Computers and Geotechnics 62 (2014) 51–60

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Computers and Geotechnics

journal homepage: www.elsevier .com/locate /compgeo

Discrete element simulation of the effect of particle size on the sizeof fracture process zone in quasi-brittle materials

http://dx.doi.org/10.1016/j.compgeo.2014.07.0020266-352X/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +1 612 625 8337.E-mail address: tarok001@umn.edu (A. Tarokh).

Ali Tarokh a,⇑, Ali Fakhimi b,c

a Department of Civil Engineering, University of Minnesota, Minneapolis, MN 55455, USAb Department of Mineral Engineering, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USAc School of Civil and Environmental Engineering, Tarbiat Modares University, Tehran, Iran

a r t i c l e i n f o a b s t r a c t

Article history:Received 28 January 2014Received in revised form 8 June 2014Accepted 1 July 2014Available online 18 July 2014

Keywords:Bonded-particle model (BPM)Particle sizeSpecimen sizeFracture process zone (FPZ)Brittleness number

Experimental tests performed on quasi-brittle materials show that a process zone develops ahead of acrack tip. This zone can affect the strength and the deformation pattern of a structure. A discrete elementapproach with a softening contact bond model is utilized to simulate the development of the fractureprocess zone in the three-point bending tests. Samples with different dimensions and particle sizes aregenerated and tested. It is shown that as the material brittleness decreases, the width of the process zonebecomes more dependent on the specimen size. Furthermore, the increase in the particle size, results inincrease in the width of the process zone. A dimensional analysis together with the numerical resultsshows that the width of process zone is a linear function of particle size (radius). This finding is discussedand compared with published experimental data in the literature.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Structures composed of quasi-brittle materials such as rock orconcrete exhibit a zone of localized microcracking when thestrength of the material is approached. The estimation of the mate-rial strength has been shown to be dependent on this localizedzone and the structure itself [1,2]. Furthermore, it has been exper-imentally observed that the path of the visible fracture is withinthis localized zone of microcracking [3,4]. Therefore, in modelingthe response of a quasi-brittle structure, the characteristics of thislocalized zone or the so-called fracture process zone should be con-sidered in predicting the tensile failure. Many researchers haveshown significant interest in experimental investigation of thegoverning parameters affecting the fracture process zone. Someof the parameters that have been reported to influence the processzone are the specimen size [3,5], crack length [6], porosity [7,8],and loading rate [9]. Microstructure is also known to have a stronginfluence on the size of process zone [10–13]. Despite the qualita-tive observations of the influence of microstructure on the size ofthe process zone, a few quantitative relationships have been pro-posed. Zietlow and Labuz [3] suggested an approximate linear rela-tionship between the width of the process zone and the logarithmof grain size. Mihashi and Nomura [14] studying the process zone

in concrete by means of acoustic emission found that the length ofprocess zone is independent of the maximum aggregate size butthe width of the process zone is strongly affected. Wang et al.[15] investigated the influence of grain size on size of the processzone using laser speckle interferometry and reported that the sizeof the process zone is influenced by the ratio of notch width to theaverage grain diameter; if this ratio decreases, the size of processzone will increase. Otsuka and Date [5] implemented three dimen-sional acoustic emission and X-rays using contrast medium onconcrete and found that with the increase of maximum aggregatesize, the width and length of the process zone increase anddecrease, respectively.

To investigate the effect of particle size on the width of the pro-cess zone, the bonded-particle model (BPM), which is based on thediscrete element method (DEM), is adopted in the present work. Atensile softening contact bond model is used to mimic the develop-ment of the process zone. By varying the particle size and thematerial brittleness, different synthetic quasi-brittle samples weregenerated. The material brittleness was modified by changing theslope of the softening line (Knp). The samples were numericallytested in the three-point bending tests. It is shown that theincrease in the particle size will increase the width of the processzone. Furthermore, it was found that the width of the process zoneis a linear function of the particle size. For a fixed specimen sizeand notch length, the increase in the particle size will result in alarger process zone making the material less brittle.

52 A. Tarokh, A. Fakhimi / Computers and Geotechnics 62 (2014) 51–60

2. Numerical model

In this paper, the CA2 computer program, which is a hybrid dis-crete-finite element program for two-dimensional analysis ofgeomaterials, was used to simulate the failure process in Bereasandstone [16–18]. The bonded-particle model [19], which is oneof the discrete element method (DEM) based particle models hasbeen used extensively for simulating rock failure [20,21]. The rockis modeled as a bonded particle system in which the rigid circularparticles (representing the grains) interact through normal andshear springs to simulate elasticity. Rigid particles mean that theymaintain their shape while they can slightly overlap at the contactpoints in reaction to the applied stresses. When rigid particles areused, only translation of the centroids and the rigid body rotationof the particles need to be considered in the simulation, i.e. thereare three degrees of freedom for a two dimensional particle. Ifthe particles could deform (not rigid), then the particle deforma-tions (strains) would need to be included in the governing differen-tial equations, effectively increasing the number of degrees offreedom in the system and the computational cost. Therefore, theassumption of the rigidity of the particles facilitates the simulationinvolving large number of particles (for the 320 � 960 mm rockbeam of the simulated Berea sandstone, more than 1 million parti-cles were used and a couple of weeks of computer time with an i7processor were needed to finish the simulation). From physicalpoint of view, application of rigid particles is justified when theparticle rigidity is much greater than the binding material. There-fore, precise modeling of the particle deformation is not necessaryto obtain a good approximation of the mechanical behavior [22].Another simplification in our model is that the particle shapesare assumed to be circular. This assumption has given good resultscompared to many experimental observations [23,24]. Hogue [25]and Houlsby [26] present a comprehensive description of theissues associated with the choice of particle geometry in DEM.Finally, in the bonded particle models, it is assumed that the failure

Fig. 1. Micro-mechanical constants involved for interaction of two circularparticles.

(a)

Fn

(Tension)

Fn

(Compression)

Un

C

D

FEB

nb

A

Kn

1

Knp

1Kn

1

Kn

1

Fig. 2. Relationships in the softening contact bond model (a) normal force an

can only occur along the particle boundaries. The relative amountof different types of microcracks appears to depend on the miner-alogy, rock type and stress state [27]. Hamil and Sriruang [28]found that the cracks propagate mostly along the grain boundariesin sedimentary rocks such as sandstone. On the other hand, in thecrystalline rocks such as granite, transgranular paths were mostfrequent and sometimes dominant.

In order to withstand tensile and deviatoric stresses, the rigidcircular particles are bonded together at the contact points. Fig. 1shows the micromechanical constants in this model. The microme-chanical constants at a contact point in this model are Kn (normalstiffness), Ks (shear stiffness), nb (normal bond), sb (shear bonds),and l (friction coefficient). In addition, the radius of the particles(R) must be specified. The genesis pressure (r0) that is the confin-ing pressure during the sample preparation (determines theamount of initial small overlap between particles) can affect thematerial behavior too. The significance of these parameters hasbeen discussed in a previous study [21].

Since quasi-brittle materials such as rock and concrete usuallydisplay tension softening during fracturing [29,30], a softeningcontact bond feature was implemented in the numerical model.In this softening model, the normal bond at a contact point isassumed to reduce linearly after the peak tensile contact load(Fig. 2a). Therefore, a new microscopic constant, the slope in thepost peak region of the normal force-normal displacementbetween two particles in contact (Knp), is introduced in the model.As shown in Fig. 2b, no modification in the shear force-relativeshear displacement of a contact is assumed in this simple model.Softening in shear is only relevant for loading under significantmean stress (more than 1/3 of uniaxial compressive strength). Dis-tinct shear failure plane forms at moderate compression in whichthe mean stress, p = (r1 + r2 + r3)/3 is in the following range rc/3 < p < rc [31]. This is not the case in the three point bending testsconducted in this study; no actual shear cracks are developed inour tests. The loading and unloading paths for both normal andshear contact forces are shown with arrows in Fig. 2.

After sample preparation, the numerical model was calibratedto obtain the mechanical properties of Berea sandstone. The proce-dures for sample preparation and calibration have been describedelsewhere [21]. The grains in this particular sandstone range from0.1 to 0.8 mm. The mechanical properties of the Berea sandstoneare E (elastic modulus) = 14 GPa, m (Poisson’s ratio) = 0.32, rc (uni-axial compressive strength) = 55–65 MPa, and rN (bending tensilestrength) = 8.6 MPa for an 80 � 240 � 30 (height � span � thick-ness) mm rock beam [3]. After calibration of the numerical model,uniaxial compressive test on an 40 � 80 mm (width � length)

Fs (Shear)

Us

Q R

Psb

O

µFn

(b)

d normal displacement and (b) shear force and shear displacement [18].

0

10

20

30

40

50

60

70

0.0 0.2 0.4 0.6 0.8 1.0

Axi

al st

ress

(MPa

)

Axial strain (%)

Numerical resultExperimental result

σc = 55 MPa

σc = 65 MPa

Fig. 3. Stress–strain curves of the numerical and experimental Berea sandstonespecimens in the uniaxial compression tests. The observed maximum andminimum values of the compressive strengths of the physical specimens areshown in the figure.

Table 1Micro-mechanical and macro-mechanical properties for samples with differentparticle sizes.

Properties R = 0.3 (mm) R = 0.6 (mm) R = 1.2 (mm)

Micro-mechanicalKn (GPa) 22 22 22Ks (GPa) 5.5 5.5 5.5Knp (GPa) 1.83 1.83 1.83nb (N/m) 2800 5600 11,200sb (N/m) 12,300 24,600 49,200l 0.5 0.5 0.5r0/Kn 0.1 0.1 0.1

Macro-mechanicalE (GPa) 13.3 13.6 13.2m 0.19 0.18 0.18rc (MPa) 60.5 63.6 71.0rt (MPa) 6.1 6.1 6.2

Kn is the normal stiffness and Ks is the shear stiffness. Knp is the slope of the soft-ening line.nb is the normal bond and sb is the shear bond.l is the friction coefficient and ro is the genesis pressure in sample preparation.E is the elastic modulus whereas m is the Possion’s ratio.rc and rt are the uniaxial compressive and tensile strengths of the material,respectively.

S

D

P

a0

Fig. 4. Three-point bending test set-up in discrete element simulation.

A. Tarokh, A. Fakhimi / Computers and Geotechnics 62 (2014) 51–60 53

rectangular sample and three-point bending test on an80 � 240 mm (height � span) beam were performed to verify theaccuracy of the numerical model. We were able to calibrate themodel for the width of the process zone as well. The ratio of Kn/Knp = 10–12 will reproduce the process zone observed in the labo-ratory testing of Berea sandstone [32]. The following mechanicalproperties were obtained: E = 13.3 GPa, m = 0.19, rc = 60.5 MPa,and rN = 8.7 MPa (three-point bending tensile strength) that withthe exception of the Poisson’s ratio are in close agreement withthe mechanical properties of Berea sandstone. The differencebetween the physical and simulated values of Poisson’s ratio isexpected to have a small impact on the numerical results; the Pois-son’s effect on stress distribution should not be significant whenlateral deformation is not constrained which is the case in thethree-point bending tests studied in this paper.

Fig. 3 demonstrates the stress–strain curves of the numericaland experimental Berea sandstones in the uniaxial compressivetests. The two curves are in good agreement if the initial deforma-tion of the physical specimen is ignored. The early portion of thecurve in the physical test is known to be caused by the closure ofthe space between the specimen and the loading platen (alsoreferred to as machine seating) as well as closure of existing micro-cracks in the specimen. In the numerical model such phenomenaare absent and therefore, the initial curvature of the stress–straincurve cannot be captured. Note that as suggested by Fig. 3, thereis a variation as great as 18% in the uniaxial compressive strengthof the physical specimens (rc = 55–65 MPa) and that the numericaltest result lies within the physical range.

The micromechanical properties that were obtained throughsample calibration are reported in Table 1. The radii of circular par-ticles (R) were assumed to have a uniform random distributionranging from 0.27 to 0.33 mm with an average radius of Rave =0.3 mm. In order to study the effect of particle size on the processzone, two other particle radii of 0.6 (with a radius range of 0.54–0.66 mm) and 1.2 mm (with a radius range of 1.08–1.32 mm) wereused. To obtain relatively similar macro-properties for the threesynthetic materials with three different particle sizes, the normaland shear bonds (nb and sb) for materials with larger particle radius(R) need to be increased in proportion to the average radius (Rave)of the particles, but the normal and shear stiffnesses (Kn and Ks) donot need any modification (see Table 1). The normal and shearstiffnesses mostly affect the elastic modulus (E) and Poisson’s ratio(m) whereas normal and shear bonds have great impact on the uni-axial compressive and tensile strengths of the material [19,21]. Theincrease in normal and shear bonds in proportion to the particleradius causes no change in normal and shear contact strengths.Normal contact strength (rn = nb/2R) and shear contact strength(rs = sb/2R) should remain constant in the three different simulatedrocks. In addition to the micro-mechanical parameters, the corre-sponding macro-mechanical properties obtained for differentsynthetic materials are reported in Table 1.

Beams with different sizes were generated from these threesynthetic materials with different particle sizes. Numericalthree-point bending tests (Fig. 4) were conducted on five differentbeam sizes of 20 � 60, 40 � 120, 80 � 240, 160 � 480 and320 � 960 mm. The first number in each beam size shows thebeam height and the second number is its span. A small appliedvertical velocity at the top center of each beam (2.5 � 10�10 meterper numerical cycle) and a damping force proportional to theunbalanced force or moment of each particle was used in orderto achieve a quasi-static solution [33]. The ratio of notch lengthto beam height (a0/D = 0.375) was assumed to be constant for allthree-point bending tests. The slope of the softening line (Knp)was modified to obtain different quasi-brittle synthetic materials.Five different synthetic materials ranging from perfectly brittle(Kn/Knp = 0) to less brittle (Kn/Knp = 100) were used in this analysis.

Fig. 5 shows the different Kn/Knp values used on the normal force-normal displacement graph. Other micromechanical constantswere left unchanged for each particle size.

Inspection of the rc values in Table 1 suggests a 15% increase inthe uniaxial compressive strength from R = 0.3 mm to R = 1.2 mm.

Kn/K

np= 0 ---Perfectly Brittle

Kn/K

np= 10

Kn/K

np= 20 Quasi-brittle

Kn/K

np= 50

Kn/K

np= 100

Kn/K

np= ---Perfectly Plastic

Knp

=

Kn/K

np= 0

Perfectly Brittle

Knp

= 0

Kn/K

np=

Perfectly Plastic

Fn (Tension)

Fn (Compression)

Un

C

F B

nb

A

Knp

1 Kn

1

Kn

1

Fig. 5. Demonstration of the effect of Kn/Knp values on the material behavior. Byincreasing Kn/Knp value, we move from perfectly brittle to perfectly plasticmaterials.

54 A. Tarokh, A. Fakhimi / Computers and Geotechnics 62 (2014) 51–60

This difference is not considered significant as even in the labora-tory testing, the compressive strength of one specific type of rockfrom the same block could vary noticeably. The Berea sandstonestrength used in our calibration varied by 18% (rc = 55–65 MPa).Serena sandstone has been reported to have a rc = 100–120 MPawhich shows a 20% difference in its strength [34].

Note that the bending tensile strength of the simulated speci-men (rN = 8.7 MPa) is greater than that in uniaxial testing(rt = 6.1 MPa in Table 1). The higher value of the bending tensilestrength (modulus of rupture) is consistent with the physicalobservation [35].

3. Numerical results

3.1. Uniaxial compressive and tensile tests

Uniaxial compressive strength (rc) and tensile strength (rt) arethe most common parameters used to describe the strength ofrocks. Several numerical uniaxial compressive and tensile testswith different particle size and various material ductilities (i.e. dif-ferent Kn/Knp values) were conducted. The size of the rectangularspecimen in the uniaxial tests was 40 � 80 (width � length) mm.Fig. 6a shows a numerical specimen under uniaxial compressionloading. The upper and lower finite element grids are used as theloading platens. The interface between the finite element gridand the particles were modeled by using normal and shear springs(Kn = Ks = 100 GPa). The friction coefficient of this interface wasassumed to be zero. Details of the mathematical description of

Fig. 6. Demonstration of the numerical tests for (a) uniaxial compressive test and (b) unitop represent the fixed displacement in x and y directions. The two finite element grids

the interface between the grid and the particles have been dis-cussed in [17]. The lower platen moves with a constant quasi-staticupward velocity of 0.2 � 10�8 meter per cycle whereas the top ofthe upper platen is fixed in the vertical direction. The axial stress,axial and lateral deformation are recorded during loading. The lat-eral deformation is simply calculated by means of two finite ele-ment grids that are glued to the lateral sides of the numericalspecimen. The use of these finite element grids facilitates the mea-surement of lateral displacement compared to calculating the indi-vidual ball displacements from the model. In order to avoid anykind of unreal resistance of the material, a low elastic modulus isused for the two finite element grids glued to the lateral sides ofthe specimen. The lateral deformation is used to calculate the Pois-son’s ratio. Fig. 6b illustrates a numerical specimen under uniaxialtensile loading. Similar to the uniaxial compression loading, theupper and lower finite element grids are used as the loading plat-ens. The top of the upper platen is fixed in the vertical directionwhile the lower platen moves with a constant downward quasi-static velocity of 0.2 � 10�8 meter per cycle. The interfacesbetween the circular particles in contact with the platens arebonded so that no cracks could develop along these interfaces.The axial stress and axial deformation were recorded during theloading.

Fig. 7 depicts the stress–strain curves for the synthetic materi-als with different material ductilities (different Kn/Knp values) forthe particle size (Rave) of 0.3 mm. The values for the compressiveand tensile strengths are reported in Table 2.

From these results it is observed that the Kn/Knp ratio has littleinfluence on the compressive strength of the simulated materials(for different R values) whereas it has affected the tensile strength;in the uniaxial compression test, the failure mechanism is gov-erned by a combination of the effects of shear and tensile crackswhereas in the uniaxial tension test, majority of the induced dam-ages are tensile cracks. This means that changing the tension soft-ening parameters (i.e. the ratio of Kn/Knp) will have a strongerimpact on the tensile strength compared to the compressivestrength of the simulated material. Therefore, the value of Kn/Knp

has an impact on the rc/rt (compressive to tensile strength) ratio;as Kn/Knp increases, rc/rt decreases for all particle sizes. This isexpected because with increase in Kn/Knp, the simulated materialbecomes less brittle. This is consistent with the general behaviorof materials. More brittle materials such as rock typically showhigh ratio of uniaxial compressive strength to tensile strength.On the other hand, for ductile metals, the ratio of compressive totensile strength is normally close to one.

axial tensile test. The arrows represent the loading direction while the crosses at theused to measure the lateral displacement are shown in Fig. 6a.

0

10

20

30

40

50

60

70

80

0.0 0.2 0.4 0.6 0.8

Axi

al st

ress

(M

Pa)

Axial strain (%)

BrittleKn/Knp=10Kn/Knp=20Kn/Knp=50Kn/Knp=100

0

2

4

6

8

10

12

0.0 0.1 0.2 0.3 0.4 0.5

Axi

al st

ress

(M

Pa)

Axial strain (%)

BrittleKn/Knp=10Kn/Knp=20Kn/Knp=50Kn/Knp=100

(a) (b) Fig. 7. Axial stress vs. axial strain for numerical specimens with Kn/Knp = 0 (perfectly brittle), 10, 20, 50, 100 and R = 0.3 mm. (a) Uniaxial compressive test. (b) Uniaxial tensiletest.

Table 2Uniaxial compressive and tensile strengths of the numerical specimens with different particle sizes.

Kn/Knp rc (MPa) rt (MPa) rc/rt

R = 0.3 (mm) R = 0.6 (mm) R = 1.2 (mm) R = 0.3 (mm) R = 0.6 (mm) R = 1.2 (mm) R = 0.3 (mm) R = 0.6 (mm) R = 1.2 (mm)

0 57.9 57.5 63.1 4.2 4.9 5.4 13.8 11.7 11.710 63.1 64.2 70.6 5.6 6.1 6.1 11.3 10.5 11.620 64.8 64.4 71.2 7.0 6.5 6.8 9.3 9.9 10.550 66.3 74.5 68.5 8.6 8.1 8.7 7.7 9.2 7.9

100 73.1 82.4 73.0 10.7 9.8 10.6 6.8 8.4 6.9

Fig. 8. The width of the fracture process zone at the crack tip for the specimen sizeof 40 � 120 mm with R = 0.6 mm, a0/D = 0.375, and Kn/Knp = 20 at (a) the peak load(b) 80% of the peak load in the post peak.

A. Tarokh, A. Fakhimi / Computers and Geotechnics 62 (2014) 51–60 55

3.2. Fracture process zone width

The main goal of this work was to study the effect of particle sizeon the width of the fracture process zone in the vicinity of the notchtip. The process zone in the numerical model is defined with contactpoints between circular particles that are in the post peak regime(e.g. point D in Fig. 2a). These damaged contact points for a speci-men size of 40 � 120 mm and Kn/Knp = 20 with particle size ofRave = 0.6 mm at two different loading stages are shown in Fig. 8.The damaged contacts are shown in blue while the actual sharpcrack inside the process zone has been shown in red. It is interestingto note that the width of the process zone at the tip of the propagat-ing crack remains constant. Furthermore, the width of the processzone at the crack tip is almost identical to the width of the damagezone surrounding the crack. This is expected as with the extensionof the main crack, unloading of the damaged zone around the crackprevails. This prevents further widening of the damaged zone.

Fig. 9 illustrates the width of the process zone vs. the particlesize for the synthetic materials with different beam sizes. This fig-ure suggests that:

(i) For a fixed Kn/Knp value and a fixed beam size (D), the widthof the process zone (W) increases when the particle size (R)increases.

(ii) For a fixed Kn/Knp value and a fixed particle size (R), thewidth of the process zone (W) increases as the beam sizes(D) increases.

In a previous work, Fakhimi and Tarokh [36] suggested the fol-lowing equation for the width of fracture process zone (W):

W ¼ W1bð1þ bÞ ð1Þ

in which b = D/D0 is the brittleness number [30] and W1 is thewidth of the process zone for very large specimens. The structuralresponse is not physically similar when varying the size scale of abody. As the size of a structure increases, it has been documentedexperimentally that the failure mode changes from the plastic col-lapse to the brittle failure. To characterize the brittleness of thestructural response quantitatively, various definitions of the so-called brittleness numbers have been proposed [37–39]. The D inthe brittleness number (b) defined by Bazant [39] stands for theeffective structural dimension (e.g. the specimen height) and D0 isa constant with the dimension of length. D0 depends on the fractureproperties of the material and on the geometry (shape) of the struc-ture, but not on the structure size [30]. Eq. (1) suggests that as thebrittleness of the material increases (i.e. lower Kn/Knp), the width of

0

4

8

12

16

0.0 0.3 0.6 0.9 1.2 1.5

W(m

m)

R (mm)

D=20 mmD=40 mmD=80 mmD=160 mmD=320 mm

0

20

40

60

80

0.0 0.3 0.6 0.9 1.2 1.5

W(m

m)

R (mm)

D=20 mmD=40 mmD=80 mmD=160 mmD=320 mm

(a) (b)Fig. 9. Fracture process zone width vs. particle radius for different beam sizes for (a) Kn/Knp = 20 and (b) Kn/Knp = 50.

0

20

40

60

80

100

120

0 40 80 120 160 200 240 280 320 360

W(m

m)

D (mm)

Kn/Knp=20Kn/Knp=50Kn/Knp=100Fitting Kn/Knp=20Fitting Kn/Knp=50Fitting Kn/Knp=100y = 3.6313x + 0.1047

R² = 0.9914

y = 1.9176x + 0.0160R² = 0.9989

y = 1.3382x + 0.0069R² = 0.9985

0.0

0.1

0.2

0.3

0.00 0.01 0.02 0.03 0.04 0.05 0.06

W-1

(mm

-1)

D-1 (mm-1)

Kn/Knp=20Kn/Knp=50Kn/Knp=100

(a) (b) Fig. 10. The relationships between (a) W�1 and D�1, (b) W and D for R = 0.6 mm.

Table 3The data from linear regression analysis of Eq. (1) for different particle sizes.

R (mm) Kn/Knp Slope y-Intercept (mm�1) W1 (mm) D0 (mm)

0.3 20 3.3947 0.2181 4.6 15.650 2.1531 0.0197 50.8 109.3

100 1.009 0.0099 101.0 141.3

0.6 20 3.6313 0.1047 9.6 34.750 1.9176 0.0160 62.5 119.9

100 1.3382 0.0069 144.9 193.9

1.2 20 2.8500 0.0615 16.3 46.350 1.7182 0.0143 69.9 120.2

100 1.3487 0.0050 200.0 269.7

56 A. Tarokh, A. Fakhimi / Computers and Geotechnics 62 (2014) 51–60

the process zone becomes less dependent on the specimen size; fora high brittleness number, the width of the process zone can be con-sidered an intrinsic material property which will not change byvarying the specimen size. This finding is consistent with someexperimental evidences [32]. In order to find D0 and W1, Eq. (1)can be written in the linear form by

1W¼ 1

W1þ D0

W1

1D

ð2Þ

The variations of W�1 vs. D�1 for R = 0.6 mm and different Kn/Knp

values are shown in Fig. 10a. The slopes (D0/W1) and y-intercepts(1/W1) obtained from these plots were used to calculate the D0

and W1 reported in Table 3. The linear trend observed in the var-iation of W�1 vs. D�1 confirms the ability of Eq. (1) to predict thevariation of the width of the process zone with the specimen size.In Fig. 10b, the width of the process zone (W) as a function of spec-imen height (D) is shown. The fitting curves predicted by Eq. (1) arein close agreement with the numerical data; Eq. (1) can closelymodel the numerical data. This equation has been shown to closelymodel the experimental data as well [32].

Bazant and Planas [30] suggested two linear and a non-linearregression method in finding the D0 values. Depending on themethod used, different D0 values are obtained. The use of the widthof process zone is yet another approach in finding the D0 values.The D0 values from the table suggest that:

(i) For a fixed Kn/Knp value, D0 increases when the particle size (R)increases. Therefore, for a specific beam size (i.e. fixed D), thebrittleness number (b = D/D0) reduces as the particle sizeincreases. When the brittleness number decreases, less brittlebehavior should be expected [30]. This implies that as the par-ticle size increases, the behavior becomes less brittle.

(ii) For fixed particle (R) and beam sizes (D), with increase in Kn/Knp, D0 increases which results in the decrease in the brittle-ness number. This is also expected because when Kn/Knp

increases the material will have a larger process zone. Thelarger the process zone, the less brittle the materialbehavior.

0

20

40

60

80

100

120

0.0 0.1 0.2

Loa

d (k

N)

Displacement (mm)

Rock Beam 1Rock Beam 2

Fig. 12. Load vs. displacement for rock beam 1 (rc = 64.4 MPa) and rock beam 2(rc = 130 MPa).

A. Tarokh, A. Fakhimi / Computers and Geotechnics 62 (2014) 51–60 57

3.3. Effect of rock strength on size of fracture process zone

The width of the process zone is a function of Kn/Knp, specimensize [36] and particle size. By increasing the compressive strengthof the simulated rock (i.e. only increasing the normal and shearbonds nb and sb in the numerical model) and leaving the rest ofthe micro-mechanical parameters, specimen size (D) and particlesize (R) constant, the width of the process zone will not change.For example if we consider two specimens of 80 � 240 mm withKn/Knp = 20 and Rave = 0.6 mm, but with different uniaxial compres-sive strengths (for rock beam 1, rc = 64.4 MPa and for rock beam 2,rc = 130 MPa as nb and sb are doubled in specimen 2), the width ofthe process zone will be about 6 mm for both cases (Fig. 11). There-fore, the compressive strength does not have an effect on the pro-cess zone dimensions. This is also true for the applieddisplacement. In this example, the amount of displacement inthe second rock is higher than the displacement in the first rockwhile the size of the process zone is identical in both cases.Fig. 12 shows the load–displacement curves for these two differentcases.

3.4. Relationship between fracture parameters and particle size

In this section, dimensional analysis [40,41] is used to obtainrelationships between the fracture parameters and the particleradius (R). The fracture toughness of the synthetic material isassumed to be a function of the following parameters: Kn, Ks, nb,sb, r0, Knp, R, and l, i.e.

KIC ¼ f1ðKn;Ks;nb; sb;r0;Knp;R;lÞ ð3Þ

The effect of sb is ignored. This is due to the fact that only mode Iloading (opening mode) is considered in this study. Therefore, onlyseven parameters influencing the fracture toughness will remain.Considering the two independent dimensions, i.e. length and forcethat are used to describe the parameters in Eq. (3), five (7 � 2 = 5)dimensionless parameters will be required to fully describe therelationship between the fracture toughness and the microme-chanical parameters. These five dimensionless parameters areintroduced in the following:

Kn

Ks;

Kn

Knp;

nb

RKn;r0

Kn;l

The four parameters nb/RKn, Kn/Ks, r0/Kn, and l are fixed in ourstudy. Therefore, the following dimensionless equation could bewritten:

KIC

rn

ffiffiffiRp ¼ f2

Kn

Knp

� �ð4Þ

Fig. 11. Process zone for a 80 � 240 mm notched beam at 80% of peak load in thepost peak with Kn/Knp = 20 and Rave = 0.6 mm for a synthetic rock with (a)rc = 64.4 MPa and (b) rc = 130 MPa.

in which rn = nb/2R is the normal contact strength. The data pointsfrom numerical analysis are shown in Fig. 13 which indicates thatthe dimensionless fracture toughness has a linear relationship withthe Kn/Knp parameter for the range of Kn/Knp values used in thisstudy. The apparent fracture toughness (KICA) for a single-edgecracked specimen under three-point bending for arbitrary depthto span ratio, could be calculated by

KICA ¼ rN

ffiffiffiffiDp ffiffiffi

ap

ð1þ 2aÞð1� aÞ1:5PS=DðaÞ ð5Þ

in which rN is the nominal tensile strength, D is the specimendepth, S is the specimen span, a is the ratio of notch length to spec-imen depth (a0/D), and PS/D(a) is a shape factor. The expression forPS/D(a) can be determined by [42]

PS=DðaÞ ¼ P1ðaÞ þ 4DS

� �ðP4ðaÞ � P1ðaÞÞ ð6Þ

in which the expressions for P4(a) and P1(a) are:

P4ðaÞ ¼ 1:900� a½�0:089þ 0:603ð1� aÞ � 0:441ð1� aÞ2

þ 1:223ð1� aÞ3� ð7Þ

P1ðaÞ ¼ 1:989� að1� aÞ½0:448� 0:458ð1� aÞ

þ 1:226ð1� aÞ2� ð8Þ

y = 0.45x + 5.09R² = 0.99

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

K IC

/( σn.R

0.5 )

Kn /Knp

All particle sizesLinear (All particle sizes)

Fig. 13. The relationship between dimensionless fracture toughness and Kn/Knp.

y = 12.72x + 1.23R² = 0.99

y = 20.02x + 47.05R² = 0.90

y = 107.39x + 73.47R² = 0.98

0

50

100

150

200

250

0.0 0.3 0.6 0.9 1.2 1.5

W∞

(mm

)

R (mm)

Kn/Knp=20Kn/Knp=50Kn/Knp=100Linear (Kn/Knp=20)Linear (Kn/Knp=50)Linear (Kn/Knp=100)

Fig. 15. Variation of W1 vs. particle radius (R) for different synthetic materials(different Kn/Knp values).

58 A. Tarokh, A. Fakhimi / Computers and Geotechnics 62 (2014) 51–60

To obtain the fracture toughness values in Fig. 13 using theapparent fracture toughness KICA, (1/KICA)2 is plotted vs. 1/D.According to Eq. (5), a linear relationship should be expected. Fromthis plot, the KIC value corresponding to very large specimens (1/Dequal to zero) is obtained [36].

The dependence of the fracture toughness to the square root ofparticle radius in Eq. (4) is consistent with that reported by Pot-yondy and Cundall [43] and Huang et al. [41] for an ideally brittlematerial (Kn/Knp = 0).

A dimensional analysis similar to that used for the fracturetoughness for the direct tensile strength of the synthetic materialshows that:

rt

rn¼ f3

Kn

Knp

� �ð9Þ

Fig. 14 suggests a linear relationship between dimensionlesstensile strength with the Kn/Knp parameter for the range of Kn/Knp

values used in this study.In Fig. 15, the variation of W1 vs. the radius of particles for dif-

ferent synthetic materials is shown. The values of W1 are obtainedfrom Table 3. Notice that a linear relationship between W1 and theparticle size (R) exists for a given value of Kn/Knp; by increasing theradius of particles in the material, W1 increases which results in aless brittle behavior of the material as for an ideal brittle material,the thickness of the process zone is zero.

It has been claimed that the size of the process zone in the modeI loading for a very large structure is proportional to the character-istic size [30], i.e.

W1 ¼ nlch ð10Þ

in which n is the coefficient of proportionality. The characteristicsize is defined by

lch ¼KIC

rt

� �2

ð11Þ

If Eqs. (4) and (9) are substituted in the above equations, weobtain

W1 ¼ nlch ¼ nKIC

rt

� �2

¼ nrnf2

ffiffiffiRp

rnf3

!2

¼ Rnf2

KnKnp

� �f3

KnKnp

� �0@

1A

2

ð12Þ

Eq. (12) suggests a linear relationship between the width of theprocess zone and the size (radius) of particles. It is important tonote that the bond strength between particles (rn) that is themajor factor in controlling the material tensile or compressivestrength is cancelled in Eq. (12). This indicates that the width of

y = 0.01x + 1.15R² = 0.94

0

1

2

3

0 10 20 30 40 50 60 70 80 90 100

σ t/σ

n

Kn /Knp

All particle sizesLinear (All particle sizes)

Fig. 14. The relationship between dimensionless tensile strength and Kn/Knp.

process zone is independent of the material strength. This observa-tion is consistent with the results of numerical simulation reportedin Section 3.3 of the paper.

4. Discussion of the results

Preceding discussions demonstrate that the width of the pro-cess zone increases as the radius of particles increases and that thisincrease is a linear function of the particle size. To display physicalsupports for findings of this study, some experimental observa-tions regarding the effect of grain size in rock or maximum aggre-gate size in concrete have been reviewed. The work of Otsuka andDate [5] was concentrated on concrete. They performed tensiletests on different sample sizes. X-ray and three-dimensionalAcoustic Emission (AE) techniques were used to investigate theprocess zone around the notch tip. They calculated the energy ofindividual events from the square of the amplitude of the wavemultiplied by the incremental duration time. They considered theregion associated with over 95% of the total AE energy to be relateddirectly to the fracture of concrete and called this area the fractureprocess zone (FPZ). In this area, more densely distributed AE eventswere observed. The area that contains 70% of the total AE energy isreferred to the fracture core zone (FCZ). The shape and the size ofFCZ are identical to the microcrack zone obtained by X-ray inspec-tions. It is important to note here that the same trend is observedfor both the FPZ and FCZ, i.e. if the size of one of them increases, thesize of the other one will increase as well. Fig. 16 demonstrates theFPZ and FCZ in the work of Otsuka and Date [5].

Otsuka and Date [5] used different concrete specimens with dif-ferent maximum aggregate sizes. All specimens had a constantuniaxial compressive strength of 20 MPa. They observed that whenthe specimen size is identical, the width of FPZ and FCZ bothincrease with the increase of the maximum aggregate size. Theydemonstrated a linear relationship between the width of FCZ andthe maximum aggregate size (Fig. 17).

Apart from some data scatter, Fig. 17 displays an approximatelinear relationship between the width of FCZ and the aggregate sizewhich supports the numerical simulation finding in this paper. It isimportant to realize that in the work of these authors, the samematerial with a constant strength was used and only the aggregatesizes were modified. This is consistent with the assumptions madein this study as the tensile and compressive strengths of the syn-thetic materials used in the numerical modeling were almost con-stant (see Table 2). It should be emphasized that typicallyaggregates do not fail and their action is to arrest the tensile crackswhich are developed through the matrix (i.e. cement). This fact is

Fig. 16. Fracture process zone and fracture core zone (after Otsuka and Date [5]).

A. Tarokh, A. Fakhimi / Computers and Geotechnics 62 (2014) 51–60 59

also consistent with the assumption made in the model that thecracks develop along the contact interfaces and the particlesremain intact. The scatter of the data in Fig. 17 suggests that thewidth of the process zone can vary even if tests are performedon the apparently similar quasi-brittle materials with the sameparticle size distribution. This can be caused by the random distri-bution of the location of the particles and initial microscopicdefects. We expect that as the intensity of initial defects and therock porosity are increased, a wider process zone to be obtained.In fact it has been recently shown through some physical tests thatgreater porosity can result in a greater process zone within a quasi-brittle material [44]. This issue needs to be addressed in futurenumerical studies.

Zietlow and Labuz [3] studied the fracture process zone in dif-ferent rock types using acoustic emission measurements. Theysuggested a linear relation between the normalized process zonewidth (x = W/D) and the logarithm of the normalized averagegrain size (d = dave/D); a non-linear relationship between the grainsize and the size of the process zone was suggested. What shouldbe noted here from the work of Zietlow and Labuz [3] is that theirproposed equation was based on the test results on four differentrock types which in general can have different bonding materialand initial defects. The interfacial material between the particlesand the initial defects have proven to have strong influences onthe properties of composite materials like concrete and rock.Therefore, to study the effect of grain size or aggregate size on frac-ture process zone dimensions, one particular rock or concrete butwith different grain size or aggregate size should be studied; theeffect of different contact bond material and initial micro-crackshas possibly caused a non-linear relationship between the width

0

20

40

60

80

0 10 20 30

Wid

th o

f FC

Z (m

m)

dmax (mm)

Concrete, after Otsuka & Date (2000)

Fig. 17. Relationship between maximum aggregate size and the width of the FCZ(Otsuka and Date [5]).

of the process zone and the grain size. This can be the reason forinconsistency of the current results with those reported by theseauthors.

Brooks et al. [13] investigated the role of grain size in fractureprocess zone development of two different marbles with differentgrain sizes using the nano-indentation technique. Although theextent of the fracture process zone defined by these authors wasbased on the observation of the reduction of nanomechanical prop-erties, their work provided a support for the increase in the size ofthe fracture process zone (distance of nanomechanical propertyreduction) with the grain size. The uniform mineralogy of bothmarbles (mainly calcite) made the grain size the sole controllingparameter in their study.

5. Conclusions

A two-dimensional discrete element model with tension soften-ing was used to study the effect of particle size on the width of thefracture process zone in quasi-brittle materials like rock. The rockwas idealized as an assembly of unbreakable grains that interactthrough the contact points. The contact points can break if theapplied normal or shear force exceeds the normal or shear bonds(contact strengths). It was found that the width of the process zoneis in general a function of both the specimen and particle sizes. Itwas also shown that the width of the process zone is a linear func-tion of the radius of particles. The discrepancy in the literatureregarding the effect of particle size on the size of fracture processzone in quasi-brittle materials could be a result of conductingexperimental tests on materials with different bonding character-istics and initial defects. With the numerical tests such as themethod implemented in this study, all the relevant parameters infracture of a material can be held unchanged while the particle sizeis modified. This allows the effect of this single parameter to bestudied in details. The analysis of the numerical results indicatesthat the discrete element method with a tension softening contactbond model is able to mimic the effect of particle size on the widthof the process zone in quasi-brittle materials and that it can beused as a reliable tool to study crack initiation and propagationin mode I (opening mode) loading condition.

Acknowledgment

The authors acknowledge the valuable discussions and com-ments provided by professors Otto D.L Strack and Joseph F. Labuzat the University of Minnesota.

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