcr results in a higher strength and a higher macroscopic internal frictional angle. Effects of microscopic tensile strength T are shown in Fig. 24. Minor variations are observed by varying T under unconfined compressive condition. But this effect vanishes as confining pressure increases. Generally speaking, parameter T has little effects on the stress envelope. However, this parameter has great impact on the macroscopic tensile strength, which is shown in Fig. 25. The macroscopic tensile strength is almost proportional to the microscopic tensile strength, and Tmacro/T  0.64, Tmacro is the macroscopic tensile strength. Another issue worth noticing is that, the tensile strength calculated from 2000 blocks is 10.89 MPa with parameters listed in Table 3. This value is very close to 10.81 MPa from 10,000 blocks, indicating that the tensile strength is also mesh independent. It may seem to be puzzling to see that the microscopic tensile strength has a little influence on the macroscopic compressive strength envelope while the tensile splitting is a major failure mechanism at all tested confining pressures, as shown in Fig. 18. In order to clarify this ‘‘puzzling’’ phenomenon, a typical structure of Voronoi mesh is shown in Fig. 26. There are three blocks, A, B and C, the interfaces between which are respectively I-1, I-2 and I-3. Before any failure, the normal tensile stress on I-1 should be close to the loading stress. Note that the microscopic tensile strength T is much smaller than the macroscopic compressive strength (though T varies from 0.1 C to 0.6 C). As a result, the failure will firstly occur on I-1 by the tensile splitting, and then on I-2 and I-3 by the shear failure as the loading increases. The overall failure is controlled by I-2 and I-3. The final failure stress is little influenced by the tensile strength of I-2. What happened in the whole is much more complex, but this is the basic mechanism. The overall failure is mainly governed by the shear strength of interfaces. Currently a great deal of attention focused on whether or not DEM can simulate different levels of brittleness (tensile to uniaxial compressive strengths ratio). In our model, it seems not be a problem. Consider the results shown in Figs. 24 and 25 as an example, the tensile strength to UCS ratio are 1:35.3, 1:12.4 and 1:6.4 respectively for T = 0.1 C, T = 0.3 C and T = 0.6 C. Our model can easily cover a broad range of brittleness levels. Note that the brittleness of granite usually lies between 1:10 and 1:20. We adopted T = 0.3 C in this work.

6. Conclusion In this study, we proposed a modified rigid block spring method based on uniformly distributed Voronoi diagram to describe the mechanical behavior of brittle rocks. The specimen of cohesive brittle rocks is represented by an assembly of Voronoi blocks. The macroscopic deformation and failure behaviors are controlled by the local elastic property and failure criterion of interfaces between blocks. The modified RBSM improves the original one by introducing a uniform distribution of normal and tangential spring along each interface and allows the description of non-uniform distribution of stress and strain and the progressive failure of interfaces. Two failure modes are considered for interfaces, the tensile mode controlled by the normal tensile strength and the shear mode described by a Mohr Column type failure criterion. The proposed model was applied to describe the basic mechanical behavior of a brittle rock, granite. Based on a series of sensitivity studies, it is found that there exists an approximately fixed

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relationship between the local elastic parameters of interfaces and the macroscopic elastic parameters. Two empirical laws were derived for the determination of local elastic parameters as functions of macroscopic elastic parameters. Comparisons between numerical results and experimental data were presented for a series of uniaxial and bi-axial compression tests with different confining pressures. It is observed that the proposed model is able to describe the main features of mechanical behaviors of brittle rocks, in particular the transition from diffused microcracks to localized fractures. Moreover, the numerical results, for example the peak compression strength, are nearly independent on the element size and mesh arrangement when the number of elements is high enough. As extension, the proposed model will be applied to initially anisotropic materials and extended to 3D conditions. Acknowledgements This work was financially supported by Chinese National Natural Science Foundation (51409138, 51179137) and National Program on Key Basic Research Project of China (2011CB013506). These supports are gratefully acknowledged. References [1] Munjiza A, Xiang J, Garcia X, Latham JP, D’Albano GGS, John NWM. The virtual geoscience workbench, VGW: open source tools for discontinuous systems. Particuology 2010;8:100–5. [2] Kozicki J, Donzé FV. A new open-source software developed for numerical simulations using discrete modeling methods. Comput Methods Appl Mech Eng 2008;197:4429–43. [3] Jean M. The non-smooth contact dynamics method. Comput Methods Appl Mech Eng 1999;177:235–57. [4] Potyondy DO, Cundall PA. A bonded-particle model for rock. Int J Rock Mech Min Sci 2004;41:1329–64. [5] Lan H, Martin CD, Hu B. Effect of heterogeneity of brittle rock on micromechanical extensile behavior during compression loading. J Geophys Res: Solid Earth 2010;115:B01202. [6] Kazerani T, Zhao J. A microstructure-based model to characterize micromechanical parameters controlling compressive and tensile failure in crystallized rock. Rock Mech Rock Eng 2014;2014(47):435–52. [7] Wang Y, Tonon F. Modeling Lac du Bonnet granite using a discrete element model. Int J Rock Mech Min Sci 2009;46:1124–35. [8] Hentz S, Daudeville L, Donzé F. Identification and validation of a discrete element model for concrete. J Eng Mech 2004;130:709–19.

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