Rock Engineering and Rock Mechanics: Structures in and onRock Masses – Alejano, Perucho, Olalla & Jiménez (Eds)

© 2014 Taylor & Francis Group, London, 978-1-138-00149-7

Effect of loading rate on fracture behaviour of rock materials

Q.B. Zhang & Y. ZouÉcole Polytechnique Fédérale de Lausanne (EPFL), School of Architecture, Civil and Environmental Engineering,Laboratory of Rock Mechanics (LMR), CH-1015 Lausanne, Switzerland

ABSTRACT: In this paper, experiments are carried out to study the quasi-static and dynamic fracture behaviourof sedimentary, igneous and metamorphic rocks. The notched semi-circular bending method has been employedto determine fracture parameters over a wide range of loading rates using both a servo-hydraulic machine anda split Hopkinson pressure bar (SHPB). The time to fracture, crack speed and velocity of the moving specimenhalves are measured by strain gauge, crack propagation gauge and high-speed photography on the macroscopiclevel. This study reveals clearly that: dynamic crack initiation and growth toughness increase with the increaseof loading rate and crack speed; the kinetic energy of the moving specimen halves increases with increasingimpact speed of the striker; and dynamic fracture energy and increases rapidly with increasing crack speed.

1 INTRODUCTION

Mechanical properties and fracture behaviour of rockmaterials are influenced by loading rate, and in par-ticular the responses distinguishably change after theloading rate exceeds a critical value (Zhang & Zhao,2013d). The significant differences between quasi-static and dynamic fracture behaviour are traced tospecific fracturing process and failure mechanisms.

The subcritical and quasi-static fracture behavioursof rocks are quite well understood (Atkinson, 1987).The International Society for Rock Mechanics (ISRM)has also suggested four standardized methods: thechevron bend (CB) and short rod (SR) methods(Ouchterlony, 1988), the cracked chevron notchedBrazilian disc (CCNBD) method (Fowell, 1995),and notched semi-circular bending (NSCB) method(Kuruppu et al., 2013), for the determination of quasi-static fracture toughness. However, the studies onfracture behaviours under dynamic loading have beenless investigated. The testing methods are basicallyextended from the corresponding quasi-static ones andcategorized approximately into Brazilian disc (BD),compact tension (CT) and bending type methods(Zhang & Zhao, 2013d). To the best of our knowl-edge, no systematic experimental studies have beenperformed to study fracture behaviour of rocks over awide range of loading rates. The following issues stillneed to be addressed: (1) detection of the time to frac-ture, crack speed and velocity of the flying fragments;(2) determination of the dynamic crack initiationand propagation toughness; (3) measurement of crackopening displacement using the HS-DIC technique;and (4) estimate of dynamic fracture energy.

2 EXPERIMENTAL PROCEDURES

2.1 Material characterizations

In order to compare with the previous studies, fourtypes of rock materials that were well studied indynamic fracture tests were selected for experiments,namely one sedimentary rock (calcareous sandstone(Yin et al., 2012, Gong & Zhao, 2013)), one igneousrock (Fangshan gabbro (Zhang et al., 1999, Zhanget al., 2001)), and two metamorphic rocks (Ya’ancoarse-grained marble (Wang et al., 2010, Wang et al.,2011) and Fangshan fine-grained marble (Zhang et al.,1999, Zhang et al., 2000, Zhang & Zhao, 2013b,Zhang & Zhao, 2013a).

2.2 Experimental techniques

The ISRM suggested NSCB method was employedto investigate the quasi-static and dynamic fracturebehaviour of rocks (Kuruppu et al., 2013, Zhou et al.,2012). The specimens of each rock were drilled fromone large block. Rock cores with a nominal diame-ter of 50 mm were drilled and sliced to obtain discswith an average thickness of 20 mm. The disk was splitalong the diameter into two semi-circular specimensand then a 5 mm long edge notch was cut using a high-speed diamond impregnated circular blade (∼0.3 mmthickness). The notch tip was sharpened using a dia-mond wire saw (∼0.1 mm thickness) to achieve a sharpcrack tip.

The quasi-static tests were performed using a servo-hydraulic machine at the loading rate of 0.002 mm/s.The dynamic fracture tests were carried out by means

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Figure 1. Close-up view of the partial SHPB bars and aspecimen with random speckle patterns.

of a SHPB system. Figure 1 shows the photograph ofloading configurations and a NSCB specimen withrandom speckle patterns on the surface that wasapplied to ensure good contrast of the images for thecalculation of strain fields. To study systematically theeffect of loading rate, the striker was launched by agas gun at speeds ranging from 2.0 to 5.0 m/s. A high-speed camera (Photron SA1.1) was located on the frontside of the specimen, and strain gauges or a designedelectrical circuit with crack propagation gauge werepositioned on the back side for calibration. The high-speed camera was operated at the setting: 192 × 224pixels for the size of 26 × 16 mm2, and 125,000 framesper second. All the signals of strain gauges and thehigh-speed camera were synchronized with a thresh-old of the signal captured from the strain gauge on theincident bar. A detailed description of experimentalprocedures was presented in the previous publication(Zhang & Zhao, 2013b).

3 QUASI-STATIC FRACTURE BEHAVIOUR

The equation to calculate the SIF (stress intensityfactor), KI, of the NSCB specimen is given by (Limet al., 1993)

where 2S is the specimen span (33 mm), R is theradius (25 mm), a is the notch length (5 mm), B isthe thickness (20 mm), and P is the load applied onthe specimen.

The time to fracture tf and the displacement of frac-ture δf are corresponded normally to the peak appliedload Pmax. The mean values of quasi-static fracturetoughness for sandstone, gabbro, CG marble and FGmarble are 0.93, 1.64, 0.82 and 1.5 MPa

√m, respec-

tively. For a plane-strain condition, the quasi-staticfracture energy is calculated by the Irwin’s correlationGC = (1 − ν2)K2

IC/E.

4 DYNAMIC FRACTURE BEHAVIOUR

The dynamic crack initiation toughness, KId, is the crit-ical dynamic SIF at the time to fracture tf, and the

Figure 2. High-speed photographs showing dynamic frac-turing process of gabbro at the striking velocity of 3 m/s.

dynamic crack growth toughness KID is the criticalSIF at a specific crack speed v, which are given by thefollowing equations (Ravi-Chandar, 2004)

where the dynamic loading rate is generally expressedas K̇dyn

I = KId/tf.To determine the dynamic SIF, the optical methods

in combination with high-speed photographs, numer-ical simulations, strain gauges, quasi-static analy-sis, and the combined methods have been used.These methods have been heavily described by (Ravi-Chandar, 2004, Jiang & Vecchio, 2009), hence notrepeated here.

4.1 Time to fracture

The time to fracture is defined as the time intervalfrom the beginning of the loading phase to the onsetof rapid crack propagation. Once the dynamic SIF isdetermined, the time to fracture is the most importantfactor (Ravi-Chandar & Knauss, 1984).

In this study, the time to fracture tf was mea-sured primarily by high-speed photographs and mean-time calibrated by strain gauges or crack propagationgauges. Figure 2 shows the a sequence of high-speedphotographs from dynamic fracturing test in gabbroat a striking speed of 3 m/s, with the time 0 µs cor-responding to the onset of loading. The experimentaldata on the time to fracture are presented together withthe results of dynamic crack initiation toughness inFigure 3.

4.2 Dynamic crack initiation toughness

It has been reported that the specimen is in a stateof force equilibrium through the time to fracture tf,and the dynamic crack initiation toughness KId is thusdetermined by Eq. 1 using the mean force applied onthe specimen and tf. The experimental data of KId/KICand tf are presented in Figure 3, and the analytical solu-tions with various critical distances (Liu et al., 1998)

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Figure 3. Normalized dynamic crack initiation toughnessKId/KIC and time to fracture tf of: (a) theoretical model (Liuet al., 1998) and experimental data, (b) the magnified view.

are also shown. The differences are probably inducedby the indirect tension method; however, the generaltrend of the experimental data corresponds well withthe analytical prediction. It is also observed that thevalues of KId/KIC in gabbro and FG marble are appar-ently higher than those in sandstone and CG marble,which are mainly governed by the time of stress waverequired to travel through the specimen, in other wordsby the longitudinal wave speed of the rock (The fasterthe speed is, the shorter the time is required to reachforce equilibrium), and by the interactions of multiplemicrocracks with the main crack tip to some extentdelay the onset of crack initiation.

It should be noted that in Figure 4, althoughthe general trend of KId/KIC increases almost lin-early with increasing normalized loading rates in therange of K̇dyn

I /KIC = 1 × 104−4 × 104 s−1, the valuesof KId/KIC in gabbro and FG marble are apparentlyhigher than those in sandstone and CG marble, whichare governed primarily by the time of stress waverequired to travel through the specimen.

4.3 Crack speed

The ranges of crack speed in sandstone, gabbro,CG marble and FG marble are 300–650, 430–1120,

Figure 4. Normalized dynamic crack initiation toughnessKId/KIC as a function of normalized loading rate K̇dyn

I /KICfor selected rock materials.

280–480, and 320–1000 m/s, respectively, and the cor-responding ratio vmax/CR are 0.36, 0.35, 0.33, and0.38. The interested reader is referred to (Zhang &Zhao, 2013b) for details. The phenomena of macro-branches were not observed during tests, perhaps dueto the small size of specimen and the indirect tensiontesting method.

4.4 Dynamic fracture energy

The following factors should be carefully considered:the forces on both sides of the specimen are non-equilibrium during wave propagation; the effect ofmultiple pulse waves needs to be eliminated; and thefriction between the specimen and the bars should besmall enough. In the present study, the single pulsewave is achieved by means of a longer length ofthe incident bar and a momentum trap. There arethree-point contacts in the NSCB method to transferdynamic loads: one contact is between the incident barand the top point of the specimen, and the other twocontacts are formed by two supporting pins and thespecimen, as shown in Figure 1. During the test, thenotch of the specimen opens up and meantime resultsin frictional force by the pins to resist it separation.Lubricants are used on the bar/specimen interfaces toreduce the friction effect. (Xia et al., 2013) also pointedout that the coefficient of friction of the NSCB methodwas about 0.02 in a well-lubricated SHPB test. Thefriction can be ignored, and thus the energy absorbedby the specimen WS is given by

where WIn., WRe., and WTr. are the energies of the inci-dent wave, the reflected wave, and the transmitted waverespectively.

Continued to Figure 2, Figure 5(a) shows high-speed photographs of the gabbro until the NSCBspecimen is split into two almost equal fragments.Each flying fragment has a combined rotational and

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translational motion, and the angular velocity ω andtranslational velocity vT can be estimated from its localcoordinates at each time step, as defined schemati-cally in Figure 5(b). At the initial stage, ω increaseswith increasing time, but vT shows a decreasing trend,which indicate obviously that the increase of ω is onthe expense of the reduction of vT. Then both veloc-ities become constant, and thus there is no change inthe total kinetic energy T .The sum of rotational kineticenergy TRot. and translational kinetic energy TTra. canbe written as

where m is the mass of one fragment, and I =r2

36π2 (9π2 + 18π − 128)m is the moment of inertiaaround the axis of rotation.

The absorbed energy WS is partitioned primarilyinto the kinetic energy of the flying fragments andthe dissipated energy � is associated with crack-ing. The dissipated energy is then obtained by � =WS − 2T , and the dynamic fracture energy (dissipatedper unit area ∂A created) is written as GdC = ∂�/∂A.The dynamic fracture energy approaches to a lowlevel when the crack speed v is small and increasesrapidly with the increase of v, as shown in Figure5(c), reflecting the crack speed-toughening effect ofthe material. A semi-empirical rate-dependent modelis proposed for the simulation of crack propagation inrock materials,

where b is a material constant, b ∝ 1/CR. The valuesof b for sandstone, gabbro, coarse-grained marble andfine-grained marble are 0.011, 0.006, 0.012 and 0.008,respectively.

4.5 Dynamic crack growth toughness

Pioneering research on dynamic crack growth tough-ness KID was performed on Solnhofen limestoneusing a projectile impact technique and an on-specimen strain gauge (Bertram & Kalthoff, 2003).We attempted to use the method of strain gauge todetermine the dynamic SIF, but the results were hardlyrepeatable, perhaps due to the heterogeneity of rocks.In the SHPB test, the forces on both sides of a specimenare non-equilibrium after the time to fracture (Zhang &Zhao, 2013, Foster et al., 2011), and the dynamicSIF becomes a function of crack speed. Due to thelimitation of the speed and resolution of high-speedcamera, it is still challenging to measure the dynamicdisplacement fields in quasi-brittle opaque materials.Thus, indirect method were applied to estimate the KID.For example, on the basis of well-known Irwin’s cor-relation KID = √

GdCE/(1 − ν2), (Chen et al., 2009)determined KID using a laser gap gauge and high-speed photography. However, it should be noted that,

Figure 5. (a) High-speed photographs of the movement ofthe flying fragments after impact (continued to Figure 2), (b)schematic of the fracturing specimen (O is the centre of mass,rOO′ is the distance of translational movement), (c) dynamicfracture energy as a function of crack speed.

when a crack propagated dynamically, the dynamicenergy release rate Gd(t, v) is related to the dynamicSIF Kdyn

I (t, v) (Ravi-Chandar, 2004)

Thus, KID at a specific crack speed is rated to thecritical GdC via

where AI(v) = ν2αd(1−ν)C2

SR(v), αd =

√1 − v2

C2L

,

αs =√

1 − v2

C2S

, R(v) = 4αdαs − (1 + α2s )2

Figure 6 shows the normalized KID/KIC increaseswith the increase of v/CR. It is interesting to note thatthe rate-sensitivity of growth toughness is more evi-dent than that of initiation toughness. The degree ofKID/KIC in CG marble is apparently lower than that inthe other three rocks, which depends on the dissipatedenergy to create the crack surface area.

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Figure 6. Normalized dynamic crack growth toughnessKID/KIC as a function of normalized crack speed v/CR.

5 CONCLUSIONS

Notched semi-circular bending tests were performedto study quasi-static and dynamic fracture behaviourof four well-studied rock types. On-specimen straingauges and high-speed photography were used todetermine dynamic fracture parameters at the macro-scopic scale. The main conclusions of this study wereas follows:

(1) The dynamic crack initiation toughness obtainedfrom the quasi-static analysis was evidenced bythe force equilibrium until the time to fracture.The crack speed and velocity of flying fragmentswere quantitatively determined from high-speedphotographs. The dynamic fracture energy wasestimated from the velocity of flying fragments,the absorbed energy and fracture surface areaof the specimen, which increased rapidly withthe increase of crack speed. The dynamic crackgrowth toughness was then derived from well-known Irwin’s correlation at a specific crackspeed.

(2) The dynamic crack initiation and growth tough-ness of four rock types were dependent on theloading rate, and the rate-sensitivity of growthtoughness was more evident than that of the for-mer. The degrees of normalized crack initiationtoughness in gabbro and fine-grained marble wereapparently higher than those in sandstone andcoarse-grained marble, which were mainly gov-erned by the time of stress wave required totravel through the specimen. The higher growthtoughness was associated with a greater degree oftransgranular microcracks.

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