Micromechanical model for the rate dependence of the fracturetoughness anisotropy of Barre granite

F. Dai a, K. Xia b,c,n, M.H.B. Nasseri c

a State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resources and Hydropower, Sichuan University, Chengdu,Sichuan 610065, Chinab State Key Laboratory of Hydraulic Engineering Simulation and Safety, School of Civil Engineering, Tianjin University, Tianjin 300072, Chinac Department of Civil Engineering, University of Toronto, ON, Canada M5S 1A4

a r t i c l e i n f o

Article history:Received 3 July 2012Received in revised form19 June 2013Accepted 4 August 2013Available online 31 August 2013

Keywords:Barre graniteMicromechanics modelFracture toughness anisotropyNotched semi-circular bend (NSCB)Crack–microcrack interaction

a b s t r a c t

Laboratory measurements of mode-I fracture toughness of Barre granite under a wide range of loadingrates were carried out with an MTS machine and a split Hopkinson pressure bar (SHPB) system using thenotched semi-circular bend (NSCB) specimen. The fracture toughness anisotropy was found to decreasewith the increase of the loading rate. A micromechanics model is utilized in this work to understand thisexperimental observation, invoking crack–microcrack interactions. Two micromechanics models areconstructed based on the microstructural investigation of Barre granite samples using the thin-sectionmethod. In both models, the rock material is assumed to be homogenous and isotropic. The main crack(i.e., the pre-crack in the NSCB specimen) and the closest microcracks are included in the numericalanalysis. Numerical results show that stress shielding occurs in the model where the two microcracksform an acute angel with the main crack and the nominal fracture toughness is bigger than the intrinsicone, while stress amplification occurs in the model where the microcrack is collinear to the main crackand the nominal fracture toughness is smaller than the intrinsic one. Assuming that the intrinsic fracturetoughness of the rock material has the usual loading rate dependency, we are able to reproduce thedecreasing trend of the fracture toughness anisotropy as observed from experiments.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Due to long-term tectonic loadings, granites are abundant withmicrocracks and pores, and these microstructural discontinuitiesare believed to be responsible for the apparent mechanicalproperties anisotropy including compressive strength [1], tensilestrength [2] and fracture toughness [3,4]. Barre granite, as astandard rock suite by the U.S. Bureau of Mines, has been wellknown for its anisotropy in mechanical properties [5] and themicrostructure of it has been thoroughly investigated [3,4].

The mode-I fracture toughness (KIC) of rocks is a criticalparameter in rock mechanics and rock engineering applicationsinvolving fracture. It is considered to be an intrinsic materialproperty of rocks to resist crack initiation and propagation andthus has been widely investigated in the rock community. Inter-national Society of Rock Mechanics (ISRM) proposed short rod (SR)and chevron bending (CB) method in 1988 [6] and crackedchevron notched Brazilian disc (CCNBD) method in 1995 [7] to

standardize the methods for static fracture toughness measure-ments. For the dynamic measurement, ISRM recently adoptednotched semi-circular bend (NSCB) method for characterizingthe dynamic mode-I fracture toughness of rocks [8]. Because mostrocks are anisotropic due to tectonic stresses, a thorough researchof mode-I fracture toughness on its microstructure related aniso-tropy is necessary.

In our previous work [9], laboratory measurements of mode-Ifracture toughness of Barre granite under a wide range of loadingrates were carried out with an MTS machine and a split Hopkinsonpressure bar (SHPB) system using the NSCB specimen. To quantifythe anisotropy, the NSCB fracture samples were fabricated alongthree pre-determined material symmetrical planes, resulting in sixsample groups. A clear loading rate dependence of the fracturetoughness anisotropy of Barre granite was observed. The fracturetoughness anisotropy was found to decrease with the increase ofthe loading rate. The question remains to be answered is thephysical reason for this observed trend of the rate dependence ofmode-I fracture toughness anisotropy.

Like other granitic rocks, Barre granite has lots of pre-existingmicrocracks and its mechanical responses should be controlled byits microscopic structures. The phenomenon of microcrackingzone near the main crack tip and its effects on the propagation

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International Journal ofRock Mechanics & Mining Sciences

1365-1609/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijrmms.2013.08.011

n Corresponding author at: Department of Civil Engineering, University ofToronto, ON, Canada M5S 1A4. Tel.: þ1 416 978 5942; fax: þ1 416 978 6813.

E-mail address: kaiwen.xia@utoronto.ca (K. Xia).

International Journal of Rock Mechanics & Mining Sciences 63 (2013) 113–121

of main crack in brittle materials such as ceramics, rocks andconcretes, have been discussed by many researchers [10–12].Existing theoretical analysis has focused on the effect of micro-cracking on the stress field near the main crack tip. Due to thecomplexity of the problem, except for a few simple cases, closed-form solutions are not available. Researchers attempted to modelthis problem in two perspectives. One is the continuum mechanicsmodel, which aims at building a constitutive framework in thecontinuously damaged material mimicking the overall effect ofmicrocracking [13,14]. But due to the complexity of the interactingproblem, no agreement has been reached among researchers onwhat is the best material model to simulate the configuration ofcrack–microcrack interaction. The second approach considers themultiple microcracks in the microcracking zone near the tip of amain crack as discrete entities; in which case, interaction isaddressed using the stress function or assumed stress stateinvoking the superposition principle [15–17].

Although there are many interesting discussions on the effectsof stress shielding and amplification of the main crack due to thepresence of microcracks, few of them have strong experimentalbasis. Mode-I fracture toughness (KIC) measurements on four typesof granites [4,18] were carried out under the standard procedureoutlined by ISRM, demonstrating different levels of anisotropy.However in the above studies, an isotropic material model wasborrowed to determine the fracture toughness values. In ourprevious work [9], the mode-I fracture toughness of anisotropicBarre granite was systematically measured using NSCB method [8],where an orthotropic material model with material constantsidentified from literature was used to accurately characterize thetrue stress intensity near the crack tip from a macroscopic pointof view.

We believe that the physical reason for the observed fracturetoughness anisotropy is due to pre-existing microcracks and thuswe use micromechanics modeling in the current study to explicitlyconsider microcracks. Microstructural observation with a newlydeveloped technique of computer-aided image analysis programclearly is used to examine the density and orientation of micro-cracks around the tip of the main crack. From two images of thinsections corresponding to the two contrasting cases of fracturetoughness measurements, two physical micromechanics modelsare constructed. Finite element analysis is then conducted todetermine the effects of embedded microcracks on the distur-bance of the stress field at the tip of the main crack. The numericalresults are consistent with the experimental results, supportingthe postulation that the preferred distribution and orientation ofpre-existing mircocracks in the otherwise homogeneous andisotropic rock material are responsible for the rate dependenceof the fracture toughness anisotropy.

2. Crack–microcrack interaction

The method of pseudo-tractions, proposed by Horii and Nemat-Nasser [19], further improved by Gong and Horii [20], has beenproven to be an effective approach to the analysis of the crack–microcrack interaction problem. Based on the complex potentials byMuskhelishvili [21] and the principle of superposition, this methodcan treat general problems with any number of interacting cracks orother inhomogeneities [20]. Herein, the concept of crack–microcrackinteraction is demonstrated using the 0th-order and 1st-ordersolution of the pseudo-traction method by Gong and Horri [20].

Consider a general problem of a semi-infinite main crack and anarbitrarily located and oriented microcrack, as shown in Fig. 1.Denote the distance between the main crack tip and the center ofthe microcrack by d and the length of the microcrack by 2 c. Theangle measured from the x-axis to the line connecting the tip of the

main crack and the center of the microcrack is θ and the microcrackorientation is defined by the angle φ from x-axis to the x0-axis.

In this paper, K0I denotes the stress intensity factor of the main

crack without considering microcracks, i.e., the far-field stressintensity factor or the load, KMA

I denotes the local stress intensityfactor of the main crack. The method proposed by Gong and Horii[20] is used here to calculate the ratio between KMA

I and K0I with both

0th order approximation and 1st order approximation for the casewhere d=c¼ 2. Because in our fracture tests on Barre granite [9], theresulting fracture mode is pure mode-I, thus in the current analysis,the far-field loading of K0

II is set to be zero. Using the first two termsof the formula given by Gong and Horii [20], the stress intensityfactors of the main crack is calculated and showed in Fig. 2, in whichthe stress intensity factor at the tip of the main crack may beeither increased (amplification, KMA

I =K0I 41) or decreased (shielding,

KMAI =K0

I o1) with respect to that of the far-field, depending on thelocation and orientation of the microcracks. This effect of amplifica-tion or shielding of the stress intensity factor of the main crack due tothe presence of the microcracks can be of vital significance to theanisotropy of rocks and hence can result in sizeable variations of themeasured fracture toughness in the experiments, which will be fullyexplored in the following sections.

3. Construction of micromechanics models

3.1. Sample configurations in the NSCB tests

Fig. 3a illustrates the 3D block diagram showing longitudinalwave velocities. The three orthogonal material directions are

y

xθ

xy

2c

φ

d

Fig. 1. One arbitrarily located microcrack near the tip of a semi-infinite crack.

0 20 40 60 80 100 120 140 160 180

20

40

60

80

100

120

140

160

180

KMAI

/K0I<1

(shielding) KMA

I/K0

I>1

(amplification)

0 order 1st order

θ (d

egre

e)

φ (degree)

KMAI

/K0I<1

(shielding)

Fig. 2. The phase diagram of amplification and shielding effects of main crack dueto the presence of a unique microcrack using solutions of 0-order and 1st-orderapproximation.

F. Dai et al. / International Journal of Rock Mechanics & Mining Sciences 63 (2013) 113–121114

identified by P-wave velocities measurements, labeled as X(3.57 km/s), Y (4.00 km/s) and Z (4.75 km/s). The sampling andnaming scheme of six NSCB sample sets are also shown in Fig. 3aand b. The first index of the sample represents the directionnormal to the splitting plane, and the second index indicates thepropagation direction of the crack. The three orthogonal material

directions are the loading directions, as shown in Fig. 3b, with thedashed line depicting the potential failure path.

3.2. Micromechanics models

The recently developed computer aided image analysis techni-ques have been widely adopted to investigate microstructuralcharacteristics of rocks through digital images analysis of thinsections. These techniques can facilitate direct observation ofmicro-details of rock thin sections involving microcrack lengthand orientation [3,4].

In order to understand the physical mechanism of the mode-Ifracture toughness anisotropy, two digital images of petrographicthin sections of rock samples corresponding to the two contrastingmeasurements of fracture toughness of Barre granite were cap-tured with a camera mounted on a standard petrographic micro-scope as shown in Fig. 4a (Case 1) and b (Case 2). A quantitativeanalysis of microcracks consists of the following steps: imageacquisition, image pre-processing, microcracks tracing, and mea-surements with automated image analysis programs. In ouranalysis, one screen pixel represents 1 μm, the shortest resolvablemicrocrack thus has a cut-off limit of approximately 1 μm. Fig. 4acorresponds to the crack–microcracks orientation in Barre granitefor the situation in which the fracture toughness is the highest(i.e., ZX). In this case named Case 1, the main crack propagatesfrom the tip of the notch with an acute angle to most microcracks.Fig. 4b shows the microcracks density and orientation in Barregranite for the case in which the fracture toughness is the lowest(i.e., XY). In this case named Case 2, the main crack is collinear tomost microcracks.

Fig. 4c (Model 1) and d (Model 2) depicts two conceptual modelsbased on Fig. 4a and b respectively, taking only into account a fewmicrocracks closest to the tip of the main crack. The nearestmicrocracks are considered because it has been shown from bothcontinuum and discrete models that the interaction is dominated bythe nearest microcracks [22]. This is understandable because stressfield around the macroscopic fracture is singular and thus theinfluence of fracture to the stress field is localized. Based on thestatistics of the geometrical distribution of microcracks in Fig. 4a,

Vp = 3.57km/s

Vp = 4.00km/s

Vp = 4.75km/s

X

Y

Z

ZY

YX

XY

XZ (BD)

YZ1st set parallel to YZ plane

2nd set parallel to XZ plane

3rd set parallel to XY plane

P/2

P/2

PX

Y Sample YX

NSCB

(BD)

(BD)

(BD)

(BD)

ZX (BD)

Fig. 3. (a) 3D block diagram showing longitudinal wave velocities and the samplinglocation of NSCB samples prepared along each plane with respect to microcrackorientations in Barre granite; the first index for sample numbering represents thedirection normal to the splitting plane, and the second index indicates thepropagation direction of the crack. (b) YX sample with the dashed line depictsthe failure plane.

1 mm 1 mm

Fig. 4. (a) Photo of microscopic thin section showing microcracks in a tested Barre granite sample with highest fracture toughness; Case 1: the main crack inclines at anangle of 451 to microcracks; (b) Photo of microscopic thin section showing microcracks in a tested Barre granite sample with lowest fracture toughness; Case 2: The maincrack is collinear to microcracks; (c) Model 1: the crack–microcracks configuration for Case 1; (d) Model 2: the crack–microcracks configuration for Case 2.

F. Dai et al. / International Journal of Rock Mechanics & Mining Sciences 63 (2013) 113–121 115

Model 1 is constructed (Fig. 4c), where two symmetric microcracksnear the notch tip are oriented at an angle of 451 to the horizontalmain crack. Denote the length of the microcrack by 2c, the distancefrom the right tip of the microcrack to the tip of the main crack is0.2c. The conceptual model shown in Fig. 4d is based on the thinsection photo of Fig. 4b. In this case, existing microcracks are mostlyparallel to the propagation path of the main fracture. For comparingpurpose, the same length of the microcrack of 2c is used and thesame distance from the main crack tip to the closest microcrack tip of0.2c. It is noted that these orientation and dimension parameters ofmicrocracks are not exact and they are used to illustrate theirinfluence to the macroscopic response of the rock material.

4. Analyses of micromechanics models

4.1. Evaluation of numerical method

The approximate solutions proposed in the literature [16,20]were developed based on the assumption of d=c41, when d=cr1,the results are invalid. If d=c is close to 1, these methods becomeimpractical because many higher orders of expansion are neces-sary to achieve reasonable accuracy. In addition, these analyticsolutions are derived in the situation that the elastic solid contain-ing the main crack and microcracks is infinite. However, the twomodels feature a specific geometry of a finite half disc. In thesecases, finite element analysis can be a good alternative and itsrobustness had been confirmed by Meguid and Gong [23], espe-cially when the microcrack is very close to the main crack.

Herein, finite element analysis is carried out to solve our micro-mechanics models using ANSYS software. Taking advantage of thesymmetry of both models, half-crack model is used to build the finiteelement meshes. PLANE82 (eight-node) element is used in theanalysis. To better simulate the stress singularity r�1/2 near the cracktip (r is the radius to the crack tip), 1/4 nodal element (singularelement) [24] is applied to mesh the vicinity of the crack tip. Thestress intensity factor is calculated based on the displacement field inthe vicinity of the crack tip. A loading force of 200 N is applied to thefinite element model. In our finite element analysis, Young's modulusis taken as 82 GPa and Possion's ratio as 0.25 for the rock material.

To further verify the accuracy of the numerical calculation,a semi-infinite main crack with one collinear microcrack undermode-I loading (correspond to the case of ɸ¼ θ¼ 01 in Fig. 1) isconsidered as a testing problem. The exact solution of this problem[16] is given below, whereΚðkÞ and ΕðkÞ denote the complete ellipticintegrals of the first kind and the second kind respectively [25]:

KMAI

K0I

¼ ΕðkÞffiffiffik

pUΚðk0Þ

ð4Þ

where

k¼ffiffiffiffiffiffiffiffiffiffid�cdþc

sand k0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffi1�k2

q

The calculated stress intensity factor via ANSYS is normalizedby K0

I and compared with the exact results [20] for different casesof d/c in Table 1. The meshed elements are between 1663 and

2139; the nodes are between 5032 and 6486. As tabulated inTable 1, the maximum error is less than 1.03%, which occurs whend/c¼1.1.

4.2. Numerical analysis of micromechanics models

With confidence gained in the finite element analysis of abovemicromechanics models, the phenomena of stress shielding andstress amplification due to crack–microcrack interactions are quanti-fied. Three numerical models are built to achieve this goal: (1) Model0, the reference model corresponding to the microcrack-free case(Fig. 5a and b), (2) Model 1 corresponding to Case 1 (Fig. 5c and d)and (3) Model 2 corresponding to Case 2 (Fig. 5e and f). With thesame loading boundary condition, the stress intensity factors, K0

I forModel 0, KM1

I for Model 1 and KM2I for Model 2 are calculated. Here

K0I is recognized as the far-field loading.For the microcrack-free case (Model 0), a similar meshing

scheme is used as reported in [9] (Fig. 5a and b). Fig. 5c and eshows the global meshing scheme of our finite element models onModel 1 and Model 2 respectively, featuring increasing gridsdensity towards the main crack tip. Fig. 5d and f shows themeshing scheme at the vicinity of the main crack and themicrocracks for Model 1 and Model 2 respectively; a close-viewof the microcrack for either model is also marked.

Fig. 6a–c compares the stress intensity (i.e., the stress differ-ence between the maximum and minimum principal stress)contours around the main crack tip. Fig. 6a shows the stressintensity contours near the tip of the main crack in the absence ofmicrocracks, from which K0

I is evaluated. Fig. 6b shows the stressintensity contours due to the presence of two symmetric micro-cracks in Case 1 and Fig. 6c shows the stress intensity contours dueto the presence of a collinear microcrack in Case 2. Nine contourlines are denoted from A to I, representing the stress intensityvalues from 40,000 Pa to 200,000 Pa with a stress increment of20,000 Pa. These contour lines indicate the concentrated stressdistribution of stress field near the tip of the crack: the closer tothe tip of the crack, the higher the stress intensity. From thesecontours, one can see that the microcracks in Model 1 tend todecrease the intensity of the stress field of the main crack, thusyielding a stress shielding effect of the main crack; whereas forModel 2, the microcrack tends to increase the intensity of thestress field of the main crack, thus yielding stress amplification ofthe main crack.

The finite element calculations have quantified the shielding oramplification effects for both models. Within the framework oflinear elastic fracture mechanics (LEFM), only the ratio betweenstress intensity factors is needed: KM1

I =K0I is 0.880 (shielding) for

Model 1 and KM2I =K0

I is 1.233 (amplification) for Model 2. Thequantification of shielding or amplification effects using finiteelement analysis can help interpret the anisotropy of measuredfracture toughness in the experiments.

4.3. Dynamic evaluation

For the dynamic fracture tests conducted in the modified SHPBsystem with careful pulse shaping, it has been proven that forModel 0 (i.e., the microcrack-free case), as long as the dynamicforce balance has been achieved on both ends of the NSCB sample,the time-varying stress intensity factor (SIF) deduced from thequasi-static data analysis is the same as that from a full dynamicanalysis [26]. Following the same strategy, quasi-static equationsfor Model 1 and Model 2 under dynamic loading will beverified below.

Using dynamic finite element analysis, SIF evolution of themain crack for Model 1 and Model 2 are determined by solving theequation of motion with Newmark time integration method in

Table 1Stress intensity factor at the tip of the main crack with one collinear microcrack.

d/c 1.1 1.2 1.3 1.4 1.5 2 3

Exact value 1.652 1.387 1.274 1.209 1.167 1.076 1.030ANSYS value 1.635 1.398 1.287 1.221 1.176 1.086 1.024Error (%) �1.029 0.721 1.020 0.992 0.771 0.999 �0.582

Error (%) ¼100� (ANSYS value � Exact value)/Exact value.

F. Dai et al. / International Journal of Rock Mechanics & Mining Sciences 63 (2013) 113–121116

crack crack tipcrack

crack crack tipmicrocrack

crack

crack microcrack crack microcrack

crack tip

Fig. 5. Finite element meshes of the Model 0, Model 1 and Model 2: (a) global mesh of Model 0; (b) close-view of the mesh at the vicinity of the main crack in Model 0; themain crack and its tip are indicated with arrows; (c) global mesh of Model 1, Case 1 (d) close-view of the mesh at the vicinity of the main crack and the inclined microcrack inModel 1; the main crack and its tip are indicated with arrows; (e) global mesh of Model 2, Case 2 (f) close-view of the mesh at the vicinity of the main crack and the collinearmicrocrack in Model 2. The main crack and its tip are indicated with arrows and the collinear microcrack is also marked.

F. Dai et al. / International Journal of Rock Mechanics & Mining Sciences 63 (2013) 113–121 117

ANSYS [27]. The finite element meshing for Model 1 and Model2 is the same as that for Fig. 5c and e, while taking the inputloading F1 and F2 as half of the dynamic loading forces exerted onthe incident and transmitted side of the sample respectively. For

pulse shaped SHPB tests, dynamic force balance can also beachieved (i.e., F1¼F2¼P/2), thus in the finite element analysis,the bearing load (i.e., F2) is applied on both loading ends of thesample [26]. As demonstrated, a typical dynamic load shown in

Fig. 6. The deformation and stress intensity trajectories at the vicinity of the main crack for the semi-circular band specimen in (a) Model 0, (b) Model 1, and (c) Model 2.

F. Dai et al. / International Journal of Rock Mechanics & Mining Sciences 63 (2013) 113–121118

Fig. 7 has been applied to our finite element models (Model 0, Model1 and Model 2). This load comes from a typical dynamic NSCB testwith force balance achieved on both ends of the sample.

For all three finite element models (Model 0, Model 1 andModel 2), the SIF evolutions calculated by dynamic finite elementanalysis are compared with those from quasi-static analysis. Theevolutions of SIF's from both static and dynamic methods matchreasonably well. Therefore, with remote force balance, the staticanalysis can fully reproduce the transient SIF evolution of the maincrack tip for all three models. The static SIF evolution is proportionalto the bearing load (e.g. P2) according to the static equationwhich canbe calibrated with static finite element analysis (Fig. 5c and e). Thus,

at any instant throughout the loading, KM1I =K0

I is 0.880 (shielding) for

Model 1, and KM2I =K0

I is 1.233 (amplification) for Model 2. It is alsonoted that for each test, the loading rate is determined as the slope ofthe pre-peak linear portion of the SIF evolution. Thus, the ratio of theloading rate for Model 1 and Model 2 to the loading rate for the

Model 0 should be KM1I

�=K0

I

�is 0.880 (shielding) for Model 1 and

KM2I

�=K0

I

�¼1.233 (amplification) for Model 2. This is because the rock

material is assumed to be linear elastic.

5. Interpretation of the rate dependence of fracture toughnessanisotropy

All rock fracture toughness measurement methods includingthese proposed by ISRM are based on the linear elastic fracturemechanics (LEFM) theory, in which the fracture toughness is

considered to be unique and the crack of the rock sample initiateswhen the stress intensity factor at the main crack reaches thefracture toughness. From the critical load recorded from tests,fracture toughness can be calculated based on the proposedformulas. For a cracked solid with microcracks, denote the ratioof the local stress intensity factor at the main crack tip and the

far-field loading by ξ, i.e., KMAI =K0

I ¼ ξ. Thus ξ¼ 1 corresponds tothe microcrack-free case, ξ41 corresponds to the stress ampli-fication where the stress intensity factor of the main crack isincreased due to the presence of microcracks, and ξo1 corre-sponds to the stress shielding where the stress intensity factor ofthe main crack decreased due to the presence of microcracks. Thestatic fracture toughness anisotropy can be readily explainedwith the crack–microcrack interaction.

The interpretation of the dynamic fracture toughness aniso-tropy is rather complicated. This is because the dynamic fracturetoughness depends on the loading rate. As shown before, the ratioof the loading rate at the main crack tip and the far-field loading is

also ξ, i.e., KMAI

�=K0

I

�¼ ξ. Let KLoad

IC be the measured mode-I fracturetoughness (the global fracture toughness) of the rock from thematerial testing device based on the maximum value of the load,

KLoadIC is the maximum apparent value of K0

I measured duringmaterial testing, which is usually considered to be KIC . However, if

ξa1, the local stress intensity factor KMAI (¼ ξUK0

I ) increases with

the loading K0I until the KMA

I is equal to the fracture toughness KIC .Under this circumstance, if the corresponding loading rate at far

field is K0I

�, then the local stress intensity factor KMA

I (¼ ξUK0I )

0

1

2

3

4

Forc

e (k

N)

Time (μs) Time (μs)0 50 100 150 200 0 50 100 150 200

0

1

2

3

4

5

SIF

(MP

a m

1/2 )

Model 0_Dynamic Model 0_Static Model 1_Dynamic Model 1_Static Model 2_Dynamic Model 2_Static

Fig. 7. (a) The dynamic load exerted on both ends of the NSCB specimen for three configurations (Model 0, Model 1 and Model 2), the load comes from a typicalmeasurement with force balance achieved on both ends of the sample. (b) The evolution of SIF of the NSCB specimen for three configurations (Model 0, Model 1 andModel 2) from both quasi-static analysis and dynamic analysis with dynamic force balance.

Table 2The fracture toughness and corresponding loading rates for three models (Model 0, Model 1 and Model 2).

Model 0 Model 1 Model 2

Loading rates(GPa m1/2/s)

Fracture toughness(MPa m1/2)

Loading rates(GPa m1/2/s)

Fracture toughness(MPa m1/2)

Loading rates(GPa m1/2/s)

Fracture toughness(MPa m1/2)

�0 1.2 �0 1.4 �0 1.040 3.2 45.5 3.6 32.4 2.660 4.0 68.2 4.5 48.7 3.280 4.7 90.9 5.3 64.9 3.8

100 5.5 113.6 6.3 81.1 4.5120 6.2 136.4 7.0 97.3 5.0140 7.0 159.1 8.0 113.5 5.7160 7.7 181.8 8.8 129.8 6.2180 8.5 204.5 9.7 146.0 6.9

F. Dai et al. / International Journal of Rock Mechanics & Mining Sciences 63 (2013) 113–121 119

actually varies with a local loading rate of ξUK0I

�. Thus,

KLoadIC ¼maxðK0

I Þ ¼maxðKMAI =ξÞ ¼ KIC=ξ ð5aÞ

KLoadIC

�¼ K0

I

�¼ KMA

I

�=ξ¼ KIC

�=ξ ð5bÞ

This suggests that the intrinsic material toughness (microcrack-free case) and the measured fracture toughness (microcracksembedded) are related as

KIC ¼ ξUKLoadIC ð6aÞ

KIC

�¼ ξUKLoad

IC

�ð6bÞ

Using these two equations, the measured fracture toughnessfor Model 1 and Model 2 can be calculated by assuming theintrinsic rock dynamic mode-I fracture toughness values at differ-ent loading rate as shown in Table 2 (Model 0). The static value forModel 0 is assumed to be 1.2 MPa m1/2, which is about the averageof the highest and lowest static values from measurementspresented in [9]. The intrinsic rock fracture toughness is thenassumed to be linearly rate dependent, as observed in the testingresults presented in [9]. Similar slope as in the experimental datais used for the assumed fracture toughness values. The measuredtoughness values and corresponding loading rates for Model 1 andModel 2 are then calculated and tabulated in Table 2; and alsoillustrated in Fig. 8. The calculated fracture toughness values for

these three models have clearly reproduced what have beenobserved in the experiments.

In our previous paper [9], an anisotropic index of mode-Ifracture toughness, αk was defined as the ratio of the maximumfracture toughness to the minimum of fracture toughness. Usingour micromechanics models for static cases, KM1

I =K0I ¼0.880 for

Model 1 and KM2I =K0

I ¼1.233 for Model 2. The measured fracturetoughness for Model 1 should be KM1

IC ¼ KIC=0:880¼ 1:136KIC andthat for Model 2 should be KM2

IC ¼ KIC=1:233¼ 0:811KIC . Thereforefor static loading, the anisotropic index of mode-I fracture tough-ness, αk is KM1

IC =KM2IC ¼1.136KIC/0.811KIC ¼1.40. For dynamic cases,

a loading rate is first picked and the corresponding fracturetoughness values for Model 1 and Model 2 are then obtained fromFig. 8 using interpolation method. Dynamic αk is thus calculated(Table 3). The variation of the anisotropic index of the simulatedMode-I fracture toughness, αk with the loading rate is plotted inFig. 9a. Our micromechanics model produced the similar trend ofthe loading rate dependence of mode-I fracture toughness aniso-tropy as reported in [9] (reproduced here in Fig. 9b).

6. Conclusion

The laboratory measurements of KIC of Barre granite under a widerange of loading rate exhibited a decreasing anisotropy with theincrease of the loading rate. Micromechanics modeling was carried outto explain this loading rate dependence of mode-I fracture toughnessanisotropy. Two models were constructed to investigate the influenceof the stress intensity factor of the main crack due to the presence of

Fig. 8. The simulated dynamic fracture toughness of Barre granite with the loadingrate for three configurations (Model 0, Model 1 and Model 2).

Table 3The mode-I fracture toughness anisotropic index (αk) of Barre granite frommicromechanics modeling.

Loading rates(GPa m1/2/s)

Fracture toughness(MPa m1/2)

Anisotropy Index αk

Model 0 Model 1 Model 2

�0 1.2 1.400 1.000 1.40040 3.2 3.449 2.892 1.19360 4.0 4.201 3.644 1.15380 4.7 4.953 4.396 1.127

100 5.5 5.705 5.148 1.108120 6.2 6.457 5.900 1.094140 7.0 7.209 6.652 1.084160 7.7 7.961 7.404 1.075180 8.5 8.713 8.156 1.068200 9.2 9.465 8.908 1.063

0 50 100 150 200 250

1.2

1.3

1.4

1.5

1.6

1.7

Fig. 9. The mode-I fracture toughness anisotropic index (αk) of Barre granitefrom (a) micromechanics modeling and (b) experiments.

F. Dai et al. / International Journal of Rock Mechanics & Mining Sciences 63 (2013) 113–121120

microcracks, showing the effects of shielding and amplification of thestress intensity at the main crack due to the presence of the nearestmicrocracks. Using our micromechanics models, we reproduced theexperimental observation on the mode-I fracture toughness of Barregranite: (1) the mode-I fracture toughness of Barre granite for each ofthe six sample groups increases with the loading rate and (2) themode-I fracture toughness anisotropy of Barre granite decreases withthe increase of the loading rate. Our crack–microcrack model thusreveals that the pre-existing microcracks in the Barre granite areresponsible for the anisotropy of Barre granite observed in theexperiments and its loading rate dependence.

Acknowledgments

We acknowledge the support by the Innovative Research Groups ofthe NSFC under Grant #51021004, and the National Basic ResearchProgram of China under Grant #2013CB035900. F. Dai acknowledgesthe support of Excellent Young Scholar Plan of Sichuan Universitythrough Grant no.2012SCU04A07. K.X.'s research is partially supportedby the National Science and Engineering Research Counsel of Canadathrough Discovery Grant no. 72031326,

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