(bim) rocks by computational homogenization

Engineering Geology 200 (2016) 58–65

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Engineering Geology journal homepage: www.elsevier.com/locate/enggeo

Technical note

Estimating the elastic moduli and isotropy of block in matrix (bim) rocks by computational homogenization Michael Tsesarsky a,b,⁎, Moria Hazan b, Erez Gal a a b

Department of Structural Engineering, Ben Gurion University of the Negev, Beer-Sheva, Israel Department of Geological and Environmental Science, Ben Gurion University of the Negev, Beer-Sheva, Israel

a r t i c l e

i n f o

Article history: Received 2 April 2015 Received in revised form 15 October 2015 Accepted 5 December 2015 Available online 9 December 2015 Keywords: Bim-rocks Elastic moduli Isotropy Computational homogenization Multi-scale analysis

a b s t r a c t Block-in-matrix (bim) rocks are comprised of stiff inclusions (blocks) in softer matrix. The heterogeneity and multi-scale nature of bim-rocks makes the evaluation of their mechanical properties a challenging task. In this research we study the elastic moduli (E, ν, K. μ) and isotropy of bim-rocks using finite elements based computational homogenization and multi-scale approach. The influence of volumetric block proportion (VBP), block shape and block orientation is presented. Application of multi-scale homogenization method to estimate the elastic moduli of a naturally occurring conglomerate is also reported. When compared to close form solution of Hashin–Shtrikman the computational homogenization approach shows good agreement for bim-rocks with spherical blocks. For bim-rocks with elliptical blocks the elastic moduli show tangible difference from the closed form solution, with deviation from isotropy increasing with VBP and preferred orientation of blocks. Based on the results of our numerical study we propose a simplified approximation for estimation of elastic moduli of bim-rocks, together with estimates of deviation from isotropy. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Various geological materials are comprised of stiff inclusions embedded in softer matrix, e.g. conglomerates, agglomerates, brecias (volcanic or tectonic), shear zones and mélanges, among other rock types. These rocks are considered problematic in most geo-engineering projects due to their spatial and compositional variability. The term block in matrix (bim) rocks was introduced by Raymond (1984) and abbreviated to “bim-rocks” by Medley (1994). The definition of a bim-rock, according to Medley, is “a mixture of rocks, composed of geotechnically significant blocks within a bonded matrix of finer texture”. Similarly to other rock/soil mixtures the size distribution of bimrocks tends to be fractal (or well graded in geotechnical notation). Medley and Lindquist (1995) argued that this distribution is scale independent, as blocks are always found, regardless of the scale of interest or observation. They have defined block–matrix threshold of 0.05Lc, where Lc is the characteristic length of blocks in the assemblage. Mechanical contrast between blocks and matrix is typically defined using shear strength ratio (friction angles) or stiffness ratio. Sufficient contrast is afforded by a friction angle ratio (tan φ of weakest block)/(tan φ of matrix) of between 1.5 and 2, as suggested by the works of Lindquist (Lindquist, 1994b; Lindquist and Goodman, 1994) ⁎ Corresponding author at: Department of Structural Engineering, Ben Gurion University of the Negev, POB653, Beer-Sheva 84905, Israel. E-mail address: [email protected] (M. Tsesarsky).

http://dx.doi.org/10.1016/j.enggeo.2015.12.003 0013-7952/© 2015 Elsevier B.V. All rights reserved.

and Volpe et al. (1991). Another means of identifying strength contrasts is to use rock stiffness. Lindquist (1994a) used a ratio of block to matrix stiffness (Eb/Em), of 2.0 to generate block/matrix contrasts matrix for physical models of mélange. Similar, but not exclusive values for stiffness ratios were also reported by other researchers (e.g. Afifipour and Moarefvand, 2014; Sonmez et al., 2006b, 2004). The overall mechanical properties of bim-rocks are affected by the mechanical properties of the matrix and the blocks, the volumetric block proportion (VBP), the block shapes, the block size distributions and the orientation of the blocks. Different techniques to determine the VBP from outcrop and boring data were reported in the literature, from simple scan lines to state of the art image processing techniques (Coli et al., 2012; Xu et al., 2008). Many researchers have shown that the VBP is the single most important parameter determining the physical and mechanical properties of bim-rocks (e.g. Afifipour and Moarefvand, 2014; Coli et al., 2012; Kahraman et al., 2015; Lindquist and Goodman, 1994; Sonmez et al., 2006a, 2004). When the VBP ranges from 0.25 and 0.7, the increase in the overall mechanical properties of bim-rocks are directly related to the volumetric block proportion of blocks in the rock mass (Lindquist and Goodman, 1994). For VBP N 0.7 the rock mass cannot be treated as a bim-rock but rather as “blocky rock mass with infilled joints” (Medley, 1994), as blocks tend to develop “contact to contact” geometry. Recently Kahraman et al. (2015) showed linear correlation between VBP and pressure wave velocity for VBP range from 0.02 to 0.9 for the Misis fault breccia.

M. Tsesarsky et al. / Engineering Geology 200 (2016) 58–65

Many mechanical problems in geo-engineering are analyzed within the framework of elasto-plasticity, for which the elastic constants (i.e. E, ν, K, and μ) of the rock material and the rock mass are essential parameters. In many cases homogeneity and isotropy are also assumed. While these assumptions may be safely made for continuous, fine-grained rocks, they cannot be assured for bim-rocks due to their inherent nature, a mix of different constituents with distinct mechanical properties and variable proportions. As pointed out by Sonmez et al. (2006b) in many rock-engineering applications, the elastic modulus of intact rock is not actually determined by laboratory tests—due to the requirements of high-quality core samples and sophisticated test equipment. Moreover it is sometimes too difficult to obtain standard core samples from weak, stratified, highly fractured and block-in-matrix rocks. Laboratory determination of the bim-rocks elastic moduli is difficult due to the variability of block size distribution in an outcrop, which in many cases exceeds the standard sample sizes, such as the ISRM suggested NX size (Ulusay and Hudson, 2006). Lindquist and Goodman (1994) proposed the use of laboratory scale models in order to determine the strength and deformability of mélanges. Their main reasoning was the scale invariability of the block distribution of the mélanges. This approach however is not always practical, as fabrication of physical models and their testing are time consuming. A large number of specimens should be prepared to ensure reproducibility, and moreover to test various stress paths and loading configurations. The physical model should also conform to the mechanical properties of blocks, in many cases of various origins and rock types, and matrix. Sonmez et al. (2006a) used large diameter (150 mm) cores to study the uniaxial compressive strength (UCS) of the Ankara agglomerate as a complementary measure to studies based on statistical evaluation of the elastic constants and strength of bim-rock properties (Sonmez et al., 2006b, 2004). Wen-Jie et al. (2011) and Coli et al. (2011) performed unconventional large scale shear tests on soil rock mixture (SRM) and Shale–Limestone Chaotic Complex bim-rock (SLCC) respectively. These tests focused on shear strength rather than elastic properties. In general the main findings are that for VBP b 0.25 the unconfined compressive strength (UCS) reflects the properties of the matrix, whereas for VBP N 0.7 UCS reflects the properties of the blocks. UCS for VBP in the range of 0.25 to 0.7 proportionally increases with VBP. A recent study by Afifipour and Moarefvand (2014) utilized large diameter samples (100 mm and 150 mm) to study the UCS and Young's modulus of high volumetric block proportion (VBP = 70%–90%) bim-rocks. It is worthy to note the work of Kahraman and Alber (2006) who have studied, on laboratory size samples, the mechanical properties of an “inverse” bim-rock, with soft blocks and stiff matrix. They have reported an inverse trend for this bim-rock, decreasing UCS with increasing VBP, however this trend was found continuous over the range of VBP studied, 0.1 to 0.95. Elastic modulus was also found to decrease with VBP over the range of 0.2 to 0.7. It should be noted that in studies reported herein, with the exception of Lindquist and Goodman (1994), the strength and elasticity modulus were determined in uniaxial compression. Moreover, radial strains were not measured and therefore Poisson's ratio was not determined. In practice the rock, and structures within, will be subjected to a tri-axial stress field. Moreover, an assumption of rock isotropy cannot be assumed without tri-axial mechanical testing. An alternative approach to physical modeling is the use of numerical tools such as the finite elements method to calculate the relevant mechanical properties of bim-rocks using the basic properties of its block and matrix constituents. Xu et al. (2008) combined digital image processing and finite element modeling (FE) to simulate the large shear tests of SRM. The FE modeling also provides the capability to study the elastic properties and their variation in the sample. Furthermore it allows analyzing the macro-structure, i.e. engineering scale analysis, based on the microstructure of the material. Numerous publications were dedicated to the verification and validation of this

59

approach, see Zohdi and Wriggers (2005); Geers et al. (2010) and references therein. The multi-scale approaches were also adopted and validated for the elastic analysis of concrete, an artificial bim-rock (e.g. Gal and Kryvoruk, 2011; Häfner et al., 2006; Wriggers and Moftah, 2006). In this paper we present a parametric study of the elastic properties of bim-rocks using the theory of homogenization. We adopt the FE implementation (Gal et al., 2008), of the theory of homogenization, in which the heterogeneous material is replaced by an equivalent homogeneous continuum to develop a unit-cell for bim-rocks. We parametrically study the variation of the elastic moduli as function of VBP, block geometry and block–matrix stiffness ratio, focusing on the deviation from the initial assumption of isotropy. The results of the parametric study are compared with results of closed form solution (Hashin–Shtrikman) for multi-phase elastic solids. We then continue with an example of the multi-scale homogenization approach: estimating the elastic constants of the Zin conglomerate. 2. Methods 2.1. Asymptotic theory of homogenization The method for obtaining the macroscopic behavior of a material, based on its microstructure is referred to as the theory of homogenization, by which the heterogeneous material is replaced by an equivalent homogeneous continuum. The method is performed on a statistically representative sample of material, referred to as a material unit cell. Numerous theories have been developed to predict the behavior of composite materials. Starting from the various effective properties obtained by the models of Eshelby (1957); Hashin (1962); Mori and Tanaka (1973), the self-consistent approaches of Hill (1965) and various mathematical homogenization methods (e.g. Christensen, 1979) pioneered by Benssousan et al. (1978) and Sanchez-Palencia (1980). Unfortunately, most of these analytical models can only give estimates or boundaries for the macroscopic properties, and the simplifying assumptions used, result in, the major differences obtained. Computational procedures for implementing homogenization have been an active area of research starting with the contribution by Guedes and Kikuchi (1990) for linear elasticity problems. Over the past two decades major contributions have been made to extend the theory of computational homogenization and improving fidelity and computational efficiency of numerical simulations, refer to Gal and Krivoruk (2011) and references therein. These developments established the finite element method (FEM) as one of the most efficient numerical methods, whereby the macroscopic responses can be obtained by volumetrically averaging numerical solutions of unit cells (e.g. Zohdi and Wriggers, 2005). The asymptotic theory of homogenization is based on the following asymptotic expansion of the displacement field: 1

2

uðx; yÞ ¼ u0 ðx; yÞ þ ξ u1 ðx; yÞ þ ξ u2 ðx; yÞ þ OðhÞ

ð1Þ

where x and y are the position vectors in macroscopic and microscopic scales respectively. The scales are related through a scale factor 0 b ξ ≪ 1, such that x = y/ξ . Assuming u0 ðx; yÞ ¼ uðxÞ

ð2Þ

u1 ðx; yÞ ¼ ε 0 ðxÞχ imn ðyÞ

ð3Þ

where ε0(x) is the macro-scale strain vector and χimn(y) is the micro-scale influence function, we obtain two coupled problems.

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The macroscopic problem is formulated using the following boundary volume problem: Find u(x) on the macroscopic domain Ω, with boundaries Γ, such that

and shear moduli, K and μ respectively, can be readily calculated from the following:

Lijlm ϵlm;x j þ f i ¼ 0 on Ω

ð4Þ

3K ¼

ui ¼ ui on Γ u and σ ij n j ¼ T j on Γ T

ð5Þ

where Lijlm is the effective material matrix, σ ij and ϵlm and is the macroscale stress and strain components and f i is body force. The macro-scale strain component is defined as follows: ϵmn ¼ um;xm

 1
¼ um;xn þ un;xm : 2

ð6Þ

σ kk δij σ ij −σ kk δij and 2μ ¼ : εkk δij εij −ε kk δij

ð13Þ

The general solution of the unit cell problem using the finite element method is obtained by resolving the unit cell problem to six RHS vectors (six in a 3D case due to symmetry of indices mn). In the matrix implementation, Lijmn is a 6 × 6 matrix where ij represents six rows and mn six columns. Each column in Lijmn can be extracted by multiplying Lijmn with a unit overall strain, εmn ¼ 1. 2.2. Unit cell generator

The micro-scale problem with periodic boundary conditions is formulated using the following boundary volume problem: Find the micro-scale influence function χ on the microscopic domain Θ, of a unit cell sized Y, such that h

Lijlm χ ðk;yi Þmn þ I klmn

i ;y j

¼ 0 on Θ

ð7Þ

χ imn ðyÞ ¼ χ imn ðy þ Y Þ on∂Θ

ð8Þ

χ imn ðyÞ ¼ 0 on∂Θver

ð9Þ

where ∂ Θ represents the domain boundary and ∂ Θver the unit cell vertices, Iklmn is 4th order identity matrix. The influence functions are defined as follows: χ ðk;yi Þmn ¼

 1 χ kmn;yl þ χ kml;yn : 2

ð10Þ

The homogenized constitutive tensor matrix that represents the macroscopic material properties is as follows: Lijlm ¼

1 jΘj

Z Θ

σ mn ij dΘ

ð11Þ

where σmn ij are the stress influence functions induced by applying an overall unit strain εmn , defined as h i σ mn ij ¼ Lijkl χ ðk;yi Þmn þ Iklmn :

ð12Þ

Here the assumption of linear elasticity is required to satisfy the condition that the unit cell is a representative elementary volume (REV). Solution of the unit cell problem using the finite element method is obtained by resolving the unit cell problem to multiple right hand system (RHS) vectors (six in a 3D case due to symmetry of indices mn). In the matrix implementation (Voigt notation), Lijmn is a 6 × 6 matrix where ij represents six rows and mn six columns. Each column in Lijmn can be extracted by multiplying L ijmn with a unit overall strain εmn . For implementation in the ABAQUS commercial package it is convenient to apply the needed unit strain, εmn through a unit thermal strain to each finite element in the unit cell model: ε mn ¼ kmn ΔT. kmn is the appropriate thermal expansion component and ΔT = 1 is the unit temperature change. For example, to obtain the first column of the material matrix Lijmn , we write k ¼ ½ 1

0 0 0

0

0 T and therefore

the applied unit strain in x1 direction is ε ¼ ½ 1 0 0 0 0 0 T . In a similar manner all the six columns of the material matrix can be readily extracted. For statistically isotropic material one loading state is sufficient for describing its overall linear elastic behavior. Thus, the effective bulk

The suggested framework executes the multi-scale analysis for bimrock assemblage by employing the original concrete unit cell generator developed by Gal et al. (2008). The unit cell is incorporated into a commercial finite element software package ABAQUS (Simulia, 2008). The finite element model, a three-dimensional array of cubic elements, includes blocks of different sizes and geometry and a homogenized matrix. The blocks, which may come from different sources, are distributed within the matrix so that each element that does not represent block material is assigned to be matrix element. The unit cell generator algorithm manipulates the VBP and block size distribution in order to create a finite element unit cell model of a bim-rock. The algorithm adjusts the finite element meshing with respect to the physical unit cell size. In this research we employed a cube with edge size comprised of 70 elements. Eight-node quadrilateral solid elements with reduced integration were used, yielding approximately 1.0.10 7 degrees of freedom for each model. The physical size of each element was determined according to the scale of the problem, however the smallest possible block model is discretized using 43 elements in order to adhere to the assumption that block–matrix threshold is 0.05Lc. Essential information needed for creating a bim-rock unit cell is the block distribution. This information is used to monitor the size and amount of the different blocks within the bim-rock. The obtained block distribution is placed into the computational volume from largest to smallest in diameter such that blocks don't overlap (no contact–contact condition). The blocks are modeled either spherical or elliptic, which can be rotated in space to any prescribed inclination degree (α). Our numerical study was limited to assemblages with VBP ≤ 0.4, and further extrapolated by regression to VBP ≤ 0.7. Effects of blocks with contact–contact relations, typical for VBP N 0.7, were not studied. Periodic boundaries are used to eliminate edge effects. If a portion of a particle extends beyond one or more faces of the 3D box, the remainder of its volume is protruded into the opposite face. 2.3. Measures for deviation from isotropy One of the major goals of this research was to study to what extent a certain bim-rock assemblage deviates from elastic isotropy. Each bim-rock assemblage was modeled twice: first the assemblage was assumed statistically isotropic, modeled using a single step unit strain (ε0 ¼ ½ 1 0 0 0 0 0 T ) loading to attain the elastic constants; then we assumed that the same assemblage is anisotropic and applied six unit strains to attain the general elasticity matrix. In order to compare between the results of the two different approaches we utilize the following approach. First, we compute the element-wise difference eij in the following manner:

eij ð%Þ ¼

aniso ciso ij −cij

caniso ij

 100:

ð14Þ

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Table 1 Material matrices (Lijkl) and element-wise difference matrix (eij) for two bim-rock unit cells: Spherical blocks (upper rows) and elliptical blocks (lower rows). Spherical blocks. Volumetric block proportion, 0.3 Liso ijkl 20.64 6.40 6.40 0.00 0.00 0.00

Laniso ijkl 6.40 20.64 6.40 0.00 0.00 0.00

6.40 6.40 20.64 0.00 0.00 0.00

0.00 0.00 0.00 7.12 0.00 0.00

0.00 0.00 0.00 0.00 7.12 0.00

0.00 0.00 0.00 0.00 0.00 7.12

20.64 6.39 6.41 0.02 0.00 0.01

eij (%) 6.39 20.88 6.41 0.02 0.03 0.05

6.41 6.41 20.83 0.01 0.00 0.04

0.02 0.02 0.01 7.11 0.00 0.02

0.00 0.03 0.00 0.00 7.13 0.01

0.01 0.05 0.04 0.02 0.01 7.15

6.28 20.28 6.68 0.01 0.00 0.51

6.47 6.68 23.61 0.02 0.01 0.91

0.04 0.01 0.02 6.84 0.20 0.00

0.01 0.00 0.01 0.20 7.31 0.03

0.07 0.51 0.91 0.00 0.03 7.46

0.00 0.17 0.17 – – –

0.17 1.13 0.12 – – –

0.17 0.12 0.90 – – –

– – – 0.09 – –

– – – – 0.13 –

– – – – – 0.47

1.44 0.26 4.56 – – –

1.41 4.56 13.9 – – –

– – – 1.97 – –

– – – – 4.53 –

– – – – – 6.49

Elliptical blocks. Volumetric block proportion, 0.3; α = 22° Liso ijkl 20.33 6.37 6.37 0.00 0.00 0.00

Laniso ijkl 6.37 20.33 6.37 0.00 0.00 0.00

6.37 6.37 20.33 0.00 0.00 0.00

0.00 0.00 0.00 6.98 0.00 0.00

0.00 0.00 0.00 0.00 6.98 0.00

0.00 0.00 0.00 0.00 0.00 6.98

20.33 6.28 6.47 0.04 0.01 0.07

eij (%)

Note that eij is computed only for the twelve non-zero elements of the isotropic material matrix. We proceed and compute the average element-wise difference eij . An example of the procedure is presented

0.01 1.44 1.41 – – –

in Table 1, where results of two different assemblages with VBP = 0.3 (Fig. 1) are presented. The upper row of Table 1 is for a bim-rock assemblage of spherical inclusions with VBP of 30%. The lower row is

Fig. 1. Visualization of spherical (upper row) and elliptical (lower row) unit cells with VBP of 30%. Left column are matrix and blocks combined, right column are blocks only.

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Fig. 2. Maximum element-wise difference vs. average difference for all bim-rock unit cells modeled.

for a bim-rock assemblage of elliptical inclusions with VBP of 30%, oriented with inclination of α = 22°. The maximum value of each element-wise difference is marked bold. When plotting the maximum of element-wise difference vs. the average difference (Fig. 2) for the configurations modeled in this research a positive linear dependency of the two parameters is evident. Hence the single value of maximum element-wise difference can be used as a proxy for studying the deviation of a certain bim-rock assemblage from isotropy. 3. Parametric study We studied two different block geometries: spherical and elliptical with aspect ratio of 2:1. For each of the blocks the following VBP was modeled: 5%, 10%, 20%, 30% and 40%. For the elliptical block random orientation was assumed at the VBPs studied. For the case of VBP = 30% we have studied the influence of the ellipsoid orientation: horizontal, 22° and 45° from horizontal. We also studied the influence of block radii at constant VBP. For all the simulations a constant block to matrix stiffness ratio Eb/Em = 10 was assumed. Constant value of Poisson's ratio, ν = 0.25, was assumed for both matrix and blocks. Fig. 3a presents the stiffness ratio of the bim-rock to matrix, Ebim/Em, as a function of VBP for both spherical and elliptical blocks with random orientation. All data points are for the isotropic case. Fig. 3b presents the maximum element-wise difference, eij, between the isotropic and anisotropic material matrices as function of VBP for spherical and elliptical blocks. The two types of blocks show a similar trend of increase in stiffens as a function of VBP, as expected, with the ellipsoids showing

slightly higher values, up to 8% difference at VBP = 0.4. Poisson's ratio (not shown) was found to decrease slightly with increase of VBP: from 0.246 to 0.22 for spherical inclusions and from 0.246 to 0.20 for elliptical inclusions. The maximum deviation from isotropy (Fig. 3b) however, exhibits a different trend: whereas for the spherical blocks the deviation never exceeds 1.6%, for the elliptical blocks the deviation is linearly increasing with VBP reaching a value of 11% at VBP of 0.4. It should be noted that the error is concentrated on the diagonal of the difference matrix and that the individual values on the diagonal are similar. Fig. 4 presents the results of composite to matrix stiffness ratio, Ebim/Em (Fig. 4a) and deviation from isotropy (Fig. 4b) for different orientations of elliptical blocks. Whereas the stiffness ratio shows lesser dependence on the orientation of the blocks, ranging from 1.7 to 2.1, the deviation from anisotropy is considerable. Maximal deviation, of 26%, is attained when the inclusions are parallel to one of the unit cell faces (α = 0°), thus creating a planar arrangement resulting in transverse isotropic fabric. With increase of inclination angle the deviation lessens reaching the value of random orientation, 10%, at α = 45°. For low VBP we have also studied the dependence of the elastic properties of the bim-rock on block size. For the case of VBP = 10% we have studied three different radii of spherical blocks, R/Luc = 0.05, 0.13 and 0.27, Luc is the edge length of the unit cell; and three different major axis for elliptical blocks, Rmax/Luc = 0.08, 0.13 and 0.27 with constant axis ratio of Rmax/Rmin = 2. It was found that for all values studied the influence of block size is negligible: Ebim/Em ranges from 1.20 to 1.24 with maximum deviation from isotropy eij ranging from 0.37% to 3.12%. For high VBP, 30%, we have also studied the dependence of the elastic properties of the bim-rock on block elastic modulus, for both R/Lc = 0.05 and 0.27. It was found that increasing the stiffness of the blocks by 3 orders of magnitude, Eb/Em from 101 to 103 had only secondary influence on the Ebim/Em ratio. For example, changing Eb/Em from 101 to 102 changed Ebim/Em from 1.76 to 2.36 with deviation from isotropy eij b 5%. Clearly values of Eb/Em significantly larger than 101 are not typical for most rocks and are only presented as reductio ad absurdum with a purpose to demonstrate its secondary effect. 4. Multi-scale analysis of the Zin conglomerate The Pleistocene Zin conglomerate formation (30°50′N 34°36′E) is a typical example of a bim-rock, comprised mainly from carbonate matrix and sub-rounded stiff carbonate blocks. The matrix is comprised from fine-grained carbonate with primary porosity of 0.3, cemented with coarse calcite with secondary porosity. The blocks (clasts) are mainly derived from the Eocene Nitzana formation (Benjamini, 1979) limestone, with a minor contribution of chert blocks.

Fig. 3. a) Stiffness ratio (Ebim/Em) and b) maximum element-wise difference as function of VBP for spherical and randomly oriented elliptical blocks. Full symbols represent extrapolation of the parametric study results to 0.4 b VBP b 0.7.

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Fig. 4. a) Stiffness ratio (Ebim/Em) and b) maximum element-wise deviation as a function of block orientation (α) for bim-rock with elliptical block VBP = 30%.

Block sizes vary considerably over 3 orders of magnitude from sub-centimeter scale (10− 3 m) to decimeter scale (10−1 m). Matrix and cement present an additional scale, partly overlapping the lower size blocks. It should be stressed out that the characteristic engineering dimension, Lc, was not determined for this study as it is case dependent. However given the invariant nature of block sizes the results presented hereafter can be extrapolated to a desired engineering scale. Limestone blocks from the Nitzana Fm. were sampled and their small strain elastic properties were determined using non-destructive ultrasonic technique to be E = 64.07 GPa, μ = 24.89 GPa and ν = 0.29. The dynamic to static stiffness ratio in carbonates ranges from 0.9–2 (Martínez-Martínez et al., 2012), increasing with porosity, fracturing and weathering. The low porosity of the Nitzana limestone, 6% (ρ = 2.55 g/cm3) and apparent lack of fracturing render stiffness ratio close to unity. Block size distribution were determined using 2D methods and transformed to 3D by application of stereological methods, i.e. Sahagian and Proussevitch (1998). A typical example of Zin conglomerate and block size distribution curve is presented in Fig. 5a and b respectively. Clearly the geometry of the blocks is different on the different scales; whereas on the finer scale (10 − 3 to 10− 2 m) the blocks are spherical on the coarser scale (10 − 1 m) the blocks are elliptical. Given the multi-scale nature of the Zin conglomerate, the shape dependency of blocks and the numerical resolution of our unit cell generator we used a sequential homogenization approach to find the effective elastic properties. We used four different scales, where at each consecutive homogenization the elastic constants of the previous homogenization are prescribed to matrix. The four scales are the following: 1) matrix, contains matrix and blocks with Rb b 0.05 cm (includes secondary porosity), VBP =

33%; 2) fine, block radius Rb b 1.5 cm, VBP = 46%; 3) medium, block radius 1.5 b Rb b 4 cm, VBP = 40%; and 4) coarse, radius Rb N 4 cm, VBP = 40%. The elastic constants of the carbonate cement where evaluated using the stiffness-porosity relation suggested by Fabricius et al. (2007) and Fabricius et al. (2010) for porosity of 0.3: E = 32 GPa and ν = 0.31. The homogenization procedure was performed twice, isotropic and anisotropic, to study the deviations from isotropy. The final isotropic values attained for the Zin conglomerate are elastic modulus E = 40.06 GPa and Poisson's ratio of ν = 0.26. The maximum deviation from assumption of isotropy is eij = 0.35%.

5. Discussion Computational homogenization is computationally expensive technique that naturally calls for an adaptive and parallelized solution strategy (Geers et al., 2010). With the constant increase of computational power and decrease of hardware costs this technique becomes advantageous over classic, typically close form solutions. In this study we have applied the computational homogenization method to study the elastic constants of bim-rocks and the deviation of bim-rocks assemblages from the assumption of isotropy. Clearly the single most important factor determining the stiffness of the bim-rock is the volumetric block proportion (VBP). The isotropy of bim-rock assemblage is determined by the shape of the blocks, as well as the VBP. Whereas for bim-rocks with spherical blocks the deviation from isotropy, eij, is less than 2%, for the elliptical blocks the deviation linearly increase with VBP, up to a value of 11% at VBP = 0.4. The inclination of the elliptical blocks adds more complexity, increasing

Fig. 5. a) Typical sample of the Zin conglomerate; b) cumulative block size distribution of Zin conglomerate.

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the deviation from isotropy. For VBP = 0.3 and horizontal inclination (α = 0°) the deviation is 26%. In general bim-rocks with elliptical blocks are stiffer then bim-rocks with spherical blocks, with difference increasing with VBP (refer to Fig. 3), up to 10% stiffer at high VBP values. Preferred orientation of the elliptical blocks further increases the stiffness. Whereas for randomly oriented blocks, at VBP of 0.3, the stiffness is higher by 7% rising to 19% for α = 0°. The VBP range modeled in this study, 0.05 to 0.4, was limited by the computational facilities employed in this research. The data in Fig. 3, can be accurately fitted (R2 = 0.99) with exponentially growing function in the form Ebim =Em ¼ expða  VBP Þ:

ð15Þ

For the data set presented here a = 2. By using this function we further extrapolate our results to higher VBP, up to 0.7, marked by full symbols in Fig. 3. It should be noted that assuming linear extrapolation of max[eij] for elliptical blocks will yield a large deviation from isotropy: up to 20% at VBP = 0.7. Using the values of Young's modulus and Poisson's ratio attained from computational homogenization we further compute the bulk and shear moduli, K and G respectively, for the bim-rock assemblages, modeled and extrapolated. The results are plotted in Fig. 6 together with Reuss average and lower Hashin–Shtrikman boundary (1963), calculated using the assumptions of Eb/Em = 10 and ν = 0.25. With respect to Hashin–Shtrikman boundary our models with spherical inclusions show good accuracy. The root mean square of residuals, RMSr for the bulk modulus (K) data set is 0.73 GPa and the normalized (to the range of data) RMSr (nRMSr) is 4.4%. The RMSr and nRMSr for shear modulus are 0.93 GPa and 8.5%, respectively. For elliptical inclusions the RMSr for bulk and shear moduli are 1.5 GPa and 2.2 GPa, respectively and the nRMSr values are 9.3% and 20.1% respectively. This clearly shows that for bim-rocks with elliptical blocks the deviation from isotropy is considerable. This is due to preferable orientation of blocks, which is not accounted by Hashin–Shtrikman, that typically increases the stiffness of the bim-rock; and due to higher sensitivity of our solution to variation of Poisson's ratio, which was to found to slightly decrease with VBP. In our research we have assumed that the matrix-block interface is displacement compatible, i.e. there is no shear displacement or separation over the interface. This assumption limits our results to lithologies such as mature conglomerates, fault breccias, gneisses and welded pyroclastic tuffs (and lapilli) among others. These results are bounded

by VBP of 0.7 and should not be extrapolated beyond, as contact–contact conditions, typical for high VBP bim-rocks, were not modeled in our simulations. In this research we have assumed that all blocks in the bim-rock are of derived from the same parent rock. While this assumption is true for various bim-rocks, it is far from being inclusive. Many conglomerates are polymictic, containing blocks from various lithologies with different properties. Sonmez et al. (2006b) showed, based on an elaborate data base, that the average modulus of elasticity for various lithologies is 35 GPa with a standard deviation of 26 GPa. Excluding some of the very soft volcanic rocks and sediments the average modulus of elasticity becomes 40 GPa with standard deviation of 25 GPa. Hence the majority of bim-rocks will have block to matrix stiffness, Eb/Em, within the same order of magnitude (b 10). High stiffness ratios, as 102, are not typical to rocks, but rather to gravely soils that are beyond the scope of this research. We have found that the stiffness ratio of block to matrix, Eb/Em, even when assumed extreme (e.g. 102) bear only a secondary effect on the elastic modulus of the bim-rock. Thus, even for very high block to matrix stiffness ratio bim-rocks our results can be used, with appropriate caution. 6. Conclusions In this research we have employed the computational homogenization approach to study the elastic moduli of block-in-matrix (bim) rocks for various combinations of VBP, block shape, block orientation and material properties. For bim-rocks comprised of rounded blocks the application of computational homogenization yields accurate results. Comparison to the close form Hashin–Shtrikman solution yields estimation errors of less than 5% for bulk modulus and less than 9% in shear modulus. Errors are increasing with VBP. For bim-rocks with non-spherical blocks the traditional approaches, such as Hashin–Shtrikman, yield a considerable estimation error: 9% and 21% for bulk and shear moduli, respectively. The error increases with both VBP and preferred orientation of blocks. Based on the findings of our research we propose a simple and robust approximation for estimation of elastic modulus of bim-rocks with spherical or elliptical blocks: Ebim/Em = exp (2 ⋅ VBP). Poisson's ratio showed little sensitivity to VBP and may be considered as constant. Bulk and shear moduli can be further computed from these values. For bulk modulus the estimate difference, relative to Hashin–Shtrikman lower bound values, will be up to 10% and 15% for spherical and elliptical blocks, respectively, increasing with VBP. For shear modulus the estimate difference, relative to Hashin–Shtrikman lower bound

Fig. 6. Bulk modulus (left) and shear modulus (right) of the bim-rocks modeled. Hashin–Shtrikman lower boundary and Reuss average are plotted for reference. VBP = volumetric block proportion. Crosses at VBP = 0.3 are for bim-rocks with elliptical blocks at different orientations.

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values, will be up to 10% and 25% for spherical and elliptical blocks, respectively, increasing with VBP. For bim-rocks with composition and structure not within the bounds of this research full material matrix can be readily attained by direct application of the computational homogenization approach and the unit cell generator. References Afifipour, M., Moarefvand, P., 2014. Mechanical behavior of bimrocks having high rock block proportion. Int. J. Rock Mech. Min. Sci. 65, 40–48. http://dx.doi.org/10.1016/j. ijrmms.2013.11.008. Benjamini, C., 1979. Facies relationships in the Avedat group (Eocene) in the Northern Negev, Israel. Isr. J. Earth Sci. 28, 47–69. Benssousan, A., Lions, J.L., Papanicoulau, G., 1978. Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam. Christensen, R.M., 1979. Mechanics of Composite Materials. Wiley-Interscience, New York. Coli, N., Berry, P., Boldini, D., 2011. In situ non-conventional shear tests for the mechanical characterisation of a bimrock. Int. J. Rock Mech. Min. Sci. 48, 95–102. http://dx.doi. org/10.1016/j.ijrmms.2010.09.012. Coli, N., Berry, P., Boldini, D., Bruno, R., 2012. The contribution of geostatistics to the characterisation of some bimrock properties. Eng. Geol. 137–138, 53–63. http://dx. doi.org/10.1016/j.enggeo.2012.03.015. Eshelby, J.D., 1957. The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems. Fabricius, I.L., Bächle, G.T., Eberli, G.P., 2010. Elastic moduli of dry and water-saturated carbonates—effect of depositional texture, porosity, and permeability. Geophysics 75, N65–N78. Fabricius, I.L., Baechle, G., Eberli, G.P., Weger, R., 2007. Estimating permeability of carbonate rocks from porosity and. Geophysics 72, E185–E191. http://dx.doi.org/10. 1190/1.2756081. Gal, E., Krivoruk, R., 2011. FRC material properties — a multi-scale approach. Comput. Concr. 8, 525–539. Gal, E., Kryvoruk, R., 2011. Meso-scale analysis of FRC using a two-step homogenization approach. Comput. Struct. 89, 921–929. http://dx.doi.org/10.1016/j. compstruc.2011.02.006. Gal, E., Ganz, A., Hadad, L., Kryvoruk, R., 2008. Development of a concrete unit cell. Int. J. Multiscale Comput. Eng. 6, 499–510. Geers, M.G.D., Kouznetsova, V.G., Brekelmans, W.A.M., 2010. Multi-scale computational homogenization: trends and challenges. J. Comput. Appl. Math. 234, 2175–2182. http://dx.doi.org/10.1016/j.cam.2009.08.077. Guedes, J., Kikuchi, N., 1990. Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods. Comput. Methods Appl. Mech. Eng. 83, 143–198. http://dx.doi.org/10.1016/0045-7825(90)90148-F. Häfner, S., Eckardt, S., Luther, T., Könke, C., 2006. Mesoscale modeling of concrete: geometry and numerics. Comput. Struct. 84, 450–461. http://dx.doi.org/10.1016/j. compstruc.2005.10.003. Hashin, Z., 1962. The elastic moduli of heterogeneous materials. J. Appl. Mech. 29, 143–150. http://dx.doi.org/10.1115/1.3636446. Hashin, Z., Shtrikman, S., 1963. A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11, 127–140. http://dx.doi. org/10.1016/0022-5096(63)90060-7. Hill, R., 1965. A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 13, 213–222. http://dx.doi.org/10.1016/0022-5096(65)90010-4. Kahraman, S., Alber, M., 2006. Estimating unconfined compressive strength and elastic modulus of a fault breccia mixture of weak blocks and strong matrix. Int. J. Rock Mech. Min. Sci. 43, 1277–1287. http://dx.doi.org/10.1016/j.ijrmms.2006.03.017. Kahraman, S., Alber, M., Fener, M., Gunaydin, O., 2015. An assessment on the indirect determination of the volumetric block proportion of Misis fault breccia

65

(Adana, Turkey). Bull. Eng. Geol. Environ. 74, 899–907. http://dx.doi.org/10. 1007/s10064-014-0666-9. Lindquist, E.S., 1994a. The mechanical properties of a physical model melange. Proceedings of 7th Congress of the International Association Engineering Geology. Balkema, Lisbon, pp. 819–826. Lindquist, E.S., 1994b. The Strength and Deformation Properties of Melange (Ph.D.) University of California, Berkeley. Lindquist, E.S., Goodman, R.E., 1994. The strength and deformation properties of a physical model melange. In: Nelson, P.P., Laubach, S.E. (Eds.), 1st North American Rock Mechanics Conference (NARMS). Balkema, Rotterdam, Austin, Texas. Martínez-Martínez, J., Benavente, D., García-del-Cura, M.A., 2012. Comparison of the static and dynamic elastic modulus in carbonate rocks. Bull. Eng. Geol. Environ. 71, 263–268. http://dx.doi.org/10.1007/s10064-011-0399-y. Medley, E., 1994. The Engineering Characterization of Melanges and Similar Block-inMatrix Rocks (bimrocks) (Ph.D.) University of Califronia, Berkeley. Medley, E.W., Lindquist, E.S., 1995. The engineering significance of the scaleindependence of some Franciscan melanges in California, USA. In: Daemen, J.K., Schultz, R.A. (Eds.), Proceedings of the 35th US Rock Mechanics Symposium. Balkema, Rotterdam, pp. 907–914. Mori, T., Tanaka, K., 1973. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 21, 571–574. http://dx.doi.org/10.1016/00016160(73)90064-3. Raymond, L.A., 1984. Classification of Melanges: In Melanges: Their Nature, Origin and Significance. Boulder, CO., Geological Society of America. Sahagian, D.L., Proussevitch, A.A., 1998. 3D particle size distributions from 2D observations: stereology for natural applications. J. Volcanol. Geotherm. Res. 84, 173–196. http://dx.doi.org/10.1016/S0377-0273(98)00043-2. Sanchez-Palencia, E., 1980. Non-Homogeneous Media and Vibration Theory: Lecture Notes in Physics. Springer, NJ. Simulia, 2008. ABAQUS ver. 6.8. Unified finite element analysis. Dassault Systemes. Sonmez, H., Gokceoglu, C., Medley, E.W., Tuncay, E., Nefeslioglu, H.A., 2006a. Estimating the uniaxial compressive strength of a volcanic bimrock. Int. J. Rock Mech. Min. Sci. 43, 554–561. http://dx.doi.org/10.1016/j.ijrmms.2005.09.014. Sonmez, H., Gokceoglu, C., Nefeslioglu, H.A., Kayabasi, A., 2006b. Estimation of rock modulus: for intact rocks with an artificial neural network and for rock masses with a new empirical equation. Int. J. Rock Mech. Min. Sci. 43, 224–235. http://dx. doi.org/10.1016/j.ijrmms.2005.06.007. Sonmez, H., Tuncay, E., Gokceoglu, C., 2004. Models to predict the uniaxial compressive strength and the modulus of elasticity for Ankara agglomerate. Int. J. Rock Mech. Min. Sci. 41, 717–729. http://dx.doi.org/10.1016/j.ijrmms. 2004.01.011. Ulusay, R., Hudson, J.A., 2006. The Complete ISRM Suggested Methods for Rock Characterization, Testing and Monitoring:1974–2006. International Society for Rock Mechanics. Volpe, R.L., Ahlgren, C.S., Goodman, R.E., 1991. Selection of engineering properties for geologically variable foundations. Proceedings of the 17th International Congress on Large Dams. ICOLD — International Committee On Large Dams, Vienna, pp. 1087–1101. Wen-Jie, X., Qiang, X., Rui-Lin, H., 2011. Study on the shear strength of soil–rock mixture by large scale direct shear test. Int. J. Rock Mech. Min. Sci. 48, 1235–1247. http://dx. doi.org/10.1016/j.ijrmms.2011.09.018. Wriggers, P., Moftah, S.O., 2006. Mesoscale models for concrete: homogenisation and damage behaviour. Finite Elem. Anal. Des. 42, 623–636. http://dx.doi.org/10.1016/j. finel.2005.11.008. Xu, W.-j., Yue, Z.-q., Hu, R.-l., 2008. Study on the mesostructure and mesomechanical characteristics of the soil–rock mixture using digital image processing based finite element method. Int. J. Rock Mech. Min. Sci. 45, 749–762. http://dx.doi.org/10.1016/j. ijrmms.2007.09.003. Zohdi, T.I., Wriggers, P., 2005. An Introduction to Computational Micromechanics. Springer, Berlin.