Research ArticleEffect of Material Properties and Strain Rate on Fragmentation ofAnisotropic Rock

R J Wu 12 and H B Li 12

1State Key Laboratory of Geomechanics and Geotechnical Engineering Institute of Rock and Soil MechanicsChinese Academy of Sciences Wuhan 430071 China2University of Chinese Academy of Sciences Beijing 100049 China

Correspondence should be addressed to H B Li hbliwhrsmaccn

Received 6 July 2020 Revised 6 September 2020 Accepted 21 September 2020 Published 13 October 2020

Academic Editor Fengqiang Gong

Copyright copy 2020 R J Wu and H B Li +is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In order to investigate the effect of the microproperties of bedding and strain rate on the fragment size distributions of layeredphyllite with different bedding dip angles a split Hopkinson pressure numerical model was established and verified by comparingwith the experimental results A newmethod to obtain reasonable layered rock dynamic simulation result was proposed+en thecumulative distribution curve and average fragment size of layered rocks were calculated after changing the strain rate andmicroparameters of bedding in the model +e results showed that the samples tend to become pulverized under high strain rateand it was harder for the samples with low dip angle to be damaged if the bedding shear strength is added while the fragmentationof high angle samples did not change significantly Furthermore the failure of layered specimens was not affected by the tensilestrength and stiffness +e wider bedding and narrower space promoted the crack initiation and propagation

1 Introduction

Rock splintering under high rate load could be observed inblasting [1] tunneling [2] earthquakes [3] rock bursts andshale gas well fracturing [4 5] +erefore it is essential toinvestigate the rock fragmentation behavior for solvingmany practical engineering problems such as blasting andexcavation could be achieved more safely and economicallyefficient rock disintegration and reliable architectural design[6ndash8]

During a brittle fragmentation event microcracksoriginate under stress concentrations open propagaterapidly to interact with another and eventually coalesce toform separated fragments Due to the complexity of dynamicrock fracture a range of experiments theoretical modelsand computational simulations have been pursued to enrichthe knowledge of the fragmentation over the past severaldecades [9ndash12] Rosin and Rammler [9] constructed afragment size distributions function which is quoted toform the widely used blasting prediction fragmentation

model [1] Besides a series of theoretical analysis [10 11] avariety of simulation methods have been done to provideinsights into the rock pulverization process Zhou andMolinari [12] used a finite element ring model to study thedynamic fragmentation properties of materials under tensileimpact load +e effects of ring size the energy conversionsand related properties are discussed With the help of thegrain-based discrete element method (GB-DEM) Li et al[13] put forward that there exist two classes of broken be-havior corresponding to the different rock pulverizationphenomenon under dynamic loads

Recent investigations have reported that the debris size isdetermined by initial defect properties [14] and thus moreattention has been paid to the effect of heterogeneity on rockfragmentation Levy and Molinari [15] extended the work ofZhou and Molinari [12] by including defect distributions toanalyze the influence of defects on average fragment size+e results showed that the distribution tail has a criticalinfluence on the fragmentation process Paliwal and Ramesh[16] concluded that the spread of the preexisting flaw

HindawiAdvances in Civil EngineeringVolume 2020 Article ID 8841796 21 pageshttpsdoiorg10115520208841796

distribution is critical at low rates while the flaw density iscrucial under high strain rate load +e rock appearstransversely isotropic as the microstructural defects arearranged in a preferred orientation which means the bed-ding plane dominates the fragment size distributionsHowever on the one hand defects and fragmentation re-main an open issue [16] especially the effect of rock het-erogeneity on fragment is not clear On the other handalthough there are some reports on the mechanical prop-erties of layered rocks [17 18] few studies have involvedhow the bedding properties affect rock fragmentation

+e discrete element model not only could simulate theinitiation growth and coalescence of microcracks duringrock failure but also directly reproduce the fragmentation ofthe rock [18ndash20] Defects in rock such as bedding could alsobe simulated [21] +ree-dimensional discrete elementmodel has obtained fragment with high fidelity [19]Nonetheless 3D simulations of rock fragmenting structuresshow technical limitations related to the computational costwhen analyzing an extensive amount of data +e 2D nu-merical model could be utilized to observe a number ofphenomena for example fragmentation spallation anddamage evolution [22 23] Considering that numerousmodel calculations are needed in the study of layered rocksthe 2D model approach offers an efficient compromisebetween accuracy and complexity +erefore 2D discreteelement model is selected to explore the effect of differentparameters on the layered rock fragmentation

In this study quasistatic and dynamic tests for layeredphyllite with different bedding dip angle were done tocalibrate themicroscopic parameters of the two-dimensionaldiscrete element model Experimental and numerical stress-time curves and ultimate fragment states were compared tovalidate the model +en the parameters of the beddingplane and the strain rate were changed to investigate thedynamic fragmentation +e numerical results were ana-lyzed to determine how the microscopic properties ofbedding plane and strain rate affect the fragment size dis-tributions of layered rock with different bedding dip angle

2 Specimen Preparation andExperimental Setup

21 Specimen Preparation Since there is no well theoreticalmethod to define the parameters of the discrete elementmodel through the macroparameters of rock and also toprovide reference for the verification of the model thelayered phyllite is selected as the sample of dynamic ex-periment +e X-ray analysis indicates that the sample ismainly composed of three kinds of mineral muscovite(2894) chlorite (2082) and albite (1947) respec-tively +e rest of the minerals are quartz (1916) calcite(955) and others (206) +e typical layered structurehas been inspected by a scanning electron microscope(SEM) as shown in Figure 1 According to the InternationalSociety for Rock Mechanics (ISRM) suggestion [24] thediameter and height of samples are 50mm and 25mmrespectively All 30 samples are divided equally into qua-sistatic and dynamic group which are cut from a block as

shown in Figure 2(a) Five bedding dip angles (0deg 225deg 45deg675deg and 90deg) are included in each group and the definitionof bedding dip angle β is presented in Figure 2(b)

22 Split Hopkinson Pressure Bar (SHPB) +e SHPB testssystem located at the State Key Laboratory of Wuhan In-stitute of geotechnical mechanics is employed to obtain rockfragment +e apparatus comprises a striker(400mmtimesΦ50mm) an incident bar (2500mmtimesΦ50mm)a transmitted bar (2500mmtimesΦ50mm) and an absorber(1000mmtimesΦ50mm) as shown in Figure 3 All thesecomponents are constructed by alloy steel with elasticmodulus of 210GPa P-wave velocity 5189ms and densityof 7800 kgm3 +e layered phyllite is placed between theincident bar and the transmitted bar In order to satisfy theassumptions of stress equilibrium in the sample and stresswave propagation in one dimension [25 26] a bullet-shapestriker and pulse shaper are used to provide a smooth in-cident wave stress [27] +e incident strain wave εI (t) thereflected strain wave εR (t) and the transmitted strain waveεT (t) are recorded by the dynamic strain gauges mounted onthe incident bar and transmitted bar to calculate the strainrate _ε (t) strain ε(t) and stress σ(t)[28]

_ε(t) c

l0εI minus εR minus εT( 1113857

ε(t) c

l01113946

t

0εI minus εR minus εT( 1113857dt

σ(t) A

2A0E εI + εR + εT( 1113857

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(1)

where c is the P-wave velocity of the bar material l0 is thelength of the specimen E is the Young modulus of the barmaterial and A and A0 are the cross-sectional area of thebars and sample respectively

23 Experimental Results +e three-wave method should bedone to judge whether the stress at both ends of the sample isthe same to guarantee stress equilibrium during the appli-cation of impact load [29] Figure 4 shows the dynamic stressat the incident and the transmission bar ends of samples+eagreement between σt and σi+ σr indicates that stressequilibrium is achieved

Quasistatic tests of layered phyllite were carried out onthe rock mechanics experimental (RMT) device +e qua-sistatic and dynamic mechanical properties of layeredphyllite are listed in Table 1 which would be utilized tocalibrate parameters of numerical model

3 Numerical Model in DEM

31 Model Setup +e bonded-particle model (BPM) [30] isperformed to build the SHPB numerical simulation testsystem+e SHPB bars and rock matrix are characterized bythe parallel bond model and the smooth joint model isinserted to solve the inherent roughness problems [31] As

2 Advances in Civil Engineering

illustrated in Figure 5 the incident bar and the transmittedbar are all set to 1500mm which is long enough to avoidsuperposition between the stress wave [20] +e microscopicproperties of particles and bonds assigned to the SHPB barsare tabulated in Table 2+e Young modulus and density are

consistent with the laboratory apparatus Using regular-distributed particles to construct the SHPB simulation de-vice model could only reproduce the homogeneous of alloysteel but also effectively ensure the one-dimensional wavepropagation [13] thus the regular distribution of 5mm

(a)

50mm

Bedding plane

50mmβ

(b)

Figure 2 Preparation of layered phyllite specimens with different bedding angles (a) Schematic diagram of sample acquisition (b)Definition of bedding angle and specimen size

Transmitted barSampleIncident bar

Ultrahigh dynamic strainometer

Striker

Tim

e

Gauge GaugePulse shaper

Absorber

Incident wave

Transmission wave

Reflected wave

Figure 3 SHPB system and stress wave propagation situation

Bedding plane

Rock matrix

Figure 1 Layered structure observed under SEM

Advances in Civil Engineering 3

Stre

ss (M

Pa)

σiσt

σrσi + σr

β = 0degndash500

ndash250

0

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500

50 100 150 2000Time (micros)

(a)

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σrσi + σr

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ss (M

Pa)

ndash500

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(b)

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σrσi + σr

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Stre

ss (M

Pa)

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σrσi + σr

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ss (M

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Figure 4 +ree-wave method to check experimental stress equilibrium

4 Advances in Civil Engineering

particles is employed to form the bar model +e tensilestrength and shear strength of the SHPB bars are both set as1e100 Pa to ensure that they would not be damaged duringwave propagation

Calibration for sample parameters is cautious to ac-complish several steps to reproduce the real experimentalprocess +e 0deg sample and 90deg sample under quasistatic loadare employed to preliminary determine the strength andYoungrsquos modulus of numerical model respectively+e 675degsample is selected to identify the smooth joint bond strengthconsidering the layered rock with this angle failed along thebedding planes +en the stiffness of the bedding are ob-tained by using the equivalent continuous model [22] Be-fore completing the model dynamic test results are utilizedto correct the parameters +e microscopic properties ofparticles and bonds of the sample are listed in Table 3

32 Validation of the DEM Model

321 Dynamic Stress Equilibrium A successful SHPB ex-periment must meet two requirements the stress wavepropagates in the form of one-dimensional wave in the bar

and the dynamic stress at both ends of the specimen remainsequal during the loading process Li et al [20] have provedthat homogeneous particles could well maintain the one-dimensional propagation of stress waves in the DEMmodel+e three-wave method as an indirect approach isemployed to compare the force of the specimen on two endsin laboratory experiments after considering that the axialforce of the specimen is hard to measure directly [26]However the forces at both ends of specimen could bedirectly obtained to judge whether it is in a stress equilib-rium state in the numerical model+erefore the three-wavemethod and the direct measurement method are both ap-plied to guarantee that the sample be in the dynamic stressequilibrium state during the impact loading process +eresult of the three-wave method is presented in Figure 6(a)which demonstrates that forces at both ends of the specimencalculated by the incident bar and the transmission bar arebasically equal

For the direct measurement method stress on the leftand right end of the sample is directly recorded +e stressequilibrium coefficient η 2(σright minus σ left)(σright + σ left) iscalculated to compare the stresses at both ends of the sampleAs shown in Figure 6(b) η remains around zero near thepeak sections which directly indicates that the axial force ofthe specimen is equal during the loading process

322 Ultimate Fragment States To further validate theBPM numerical model dynamic tests on the 45deg samples aresimulated +e incident stress wave of 45deg sample is derivedfrom the experiment to ensure the numerical dynamic stressis comparable with the experimental as shown inFigure 7(a) Figures 7(b)sim7(f) illustrate the experimental andnumerical dynamic uniaxial compression stressndashtime curvefor five angle samples whose values and change trends arebasically consistent +e dynamic compressive strength oflayered rock changes in U-shape with the increase of bed-ding angle and the strength is the minimum at 675degFigure 7(g) depicts ultimate fragment state of five angle

1000mm 1000mm500mm500mm

Incident bar Transmitted barSpecimenMeasure Measure

kn

ks

Measure circle in bar Layered specimen Smooth joint model

Figure 5 +e SHPB numerical simulation test system

Table 1 Quasistatic and dynamic mechanical properties of layered phyllite

Bedding angle β 0 225 45 675 90Quasistatic elastic modulus (GPa) 171 291 256 277 194Quasistatic compression strength (MPa) 1522 1277 736 631 1157Dynamic strain rate (sminus1) 948 994 1382 1426 1372Dynamic compression strength (MPa) 2481 2146 1432 1225 1870

Table 2 Microscopic properties of particles and bonds of the SHPBbars

Properties Material parametersrsquo valueParticlesBall density (kgm3) 7900Ball radius (m) 05e-3Particle-based material parametersYoungrsquos modulus (Ec Pa) 182e9Stiffness ratio (knks) 235Parallel bond model parametersYoungrsquos modulus (Ec Pa) 182e9Stiffness ratio (knks) 235Normal strength (σc Pa) 1e100Shear strength (τc Pa) 1e100

Advances in Civil Engineering 5

samples after the experimental and numerical tests at thestrain rate asymp140 sminus1 +e simulation results of low beddingangle samples show that the bedding plane affects the localfailure of the samples High bedding angle samples arebroken into schistose along the bedding and partially pul-verized in both situations +ese results indicate that theSHPB numerical simulation test system is reasonable tocharacterize the realistic behaviors of layered phyllite

4 Results and Discussion

+e particles that are still bonded together after loading aredefined as a fragment +e fragment that is composed byindividual particle is not counted to simplify the calculation+e cumulative distribution curve of layered rocks with five

bedding angles is counted in different situation What ismore the average fragment size is calculated to quantita-tively compare the damage

δ 1113944

n

i1ηiδi

1113944n

i1ηi

(2)

where n is the total fragment number δ is the area of the ithfragment and η represents the percentage of the ith frag-ment area

41 Effect of Strain Rate Five impact velocities were appliedto the simulation layered samples to obtain the fragmen-tation under different strain rates As shown in Figure 8 theloading strain rate has a significant effect on the strength and

Table 3 Microscopic properties of particles and bonds of the sample

Properties Material parametersrsquo valueParticlesBall density (kgm3) 2620Minimum ball radius (m) 02eminus 3Maximum ball radius (m) 0332eminus 3Youngrsquos modulus (Ec Pa) 164e9Stiffness ratio (knks) 115Parallel bond modelYoungrsquos modulus (Ec Pa) 164e9Stiffness ratio (pb knpb ks) 115Normal strength (pb σc Pa) 52e6plusmn 13e6Shear strength (pb_τc Pa) 140e6plusmn 35e6Smooth joint modelNormal stiffness (sj kn (Pam)) 1573e12Shear stiffness (sj ks (Pam)) 638e12Friction coefficient (φ) 35degCohesion (sj_τc Pa) 36e6plusmn 3e6Tensile strength (sj σc Pa) 3e6plusmn 05e6

Stre

ss (M

Pa)

ndash500

ndash250

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250

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35 70 105 1400Time (micros)

σiσt

σrσi + σr

(a)

Stre

ss (M

Pa)

0

50

100

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ndash2

ndash1

0

1

2

Stre

ss eq

uilib

rium

coef

ficie

nt

40 80 1200Time (μs)

σrightσleη

(b)

Figure 6+e dynamic stress equilibrium state of SHPB numerical model (a)+ree-wavemethod to check numerical stress equilibrium (b)Direct measurement method to check stress equilibrium

6 Advances in Civil Engineering

Stre

ss (M

Pa)

Experimental incident waveNumerical incident wave

0

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Experimental stressndashtimeNumerical stressndashtime

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Experimental stressndashtimeNumerical stressndashtime

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ress

(MPa

)

Experimental stressndashtimeNumerical stressndashtime

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β = 675deg

Stre

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Experimental stressndashtimeNumerical stressndashtime

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β = 90deg

Stre

ss (M

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Experimental stressndashtimeNumerical stressndashtime

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80

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20 40 60 80 100 120 1400Time (μs)

(f )

Figure 7 Continued

Advances in Civil Engineering 7

(g)

Figure 7 SHPB numerical model and simulation results (a) Numerical and experimental incident stress wave of 45deg sample (bndashf)Numerical and experimental dynamic compression stress of five angle samples (g) +e ultimate fragment state of five angle samples afterexperiment and simulation

Perc

ent p

assin

g

β = 0deg0

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Fragment size (mm2)

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assin

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Fragment size (mm2)

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Figure 8 Continued

8 Advances in Civil Engineering

failure of the specimen Under any dynamic load thestrength of the specimen always presents a U-shaped changewith the increase of bedding angle and the minimumstrength occurs at 45degndash675deg

+e fragment size decreases gradually as the strain rateincreases while the strength presents an opposite trend +esamples with low bedding angle show minor damage undersmaller impact load and that changes at higher strain rateWith the increase of strain rate more cracks initiate alongthe loading direction and propagate to intersect with the

cracks on the weak bedding plane of 0degndash225deg sample whichleads to further failure of the specimen

Although the impact splitting effect is increased theweak bedding plane limits the crack to the local part of45degndash675deg specimen and hinders its further expansioncausing specimensrsquo difficulty to be further destroyed Asshown in Figure 8 when reaching a certain strain rate theshiver of rocks with β 45degndash675deg tends to be stable +eresults explored that in order to achieve better fragmentationunder high strain rate the relationship between the loading

β = 90deg

Perc

ent p

assin

g

0

20

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Fragment size (mm2)

(e)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

15

20

25

30

35

lgδ_

120 18060Strain rate (sndash1)

(f )

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

Stre

ngth

(MPa

)

100

150

200

250

300

100 150 200 25050Strain rate (sndash1)

(g)

Figure 8 +e cumulative distribution curve and average fragment size of layered rocks under different strain rates (andashe) +e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red _ε asymp 80 sminus1 blue _ε asymp 140 sminus1 yellow _ε asymp 170 sminus1 green _ε asymp 200 sminus1 and gray _ε asymp 215 sminus1 (f )Average fragment size vs strain rate under different bedding angles (g) Dynamic strength vs strain rate under different bedding angles

Advances in Civil Engineering 9

β = 0deg

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assin

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Fragment size (mm2)

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Fragment size (mm2)

(e)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

15

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lgδ_

50 100 1500Coh (MPa)

(f )

Figure 9 Continued

10 Advances in Civil Engineering

direction and the bedding dip angle should be changedunder different impact forces

42 Effect of Weak Bedding Strength

421 Effect of Weak Bedding Shear Strength +e weakbedding plane is endowed with cohesion strength for 4MPa24MPa 36Mpa 72Mpa and 140MPa respectively asshown in Figure 9 +e minimum strength corresponds tothe weak bedding immediately destroyed the last is the bondstrength of matrix

Tien et al [32] studied the properties of homogeneousrock-like materials and layered materials under staticloading It is considered that the failure of β 0deg samples isnot affected by the bedding as the compressive strength issimilar to that of isotropic rock However from the view ofrock failure it was found that the bedding would disturb thecrack propagation of the specimen When the shear strengthof the bedding plane is minor the bedding as a weak planeleads to the local fracture of 0degndash225deg specimen And there isno preferential defect in the sample as the shear strengthincreases so the degree of fragmentation decreases

+e situation becomes complicated for 45deg specimens+e bedding plane was destroyed first at low bedding shearstrength and the failure that was perpendicular to the weakplane of bedding has happened which leads to the furtherfracture of the specimen However in the case of highbedding shear strength the bedding no longer hinders thepropagation of the splitting crack resulting in smallerfragments of the specimen At the same time the bedding

shear strength makes no contribution to the crack growthand dynamic strength of the 90deg samples

422 Effect of Weak Bedding Tension Strength At eachbedding angle five kinds of tensile strength weak planespecimens are loaded from the weakest 001MPa to 52MPaas same as with the matrix +e results listed in Figure 10illustrate that the tensile strength of the weak plane has littleeffect on the ultimate fragment state and the dynamiccompression strength In most cases the upper part of thecumulative curve causes differentiation pointing that thetensile strength mainly limits the generation of large pieces

Only for 0deg specimen the crack was harder to develop asthe strength enhanced Compared with the bedding shearstrength situation it could be seen that the bedding shearstrength influences the failure of 225deg specimen while the 0degspecimen is mainly damaged by tensile strength at strainrateasymp120 sminus1 +e average size of high bedding angle spec-imen is kept at a small level and near-zero strength results inless fragmentation

43 Effect of Weak Bedding Stiffness +e initial knks ofbedding is 246 which is changed to explore the influenceof stiffness on rock fragmentation as shown in Figure 11 Itis obvious that the bedding stiffness has no effect on thefailure of layered rock However the average fragment sizeof high angle samples tends to be consistent when the kngradually increase +e failure of bedding plane occurs inthe initial loading stage which results in its ability to resist

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

Stre

ngth

(MPa

)90

180

270

50 100 1500Coh (MPa)

(g)

Figure 9 +e cumulative distribution curve and average fragment size of layered rocks with different bedding shear strength (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red coh 4MPa blue coh 24MPa yellow coh 36MPa green coh 72MPaand gray coh 140MPa (f ) Average fragment size vs bedding shear strength under different bedding angles (g) Dynamic strength vsbedding shear strength under different bedding angles

Advances in Civil Engineering 11

Perc

ent p

assin

g

β = 0deg0

20

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101 102 103100

Fragment size (mm2)

(a)

β = 225deg

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ent p

assin

g

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Fragment size (mm2)

(b)

β = 45deg

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(d)

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Fragment size (mm2)

(e)

20

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lgδ_

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 10 Continued

12 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 10 +e cumulative distribution curve and average fragment size of layered rocks with different bedding tensile strength (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red ten 001MPa blue ten 3MPa yellow ten 18MPa green ten 42MPaand gray ten 52MPa (f ) Average fragment size vs bedding tension strength under different bedding angles (g) Dynamic strength vsbedding tension strength under different bedding angles

Perc

ent p

assin

g

β = 0deg0

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Fragment size (mm2)

(a)

β = 225deg

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assin

g

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Fragment size (mm2)

(b)

Figure 11 Continued

Advances in Civil Engineering 13

β = 45deg

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assin

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Fragment size (mm2)

(c)

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(d)

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Fragment size (mm2)

(e)

20

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lgδ_

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 11 Continued

14 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 11 +e cumulative distribution curve and average fragment size of layered rocks with different bedding stiffness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red kskn 10 blue kskn 246 yellow knks 1 green knks 246 and grayknks 10 (f ) Average fragment size vs bedding stiffness under different bedding angles (g) Dynamic strength vs bedding stiffness underdifferent bedding angles

Perc

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β = 0deg0

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Fragment size (mm2)

(a)

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Fragment size (mm2)

(b)

Figure 12 Continued

Advances in Civil Engineering 15

β = 45deg

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Fragment size (mm2)

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Fragment size (mm2)

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g

0

20

40

60

80

100

101 102100

Fragment size (mm2)

(e)

15

20

25

30

1 2 3 4 50Thickness (mm)

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 12 Continued

16 Advances in Civil Engineering

Stre

ngth

(MPa

)

120

160

200

240

1 2 3 4 50Thickness (mm)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 12 +e cumulative distribution curve and average fragment size of layered rocks with different bedding thickness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red thickness 08mm blue thickness 14mm yellow thickness 20mmgreen thickness 36mm and gray thickness 50mm (f) Average fragment size vs thickness under different bedding angles (g)Dynamic strength vs thickness under different bedding angles

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 13 Continued

Advances in Civil Engineering 17

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

15

20

25

30

35

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(f )

Figure 13 Continued

18 Advances in Civil Engineering

deformation having no longer an effect on the crackpropagation

44Effect ofWeakBeddingAickness As shown in Figure 12weak plane models with thicknesses of 08mm 14mm20mm 36mm and 50mm are constructed +e beddingstrength is half of the original matrix rather than using SJmodel to avoid the distortion of the simulation results led byall the SJ model damage during loading +e smallestthickness represents the existence of a weak surface In thecase of 50mm half sample is strong matrix and the other isweak It could be seen from Figure 12 that when there existweak planes in the 45deg specimen the central bedding planedivides the specimen into two parts bearing the load sep-arately As a result the bearing capacity of the 45deg specimenschanged little under different conditions which lead tosimilar failure patterns for 45deg specimens

However the bearing capacity and failure mode ofspecimens with other bedding angles change greatly when theproportion of weak plane increases +e simulation revealsthat more defects lead to greater fragmentation and the sizehas a significant downward trend for all angles except 45degwhich means most angle rocks are sensitive to it At the sametime the distribution curves for 45deg specimen have littledifferences and others move left as a whole +e weak planealways makes the specimen with high angle easier to developmore cracks regardless of its width compared to 225degspecimen being the most difficult to be further damaged

45 Effect of Weak Bedding Space A smaller spacing meansthat the number of laminations increases which changesfrom 10 to 1 It could be analyzed from Figure 13 that whenthe bedding angle is greater than 45deg the failure of thespecimens under any spacing is influenced by the beddingplane And more split cracks begin to appear due to thebarrier effect of weak plane being weakened when there isonly one weak plane +e smaller fragment would emerge asthe split crack intersects with the weak plane +e localfailure of specimens is controlled by the bedding planeswhen the spacing is less than 12mm for the specimens withlow bedding angle However the effect of bedding planes for0deg specimens is no longer significant with the space in-creasing while that of 225 specimens still affects localfracture As a result the fragment distribution of otherbedding dip angle samples except 225deg correspondinglyshifts to the left when the bedding space decreases Nev-ertheless the average fragment size rises slightly when thespace exceeds a quarter

5 Conclusions

+e effect of the microscopic properties of bedding andstrain rate on the fragment size distributions of layered rockwith different bedding dip angles was investigated by usingthe 2D discrete element model Several conclusions weredrawn from the experiment

(1) A SHPBnumerical model was established and validatedA new method to obtain reasonable layered rock

Stre

ngth

(MPa

)90

180

270

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(g)

Figure 13+e cumulative distribution curve and average fragment size of layered rocks with different bedding space (andashe)+e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red space 5mm blue space 65mm yellow space 10mm green space 125mm andgray space 25mm (f) Average fragment size vs bedding space under different bedding angles (g) Dynamic strength vs bedding spaceunder different bedding angles

Advances in Civil Engineering 19

dynamic simulation result is proposed +e quasistaticand dynamic test results are combined to calibrate theelastic modulus and strength of layered rock to repro-duce the stressndashstrain curve and ultimate fragment state

(2) +e average size of samples with different beddingangles decreases as the strain rate increases while thestrength presents an opposite trend +e samplestend to become pulverized under high strain rateespecially for the low bedding angle samples as morecracks initiate along the loading direction andpropagate to intersect with the cracks

(3) It is harder for the samples with low dip angle to bedamaged if the bedding shear strength is added whilethe fragmentation of high angle samples does notchange significantly +e average fragment size of 45degsamples varies greatly with the shear strength becausethe failure mode of the specimens changed And thefailure of layered specimens with all dip angles is notaffected by the tensile strength and stiffness

(4) +e wider bedding may promote the crack initiationand propagation under the same impact load whichmeans large fragments develop into small ones It wasinitially considered that the effect of bedding plane onlowdip angle samples is not considerableHowever thebedding has a dramatic impact on the final fragment

(5) Because the effect of hindering splitting crack hasalways existed for the bedding plane of 225deg sampleand more cracks initiate along the loading directionthe fragment size distribution curve of all beddingangle samples gradually moves to the right when thebedding space increases except for the 225deg samplesNevertheless the average fragment size rises slightlywhen the space exceeds a range

Data Availability

Some data models or codes generated or used during thestudy are available from the corresponding author uponrequest static mechanical properties of layered phyllite withdifferent bedding angles and all results of the dynamiccompression test

Conflicts of Interest

+e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

+e constructive comments of three anonymous reviewersare appreciated +is work received funding from the Na-tional Natural Science Foundation of China (Nos 51439008and 51679231) and China Scholarship Council (to the firstauthor No 202004910629)

References

[1] C V B Cunningham ldquoFragmentation estimations and theKuz-Ram modelndashfour years onrdquo in Proceedings of the 2nd

International Symposium on Rock Fragmentation by Blastingpp 475ndash487 Keystone CO USA August 1987

[2] X F Li H B Li Y Q Liu Q C Zhou and X Xia ldquoNumericalsimulation of rock fragmentation mechanisms subject towedge penetration for TBMsrdquo Tunnelling and UndergroundSpace Technology vol 53 pp 96ndash108 2016

[3] J C Li H B Li and J Zhao ldquoAn improved equivalentviscoelastic medium method for wave propagation acrosslayered rock massesrdquo International Journal of Rock Mechanicsand Mining Sciences vol 73 no 1 pp 62ndash69 2015

[4] F Gong W Wuxing Wu T Tianbin Li and X Xuefeng SildquoExperimental simulation and investigation of spalling failureof rectangular tunnel under different three-dimensional stressstatesrdquo International Journal of Rock Mechanics and MiningSciences vol 122 p 104081 2019

[5] D A Shockey D R Curran L Seaman J T Rosenberg andC F Petersen ldquoFragmentation of rock under dynamic loadsrdquoInternational Journal of Rock Mechanics and Mining Sciencesamp Geomechanics Abstracts vol 11 no 8 pp 303ndash317 1974

[6] F Q Gong S Luo G Lin and X B Li ldquoEvaluation of shearstrength parameters of rocks by preset angle shear directshear and triaxial compression testsrdquo Rock Mechanics andRock Engineering vol 53 no 5 pp 2505ndash2519 2020

[7] X F Li H B Li Q B Zhang J L Jiang and J ZhaoldquoDynamic fragmentation of rock material characte-ristic sizefragment distribution and pulverization lawrdquo EngineeringFracture Mechanics vol 199 pp 739ndash759 2018

[8] J D Hogan R J Rogers J G Spray and S Boonsue ldquoDynamicfragmentation of granite for impact energies of 6ndash28 Jrdquo En-gineering Fracture Mechanics vol 79 pp 103ndash125 2012

[9] P Rosin and E Rammler ldquo+e laws governing the fineness ofpowdered coalrdquo Journal of Institute of Fuel vol 7 pp 29ndash36 1933

[10] N F Mott and E H Linfoot ldquoMinistry of supply +e rawmaterial guiderdquo Ae Analyst vol 68 no 806 1943

[11] D E Grady ldquoLocal inertial effects in dynamic fragmentationrdquoJournal of Applied Physics vol 53 no 1 pp 322ndash325 1982

[12] F Zhou and J F Molinari ldquoOn the rate-dependency ofdynamic tensile strength of a model ceramic systemrdquo Com-puter Methods in Applied Mechanics and Engineering vol 194no 12-16 pp 1693ndash1709 2005

[13] X F Li X Li H B Li Q B Zhang and J Zhao ldquoDynamictensile behaviours of heterogeneous rocks the grain scalefracturing characteristics on strength and fragmentationrdquoInternational Journal of Impact Engineering vol 118pp 98ndash118 2018

[14] F Q Gong X B Li and X L Liu ldquoPreliminary experimentalstudy of characteristics of rock subjected to 3D coupled staticand dynamic loadsrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 6 pp 1179ndash1190 2011

[15] S Levy and J F Molinari ldquoDynamic fragmentation of ce-ramics signature of defects and scaling of fragment sizesrdquoJournal of the Mechanics and Physics of Solids vol 58 no 1pp 12ndash26 2010

[16] B Paliwal and K T Ramesh ldquoAn interacting micro-crackdamage model for failure of brittle materials under com-pressionrdquo Journal of the Mechanics and Physics of Solidsvol 56 no 3 pp 896ndash923 2008

[17] R J Wu H B Li X F Li X Xia and L W Liu ldquoExperimentalstudy and numerical simulation of the dynamic behavior oftransversely isotropic phylliterdquo International Journal of Geo-mechanics vol 20 no 8 Article ID 04020105 2020

[18] X Ou X Zhang H Feng C Zhang and J Yang ldquoEffect of theconfining pressure on the dynamic compression properties of

20 Advances in Civil Engineering

transversely isotropic rocksrdquo Advances in Civil Engineeringvol 6 p 11 2019

[19] H-B Du F Dai Y Xu Y Liu and H-n Xu ldquoNumericalinvestigation on the dynamic strength and failure behavior ofrocks under hydrostatic confinement in SHPB testingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 108 pp 43ndash57 2018

[20] X F Li Q B Zhang H B Li and J Zhao ldquoGrain-baseddiscrete element method (GB-DEM) modelling of multi-scalefracturing in rocks under dynamic loadingrdquo Rock Mechanicsand Rock Engineering vol 51 no 12 pp 3785ndash3817 2018

[21] F Dai M D Wei N W Xu Y Ma and D S Yang ldquoNu-merical assessment of the progressive rock fracture mecha-nism of cracked chevron notched Brazilian disc specimensrdquoRock Mechanics and Rock Engineering vol 48 no 2pp 463ndash479 2015

[22] B Park and K-B Min ldquoBonded-particle discrete elementmodeling of mechanical behavior of transversely isotropicrockrdquo International Journal of Rock Mechanics and MiningSciences vol 76 pp 243ndash255 2015

[23] E Cadoni ldquoDynamic characterization of orthogneiss rocksubjected to intermediate and high strain rates in tensionrdquoRock Mechanics and Rock Engineering vol 43 no 6pp 667ndash676 2010

[24] Y X Zhou K Xia X B Li et al ldquoSuggested methods fordetermining the dynamic strength parameters and Mode-Ifracture toughness of rock materialsrdquo International Journal ofRock Mechanics and Mining Sciences vol 49 no 1pp 105ndash112 2011

[25] F Dai K Xia J P Zuo R Zhang and N W Xu ldquoStatic anddynamic flexural strength anisotropy of Barre graniterdquo Rock Me-chanics and Rock Engineering vol 46 no 6 pp 1589ndash1602 2013

[26] F-Q Gong X-F Si X-B Li and S-Y Wang ldquoDynamictriaxial compression tests on sandstone at high strain ratesand low confining pressures with split Hopkinson pressurebarrdquo International Journal of Rock Mechanics and MiningSciences vol 113 pp 211ndash219 2019

[27] X F Li H B Li L W Liu et al ldquoInvestigating the crackinitiation and propagation mechanism in brittle rocks usinggrain-based finite-discrete element methodrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 127p 104219 2020

[28] F Gong and J Hu ldquoEnergy dissipation characteristic of redsandstone in the dynamic Brazilian disc test with SHPBsetuprdquo Advances in Civil Engineering vol 2020 Article ID7160937 10 pages 2020

[29] J C Li H B Li G W Ma and J Zhao ldquoA time-domainrecursive method to analyse transient wave propagationacross rock jointsrdquo Geophysical Journal Internationalvol 188 no 2 pp 631ndash644 2012

[30] P A Cundall ldquoA computer model for simulating progressivelarge-scale movement in blocky rock systemrdquo in Proceedingsof the International Symposium on Rock Mechanics NancyFrance October 1971

[31] B Park K-B Min N +ompson and P Horsrud ldquo+ree-dimensional bonded-particle discrete element modeling ofmechanical behavior of transversely isotropic rockrdquo Inter-national Journal of Rock Mechanics and Mining Sciencesvol 110 pp 120ndash132 2018

[32] Y M Tien M C Kuo and C H Juang ldquoAn experimentalinvestigation of the failure mechanism of simulated transverselyisotropic rocksrdquo International Journal of Rock Mechanics andMining Sciences vol 43 no 8 pp 1163ndash1181 2006

Advances in Civil Engineering 21

illustrated in Figure 5 the incident bar and the transmittedbar are all set to 1500mm which is long enough to avoidsuperposition between the stress wave [20] +e microscopicproperties of particles and bonds assigned to the SHPB barsare tabulated in Table 2+e Young modulus and density are

consistent with the laboratory apparatus Using regular-distributed particles to construct the SHPB simulation de-vice model could only reproduce the homogeneous of alloysteel but also effectively ensure the one-dimensional wavepropagation [13] thus the regular distribution of 5mm

(a)

50mm

Bedding plane

50mmβ

(b)

Figure 2 Preparation of layered phyllite specimens with different bedding angles (a) Schematic diagram of sample acquisition (b)Definition of bedding angle and specimen size

Transmitted barSampleIncident bar

Ultrahigh dynamic strainometer

Striker

Tim

e

Gauge GaugePulse shaper

Absorber

Incident wave

Transmission wave

Reflected wave

Figure 3 SHPB system and stress wave propagation situation

Bedding plane

Rock matrix

Figure 1 Layered structure observed under SEM

Advances in Civil Engineering 3

Stre

ss (M

Pa)

σiσt

σrσi + σr

β = 0degndash500

ndash250

0

250

500

50 100 150 2000Time (micros)

(a)

σiσt

σrσi + σr

β = 225deg

Stre

ss (M

Pa)

ndash500

ndash250

0

250

500

50 100 150 2000Time (micros)

(b)

σiσt

σrσi + σr

β = 45deg

Stre

ss (M

Pa)

ndash500

ndash250

0

250

500

35 70 105 1400Time (micros)

(c)

σiσt

σrσi + σr

β = 675deg

Stre

ss (M

Pa)

ndash500

ndash250

0

250

500

35 70 105 1400Time (micros)

(d)

β = 90deg

Stre

ss (M

Pa)

ndash500

ndash250

0

250

500

σiσt

σrσi + σr

35 70 105 1400Time (micros)

(e)

Figure 4 +ree-wave method to check experimental stress equilibrium

4 Advances in Civil Engineering

particles is employed to form the bar model +e tensilestrength and shear strength of the SHPB bars are both set as1e100 Pa to ensure that they would not be damaged duringwave propagation

Calibration for sample parameters is cautious to ac-complish several steps to reproduce the real experimentalprocess +e 0deg sample and 90deg sample under quasistatic loadare employed to preliminary determine the strength andYoungrsquos modulus of numerical model respectively+e 675degsample is selected to identify the smooth joint bond strengthconsidering the layered rock with this angle failed along thebedding planes +en the stiffness of the bedding are ob-tained by using the equivalent continuous model [22] Be-fore completing the model dynamic test results are utilizedto correct the parameters +e microscopic properties ofparticles and bonds of the sample are listed in Table 3

32 Validation of the DEM Model

321 Dynamic Stress Equilibrium A successful SHPB ex-periment must meet two requirements the stress wavepropagates in the form of one-dimensional wave in the bar

and the dynamic stress at both ends of the specimen remainsequal during the loading process Li et al [20] have provedthat homogeneous particles could well maintain the one-dimensional propagation of stress waves in the DEMmodel+e three-wave method as an indirect approach isemployed to compare the force of the specimen on two endsin laboratory experiments after considering that the axialforce of the specimen is hard to measure directly [26]However the forces at both ends of specimen could bedirectly obtained to judge whether it is in a stress equilib-rium state in the numerical model+erefore the three-wavemethod and the direct measurement method are both ap-plied to guarantee that the sample be in the dynamic stressequilibrium state during the impact loading process +eresult of the three-wave method is presented in Figure 6(a)which demonstrates that forces at both ends of the specimencalculated by the incident bar and the transmission bar arebasically equal

For the direct measurement method stress on the leftand right end of the sample is directly recorded +e stressequilibrium coefficient η 2(σright minus σ left)(σright + σ left) iscalculated to compare the stresses at both ends of the sampleAs shown in Figure 6(b) η remains around zero near thepeak sections which directly indicates that the axial force ofthe specimen is equal during the loading process

322 Ultimate Fragment States To further validate theBPM numerical model dynamic tests on the 45deg samples aresimulated +e incident stress wave of 45deg sample is derivedfrom the experiment to ensure the numerical dynamic stressis comparable with the experimental as shown inFigure 7(a) Figures 7(b)sim7(f) illustrate the experimental andnumerical dynamic uniaxial compression stressndashtime curvefor five angle samples whose values and change trends arebasically consistent +e dynamic compressive strength oflayered rock changes in U-shape with the increase of bed-ding angle and the strength is the minimum at 675degFigure 7(g) depicts ultimate fragment state of five angle

1000mm 1000mm500mm500mm

Incident bar Transmitted barSpecimenMeasure Measure

kn

ks

Measure circle in bar Layered specimen Smooth joint model

Figure 5 +e SHPB numerical simulation test system

Table 1 Quasistatic and dynamic mechanical properties of layered phyllite

Bedding angle β 0 225 45 675 90Quasistatic elastic modulus (GPa) 171 291 256 277 194Quasistatic compression strength (MPa) 1522 1277 736 631 1157Dynamic strain rate (sminus1) 948 994 1382 1426 1372Dynamic compression strength (MPa) 2481 2146 1432 1225 1870

Table 2 Microscopic properties of particles and bonds of the SHPBbars

Properties Material parametersrsquo valueParticlesBall density (kgm3) 7900Ball radius (m) 05e-3Particle-based material parametersYoungrsquos modulus (Ec Pa) 182e9Stiffness ratio (knks) 235Parallel bond model parametersYoungrsquos modulus (Ec Pa) 182e9Stiffness ratio (knks) 235Normal strength (σc Pa) 1e100Shear strength (τc Pa) 1e100

Advances in Civil Engineering 5

samples after the experimental and numerical tests at thestrain rate asymp140 sminus1 +e simulation results of low beddingangle samples show that the bedding plane affects the localfailure of the samples High bedding angle samples arebroken into schistose along the bedding and partially pul-verized in both situations +ese results indicate that theSHPB numerical simulation test system is reasonable tocharacterize the realistic behaviors of layered phyllite

4 Results and Discussion

+e particles that are still bonded together after loading aredefined as a fragment +e fragment that is composed byindividual particle is not counted to simplify the calculation+e cumulative distribution curve of layered rocks with five

bedding angles is counted in different situation What ismore the average fragment size is calculated to quantita-tively compare the damage

δ 1113944

n

i1ηiδi

1113944n

i1ηi

(2)

where n is the total fragment number δ is the area of the ithfragment and η represents the percentage of the ith frag-ment area

41 Effect of Strain Rate Five impact velocities were appliedto the simulation layered samples to obtain the fragmen-tation under different strain rates As shown in Figure 8 theloading strain rate has a significant effect on the strength and

Table 3 Microscopic properties of particles and bonds of the sample

Properties Material parametersrsquo valueParticlesBall density (kgm3) 2620Minimum ball radius (m) 02eminus 3Maximum ball radius (m) 0332eminus 3Youngrsquos modulus (Ec Pa) 164e9Stiffness ratio (knks) 115Parallel bond modelYoungrsquos modulus (Ec Pa) 164e9Stiffness ratio (pb knpb ks) 115Normal strength (pb σc Pa) 52e6plusmn 13e6Shear strength (pb_τc Pa) 140e6plusmn 35e6Smooth joint modelNormal stiffness (sj kn (Pam)) 1573e12Shear stiffness (sj ks (Pam)) 638e12Friction coefficient (φ) 35degCohesion (sj_τc Pa) 36e6plusmn 3e6Tensile strength (sj σc Pa) 3e6plusmn 05e6

Stre

ss (M

Pa)

ndash500

ndash250

0

250

500

35 70 105 1400Time (micros)

σiσt

σrσi + σr

(a)

Stre

ss (M

Pa)

0

50

100

150

ndash2

ndash1

0

1

2

Stre

ss eq

uilib

rium

coef

ficie

nt

40 80 1200Time (μs)

σrightσleη

(b)

Figure 6+e dynamic stress equilibrium state of SHPB numerical model (a)+ree-wavemethod to check numerical stress equilibrium (b)Direct measurement method to check stress equilibrium

6 Advances in Civil Engineering

Stre

ss (M

Pa)

Experimental incident waveNumerical incident wave

0

125

250

375

500

50 100 1500Time (micros)

(a)

β = 0deg

Stre

ss (M

Pa)

Experimental stressndashtimeNumerical stressndashtime

0

100

200

300

40 80 120 1600Time (μs)

(b)

β = 225deg

Stre

ss (M

Pa)

Experimental stressndashtimeNumerical stressndashtime

0

90

180

270

50 100 1500Time (μs)

(c)

β = 45degSt

ress

(MPa

)

Experimental stressndashtimeNumerical stressndashtime

0

40

80

120

160

25 50 75 1000Time (μs)

(d)

β = 675deg

Stre

ss (M

Pa)

Experimental stressndashtimeNumerical stressndashtime

0

50

100

150

35 70 105 1400Time (μs)

(e)

β = 90deg

Stre

ss (M

Pa)

Experimental stressndashtimeNumerical stressndashtime

0

80

160

240

20 40 60 80 100 120 1400Time (μs)

(f )

Figure 7 Continued

Advances in Civil Engineering 7

(g)

Figure 7 SHPB numerical model and simulation results (a) Numerical and experimental incident stress wave of 45deg sample (bndashf)Numerical and experimental dynamic compression stress of five angle samples (g) +e ultimate fragment state of five angle samples afterexperiment and simulation

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

100 101 102 10310ndash1

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 10310ndash1

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 103

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 103

Fragment size (mm2)

(d)

Figure 8 Continued

8 Advances in Civil Engineering

failure of the specimen Under any dynamic load thestrength of the specimen always presents a U-shaped changewith the increase of bedding angle and the minimumstrength occurs at 45degndash675deg

+e fragment size decreases gradually as the strain rateincreases while the strength presents an opposite trend +esamples with low bedding angle show minor damage undersmaller impact load and that changes at higher strain rateWith the increase of strain rate more cracks initiate alongthe loading direction and propagate to intersect with the

cracks on the weak bedding plane of 0degndash225deg sample whichleads to further failure of the specimen

Although the impact splitting effect is increased theweak bedding plane limits the crack to the local part of45degndash675deg specimen and hinders its further expansioncausing specimensrsquo difficulty to be further destroyed Asshown in Figure 8 when reaching a certain strain rate theshiver of rocks with β 45degndash675deg tends to be stable +eresults explored that in order to achieve better fragmentationunder high strain rate the relationship between the loading

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 10310ndash1

Fragment size (mm2)

(e)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

15

20

25

30

35

lgδ_

120 18060Strain rate (sndash1)

(f )

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

Stre

ngth

(MPa

)

100

150

200

250

300

100 150 200 25050Strain rate (sndash1)

(g)

Figure 8 +e cumulative distribution curve and average fragment size of layered rocks under different strain rates (andashe) +e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red _ε asymp 80 sminus1 blue _ε asymp 140 sminus1 yellow _ε asymp 170 sminus1 green _ε asymp 200 sminus1 and gray _ε asymp 215 sminus1 (f )Average fragment size vs strain rate under different bedding angles (g) Dynamic strength vs strain rate under different bedding angles

Advances in Civil Engineering 9

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

15

20

25

30

35

lgδ_

50 100 1500Coh (MPa)

(f )

Figure 9 Continued

10 Advances in Civil Engineering

direction and the bedding dip angle should be changedunder different impact forces

42 Effect of Weak Bedding Strength

421 Effect of Weak Bedding Shear Strength +e weakbedding plane is endowed with cohesion strength for 4MPa24MPa 36Mpa 72Mpa and 140MPa respectively asshown in Figure 9 +e minimum strength corresponds tothe weak bedding immediately destroyed the last is the bondstrength of matrix

Tien et al [32] studied the properties of homogeneousrock-like materials and layered materials under staticloading It is considered that the failure of β 0deg samples isnot affected by the bedding as the compressive strength issimilar to that of isotropic rock However from the view ofrock failure it was found that the bedding would disturb thecrack propagation of the specimen When the shear strengthof the bedding plane is minor the bedding as a weak planeleads to the local fracture of 0degndash225deg specimen And there isno preferential defect in the sample as the shear strengthincreases so the degree of fragmentation decreases

+e situation becomes complicated for 45deg specimens+e bedding plane was destroyed first at low bedding shearstrength and the failure that was perpendicular to the weakplane of bedding has happened which leads to the furtherfracture of the specimen However in the case of highbedding shear strength the bedding no longer hinders thepropagation of the splitting crack resulting in smallerfragments of the specimen At the same time the bedding

shear strength makes no contribution to the crack growthand dynamic strength of the 90deg samples

422 Effect of Weak Bedding Tension Strength At eachbedding angle five kinds of tensile strength weak planespecimens are loaded from the weakest 001MPa to 52MPaas same as with the matrix +e results listed in Figure 10illustrate that the tensile strength of the weak plane has littleeffect on the ultimate fragment state and the dynamiccompression strength In most cases the upper part of thecumulative curve causes differentiation pointing that thetensile strength mainly limits the generation of large pieces

Only for 0deg specimen the crack was harder to develop asthe strength enhanced Compared with the bedding shearstrength situation it could be seen that the bedding shearstrength influences the failure of 225deg specimen while the 0degspecimen is mainly damaged by tensile strength at strainrateasymp120 sminus1 +e average size of high bedding angle spec-imen is kept at a small level and near-zero strength results inless fragmentation

43 Effect of Weak Bedding Stiffness +e initial knks ofbedding is 246 which is changed to explore the influenceof stiffness on rock fragmentation as shown in Figure 11 Itis obvious that the bedding stiffness has no effect on thefailure of layered rock However the average fragment sizeof high angle samples tends to be consistent when the kngradually increase +e failure of bedding plane occurs inthe initial loading stage which results in its ability to resist

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

Stre

ngth

(MPa

)90

180

270

50 100 1500Coh (MPa)

(g)

Figure 9 +e cumulative distribution curve and average fragment size of layered rocks with different bedding shear strength (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red coh 4MPa blue coh 24MPa yellow coh 36MPa green coh 72MPaand gray coh 140MPa (f ) Average fragment size vs bedding shear strength under different bedding angles (g) Dynamic strength vsbedding shear strength under different bedding angles

Advances in Civil Engineering 11

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(e)

20

25

30

35

lgδ_

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 10 Continued

12 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 10 +e cumulative distribution curve and average fragment size of layered rocks with different bedding tensile strength (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red ten 001MPa blue ten 3MPa yellow ten 18MPa green ten 42MPaand gray ten 52MPa (f ) Average fragment size vs bedding tension strength under different bedding angles (g) Dynamic strength vsbedding tension strength under different bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 11 Continued

Advances in Civil Engineering 13

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(e)

20

25

30

35

lgδ_

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 11 Continued

14 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 11 +e cumulative distribution curve and average fragment size of layered rocks with different bedding stiffness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red kskn 10 blue kskn 246 yellow knks 1 green knks 246 and grayknks 10 (f ) Average fragment size vs bedding stiffness under different bedding angles (g) Dynamic strength vs bedding stiffness underdifferent bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 12 Continued

Advances in Civil Engineering 15

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

10ndash1 101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102100

Fragment size (mm2)

(e)

15

20

25

30

1 2 3 4 50Thickness (mm)

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 12 Continued

16 Advances in Civil Engineering

Stre

ngth

(MPa

)

120

160

200

240

1 2 3 4 50Thickness (mm)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 12 +e cumulative distribution curve and average fragment size of layered rocks with different bedding thickness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red thickness 08mm blue thickness 14mm yellow thickness 20mmgreen thickness 36mm and gray thickness 50mm (f) Average fragment size vs thickness under different bedding angles (g)Dynamic strength vs thickness under different bedding angles

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 13 Continued

Advances in Civil Engineering 17

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

15

20

25

30

35

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(f )

Figure 13 Continued

18 Advances in Civil Engineering

deformation having no longer an effect on the crackpropagation

44Effect ofWeakBeddingAickness As shown in Figure 12weak plane models with thicknesses of 08mm 14mm20mm 36mm and 50mm are constructed +e beddingstrength is half of the original matrix rather than using SJmodel to avoid the distortion of the simulation results led byall the SJ model damage during loading +e smallestthickness represents the existence of a weak surface In thecase of 50mm half sample is strong matrix and the other isweak It could be seen from Figure 12 that when there existweak planes in the 45deg specimen the central bedding planedivides the specimen into two parts bearing the load sep-arately As a result the bearing capacity of the 45deg specimenschanged little under different conditions which lead tosimilar failure patterns for 45deg specimens

However the bearing capacity and failure mode ofspecimens with other bedding angles change greatly when theproportion of weak plane increases +e simulation revealsthat more defects lead to greater fragmentation and the sizehas a significant downward trend for all angles except 45degwhich means most angle rocks are sensitive to it At the sametime the distribution curves for 45deg specimen have littledifferences and others move left as a whole +e weak planealways makes the specimen with high angle easier to developmore cracks regardless of its width compared to 225degspecimen being the most difficult to be further damaged

45 Effect of Weak Bedding Space A smaller spacing meansthat the number of laminations increases which changesfrom 10 to 1 It could be analyzed from Figure 13 that whenthe bedding angle is greater than 45deg the failure of thespecimens under any spacing is influenced by the beddingplane And more split cracks begin to appear due to thebarrier effect of weak plane being weakened when there isonly one weak plane +e smaller fragment would emerge asthe split crack intersects with the weak plane +e localfailure of specimens is controlled by the bedding planeswhen the spacing is less than 12mm for the specimens withlow bedding angle However the effect of bedding planes for0deg specimens is no longer significant with the space in-creasing while that of 225 specimens still affects localfracture As a result the fragment distribution of otherbedding dip angle samples except 225deg correspondinglyshifts to the left when the bedding space decreases Nev-ertheless the average fragment size rises slightly when thespace exceeds a quarter

5 Conclusions

+e effect of the microscopic properties of bedding andstrain rate on the fragment size distributions of layered rockwith different bedding dip angles was investigated by usingthe 2D discrete element model Several conclusions weredrawn from the experiment

(1) A SHPBnumerical model was established and validatedA new method to obtain reasonable layered rock

Stre

ngth

(MPa

)90

180

270

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(g)

Figure 13+e cumulative distribution curve and average fragment size of layered rocks with different bedding space (andashe)+e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red space 5mm blue space 65mm yellow space 10mm green space 125mm andgray space 25mm (f) Average fragment size vs bedding space under different bedding angles (g) Dynamic strength vs bedding spaceunder different bedding angles

Advances in Civil Engineering 19

dynamic simulation result is proposed +e quasistaticand dynamic test results are combined to calibrate theelastic modulus and strength of layered rock to repro-duce the stressndashstrain curve and ultimate fragment state

(2) +e average size of samples with different beddingangles decreases as the strain rate increases while thestrength presents an opposite trend +e samplestend to become pulverized under high strain rateespecially for the low bedding angle samples as morecracks initiate along the loading direction andpropagate to intersect with the cracks

(3) It is harder for the samples with low dip angle to bedamaged if the bedding shear strength is added whilethe fragmentation of high angle samples does notchange significantly +e average fragment size of 45degsamples varies greatly with the shear strength becausethe failure mode of the specimens changed And thefailure of layered specimens with all dip angles is notaffected by the tensile strength and stiffness

(4) +e wider bedding may promote the crack initiationand propagation under the same impact load whichmeans large fragments develop into small ones It wasinitially considered that the effect of bedding plane onlowdip angle samples is not considerableHowever thebedding has a dramatic impact on the final fragment

(5) Because the effect of hindering splitting crack hasalways existed for the bedding plane of 225deg sampleand more cracks initiate along the loading directionthe fragment size distribution curve of all beddingangle samples gradually moves to the right when thebedding space increases except for the 225deg samplesNevertheless the average fragment size rises slightlywhen the space exceeds a range

Data Availability

Some data models or codes generated or used during thestudy are available from the corresponding author uponrequest static mechanical properties of layered phyllite withdifferent bedding angles and all results of the dynamiccompression test

Conflicts of Interest

+e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

+e constructive comments of three anonymous reviewersare appreciated +is work received funding from the Na-tional Natural Science Foundation of China (Nos 51439008and 51679231) and China Scholarship Council (to the firstauthor No 202004910629)

References

[1] C V B Cunningham ldquoFragmentation estimations and theKuz-Ram modelndashfour years onrdquo in Proceedings of the 2nd

International Symposium on Rock Fragmentation by Blastingpp 475ndash487 Keystone CO USA August 1987

[2] X F Li H B Li Y Q Liu Q C Zhou and X Xia ldquoNumericalsimulation of rock fragmentation mechanisms subject towedge penetration for TBMsrdquo Tunnelling and UndergroundSpace Technology vol 53 pp 96ndash108 2016

[3] J C Li H B Li and J Zhao ldquoAn improved equivalentviscoelastic medium method for wave propagation acrosslayered rock massesrdquo International Journal of Rock Mechanicsand Mining Sciences vol 73 no 1 pp 62ndash69 2015

[4] F Gong W Wuxing Wu T Tianbin Li and X Xuefeng SildquoExperimental simulation and investigation of spalling failureof rectangular tunnel under different three-dimensional stressstatesrdquo International Journal of Rock Mechanics and MiningSciences vol 122 p 104081 2019

[5] D A Shockey D R Curran L Seaman J T Rosenberg andC F Petersen ldquoFragmentation of rock under dynamic loadsrdquoInternational Journal of Rock Mechanics and Mining Sciencesamp Geomechanics Abstracts vol 11 no 8 pp 303ndash317 1974

[6] F Q Gong S Luo G Lin and X B Li ldquoEvaluation of shearstrength parameters of rocks by preset angle shear directshear and triaxial compression testsrdquo Rock Mechanics andRock Engineering vol 53 no 5 pp 2505ndash2519 2020

[7] X F Li H B Li Q B Zhang J L Jiang and J ZhaoldquoDynamic fragmentation of rock material characte-ristic sizefragment distribution and pulverization lawrdquo EngineeringFracture Mechanics vol 199 pp 739ndash759 2018

[8] J D Hogan R J Rogers J G Spray and S Boonsue ldquoDynamicfragmentation of granite for impact energies of 6ndash28 Jrdquo En-gineering Fracture Mechanics vol 79 pp 103ndash125 2012

[9] P Rosin and E Rammler ldquo+e laws governing the fineness ofpowdered coalrdquo Journal of Institute of Fuel vol 7 pp 29ndash36 1933

[10] N F Mott and E H Linfoot ldquoMinistry of supply +e rawmaterial guiderdquo Ae Analyst vol 68 no 806 1943

[11] D E Grady ldquoLocal inertial effects in dynamic fragmentationrdquoJournal of Applied Physics vol 53 no 1 pp 322ndash325 1982

[12] F Zhou and J F Molinari ldquoOn the rate-dependency ofdynamic tensile strength of a model ceramic systemrdquo Com-puter Methods in Applied Mechanics and Engineering vol 194no 12-16 pp 1693ndash1709 2005

[13] X F Li X Li H B Li Q B Zhang and J Zhao ldquoDynamictensile behaviours of heterogeneous rocks the grain scalefracturing characteristics on strength and fragmentationrdquoInternational Journal of Impact Engineering vol 118pp 98ndash118 2018

[14] F Q Gong X B Li and X L Liu ldquoPreliminary experimentalstudy of characteristics of rock subjected to 3D coupled staticand dynamic loadsrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 6 pp 1179ndash1190 2011

[15] S Levy and J F Molinari ldquoDynamic fragmentation of ce-ramics signature of defects and scaling of fragment sizesrdquoJournal of the Mechanics and Physics of Solids vol 58 no 1pp 12ndash26 2010

[16] B Paliwal and K T Ramesh ldquoAn interacting micro-crackdamage model for failure of brittle materials under com-pressionrdquo Journal of the Mechanics and Physics of Solidsvol 56 no 3 pp 896ndash923 2008

[17] R J Wu H B Li X F Li X Xia and L W Liu ldquoExperimentalstudy and numerical simulation of the dynamic behavior oftransversely isotropic phylliterdquo International Journal of Geo-mechanics vol 20 no 8 Article ID 04020105 2020

[18] X Ou X Zhang H Feng C Zhang and J Yang ldquoEffect of theconfining pressure on the dynamic compression properties of

20 Advances in Civil Engineering

transversely isotropic rocksrdquo Advances in Civil Engineeringvol 6 p 11 2019

[19] H-B Du F Dai Y Xu Y Liu and H-n Xu ldquoNumericalinvestigation on the dynamic strength and failure behavior ofrocks under hydrostatic confinement in SHPB testingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 108 pp 43ndash57 2018

[20] X F Li Q B Zhang H B Li and J Zhao ldquoGrain-baseddiscrete element method (GB-DEM) modelling of multi-scalefracturing in rocks under dynamic loadingrdquo Rock Mechanicsand Rock Engineering vol 51 no 12 pp 3785ndash3817 2018

[21] F Dai M D Wei N W Xu Y Ma and D S Yang ldquoNu-merical assessment of the progressive rock fracture mecha-nism of cracked chevron notched Brazilian disc specimensrdquoRock Mechanics and Rock Engineering vol 48 no 2pp 463ndash479 2015

[22] B Park and K-B Min ldquoBonded-particle discrete elementmodeling of mechanical behavior of transversely isotropicrockrdquo International Journal of Rock Mechanics and MiningSciences vol 76 pp 243ndash255 2015

[23] E Cadoni ldquoDynamic characterization of orthogneiss rocksubjected to intermediate and high strain rates in tensionrdquoRock Mechanics and Rock Engineering vol 43 no 6pp 667ndash676 2010

[24] Y X Zhou K Xia X B Li et al ldquoSuggested methods fordetermining the dynamic strength parameters and Mode-Ifracture toughness of rock materialsrdquo International Journal ofRock Mechanics and Mining Sciences vol 49 no 1pp 105ndash112 2011

[25] F Dai K Xia J P Zuo R Zhang and N W Xu ldquoStatic anddynamic flexural strength anisotropy of Barre graniterdquo Rock Me-chanics and Rock Engineering vol 46 no 6 pp 1589ndash1602 2013

[26] F-Q Gong X-F Si X-B Li and S-Y Wang ldquoDynamictriaxial compression tests on sandstone at high strain ratesand low confining pressures with split Hopkinson pressurebarrdquo International Journal of Rock Mechanics and MiningSciences vol 113 pp 211ndash219 2019

[27] X F Li H B Li L W Liu et al ldquoInvestigating the crackinitiation and propagation mechanism in brittle rocks usinggrain-based finite-discrete element methodrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 127p 104219 2020

[28] F Gong and J Hu ldquoEnergy dissipation characteristic of redsandstone in the dynamic Brazilian disc test with SHPBsetuprdquo Advances in Civil Engineering vol 2020 Article ID7160937 10 pages 2020

[29] J C Li H B Li G W Ma and J Zhao ldquoA time-domainrecursive method to analyse transient wave propagationacross rock jointsrdquo Geophysical Journal Internationalvol 188 no 2 pp 631ndash644 2012

[30] P A Cundall ldquoA computer model for simulating progressivelarge-scale movement in blocky rock systemrdquo in Proceedingsof the International Symposium on Rock Mechanics NancyFrance October 1971

[31] B Park K-B Min N +ompson and P Horsrud ldquo+ree-dimensional bonded-particle discrete element modeling ofmechanical behavior of transversely isotropic rockrdquo Inter-national Journal of Rock Mechanics and Mining Sciencesvol 110 pp 120ndash132 2018

[32] Y M Tien M C Kuo and C H Juang ldquoAn experimentalinvestigation of the failure mechanism of simulated transverselyisotropic rocksrdquo International Journal of Rock Mechanics andMining Sciences vol 43 no 8 pp 1163ndash1181 2006

Advances in Civil Engineering 21

Stre

ss (M

Pa)

σiσt

σrσi + σr

β = 0degndash500

ndash250

0

250

500

50 100 150 2000Time (micros)

(a)

σiσt

σrσi + σr

β = 225deg

Stre

ss (M

Pa)

ndash500

ndash250

0

250

500

50 100 150 2000Time (micros)

(b)

σiσt

σrσi + σr

β = 45deg

Stre

ss (M

Pa)

ndash500

ndash250

0

250

500

35 70 105 1400Time (micros)

(c)

σiσt

σrσi + σr

β = 675deg

Stre

ss (M

Pa)

ndash500

ndash250

0

250

500

35 70 105 1400Time (micros)

(d)

β = 90deg

Stre

ss (M

Pa)

ndash500

ndash250

0

250

500

σiσt

σrσi + σr

35 70 105 1400Time (micros)

(e)

Figure 4 +ree-wave method to check experimental stress equilibrium

4 Advances in Civil Engineering

particles is employed to form the bar model +e tensilestrength and shear strength of the SHPB bars are both set as1e100 Pa to ensure that they would not be damaged duringwave propagation

Calibration for sample parameters is cautious to ac-complish several steps to reproduce the real experimentalprocess +e 0deg sample and 90deg sample under quasistatic loadare employed to preliminary determine the strength andYoungrsquos modulus of numerical model respectively+e 675degsample is selected to identify the smooth joint bond strengthconsidering the layered rock with this angle failed along thebedding planes +en the stiffness of the bedding are ob-tained by using the equivalent continuous model [22] Be-fore completing the model dynamic test results are utilizedto correct the parameters +e microscopic properties ofparticles and bonds of the sample are listed in Table 3

32 Validation of the DEM Model

321 Dynamic Stress Equilibrium A successful SHPB ex-periment must meet two requirements the stress wavepropagates in the form of one-dimensional wave in the bar

and the dynamic stress at both ends of the specimen remainsequal during the loading process Li et al [20] have provedthat homogeneous particles could well maintain the one-dimensional propagation of stress waves in the DEMmodel+e three-wave method as an indirect approach isemployed to compare the force of the specimen on two endsin laboratory experiments after considering that the axialforce of the specimen is hard to measure directly [26]However the forces at both ends of specimen could bedirectly obtained to judge whether it is in a stress equilib-rium state in the numerical model+erefore the three-wavemethod and the direct measurement method are both ap-plied to guarantee that the sample be in the dynamic stressequilibrium state during the impact loading process +eresult of the three-wave method is presented in Figure 6(a)which demonstrates that forces at both ends of the specimencalculated by the incident bar and the transmission bar arebasically equal

For the direct measurement method stress on the leftand right end of the sample is directly recorded +e stressequilibrium coefficient η 2(σright minus σ left)(σright + σ left) iscalculated to compare the stresses at both ends of the sampleAs shown in Figure 6(b) η remains around zero near thepeak sections which directly indicates that the axial force ofthe specimen is equal during the loading process

322 Ultimate Fragment States To further validate theBPM numerical model dynamic tests on the 45deg samples aresimulated +e incident stress wave of 45deg sample is derivedfrom the experiment to ensure the numerical dynamic stressis comparable with the experimental as shown inFigure 7(a) Figures 7(b)sim7(f) illustrate the experimental andnumerical dynamic uniaxial compression stressndashtime curvefor five angle samples whose values and change trends arebasically consistent +e dynamic compressive strength oflayered rock changes in U-shape with the increase of bed-ding angle and the strength is the minimum at 675degFigure 7(g) depicts ultimate fragment state of five angle

1000mm 1000mm500mm500mm

Incident bar Transmitted barSpecimenMeasure Measure

kn

ks

Measure circle in bar Layered specimen Smooth joint model

Figure 5 +e SHPB numerical simulation test system

Table 1 Quasistatic and dynamic mechanical properties of layered phyllite

Bedding angle β 0 225 45 675 90Quasistatic elastic modulus (GPa) 171 291 256 277 194Quasistatic compression strength (MPa) 1522 1277 736 631 1157Dynamic strain rate (sminus1) 948 994 1382 1426 1372Dynamic compression strength (MPa) 2481 2146 1432 1225 1870

Table 2 Microscopic properties of particles and bonds of the SHPBbars

Properties Material parametersrsquo valueParticlesBall density (kgm3) 7900Ball radius (m) 05e-3Particle-based material parametersYoungrsquos modulus (Ec Pa) 182e9Stiffness ratio (knks) 235Parallel bond model parametersYoungrsquos modulus (Ec Pa) 182e9Stiffness ratio (knks) 235Normal strength (σc Pa) 1e100Shear strength (τc Pa) 1e100

Advances in Civil Engineering 5

samples after the experimental and numerical tests at thestrain rate asymp140 sminus1 +e simulation results of low beddingangle samples show that the bedding plane affects the localfailure of the samples High bedding angle samples arebroken into schistose along the bedding and partially pul-verized in both situations +ese results indicate that theSHPB numerical simulation test system is reasonable tocharacterize the realistic behaviors of layered phyllite

4 Results and Discussion

+e particles that are still bonded together after loading aredefined as a fragment +e fragment that is composed byindividual particle is not counted to simplify the calculation+e cumulative distribution curve of layered rocks with five

bedding angles is counted in different situation What ismore the average fragment size is calculated to quantita-tively compare the damage

δ 1113944

n

i1ηiδi

1113944n

i1ηi

(2)

where n is the total fragment number δ is the area of the ithfragment and η represents the percentage of the ith frag-ment area

41 Effect of Strain Rate Five impact velocities were appliedto the simulation layered samples to obtain the fragmen-tation under different strain rates As shown in Figure 8 theloading strain rate has a significant effect on the strength and

Table 3 Microscopic properties of particles and bonds of the sample

Properties Material parametersrsquo valueParticlesBall density (kgm3) 2620Minimum ball radius (m) 02eminus 3Maximum ball radius (m) 0332eminus 3Youngrsquos modulus (Ec Pa) 164e9Stiffness ratio (knks) 115Parallel bond modelYoungrsquos modulus (Ec Pa) 164e9Stiffness ratio (pb knpb ks) 115Normal strength (pb σc Pa) 52e6plusmn 13e6Shear strength (pb_τc Pa) 140e6plusmn 35e6Smooth joint modelNormal stiffness (sj kn (Pam)) 1573e12Shear stiffness (sj ks (Pam)) 638e12Friction coefficient (φ) 35degCohesion (sj_τc Pa) 36e6plusmn 3e6Tensile strength (sj σc Pa) 3e6plusmn 05e6

Stre

ss (M

Pa)

ndash500

ndash250

0

250

500

35 70 105 1400Time (micros)

σiσt

σrσi + σr

(a)

Stre

ss (M

Pa)

0

50

100

150

ndash2

ndash1

0

1

2

Stre

ss eq

uilib

rium

coef

ficie

nt

40 80 1200Time (μs)

σrightσleη

(b)

Figure 6+e dynamic stress equilibrium state of SHPB numerical model (a)+ree-wavemethod to check numerical stress equilibrium (b)Direct measurement method to check stress equilibrium

6 Advances in Civil Engineering

Stre

ss (M

Pa)

Experimental incident waveNumerical incident wave

0

125

250

375

500

50 100 1500Time (micros)

(a)

β = 0deg

Stre

ss (M

Pa)

Experimental stressndashtimeNumerical stressndashtime

0

100

200

300

40 80 120 1600Time (μs)

(b)

β = 225deg

Stre

ss (M

Pa)

Experimental stressndashtimeNumerical stressndashtime

0

90

180

270

50 100 1500Time (μs)

(c)

β = 45degSt

ress

(MPa

)

Experimental stressndashtimeNumerical stressndashtime

0

40

80

120

160

25 50 75 1000Time (μs)

(d)

β = 675deg

Stre

ss (M

Pa)

Experimental stressndashtimeNumerical stressndashtime

0

50

100

150

35 70 105 1400Time (μs)

(e)

β = 90deg

Stre

ss (M

Pa)

Experimental stressndashtimeNumerical stressndashtime

0

80

160

240

20 40 60 80 100 120 1400Time (μs)

(f )

Figure 7 Continued

Advances in Civil Engineering 7

(g)

Figure 7 SHPB numerical model and simulation results (a) Numerical and experimental incident stress wave of 45deg sample (bndashf)Numerical and experimental dynamic compression stress of five angle samples (g) +e ultimate fragment state of five angle samples afterexperiment and simulation

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

100 101 102 10310ndash1

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 10310ndash1

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 103

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 103

Fragment size (mm2)

(d)

Figure 8 Continued

8 Advances in Civil Engineering

failure of the specimen Under any dynamic load thestrength of the specimen always presents a U-shaped changewith the increase of bedding angle and the minimumstrength occurs at 45degndash675deg

+e fragment size decreases gradually as the strain rateincreases while the strength presents an opposite trend +esamples with low bedding angle show minor damage undersmaller impact load and that changes at higher strain rateWith the increase of strain rate more cracks initiate alongthe loading direction and propagate to intersect with the

cracks on the weak bedding plane of 0degndash225deg sample whichleads to further failure of the specimen

Although the impact splitting effect is increased theweak bedding plane limits the crack to the local part of45degndash675deg specimen and hinders its further expansioncausing specimensrsquo difficulty to be further destroyed Asshown in Figure 8 when reaching a certain strain rate theshiver of rocks with β 45degndash675deg tends to be stable +eresults explored that in order to achieve better fragmentationunder high strain rate the relationship between the loading

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 10310ndash1

Fragment size (mm2)

(e)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

15

20

25

30

35

lgδ_

120 18060Strain rate (sndash1)

(f )

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

Stre

ngth

(MPa

)

100

150

200

250

300

100 150 200 25050Strain rate (sndash1)

(g)

Figure 8 +e cumulative distribution curve and average fragment size of layered rocks under different strain rates (andashe) +e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red _ε asymp 80 sminus1 blue _ε asymp 140 sminus1 yellow _ε asymp 170 sminus1 green _ε asymp 200 sminus1 and gray _ε asymp 215 sminus1 (f )Average fragment size vs strain rate under different bedding angles (g) Dynamic strength vs strain rate under different bedding angles

Advances in Civil Engineering 9

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

15

20

25

30

35

lgδ_

50 100 1500Coh (MPa)

(f )

Figure 9 Continued

10 Advances in Civil Engineering

direction and the bedding dip angle should be changedunder different impact forces

42 Effect of Weak Bedding Strength

421 Effect of Weak Bedding Shear Strength +e weakbedding plane is endowed with cohesion strength for 4MPa24MPa 36Mpa 72Mpa and 140MPa respectively asshown in Figure 9 +e minimum strength corresponds tothe weak bedding immediately destroyed the last is the bondstrength of matrix

Tien et al [32] studied the properties of homogeneousrock-like materials and layered materials under staticloading It is considered that the failure of β 0deg samples isnot affected by the bedding as the compressive strength issimilar to that of isotropic rock However from the view ofrock failure it was found that the bedding would disturb thecrack propagation of the specimen When the shear strengthof the bedding plane is minor the bedding as a weak planeleads to the local fracture of 0degndash225deg specimen And there isno preferential defect in the sample as the shear strengthincreases so the degree of fragmentation decreases

+e situation becomes complicated for 45deg specimens+e bedding plane was destroyed first at low bedding shearstrength and the failure that was perpendicular to the weakplane of bedding has happened which leads to the furtherfracture of the specimen However in the case of highbedding shear strength the bedding no longer hinders thepropagation of the splitting crack resulting in smallerfragments of the specimen At the same time the bedding

shear strength makes no contribution to the crack growthand dynamic strength of the 90deg samples

422 Effect of Weak Bedding Tension Strength At eachbedding angle five kinds of tensile strength weak planespecimens are loaded from the weakest 001MPa to 52MPaas same as with the matrix +e results listed in Figure 10illustrate that the tensile strength of the weak plane has littleeffect on the ultimate fragment state and the dynamiccompression strength In most cases the upper part of thecumulative curve causes differentiation pointing that thetensile strength mainly limits the generation of large pieces

Only for 0deg specimen the crack was harder to develop asthe strength enhanced Compared with the bedding shearstrength situation it could be seen that the bedding shearstrength influences the failure of 225deg specimen while the 0degspecimen is mainly damaged by tensile strength at strainrateasymp120 sminus1 +e average size of high bedding angle spec-imen is kept at a small level and near-zero strength results inless fragmentation

43 Effect of Weak Bedding Stiffness +e initial knks ofbedding is 246 which is changed to explore the influenceof stiffness on rock fragmentation as shown in Figure 11 Itis obvious that the bedding stiffness has no effect on thefailure of layered rock However the average fragment sizeof high angle samples tends to be consistent when the kngradually increase +e failure of bedding plane occurs inthe initial loading stage which results in its ability to resist

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

Stre

ngth

(MPa

)90

180

270

50 100 1500Coh (MPa)

(g)

Figure 9 +e cumulative distribution curve and average fragment size of layered rocks with different bedding shear strength (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red coh 4MPa blue coh 24MPa yellow coh 36MPa green coh 72MPaand gray coh 140MPa (f ) Average fragment size vs bedding shear strength under different bedding angles (g) Dynamic strength vsbedding shear strength under different bedding angles

Advances in Civil Engineering 11

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(e)

20

25

30

35

lgδ_

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 10 Continued

12 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 10 +e cumulative distribution curve and average fragment size of layered rocks with different bedding tensile strength (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red ten 001MPa blue ten 3MPa yellow ten 18MPa green ten 42MPaand gray ten 52MPa (f ) Average fragment size vs bedding tension strength under different bedding angles (g) Dynamic strength vsbedding tension strength under different bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 11 Continued

Advances in Civil Engineering 13

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(e)

20

25

30

35

lgδ_

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 11 Continued

14 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 11 +e cumulative distribution curve and average fragment size of layered rocks with different bedding stiffness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red kskn 10 blue kskn 246 yellow knks 1 green knks 246 and grayknks 10 (f ) Average fragment size vs bedding stiffness under different bedding angles (g) Dynamic strength vs bedding stiffness underdifferent bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 12 Continued

Advances in Civil Engineering 15

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

10ndash1 101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102100

Fragment size (mm2)

(e)

15

20

25

30

1 2 3 4 50Thickness (mm)

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 12 Continued

16 Advances in Civil Engineering

Stre

ngth

(MPa

)

120

160

200

240

1 2 3 4 50Thickness (mm)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 12 +e cumulative distribution curve and average fragment size of layered rocks with different bedding thickness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red thickness 08mm blue thickness 14mm yellow thickness 20mmgreen thickness 36mm and gray thickness 50mm (f) Average fragment size vs thickness under different bedding angles (g)Dynamic strength vs thickness under different bedding angles

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 13 Continued

Advances in Civil Engineering 17

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

15

20

25

30

35

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(f )

Figure 13 Continued

18 Advances in Civil Engineering

deformation having no longer an effect on the crackpropagation

44Effect ofWeakBeddingAickness As shown in Figure 12weak plane models with thicknesses of 08mm 14mm20mm 36mm and 50mm are constructed +e beddingstrength is half of the original matrix rather than using SJmodel to avoid the distortion of the simulation results led byall the SJ model damage during loading +e smallestthickness represents the existence of a weak surface In thecase of 50mm half sample is strong matrix and the other isweak It could be seen from Figure 12 that when there existweak planes in the 45deg specimen the central bedding planedivides the specimen into two parts bearing the load sep-arately As a result the bearing capacity of the 45deg specimenschanged little under different conditions which lead tosimilar failure patterns for 45deg specimens

However the bearing capacity and failure mode ofspecimens with other bedding angles change greatly when theproportion of weak plane increases +e simulation revealsthat more defects lead to greater fragmentation and the sizehas a significant downward trend for all angles except 45degwhich means most angle rocks are sensitive to it At the sametime the distribution curves for 45deg specimen have littledifferences and others move left as a whole +e weak planealways makes the specimen with high angle easier to developmore cracks regardless of its width compared to 225degspecimen being the most difficult to be further damaged

45 Effect of Weak Bedding Space A smaller spacing meansthat the number of laminations increases which changesfrom 10 to 1 It could be analyzed from Figure 13 that whenthe bedding angle is greater than 45deg the failure of thespecimens under any spacing is influenced by the beddingplane And more split cracks begin to appear due to thebarrier effect of weak plane being weakened when there isonly one weak plane +e smaller fragment would emerge asthe split crack intersects with the weak plane +e localfailure of specimens is controlled by the bedding planeswhen the spacing is less than 12mm for the specimens withlow bedding angle However the effect of bedding planes for0deg specimens is no longer significant with the space in-creasing while that of 225 specimens still affects localfracture As a result the fragment distribution of otherbedding dip angle samples except 225deg correspondinglyshifts to the left when the bedding space decreases Nev-ertheless the average fragment size rises slightly when thespace exceeds a quarter

5 Conclusions

+e effect of the microscopic properties of bedding andstrain rate on the fragment size distributions of layered rockwith different bedding dip angles was investigated by usingthe 2D discrete element model Several conclusions weredrawn from the experiment

(1) A SHPBnumerical model was established and validatedA new method to obtain reasonable layered rock

Stre

ngth

(MPa

)90

180

270

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(g)

Figure 13+e cumulative distribution curve and average fragment size of layered rocks with different bedding space (andashe)+e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red space 5mm blue space 65mm yellow space 10mm green space 125mm andgray space 25mm (f) Average fragment size vs bedding space under different bedding angles (g) Dynamic strength vs bedding spaceunder different bedding angles

Advances in Civil Engineering 19

dynamic simulation result is proposed +e quasistaticand dynamic test results are combined to calibrate theelastic modulus and strength of layered rock to repro-duce the stressndashstrain curve and ultimate fragment state

(2) +e average size of samples with different beddingangles decreases as the strain rate increases while thestrength presents an opposite trend +e samplestend to become pulverized under high strain rateespecially for the low bedding angle samples as morecracks initiate along the loading direction andpropagate to intersect with the cracks

(3) It is harder for the samples with low dip angle to bedamaged if the bedding shear strength is added whilethe fragmentation of high angle samples does notchange significantly +e average fragment size of 45degsamples varies greatly with the shear strength becausethe failure mode of the specimens changed And thefailure of layered specimens with all dip angles is notaffected by the tensile strength and stiffness

(4) +e wider bedding may promote the crack initiationand propagation under the same impact load whichmeans large fragments develop into small ones It wasinitially considered that the effect of bedding plane onlowdip angle samples is not considerableHowever thebedding has a dramatic impact on the final fragment

(5) Because the effect of hindering splitting crack hasalways existed for the bedding plane of 225deg sampleand more cracks initiate along the loading directionthe fragment size distribution curve of all beddingangle samples gradually moves to the right when thebedding space increases except for the 225deg samplesNevertheless the average fragment size rises slightlywhen the space exceeds a range

Data Availability

Some data models or codes generated or used during thestudy are available from the corresponding author uponrequest static mechanical properties of layered phyllite withdifferent bedding angles and all results of the dynamiccompression test

Conflicts of Interest

+e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

+e constructive comments of three anonymous reviewersare appreciated +is work received funding from the Na-tional Natural Science Foundation of China (Nos 51439008and 51679231) and China Scholarship Council (to the firstauthor No 202004910629)

References

[1] C V B Cunningham ldquoFragmentation estimations and theKuz-Ram modelndashfour years onrdquo in Proceedings of the 2nd

International Symposium on Rock Fragmentation by Blastingpp 475ndash487 Keystone CO USA August 1987

[2] X F Li H B Li Y Q Liu Q C Zhou and X Xia ldquoNumericalsimulation of rock fragmentation mechanisms subject towedge penetration for TBMsrdquo Tunnelling and UndergroundSpace Technology vol 53 pp 96ndash108 2016

[3] J C Li H B Li and J Zhao ldquoAn improved equivalentviscoelastic medium method for wave propagation acrosslayered rock massesrdquo International Journal of Rock Mechanicsand Mining Sciences vol 73 no 1 pp 62ndash69 2015

[4] F Gong W Wuxing Wu T Tianbin Li and X Xuefeng SildquoExperimental simulation and investigation of spalling failureof rectangular tunnel under different three-dimensional stressstatesrdquo International Journal of Rock Mechanics and MiningSciences vol 122 p 104081 2019

[5] D A Shockey D R Curran L Seaman J T Rosenberg andC F Petersen ldquoFragmentation of rock under dynamic loadsrdquoInternational Journal of Rock Mechanics and Mining Sciencesamp Geomechanics Abstracts vol 11 no 8 pp 303ndash317 1974

[6] F Q Gong S Luo G Lin and X B Li ldquoEvaluation of shearstrength parameters of rocks by preset angle shear directshear and triaxial compression testsrdquo Rock Mechanics andRock Engineering vol 53 no 5 pp 2505ndash2519 2020

[7] X F Li H B Li Q B Zhang J L Jiang and J ZhaoldquoDynamic fragmentation of rock material characte-ristic sizefragment distribution and pulverization lawrdquo EngineeringFracture Mechanics vol 199 pp 739ndash759 2018

[8] J D Hogan R J Rogers J G Spray and S Boonsue ldquoDynamicfragmentation of granite for impact energies of 6ndash28 Jrdquo En-gineering Fracture Mechanics vol 79 pp 103ndash125 2012

[9] P Rosin and E Rammler ldquo+e laws governing the fineness ofpowdered coalrdquo Journal of Institute of Fuel vol 7 pp 29ndash36 1933

[10] N F Mott and E H Linfoot ldquoMinistry of supply +e rawmaterial guiderdquo Ae Analyst vol 68 no 806 1943

[11] D E Grady ldquoLocal inertial effects in dynamic fragmentationrdquoJournal of Applied Physics vol 53 no 1 pp 322ndash325 1982

[12] F Zhou and J F Molinari ldquoOn the rate-dependency ofdynamic tensile strength of a model ceramic systemrdquo Com-puter Methods in Applied Mechanics and Engineering vol 194no 12-16 pp 1693ndash1709 2005

[13] X F Li X Li H B Li Q B Zhang and J Zhao ldquoDynamictensile behaviours of heterogeneous rocks the grain scalefracturing characteristics on strength and fragmentationrdquoInternational Journal of Impact Engineering vol 118pp 98ndash118 2018

[14] F Q Gong X B Li and X L Liu ldquoPreliminary experimentalstudy of characteristics of rock subjected to 3D coupled staticand dynamic loadsrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 6 pp 1179ndash1190 2011

[15] S Levy and J F Molinari ldquoDynamic fragmentation of ce-ramics signature of defects and scaling of fragment sizesrdquoJournal of the Mechanics and Physics of Solids vol 58 no 1pp 12ndash26 2010

[16] B Paliwal and K T Ramesh ldquoAn interacting micro-crackdamage model for failure of brittle materials under com-pressionrdquo Journal of the Mechanics and Physics of Solidsvol 56 no 3 pp 896ndash923 2008

[17] R J Wu H B Li X F Li X Xia and L W Liu ldquoExperimentalstudy and numerical simulation of the dynamic behavior oftransversely isotropic phylliterdquo International Journal of Geo-mechanics vol 20 no 8 Article ID 04020105 2020

[18] X Ou X Zhang H Feng C Zhang and J Yang ldquoEffect of theconfining pressure on the dynamic compression properties of

20 Advances in Civil Engineering

transversely isotropic rocksrdquo Advances in Civil Engineeringvol 6 p 11 2019

[19] H-B Du F Dai Y Xu Y Liu and H-n Xu ldquoNumericalinvestigation on the dynamic strength and failure behavior ofrocks under hydrostatic confinement in SHPB testingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 108 pp 43ndash57 2018

[20] X F Li Q B Zhang H B Li and J Zhao ldquoGrain-baseddiscrete element method (GB-DEM) modelling of multi-scalefracturing in rocks under dynamic loadingrdquo Rock Mechanicsand Rock Engineering vol 51 no 12 pp 3785ndash3817 2018

[21] F Dai M D Wei N W Xu Y Ma and D S Yang ldquoNu-merical assessment of the progressive rock fracture mecha-nism of cracked chevron notched Brazilian disc specimensrdquoRock Mechanics and Rock Engineering vol 48 no 2pp 463ndash479 2015

[22] B Park and K-B Min ldquoBonded-particle discrete elementmodeling of mechanical behavior of transversely isotropicrockrdquo International Journal of Rock Mechanics and MiningSciences vol 76 pp 243ndash255 2015

[23] E Cadoni ldquoDynamic characterization of orthogneiss rocksubjected to intermediate and high strain rates in tensionrdquoRock Mechanics and Rock Engineering vol 43 no 6pp 667ndash676 2010

[24] Y X Zhou K Xia X B Li et al ldquoSuggested methods fordetermining the dynamic strength parameters and Mode-Ifracture toughness of rock materialsrdquo International Journal ofRock Mechanics and Mining Sciences vol 49 no 1pp 105ndash112 2011

[25] F Dai K Xia J P Zuo R Zhang and N W Xu ldquoStatic anddynamic flexural strength anisotropy of Barre graniterdquo Rock Me-chanics and Rock Engineering vol 46 no 6 pp 1589ndash1602 2013

[26] F-Q Gong X-F Si X-B Li and S-Y Wang ldquoDynamictriaxial compression tests on sandstone at high strain ratesand low confining pressures with split Hopkinson pressurebarrdquo International Journal of Rock Mechanics and MiningSciences vol 113 pp 211ndash219 2019

[27] X F Li H B Li L W Liu et al ldquoInvestigating the crackinitiation and propagation mechanism in brittle rocks usinggrain-based finite-discrete element methodrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 127p 104219 2020

[28] F Gong and J Hu ldquoEnergy dissipation characteristic of redsandstone in the dynamic Brazilian disc test with SHPBsetuprdquo Advances in Civil Engineering vol 2020 Article ID7160937 10 pages 2020

[29] J C Li H B Li G W Ma and J Zhao ldquoA time-domainrecursive method to analyse transient wave propagationacross rock jointsrdquo Geophysical Journal Internationalvol 188 no 2 pp 631ndash644 2012

[30] P A Cundall ldquoA computer model for simulating progressivelarge-scale movement in blocky rock systemrdquo in Proceedingsof the International Symposium on Rock Mechanics NancyFrance October 1971

[31] B Park K-B Min N +ompson and P Horsrud ldquo+ree-dimensional bonded-particle discrete element modeling ofmechanical behavior of transversely isotropic rockrdquo Inter-national Journal of Rock Mechanics and Mining Sciencesvol 110 pp 120ndash132 2018

[32] Y M Tien M C Kuo and C H Juang ldquoAn experimentalinvestigation of the failure mechanism of simulated transverselyisotropic rocksrdquo International Journal of Rock Mechanics andMining Sciences vol 43 no 8 pp 1163ndash1181 2006

Advances in Civil Engineering 21

particles is employed to form the bar model +e tensilestrength and shear strength of the SHPB bars are both set as1e100 Pa to ensure that they would not be damaged duringwave propagation

Calibration for sample parameters is cautious to ac-complish several steps to reproduce the real experimentalprocess +e 0deg sample and 90deg sample under quasistatic loadare employed to preliminary determine the strength andYoungrsquos modulus of numerical model respectively+e 675degsample is selected to identify the smooth joint bond strengthconsidering the layered rock with this angle failed along thebedding planes +en the stiffness of the bedding are ob-tained by using the equivalent continuous model [22] Be-fore completing the model dynamic test results are utilizedto correct the parameters +e microscopic properties ofparticles and bonds of the sample are listed in Table 3

32 Validation of the DEM Model

321 Dynamic Stress Equilibrium A successful SHPB ex-periment must meet two requirements the stress wavepropagates in the form of one-dimensional wave in the bar

and the dynamic stress at both ends of the specimen remainsequal during the loading process Li et al [20] have provedthat homogeneous particles could well maintain the one-dimensional propagation of stress waves in the DEMmodel+e three-wave method as an indirect approach isemployed to compare the force of the specimen on two endsin laboratory experiments after considering that the axialforce of the specimen is hard to measure directly [26]However the forces at both ends of specimen could bedirectly obtained to judge whether it is in a stress equilib-rium state in the numerical model+erefore the three-wavemethod and the direct measurement method are both ap-plied to guarantee that the sample be in the dynamic stressequilibrium state during the impact loading process +eresult of the three-wave method is presented in Figure 6(a)which demonstrates that forces at both ends of the specimencalculated by the incident bar and the transmission bar arebasically equal

For the direct measurement method stress on the leftand right end of the sample is directly recorded +e stressequilibrium coefficient η 2(σright minus σ left)(σright + σ left) iscalculated to compare the stresses at both ends of the sampleAs shown in Figure 6(b) η remains around zero near thepeak sections which directly indicates that the axial force ofthe specimen is equal during the loading process

322 Ultimate Fragment States To further validate theBPM numerical model dynamic tests on the 45deg samples aresimulated +e incident stress wave of 45deg sample is derivedfrom the experiment to ensure the numerical dynamic stressis comparable with the experimental as shown inFigure 7(a) Figures 7(b)sim7(f) illustrate the experimental andnumerical dynamic uniaxial compression stressndashtime curvefor five angle samples whose values and change trends arebasically consistent +e dynamic compressive strength oflayered rock changes in U-shape with the increase of bed-ding angle and the strength is the minimum at 675degFigure 7(g) depicts ultimate fragment state of five angle

1000mm 1000mm500mm500mm

Incident bar Transmitted barSpecimenMeasure Measure

kn

ks

Measure circle in bar Layered specimen Smooth joint model

Figure 5 +e SHPB numerical simulation test system

Table 1 Quasistatic and dynamic mechanical properties of layered phyllite

Bedding angle β 0 225 45 675 90Quasistatic elastic modulus (GPa) 171 291 256 277 194Quasistatic compression strength (MPa) 1522 1277 736 631 1157Dynamic strain rate (sminus1) 948 994 1382 1426 1372Dynamic compression strength (MPa) 2481 2146 1432 1225 1870

Table 2 Microscopic properties of particles and bonds of the SHPBbars

Properties Material parametersrsquo valueParticlesBall density (kgm3) 7900Ball radius (m) 05e-3Particle-based material parametersYoungrsquos modulus (Ec Pa) 182e9Stiffness ratio (knks) 235Parallel bond model parametersYoungrsquos modulus (Ec Pa) 182e9Stiffness ratio (knks) 235Normal strength (σc Pa) 1e100Shear strength (τc Pa) 1e100

Advances in Civil Engineering 5

samples after the experimental and numerical tests at thestrain rate asymp140 sminus1 +e simulation results of low beddingangle samples show that the bedding plane affects the localfailure of the samples High bedding angle samples arebroken into schistose along the bedding and partially pul-verized in both situations +ese results indicate that theSHPB numerical simulation test system is reasonable tocharacterize the realistic behaviors of layered phyllite

4 Results and Discussion

+e particles that are still bonded together after loading aredefined as a fragment +e fragment that is composed byindividual particle is not counted to simplify the calculation+e cumulative distribution curve of layered rocks with five

bedding angles is counted in different situation What ismore the average fragment size is calculated to quantita-tively compare the damage

δ 1113944

n

i1ηiδi

1113944n

i1ηi

(2)

where n is the total fragment number δ is the area of the ithfragment and η represents the percentage of the ith frag-ment area

41 Effect of Strain Rate Five impact velocities were appliedto the simulation layered samples to obtain the fragmen-tation under different strain rates As shown in Figure 8 theloading strain rate has a significant effect on the strength and

Table 3 Microscopic properties of particles and bonds of the sample

Properties Material parametersrsquo valueParticlesBall density (kgm3) 2620Minimum ball radius (m) 02eminus 3Maximum ball radius (m) 0332eminus 3Youngrsquos modulus (Ec Pa) 164e9Stiffness ratio (knks) 115Parallel bond modelYoungrsquos modulus (Ec Pa) 164e9Stiffness ratio (pb knpb ks) 115Normal strength (pb σc Pa) 52e6plusmn 13e6Shear strength (pb_τc Pa) 140e6plusmn 35e6Smooth joint modelNormal stiffness (sj kn (Pam)) 1573e12Shear stiffness (sj ks (Pam)) 638e12Friction coefficient (φ) 35degCohesion (sj_τc Pa) 36e6plusmn 3e6Tensile strength (sj σc Pa) 3e6plusmn 05e6

Stre

ss (M

Pa)

ndash500

ndash250

0

250

500

35 70 105 1400Time (micros)

σiσt

σrσi + σr

(a)

Stre

ss (M

Pa)

0

50

100

150

ndash2

ndash1

0

1

2

Stre

ss eq

uilib

rium

coef

ficie

nt

40 80 1200Time (μs)

σrightσleη

(b)

Figure 6+e dynamic stress equilibrium state of SHPB numerical model (a)+ree-wavemethod to check numerical stress equilibrium (b)Direct measurement method to check stress equilibrium

6 Advances in Civil Engineering

Stre

ss (M

Pa)

Experimental incident waveNumerical incident wave

0

125

250

375

500

50 100 1500Time (micros)

(a)

β = 0deg

Stre

ss (M

Pa)

Experimental stressndashtimeNumerical stressndashtime

0

100

200

300

40 80 120 1600Time (μs)

(b)

β = 225deg

Stre

ss (M

Pa)

Experimental stressndashtimeNumerical stressndashtime

0

90

180

270

50 100 1500Time (μs)

(c)

β = 45degSt

ress

(MPa

)

Experimental stressndashtimeNumerical stressndashtime

0

40

80

120

160

25 50 75 1000Time (μs)

(d)

β = 675deg

Stre

ss (M

Pa)

Experimental stressndashtimeNumerical stressndashtime

0

50

100

150

35 70 105 1400Time (μs)

(e)

β = 90deg

Stre

ss (M

Pa)

Experimental stressndashtimeNumerical stressndashtime

0

80

160

240

20 40 60 80 100 120 1400Time (μs)

(f )

Figure 7 Continued

Advances in Civil Engineering 7

(g)

Figure 7 SHPB numerical model and simulation results (a) Numerical and experimental incident stress wave of 45deg sample (bndashf)Numerical and experimental dynamic compression stress of five angle samples (g) +e ultimate fragment state of five angle samples afterexperiment and simulation

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

100 101 102 10310ndash1

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 10310ndash1

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 103

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 103

Fragment size (mm2)

(d)

Figure 8 Continued

8 Advances in Civil Engineering

failure of the specimen Under any dynamic load thestrength of the specimen always presents a U-shaped changewith the increase of bedding angle and the minimumstrength occurs at 45degndash675deg

+e fragment size decreases gradually as the strain rateincreases while the strength presents an opposite trend +esamples with low bedding angle show minor damage undersmaller impact load and that changes at higher strain rateWith the increase of strain rate more cracks initiate alongthe loading direction and propagate to intersect with the

cracks on the weak bedding plane of 0degndash225deg sample whichleads to further failure of the specimen

Although the impact splitting effect is increased theweak bedding plane limits the crack to the local part of45degndash675deg specimen and hinders its further expansioncausing specimensrsquo difficulty to be further destroyed Asshown in Figure 8 when reaching a certain strain rate theshiver of rocks with β 45degndash675deg tends to be stable +eresults explored that in order to achieve better fragmentationunder high strain rate the relationship between the loading

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 10310ndash1

Fragment size (mm2)

(e)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

15

20

25

30

35

lgδ_

120 18060Strain rate (sndash1)

(f )

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

Stre

ngth

(MPa

)

100

150

200

250

300

100 150 200 25050Strain rate (sndash1)

(g)

Figure 8 +e cumulative distribution curve and average fragment size of layered rocks under different strain rates (andashe) +e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red _ε asymp 80 sminus1 blue _ε asymp 140 sminus1 yellow _ε asymp 170 sminus1 green _ε asymp 200 sminus1 and gray _ε asymp 215 sminus1 (f )Average fragment size vs strain rate under different bedding angles (g) Dynamic strength vs strain rate under different bedding angles

Advances in Civil Engineering 9

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

15

20

25

30

35

lgδ_

50 100 1500Coh (MPa)

(f )

Figure 9 Continued

10 Advances in Civil Engineering

direction and the bedding dip angle should be changedunder different impact forces

42 Effect of Weak Bedding Strength

421 Effect of Weak Bedding Shear Strength +e weakbedding plane is endowed with cohesion strength for 4MPa24MPa 36Mpa 72Mpa and 140MPa respectively asshown in Figure 9 +e minimum strength corresponds tothe weak bedding immediately destroyed the last is the bondstrength of matrix

Tien et al [32] studied the properties of homogeneousrock-like materials and layered materials under staticloading It is considered that the failure of β 0deg samples isnot affected by the bedding as the compressive strength issimilar to that of isotropic rock However from the view ofrock failure it was found that the bedding would disturb thecrack propagation of the specimen When the shear strengthof the bedding plane is minor the bedding as a weak planeleads to the local fracture of 0degndash225deg specimen And there isno preferential defect in the sample as the shear strengthincreases so the degree of fragmentation decreases

+e situation becomes complicated for 45deg specimens+e bedding plane was destroyed first at low bedding shearstrength and the failure that was perpendicular to the weakplane of bedding has happened which leads to the furtherfracture of the specimen However in the case of highbedding shear strength the bedding no longer hinders thepropagation of the splitting crack resulting in smallerfragments of the specimen At the same time the bedding

shear strength makes no contribution to the crack growthand dynamic strength of the 90deg samples

422 Effect of Weak Bedding Tension Strength At eachbedding angle five kinds of tensile strength weak planespecimens are loaded from the weakest 001MPa to 52MPaas same as with the matrix +e results listed in Figure 10illustrate that the tensile strength of the weak plane has littleeffect on the ultimate fragment state and the dynamiccompression strength In most cases the upper part of thecumulative curve causes differentiation pointing that thetensile strength mainly limits the generation of large pieces

Only for 0deg specimen the crack was harder to develop asthe strength enhanced Compared with the bedding shearstrength situation it could be seen that the bedding shearstrength influences the failure of 225deg specimen while the 0degspecimen is mainly damaged by tensile strength at strainrateasymp120 sminus1 +e average size of high bedding angle spec-imen is kept at a small level and near-zero strength results inless fragmentation

43 Effect of Weak Bedding Stiffness +e initial knks ofbedding is 246 which is changed to explore the influenceof stiffness on rock fragmentation as shown in Figure 11 Itis obvious that the bedding stiffness has no effect on thefailure of layered rock However the average fragment sizeof high angle samples tends to be consistent when the kngradually increase +e failure of bedding plane occurs inthe initial loading stage which results in its ability to resist

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

Stre

ngth

(MPa

)90

180

270

50 100 1500Coh (MPa)

(g)

Figure 9 +e cumulative distribution curve and average fragment size of layered rocks with different bedding shear strength (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red coh 4MPa blue coh 24MPa yellow coh 36MPa green coh 72MPaand gray coh 140MPa (f ) Average fragment size vs bedding shear strength under different bedding angles (g) Dynamic strength vsbedding shear strength under different bedding angles

Advances in Civil Engineering 11

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(e)

20

25

30

35

lgδ_

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 10 Continued

12 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 10 +e cumulative distribution curve and average fragment size of layered rocks with different bedding tensile strength (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red ten 001MPa blue ten 3MPa yellow ten 18MPa green ten 42MPaand gray ten 52MPa (f ) Average fragment size vs bedding tension strength under different bedding angles (g) Dynamic strength vsbedding tension strength under different bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 11 Continued

Advances in Civil Engineering 13

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(e)

20

25

30

35

lgδ_

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 11 Continued

14 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 11 +e cumulative distribution curve and average fragment size of layered rocks with different bedding stiffness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red kskn 10 blue kskn 246 yellow knks 1 green knks 246 and grayknks 10 (f ) Average fragment size vs bedding stiffness under different bedding angles (g) Dynamic strength vs bedding stiffness underdifferent bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 12 Continued

Advances in Civil Engineering 15

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

10ndash1 101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102100

Fragment size (mm2)

(e)

15

20

25

30

1 2 3 4 50Thickness (mm)

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 12 Continued

16 Advances in Civil Engineering

Stre

ngth

(MPa

)

120

160

200

240

1 2 3 4 50Thickness (mm)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 12 +e cumulative distribution curve and average fragment size of layered rocks with different bedding thickness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red thickness 08mm blue thickness 14mm yellow thickness 20mmgreen thickness 36mm and gray thickness 50mm (f) Average fragment size vs thickness under different bedding angles (g)Dynamic strength vs thickness under different bedding angles

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 13 Continued

Advances in Civil Engineering 17

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

15

20

25

30

35

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(f )

Figure 13 Continued

18 Advances in Civil Engineering

deformation having no longer an effect on the crackpropagation

44Effect ofWeakBeddingAickness As shown in Figure 12weak plane models with thicknesses of 08mm 14mm20mm 36mm and 50mm are constructed +e beddingstrength is half of the original matrix rather than using SJmodel to avoid the distortion of the simulation results led byall the SJ model damage during loading +e smallestthickness represents the existence of a weak surface In thecase of 50mm half sample is strong matrix and the other isweak It could be seen from Figure 12 that when there existweak planes in the 45deg specimen the central bedding planedivides the specimen into two parts bearing the load sep-arately As a result the bearing capacity of the 45deg specimenschanged little under different conditions which lead tosimilar failure patterns for 45deg specimens

However the bearing capacity and failure mode ofspecimens with other bedding angles change greatly when theproportion of weak plane increases +e simulation revealsthat more defects lead to greater fragmentation and the sizehas a significant downward trend for all angles except 45degwhich means most angle rocks are sensitive to it At the sametime the distribution curves for 45deg specimen have littledifferences and others move left as a whole +e weak planealways makes the specimen with high angle easier to developmore cracks regardless of its width compared to 225degspecimen being the most difficult to be further damaged

45 Effect of Weak Bedding Space A smaller spacing meansthat the number of laminations increases which changesfrom 10 to 1 It could be analyzed from Figure 13 that whenthe bedding angle is greater than 45deg the failure of thespecimens under any spacing is influenced by the beddingplane And more split cracks begin to appear due to thebarrier effect of weak plane being weakened when there isonly one weak plane +e smaller fragment would emerge asthe split crack intersects with the weak plane +e localfailure of specimens is controlled by the bedding planeswhen the spacing is less than 12mm for the specimens withlow bedding angle However the effect of bedding planes for0deg specimens is no longer significant with the space in-creasing while that of 225 specimens still affects localfracture As a result the fragment distribution of otherbedding dip angle samples except 225deg correspondinglyshifts to the left when the bedding space decreases Nev-ertheless the average fragment size rises slightly when thespace exceeds a quarter

5 Conclusions

+e effect of the microscopic properties of bedding andstrain rate on the fragment size distributions of layered rockwith different bedding dip angles was investigated by usingthe 2D discrete element model Several conclusions weredrawn from the experiment

(1) A SHPBnumerical model was established and validatedA new method to obtain reasonable layered rock

Stre

ngth

(MPa

)90

180

270

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(g)

Figure 13+e cumulative distribution curve and average fragment size of layered rocks with different bedding space (andashe)+e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red space 5mm blue space 65mm yellow space 10mm green space 125mm andgray space 25mm (f) Average fragment size vs bedding space under different bedding angles (g) Dynamic strength vs bedding spaceunder different bedding angles

Advances in Civil Engineering 19

dynamic simulation result is proposed +e quasistaticand dynamic test results are combined to calibrate theelastic modulus and strength of layered rock to repro-duce the stressndashstrain curve and ultimate fragment state

(2) +e average size of samples with different beddingangles decreases as the strain rate increases while thestrength presents an opposite trend +e samplestend to become pulverized under high strain rateespecially for the low bedding angle samples as morecracks initiate along the loading direction andpropagate to intersect with the cracks

(3) It is harder for the samples with low dip angle to bedamaged if the bedding shear strength is added whilethe fragmentation of high angle samples does notchange significantly +e average fragment size of 45degsamples varies greatly with the shear strength becausethe failure mode of the specimens changed And thefailure of layered specimens with all dip angles is notaffected by the tensile strength and stiffness

(4) +e wider bedding may promote the crack initiationand propagation under the same impact load whichmeans large fragments develop into small ones It wasinitially considered that the effect of bedding plane onlowdip angle samples is not considerableHowever thebedding has a dramatic impact on the final fragment

(5) Because the effect of hindering splitting crack hasalways existed for the bedding plane of 225deg sampleand more cracks initiate along the loading directionthe fragment size distribution curve of all beddingangle samples gradually moves to the right when thebedding space increases except for the 225deg samplesNevertheless the average fragment size rises slightlywhen the space exceeds a range

Data Availability

Some data models or codes generated or used during thestudy are available from the corresponding author uponrequest static mechanical properties of layered phyllite withdifferent bedding angles and all results of the dynamiccompression test

Conflicts of Interest

+e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

+e constructive comments of three anonymous reviewersare appreciated +is work received funding from the Na-tional Natural Science Foundation of China (Nos 51439008and 51679231) and China Scholarship Council (to the firstauthor No 202004910629)

References

[1] C V B Cunningham ldquoFragmentation estimations and theKuz-Ram modelndashfour years onrdquo in Proceedings of the 2nd

International Symposium on Rock Fragmentation by Blastingpp 475ndash487 Keystone CO USA August 1987

[2] X F Li H B Li Y Q Liu Q C Zhou and X Xia ldquoNumericalsimulation of rock fragmentation mechanisms subject towedge penetration for TBMsrdquo Tunnelling and UndergroundSpace Technology vol 53 pp 96ndash108 2016

[3] J C Li H B Li and J Zhao ldquoAn improved equivalentviscoelastic medium method for wave propagation acrosslayered rock massesrdquo International Journal of Rock Mechanicsand Mining Sciences vol 73 no 1 pp 62ndash69 2015

[4] F Gong W Wuxing Wu T Tianbin Li and X Xuefeng SildquoExperimental simulation and investigation of spalling failureof rectangular tunnel under different three-dimensional stressstatesrdquo International Journal of Rock Mechanics and MiningSciences vol 122 p 104081 2019

[5] D A Shockey D R Curran L Seaman J T Rosenberg andC F Petersen ldquoFragmentation of rock under dynamic loadsrdquoInternational Journal of Rock Mechanics and Mining Sciencesamp Geomechanics Abstracts vol 11 no 8 pp 303ndash317 1974

[6] F Q Gong S Luo G Lin and X B Li ldquoEvaluation of shearstrength parameters of rocks by preset angle shear directshear and triaxial compression testsrdquo Rock Mechanics andRock Engineering vol 53 no 5 pp 2505ndash2519 2020

[7] X F Li H B Li Q B Zhang J L Jiang and J ZhaoldquoDynamic fragmentation of rock material characte-ristic sizefragment distribution and pulverization lawrdquo EngineeringFracture Mechanics vol 199 pp 739ndash759 2018

[8] J D Hogan R J Rogers J G Spray and S Boonsue ldquoDynamicfragmentation of granite for impact energies of 6ndash28 Jrdquo En-gineering Fracture Mechanics vol 79 pp 103ndash125 2012

[9] P Rosin and E Rammler ldquo+e laws governing the fineness ofpowdered coalrdquo Journal of Institute of Fuel vol 7 pp 29ndash36 1933

[10] N F Mott and E H Linfoot ldquoMinistry of supply +e rawmaterial guiderdquo Ae Analyst vol 68 no 806 1943

[11] D E Grady ldquoLocal inertial effects in dynamic fragmentationrdquoJournal of Applied Physics vol 53 no 1 pp 322ndash325 1982

[12] F Zhou and J F Molinari ldquoOn the rate-dependency ofdynamic tensile strength of a model ceramic systemrdquo Com-puter Methods in Applied Mechanics and Engineering vol 194no 12-16 pp 1693ndash1709 2005

[13] X F Li X Li H B Li Q B Zhang and J Zhao ldquoDynamictensile behaviours of heterogeneous rocks the grain scalefracturing characteristics on strength and fragmentationrdquoInternational Journal of Impact Engineering vol 118pp 98ndash118 2018

[14] F Q Gong X B Li and X L Liu ldquoPreliminary experimentalstudy of characteristics of rock subjected to 3D coupled staticand dynamic loadsrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 6 pp 1179ndash1190 2011

[15] S Levy and J F Molinari ldquoDynamic fragmentation of ce-ramics signature of defects and scaling of fragment sizesrdquoJournal of the Mechanics and Physics of Solids vol 58 no 1pp 12ndash26 2010

[16] B Paliwal and K T Ramesh ldquoAn interacting micro-crackdamage model for failure of brittle materials under com-pressionrdquo Journal of the Mechanics and Physics of Solidsvol 56 no 3 pp 896ndash923 2008

[17] R J Wu H B Li X F Li X Xia and L W Liu ldquoExperimentalstudy and numerical simulation of the dynamic behavior oftransversely isotropic phylliterdquo International Journal of Geo-mechanics vol 20 no 8 Article ID 04020105 2020

[18] X Ou X Zhang H Feng C Zhang and J Yang ldquoEffect of theconfining pressure on the dynamic compression properties of

20 Advances in Civil Engineering

transversely isotropic rocksrdquo Advances in Civil Engineeringvol 6 p 11 2019

[19] H-B Du F Dai Y Xu Y Liu and H-n Xu ldquoNumericalinvestigation on the dynamic strength and failure behavior ofrocks under hydrostatic confinement in SHPB testingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 108 pp 43ndash57 2018

[20] X F Li Q B Zhang H B Li and J Zhao ldquoGrain-baseddiscrete element method (GB-DEM) modelling of multi-scalefracturing in rocks under dynamic loadingrdquo Rock Mechanicsand Rock Engineering vol 51 no 12 pp 3785ndash3817 2018

[21] F Dai M D Wei N W Xu Y Ma and D S Yang ldquoNu-merical assessment of the progressive rock fracture mecha-nism of cracked chevron notched Brazilian disc specimensrdquoRock Mechanics and Rock Engineering vol 48 no 2pp 463ndash479 2015

[22] B Park and K-B Min ldquoBonded-particle discrete elementmodeling of mechanical behavior of transversely isotropicrockrdquo International Journal of Rock Mechanics and MiningSciences vol 76 pp 243ndash255 2015

[23] E Cadoni ldquoDynamic characterization of orthogneiss rocksubjected to intermediate and high strain rates in tensionrdquoRock Mechanics and Rock Engineering vol 43 no 6pp 667ndash676 2010

[24] Y X Zhou K Xia X B Li et al ldquoSuggested methods fordetermining the dynamic strength parameters and Mode-Ifracture toughness of rock materialsrdquo International Journal ofRock Mechanics and Mining Sciences vol 49 no 1pp 105ndash112 2011

[25] F Dai K Xia J P Zuo R Zhang and N W Xu ldquoStatic anddynamic flexural strength anisotropy of Barre graniterdquo Rock Me-chanics and Rock Engineering vol 46 no 6 pp 1589ndash1602 2013

[26] F-Q Gong X-F Si X-B Li and S-Y Wang ldquoDynamictriaxial compression tests on sandstone at high strain ratesand low confining pressures with split Hopkinson pressurebarrdquo International Journal of Rock Mechanics and MiningSciences vol 113 pp 211ndash219 2019

[27] X F Li H B Li L W Liu et al ldquoInvestigating the crackinitiation and propagation mechanism in brittle rocks usinggrain-based finite-discrete element methodrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 127p 104219 2020

[28] F Gong and J Hu ldquoEnergy dissipation characteristic of redsandstone in the dynamic Brazilian disc test with SHPBsetuprdquo Advances in Civil Engineering vol 2020 Article ID7160937 10 pages 2020

[29] J C Li H B Li G W Ma and J Zhao ldquoA time-domainrecursive method to analyse transient wave propagationacross rock jointsrdquo Geophysical Journal Internationalvol 188 no 2 pp 631ndash644 2012

[30] P A Cundall ldquoA computer model for simulating progressivelarge-scale movement in blocky rock systemrdquo in Proceedingsof the International Symposium on Rock Mechanics NancyFrance October 1971

[31] B Park K-B Min N +ompson and P Horsrud ldquo+ree-dimensional bonded-particle discrete element modeling ofmechanical behavior of transversely isotropic rockrdquo Inter-national Journal of Rock Mechanics and Mining Sciencesvol 110 pp 120ndash132 2018

[32] Y M Tien M C Kuo and C H Juang ldquoAn experimentalinvestigation of the failure mechanism of simulated transverselyisotropic rocksrdquo International Journal of Rock Mechanics andMining Sciences vol 43 no 8 pp 1163ndash1181 2006

Advances in Civil Engineering 21

samples after the experimental and numerical tests at thestrain rate asymp140 sminus1 +e simulation results of low beddingangle samples show that the bedding plane affects the localfailure of the samples High bedding angle samples arebroken into schistose along the bedding and partially pul-verized in both situations +ese results indicate that theSHPB numerical simulation test system is reasonable tocharacterize the realistic behaviors of layered phyllite

4 Results and Discussion

+e particles that are still bonded together after loading aredefined as a fragment +e fragment that is composed byindividual particle is not counted to simplify the calculation+e cumulative distribution curve of layered rocks with five

bedding angles is counted in different situation What ismore the average fragment size is calculated to quantita-tively compare the damage

δ 1113944

n

i1ηiδi

1113944n

i1ηi

(2)

where n is the total fragment number δ is the area of the ithfragment and η represents the percentage of the ith frag-ment area

41 Effect of Strain Rate Five impact velocities were appliedto the simulation layered samples to obtain the fragmen-tation under different strain rates As shown in Figure 8 theloading strain rate has a significant effect on the strength and

Table 3 Microscopic properties of particles and bonds of the sample

Properties Material parametersrsquo valueParticlesBall density (kgm3) 2620Minimum ball radius (m) 02eminus 3Maximum ball radius (m) 0332eminus 3Youngrsquos modulus (Ec Pa) 164e9Stiffness ratio (knks) 115Parallel bond modelYoungrsquos modulus (Ec Pa) 164e9Stiffness ratio (pb knpb ks) 115Normal strength (pb σc Pa) 52e6plusmn 13e6Shear strength (pb_τc Pa) 140e6plusmn 35e6Smooth joint modelNormal stiffness (sj kn (Pam)) 1573e12Shear stiffness (sj ks (Pam)) 638e12Friction coefficient (φ) 35degCohesion (sj_τc Pa) 36e6plusmn 3e6Tensile strength (sj σc Pa) 3e6plusmn 05e6

Stre

ss (M

Pa)

ndash500

ndash250

0

250

500

35 70 105 1400Time (micros)

σiσt

σrσi + σr

(a)

Stre

ss (M

Pa)

0

50

100

150

ndash2

ndash1

0

1

2

Stre

ss eq

uilib

rium

coef

ficie

nt

40 80 1200Time (μs)

σrightσleη

(b)

Figure 6+e dynamic stress equilibrium state of SHPB numerical model (a)+ree-wavemethod to check numerical stress equilibrium (b)Direct measurement method to check stress equilibrium

6 Advances in Civil Engineering

Stre

ss (M

Pa)

Experimental incident waveNumerical incident wave

0

125

250

375

500

50 100 1500Time (micros)

(a)

β = 0deg

Stre

ss (M

Pa)

Experimental stressndashtimeNumerical stressndashtime

0

100

200

300

40 80 120 1600Time (μs)

(b)

β = 225deg

Stre

ss (M

Pa)

Experimental stressndashtimeNumerical stressndashtime

0

90

180

270

50 100 1500Time (μs)

(c)

β = 45degSt

ress

(MPa

)

Experimental stressndashtimeNumerical stressndashtime

0

40

80

120

160

25 50 75 1000Time (μs)

(d)

β = 675deg

Stre

ss (M

Pa)

Experimental stressndashtimeNumerical stressndashtime

0

50

100

150

35 70 105 1400Time (μs)

(e)

β = 90deg

Stre

ss (M

Pa)

Experimental stressndashtimeNumerical stressndashtime

0

80

160

240

20 40 60 80 100 120 1400Time (μs)

(f )

Figure 7 Continued

Advances in Civil Engineering 7

(g)

Figure 7 SHPB numerical model and simulation results (a) Numerical and experimental incident stress wave of 45deg sample (bndashf)Numerical and experimental dynamic compression stress of five angle samples (g) +e ultimate fragment state of five angle samples afterexperiment and simulation

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

100 101 102 10310ndash1

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 10310ndash1

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 103

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 103

Fragment size (mm2)

(d)

Figure 8 Continued

8 Advances in Civil Engineering

failure of the specimen Under any dynamic load thestrength of the specimen always presents a U-shaped changewith the increase of bedding angle and the minimumstrength occurs at 45degndash675deg

+e fragment size decreases gradually as the strain rateincreases while the strength presents an opposite trend +esamples with low bedding angle show minor damage undersmaller impact load and that changes at higher strain rateWith the increase of strain rate more cracks initiate alongthe loading direction and propagate to intersect with the

cracks on the weak bedding plane of 0degndash225deg sample whichleads to further failure of the specimen

Although the impact splitting effect is increased theweak bedding plane limits the crack to the local part of45degndash675deg specimen and hinders its further expansioncausing specimensrsquo difficulty to be further destroyed Asshown in Figure 8 when reaching a certain strain rate theshiver of rocks with β 45degndash675deg tends to be stable +eresults explored that in order to achieve better fragmentationunder high strain rate the relationship between the loading

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 10310ndash1

Fragment size (mm2)

(e)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

15

20

25

30

35

lgδ_

120 18060Strain rate (sndash1)

(f )

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

Stre

ngth

(MPa

)

100

150

200

250

300

100 150 200 25050Strain rate (sndash1)

(g)

Figure 8 +e cumulative distribution curve and average fragment size of layered rocks under different strain rates (andashe) +e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red _ε asymp 80 sminus1 blue _ε asymp 140 sminus1 yellow _ε asymp 170 sminus1 green _ε asymp 200 sminus1 and gray _ε asymp 215 sminus1 (f )Average fragment size vs strain rate under different bedding angles (g) Dynamic strength vs strain rate under different bedding angles

Advances in Civil Engineering 9

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

15

20

25

30

35

lgδ_

50 100 1500Coh (MPa)

(f )

Figure 9 Continued

10 Advances in Civil Engineering

direction and the bedding dip angle should be changedunder different impact forces

42 Effect of Weak Bedding Strength

421 Effect of Weak Bedding Shear Strength +e weakbedding plane is endowed with cohesion strength for 4MPa24MPa 36Mpa 72Mpa and 140MPa respectively asshown in Figure 9 +e minimum strength corresponds tothe weak bedding immediately destroyed the last is the bondstrength of matrix

Tien et al [32] studied the properties of homogeneousrock-like materials and layered materials under staticloading It is considered that the failure of β 0deg samples isnot affected by the bedding as the compressive strength issimilar to that of isotropic rock However from the view ofrock failure it was found that the bedding would disturb thecrack propagation of the specimen When the shear strengthof the bedding plane is minor the bedding as a weak planeleads to the local fracture of 0degndash225deg specimen And there isno preferential defect in the sample as the shear strengthincreases so the degree of fragmentation decreases

+e situation becomes complicated for 45deg specimens+e bedding plane was destroyed first at low bedding shearstrength and the failure that was perpendicular to the weakplane of bedding has happened which leads to the furtherfracture of the specimen However in the case of highbedding shear strength the bedding no longer hinders thepropagation of the splitting crack resulting in smallerfragments of the specimen At the same time the bedding

shear strength makes no contribution to the crack growthand dynamic strength of the 90deg samples

422 Effect of Weak Bedding Tension Strength At eachbedding angle five kinds of tensile strength weak planespecimens are loaded from the weakest 001MPa to 52MPaas same as with the matrix +e results listed in Figure 10illustrate that the tensile strength of the weak plane has littleeffect on the ultimate fragment state and the dynamiccompression strength In most cases the upper part of thecumulative curve causes differentiation pointing that thetensile strength mainly limits the generation of large pieces

Only for 0deg specimen the crack was harder to develop asthe strength enhanced Compared with the bedding shearstrength situation it could be seen that the bedding shearstrength influences the failure of 225deg specimen while the 0degspecimen is mainly damaged by tensile strength at strainrateasymp120 sminus1 +e average size of high bedding angle spec-imen is kept at a small level and near-zero strength results inless fragmentation

43 Effect of Weak Bedding Stiffness +e initial knks ofbedding is 246 which is changed to explore the influenceof stiffness on rock fragmentation as shown in Figure 11 Itis obvious that the bedding stiffness has no effect on thefailure of layered rock However the average fragment sizeof high angle samples tends to be consistent when the kngradually increase +e failure of bedding plane occurs inthe initial loading stage which results in its ability to resist

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

Stre

ngth

(MPa

)90

180

270

50 100 1500Coh (MPa)

(g)

Figure 9 +e cumulative distribution curve and average fragment size of layered rocks with different bedding shear strength (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red coh 4MPa blue coh 24MPa yellow coh 36MPa green coh 72MPaand gray coh 140MPa (f ) Average fragment size vs bedding shear strength under different bedding angles (g) Dynamic strength vsbedding shear strength under different bedding angles

Advances in Civil Engineering 11

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(e)

20

25

30

35

lgδ_

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 10 Continued

12 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 10 +e cumulative distribution curve and average fragment size of layered rocks with different bedding tensile strength (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red ten 001MPa blue ten 3MPa yellow ten 18MPa green ten 42MPaand gray ten 52MPa (f ) Average fragment size vs bedding tension strength under different bedding angles (g) Dynamic strength vsbedding tension strength under different bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 11 Continued

Advances in Civil Engineering 13

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(e)

20

25

30

35

lgδ_

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 11 Continued

14 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 11 +e cumulative distribution curve and average fragment size of layered rocks with different bedding stiffness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red kskn 10 blue kskn 246 yellow knks 1 green knks 246 and grayknks 10 (f ) Average fragment size vs bedding stiffness under different bedding angles (g) Dynamic strength vs bedding stiffness underdifferent bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 12 Continued

Advances in Civil Engineering 15

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

10ndash1 101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102100

Fragment size (mm2)

(e)

15

20

25

30

1 2 3 4 50Thickness (mm)

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 12 Continued

16 Advances in Civil Engineering

Stre

ngth

(MPa

)

120

160

200

240

1 2 3 4 50Thickness (mm)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 12 +e cumulative distribution curve and average fragment size of layered rocks with different bedding thickness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red thickness 08mm blue thickness 14mm yellow thickness 20mmgreen thickness 36mm and gray thickness 50mm (f) Average fragment size vs thickness under different bedding angles (g)Dynamic strength vs thickness under different bedding angles

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 13 Continued

Advances in Civil Engineering 17

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

15

20

25

30

35

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(f )

Figure 13 Continued

18 Advances in Civil Engineering

deformation having no longer an effect on the crackpropagation

44Effect ofWeakBeddingAickness As shown in Figure 12weak plane models with thicknesses of 08mm 14mm20mm 36mm and 50mm are constructed +e beddingstrength is half of the original matrix rather than using SJmodel to avoid the distortion of the simulation results led byall the SJ model damage during loading +e smallestthickness represents the existence of a weak surface In thecase of 50mm half sample is strong matrix and the other isweak It could be seen from Figure 12 that when there existweak planes in the 45deg specimen the central bedding planedivides the specimen into two parts bearing the load sep-arately As a result the bearing capacity of the 45deg specimenschanged little under different conditions which lead tosimilar failure patterns for 45deg specimens

However the bearing capacity and failure mode ofspecimens with other bedding angles change greatly when theproportion of weak plane increases +e simulation revealsthat more defects lead to greater fragmentation and the sizehas a significant downward trend for all angles except 45degwhich means most angle rocks are sensitive to it At the sametime the distribution curves for 45deg specimen have littledifferences and others move left as a whole +e weak planealways makes the specimen with high angle easier to developmore cracks regardless of its width compared to 225degspecimen being the most difficult to be further damaged

45 Effect of Weak Bedding Space A smaller spacing meansthat the number of laminations increases which changesfrom 10 to 1 It could be analyzed from Figure 13 that whenthe bedding angle is greater than 45deg the failure of thespecimens under any spacing is influenced by the beddingplane And more split cracks begin to appear due to thebarrier effect of weak plane being weakened when there isonly one weak plane +e smaller fragment would emerge asthe split crack intersects with the weak plane +e localfailure of specimens is controlled by the bedding planeswhen the spacing is less than 12mm for the specimens withlow bedding angle However the effect of bedding planes for0deg specimens is no longer significant with the space in-creasing while that of 225 specimens still affects localfracture As a result the fragment distribution of otherbedding dip angle samples except 225deg correspondinglyshifts to the left when the bedding space decreases Nev-ertheless the average fragment size rises slightly when thespace exceeds a quarter

5 Conclusions

+e effect of the microscopic properties of bedding andstrain rate on the fragment size distributions of layered rockwith different bedding dip angles was investigated by usingthe 2D discrete element model Several conclusions weredrawn from the experiment

(1) A SHPBnumerical model was established and validatedA new method to obtain reasonable layered rock

Stre

ngth

(MPa

)90

180

270

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(g)

Figure 13+e cumulative distribution curve and average fragment size of layered rocks with different bedding space (andashe)+e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red space 5mm blue space 65mm yellow space 10mm green space 125mm andgray space 25mm (f) Average fragment size vs bedding space under different bedding angles (g) Dynamic strength vs bedding spaceunder different bedding angles

Advances in Civil Engineering 19

dynamic simulation result is proposed +e quasistaticand dynamic test results are combined to calibrate theelastic modulus and strength of layered rock to repro-duce the stressndashstrain curve and ultimate fragment state

(2) +e average size of samples with different beddingangles decreases as the strain rate increases while thestrength presents an opposite trend +e samplestend to become pulverized under high strain rateespecially for the low bedding angle samples as morecracks initiate along the loading direction andpropagate to intersect with the cracks

(3) It is harder for the samples with low dip angle to bedamaged if the bedding shear strength is added whilethe fragmentation of high angle samples does notchange significantly +e average fragment size of 45degsamples varies greatly with the shear strength becausethe failure mode of the specimens changed And thefailure of layered specimens with all dip angles is notaffected by the tensile strength and stiffness

(4) +e wider bedding may promote the crack initiationand propagation under the same impact load whichmeans large fragments develop into small ones It wasinitially considered that the effect of bedding plane onlowdip angle samples is not considerableHowever thebedding has a dramatic impact on the final fragment

(5) Because the effect of hindering splitting crack hasalways existed for the bedding plane of 225deg sampleand more cracks initiate along the loading directionthe fragment size distribution curve of all beddingangle samples gradually moves to the right when thebedding space increases except for the 225deg samplesNevertheless the average fragment size rises slightlywhen the space exceeds a range

Data Availability

Some data models or codes generated or used during thestudy are available from the corresponding author uponrequest static mechanical properties of layered phyllite withdifferent bedding angles and all results of the dynamiccompression test

Conflicts of Interest

+e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

+e constructive comments of three anonymous reviewersare appreciated +is work received funding from the Na-tional Natural Science Foundation of China (Nos 51439008and 51679231) and China Scholarship Council (to the firstauthor No 202004910629)

References

[1] C V B Cunningham ldquoFragmentation estimations and theKuz-Ram modelndashfour years onrdquo in Proceedings of the 2nd

International Symposium on Rock Fragmentation by Blastingpp 475ndash487 Keystone CO USA August 1987

[2] X F Li H B Li Y Q Liu Q C Zhou and X Xia ldquoNumericalsimulation of rock fragmentation mechanisms subject towedge penetration for TBMsrdquo Tunnelling and UndergroundSpace Technology vol 53 pp 96ndash108 2016

[3] J C Li H B Li and J Zhao ldquoAn improved equivalentviscoelastic medium method for wave propagation acrosslayered rock massesrdquo International Journal of Rock Mechanicsand Mining Sciences vol 73 no 1 pp 62ndash69 2015

[4] F Gong W Wuxing Wu T Tianbin Li and X Xuefeng SildquoExperimental simulation and investigation of spalling failureof rectangular tunnel under different three-dimensional stressstatesrdquo International Journal of Rock Mechanics and MiningSciences vol 122 p 104081 2019

[5] D A Shockey D R Curran L Seaman J T Rosenberg andC F Petersen ldquoFragmentation of rock under dynamic loadsrdquoInternational Journal of Rock Mechanics and Mining Sciencesamp Geomechanics Abstracts vol 11 no 8 pp 303ndash317 1974

[6] F Q Gong S Luo G Lin and X B Li ldquoEvaluation of shearstrength parameters of rocks by preset angle shear directshear and triaxial compression testsrdquo Rock Mechanics andRock Engineering vol 53 no 5 pp 2505ndash2519 2020

[7] X F Li H B Li Q B Zhang J L Jiang and J ZhaoldquoDynamic fragmentation of rock material characte-ristic sizefragment distribution and pulverization lawrdquo EngineeringFracture Mechanics vol 199 pp 739ndash759 2018

[8] J D Hogan R J Rogers J G Spray and S Boonsue ldquoDynamicfragmentation of granite for impact energies of 6ndash28 Jrdquo En-gineering Fracture Mechanics vol 79 pp 103ndash125 2012

[9] P Rosin and E Rammler ldquo+e laws governing the fineness ofpowdered coalrdquo Journal of Institute of Fuel vol 7 pp 29ndash36 1933

[10] N F Mott and E H Linfoot ldquoMinistry of supply +e rawmaterial guiderdquo Ae Analyst vol 68 no 806 1943

[11] D E Grady ldquoLocal inertial effects in dynamic fragmentationrdquoJournal of Applied Physics vol 53 no 1 pp 322ndash325 1982

[12] F Zhou and J F Molinari ldquoOn the rate-dependency ofdynamic tensile strength of a model ceramic systemrdquo Com-puter Methods in Applied Mechanics and Engineering vol 194no 12-16 pp 1693ndash1709 2005

[13] X F Li X Li H B Li Q B Zhang and J Zhao ldquoDynamictensile behaviours of heterogeneous rocks the grain scalefracturing characteristics on strength and fragmentationrdquoInternational Journal of Impact Engineering vol 118pp 98ndash118 2018

[14] F Q Gong X B Li and X L Liu ldquoPreliminary experimentalstudy of characteristics of rock subjected to 3D coupled staticand dynamic loadsrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 6 pp 1179ndash1190 2011

[15] S Levy and J F Molinari ldquoDynamic fragmentation of ce-ramics signature of defects and scaling of fragment sizesrdquoJournal of the Mechanics and Physics of Solids vol 58 no 1pp 12ndash26 2010

[16] B Paliwal and K T Ramesh ldquoAn interacting micro-crackdamage model for failure of brittle materials under com-pressionrdquo Journal of the Mechanics and Physics of Solidsvol 56 no 3 pp 896ndash923 2008

[17] R J Wu H B Li X F Li X Xia and L W Liu ldquoExperimentalstudy and numerical simulation of the dynamic behavior oftransversely isotropic phylliterdquo International Journal of Geo-mechanics vol 20 no 8 Article ID 04020105 2020

[18] X Ou X Zhang H Feng C Zhang and J Yang ldquoEffect of theconfining pressure on the dynamic compression properties of

20 Advances in Civil Engineering

transversely isotropic rocksrdquo Advances in Civil Engineeringvol 6 p 11 2019

[19] H-B Du F Dai Y Xu Y Liu and H-n Xu ldquoNumericalinvestigation on the dynamic strength and failure behavior ofrocks under hydrostatic confinement in SHPB testingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 108 pp 43ndash57 2018

[20] X F Li Q B Zhang H B Li and J Zhao ldquoGrain-baseddiscrete element method (GB-DEM) modelling of multi-scalefracturing in rocks under dynamic loadingrdquo Rock Mechanicsand Rock Engineering vol 51 no 12 pp 3785ndash3817 2018

[21] F Dai M D Wei N W Xu Y Ma and D S Yang ldquoNu-merical assessment of the progressive rock fracture mecha-nism of cracked chevron notched Brazilian disc specimensrdquoRock Mechanics and Rock Engineering vol 48 no 2pp 463ndash479 2015

[22] B Park and K-B Min ldquoBonded-particle discrete elementmodeling of mechanical behavior of transversely isotropicrockrdquo International Journal of Rock Mechanics and MiningSciences vol 76 pp 243ndash255 2015

[23] E Cadoni ldquoDynamic characterization of orthogneiss rocksubjected to intermediate and high strain rates in tensionrdquoRock Mechanics and Rock Engineering vol 43 no 6pp 667ndash676 2010

[24] Y X Zhou K Xia X B Li et al ldquoSuggested methods fordetermining the dynamic strength parameters and Mode-Ifracture toughness of rock materialsrdquo International Journal ofRock Mechanics and Mining Sciences vol 49 no 1pp 105ndash112 2011

[25] F Dai K Xia J P Zuo R Zhang and N W Xu ldquoStatic anddynamic flexural strength anisotropy of Barre graniterdquo Rock Me-chanics and Rock Engineering vol 46 no 6 pp 1589ndash1602 2013

[26] F-Q Gong X-F Si X-B Li and S-Y Wang ldquoDynamictriaxial compression tests on sandstone at high strain ratesand low confining pressures with split Hopkinson pressurebarrdquo International Journal of Rock Mechanics and MiningSciences vol 113 pp 211ndash219 2019

[27] X F Li H B Li L W Liu et al ldquoInvestigating the crackinitiation and propagation mechanism in brittle rocks usinggrain-based finite-discrete element methodrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 127p 104219 2020

[28] F Gong and J Hu ldquoEnergy dissipation characteristic of redsandstone in the dynamic Brazilian disc test with SHPBsetuprdquo Advances in Civil Engineering vol 2020 Article ID7160937 10 pages 2020

[29] J C Li H B Li G W Ma and J Zhao ldquoA time-domainrecursive method to analyse transient wave propagationacross rock jointsrdquo Geophysical Journal Internationalvol 188 no 2 pp 631ndash644 2012

[30] P A Cundall ldquoA computer model for simulating progressivelarge-scale movement in blocky rock systemrdquo in Proceedingsof the International Symposium on Rock Mechanics NancyFrance October 1971

[31] B Park K-B Min N +ompson and P Horsrud ldquo+ree-dimensional bonded-particle discrete element modeling ofmechanical behavior of transversely isotropic rockrdquo Inter-national Journal of Rock Mechanics and Mining Sciencesvol 110 pp 120ndash132 2018

[32] Y M Tien M C Kuo and C H Juang ldquoAn experimentalinvestigation of the failure mechanism of simulated transverselyisotropic rocksrdquo International Journal of Rock Mechanics andMining Sciences vol 43 no 8 pp 1163ndash1181 2006

Advances in Civil Engineering 21

Stre

ss (M

Pa)

Experimental incident waveNumerical incident wave

0

125

250

375

500

50 100 1500Time (micros)

(a)

β = 0deg

Stre

ss (M

Pa)

Experimental stressndashtimeNumerical stressndashtime

0

100

200

300

40 80 120 1600Time (μs)

(b)

β = 225deg

Stre

ss (M

Pa)

Experimental stressndashtimeNumerical stressndashtime

0

90

180

270

50 100 1500Time (μs)

(c)

β = 45degSt

ress

(MPa

)

Experimental stressndashtimeNumerical stressndashtime

0

40

80

120

160

25 50 75 1000Time (μs)

(d)

β = 675deg

Stre

ss (M

Pa)

Experimental stressndashtimeNumerical stressndashtime

0

50

100

150

35 70 105 1400Time (μs)

(e)

β = 90deg

Stre

ss (M

Pa)

Experimental stressndashtimeNumerical stressndashtime

0

80

160

240

20 40 60 80 100 120 1400Time (μs)

(f )

Figure 7 Continued

Advances in Civil Engineering 7

(g)

Figure 7 SHPB numerical model and simulation results (a) Numerical and experimental incident stress wave of 45deg sample (bndashf)Numerical and experimental dynamic compression stress of five angle samples (g) +e ultimate fragment state of five angle samples afterexperiment and simulation

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

100 101 102 10310ndash1

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 10310ndash1

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 103

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 103

Fragment size (mm2)

(d)

Figure 8 Continued

8 Advances in Civil Engineering

failure of the specimen Under any dynamic load thestrength of the specimen always presents a U-shaped changewith the increase of bedding angle and the minimumstrength occurs at 45degndash675deg

+e fragment size decreases gradually as the strain rateincreases while the strength presents an opposite trend +esamples with low bedding angle show minor damage undersmaller impact load and that changes at higher strain rateWith the increase of strain rate more cracks initiate alongthe loading direction and propagate to intersect with the

cracks on the weak bedding plane of 0degndash225deg sample whichleads to further failure of the specimen

Although the impact splitting effect is increased theweak bedding plane limits the crack to the local part of45degndash675deg specimen and hinders its further expansioncausing specimensrsquo difficulty to be further destroyed Asshown in Figure 8 when reaching a certain strain rate theshiver of rocks with β 45degndash675deg tends to be stable +eresults explored that in order to achieve better fragmentationunder high strain rate the relationship between the loading

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 10310ndash1

Fragment size (mm2)

(e)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

15

20

25

30

35

lgδ_

120 18060Strain rate (sndash1)

(f )

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

Stre

ngth

(MPa

)

100

150

200

250

300

100 150 200 25050Strain rate (sndash1)

(g)

Figure 8 +e cumulative distribution curve and average fragment size of layered rocks under different strain rates (andashe) +e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red _ε asymp 80 sminus1 blue _ε asymp 140 sminus1 yellow _ε asymp 170 sminus1 green _ε asymp 200 sminus1 and gray _ε asymp 215 sminus1 (f )Average fragment size vs strain rate under different bedding angles (g) Dynamic strength vs strain rate under different bedding angles

Advances in Civil Engineering 9

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

15

20

25

30

35

lgδ_

50 100 1500Coh (MPa)

(f )

Figure 9 Continued

10 Advances in Civil Engineering

direction and the bedding dip angle should be changedunder different impact forces

42 Effect of Weak Bedding Strength

421 Effect of Weak Bedding Shear Strength +e weakbedding plane is endowed with cohesion strength for 4MPa24MPa 36Mpa 72Mpa and 140MPa respectively asshown in Figure 9 +e minimum strength corresponds tothe weak bedding immediately destroyed the last is the bondstrength of matrix

Tien et al [32] studied the properties of homogeneousrock-like materials and layered materials under staticloading It is considered that the failure of β 0deg samples isnot affected by the bedding as the compressive strength issimilar to that of isotropic rock However from the view ofrock failure it was found that the bedding would disturb thecrack propagation of the specimen When the shear strengthof the bedding plane is minor the bedding as a weak planeleads to the local fracture of 0degndash225deg specimen And there isno preferential defect in the sample as the shear strengthincreases so the degree of fragmentation decreases

+e situation becomes complicated for 45deg specimens+e bedding plane was destroyed first at low bedding shearstrength and the failure that was perpendicular to the weakplane of bedding has happened which leads to the furtherfracture of the specimen However in the case of highbedding shear strength the bedding no longer hinders thepropagation of the splitting crack resulting in smallerfragments of the specimen At the same time the bedding

shear strength makes no contribution to the crack growthand dynamic strength of the 90deg samples

422 Effect of Weak Bedding Tension Strength At eachbedding angle five kinds of tensile strength weak planespecimens are loaded from the weakest 001MPa to 52MPaas same as with the matrix +e results listed in Figure 10illustrate that the tensile strength of the weak plane has littleeffect on the ultimate fragment state and the dynamiccompression strength In most cases the upper part of thecumulative curve causes differentiation pointing that thetensile strength mainly limits the generation of large pieces

Only for 0deg specimen the crack was harder to develop asthe strength enhanced Compared with the bedding shearstrength situation it could be seen that the bedding shearstrength influences the failure of 225deg specimen while the 0degspecimen is mainly damaged by tensile strength at strainrateasymp120 sminus1 +e average size of high bedding angle spec-imen is kept at a small level and near-zero strength results inless fragmentation

43 Effect of Weak Bedding Stiffness +e initial knks ofbedding is 246 which is changed to explore the influenceof stiffness on rock fragmentation as shown in Figure 11 Itis obvious that the bedding stiffness has no effect on thefailure of layered rock However the average fragment sizeof high angle samples tends to be consistent when the kngradually increase +e failure of bedding plane occurs inthe initial loading stage which results in its ability to resist

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

Stre

ngth

(MPa

)90

180

270

50 100 1500Coh (MPa)

(g)

Figure 9 +e cumulative distribution curve and average fragment size of layered rocks with different bedding shear strength (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red coh 4MPa blue coh 24MPa yellow coh 36MPa green coh 72MPaand gray coh 140MPa (f ) Average fragment size vs bedding shear strength under different bedding angles (g) Dynamic strength vsbedding shear strength under different bedding angles

Advances in Civil Engineering 11

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(e)

20

25

30

35

lgδ_

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 10 Continued

12 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 10 +e cumulative distribution curve and average fragment size of layered rocks with different bedding tensile strength (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red ten 001MPa blue ten 3MPa yellow ten 18MPa green ten 42MPaand gray ten 52MPa (f ) Average fragment size vs bedding tension strength under different bedding angles (g) Dynamic strength vsbedding tension strength under different bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 11 Continued

Advances in Civil Engineering 13

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(e)

20

25

30

35

lgδ_

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 11 Continued

14 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 11 +e cumulative distribution curve and average fragment size of layered rocks with different bedding stiffness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red kskn 10 blue kskn 246 yellow knks 1 green knks 246 and grayknks 10 (f ) Average fragment size vs bedding stiffness under different bedding angles (g) Dynamic strength vs bedding stiffness underdifferent bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 12 Continued

Advances in Civil Engineering 15

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

10ndash1 101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102100

Fragment size (mm2)

(e)

15

20

25

30

1 2 3 4 50Thickness (mm)

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 12 Continued

16 Advances in Civil Engineering

Stre

ngth

(MPa

)

120

160

200

240

1 2 3 4 50Thickness (mm)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 12 +e cumulative distribution curve and average fragment size of layered rocks with different bedding thickness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red thickness 08mm blue thickness 14mm yellow thickness 20mmgreen thickness 36mm and gray thickness 50mm (f) Average fragment size vs thickness under different bedding angles (g)Dynamic strength vs thickness under different bedding angles

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 13 Continued

Advances in Civil Engineering 17

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

15

20

25

30

35

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(f )

Figure 13 Continued

18 Advances in Civil Engineering

deformation having no longer an effect on the crackpropagation

44Effect ofWeakBeddingAickness As shown in Figure 12weak plane models with thicknesses of 08mm 14mm20mm 36mm and 50mm are constructed +e beddingstrength is half of the original matrix rather than using SJmodel to avoid the distortion of the simulation results led byall the SJ model damage during loading +e smallestthickness represents the existence of a weak surface In thecase of 50mm half sample is strong matrix and the other isweak It could be seen from Figure 12 that when there existweak planes in the 45deg specimen the central bedding planedivides the specimen into two parts bearing the load sep-arately As a result the bearing capacity of the 45deg specimenschanged little under different conditions which lead tosimilar failure patterns for 45deg specimens

However the bearing capacity and failure mode ofspecimens with other bedding angles change greatly when theproportion of weak plane increases +e simulation revealsthat more defects lead to greater fragmentation and the sizehas a significant downward trend for all angles except 45degwhich means most angle rocks are sensitive to it At the sametime the distribution curves for 45deg specimen have littledifferences and others move left as a whole +e weak planealways makes the specimen with high angle easier to developmore cracks regardless of its width compared to 225degspecimen being the most difficult to be further damaged

45 Effect of Weak Bedding Space A smaller spacing meansthat the number of laminations increases which changesfrom 10 to 1 It could be analyzed from Figure 13 that whenthe bedding angle is greater than 45deg the failure of thespecimens under any spacing is influenced by the beddingplane And more split cracks begin to appear due to thebarrier effect of weak plane being weakened when there isonly one weak plane +e smaller fragment would emerge asthe split crack intersects with the weak plane +e localfailure of specimens is controlled by the bedding planeswhen the spacing is less than 12mm for the specimens withlow bedding angle However the effect of bedding planes for0deg specimens is no longer significant with the space in-creasing while that of 225 specimens still affects localfracture As a result the fragment distribution of otherbedding dip angle samples except 225deg correspondinglyshifts to the left when the bedding space decreases Nev-ertheless the average fragment size rises slightly when thespace exceeds a quarter

5 Conclusions

+e effect of the microscopic properties of bedding andstrain rate on the fragment size distributions of layered rockwith different bedding dip angles was investigated by usingthe 2D discrete element model Several conclusions weredrawn from the experiment

(1) A SHPBnumerical model was established and validatedA new method to obtain reasonable layered rock

Stre

ngth

(MPa

)90

180

270

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(g)

Figure 13+e cumulative distribution curve and average fragment size of layered rocks with different bedding space (andashe)+e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red space 5mm blue space 65mm yellow space 10mm green space 125mm andgray space 25mm (f) Average fragment size vs bedding space under different bedding angles (g) Dynamic strength vs bedding spaceunder different bedding angles

Advances in Civil Engineering 19

dynamic simulation result is proposed +e quasistaticand dynamic test results are combined to calibrate theelastic modulus and strength of layered rock to repro-duce the stressndashstrain curve and ultimate fragment state

(2) +e average size of samples with different beddingangles decreases as the strain rate increases while thestrength presents an opposite trend +e samplestend to become pulverized under high strain rateespecially for the low bedding angle samples as morecracks initiate along the loading direction andpropagate to intersect with the cracks

(3) It is harder for the samples with low dip angle to bedamaged if the bedding shear strength is added whilethe fragmentation of high angle samples does notchange significantly +e average fragment size of 45degsamples varies greatly with the shear strength becausethe failure mode of the specimens changed And thefailure of layered specimens with all dip angles is notaffected by the tensile strength and stiffness

(4) +e wider bedding may promote the crack initiationand propagation under the same impact load whichmeans large fragments develop into small ones It wasinitially considered that the effect of bedding plane onlowdip angle samples is not considerableHowever thebedding has a dramatic impact on the final fragment

(5) Because the effect of hindering splitting crack hasalways existed for the bedding plane of 225deg sampleand more cracks initiate along the loading directionthe fragment size distribution curve of all beddingangle samples gradually moves to the right when thebedding space increases except for the 225deg samplesNevertheless the average fragment size rises slightlywhen the space exceeds a range

Data Availability

Some data models or codes generated or used during thestudy are available from the corresponding author uponrequest static mechanical properties of layered phyllite withdifferent bedding angles and all results of the dynamiccompression test

Conflicts of Interest

+e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

+e constructive comments of three anonymous reviewersare appreciated +is work received funding from the Na-tional Natural Science Foundation of China (Nos 51439008and 51679231) and China Scholarship Council (to the firstauthor No 202004910629)

References

[1] C V B Cunningham ldquoFragmentation estimations and theKuz-Ram modelndashfour years onrdquo in Proceedings of the 2nd

International Symposium on Rock Fragmentation by Blastingpp 475ndash487 Keystone CO USA August 1987

[2] X F Li H B Li Y Q Liu Q C Zhou and X Xia ldquoNumericalsimulation of rock fragmentation mechanisms subject towedge penetration for TBMsrdquo Tunnelling and UndergroundSpace Technology vol 53 pp 96ndash108 2016

[3] J C Li H B Li and J Zhao ldquoAn improved equivalentviscoelastic medium method for wave propagation acrosslayered rock massesrdquo International Journal of Rock Mechanicsand Mining Sciences vol 73 no 1 pp 62ndash69 2015

[4] F Gong W Wuxing Wu T Tianbin Li and X Xuefeng SildquoExperimental simulation and investigation of spalling failureof rectangular tunnel under different three-dimensional stressstatesrdquo International Journal of Rock Mechanics and MiningSciences vol 122 p 104081 2019

[5] D A Shockey D R Curran L Seaman J T Rosenberg andC F Petersen ldquoFragmentation of rock under dynamic loadsrdquoInternational Journal of Rock Mechanics and Mining Sciencesamp Geomechanics Abstracts vol 11 no 8 pp 303ndash317 1974

[6] F Q Gong S Luo G Lin and X B Li ldquoEvaluation of shearstrength parameters of rocks by preset angle shear directshear and triaxial compression testsrdquo Rock Mechanics andRock Engineering vol 53 no 5 pp 2505ndash2519 2020

[7] X F Li H B Li Q B Zhang J L Jiang and J ZhaoldquoDynamic fragmentation of rock material characte-ristic sizefragment distribution and pulverization lawrdquo EngineeringFracture Mechanics vol 199 pp 739ndash759 2018

[8] J D Hogan R J Rogers J G Spray and S Boonsue ldquoDynamicfragmentation of granite for impact energies of 6ndash28 Jrdquo En-gineering Fracture Mechanics vol 79 pp 103ndash125 2012

[9] P Rosin and E Rammler ldquo+e laws governing the fineness ofpowdered coalrdquo Journal of Institute of Fuel vol 7 pp 29ndash36 1933

[10] N F Mott and E H Linfoot ldquoMinistry of supply +e rawmaterial guiderdquo Ae Analyst vol 68 no 806 1943

[11] D E Grady ldquoLocal inertial effects in dynamic fragmentationrdquoJournal of Applied Physics vol 53 no 1 pp 322ndash325 1982

[12] F Zhou and J F Molinari ldquoOn the rate-dependency ofdynamic tensile strength of a model ceramic systemrdquo Com-puter Methods in Applied Mechanics and Engineering vol 194no 12-16 pp 1693ndash1709 2005

[13] X F Li X Li H B Li Q B Zhang and J Zhao ldquoDynamictensile behaviours of heterogeneous rocks the grain scalefracturing characteristics on strength and fragmentationrdquoInternational Journal of Impact Engineering vol 118pp 98ndash118 2018

[14] F Q Gong X B Li and X L Liu ldquoPreliminary experimentalstudy of characteristics of rock subjected to 3D coupled staticand dynamic loadsrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 6 pp 1179ndash1190 2011

[15] S Levy and J F Molinari ldquoDynamic fragmentation of ce-ramics signature of defects and scaling of fragment sizesrdquoJournal of the Mechanics and Physics of Solids vol 58 no 1pp 12ndash26 2010

[16] B Paliwal and K T Ramesh ldquoAn interacting micro-crackdamage model for failure of brittle materials under com-pressionrdquo Journal of the Mechanics and Physics of Solidsvol 56 no 3 pp 896ndash923 2008

[17] R J Wu H B Li X F Li X Xia and L W Liu ldquoExperimentalstudy and numerical simulation of the dynamic behavior oftransversely isotropic phylliterdquo International Journal of Geo-mechanics vol 20 no 8 Article ID 04020105 2020

[18] X Ou X Zhang H Feng C Zhang and J Yang ldquoEffect of theconfining pressure on the dynamic compression properties of

20 Advances in Civil Engineering

transversely isotropic rocksrdquo Advances in Civil Engineeringvol 6 p 11 2019

[19] H-B Du F Dai Y Xu Y Liu and H-n Xu ldquoNumericalinvestigation on the dynamic strength and failure behavior ofrocks under hydrostatic confinement in SHPB testingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 108 pp 43ndash57 2018

[20] X F Li Q B Zhang H B Li and J Zhao ldquoGrain-baseddiscrete element method (GB-DEM) modelling of multi-scalefracturing in rocks under dynamic loadingrdquo Rock Mechanicsand Rock Engineering vol 51 no 12 pp 3785ndash3817 2018

[21] F Dai M D Wei N W Xu Y Ma and D S Yang ldquoNu-merical assessment of the progressive rock fracture mecha-nism of cracked chevron notched Brazilian disc specimensrdquoRock Mechanics and Rock Engineering vol 48 no 2pp 463ndash479 2015

[22] B Park and K-B Min ldquoBonded-particle discrete elementmodeling of mechanical behavior of transversely isotropicrockrdquo International Journal of Rock Mechanics and MiningSciences vol 76 pp 243ndash255 2015

[23] E Cadoni ldquoDynamic characterization of orthogneiss rocksubjected to intermediate and high strain rates in tensionrdquoRock Mechanics and Rock Engineering vol 43 no 6pp 667ndash676 2010

[24] Y X Zhou K Xia X B Li et al ldquoSuggested methods fordetermining the dynamic strength parameters and Mode-Ifracture toughness of rock materialsrdquo International Journal ofRock Mechanics and Mining Sciences vol 49 no 1pp 105ndash112 2011

[25] F Dai K Xia J P Zuo R Zhang and N W Xu ldquoStatic anddynamic flexural strength anisotropy of Barre graniterdquo Rock Me-chanics and Rock Engineering vol 46 no 6 pp 1589ndash1602 2013

[26] F-Q Gong X-F Si X-B Li and S-Y Wang ldquoDynamictriaxial compression tests on sandstone at high strain ratesand low confining pressures with split Hopkinson pressurebarrdquo International Journal of Rock Mechanics and MiningSciences vol 113 pp 211ndash219 2019

[27] X F Li H B Li L W Liu et al ldquoInvestigating the crackinitiation and propagation mechanism in brittle rocks usinggrain-based finite-discrete element methodrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 127p 104219 2020

[28] F Gong and J Hu ldquoEnergy dissipation characteristic of redsandstone in the dynamic Brazilian disc test with SHPBsetuprdquo Advances in Civil Engineering vol 2020 Article ID7160937 10 pages 2020

[29] J C Li H B Li G W Ma and J Zhao ldquoA time-domainrecursive method to analyse transient wave propagationacross rock jointsrdquo Geophysical Journal Internationalvol 188 no 2 pp 631ndash644 2012

[30] P A Cundall ldquoA computer model for simulating progressivelarge-scale movement in blocky rock systemrdquo in Proceedingsof the International Symposium on Rock Mechanics NancyFrance October 1971

[31] B Park K-B Min N +ompson and P Horsrud ldquo+ree-dimensional bonded-particle discrete element modeling ofmechanical behavior of transversely isotropic rockrdquo Inter-national Journal of Rock Mechanics and Mining Sciencesvol 110 pp 120ndash132 2018

[32] Y M Tien M C Kuo and C H Juang ldquoAn experimentalinvestigation of the failure mechanism of simulated transverselyisotropic rocksrdquo International Journal of Rock Mechanics andMining Sciences vol 43 no 8 pp 1163ndash1181 2006

Advances in Civil Engineering 21

(g)

Figure 7 SHPB numerical model and simulation results (a) Numerical and experimental incident stress wave of 45deg sample (bndashf)Numerical and experimental dynamic compression stress of five angle samples (g) +e ultimate fragment state of five angle samples afterexperiment and simulation

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

100 101 102 10310ndash1

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 10310ndash1

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 103

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 103

Fragment size (mm2)

(d)

Figure 8 Continued

8 Advances in Civil Engineering

failure of the specimen Under any dynamic load thestrength of the specimen always presents a U-shaped changewith the increase of bedding angle and the minimumstrength occurs at 45degndash675deg

+e fragment size decreases gradually as the strain rateincreases while the strength presents an opposite trend +esamples with low bedding angle show minor damage undersmaller impact load and that changes at higher strain rateWith the increase of strain rate more cracks initiate alongthe loading direction and propagate to intersect with the

cracks on the weak bedding plane of 0degndash225deg sample whichleads to further failure of the specimen

Although the impact splitting effect is increased theweak bedding plane limits the crack to the local part of45degndash675deg specimen and hinders its further expansioncausing specimensrsquo difficulty to be further destroyed Asshown in Figure 8 when reaching a certain strain rate theshiver of rocks with β 45degndash675deg tends to be stable +eresults explored that in order to achieve better fragmentationunder high strain rate the relationship between the loading

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 10310ndash1

Fragment size (mm2)

(e)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

15

20

25

30

35

lgδ_

120 18060Strain rate (sndash1)

(f )

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

Stre

ngth

(MPa

)

100

150

200

250

300

100 150 200 25050Strain rate (sndash1)

(g)

Figure 8 +e cumulative distribution curve and average fragment size of layered rocks under different strain rates (andashe) +e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red _ε asymp 80 sminus1 blue _ε asymp 140 sminus1 yellow _ε asymp 170 sminus1 green _ε asymp 200 sminus1 and gray _ε asymp 215 sminus1 (f )Average fragment size vs strain rate under different bedding angles (g) Dynamic strength vs strain rate under different bedding angles

Advances in Civil Engineering 9

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

15

20

25

30

35

lgδ_

50 100 1500Coh (MPa)

(f )

Figure 9 Continued

10 Advances in Civil Engineering

direction and the bedding dip angle should be changedunder different impact forces

42 Effect of Weak Bedding Strength

421 Effect of Weak Bedding Shear Strength +e weakbedding plane is endowed with cohesion strength for 4MPa24MPa 36Mpa 72Mpa and 140MPa respectively asshown in Figure 9 +e minimum strength corresponds tothe weak bedding immediately destroyed the last is the bondstrength of matrix

Tien et al [32] studied the properties of homogeneousrock-like materials and layered materials under staticloading It is considered that the failure of β 0deg samples isnot affected by the bedding as the compressive strength issimilar to that of isotropic rock However from the view ofrock failure it was found that the bedding would disturb thecrack propagation of the specimen When the shear strengthof the bedding plane is minor the bedding as a weak planeleads to the local fracture of 0degndash225deg specimen And there isno preferential defect in the sample as the shear strengthincreases so the degree of fragmentation decreases

+e situation becomes complicated for 45deg specimens+e bedding plane was destroyed first at low bedding shearstrength and the failure that was perpendicular to the weakplane of bedding has happened which leads to the furtherfracture of the specimen However in the case of highbedding shear strength the bedding no longer hinders thepropagation of the splitting crack resulting in smallerfragments of the specimen At the same time the bedding

shear strength makes no contribution to the crack growthand dynamic strength of the 90deg samples

422 Effect of Weak Bedding Tension Strength At eachbedding angle five kinds of tensile strength weak planespecimens are loaded from the weakest 001MPa to 52MPaas same as with the matrix +e results listed in Figure 10illustrate that the tensile strength of the weak plane has littleeffect on the ultimate fragment state and the dynamiccompression strength In most cases the upper part of thecumulative curve causes differentiation pointing that thetensile strength mainly limits the generation of large pieces

Only for 0deg specimen the crack was harder to develop asthe strength enhanced Compared with the bedding shearstrength situation it could be seen that the bedding shearstrength influences the failure of 225deg specimen while the 0degspecimen is mainly damaged by tensile strength at strainrateasymp120 sminus1 +e average size of high bedding angle spec-imen is kept at a small level and near-zero strength results inless fragmentation

43 Effect of Weak Bedding Stiffness +e initial knks ofbedding is 246 which is changed to explore the influenceof stiffness on rock fragmentation as shown in Figure 11 Itis obvious that the bedding stiffness has no effect on thefailure of layered rock However the average fragment sizeof high angle samples tends to be consistent when the kngradually increase +e failure of bedding plane occurs inthe initial loading stage which results in its ability to resist

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

Stre

ngth

(MPa

)90

180

270

50 100 1500Coh (MPa)

(g)

Figure 9 +e cumulative distribution curve and average fragment size of layered rocks with different bedding shear strength (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red coh 4MPa blue coh 24MPa yellow coh 36MPa green coh 72MPaand gray coh 140MPa (f ) Average fragment size vs bedding shear strength under different bedding angles (g) Dynamic strength vsbedding shear strength under different bedding angles

Advances in Civil Engineering 11

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(e)

20

25

30

35

lgδ_

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 10 Continued

12 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 10 +e cumulative distribution curve and average fragment size of layered rocks with different bedding tensile strength (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red ten 001MPa blue ten 3MPa yellow ten 18MPa green ten 42MPaand gray ten 52MPa (f ) Average fragment size vs bedding tension strength under different bedding angles (g) Dynamic strength vsbedding tension strength under different bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 11 Continued

Advances in Civil Engineering 13

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(e)

20

25

30

35

lgδ_

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 11 Continued

14 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 11 +e cumulative distribution curve and average fragment size of layered rocks with different bedding stiffness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red kskn 10 blue kskn 246 yellow knks 1 green knks 246 and grayknks 10 (f ) Average fragment size vs bedding stiffness under different bedding angles (g) Dynamic strength vs bedding stiffness underdifferent bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 12 Continued

Advances in Civil Engineering 15

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

10ndash1 101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102100

Fragment size (mm2)

(e)

15

20

25

30

1 2 3 4 50Thickness (mm)

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 12 Continued

16 Advances in Civil Engineering

Stre

ngth

(MPa

)

120

160

200

240

1 2 3 4 50Thickness (mm)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 12 +e cumulative distribution curve and average fragment size of layered rocks with different bedding thickness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red thickness 08mm blue thickness 14mm yellow thickness 20mmgreen thickness 36mm and gray thickness 50mm (f) Average fragment size vs thickness under different bedding angles (g)Dynamic strength vs thickness under different bedding angles

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 13 Continued

Advances in Civil Engineering 17

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

15

20

25

30

35

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(f )

Figure 13 Continued

18 Advances in Civil Engineering

deformation having no longer an effect on the crackpropagation

44Effect ofWeakBeddingAickness As shown in Figure 12weak plane models with thicknesses of 08mm 14mm20mm 36mm and 50mm are constructed +e beddingstrength is half of the original matrix rather than using SJmodel to avoid the distortion of the simulation results led byall the SJ model damage during loading +e smallestthickness represents the existence of a weak surface In thecase of 50mm half sample is strong matrix and the other isweak It could be seen from Figure 12 that when there existweak planes in the 45deg specimen the central bedding planedivides the specimen into two parts bearing the load sep-arately As a result the bearing capacity of the 45deg specimenschanged little under different conditions which lead tosimilar failure patterns for 45deg specimens

However the bearing capacity and failure mode ofspecimens with other bedding angles change greatly when theproportion of weak plane increases +e simulation revealsthat more defects lead to greater fragmentation and the sizehas a significant downward trend for all angles except 45degwhich means most angle rocks are sensitive to it At the sametime the distribution curves for 45deg specimen have littledifferences and others move left as a whole +e weak planealways makes the specimen with high angle easier to developmore cracks regardless of its width compared to 225degspecimen being the most difficult to be further damaged

45 Effect of Weak Bedding Space A smaller spacing meansthat the number of laminations increases which changesfrom 10 to 1 It could be analyzed from Figure 13 that whenthe bedding angle is greater than 45deg the failure of thespecimens under any spacing is influenced by the beddingplane And more split cracks begin to appear due to thebarrier effect of weak plane being weakened when there isonly one weak plane +e smaller fragment would emerge asthe split crack intersects with the weak plane +e localfailure of specimens is controlled by the bedding planeswhen the spacing is less than 12mm for the specimens withlow bedding angle However the effect of bedding planes for0deg specimens is no longer significant with the space in-creasing while that of 225 specimens still affects localfracture As a result the fragment distribution of otherbedding dip angle samples except 225deg correspondinglyshifts to the left when the bedding space decreases Nev-ertheless the average fragment size rises slightly when thespace exceeds a quarter

5 Conclusions

+e effect of the microscopic properties of bedding andstrain rate on the fragment size distributions of layered rockwith different bedding dip angles was investigated by usingthe 2D discrete element model Several conclusions weredrawn from the experiment

(1) A SHPBnumerical model was established and validatedA new method to obtain reasonable layered rock

Stre

ngth

(MPa

)90

180

270

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(g)

Figure 13+e cumulative distribution curve and average fragment size of layered rocks with different bedding space (andashe)+e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red space 5mm blue space 65mm yellow space 10mm green space 125mm andgray space 25mm (f) Average fragment size vs bedding space under different bedding angles (g) Dynamic strength vs bedding spaceunder different bedding angles

Advances in Civil Engineering 19

dynamic simulation result is proposed +e quasistaticand dynamic test results are combined to calibrate theelastic modulus and strength of layered rock to repro-duce the stressndashstrain curve and ultimate fragment state

(2) +e average size of samples with different beddingangles decreases as the strain rate increases while thestrength presents an opposite trend +e samplestend to become pulverized under high strain rateespecially for the low bedding angle samples as morecracks initiate along the loading direction andpropagate to intersect with the cracks

(3) It is harder for the samples with low dip angle to bedamaged if the bedding shear strength is added whilethe fragmentation of high angle samples does notchange significantly +e average fragment size of 45degsamples varies greatly with the shear strength becausethe failure mode of the specimens changed And thefailure of layered specimens with all dip angles is notaffected by the tensile strength and stiffness

(4) +e wider bedding may promote the crack initiationand propagation under the same impact load whichmeans large fragments develop into small ones It wasinitially considered that the effect of bedding plane onlowdip angle samples is not considerableHowever thebedding has a dramatic impact on the final fragment

(5) Because the effect of hindering splitting crack hasalways existed for the bedding plane of 225deg sampleand more cracks initiate along the loading directionthe fragment size distribution curve of all beddingangle samples gradually moves to the right when thebedding space increases except for the 225deg samplesNevertheless the average fragment size rises slightlywhen the space exceeds a range

Data Availability

Some data models or codes generated or used during thestudy are available from the corresponding author uponrequest static mechanical properties of layered phyllite withdifferent bedding angles and all results of the dynamiccompression test

Conflicts of Interest

+e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

+e constructive comments of three anonymous reviewersare appreciated +is work received funding from the Na-tional Natural Science Foundation of China (Nos 51439008and 51679231) and China Scholarship Council (to the firstauthor No 202004910629)

References

[1] C V B Cunningham ldquoFragmentation estimations and theKuz-Ram modelndashfour years onrdquo in Proceedings of the 2nd

International Symposium on Rock Fragmentation by Blastingpp 475ndash487 Keystone CO USA August 1987

[2] X F Li H B Li Y Q Liu Q C Zhou and X Xia ldquoNumericalsimulation of rock fragmentation mechanisms subject towedge penetration for TBMsrdquo Tunnelling and UndergroundSpace Technology vol 53 pp 96ndash108 2016

[3] J C Li H B Li and J Zhao ldquoAn improved equivalentviscoelastic medium method for wave propagation acrosslayered rock massesrdquo International Journal of Rock Mechanicsand Mining Sciences vol 73 no 1 pp 62ndash69 2015

[4] F Gong W Wuxing Wu T Tianbin Li and X Xuefeng SildquoExperimental simulation and investigation of spalling failureof rectangular tunnel under different three-dimensional stressstatesrdquo International Journal of Rock Mechanics and MiningSciences vol 122 p 104081 2019

[5] D A Shockey D R Curran L Seaman J T Rosenberg andC F Petersen ldquoFragmentation of rock under dynamic loadsrdquoInternational Journal of Rock Mechanics and Mining Sciencesamp Geomechanics Abstracts vol 11 no 8 pp 303ndash317 1974

[6] F Q Gong S Luo G Lin and X B Li ldquoEvaluation of shearstrength parameters of rocks by preset angle shear directshear and triaxial compression testsrdquo Rock Mechanics andRock Engineering vol 53 no 5 pp 2505ndash2519 2020

[7] X F Li H B Li Q B Zhang J L Jiang and J ZhaoldquoDynamic fragmentation of rock material characte-ristic sizefragment distribution and pulverization lawrdquo EngineeringFracture Mechanics vol 199 pp 739ndash759 2018

[8] J D Hogan R J Rogers J G Spray and S Boonsue ldquoDynamicfragmentation of granite for impact energies of 6ndash28 Jrdquo En-gineering Fracture Mechanics vol 79 pp 103ndash125 2012

[9] P Rosin and E Rammler ldquo+e laws governing the fineness ofpowdered coalrdquo Journal of Institute of Fuel vol 7 pp 29ndash36 1933

[10] N F Mott and E H Linfoot ldquoMinistry of supply +e rawmaterial guiderdquo Ae Analyst vol 68 no 806 1943

[11] D E Grady ldquoLocal inertial effects in dynamic fragmentationrdquoJournal of Applied Physics vol 53 no 1 pp 322ndash325 1982

[12] F Zhou and J F Molinari ldquoOn the rate-dependency ofdynamic tensile strength of a model ceramic systemrdquo Com-puter Methods in Applied Mechanics and Engineering vol 194no 12-16 pp 1693ndash1709 2005

[13] X F Li X Li H B Li Q B Zhang and J Zhao ldquoDynamictensile behaviours of heterogeneous rocks the grain scalefracturing characteristics on strength and fragmentationrdquoInternational Journal of Impact Engineering vol 118pp 98ndash118 2018

[14] F Q Gong X B Li and X L Liu ldquoPreliminary experimentalstudy of characteristics of rock subjected to 3D coupled staticand dynamic loadsrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 6 pp 1179ndash1190 2011

[15] S Levy and J F Molinari ldquoDynamic fragmentation of ce-ramics signature of defects and scaling of fragment sizesrdquoJournal of the Mechanics and Physics of Solids vol 58 no 1pp 12ndash26 2010

[16] B Paliwal and K T Ramesh ldquoAn interacting micro-crackdamage model for failure of brittle materials under com-pressionrdquo Journal of the Mechanics and Physics of Solidsvol 56 no 3 pp 896ndash923 2008

[17] R J Wu H B Li X F Li X Xia and L W Liu ldquoExperimentalstudy and numerical simulation of the dynamic behavior oftransversely isotropic phylliterdquo International Journal of Geo-mechanics vol 20 no 8 Article ID 04020105 2020

[18] X Ou X Zhang H Feng C Zhang and J Yang ldquoEffect of theconfining pressure on the dynamic compression properties of

20 Advances in Civil Engineering

transversely isotropic rocksrdquo Advances in Civil Engineeringvol 6 p 11 2019

[19] H-B Du F Dai Y Xu Y Liu and H-n Xu ldquoNumericalinvestigation on the dynamic strength and failure behavior ofrocks under hydrostatic confinement in SHPB testingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 108 pp 43ndash57 2018

[20] X F Li Q B Zhang H B Li and J Zhao ldquoGrain-baseddiscrete element method (GB-DEM) modelling of multi-scalefracturing in rocks under dynamic loadingrdquo Rock Mechanicsand Rock Engineering vol 51 no 12 pp 3785ndash3817 2018

[21] F Dai M D Wei N W Xu Y Ma and D S Yang ldquoNu-merical assessment of the progressive rock fracture mecha-nism of cracked chevron notched Brazilian disc specimensrdquoRock Mechanics and Rock Engineering vol 48 no 2pp 463ndash479 2015

[22] B Park and K-B Min ldquoBonded-particle discrete elementmodeling of mechanical behavior of transversely isotropicrockrdquo International Journal of Rock Mechanics and MiningSciences vol 76 pp 243ndash255 2015

[23] E Cadoni ldquoDynamic characterization of orthogneiss rocksubjected to intermediate and high strain rates in tensionrdquoRock Mechanics and Rock Engineering vol 43 no 6pp 667ndash676 2010

[24] Y X Zhou K Xia X B Li et al ldquoSuggested methods fordetermining the dynamic strength parameters and Mode-Ifracture toughness of rock materialsrdquo International Journal ofRock Mechanics and Mining Sciences vol 49 no 1pp 105ndash112 2011

[25] F Dai K Xia J P Zuo R Zhang and N W Xu ldquoStatic anddynamic flexural strength anisotropy of Barre graniterdquo Rock Me-chanics and Rock Engineering vol 46 no 6 pp 1589ndash1602 2013

[26] F-Q Gong X-F Si X-B Li and S-Y Wang ldquoDynamictriaxial compression tests on sandstone at high strain ratesand low confining pressures with split Hopkinson pressurebarrdquo International Journal of Rock Mechanics and MiningSciences vol 113 pp 211ndash219 2019

[27] X F Li H B Li L W Liu et al ldquoInvestigating the crackinitiation and propagation mechanism in brittle rocks usinggrain-based finite-discrete element methodrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 127p 104219 2020

[28] F Gong and J Hu ldquoEnergy dissipation characteristic of redsandstone in the dynamic Brazilian disc test with SHPBsetuprdquo Advances in Civil Engineering vol 2020 Article ID7160937 10 pages 2020

[29] J C Li H B Li G W Ma and J Zhao ldquoA time-domainrecursive method to analyse transient wave propagationacross rock jointsrdquo Geophysical Journal Internationalvol 188 no 2 pp 631ndash644 2012

[30] P A Cundall ldquoA computer model for simulating progressivelarge-scale movement in blocky rock systemrdquo in Proceedingsof the International Symposium on Rock Mechanics NancyFrance October 1971

[31] B Park K-B Min N +ompson and P Horsrud ldquo+ree-dimensional bonded-particle discrete element modeling ofmechanical behavior of transversely isotropic rockrdquo Inter-national Journal of Rock Mechanics and Mining Sciencesvol 110 pp 120ndash132 2018

[32] Y M Tien M C Kuo and C H Juang ldquoAn experimentalinvestigation of the failure mechanism of simulated transverselyisotropic rocksrdquo International Journal of Rock Mechanics andMining Sciences vol 43 no 8 pp 1163ndash1181 2006

Advances in Civil Engineering 21

failure of the specimen Under any dynamic load thestrength of the specimen always presents a U-shaped changewith the increase of bedding angle and the minimumstrength occurs at 45degndash675deg

+e fragment size decreases gradually as the strain rateincreases while the strength presents an opposite trend +esamples with low bedding angle show minor damage undersmaller impact load and that changes at higher strain rateWith the increase of strain rate more cracks initiate alongthe loading direction and propagate to intersect with the

cracks on the weak bedding plane of 0degndash225deg sample whichleads to further failure of the specimen

Although the impact splitting effect is increased theweak bedding plane limits the crack to the local part of45degndash675deg specimen and hinders its further expansioncausing specimensrsquo difficulty to be further destroyed Asshown in Figure 8 when reaching a certain strain rate theshiver of rocks with β 45degndash675deg tends to be stable +eresults explored that in order to achieve better fragmentationunder high strain rate the relationship between the loading

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

100 101 102 10310ndash1

Fragment size (mm2)

(e)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

15

20

25

30

35

lgδ_

120 18060Strain rate (sndash1)

(f )

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

Stre

ngth

(MPa

)

100

150

200

250

300

100 150 200 25050Strain rate (sndash1)

(g)

Figure 8 +e cumulative distribution curve and average fragment size of layered rocks under different strain rates (andashe) +e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red _ε asymp 80 sminus1 blue _ε asymp 140 sminus1 yellow _ε asymp 170 sminus1 green _ε asymp 200 sminus1 and gray _ε asymp 215 sminus1 (f )Average fragment size vs strain rate under different bedding angles (g) Dynamic strength vs strain rate under different bedding angles

Advances in Civil Engineering 9

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

15

20

25

30

35

lgδ_

50 100 1500Coh (MPa)

(f )

Figure 9 Continued

10 Advances in Civil Engineering

direction and the bedding dip angle should be changedunder different impact forces

42 Effect of Weak Bedding Strength

421 Effect of Weak Bedding Shear Strength +e weakbedding plane is endowed with cohesion strength for 4MPa24MPa 36Mpa 72Mpa and 140MPa respectively asshown in Figure 9 +e minimum strength corresponds tothe weak bedding immediately destroyed the last is the bondstrength of matrix

Tien et al [32] studied the properties of homogeneousrock-like materials and layered materials under staticloading It is considered that the failure of β 0deg samples isnot affected by the bedding as the compressive strength issimilar to that of isotropic rock However from the view ofrock failure it was found that the bedding would disturb thecrack propagation of the specimen When the shear strengthof the bedding plane is minor the bedding as a weak planeleads to the local fracture of 0degndash225deg specimen And there isno preferential defect in the sample as the shear strengthincreases so the degree of fragmentation decreases

+e situation becomes complicated for 45deg specimens+e bedding plane was destroyed first at low bedding shearstrength and the failure that was perpendicular to the weakplane of bedding has happened which leads to the furtherfracture of the specimen However in the case of highbedding shear strength the bedding no longer hinders thepropagation of the splitting crack resulting in smallerfragments of the specimen At the same time the bedding

shear strength makes no contribution to the crack growthand dynamic strength of the 90deg samples

422 Effect of Weak Bedding Tension Strength At eachbedding angle five kinds of tensile strength weak planespecimens are loaded from the weakest 001MPa to 52MPaas same as with the matrix +e results listed in Figure 10illustrate that the tensile strength of the weak plane has littleeffect on the ultimate fragment state and the dynamiccompression strength In most cases the upper part of thecumulative curve causes differentiation pointing that thetensile strength mainly limits the generation of large pieces

Only for 0deg specimen the crack was harder to develop asthe strength enhanced Compared with the bedding shearstrength situation it could be seen that the bedding shearstrength influences the failure of 225deg specimen while the 0degspecimen is mainly damaged by tensile strength at strainrateasymp120 sminus1 +e average size of high bedding angle spec-imen is kept at a small level and near-zero strength results inless fragmentation

43 Effect of Weak Bedding Stiffness +e initial knks ofbedding is 246 which is changed to explore the influenceof stiffness on rock fragmentation as shown in Figure 11 Itis obvious that the bedding stiffness has no effect on thefailure of layered rock However the average fragment sizeof high angle samples tends to be consistent when the kngradually increase +e failure of bedding plane occurs inthe initial loading stage which results in its ability to resist

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

Stre

ngth

(MPa

)90

180

270

50 100 1500Coh (MPa)

(g)

Figure 9 +e cumulative distribution curve and average fragment size of layered rocks with different bedding shear strength (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red coh 4MPa blue coh 24MPa yellow coh 36MPa green coh 72MPaand gray coh 140MPa (f ) Average fragment size vs bedding shear strength under different bedding angles (g) Dynamic strength vsbedding shear strength under different bedding angles

Advances in Civil Engineering 11

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(e)

20

25

30

35

lgδ_

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 10 Continued

12 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 10 +e cumulative distribution curve and average fragment size of layered rocks with different bedding tensile strength (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red ten 001MPa blue ten 3MPa yellow ten 18MPa green ten 42MPaand gray ten 52MPa (f ) Average fragment size vs bedding tension strength under different bedding angles (g) Dynamic strength vsbedding tension strength under different bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 11 Continued

Advances in Civil Engineering 13

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(e)

20

25

30

35

lgδ_

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 11 Continued

14 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 11 +e cumulative distribution curve and average fragment size of layered rocks with different bedding stiffness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red kskn 10 blue kskn 246 yellow knks 1 green knks 246 and grayknks 10 (f ) Average fragment size vs bedding stiffness under different bedding angles (g) Dynamic strength vs bedding stiffness underdifferent bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 12 Continued

Advances in Civil Engineering 15

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

10ndash1 101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102100

Fragment size (mm2)

(e)

15

20

25

30

1 2 3 4 50Thickness (mm)

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 12 Continued

16 Advances in Civil Engineering

Stre

ngth

(MPa

)

120

160

200

240

1 2 3 4 50Thickness (mm)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 12 +e cumulative distribution curve and average fragment size of layered rocks with different bedding thickness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red thickness 08mm blue thickness 14mm yellow thickness 20mmgreen thickness 36mm and gray thickness 50mm (f) Average fragment size vs thickness under different bedding angles (g)Dynamic strength vs thickness under different bedding angles

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 13 Continued

Advances in Civil Engineering 17

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

15

20

25

30

35

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(f )

Figure 13 Continued

18 Advances in Civil Engineering

deformation having no longer an effect on the crackpropagation

44Effect ofWeakBeddingAickness As shown in Figure 12weak plane models with thicknesses of 08mm 14mm20mm 36mm and 50mm are constructed +e beddingstrength is half of the original matrix rather than using SJmodel to avoid the distortion of the simulation results led byall the SJ model damage during loading +e smallestthickness represents the existence of a weak surface In thecase of 50mm half sample is strong matrix and the other isweak It could be seen from Figure 12 that when there existweak planes in the 45deg specimen the central bedding planedivides the specimen into two parts bearing the load sep-arately As a result the bearing capacity of the 45deg specimenschanged little under different conditions which lead tosimilar failure patterns for 45deg specimens

However the bearing capacity and failure mode ofspecimens with other bedding angles change greatly when theproportion of weak plane increases +e simulation revealsthat more defects lead to greater fragmentation and the sizehas a significant downward trend for all angles except 45degwhich means most angle rocks are sensitive to it At the sametime the distribution curves for 45deg specimen have littledifferences and others move left as a whole +e weak planealways makes the specimen with high angle easier to developmore cracks regardless of its width compared to 225degspecimen being the most difficult to be further damaged

45 Effect of Weak Bedding Space A smaller spacing meansthat the number of laminations increases which changesfrom 10 to 1 It could be analyzed from Figure 13 that whenthe bedding angle is greater than 45deg the failure of thespecimens under any spacing is influenced by the beddingplane And more split cracks begin to appear due to thebarrier effect of weak plane being weakened when there isonly one weak plane +e smaller fragment would emerge asthe split crack intersects with the weak plane +e localfailure of specimens is controlled by the bedding planeswhen the spacing is less than 12mm for the specimens withlow bedding angle However the effect of bedding planes for0deg specimens is no longer significant with the space in-creasing while that of 225 specimens still affects localfracture As a result the fragment distribution of otherbedding dip angle samples except 225deg correspondinglyshifts to the left when the bedding space decreases Nev-ertheless the average fragment size rises slightly when thespace exceeds a quarter

5 Conclusions

+e effect of the microscopic properties of bedding andstrain rate on the fragment size distributions of layered rockwith different bedding dip angles was investigated by usingthe 2D discrete element model Several conclusions weredrawn from the experiment

(1) A SHPBnumerical model was established and validatedA new method to obtain reasonable layered rock

Stre

ngth

(MPa

)90

180

270

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(g)

Figure 13+e cumulative distribution curve and average fragment size of layered rocks with different bedding space (andashe)+e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red space 5mm blue space 65mm yellow space 10mm green space 125mm andgray space 25mm (f) Average fragment size vs bedding space under different bedding angles (g) Dynamic strength vs bedding spaceunder different bedding angles

Advances in Civil Engineering 19

dynamic simulation result is proposed +e quasistaticand dynamic test results are combined to calibrate theelastic modulus and strength of layered rock to repro-duce the stressndashstrain curve and ultimate fragment state

(2) +e average size of samples with different beddingangles decreases as the strain rate increases while thestrength presents an opposite trend +e samplestend to become pulverized under high strain rateespecially for the low bedding angle samples as morecracks initiate along the loading direction andpropagate to intersect with the cracks

(3) It is harder for the samples with low dip angle to bedamaged if the bedding shear strength is added whilethe fragmentation of high angle samples does notchange significantly +e average fragment size of 45degsamples varies greatly with the shear strength becausethe failure mode of the specimens changed And thefailure of layered specimens with all dip angles is notaffected by the tensile strength and stiffness

(4) +e wider bedding may promote the crack initiationand propagation under the same impact load whichmeans large fragments develop into small ones It wasinitially considered that the effect of bedding plane onlowdip angle samples is not considerableHowever thebedding has a dramatic impact on the final fragment

(5) Because the effect of hindering splitting crack hasalways existed for the bedding plane of 225deg sampleand more cracks initiate along the loading directionthe fragment size distribution curve of all beddingangle samples gradually moves to the right when thebedding space increases except for the 225deg samplesNevertheless the average fragment size rises slightlywhen the space exceeds a range

Data Availability

Some data models or codes generated or used during thestudy are available from the corresponding author uponrequest static mechanical properties of layered phyllite withdifferent bedding angles and all results of the dynamiccompression test

Conflicts of Interest

+e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

+e constructive comments of three anonymous reviewersare appreciated +is work received funding from the Na-tional Natural Science Foundation of China (Nos 51439008and 51679231) and China Scholarship Council (to the firstauthor No 202004910629)

References

[1] C V B Cunningham ldquoFragmentation estimations and theKuz-Ram modelndashfour years onrdquo in Proceedings of the 2nd

International Symposium on Rock Fragmentation by Blastingpp 475ndash487 Keystone CO USA August 1987

[2] X F Li H B Li Y Q Liu Q C Zhou and X Xia ldquoNumericalsimulation of rock fragmentation mechanisms subject towedge penetration for TBMsrdquo Tunnelling and UndergroundSpace Technology vol 53 pp 96ndash108 2016

[3] J C Li H B Li and J Zhao ldquoAn improved equivalentviscoelastic medium method for wave propagation acrosslayered rock massesrdquo International Journal of Rock Mechanicsand Mining Sciences vol 73 no 1 pp 62ndash69 2015

[4] F Gong W Wuxing Wu T Tianbin Li and X Xuefeng SildquoExperimental simulation and investigation of spalling failureof rectangular tunnel under different three-dimensional stressstatesrdquo International Journal of Rock Mechanics and MiningSciences vol 122 p 104081 2019

[5] D A Shockey D R Curran L Seaman J T Rosenberg andC F Petersen ldquoFragmentation of rock under dynamic loadsrdquoInternational Journal of Rock Mechanics and Mining Sciencesamp Geomechanics Abstracts vol 11 no 8 pp 303ndash317 1974

[6] F Q Gong S Luo G Lin and X B Li ldquoEvaluation of shearstrength parameters of rocks by preset angle shear directshear and triaxial compression testsrdquo Rock Mechanics andRock Engineering vol 53 no 5 pp 2505ndash2519 2020

[7] X F Li H B Li Q B Zhang J L Jiang and J ZhaoldquoDynamic fragmentation of rock material characte-ristic sizefragment distribution and pulverization lawrdquo EngineeringFracture Mechanics vol 199 pp 739ndash759 2018

[8] J D Hogan R J Rogers J G Spray and S Boonsue ldquoDynamicfragmentation of granite for impact energies of 6ndash28 Jrdquo En-gineering Fracture Mechanics vol 79 pp 103ndash125 2012

[9] P Rosin and E Rammler ldquo+e laws governing the fineness ofpowdered coalrdquo Journal of Institute of Fuel vol 7 pp 29ndash36 1933

[10] N F Mott and E H Linfoot ldquoMinistry of supply +e rawmaterial guiderdquo Ae Analyst vol 68 no 806 1943

[11] D E Grady ldquoLocal inertial effects in dynamic fragmentationrdquoJournal of Applied Physics vol 53 no 1 pp 322ndash325 1982

[12] F Zhou and J F Molinari ldquoOn the rate-dependency ofdynamic tensile strength of a model ceramic systemrdquo Com-puter Methods in Applied Mechanics and Engineering vol 194no 12-16 pp 1693ndash1709 2005

[13] X F Li X Li H B Li Q B Zhang and J Zhao ldquoDynamictensile behaviours of heterogeneous rocks the grain scalefracturing characteristics on strength and fragmentationrdquoInternational Journal of Impact Engineering vol 118pp 98ndash118 2018

[14] F Q Gong X B Li and X L Liu ldquoPreliminary experimentalstudy of characteristics of rock subjected to 3D coupled staticand dynamic loadsrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 6 pp 1179ndash1190 2011

[15] S Levy and J F Molinari ldquoDynamic fragmentation of ce-ramics signature of defects and scaling of fragment sizesrdquoJournal of the Mechanics and Physics of Solids vol 58 no 1pp 12ndash26 2010

[16] B Paliwal and K T Ramesh ldquoAn interacting micro-crackdamage model for failure of brittle materials under com-pressionrdquo Journal of the Mechanics and Physics of Solidsvol 56 no 3 pp 896ndash923 2008

[17] R J Wu H B Li X F Li X Xia and L W Liu ldquoExperimentalstudy and numerical simulation of the dynamic behavior oftransversely isotropic phylliterdquo International Journal of Geo-mechanics vol 20 no 8 Article ID 04020105 2020

[18] X Ou X Zhang H Feng C Zhang and J Yang ldquoEffect of theconfining pressure on the dynamic compression properties of

20 Advances in Civil Engineering

transversely isotropic rocksrdquo Advances in Civil Engineeringvol 6 p 11 2019

[19] H-B Du F Dai Y Xu Y Liu and H-n Xu ldquoNumericalinvestigation on the dynamic strength and failure behavior ofrocks under hydrostatic confinement in SHPB testingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 108 pp 43ndash57 2018

[20] X F Li Q B Zhang H B Li and J Zhao ldquoGrain-baseddiscrete element method (GB-DEM) modelling of multi-scalefracturing in rocks under dynamic loadingrdquo Rock Mechanicsand Rock Engineering vol 51 no 12 pp 3785ndash3817 2018

[21] F Dai M D Wei N W Xu Y Ma and D S Yang ldquoNu-merical assessment of the progressive rock fracture mecha-nism of cracked chevron notched Brazilian disc specimensrdquoRock Mechanics and Rock Engineering vol 48 no 2pp 463ndash479 2015

[22] B Park and K-B Min ldquoBonded-particle discrete elementmodeling of mechanical behavior of transversely isotropicrockrdquo International Journal of Rock Mechanics and MiningSciences vol 76 pp 243ndash255 2015

[23] E Cadoni ldquoDynamic characterization of orthogneiss rocksubjected to intermediate and high strain rates in tensionrdquoRock Mechanics and Rock Engineering vol 43 no 6pp 667ndash676 2010

[24] Y X Zhou K Xia X B Li et al ldquoSuggested methods fordetermining the dynamic strength parameters and Mode-Ifracture toughness of rock materialsrdquo International Journal ofRock Mechanics and Mining Sciences vol 49 no 1pp 105ndash112 2011

[25] F Dai K Xia J P Zuo R Zhang and N W Xu ldquoStatic anddynamic flexural strength anisotropy of Barre graniterdquo Rock Me-chanics and Rock Engineering vol 46 no 6 pp 1589ndash1602 2013

[26] F-Q Gong X-F Si X-B Li and S-Y Wang ldquoDynamictriaxial compression tests on sandstone at high strain ratesand low confining pressures with split Hopkinson pressurebarrdquo International Journal of Rock Mechanics and MiningSciences vol 113 pp 211ndash219 2019

[27] X F Li H B Li L W Liu et al ldquoInvestigating the crackinitiation and propagation mechanism in brittle rocks usinggrain-based finite-discrete element methodrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 127p 104219 2020

[28] F Gong and J Hu ldquoEnergy dissipation characteristic of redsandstone in the dynamic Brazilian disc test with SHPBsetuprdquo Advances in Civil Engineering vol 2020 Article ID7160937 10 pages 2020

[29] J C Li H B Li G W Ma and J Zhao ldquoA time-domainrecursive method to analyse transient wave propagationacross rock jointsrdquo Geophysical Journal Internationalvol 188 no 2 pp 631ndash644 2012

[30] P A Cundall ldquoA computer model for simulating progressivelarge-scale movement in blocky rock systemrdquo in Proceedingsof the International Symposium on Rock Mechanics NancyFrance October 1971

[31] B Park K-B Min N +ompson and P Horsrud ldquo+ree-dimensional bonded-particle discrete element modeling ofmechanical behavior of transversely isotropic rockrdquo Inter-national Journal of Rock Mechanics and Mining Sciencesvol 110 pp 120ndash132 2018

[32] Y M Tien M C Kuo and C H Juang ldquoAn experimentalinvestigation of the failure mechanism of simulated transverselyisotropic rocksrdquo International Journal of Rock Mechanics andMining Sciences vol 43 no 8 pp 1163ndash1181 2006

Advances in Civil Engineering 21

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

15

20

25

30

35

lgδ_

50 100 1500Coh (MPa)

(f )

Figure 9 Continued

10 Advances in Civil Engineering

direction and the bedding dip angle should be changedunder different impact forces

42 Effect of Weak Bedding Strength

421 Effect of Weak Bedding Shear Strength +e weakbedding plane is endowed with cohesion strength for 4MPa24MPa 36Mpa 72Mpa and 140MPa respectively asshown in Figure 9 +e minimum strength corresponds tothe weak bedding immediately destroyed the last is the bondstrength of matrix

Tien et al [32] studied the properties of homogeneousrock-like materials and layered materials under staticloading It is considered that the failure of β 0deg samples isnot affected by the bedding as the compressive strength issimilar to that of isotropic rock However from the view ofrock failure it was found that the bedding would disturb thecrack propagation of the specimen When the shear strengthof the bedding plane is minor the bedding as a weak planeleads to the local fracture of 0degndash225deg specimen And there isno preferential defect in the sample as the shear strengthincreases so the degree of fragmentation decreases

+e situation becomes complicated for 45deg specimens+e bedding plane was destroyed first at low bedding shearstrength and the failure that was perpendicular to the weakplane of bedding has happened which leads to the furtherfracture of the specimen However in the case of highbedding shear strength the bedding no longer hinders thepropagation of the splitting crack resulting in smallerfragments of the specimen At the same time the bedding

shear strength makes no contribution to the crack growthand dynamic strength of the 90deg samples

422 Effect of Weak Bedding Tension Strength At eachbedding angle five kinds of tensile strength weak planespecimens are loaded from the weakest 001MPa to 52MPaas same as with the matrix +e results listed in Figure 10illustrate that the tensile strength of the weak plane has littleeffect on the ultimate fragment state and the dynamiccompression strength In most cases the upper part of thecumulative curve causes differentiation pointing that thetensile strength mainly limits the generation of large pieces

Only for 0deg specimen the crack was harder to develop asthe strength enhanced Compared with the bedding shearstrength situation it could be seen that the bedding shearstrength influences the failure of 225deg specimen while the 0degspecimen is mainly damaged by tensile strength at strainrateasymp120 sminus1 +e average size of high bedding angle spec-imen is kept at a small level and near-zero strength results inless fragmentation

43 Effect of Weak Bedding Stiffness +e initial knks ofbedding is 246 which is changed to explore the influenceof stiffness on rock fragmentation as shown in Figure 11 Itis obvious that the bedding stiffness has no effect on thefailure of layered rock However the average fragment sizeof high angle samples tends to be consistent when the kngradually increase +e failure of bedding plane occurs inthe initial loading stage which results in its ability to resist

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

Stre

ngth

(MPa

)90

180

270

50 100 1500Coh (MPa)

(g)

Figure 9 +e cumulative distribution curve and average fragment size of layered rocks with different bedding shear strength (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red coh 4MPa blue coh 24MPa yellow coh 36MPa green coh 72MPaand gray coh 140MPa (f ) Average fragment size vs bedding shear strength under different bedding angles (g) Dynamic strength vsbedding shear strength under different bedding angles

Advances in Civil Engineering 11

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(e)

20

25

30

35

lgδ_

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 10 Continued

12 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 10 +e cumulative distribution curve and average fragment size of layered rocks with different bedding tensile strength (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red ten 001MPa blue ten 3MPa yellow ten 18MPa green ten 42MPaand gray ten 52MPa (f ) Average fragment size vs bedding tension strength under different bedding angles (g) Dynamic strength vsbedding tension strength under different bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 11 Continued

Advances in Civil Engineering 13

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(e)

20

25

30

35

lgδ_

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 11 Continued

14 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 11 +e cumulative distribution curve and average fragment size of layered rocks with different bedding stiffness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red kskn 10 blue kskn 246 yellow knks 1 green knks 246 and grayknks 10 (f ) Average fragment size vs bedding stiffness under different bedding angles (g) Dynamic strength vs bedding stiffness underdifferent bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 12 Continued

Advances in Civil Engineering 15

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

10ndash1 101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102100

Fragment size (mm2)

(e)

15

20

25

30

1 2 3 4 50Thickness (mm)

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 12 Continued

16 Advances in Civil Engineering

Stre

ngth

(MPa

)

120

160

200

240

1 2 3 4 50Thickness (mm)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 12 +e cumulative distribution curve and average fragment size of layered rocks with different bedding thickness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red thickness 08mm blue thickness 14mm yellow thickness 20mmgreen thickness 36mm and gray thickness 50mm (f) Average fragment size vs thickness under different bedding angles (g)Dynamic strength vs thickness under different bedding angles

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 13 Continued

Advances in Civil Engineering 17

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

15

20

25

30

35

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(f )

Figure 13 Continued

18 Advances in Civil Engineering

deformation having no longer an effect on the crackpropagation

44Effect ofWeakBeddingAickness As shown in Figure 12weak plane models with thicknesses of 08mm 14mm20mm 36mm and 50mm are constructed +e beddingstrength is half of the original matrix rather than using SJmodel to avoid the distortion of the simulation results led byall the SJ model damage during loading +e smallestthickness represents the existence of a weak surface In thecase of 50mm half sample is strong matrix and the other isweak It could be seen from Figure 12 that when there existweak planes in the 45deg specimen the central bedding planedivides the specimen into two parts bearing the load sep-arately As a result the bearing capacity of the 45deg specimenschanged little under different conditions which lead tosimilar failure patterns for 45deg specimens

However the bearing capacity and failure mode ofspecimens with other bedding angles change greatly when theproportion of weak plane increases +e simulation revealsthat more defects lead to greater fragmentation and the sizehas a significant downward trend for all angles except 45degwhich means most angle rocks are sensitive to it At the sametime the distribution curves for 45deg specimen have littledifferences and others move left as a whole +e weak planealways makes the specimen with high angle easier to developmore cracks regardless of its width compared to 225degspecimen being the most difficult to be further damaged

45 Effect of Weak Bedding Space A smaller spacing meansthat the number of laminations increases which changesfrom 10 to 1 It could be analyzed from Figure 13 that whenthe bedding angle is greater than 45deg the failure of thespecimens under any spacing is influenced by the beddingplane And more split cracks begin to appear due to thebarrier effect of weak plane being weakened when there isonly one weak plane +e smaller fragment would emerge asthe split crack intersects with the weak plane +e localfailure of specimens is controlled by the bedding planeswhen the spacing is less than 12mm for the specimens withlow bedding angle However the effect of bedding planes for0deg specimens is no longer significant with the space in-creasing while that of 225 specimens still affects localfracture As a result the fragment distribution of otherbedding dip angle samples except 225deg correspondinglyshifts to the left when the bedding space decreases Nev-ertheless the average fragment size rises slightly when thespace exceeds a quarter

5 Conclusions

+e effect of the microscopic properties of bedding andstrain rate on the fragment size distributions of layered rockwith different bedding dip angles was investigated by usingthe 2D discrete element model Several conclusions weredrawn from the experiment

(1) A SHPBnumerical model was established and validatedA new method to obtain reasonable layered rock

Stre

ngth

(MPa

)90

180

270

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(g)

Figure 13+e cumulative distribution curve and average fragment size of layered rocks with different bedding space (andashe)+e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red space 5mm blue space 65mm yellow space 10mm green space 125mm andgray space 25mm (f) Average fragment size vs bedding space under different bedding angles (g) Dynamic strength vs bedding spaceunder different bedding angles

Advances in Civil Engineering 19

dynamic simulation result is proposed +e quasistaticand dynamic test results are combined to calibrate theelastic modulus and strength of layered rock to repro-duce the stressndashstrain curve and ultimate fragment state

(2) +e average size of samples with different beddingangles decreases as the strain rate increases while thestrength presents an opposite trend +e samplestend to become pulverized under high strain rateespecially for the low bedding angle samples as morecracks initiate along the loading direction andpropagate to intersect with the cracks

(3) It is harder for the samples with low dip angle to bedamaged if the bedding shear strength is added whilethe fragmentation of high angle samples does notchange significantly +e average fragment size of 45degsamples varies greatly with the shear strength becausethe failure mode of the specimens changed And thefailure of layered specimens with all dip angles is notaffected by the tensile strength and stiffness

(4) +e wider bedding may promote the crack initiationand propagation under the same impact load whichmeans large fragments develop into small ones It wasinitially considered that the effect of bedding plane onlowdip angle samples is not considerableHowever thebedding has a dramatic impact on the final fragment

(5) Because the effect of hindering splitting crack hasalways existed for the bedding plane of 225deg sampleand more cracks initiate along the loading directionthe fragment size distribution curve of all beddingangle samples gradually moves to the right when thebedding space increases except for the 225deg samplesNevertheless the average fragment size rises slightlywhen the space exceeds a range

Data Availability

Some data models or codes generated or used during thestudy are available from the corresponding author uponrequest static mechanical properties of layered phyllite withdifferent bedding angles and all results of the dynamiccompression test

Conflicts of Interest

+e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

+e constructive comments of three anonymous reviewersare appreciated +is work received funding from the Na-tional Natural Science Foundation of China (Nos 51439008and 51679231) and China Scholarship Council (to the firstauthor No 202004910629)

References

[1] C V B Cunningham ldquoFragmentation estimations and theKuz-Ram modelndashfour years onrdquo in Proceedings of the 2nd

International Symposium on Rock Fragmentation by Blastingpp 475ndash487 Keystone CO USA August 1987

[2] X F Li H B Li Y Q Liu Q C Zhou and X Xia ldquoNumericalsimulation of rock fragmentation mechanisms subject towedge penetration for TBMsrdquo Tunnelling and UndergroundSpace Technology vol 53 pp 96ndash108 2016

[3] J C Li H B Li and J Zhao ldquoAn improved equivalentviscoelastic medium method for wave propagation acrosslayered rock massesrdquo International Journal of Rock Mechanicsand Mining Sciences vol 73 no 1 pp 62ndash69 2015

[4] F Gong W Wuxing Wu T Tianbin Li and X Xuefeng SildquoExperimental simulation and investigation of spalling failureof rectangular tunnel under different three-dimensional stressstatesrdquo International Journal of Rock Mechanics and MiningSciences vol 122 p 104081 2019

[5] D A Shockey D R Curran L Seaman J T Rosenberg andC F Petersen ldquoFragmentation of rock under dynamic loadsrdquoInternational Journal of Rock Mechanics and Mining Sciencesamp Geomechanics Abstracts vol 11 no 8 pp 303ndash317 1974

[6] F Q Gong S Luo G Lin and X B Li ldquoEvaluation of shearstrength parameters of rocks by preset angle shear directshear and triaxial compression testsrdquo Rock Mechanics andRock Engineering vol 53 no 5 pp 2505ndash2519 2020

[7] X F Li H B Li Q B Zhang J L Jiang and J ZhaoldquoDynamic fragmentation of rock material characte-ristic sizefragment distribution and pulverization lawrdquo EngineeringFracture Mechanics vol 199 pp 739ndash759 2018

[8] J D Hogan R J Rogers J G Spray and S Boonsue ldquoDynamicfragmentation of granite for impact energies of 6ndash28 Jrdquo En-gineering Fracture Mechanics vol 79 pp 103ndash125 2012

[9] P Rosin and E Rammler ldquo+e laws governing the fineness ofpowdered coalrdquo Journal of Institute of Fuel vol 7 pp 29ndash36 1933

[10] N F Mott and E H Linfoot ldquoMinistry of supply +e rawmaterial guiderdquo Ae Analyst vol 68 no 806 1943

[11] D E Grady ldquoLocal inertial effects in dynamic fragmentationrdquoJournal of Applied Physics vol 53 no 1 pp 322ndash325 1982

[12] F Zhou and J F Molinari ldquoOn the rate-dependency ofdynamic tensile strength of a model ceramic systemrdquo Com-puter Methods in Applied Mechanics and Engineering vol 194no 12-16 pp 1693ndash1709 2005

[13] X F Li X Li H B Li Q B Zhang and J Zhao ldquoDynamictensile behaviours of heterogeneous rocks the grain scalefracturing characteristics on strength and fragmentationrdquoInternational Journal of Impact Engineering vol 118pp 98ndash118 2018

[14] F Q Gong X B Li and X L Liu ldquoPreliminary experimentalstudy of characteristics of rock subjected to 3D coupled staticand dynamic loadsrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 6 pp 1179ndash1190 2011

[15] S Levy and J F Molinari ldquoDynamic fragmentation of ce-ramics signature of defects and scaling of fragment sizesrdquoJournal of the Mechanics and Physics of Solids vol 58 no 1pp 12ndash26 2010

[16] B Paliwal and K T Ramesh ldquoAn interacting micro-crackdamage model for failure of brittle materials under com-pressionrdquo Journal of the Mechanics and Physics of Solidsvol 56 no 3 pp 896ndash923 2008

[17] R J Wu H B Li X F Li X Xia and L W Liu ldquoExperimentalstudy and numerical simulation of the dynamic behavior oftransversely isotropic phylliterdquo International Journal of Geo-mechanics vol 20 no 8 Article ID 04020105 2020

[18] X Ou X Zhang H Feng C Zhang and J Yang ldquoEffect of theconfining pressure on the dynamic compression properties of

20 Advances in Civil Engineering

transversely isotropic rocksrdquo Advances in Civil Engineeringvol 6 p 11 2019

[19] H-B Du F Dai Y Xu Y Liu and H-n Xu ldquoNumericalinvestigation on the dynamic strength and failure behavior ofrocks under hydrostatic confinement in SHPB testingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 108 pp 43ndash57 2018

[20] X F Li Q B Zhang H B Li and J Zhao ldquoGrain-baseddiscrete element method (GB-DEM) modelling of multi-scalefracturing in rocks under dynamic loadingrdquo Rock Mechanicsand Rock Engineering vol 51 no 12 pp 3785ndash3817 2018

[21] F Dai M D Wei N W Xu Y Ma and D S Yang ldquoNu-merical assessment of the progressive rock fracture mecha-nism of cracked chevron notched Brazilian disc specimensrdquoRock Mechanics and Rock Engineering vol 48 no 2pp 463ndash479 2015

[22] B Park and K-B Min ldquoBonded-particle discrete elementmodeling of mechanical behavior of transversely isotropicrockrdquo International Journal of Rock Mechanics and MiningSciences vol 76 pp 243ndash255 2015

[23] E Cadoni ldquoDynamic characterization of orthogneiss rocksubjected to intermediate and high strain rates in tensionrdquoRock Mechanics and Rock Engineering vol 43 no 6pp 667ndash676 2010

[24] Y X Zhou K Xia X B Li et al ldquoSuggested methods fordetermining the dynamic strength parameters and Mode-Ifracture toughness of rock materialsrdquo International Journal ofRock Mechanics and Mining Sciences vol 49 no 1pp 105ndash112 2011

[25] F Dai K Xia J P Zuo R Zhang and N W Xu ldquoStatic anddynamic flexural strength anisotropy of Barre graniterdquo Rock Me-chanics and Rock Engineering vol 46 no 6 pp 1589ndash1602 2013

[26] F-Q Gong X-F Si X-B Li and S-Y Wang ldquoDynamictriaxial compression tests on sandstone at high strain ratesand low confining pressures with split Hopkinson pressurebarrdquo International Journal of Rock Mechanics and MiningSciences vol 113 pp 211ndash219 2019

[27] X F Li H B Li L W Liu et al ldquoInvestigating the crackinitiation and propagation mechanism in brittle rocks usinggrain-based finite-discrete element methodrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 127p 104219 2020

[28] F Gong and J Hu ldquoEnergy dissipation characteristic of redsandstone in the dynamic Brazilian disc test with SHPBsetuprdquo Advances in Civil Engineering vol 2020 Article ID7160937 10 pages 2020

[29] J C Li H B Li G W Ma and J Zhao ldquoA time-domainrecursive method to analyse transient wave propagationacross rock jointsrdquo Geophysical Journal Internationalvol 188 no 2 pp 631ndash644 2012

[30] P A Cundall ldquoA computer model for simulating progressivelarge-scale movement in blocky rock systemrdquo in Proceedingsof the International Symposium on Rock Mechanics NancyFrance October 1971

[31] B Park K-B Min N +ompson and P Horsrud ldquo+ree-dimensional bonded-particle discrete element modeling ofmechanical behavior of transversely isotropic rockrdquo Inter-national Journal of Rock Mechanics and Mining Sciencesvol 110 pp 120ndash132 2018

[32] Y M Tien M C Kuo and C H Juang ldquoAn experimentalinvestigation of the failure mechanism of simulated transverselyisotropic rocksrdquo International Journal of Rock Mechanics andMining Sciences vol 43 no 8 pp 1163ndash1181 2006

Advances in Civil Engineering 21

direction and the bedding dip angle should be changedunder different impact forces

42 Effect of Weak Bedding Strength

421 Effect of Weak Bedding Shear Strength +e weakbedding plane is endowed with cohesion strength for 4MPa24MPa 36Mpa 72Mpa and 140MPa respectively asshown in Figure 9 +e minimum strength corresponds tothe weak bedding immediately destroyed the last is the bondstrength of matrix

Tien et al [32] studied the properties of homogeneousrock-like materials and layered materials under staticloading It is considered that the failure of β 0deg samples isnot affected by the bedding as the compressive strength issimilar to that of isotropic rock However from the view ofrock failure it was found that the bedding would disturb thecrack propagation of the specimen When the shear strengthof the bedding plane is minor the bedding as a weak planeleads to the local fracture of 0degndash225deg specimen And there isno preferential defect in the sample as the shear strengthincreases so the degree of fragmentation decreases

+e situation becomes complicated for 45deg specimens+e bedding plane was destroyed first at low bedding shearstrength and the failure that was perpendicular to the weakplane of bedding has happened which leads to the furtherfracture of the specimen However in the case of highbedding shear strength the bedding no longer hinders thepropagation of the splitting crack resulting in smallerfragments of the specimen At the same time the bedding

shear strength makes no contribution to the crack growthand dynamic strength of the 90deg samples

422 Effect of Weak Bedding Tension Strength At eachbedding angle five kinds of tensile strength weak planespecimens are loaded from the weakest 001MPa to 52MPaas same as with the matrix +e results listed in Figure 10illustrate that the tensile strength of the weak plane has littleeffect on the ultimate fragment state and the dynamiccompression strength In most cases the upper part of thecumulative curve causes differentiation pointing that thetensile strength mainly limits the generation of large pieces

Only for 0deg specimen the crack was harder to develop asthe strength enhanced Compared with the bedding shearstrength situation it could be seen that the bedding shearstrength influences the failure of 225deg specimen while the 0degspecimen is mainly damaged by tensile strength at strainrateasymp120 sminus1 +e average size of high bedding angle spec-imen is kept at a small level and near-zero strength results inless fragmentation

43 Effect of Weak Bedding Stiffness +e initial knks ofbedding is 246 which is changed to explore the influenceof stiffness on rock fragmentation as shown in Figure 11 Itis obvious that the bedding stiffness has no effect on thefailure of layered rock However the average fragment sizeof high angle samples tends to be consistent when the kngradually increase +e failure of bedding plane occurs inthe initial loading stage which results in its ability to resist

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

Stre

ngth

(MPa

)90

180

270

50 100 1500Coh (MPa)

(g)

Figure 9 +e cumulative distribution curve and average fragment size of layered rocks with different bedding shear strength (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red coh 4MPa blue coh 24MPa yellow coh 36MPa green coh 72MPaand gray coh 140MPa (f ) Average fragment size vs bedding shear strength under different bedding angles (g) Dynamic strength vsbedding shear strength under different bedding angles

Advances in Civil Engineering 11

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(e)

20

25

30

35

lgδ_

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 10 Continued

12 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 10 +e cumulative distribution curve and average fragment size of layered rocks with different bedding tensile strength (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red ten 001MPa blue ten 3MPa yellow ten 18MPa green ten 42MPaand gray ten 52MPa (f ) Average fragment size vs bedding tension strength under different bedding angles (g) Dynamic strength vsbedding tension strength under different bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 11 Continued

Advances in Civil Engineering 13

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(e)

20

25

30

35

lgδ_

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 11 Continued

14 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 11 +e cumulative distribution curve and average fragment size of layered rocks with different bedding stiffness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red kskn 10 blue kskn 246 yellow knks 1 green knks 246 and grayknks 10 (f ) Average fragment size vs bedding stiffness under different bedding angles (g) Dynamic strength vs bedding stiffness underdifferent bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 12 Continued

Advances in Civil Engineering 15

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

10ndash1 101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102100

Fragment size (mm2)

(e)

15

20

25

30

1 2 3 4 50Thickness (mm)

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 12 Continued

16 Advances in Civil Engineering

Stre

ngth

(MPa

)

120

160

200

240

1 2 3 4 50Thickness (mm)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 12 +e cumulative distribution curve and average fragment size of layered rocks with different bedding thickness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red thickness 08mm blue thickness 14mm yellow thickness 20mmgreen thickness 36mm and gray thickness 50mm (f) Average fragment size vs thickness under different bedding angles (g)Dynamic strength vs thickness under different bedding angles

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 13 Continued

Advances in Civil Engineering 17

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

15

20

25

30

35

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(f )

Figure 13 Continued

18 Advances in Civil Engineering

deformation having no longer an effect on the crackpropagation

44Effect ofWeakBeddingAickness As shown in Figure 12weak plane models with thicknesses of 08mm 14mm20mm 36mm and 50mm are constructed +e beddingstrength is half of the original matrix rather than using SJmodel to avoid the distortion of the simulation results led byall the SJ model damage during loading +e smallestthickness represents the existence of a weak surface In thecase of 50mm half sample is strong matrix and the other isweak It could be seen from Figure 12 that when there existweak planes in the 45deg specimen the central bedding planedivides the specimen into two parts bearing the load sep-arately As a result the bearing capacity of the 45deg specimenschanged little under different conditions which lead tosimilar failure patterns for 45deg specimens

However the bearing capacity and failure mode ofspecimens with other bedding angles change greatly when theproportion of weak plane increases +e simulation revealsthat more defects lead to greater fragmentation and the sizehas a significant downward trend for all angles except 45degwhich means most angle rocks are sensitive to it At the sametime the distribution curves for 45deg specimen have littledifferences and others move left as a whole +e weak planealways makes the specimen with high angle easier to developmore cracks regardless of its width compared to 225degspecimen being the most difficult to be further damaged

45 Effect of Weak Bedding Space A smaller spacing meansthat the number of laminations increases which changesfrom 10 to 1 It could be analyzed from Figure 13 that whenthe bedding angle is greater than 45deg the failure of thespecimens under any spacing is influenced by the beddingplane And more split cracks begin to appear due to thebarrier effect of weak plane being weakened when there isonly one weak plane +e smaller fragment would emerge asthe split crack intersects with the weak plane +e localfailure of specimens is controlled by the bedding planeswhen the spacing is less than 12mm for the specimens withlow bedding angle However the effect of bedding planes for0deg specimens is no longer significant with the space in-creasing while that of 225 specimens still affects localfracture As a result the fragment distribution of otherbedding dip angle samples except 225deg correspondinglyshifts to the left when the bedding space decreases Nev-ertheless the average fragment size rises slightly when thespace exceeds a quarter

5 Conclusions

+e effect of the microscopic properties of bedding andstrain rate on the fragment size distributions of layered rockwith different bedding dip angles was investigated by usingthe 2D discrete element model Several conclusions weredrawn from the experiment

(1) A SHPBnumerical model was established and validatedA new method to obtain reasonable layered rock

Stre

ngth

(MPa

)90

180

270

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(g)

Figure 13+e cumulative distribution curve and average fragment size of layered rocks with different bedding space (andashe)+e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red space 5mm blue space 65mm yellow space 10mm green space 125mm andgray space 25mm (f) Average fragment size vs bedding space under different bedding angles (g) Dynamic strength vs bedding spaceunder different bedding angles

Advances in Civil Engineering 19

dynamic simulation result is proposed +e quasistaticand dynamic test results are combined to calibrate theelastic modulus and strength of layered rock to repro-duce the stressndashstrain curve and ultimate fragment state

(2) +e average size of samples with different beddingangles decreases as the strain rate increases while thestrength presents an opposite trend +e samplestend to become pulverized under high strain rateespecially for the low bedding angle samples as morecracks initiate along the loading direction andpropagate to intersect with the cracks

(3) It is harder for the samples with low dip angle to bedamaged if the bedding shear strength is added whilethe fragmentation of high angle samples does notchange significantly +e average fragment size of 45degsamples varies greatly with the shear strength becausethe failure mode of the specimens changed And thefailure of layered specimens with all dip angles is notaffected by the tensile strength and stiffness

(4) +e wider bedding may promote the crack initiationand propagation under the same impact load whichmeans large fragments develop into small ones It wasinitially considered that the effect of bedding plane onlowdip angle samples is not considerableHowever thebedding has a dramatic impact on the final fragment

(5) Because the effect of hindering splitting crack hasalways existed for the bedding plane of 225deg sampleand more cracks initiate along the loading directionthe fragment size distribution curve of all beddingangle samples gradually moves to the right when thebedding space increases except for the 225deg samplesNevertheless the average fragment size rises slightlywhen the space exceeds a range

Data Availability

Some data models or codes generated or used during thestudy are available from the corresponding author uponrequest static mechanical properties of layered phyllite withdifferent bedding angles and all results of the dynamiccompression test

Conflicts of Interest

+e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

+e constructive comments of three anonymous reviewersare appreciated +is work received funding from the Na-tional Natural Science Foundation of China (Nos 51439008and 51679231) and China Scholarship Council (to the firstauthor No 202004910629)

References

[1] C V B Cunningham ldquoFragmentation estimations and theKuz-Ram modelndashfour years onrdquo in Proceedings of the 2nd

International Symposium on Rock Fragmentation by Blastingpp 475ndash487 Keystone CO USA August 1987

[2] X F Li H B Li Y Q Liu Q C Zhou and X Xia ldquoNumericalsimulation of rock fragmentation mechanisms subject towedge penetration for TBMsrdquo Tunnelling and UndergroundSpace Technology vol 53 pp 96ndash108 2016

[3] J C Li H B Li and J Zhao ldquoAn improved equivalentviscoelastic medium method for wave propagation acrosslayered rock massesrdquo International Journal of Rock Mechanicsand Mining Sciences vol 73 no 1 pp 62ndash69 2015

[4] F Gong W Wuxing Wu T Tianbin Li and X Xuefeng SildquoExperimental simulation and investigation of spalling failureof rectangular tunnel under different three-dimensional stressstatesrdquo International Journal of Rock Mechanics and MiningSciences vol 122 p 104081 2019

[5] D A Shockey D R Curran L Seaman J T Rosenberg andC F Petersen ldquoFragmentation of rock under dynamic loadsrdquoInternational Journal of Rock Mechanics and Mining Sciencesamp Geomechanics Abstracts vol 11 no 8 pp 303ndash317 1974

[6] F Q Gong S Luo G Lin and X B Li ldquoEvaluation of shearstrength parameters of rocks by preset angle shear directshear and triaxial compression testsrdquo Rock Mechanics andRock Engineering vol 53 no 5 pp 2505ndash2519 2020

[7] X F Li H B Li Q B Zhang J L Jiang and J ZhaoldquoDynamic fragmentation of rock material characte-ristic sizefragment distribution and pulverization lawrdquo EngineeringFracture Mechanics vol 199 pp 739ndash759 2018

[8] J D Hogan R J Rogers J G Spray and S Boonsue ldquoDynamicfragmentation of granite for impact energies of 6ndash28 Jrdquo En-gineering Fracture Mechanics vol 79 pp 103ndash125 2012

[9] P Rosin and E Rammler ldquo+e laws governing the fineness ofpowdered coalrdquo Journal of Institute of Fuel vol 7 pp 29ndash36 1933

[10] N F Mott and E H Linfoot ldquoMinistry of supply +e rawmaterial guiderdquo Ae Analyst vol 68 no 806 1943

[11] D E Grady ldquoLocal inertial effects in dynamic fragmentationrdquoJournal of Applied Physics vol 53 no 1 pp 322ndash325 1982

[12] F Zhou and J F Molinari ldquoOn the rate-dependency ofdynamic tensile strength of a model ceramic systemrdquo Com-puter Methods in Applied Mechanics and Engineering vol 194no 12-16 pp 1693ndash1709 2005

[13] X F Li X Li H B Li Q B Zhang and J Zhao ldquoDynamictensile behaviours of heterogeneous rocks the grain scalefracturing characteristics on strength and fragmentationrdquoInternational Journal of Impact Engineering vol 118pp 98ndash118 2018

[14] F Q Gong X B Li and X L Liu ldquoPreliminary experimentalstudy of characteristics of rock subjected to 3D coupled staticand dynamic loadsrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 6 pp 1179ndash1190 2011

[15] S Levy and J F Molinari ldquoDynamic fragmentation of ce-ramics signature of defects and scaling of fragment sizesrdquoJournal of the Mechanics and Physics of Solids vol 58 no 1pp 12ndash26 2010

[16] B Paliwal and K T Ramesh ldquoAn interacting micro-crackdamage model for failure of brittle materials under com-pressionrdquo Journal of the Mechanics and Physics of Solidsvol 56 no 3 pp 896ndash923 2008

[17] R J Wu H B Li X F Li X Xia and L W Liu ldquoExperimentalstudy and numerical simulation of the dynamic behavior oftransversely isotropic phylliterdquo International Journal of Geo-mechanics vol 20 no 8 Article ID 04020105 2020

[18] X Ou X Zhang H Feng C Zhang and J Yang ldquoEffect of theconfining pressure on the dynamic compression properties of

20 Advances in Civil Engineering

transversely isotropic rocksrdquo Advances in Civil Engineeringvol 6 p 11 2019

[19] H-B Du F Dai Y Xu Y Liu and H-n Xu ldquoNumericalinvestigation on the dynamic strength and failure behavior ofrocks under hydrostatic confinement in SHPB testingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 108 pp 43ndash57 2018

[20] X F Li Q B Zhang H B Li and J Zhao ldquoGrain-baseddiscrete element method (GB-DEM) modelling of multi-scalefracturing in rocks under dynamic loadingrdquo Rock Mechanicsand Rock Engineering vol 51 no 12 pp 3785ndash3817 2018

[21] F Dai M D Wei N W Xu Y Ma and D S Yang ldquoNu-merical assessment of the progressive rock fracture mecha-nism of cracked chevron notched Brazilian disc specimensrdquoRock Mechanics and Rock Engineering vol 48 no 2pp 463ndash479 2015

[22] B Park and K-B Min ldquoBonded-particle discrete elementmodeling of mechanical behavior of transversely isotropicrockrdquo International Journal of Rock Mechanics and MiningSciences vol 76 pp 243ndash255 2015

[23] E Cadoni ldquoDynamic characterization of orthogneiss rocksubjected to intermediate and high strain rates in tensionrdquoRock Mechanics and Rock Engineering vol 43 no 6pp 667ndash676 2010

[24] Y X Zhou K Xia X B Li et al ldquoSuggested methods fordetermining the dynamic strength parameters and Mode-Ifracture toughness of rock materialsrdquo International Journal ofRock Mechanics and Mining Sciences vol 49 no 1pp 105ndash112 2011

[25] F Dai K Xia J P Zuo R Zhang and N W Xu ldquoStatic anddynamic flexural strength anisotropy of Barre graniterdquo Rock Me-chanics and Rock Engineering vol 46 no 6 pp 1589ndash1602 2013

[26] F-Q Gong X-F Si X-B Li and S-Y Wang ldquoDynamictriaxial compression tests on sandstone at high strain ratesand low confining pressures with split Hopkinson pressurebarrdquo International Journal of Rock Mechanics and MiningSciences vol 113 pp 211ndash219 2019

[27] X F Li H B Li L W Liu et al ldquoInvestigating the crackinitiation and propagation mechanism in brittle rocks usinggrain-based finite-discrete element methodrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 127p 104219 2020

[28] F Gong and J Hu ldquoEnergy dissipation characteristic of redsandstone in the dynamic Brazilian disc test with SHPBsetuprdquo Advances in Civil Engineering vol 2020 Article ID7160937 10 pages 2020

[29] J C Li H B Li G W Ma and J Zhao ldquoA time-domainrecursive method to analyse transient wave propagationacross rock jointsrdquo Geophysical Journal Internationalvol 188 no 2 pp 631ndash644 2012

[30] P A Cundall ldquoA computer model for simulating progressivelarge-scale movement in blocky rock systemrdquo in Proceedingsof the International Symposium on Rock Mechanics NancyFrance October 1971

[31] B Park K-B Min N +ompson and P Horsrud ldquo+ree-dimensional bonded-particle discrete element modeling ofmechanical behavior of transversely isotropic rockrdquo Inter-national Journal of Rock Mechanics and Mining Sciencesvol 110 pp 120ndash132 2018

[32] Y M Tien M C Kuo and C H Juang ldquoAn experimentalinvestigation of the failure mechanism of simulated transverselyisotropic rocksrdquo International Journal of Rock Mechanics andMining Sciences vol 43 no 8 pp 1163ndash1181 2006

Advances in Civil Engineering 21

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(e)

20

25

30

35

lgδ_

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 10 Continued

12 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 10 +e cumulative distribution curve and average fragment size of layered rocks with different bedding tensile strength (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red ten 001MPa blue ten 3MPa yellow ten 18MPa green ten 42MPaand gray ten 52MPa (f ) Average fragment size vs bedding tension strength under different bedding angles (g) Dynamic strength vsbedding tension strength under different bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 11 Continued

Advances in Civil Engineering 13

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(e)

20

25

30

35

lgδ_

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 11 Continued

14 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 11 +e cumulative distribution curve and average fragment size of layered rocks with different bedding stiffness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red kskn 10 blue kskn 246 yellow knks 1 green knks 246 and grayknks 10 (f ) Average fragment size vs bedding stiffness under different bedding angles (g) Dynamic strength vs bedding stiffness underdifferent bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 12 Continued

Advances in Civil Engineering 15

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

10ndash1 101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102100

Fragment size (mm2)

(e)

15

20

25

30

1 2 3 4 50Thickness (mm)

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 12 Continued

16 Advances in Civil Engineering

Stre

ngth

(MPa

)

120

160

200

240

1 2 3 4 50Thickness (mm)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 12 +e cumulative distribution curve and average fragment size of layered rocks with different bedding thickness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red thickness 08mm blue thickness 14mm yellow thickness 20mmgreen thickness 36mm and gray thickness 50mm (f) Average fragment size vs thickness under different bedding angles (g)Dynamic strength vs thickness under different bedding angles

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 13 Continued

Advances in Civil Engineering 17

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

15

20

25

30

35

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(f )

Figure 13 Continued

18 Advances in Civil Engineering

deformation having no longer an effect on the crackpropagation

44Effect ofWeakBeddingAickness As shown in Figure 12weak plane models with thicknesses of 08mm 14mm20mm 36mm and 50mm are constructed +e beddingstrength is half of the original matrix rather than using SJmodel to avoid the distortion of the simulation results led byall the SJ model damage during loading +e smallestthickness represents the existence of a weak surface In thecase of 50mm half sample is strong matrix and the other isweak It could be seen from Figure 12 that when there existweak planes in the 45deg specimen the central bedding planedivides the specimen into two parts bearing the load sep-arately As a result the bearing capacity of the 45deg specimenschanged little under different conditions which lead tosimilar failure patterns for 45deg specimens

However the bearing capacity and failure mode ofspecimens with other bedding angles change greatly when theproportion of weak plane increases +e simulation revealsthat more defects lead to greater fragmentation and the sizehas a significant downward trend for all angles except 45degwhich means most angle rocks are sensitive to it At the sametime the distribution curves for 45deg specimen have littledifferences and others move left as a whole +e weak planealways makes the specimen with high angle easier to developmore cracks regardless of its width compared to 225degspecimen being the most difficult to be further damaged

45 Effect of Weak Bedding Space A smaller spacing meansthat the number of laminations increases which changesfrom 10 to 1 It could be analyzed from Figure 13 that whenthe bedding angle is greater than 45deg the failure of thespecimens under any spacing is influenced by the beddingplane And more split cracks begin to appear due to thebarrier effect of weak plane being weakened when there isonly one weak plane +e smaller fragment would emerge asthe split crack intersects with the weak plane +e localfailure of specimens is controlled by the bedding planeswhen the spacing is less than 12mm for the specimens withlow bedding angle However the effect of bedding planes for0deg specimens is no longer significant with the space in-creasing while that of 225 specimens still affects localfracture As a result the fragment distribution of otherbedding dip angle samples except 225deg correspondinglyshifts to the left when the bedding space decreases Nev-ertheless the average fragment size rises slightly when thespace exceeds a quarter

5 Conclusions

+e effect of the microscopic properties of bedding andstrain rate on the fragment size distributions of layered rockwith different bedding dip angles was investigated by usingthe 2D discrete element model Several conclusions weredrawn from the experiment

(1) A SHPBnumerical model was established and validatedA new method to obtain reasonable layered rock

Stre

ngth

(MPa

)90

180

270

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(g)

Figure 13+e cumulative distribution curve and average fragment size of layered rocks with different bedding space (andashe)+e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red space 5mm blue space 65mm yellow space 10mm green space 125mm andgray space 25mm (f) Average fragment size vs bedding space under different bedding angles (g) Dynamic strength vs bedding spaceunder different bedding angles

Advances in Civil Engineering 19

dynamic simulation result is proposed +e quasistaticand dynamic test results are combined to calibrate theelastic modulus and strength of layered rock to repro-duce the stressndashstrain curve and ultimate fragment state

(2) +e average size of samples with different beddingangles decreases as the strain rate increases while thestrength presents an opposite trend +e samplestend to become pulverized under high strain rateespecially for the low bedding angle samples as morecracks initiate along the loading direction andpropagate to intersect with the cracks

(3) It is harder for the samples with low dip angle to bedamaged if the bedding shear strength is added whilethe fragmentation of high angle samples does notchange significantly +e average fragment size of 45degsamples varies greatly with the shear strength becausethe failure mode of the specimens changed And thefailure of layered specimens with all dip angles is notaffected by the tensile strength and stiffness

(4) +e wider bedding may promote the crack initiationand propagation under the same impact load whichmeans large fragments develop into small ones It wasinitially considered that the effect of bedding plane onlowdip angle samples is not considerableHowever thebedding has a dramatic impact on the final fragment

(5) Because the effect of hindering splitting crack hasalways existed for the bedding plane of 225deg sampleand more cracks initiate along the loading directionthe fragment size distribution curve of all beddingangle samples gradually moves to the right when thebedding space increases except for the 225deg samplesNevertheless the average fragment size rises slightlywhen the space exceeds a range

Data Availability

Some data models or codes generated or used during thestudy are available from the corresponding author uponrequest static mechanical properties of layered phyllite withdifferent bedding angles and all results of the dynamiccompression test

Conflicts of Interest

+e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

+e constructive comments of three anonymous reviewersare appreciated +is work received funding from the Na-tional Natural Science Foundation of China (Nos 51439008and 51679231) and China Scholarship Council (to the firstauthor No 202004910629)

References

[1] C V B Cunningham ldquoFragmentation estimations and theKuz-Ram modelndashfour years onrdquo in Proceedings of the 2nd

International Symposium on Rock Fragmentation by Blastingpp 475ndash487 Keystone CO USA August 1987

[2] X F Li H B Li Y Q Liu Q C Zhou and X Xia ldquoNumericalsimulation of rock fragmentation mechanisms subject towedge penetration for TBMsrdquo Tunnelling and UndergroundSpace Technology vol 53 pp 96ndash108 2016

[3] J C Li H B Li and J Zhao ldquoAn improved equivalentviscoelastic medium method for wave propagation acrosslayered rock massesrdquo International Journal of Rock Mechanicsand Mining Sciences vol 73 no 1 pp 62ndash69 2015

[4] F Gong W Wuxing Wu T Tianbin Li and X Xuefeng SildquoExperimental simulation and investigation of spalling failureof rectangular tunnel under different three-dimensional stressstatesrdquo International Journal of Rock Mechanics and MiningSciences vol 122 p 104081 2019

[5] D A Shockey D R Curran L Seaman J T Rosenberg andC F Petersen ldquoFragmentation of rock under dynamic loadsrdquoInternational Journal of Rock Mechanics and Mining Sciencesamp Geomechanics Abstracts vol 11 no 8 pp 303ndash317 1974

[6] F Q Gong S Luo G Lin and X B Li ldquoEvaluation of shearstrength parameters of rocks by preset angle shear directshear and triaxial compression testsrdquo Rock Mechanics andRock Engineering vol 53 no 5 pp 2505ndash2519 2020

[7] X F Li H B Li Q B Zhang J L Jiang and J ZhaoldquoDynamic fragmentation of rock material characte-ristic sizefragment distribution and pulverization lawrdquo EngineeringFracture Mechanics vol 199 pp 739ndash759 2018

[8] J D Hogan R J Rogers J G Spray and S Boonsue ldquoDynamicfragmentation of granite for impact energies of 6ndash28 Jrdquo En-gineering Fracture Mechanics vol 79 pp 103ndash125 2012

[9] P Rosin and E Rammler ldquo+e laws governing the fineness ofpowdered coalrdquo Journal of Institute of Fuel vol 7 pp 29ndash36 1933

[10] N F Mott and E H Linfoot ldquoMinistry of supply +e rawmaterial guiderdquo Ae Analyst vol 68 no 806 1943

[11] D E Grady ldquoLocal inertial effects in dynamic fragmentationrdquoJournal of Applied Physics vol 53 no 1 pp 322ndash325 1982

[12] F Zhou and J F Molinari ldquoOn the rate-dependency ofdynamic tensile strength of a model ceramic systemrdquo Com-puter Methods in Applied Mechanics and Engineering vol 194no 12-16 pp 1693ndash1709 2005

[13] X F Li X Li H B Li Q B Zhang and J Zhao ldquoDynamictensile behaviours of heterogeneous rocks the grain scalefracturing characteristics on strength and fragmentationrdquoInternational Journal of Impact Engineering vol 118pp 98ndash118 2018

[14] F Q Gong X B Li and X L Liu ldquoPreliminary experimentalstudy of characteristics of rock subjected to 3D coupled staticand dynamic loadsrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 6 pp 1179ndash1190 2011

[15] S Levy and J F Molinari ldquoDynamic fragmentation of ce-ramics signature of defects and scaling of fragment sizesrdquoJournal of the Mechanics and Physics of Solids vol 58 no 1pp 12ndash26 2010

[16] B Paliwal and K T Ramesh ldquoAn interacting micro-crackdamage model for failure of brittle materials under com-pressionrdquo Journal of the Mechanics and Physics of Solidsvol 56 no 3 pp 896ndash923 2008

[17] R J Wu H B Li X F Li X Xia and L W Liu ldquoExperimentalstudy and numerical simulation of the dynamic behavior oftransversely isotropic phylliterdquo International Journal of Geo-mechanics vol 20 no 8 Article ID 04020105 2020

[18] X Ou X Zhang H Feng C Zhang and J Yang ldquoEffect of theconfining pressure on the dynamic compression properties of

20 Advances in Civil Engineering

transversely isotropic rocksrdquo Advances in Civil Engineeringvol 6 p 11 2019

[19] H-B Du F Dai Y Xu Y Liu and H-n Xu ldquoNumericalinvestigation on the dynamic strength and failure behavior ofrocks under hydrostatic confinement in SHPB testingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 108 pp 43ndash57 2018

[20] X F Li Q B Zhang H B Li and J Zhao ldquoGrain-baseddiscrete element method (GB-DEM) modelling of multi-scalefracturing in rocks under dynamic loadingrdquo Rock Mechanicsand Rock Engineering vol 51 no 12 pp 3785ndash3817 2018

[21] F Dai M D Wei N W Xu Y Ma and D S Yang ldquoNu-merical assessment of the progressive rock fracture mecha-nism of cracked chevron notched Brazilian disc specimensrdquoRock Mechanics and Rock Engineering vol 48 no 2pp 463ndash479 2015

[22] B Park and K-B Min ldquoBonded-particle discrete elementmodeling of mechanical behavior of transversely isotropicrockrdquo International Journal of Rock Mechanics and MiningSciences vol 76 pp 243ndash255 2015

[23] E Cadoni ldquoDynamic characterization of orthogneiss rocksubjected to intermediate and high strain rates in tensionrdquoRock Mechanics and Rock Engineering vol 43 no 6pp 667ndash676 2010

[24] Y X Zhou K Xia X B Li et al ldquoSuggested methods fordetermining the dynamic strength parameters and Mode-Ifracture toughness of rock materialsrdquo International Journal ofRock Mechanics and Mining Sciences vol 49 no 1pp 105ndash112 2011

[25] F Dai K Xia J P Zuo R Zhang and N W Xu ldquoStatic anddynamic flexural strength anisotropy of Barre graniterdquo Rock Me-chanics and Rock Engineering vol 46 no 6 pp 1589ndash1602 2013

[26] F-Q Gong X-F Si X-B Li and S-Y Wang ldquoDynamictriaxial compression tests on sandstone at high strain ratesand low confining pressures with split Hopkinson pressurebarrdquo International Journal of Rock Mechanics and MiningSciences vol 113 pp 211ndash219 2019

[27] X F Li H B Li L W Liu et al ldquoInvestigating the crackinitiation and propagation mechanism in brittle rocks usinggrain-based finite-discrete element methodrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 127p 104219 2020

[28] F Gong and J Hu ldquoEnergy dissipation characteristic of redsandstone in the dynamic Brazilian disc test with SHPBsetuprdquo Advances in Civil Engineering vol 2020 Article ID7160937 10 pages 2020

[29] J C Li H B Li G W Ma and J Zhao ldquoA time-domainrecursive method to analyse transient wave propagationacross rock jointsrdquo Geophysical Journal Internationalvol 188 no 2 pp 631ndash644 2012

[30] P A Cundall ldquoA computer model for simulating progressivelarge-scale movement in blocky rock systemrdquo in Proceedingsof the International Symposium on Rock Mechanics NancyFrance October 1971

[31] B Park K-B Min N +ompson and P Horsrud ldquo+ree-dimensional bonded-particle discrete element modeling ofmechanical behavior of transversely isotropic rockrdquo Inter-national Journal of Rock Mechanics and Mining Sciencesvol 110 pp 120ndash132 2018

[32] Y M Tien M C Kuo and C H Juang ldquoAn experimentalinvestigation of the failure mechanism of simulated transverselyisotropic rocksrdquo International Journal of Rock Mechanics andMining Sciences vol 43 no 8 pp 1163ndash1181 2006

Advances in Civil Engineering 21

Stre

ngth

(MPa

)90

180

270

20 40 600Ten (MPa)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 10 +e cumulative distribution curve and average fragment size of layered rocks with different bedding tensile strength (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red ten 001MPa blue ten 3MPa yellow ten 18MPa green ten 42MPaand gray ten 52MPa (f ) Average fragment size vs bedding tension strength under different bedding angles (g) Dynamic strength vsbedding tension strength under different bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 11 Continued

Advances in Civil Engineering 13

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(e)

20

25

30

35

lgδ_

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 11 Continued

14 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 11 +e cumulative distribution curve and average fragment size of layered rocks with different bedding stiffness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red kskn 10 blue kskn 246 yellow knks 1 green knks 246 and grayknks 10 (f ) Average fragment size vs bedding stiffness under different bedding angles (g) Dynamic strength vs bedding stiffness underdifferent bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 12 Continued

Advances in Civil Engineering 15

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

10ndash1 101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102100

Fragment size (mm2)

(e)

15

20

25

30

1 2 3 4 50Thickness (mm)

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 12 Continued

16 Advances in Civil Engineering

Stre

ngth

(MPa

)

120

160

200

240

1 2 3 4 50Thickness (mm)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 12 +e cumulative distribution curve and average fragment size of layered rocks with different bedding thickness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red thickness 08mm blue thickness 14mm yellow thickness 20mmgreen thickness 36mm and gray thickness 50mm (f) Average fragment size vs thickness under different bedding angles (g)Dynamic strength vs thickness under different bedding angles

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 13 Continued

Advances in Civil Engineering 17

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

15

20

25

30

35

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(f )

Figure 13 Continued

18 Advances in Civil Engineering

deformation having no longer an effect on the crackpropagation

44Effect ofWeakBeddingAickness As shown in Figure 12weak plane models with thicknesses of 08mm 14mm20mm 36mm and 50mm are constructed +e beddingstrength is half of the original matrix rather than using SJmodel to avoid the distortion of the simulation results led byall the SJ model damage during loading +e smallestthickness represents the existence of a weak surface In thecase of 50mm half sample is strong matrix and the other isweak It could be seen from Figure 12 that when there existweak planes in the 45deg specimen the central bedding planedivides the specimen into two parts bearing the load sep-arately As a result the bearing capacity of the 45deg specimenschanged little under different conditions which lead tosimilar failure patterns for 45deg specimens

However the bearing capacity and failure mode ofspecimens with other bedding angles change greatly when theproportion of weak plane increases +e simulation revealsthat more defects lead to greater fragmentation and the sizehas a significant downward trend for all angles except 45degwhich means most angle rocks are sensitive to it At the sametime the distribution curves for 45deg specimen have littledifferences and others move left as a whole +e weak planealways makes the specimen with high angle easier to developmore cracks regardless of its width compared to 225degspecimen being the most difficult to be further damaged

45 Effect of Weak Bedding Space A smaller spacing meansthat the number of laminations increases which changesfrom 10 to 1 It could be analyzed from Figure 13 that whenthe bedding angle is greater than 45deg the failure of thespecimens under any spacing is influenced by the beddingplane And more split cracks begin to appear due to thebarrier effect of weak plane being weakened when there isonly one weak plane +e smaller fragment would emerge asthe split crack intersects with the weak plane +e localfailure of specimens is controlled by the bedding planeswhen the spacing is less than 12mm for the specimens withlow bedding angle However the effect of bedding planes for0deg specimens is no longer significant with the space in-creasing while that of 225 specimens still affects localfracture As a result the fragment distribution of otherbedding dip angle samples except 225deg correspondinglyshifts to the left when the bedding space decreases Nev-ertheless the average fragment size rises slightly when thespace exceeds a quarter

5 Conclusions

+e effect of the microscopic properties of bedding andstrain rate on the fragment size distributions of layered rockwith different bedding dip angles was investigated by usingthe 2D discrete element model Several conclusions weredrawn from the experiment

(1) A SHPBnumerical model was established and validatedA new method to obtain reasonable layered rock

Stre

ngth

(MPa

)90

180

270

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(g)

Figure 13+e cumulative distribution curve and average fragment size of layered rocks with different bedding space (andashe)+e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red space 5mm blue space 65mm yellow space 10mm green space 125mm andgray space 25mm (f) Average fragment size vs bedding space under different bedding angles (g) Dynamic strength vs bedding spaceunder different bedding angles

Advances in Civil Engineering 19

dynamic simulation result is proposed +e quasistaticand dynamic test results are combined to calibrate theelastic modulus and strength of layered rock to repro-duce the stressndashstrain curve and ultimate fragment state

(2) +e average size of samples with different beddingangles decreases as the strain rate increases while thestrength presents an opposite trend +e samplestend to become pulverized under high strain rateespecially for the low bedding angle samples as morecracks initiate along the loading direction andpropagate to intersect with the cracks

(3) It is harder for the samples with low dip angle to bedamaged if the bedding shear strength is added whilethe fragmentation of high angle samples does notchange significantly +e average fragment size of 45degsamples varies greatly with the shear strength becausethe failure mode of the specimens changed And thefailure of layered specimens with all dip angles is notaffected by the tensile strength and stiffness

(4) +e wider bedding may promote the crack initiationand propagation under the same impact load whichmeans large fragments develop into small ones It wasinitially considered that the effect of bedding plane onlowdip angle samples is not considerableHowever thebedding has a dramatic impact on the final fragment

(5) Because the effect of hindering splitting crack hasalways existed for the bedding plane of 225deg sampleand more cracks initiate along the loading directionthe fragment size distribution curve of all beddingangle samples gradually moves to the right when thebedding space increases except for the 225deg samplesNevertheless the average fragment size rises slightlywhen the space exceeds a range

Data Availability

Some data models or codes generated or used during thestudy are available from the corresponding author uponrequest static mechanical properties of layered phyllite withdifferent bedding angles and all results of the dynamiccompression test

Conflicts of Interest

+e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

+e constructive comments of three anonymous reviewersare appreciated +is work received funding from the Na-tional Natural Science Foundation of China (Nos 51439008and 51679231) and China Scholarship Council (to the firstauthor No 202004910629)

References

[1] C V B Cunningham ldquoFragmentation estimations and theKuz-Ram modelndashfour years onrdquo in Proceedings of the 2nd

International Symposium on Rock Fragmentation by Blastingpp 475ndash487 Keystone CO USA August 1987

[2] X F Li H B Li Y Q Liu Q C Zhou and X Xia ldquoNumericalsimulation of rock fragmentation mechanisms subject towedge penetration for TBMsrdquo Tunnelling and UndergroundSpace Technology vol 53 pp 96ndash108 2016

[3] J C Li H B Li and J Zhao ldquoAn improved equivalentviscoelastic medium method for wave propagation acrosslayered rock massesrdquo International Journal of Rock Mechanicsand Mining Sciences vol 73 no 1 pp 62ndash69 2015

[4] F Gong W Wuxing Wu T Tianbin Li and X Xuefeng SildquoExperimental simulation and investigation of spalling failureof rectangular tunnel under different three-dimensional stressstatesrdquo International Journal of Rock Mechanics and MiningSciences vol 122 p 104081 2019

[5] D A Shockey D R Curran L Seaman J T Rosenberg andC F Petersen ldquoFragmentation of rock under dynamic loadsrdquoInternational Journal of Rock Mechanics and Mining Sciencesamp Geomechanics Abstracts vol 11 no 8 pp 303ndash317 1974

[6] F Q Gong S Luo G Lin and X B Li ldquoEvaluation of shearstrength parameters of rocks by preset angle shear directshear and triaxial compression testsrdquo Rock Mechanics andRock Engineering vol 53 no 5 pp 2505ndash2519 2020

[7] X F Li H B Li Q B Zhang J L Jiang and J ZhaoldquoDynamic fragmentation of rock material characte-ristic sizefragment distribution and pulverization lawrdquo EngineeringFracture Mechanics vol 199 pp 739ndash759 2018

[8] J D Hogan R J Rogers J G Spray and S Boonsue ldquoDynamicfragmentation of granite for impact energies of 6ndash28 Jrdquo En-gineering Fracture Mechanics vol 79 pp 103ndash125 2012

[9] P Rosin and E Rammler ldquo+e laws governing the fineness ofpowdered coalrdquo Journal of Institute of Fuel vol 7 pp 29ndash36 1933

[10] N F Mott and E H Linfoot ldquoMinistry of supply +e rawmaterial guiderdquo Ae Analyst vol 68 no 806 1943

[11] D E Grady ldquoLocal inertial effects in dynamic fragmentationrdquoJournal of Applied Physics vol 53 no 1 pp 322ndash325 1982

[12] F Zhou and J F Molinari ldquoOn the rate-dependency ofdynamic tensile strength of a model ceramic systemrdquo Com-puter Methods in Applied Mechanics and Engineering vol 194no 12-16 pp 1693ndash1709 2005

[13] X F Li X Li H B Li Q B Zhang and J Zhao ldquoDynamictensile behaviours of heterogeneous rocks the grain scalefracturing characteristics on strength and fragmentationrdquoInternational Journal of Impact Engineering vol 118pp 98ndash118 2018

[14] F Q Gong X B Li and X L Liu ldquoPreliminary experimentalstudy of characteristics of rock subjected to 3D coupled staticand dynamic loadsrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 6 pp 1179ndash1190 2011

[15] S Levy and J F Molinari ldquoDynamic fragmentation of ce-ramics signature of defects and scaling of fragment sizesrdquoJournal of the Mechanics and Physics of Solids vol 58 no 1pp 12ndash26 2010

[16] B Paliwal and K T Ramesh ldquoAn interacting micro-crackdamage model for failure of brittle materials under com-pressionrdquo Journal of the Mechanics and Physics of Solidsvol 56 no 3 pp 896ndash923 2008

[17] R J Wu H B Li X F Li X Xia and L W Liu ldquoExperimentalstudy and numerical simulation of the dynamic behavior oftransversely isotropic phylliterdquo International Journal of Geo-mechanics vol 20 no 8 Article ID 04020105 2020

[18] X Ou X Zhang H Feng C Zhang and J Yang ldquoEffect of theconfining pressure on the dynamic compression properties of

20 Advances in Civil Engineering

transversely isotropic rocksrdquo Advances in Civil Engineeringvol 6 p 11 2019

[19] H-B Du F Dai Y Xu Y Liu and H-n Xu ldquoNumericalinvestigation on the dynamic strength and failure behavior ofrocks under hydrostatic confinement in SHPB testingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 108 pp 43ndash57 2018

[20] X F Li Q B Zhang H B Li and J Zhao ldquoGrain-baseddiscrete element method (GB-DEM) modelling of multi-scalefracturing in rocks under dynamic loadingrdquo Rock Mechanicsand Rock Engineering vol 51 no 12 pp 3785ndash3817 2018

[21] F Dai M D Wei N W Xu Y Ma and D S Yang ldquoNu-merical assessment of the progressive rock fracture mecha-nism of cracked chevron notched Brazilian disc specimensrdquoRock Mechanics and Rock Engineering vol 48 no 2pp 463ndash479 2015

[22] B Park and K-B Min ldquoBonded-particle discrete elementmodeling of mechanical behavior of transversely isotropicrockrdquo International Journal of Rock Mechanics and MiningSciences vol 76 pp 243ndash255 2015

[23] E Cadoni ldquoDynamic characterization of orthogneiss rocksubjected to intermediate and high strain rates in tensionrdquoRock Mechanics and Rock Engineering vol 43 no 6pp 667ndash676 2010

[24] Y X Zhou K Xia X B Li et al ldquoSuggested methods fordetermining the dynamic strength parameters and Mode-Ifracture toughness of rock materialsrdquo International Journal ofRock Mechanics and Mining Sciences vol 49 no 1pp 105ndash112 2011

[25] F Dai K Xia J P Zuo R Zhang and N W Xu ldquoStatic anddynamic flexural strength anisotropy of Barre graniterdquo Rock Me-chanics and Rock Engineering vol 46 no 6 pp 1589ndash1602 2013

[26] F-Q Gong X-F Si X-B Li and S-Y Wang ldquoDynamictriaxial compression tests on sandstone at high strain ratesand low confining pressures with split Hopkinson pressurebarrdquo International Journal of Rock Mechanics and MiningSciences vol 113 pp 211ndash219 2019

[27] X F Li H B Li L W Liu et al ldquoInvestigating the crackinitiation and propagation mechanism in brittle rocks usinggrain-based finite-discrete element methodrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 127p 104219 2020

[28] F Gong and J Hu ldquoEnergy dissipation characteristic of redsandstone in the dynamic Brazilian disc test with SHPBsetuprdquo Advances in Civil Engineering vol 2020 Article ID7160937 10 pages 2020

[29] J C Li H B Li G W Ma and J Zhao ldquoA time-domainrecursive method to analyse transient wave propagationacross rock jointsrdquo Geophysical Journal Internationalvol 188 no 2 pp 631ndash644 2012

[30] P A Cundall ldquoA computer model for simulating progressivelarge-scale movement in blocky rock systemrdquo in Proceedingsof the International Symposium on Rock Mechanics NancyFrance October 1971

[31] B Park K-B Min N +ompson and P Horsrud ldquo+ree-dimensional bonded-particle discrete element modeling ofmechanical behavior of transversely isotropic rockrdquo Inter-national Journal of Rock Mechanics and Mining Sciencesvol 110 pp 120ndash132 2018

[32] Y M Tien M C Kuo and C H Juang ldquoAn experimentalinvestigation of the failure mechanism of simulated transverselyisotropic rocksrdquo International Journal of Rock Mechanics andMining Sciences vol 43 no 8 pp 1163ndash1181 2006

Advances in Civil Engineering 21

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(e)

20

25

30

35

lgδ_

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 11 Continued

14 Advances in Civil Engineering

Stre

ngth

(MPa

)90

180

270

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 11 +e cumulative distribution curve and average fragment size of layered rocks with different bedding stiffness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red kskn 10 blue kskn 246 yellow knks 1 green knks 246 and grayknks 10 (f ) Average fragment size vs bedding stiffness under different bedding angles (g) Dynamic strength vs bedding stiffness underdifferent bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 12 Continued

Advances in Civil Engineering 15

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

10ndash1 101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102100

Fragment size (mm2)

(e)

15

20

25

30

1 2 3 4 50Thickness (mm)

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 12 Continued

16 Advances in Civil Engineering

Stre

ngth

(MPa

)

120

160

200

240

1 2 3 4 50Thickness (mm)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 12 +e cumulative distribution curve and average fragment size of layered rocks with different bedding thickness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red thickness 08mm blue thickness 14mm yellow thickness 20mmgreen thickness 36mm and gray thickness 50mm (f) Average fragment size vs thickness under different bedding angles (g)Dynamic strength vs thickness under different bedding angles

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 13 Continued

Advances in Civil Engineering 17

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

15

20

25

30

35

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(f )

Figure 13 Continued

18 Advances in Civil Engineering

deformation having no longer an effect on the crackpropagation

44Effect ofWeakBeddingAickness As shown in Figure 12weak plane models with thicknesses of 08mm 14mm20mm 36mm and 50mm are constructed +e beddingstrength is half of the original matrix rather than using SJmodel to avoid the distortion of the simulation results led byall the SJ model damage during loading +e smallestthickness represents the existence of a weak surface In thecase of 50mm half sample is strong matrix and the other isweak It could be seen from Figure 12 that when there existweak planes in the 45deg specimen the central bedding planedivides the specimen into two parts bearing the load sep-arately As a result the bearing capacity of the 45deg specimenschanged little under different conditions which lead tosimilar failure patterns for 45deg specimens

However the bearing capacity and failure mode ofspecimens with other bedding angles change greatly when theproportion of weak plane increases +e simulation revealsthat more defects lead to greater fragmentation and the sizehas a significant downward trend for all angles except 45degwhich means most angle rocks are sensitive to it At the sametime the distribution curves for 45deg specimen have littledifferences and others move left as a whole +e weak planealways makes the specimen with high angle easier to developmore cracks regardless of its width compared to 225degspecimen being the most difficult to be further damaged

45 Effect of Weak Bedding Space A smaller spacing meansthat the number of laminations increases which changesfrom 10 to 1 It could be analyzed from Figure 13 that whenthe bedding angle is greater than 45deg the failure of thespecimens under any spacing is influenced by the beddingplane And more split cracks begin to appear due to thebarrier effect of weak plane being weakened when there isonly one weak plane +e smaller fragment would emerge asthe split crack intersects with the weak plane +e localfailure of specimens is controlled by the bedding planeswhen the spacing is less than 12mm for the specimens withlow bedding angle However the effect of bedding planes for0deg specimens is no longer significant with the space in-creasing while that of 225 specimens still affects localfracture As a result the fragment distribution of otherbedding dip angle samples except 225deg correspondinglyshifts to the left when the bedding space decreases Nev-ertheless the average fragment size rises slightly when thespace exceeds a quarter

5 Conclusions

+e effect of the microscopic properties of bedding andstrain rate on the fragment size distributions of layered rockwith different bedding dip angles was investigated by usingthe 2D discrete element model Several conclusions weredrawn from the experiment

(1) A SHPBnumerical model was established and validatedA new method to obtain reasonable layered rock

Stre

ngth

(MPa

)90

180

270

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(g)

Figure 13+e cumulative distribution curve and average fragment size of layered rocks with different bedding space (andashe)+e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red space 5mm blue space 65mm yellow space 10mm green space 125mm andgray space 25mm (f) Average fragment size vs bedding space under different bedding angles (g) Dynamic strength vs bedding spaceunder different bedding angles

Advances in Civil Engineering 19

dynamic simulation result is proposed +e quasistaticand dynamic test results are combined to calibrate theelastic modulus and strength of layered rock to repro-duce the stressndashstrain curve and ultimate fragment state

(2) +e average size of samples with different beddingangles decreases as the strain rate increases while thestrength presents an opposite trend +e samplestend to become pulverized under high strain rateespecially for the low bedding angle samples as morecracks initiate along the loading direction andpropagate to intersect with the cracks

(3) It is harder for the samples with low dip angle to bedamaged if the bedding shear strength is added whilethe fragmentation of high angle samples does notchange significantly +e average fragment size of 45degsamples varies greatly with the shear strength becausethe failure mode of the specimens changed And thefailure of layered specimens with all dip angles is notaffected by the tensile strength and stiffness

(4) +e wider bedding may promote the crack initiationand propagation under the same impact load whichmeans large fragments develop into small ones It wasinitially considered that the effect of bedding plane onlowdip angle samples is not considerableHowever thebedding has a dramatic impact on the final fragment

(5) Because the effect of hindering splitting crack hasalways existed for the bedding plane of 225deg sampleand more cracks initiate along the loading directionthe fragment size distribution curve of all beddingangle samples gradually moves to the right when thebedding space increases except for the 225deg samplesNevertheless the average fragment size rises slightlywhen the space exceeds a range

Data Availability

Some data models or codes generated or used during thestudy are available from the corresponding author uponrequest static mechanical properties of layered phyllite withdifferent bedding angles and all results of the dynamiccompression test

Conflicts of Interest

+e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

+e constructive comments of three anonymous reviewersare appreciated +is work received funding from the Na-tional Natural Science Foundation of China (Nos 51439008and 51679231) and China Scholarship Council (to the firstauthor No 202004910629)

References

[1] C V B Cunningham ldquoFragmentation estimations and theKuz-Ram modelndashfour years onrdquo in Proceedings of the 2nd

International Symposium on Rock Fragmentation by Blastingpp 475ndash487 Keystone CO USA August 1987

[2] X F Li H B Li Y Q Liu Q C Zhou and X Xia ldquoNumericalsimulation of rock fragmentation mechanisms subject towedge penetration for TBMsrdquo Tunnelling and UndergroundSpace Technology vol 53 pp 96ndash108 2016

[3] J C Li H B Li and J Zhao ldquoAn improved equivalentviscoelastic medium method for wave propagation acrosslayered rock massesrdquo International Journal of Rock Mechanicsand Mining Sciences vol 73 no 1 pp 62ndash69 2015

[4] F Gong W Wuxing Wu T Tianbin Li and X Xuefeng SildquoExperimental simulation and investigation of spalling failureof rectangular tunnel under different three-dimensional stressstatesrdquo International Journal of Rock Mechanics and MiningSciences vol 122 p 104081 2019

[5] D A Shockey D R Curran L Seaman J T Rosenberg andC F Petersen ldquoFragmentation of rock under dynamic loadsrdquoInternational Journal of Rock Mechanics and Mining Sciencesamp Geomechanics Abstracts vol 11 no 8 pp 303ndash317 1974

[6] F Q Gong S Luo G Lin and X B Li ldquoEvaluation of shearstrength parameters of rocks by preset angle shear directshear and triaxial compression testsrdquo Rock Mechanics andRock Engineering vol 53 no 5 pp 2505ndash2519 2020

[7] X F Li H B Li Q B Zhang J L Jiang and J ZhaoldquoDynamic fragmentation of rock material characte-ristic sizefragment distribution and pulverization lawrdquo EngineeringFracture Mechanics vol 199 pp 739ndash759 2018

[8] J D Hogan R J Rogers J G Spray and S Boonsue ldquoDynamicfragmentation of granite for impact energies of 6ndash28 Jrdquo En-gineering Fracture Mechanics vol 79 pp 103ndash125 2012

[9] P Rosin and E Rammler ldquo+e laws governing the fineness ofpowdered coalrdquo Journal of Institute of Fuel vol 7 pp 29ndash36 1933

[10] N F Mott and E H Linfoot ldquoMinistry of supply +e rawmaterial guiderdquo Ae Analyst vol 68 no 806 1943

[11] D E Grady ldquoLocal inertial effects in dynamic fragmentationrdquoJournal of Applied Physics vol 53 no 1 pp 322ndash325 1982

[12] F Zhou and J F Molinari ldquoOn the rate-dependency ofdynamic tensile strength of a model ceramic systemrdquo Com-puter Methods in Applied Mechanics and Engineering vol 194no 12-16 pp 1693ndash1709 2005

[13] X F Li X Li H B Li Q B Zhang and J Zhao ldquoDynamictensile behaviours of heterogeneous rocks the grain scalefracturing characteristics on strength and fragmentationrdquoInternational Journal of Impact Engineering vol 118pp 98ndash118 2018

[14] F Q Gong X B Li and X L Liu ldquoPreliminary experimentalstudy of characteristics of rock subjected to 3D coupled staticand dynamic loadsrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 6 pp 1179ndash1190 2011

[15] S Levy and J F Molinari ldquoDynamic fragmentation of ce-ramics signature of defects and scaling of fragment sizesrdquoJournal of the Mechanics and Physics of Solids vol 58 no 1pp 12ndash26 2010

[16] B Paliwal and K T Ramesh ldquoAn interacting micro-crackdamage model for failure of brittle materials under com-pressionrdquo Journal of the Mechanics and Physics of Solidsvol 56 no 3 pp 896ndash923 2008

[17] R J Wu H B Li X F Li X Xia and L W Liu ldquoExperimentalstudy and numerical simulation of the dynamic behavior oftransversely isotropic phylliterdquo International Journal of Geo-mechanics vol 20 no 8 Article ID 04020105 2020

[18] X Ou X Zhang H Feng C Zhang and J Yang ldquoEffect of theconfining pressure on the dynamic compression properties of

20 Advances in Civil Engineering

transversely isotropic rocksrdquo Advances in Civil Engineeringvol 6 p 11 2019

[19] H-B Du F Dai Y Xu Y Liu and H-n Xu ldquoNumericalinvestigation on the dynamic strength and failure behavior ofrocks under hydrostatic confinement in SHPB testingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 108 pp 43ndash57 2018

[20] X F Li Q B Zhang H B Li and J Zhao ldquoGrain-baseddiscrete element method (GB-DEM) modelling of multi-scalefracturing in rocks under dynamic loadingrdquo Rock Mechanicsand Rock Engineering vol 51 no 12 pp 3785ndash3817 2018

[21] F Dai M D Wei N W Xu Y Ma and D S Yang ldquoNu-merical assessment of the progressive rock fracture mecha-nism of cracked chevron notched Brazilian disc specimensrdquoRock Mechanics and Rock Engineering vol 48 no 2pp 463ndash479 2015

[22] B Park and K-B Min ldquoBonded-particle discrete elementmodeling of mechanical behavior of transversely isotropicrockrdquo International Journal of Rock Mechanics and MiningSciences vol 76 pp 243ndash255 2015

[23] E Cadoni ldquoDynamic characterization of orthogneiss rocksubjected to intermediate and high strain rates in tensionrdquoRock Mechanics and Rock Engineering vol 43 no 6pp 667ndash676 2010

[24] Y X Zhou K Xia X B Li et al ldquoSuggested methods fordetermining the dynamic strength parameters and Mode-Ifracture toughness of rock materialsrdquo International Journal ofRock Mechanics and Mining Sciences vol 49 no 1pp 105ndash112 2011

[25] F Dai K Xia J P Zuo R Zhang and N W Xu ldquoStatic anddynamic flexural strength anisotropy of Barre graniterdquo Rock Me-chanics and Rock Engineering vol 46 no 6 pp 1589ndash1602 2013

[26] F-Q Gong X-F Si X-B Li and S-Y Wang ldquoDynamictriaxial compression tests on sandstone at high strain ratesand low confining pressures with split Hopkinson pressurebarrdquo International Journal of Rock Mechanics and MiningSciences vol 113 pp 211ndash219 2019

[27] X F Li H B Li L W Liu et al ldquoInvestigating the crackinitiation and propagation mechanism in brittle rocks usinggrain-based finite-discrete element methodrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 127p 104219 2020

[28] F Gong and J Hu ldquoEnergy dissipation characteristic of redsandstone in the dynamic Brazilian disc test with SHPBsetuprdquo Advances in Civil Engineering vol 2020 Article ID7160937 10 pages 2020

[29] J C Li H B Li G W Ma and J Zhao ldquoA time-domainrecursive method to analyse transient wave propagationacross rock jointsrdquo Geophysical Journal Internationalvol 188 no 2 pp 631ndash644 2012

[30] P A Cundall ldquoA computer model for simulating progressivelarge-scale movement in blocky rock systemrdquo in Proceedingsof the International Symposium on Rock Mechanics NancyFrance October 1971

[31] B Park K-B Min N +ompson and P Horsrud ldquo+ree-dimensional bonded-particle discrete element modeling ofmechanical behavior of transversely isotropic rockrdquo Inter-national Journal of Rock Mechanics and Mining Sciencesvol 110 pp 120ndash132 2018

[32] Y M Tien M C Kuo and C H Juang ldquoAn experimentalinvestigation of the failure mechanism of simulated transverselyisotropic rocksrdquo International Journal of Rock Mechanics andMining Sciences vol 43 no 8 pp 1163ndash1181 2006

Advances in Civil Engineering 21

Stre

ngth

(MPa

)90

180

270

2 4 6 8 100knks

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 11 +e cumulative distribution curve and average fragment size of layered rocks with different bedding stiffness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red kskn 10 blue kskn 246 yellow knks 1 green knks 246 and grayknks 10 (f ) Average fragment size vs bedding stiffness under different bedding angles (g) Dynamic strength vs bedding stiffness underdifferent bedding angles

Perc

ent p

assin

g

β = 0deg0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 12 Continued

Advances in Civil Engineering 15

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

10ndash1 101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102100

Fragment size (mm2)

(e)

15

20

25

30

1 2 3 4 50Thickness (mm)

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 12 Continued

16 Advances in Civil Engineering

Stre

ngth

(MPa

)

120

160

200

240

1 2 3 4 50Thickness (mm)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 12 +e cumulative distribution curve and average fragment size of layered rocks with different bedding thickness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red thickness 08mm blue thickness 14mm yellow thickness 20mmgreen thickness 36mm and gray thickness 50mm (f) Average fragment size vs thickness under different bedding angles (g)Dynamic strength vs thickness under different bedding angles

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 13 Continued

Advances in Civil Engineering 17

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

15

20

25

30

35

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(f )

Figure 13 Continued

18 Advances in Civil Engineering

deformation having no longer an effect on the crackpropagation

44Effect ofWeakBeddingAickness As shown in Figure 12weak plane models with thicknesses of 08mm 14mm20mm 36mm and 50mm are constructed +e beddingstrength is half of the original matrix rather than using SJmodel to avoid the distortion of the simulation results led byall the SJ model damage during loading +e smallestthickness represents the existence of a weak surface In thecase of 50mm half sample is strong matrix and the other isweak It could be seen from Figure 12 that when there existweak planes in the 45deg specimen the central bedding planedivides the specimen into two parts bearing the load sep-arately As a result the bearing capacity of the 45deg specimenschanged little under different conditions which lead tosimilar failure patterns for 45deg specimens

However the bearing capacity and failure mode ofspecimens with other bedding angles change greatly when theproportion of weak plane increases +e simulation revealsthat more defects lead to greater fragmentation and the sizehas a significant downward trend for all angles except 45degwhich means most angle rocks are sensitive to it At the sametime the distribution curves for 45deg specimen have littledifferences and others move left as a whole +e weak planealways makes the specimen with high angle easier to developmore cracks regardless of its width compared to 225degspecimen being the most difficult to be further damaged

45 Effect of Weak Bedding Space A smaller spacing meansthat the number of laminations increases which changesfrom 10 to 1 It could be analyzed from Figure 13 that whenthe bedding angle is greater than 45deg the failure of thespecimens under any spacing is influenced by the beddingplane And more split cracks begin to appear due to thebarrier effect of weak plane being weakened when there isonly one weak plane +e smaller fragment would emerge asthe split crack intersects with the weak plane +e localfailure of specimens is controlled by the bedding planeswhen the spacing is less than 12mm for the specimens withlow bedding angle However the effect of bedding planes for0deg specimens is no longer significant with the space in-creasing while that of 225 specimens still affects localfracture As a result the fragment distribution of otherbedding dip angle samples except 225deg correspondinglyshifts to the left when the bedding space decreases Nev-ertheless the average fragment size rises slightly when thespace exceeds a quarter

5 Conclusions

+e effect of the microscopic properties of bedding andstrain rate on the fragment size distributions of layered rockwith different bedding dip angles was investigated by usingthe 2D discrete element model Several conclusions weredrawn from the experiment

(1) A SHPBnumerical model was established and validatedA new method to obtain reasonable layered rock

Stre

ngth

(MPa

)90

180

270

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(g)

Figure 13+e cumulative distribution curve and average fragment size of layered rocks with different bedding space (andashe)+e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red space 5mm blue space 65mm yellow space 10mm green space 125mm andgray space 25mm (f) Average fragment size vs bedding space under different bedding angles (g) Dynamic strength vs bedding spaceunder different bedding angles

Advances in Civil Engineering 19

dynamic simulation result is proposed +e quasistaticand dynamic test results are combined to calibrate theelastic modulus and strength of layered rock to repro-duce the stressndashstrain curve and ultimate fragment state

(2) +e average size of samples with different beddingangles decreases as the strain rate increases while thestrength presents an opposite trend +e samplestend to become pulverized under high strain rateespecially for the low bedding angle samples as morecracks initiate along the loading direction andpropagate to intersect with the cracks

(3) It is harder for the samples with low dip angle to bedamaged if the bedding shear strength is added whilethe fragmentation of high angle samples does notchange significantly +e average fragment size of 45degsamples varies greatly with the shear strength becausethe failure mode of the specimens changed And thefailure of layered specimens with all dip angles is notaffected by the tensile strength and stiffness

(4) +e wider bedding may promote the crack initiationand propagation under the same impact load whichmeans large fragments develop into small ones It wasinitially considered that the effect of bedding plane onlowdip angle samples is not considerableHowever thebedding has a dramatic impact on the final fragment

(5) Because the effect of hindering splitting crack hasalways existed for the bedding plane of 225deg sampleand more cracks initiate along the loading directionthe fragment size distribution curve of all beddingangle samples gradually moves to the right when thebedding space increases except for the 225deg samplesNevertheless the average fragment size rises slightlywhen the space exceeds a range

Data Availability

Some data models or codes generated or used during thestudy are available from the corresponding author uponrequest static mechanical properties of layered phyllite withdifferent bedding angles and all results of the dynamiccompression test

Conflicts of Interest

+e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

+e constructive comments of three anonymous reviewersare appreciated +is work received funding from the Na-tional Natural Science Foundation of China (Nos 51439008and 51679231) and China Scholarship Council (to the firstauthor No 202004910629)

References

[1] C V B Cunningham ldquoFragmentation estimations and theKuz-Ram modelndashfour years onrdquo in Proceedings of the 2nd

International Symposium on Rock Fragmentation by Blastingpp 475ndash487 Keystone CO USA August 1987

[2] X F Li H B Li Y Q Liu Q C Zhou and X Xia ldquoNumericalsimulation of rock fragmentation mechanisms subject towedge penetration for TBMsrdquo Tunnelling and UndergroundSpace Technology vol 53 pp 96ndash108 2016

[3] J C Li H B Li and J Zhao ldquoAn improved equivalentviscoelastic medium method for wave propagation acrosslayered rock massesrdquo International Journal of Rock Mechanicsand Mining Sciences vol 73 no 1 pp 62ndash69 2015

[4] F Gong W Wuxing Wu T Tianbin Li and X Xuefeng SildquoExperimental simulation and investigation of spalling failureof rectangular tunnel under different three-dimensional stressstatesrdquo International Journal of Rock Mechanics and MiningSciences vol 122 p 104081 2019

[5] D A Shockey D R Curran L Seaman J T Rosenberg andC F Petersen ldquoFragmentation of rock under dynamic loadsrdquoInternational Journal of Rock Mechanics and Mining Sciencesamp Geomechanics Abstracts vol 11 no 8 pp 303ndash317 1974

[6] F Q Gong S Luo G Lin and X B Li ldquoEvaluation of shearstrength parameters of rocks by preset angle shear directshear and triaxial compression testsrdquo Rock Mechanics andRock Engineering vol 53 no 5 pp 2505ndash2519 2020

[7] X F Li H B Li Q B Zhang J L Jiang and J ZhaoldquoDynamic fragmentation of rock material characte-ristic sizefragment distribution and pulverization lawrdquo EngineeringFracture Mechanics vol 199 pp 739ndash759 2018

[8] J D Hogan R J Rogers J G Spray and S Boonsue ldquoDynamicfragmentation of granite for impact energies of 6ndash28 Jrdquo En-gineering Fracture Mechanics vol 79 pp 103ndash125 2012

[9] P Rosin and E Rammler ldquo+e laws governing the fineness ofpowdered coalrdquo Journal of Institute of Fuel vol 7 pp 29ndash36 1933

[10] N F Mott and E H Linfoot ldquoMinistry of supply +e rawmaterial guiderdquo Ae Analyst vol 68 no 806 1943

[11] D E Grady ldquoLocal inertial effects in dynamic fragmentationrdquoJournal of Applied Physics vol 53 no 1 pp 322ndash325 1982

[12] F Zhou and J F Molinari ldquoOn the rate-dependency ofdynamic tensile strength of a model ceramic systemrdquo Com-puter Methods in Applied Mechanics and Engineering vol 194no 12-16 pp 1693ndash1709 2005

[13] X F Li X Li H B Li Q B Zhang and J Zhao ldquoDynamictensile behaviours of heterogeneous rocks the grain scalefracturing characteristics on strength and fragmentationrdquoInternational Journal of Impact Engineering vol 118pp 98ndash118 2018

[14] F Q Gong X B Li and X L Liu ldquoPreliminary experimentalstudy of characteristics of rock subjected to 3D coupled staticand dynamic loadsrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 6 pp 1179ndash1190 2011

[15] S Levy and J F Molinari ldquoDynamic fragmentation of ce-ramics signature of defects and scaling of fragment sizesrdquoJournal of the Mechanics and Physics of Solids vol 58 no 1pp 12ndash26 2010

[16] B Paliwal and K T Ramesh ldquoAn interacting micro-crackdamage model for failure of brittle materials under com-pressionrdquo Journal of the Mechanics and Physics of Solidsvol 56 no 3 pp 896ndash923 2008

[17] R J Wu H B Li X F Li X Xia and L W Liu ldquoExperimentalstudy and numerical simulation of the dynamic behavior oftransversely isotropic phylliterdquo International Journal of Geo-mechanics vol 20 no 8 Article ID 04020105 2020

[18] X Ou X Zhang H Feng C Zhang and J Yang ldquoEffect of theconfining pressure on the dynamic compression properties of

20 Advances in Civil Engineering

transversely isotropic rocksrdquo Advances in Civil Engineeringvol 6 p 11 2019

[19] H-B Du F Dai Y Xu Y Liu and H-n Xu ldquoNumericalinvestigation on the dynamic strength and failure behavior ofrocks under hydrostatic confinement in SHPB testingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 108 pp 43ndash57 2018

[20] X F Li Q B Zhang H B Li and J Zhao ldquoGrain-baseddiscrete element method (GB-DEM) modelling of multi-scalefracturing in rocks under dynamic loadingrdquo Rock Mechanicsand Rock Engineering vol 51 no 12 pp 3785ndash3817 2018

[21] F Dai M D Wei N W Xu Y Ma and D S Yang ldquoNu-merical assessment of the progressive rock fracture mecha-nism of cracked chevron notched Brazilian disc specimensrdquoRock Mechanics and Rock Engineering vol 48 no 2pp 463ndash479 2015

[22] B Park and K-B Min ldquoBonded-particle discrete elementmodeling of mechanical behavior of transversely isotropicrockrdquo International Journal of Rock Mechanics and MiningSciences vol 76 pp 243ndash255 2015

[23] E Cadoni ldquoDynamic characterization of orthogneiss rocksubjected to intermediate and high strain rates in tensionrdquoRock Mechanics and Rock Engineering vol 43 no 6pp 667ndash676 2010

[24] Y X Zhou K Xia X B Li et al ldquoSuggested methods fordetermining the dynamic strength parameters and Mode-Ifracture toughness of rock materialsrdquo International Journal ofRock Mechanics and Mining Sciences vol 49 no 1pp 105ndash112 2011

[25] F Dai K Xia J P Zuo R Zhang and N W Xu ldquoStatic anddynamic flexural strength anisotropy of Barre graniterdquo Rock Me-chanics and Rock Engineering vol 46 no 6 pp 1589ndash1602 2013

[26] F-Q Gong X-F Si X-B Li and S-Y Wang ldquoDynamictriaxial compression tests on sandstone at high strain ratesand low confining pressures with split Hopkinson pressurebarrdquo International Journal of Rock Mechanics and MiningSciences vol 113 pp 211ndash219 2019

[27] X F Li H B Li L W Liu et al ldquoInvestigating the crackinitiation and propagation mechanism in brittle rocks usinggrain-based finite-discrete element methodrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 127p 104219 2020

[28] F Gong and J Hu ldquoEnergy dissipation characteristic of redsandstone in the dynamic Brazilian disc test with SHPBsetuprdquo Advances in Civil Engineering vol 2020 Article ID7160937 10 pages 2020

[29] J C Li H B Li G W Ma and J Zhao ldquoA time-domainrecursive method to analyse transient wave propagationacross rock jointsrdquo Geophysical Journal Internationalvol 188 no 2 pp 631ndash644 2012

[30] P A Cundall ldquoA computer model for simulating progressivelarge-scale movement in blocky rock systemrdquo in Proceedingsof the International Symposium on Rock Mechanics NancyFrance October 1971

[31] B Park K-B Min N +ompson and P Horsrud ldquo+ree-dimensional bonded-particle discrete element modeling ofmechanical behavior of transversely isotropic rockrdquo Inter-national Journal of Rock Mechanics and Mining Sciencesvol 110 pp 120ndash132 2018

[32] Y M Tien M C Kuo and C H Juang ldquoAn experimentalinvestigation of the failure mechanism of simulated transverselyisotropic rocksrdquo International Journal of Rock Mechanics andMining Sciences vol 43 no 8 pp 1163ndash1181 2006

Advances in Civil Engineering 21

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 10310010ndash1

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

10ndash1 101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102100

Fragment size (mm2)

(e)

15

20

25

30

1 2 3 4 50Thickness (mm)

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(f )

Figure 12 Continued

16 Advances in Civil Engineering

Stre

ngth

(MPa

)

120

160

200

240

1 2 3 4 50Thickness (mm)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 12 +e cumulative distribution curve and average fragment size of layered rocks with different bedding thickness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red thickness 08mm blue thickness 14mm yellow thickness 20mmgreen thickness 36mm and gray thickness 50mm (f) Average fragment size vs thickness under different bedding angles (g)Dynamic strength vs thickness under different bedding angles

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 13 Continued

Advances in Civil Engineering 17

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

15

20

25

30

35

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(f )

Figure 13 Continued

18 Advances in Civil Engineering

deformation having no longer an effect on the crackpropagation

44Effect ofWeakBeddingAickness As shown in Figure 12weak plane models with thicknesses of 08mm 14mm20mm 36mm and 50mm are constructed +e beddingstrength is half of the original matrix rather than using SJmodel to avoid the distortion of the simulation results led byall the SJ model damage during loading +e smallestthickness represents the existence of a weak surface In thecase of 50mm half sample is strong matrix and the other isweak It could be seen from Figure 12 that when there existweak planes in the 45deg specimen the central bedding planedivides the specimen into two parts bearing the load sep-arately As a result the bearing capacity of the 45deg specimenschanged little under different conditions which lead tosimilar failure patterns for 45deg specimens

However the bearing capacity and failure mode ofspecimens with other bedding angles change greatly when theproportion of weak plane increases +e simulation revealsthat more defects lead to greater fragmentation and the sizehas a significant downward trend for all angles except 45degwhich means most angle rocks are sensitive to it At the sametime the distribution curves for 45deg specimen have littledifferences and others move left as a whole +e weak planealways makes the specimen with high angle easier to developmore cracks regardless of its width compared to 225degspecimen being the most difficult to be further damaged

45 Effect of Weak Bedding Space A smaller spacing meansthat the number of laminations increases which changesfrom 10 to 1 It could be analyzed from Figure 13 that whenthe bedding angle is greater than 45deg the failure of thespecimens under any spacing is influenced by the beddingplane And more split cracks begin to appear due to thebarrier effect of weak plane being weakened when there isonly one weak plane +e smaller fragment would emerge asthe split crack intersects with the weak plane +e localfailure of specimens is controlled by the bedding planeswhen the spacing is less than 12mm for the specimens withlow bedding angle However the effect of bedding planes for0deg specimens is no longer significant with the space in-creasing while that of 225 specimens still affects localfracture As a result the fragment distribution of otherbedding dip angle samples except 225deg correspondinglyshifts to the left when the bedding space decreases Nev-ertheless the average fragment size rises slightly when thespace exceeds a quarter

5 Conclusions

+e effect of the microscopic properties of bedding andstrain rate on the fragment size distributions of layered rockwith different bedding dip angles was investigated by usingthe 2D discrete element model Several conclusions weredrawn from the experiment

(1) A SHPBnumerical model was established and validatedA new method to obtain reasonable layered rock

Stre

ngth

(MPa

)90

180

270

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(g)

Figure 13+e cumulative distribution curve and average fragment size of layered rocks with different bedding space (andashe)+e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red space 5mm blue space 65mm yellow space 10mm green space 125mm andgray space 25mm (f) Average fragment size vs bedding space under different bedding angles (g) Dynamic strength vs bedding spaceunder different bedding angles

Advances in Civil Engineering 19

dynamic simulation result is proposed +e quasistaticand dynamic test results are combined to calibrate theelastic modulus and strength of layered rock to repro-duce the stressndashstrain curve and ultimate fragment state

(2) +e average size of samples with different beddingangles decreases as the strain rate increases while thestrength presents an opposite trend +e samplestend to become pulverized under high strain rateespecially for the low bedding angle samples as morecracks initiate along the loading direction andpropagate to intersect with the cracks

(3) It is harder for the samples with low dip angle to bedamaged if the bedding shear strength is added whilethe fragmentation of high angle samples does notchange significantly +e average fragment size of 45degsamples varies greatly with the shear strength becausethe failure mode of the specimens changed And thefailure of layered specimens with all dip angles is notaffected by the tensile strength and stiffness

(4) +e wider bedding may promote the crack initiationand propagation under the same impact load whichmeans large fragments develop into small ones It wasinitially considered that the effect of bedding plane onlowdip angle samples is not considerableHowever thebedding has a dramatic impact on the final fragment

(5) Because the effect of hindering splitting crack hasalways existed for the bedding plane of 225deg sampleand more cracks initiate along the loading directionthe fragment size distribution curve of all beddingangle samples gradually moves to the right when thebedding space increases except for the 225deg samplesNevertheless the average fragment size rises slightlywhen the space exceeds a range

Data Availability

Some data models or codes generated or used during thestudy are available from the corresponding author uponrequest static mechanical properties of layered phyllite withdifferent bedding angles and all results of the dynamiccompression test

Conflicts of Interest

+e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

+e constructive comments of three anonymous reviewersare appreciated +is work received funding from the Na-tional Natural Science Foundation of China (Nos 51439008and 51679231) and China Scholarship Council (to the firstauthor No 202004910629)

References

[1] C V B Cunningham ldquoFragmentation estimations and theKuz-Ram modelndashfour years onrdquo in Proceedings of the 2nd

International Symposium on Rock Fragmentation by Blastingpp 475ndash487 Keystone CO USA August 1987

[2] X F Li H B Li Y Q Liu Q C Zhou and X Xia ldquoNumericalsimulation of rock fragmentation mechanisms subject towedge penetration for TBMsrdquo Tunnelling and UndergroundSpace Technology vol 53 pp 96ndash108 2016

[3] J C Li H B Li and J Zhao ldquoAn improved equivalentviscoelastic medium method for wave propagation acrosslayered rock massesrdquo International Journal of Rock Mechanicsand Mining Sciences vol 73 no 1 pp 62ndash69 2015

[4] F Gong W Wuxing Wu T Tianbin Li and X Xuefeng SildquoExperimental simulation and investigation of spalling failureof rectangular tunnel under different three-dimensional stressstatesrdquo International Journal of Rock Mechanics and MiningSciences vol 122 p 104081 2019

[5] D A Shockey D R Curran L Seaman J T Rosenberg andC F Petersen ldquoFragmentation of rock under dynamic loadsrdquoInternational Journal of Rock Mechanics and Mining Sciencesamp Geomechanics Abstracts vol 11 no 8 pp 303ndash317 1974

[6] F Q Gong S Luo G Lin and X B Li ldquoEvaluation of shearstrength parameters of rocks by preset angle shear directshear and triaxial compression testsrdquo Rock Mechanics andRock Engineering vol 53 no 5 pp 2505ndash2519 2020

[7] X F Li H B Li Q B Zhang J L Jiang and J ZhaoldquoDynamic fragmentation of rock material characte-ristic sizefragment distribution and pulverization lawrdquo EngineeringFracture Mechanics vol 199 pp 739ndash759 2018

[8] J D Hogan R J Rogers J G Spray and S Boonsue ldquoDynamicfragmentation of granite for impact energies of 6ndash28 Jrdquo En-gineering Fracture Mechanics vol 79 pp 103ndash125 2012

[9] P Rosin and E Rammler ldquo+e laws governing the fineness ofpowdered coalrdquo Journal of Institute of Fuel vol 7 pp 29ndash36 1933

[10] N F Mott and E H Linfoot ldquoMinistry of supply +e rawmaterial guiderdquo Ae Analyst vol 68 no 806 1943

[11] D E Grady ldquoLocal inertial effects in dynamic fragmentationrdquoJournal of Applied Physics vol 53 no 1 pp 322ndash325 1982

[12] F Zhou and J F Molinari ldquoOn the rate-dependency ofdynamic tensile strength of a model ceramic systemrdquo Com-puter Methods in Applied Mechanics and Engineering vol 194no 12-16 pp 1693ndash1709 2005

[13] X F Li X Li H B Li Q B Zhang and J Zhao ldquoDynamictensile behaviours of heterogeneous rocks the grain scalefracturing characteristics on strength and fragmentationrdquoInternational Journal of Impact Engineering vol 118pp 98ndash118 2018

[14] F Q Gong X B Li and X L Liu ldquoPreliminary experimentalstudy of characteristics of rock subjected to 3D coupled staticand dynamic loadsrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 6 pp 1179ndash1190 2011

[15] S Levy and J F Molinari ldquoDynamic fragmentation of ce-ramics signature of defects and scaling of fragment sizesrdquoJournal of the Mechanics and Physics of Solids vol 58 no 1pp 12ndash26 2010

[16] B Paliwal and K T Ramesh ldquoAn interacting micro-crackdamage model for failure of brittle materials under com-pressionrdquo Journal of the Mechanics and Physics of Solidsvol 56 no 3 pp 896ndash923 2008

[17] R J Wu H B Li X F Li X Xia and L W Liu ldquoExperimentalstudy and numerical simulation of the dynamic behavior oftransversely isotropic phylliterdquo International Journal of Geo-mechanics vol 20 no 8 Article ID 04020105 2020

[18] X Ou X Zhang H Feng C Zhang and J Yang ldquoEffect of theconfining pressure on the dynamic compression properties of

20 Advances in Civil Engineering

transversely isotropic rocksrdquo Advances in Civil Engineeringvol 6 p 11 2019

[19] H-B Du F Dai Y Xu Y Liu and H-n Xu ldquoNumericalinvestigation on the dynamic strength and failure behavior ofrocks under hydrostatic confinement in SHPB testingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 108 pp 43ndash57 2018

[20] X F Li Q B Zhang H B Li and J Zhao ldquoGrain-baseddiscrete element method (GB-DEM) modelling of multi-scalefracturing in rocks under dynamic loadingrdquo Rock Mechanicsand Rock Engineering vol 51 no 12 pp 3785ndash3817 2018

[21] F Dai M D Wei N W Xu Y Ma and D S Yang ldquoNu-merical assessment of the progressive rock fracture mecha-nism of cracked chevron notched Brazilian disc specimensrdquoRock Mechanics and Rock Engineering vol 48 no 2pp 463ndash479 2015

[22] B Park and K-B Min ldquoBonded-particle discrete elementmodeling of mechanical behavior of transversely isotropicrockrdquo International Journal of Rock Mechanics and MiningSciences vol 76 pp 243ndash255 2015

[23] E Cadoni ldquoDynamic characterization of orthogneiss rocksubjected to intermediate and high strain rates in tensionrdquoRock Mechanics and Rock Engineering vol 43 no 6pp 667ndash676 2010

[24] Y X Zhou K Xia X B Li et al ldquoSuggested methods fordetermining the dynamic strength parameters and Mode-Ifracture toughness of rock materialsrdquo International Journal ofRock Mechanics and Mining Sciences vol 49 no 1pp 105ndash112 2011

[25] F Dai K Xia J P Zuo R Zhang and N W Xu ldquoStatic anddynamic flexural strength anisotropy of Barre graniterdquo Rock Me-chanics and Rock Engineering vol 46 no 6 pp 1589ndash1602 2013

[26] F-Q Gong X-F Si X-B Li and S-Y Wang ldquoDynamictriaxial compression tests on sandstone at high strain ratesand low confining pressures with split Hopkinson pressurebarrdquo International Journal of Rock Mechanics and MiningSciences vol 113 pp 211ndash219 2019

[27] X F Li H B Li L W Liu et al ldquoInvestigating the crackinitiation and propagation mechanism in brittle rocks usinggrain-based finite-discrete element methodrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 127p 104219 2020

[28] F Gong and J Hu ldquoEnergy dissipation characteristic of redsandstone in the dynamic Brazilian disc test with SHPBsetuprdquo Advances in Civil Engineering vol 2020 Article ID7160937 10 pages 2020

[29] J C Li H B Li G W Ma and J Zhao ldquoA time-domainrecursive method to analyse transient wave propagationacross rock jointsrdquo Geophysical Journal Internationalvol 188 no 2 pp 631ndash644 2012

[30] P A Cundall ldquoA computer model for simulating progressivelarge-scale movement in blocky rock systemrdquo in Proceedingsof the International Symposium on Rock Mechanics NancyFrance October 1971

[31] B Park K-B Min N +ompson and P Horsrud ldquo+ree-dimensional bonded-particle discrete element modeling ofmechanical behavior of transversely isotropic rockrdquo Inter-national Journal of Rock Mechanics and Mining Sciencesvol 110 pp 120ndash132 2018

[32] Y M Tien M C Kuo and C H Juang ldquoAn experimentalinvestigation of the failure mechanism of simulated transverselyisotropic rocksrdquo International Journal of Rock Mechanics andMining Sciences vol 43 no 8 pp 1163ndash1181 2006

Advances in Civil Engineering 21

Stre

ngth

(MPa

)

120

160

200

240

1 2 3 4 50Thickness (mm)

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

(g)

Figure 12 +e cumulative distribution curve and average fragment size of layered rocks with different bedding thickness (andashe) +edistribution curve of 0deg 225deg 45deg 675deg and 90deg samples red thickness 08mm blue thickness 14mm yellow thickness 20mmgreen thickness 36mm and gray thickness 50mm (f) Average fragment size vs thickness under different bedding angles (g)Dynamic strength vs thickness under different bedding angles

β = 0deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(a)

β = 225deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(b)

Figure 13 Continued

Advances in Civil Engineering 17

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

15

20

25

30

35

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(f )

Figure 13 Continued

18 Advances in Civil Engineering

deformation having no longer an effect on the crackpropagation

44Effect ofWeakBeddingAickness As shown in Figure 12weak plane models with thicknesses of 08mm 14mm20mm 36mm and 50mm are constructed +e beddingstrength is half of the original matrix rather than using SJmodel to avoid the distortion of the simulation results led byall the SJ model damage during loading +e smallestthickness represents the existence of a weak surface In thecase of 50mm half sample is strong matrix and the other isweak It could be seen from Figure 12 that when there existweak planes in the 45deg specimen the central bedding planedivides the specimen into two parts bearing the load sep-arately As a result the bearing capacity of the 45deg specimenschanged little under different conditions which lead tosimilar failure patterns for 45deg specimens

However the bearing capacity and failure mode ofspecimens with other bedding angles change greatly when theproportion of weak plane increases +e simulation revealsthat more defects lead to greater fragmentation and the sizehas a significant downward trend for all angles except 45degwhich means most angle rocks are sensitive to it At the sametime the distribution curves for 45deg specimen have littledifferences and others move left as a whole +e weak planealways makes the specimen with high angle easier to developmore cracks regardless of its width compared to 225degspecimen being the most difficult to be further damaged

45 Effect of Weak Bedding Space A smaller spacing meansthat the number of laminations increases which changesfrom 10 to 1 It could be analyzed from Figure 13 that whenthe bedding angle is greater than 45deg the failure of thespecimens under any spacing is influenced by the beddingplane And more split cracks begin to appear due to thebarrier effect of weak plane being weakened when there isonly one weak plane +e smaller fragment would emerge asthe split crack intersects with the weak plane +e localfailure of specimens is controlled by the bedding planeswhen the spacing is less than 12mm for the specimens withlow bedding angle However the effect of bedding planes for0deg specimens is no longer significant with the space in-creasing while that of 225 specimens still affects localfracture As a result the fragment distribution of otherbedding dip angle samples except 225deg correspondinglyshifts to the left when the bedding space decreases Nev-ertheless the average fragment size rises slightly when thespace exceeds a quarter

5 Conclusions

+e effect of the microscopic properties of bedding andstrain rate on the fragment size distributions of layered rockwith different bedding dip angles was investigated by usingthe 2D discrete element model Several conclusions weredrawn from the experiment

(1) A SHPBnumerical model was established and validatedA new method to obtain reasonable layered rock

Stre

ngth

(MPa

)90

180

270

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(g)

Figure 13+e cumulative distribution curve and average fragment size of layered rocks with different bedding space (andashe)+e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red space 5mm blue space 65mm yellow space 10mm green space 125mm andgray space 25mm (f) Average fragment size vs bedding space under different bedding angles (g) Dynamic strength vs bedding spaceunder different bedding angles

Advances in Civil Engineering 19

dynamic simulation result is proposed +e quasistaticand dynamic test results are combined to calibrate theelastic modulus and strength of layered rock to repro-duce the stressndashstrain curve and ultimate fragment state

(2) +e average size of samples with different beddingangles decreases as the strain rate increases while thestrength presents an opposite trend +e samplestend to become pulverized under high strain rateespecially for the low bedding angle samples as morecracks initiate along the loading direction andpropagate to intersect with the cracks

(3) It is harder for the samples with low dip angle to bedamaged if the bedding shear strength is added whilethe fragmentation of high angle samples does notchange significantly +e average fragment size of 45degsamples varies greatly with the shear strength becausethe failure mode of the specimens changed And thefailure of layered specimens with all dip angles is notaffected by the tensile strength and stiffness

(4) +e wider bedding may promote the crack initiationand propagation under the same impact load whichmeans large fragments develop into small ones It wasinitially considered that the effect of bedding plane onlowdip angle samples is not considerableHowever thebedding has a dramatic impact on the final fragment

(5) Because the effect of hindering splitting crack hasalways existed for the bedding plane of 225deg sampleand more cracks initiate along the loading directionthe fragment size distribution curve of all beddingangle samples gradually moves to the right when thebedding space increases except for the 225deg samplesNevertheless the average fragment size rises slightlywhen the space exceeds a range

Data Availability

Some data models or codes generated or used during thestudy are available from the corresponding author uponrequest static mechanical properties of layered phyllite withdifferent bedding angles and all results of the dynamiccompression test

Conflicts of Interest

+e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

+e constructive comments of three anonymous reviewersare appreciated +is work received funding from the Na-tional Natural Science Foundation of China (Nos 51439008and 51679231) and China Scholarship Council (to the firstauthor No 202004910629)

References

[1] C V B Cunningham ldquoFragmentation estimations and theKuz-Ram modelndashfour years onrdquo in Proceedings of the 2nd

International Symposium on Rock Fragmentation by Blastingpp 475ndash487 Keystone CO USA August 1987

[2] X F Li H B Li Y Q Liu Q C Zhou and X Xia ldquoNumericalsimulation of rock fragmentation mechanisms subject towedge penetration for TBMsrdquo Tunnelling and UndergroundSpace Technology vol 53 pp 96ndash108 2016

[3] J C Li H B Li and J Zhao ldquoAn improved equivalentviscoelastic medium method for wave propagation acrosslayered rock massesrdquo International Journal of Rock Mechanicsand Mining Sciences vol 73 no 1 pp 62ndash69 2015

[4] F Gong W Wuxing Wu T Tianbin Li and X Xuefeng SildquoExperimental simulation and investigation of spalling failureof rectangular tunnel under different three-dimensional stressstatesrdquo International Journal of Rock Mechanics and MiningSciences vol 122 p 104081 2019

[5] D A Shockey D R Curran L Seaman J T Rosenberg andC F Petersen ldquoFragmentation of rock under dynamic loadsrdquoInternational Journal of Rock Mechanics and Mining Sciencesamp Geomechanics Abstracts vol 11 no 8 pp 303ndash317 1974

[6] F Q Gong S Luo G Lin and X B Li ldquoEvaluation of shearstrength parameters of rocks by preset angle shear directshear and triaxial compression testsrdquo Rock Mechanics andRock Engineering vol 53 no 5 pp 2505ndash2519 2020

[7] X F Li H B Li Q B Zhang J L Jiang and J ZhaoldquoDynamic fragmentation of rock material characte-ristic sizefragment distribution and pulverization lawrdquo EngineeringFracture Mechanics vol 199 pp 739ndash759 2018

[8] J D Hogan R J Rogers J G Spray and S Boonsue ldquoDynamicfragmentation of granite for impact energies of 6ndash28 Jrdquo En-gineering Fracture Mechanics vol 79 pp 103ndash125 2012

[9] P Rosin and E Rammler ldquo+e laws governing the fineness ofpowdered coalrdquo Journal of Institute of Fuel vol 7 pp 29ndash36 1933

[10] N F Mott and E H Linfoot ldquoMinistry of supply +e rawmaterial guiderdquo Ae Analyst vol 68 no 806 1943

[11] D E Grady ldquoLocal inertial effects in dynamic fragmentationrdquoJournal of Applied Physics vol 53 no 1 pp 322ndash325 1982

[12] F Zhou and J F Molinari ldquoOn the rate-dependency ofdynamic tensile strength of a model ceramic systemrdquo Com-puter Methods in Applied Mechanics and Engineering vol 194no 12-16 pp 1693ndash1709 2005

[13] X F Li X Li H B Li Q B Zhang and J Zhao ldquoDynamictensile behaviours of heterogeneous rocks the grain scalefracturing characteristics on strength and fragmentationrdquoInternational Journal of Impact Engineering vol 118pp 98ndash118 2018

[14] F Q Gong X B Li and X L Liu ldquoPreliminary experimentalstudy of characteristics of rock subjected to 3D coupled staticand dynamic loadsrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 6 pp 1179ndash1190 2011

[15] S Levy and J F Molinari ldquoDynamic fragmentation of ce-ramics signature of defects and scaling of fragment sizesrdquoJournal of the Mechanics and Physics of Solids vol 58 no 1pp 12ndash26 2010

[16] B Paliwal and K T Ramesh ldquoAn interacting micro-crackdamage model for failure of brittle materials under com-pressionrdquo Journal of the Mechanics and Physics of Solidsvol 56 no 3 pp 896ndash923 2008

[17] R J Wu H B Li X F Li X Xia and L W Liu ldquoExperimentalstudy and numerical simulation of the dynamic behavior oftransversely isotropic phylliterdquo International Journal of Geo-mechanics vol 20 no 8 Article ID 04020105 2020

[18] X Ou X Zhang H Feng C Zhang and J Yang ldquoEffect of theconfining pressure on the dynamic compression properties of

20 Advances in Civil Engineering

transversely isotropic rocksrdquo Advances in Civil Engineeringvol 6 p 11 2019

[19] H-B Du F Dai Y Xu Y Liu and H-n Xu ldquoNumericalinvestigation on the dynamic strength and failure behavior ofrocks under hydrostatic confinement in SHPB testingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 108 pp 43ndash57 2018

[20] X F Li Q B Zhang H B Li and J Zhao ldquoGrain-baseddiscrete element method (GB-DEM) modelling of multi-scalefracturing in rocks under dynamic loadingrdquo Rock Mechanicsand Rock Engineering vol 51 no 12 pp 3785ndash3817 2018

[21] F Dai M D Wei N W Xu Y Ma and D S Yang ldquoNu-merical assessment of the progressive rock fracture mecha-nism of cracked chevron notched Brazilian disc specimensrdquoRock Mechanics and Rock Engineering vol 48 no 2pp 463ndash479 2015

[22] B Park and K-B Min ldquoBonded-particle discrete elementmodeling of mechanical behavior of transversely isotropicrockrdquo International Journal of Rock Mechanics and MiningSciences vol 76 pp 243ndash255 2015

[23] E Cadoni ldquoDynamic characterization of orthogneiss rocksubjected to intermediate and high strain rates in tensionrdquoRock Mechanics and Rock Engineering vol 43 no 6pp 667ndash676 2010

[24] Y X Zhou K Xia X B Li et al ldquoSuggested methods fordetermining the dynamic strength parameters and Mode-Ifracture toughness of rock materialsrdquo International Journal ofRock Mechanics and Mining Sciences vol 49 no 1pp 105ndash112 2011

[25] F Dai K Xia J P Zuo R Zhang and N W Xu ldquoStatic anddynamic flexural strength anisotropy of Barre graniterdquo Rock Me-chanics and Rock Engineering vol 46 no 6 pp 1589ndash1602 2013

[26] F-Q Gong X-F Si X-B Li and S-Y Wang ldquoDynamictriaxial compression tests on sandstone at high strain ratesand low confining pressures with split Hopkinson pressurebarrdquo International Journal of Rock Mechanics and MiningSciences vol 113 pp 211ndash219 2019

[27] X F Li H B Li L W Liu et al ldquoInvestigating the crackinitiation and propagation mechanism in brittle rocks usinggrain-based finite-discrete element methodrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 127p 104219 2020

[28] F Gong and J Hu ldquoEnergy dissipation characteristic of redsandstone in the dynamic Brazilian disc test with SHPBsetuprdquo Advances in Civil Engineering vol 2020 Article ID7160937 10 pages 2020

[29] J C Li H B Li G W Ma and J Zhao ldquoA time-domainrecursive method to analyse transient wave propagationacross rock jointsrdquo Geophysical Journal Internationalvol 188 no 2 pp 631ndash644 2012

[30] P A Cundall ldquoA computer model for simulating progressivelarge-scale movement in blocky rock systemrdquo in Proceedingsof the International Symposium on Rock Mechanics NancyFrance October 1971

[31] B Park K-B Min N +ompson and P Horsrud ldquo+ree-dimensional bonded-particle discrete element modeling ofmechanical behavior of transversely isotropic rockrdquo Inter-national Journal of Rock Mechanics and Mining Sciencesvol 110 pp 120ndash132 2018

[32] Y M Tien M C Kuo and C H Juang ldquoAn experimentalinvestigation of the failure mechanism of simulated transverselyisotropic rocksrdquo International Journal of Rock Mechanics andMining Sciences vol 43 no 8 pp 1163ndash1181 2006

Advances in Civil Engineering 21

β = 45deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(c)

β = 675deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(d)

β = 90deg

Perc

ent p

assin

g

0

20

40

60

80

100

101 102 103100

Fragment size (mm2)

(e)

15

20

25

30

35

lgδ_

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(f )

Figure 13 Continued

18 Advances in Civil Engineering

deformation having no longer an effect on the crackpropagation

44Effect ofWeakBeddingAickness As shown in Figure 12weak plane models with thicknesses of 08mm 14mm20mm 36mm and 50mm are constructed +e beddingstrength is half of the original matrix rather than using SJmodel to avoid the distortion of the simulation results led byall the SJ model damage during loading +e smallestthickness represents the existence of a weak surface In thecase of 50mm half sample is strong matrix and the other isweak It could be seen from Figure 12 that when there existweak planes in the 45deg specimen the central bedding planedivides the specimen into two parts bearing the load sep-arately As a result the bearing capacity of the 45deg specimenschanged little under different conditions which lead tosimilar failure patterns for 45deg specimens

However the bearing capacity and failure mode ofspecimens with other bedding angles change greatly when theproportion of weak plane increases +e simulation revealsthat more defects lead to greater fragmentation and the sizehas a significant downward trend for all angles except 45degwhich means most angle rocks are sensitive to it At the sametime the distribution curves for 45deg specimen have littledifferences and others move left as a whole +e weak planealways makes the specimen with high angle easier to developmore cracks regardless of its width compared to 225degspecimen being the most difficult to be further damaged

45 Effect of Weak Bedding Space A smaller spacing meansthat the number of laminations increases which changesfrom 10 to 1 It could be analyzed from Figure 13 that whenthe bedding angle is greater than 45deg the failure of thespecimens under any spacing is influenced by the beddingplane And more split cracks begin to appear due to thebarrier effect of weak plane being weakened when there isonly one weak plane +e smaller fragment would emerge asthe split crack intersects with the weak plane +e localfailure of specimens is controlled by the bedding planeswhen the spacing is less than 12mm for the specimens withlow bedding angle However the effect of bedding planes for0deg specimens is no longer significant with the space in-creasing while that of 225 specimens still affects localfracture As a result the fragment distribution of otherbedding dip angle samples except 225deg correspondinglyshifts to the left when the bedding space decreases Nev-ertheless the average fragment size rises slightly when thespace exceeds a quarter

5 Conclusions

+e effect of the microscopic properties of bedding andstrain rate on the fragment size distributions of layered rockwith different bedding dip angles was investigated by usingthe 2D discrete element model Several conclusions weredrawn from the experiment

(1) A SHPBnumerical model was established and validatedA new method to obtain reasonable layered rock

Stre

ngth

(MPa

)90

180

270

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(g)

Figure 13+e cumulative distribution curve and average fragment size of layered rocks with different bedding space (andashe)+e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red space 5mm blue space 65mm yellow space 10mm green space 125mm andgray space 25mm (f) Average fragment size vs bedding space under different bedding angles (g) Dynamic strength vs bedding spaceunder different bedding angles

Advances in Civil Engineering 19

dynamic simulation result is proposed +e quasistaticand dynamic test results are combined to calibrate theelastic modulus and strength of layered rock to repro-duce the stressndashstrain curve and ultimate fragment state

(2) +e average size of samples with different beddingangles decreases as the strain rate increases while thestrength presents an opposite trend +e samplestend to become pulverized under high strain rateespecially for the low bedding angle samples as morecracks initiate along the loading direction andpropagate to intersect with the cracks

(3) It is harder for the samples with low dip angle to bedamaged if the bedding shear strength is added whilethe fragmentation of high angle samples does notchange significantly +e average fragment size of 45degsamples varies greatly with the shear strength becausethe failure mode of the specimens changed And thefailure of layered specimens with all dip angles is notaffected by the tensile strength and stiffness

(4) +e wider bedding may promote the crack initiationand propagation under the same impact load whichmeans large fragments develop into small ones It wasinitially considered that the effect of bedding plane onlowdip angle samples is not considerableHowever thebedding has a dramatic impact on the final fragment

(5) Because the effect of hindering splitting crack hasalways existed for the bedding plane of 225deg sampleand more cracks initiate along the loading directionthe fragment size distribution curve of all beddingangle samples gradually moves to the right when thebedding space increases except for the 225deg samplesNevertheless the average fragment size rises slightlywhen the space exceeds a range

Data Availability

Some data models or codes generated or used during thestudy are available from the corresponding author uponrequest static mechanical properties of layered phyllite withdifferent bedding angles and all results of the dynamiccompression test

Conflicts of Interest

+e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

+e constructive comments of three anonymous reviewersare appreciated +is work received funding from the Na-tional Natural Science Foundation of China (Nos 51439008and 51679231) and China Scholarship Council (to the firstauthor No 202004910629)

References

[1] C V B Cunningham ldquoFragmentation estimations and theKuz-Ram modelndashfour years onrdquo in Proceedings of the 2nd

International Symposium on Rock Fragmentation by Blastingpp 475ndash487 Keystone CO USA August 1987

[2] X F Li H B Li Y Q Liu Q C Zhou and X Xia ldquoNumericalsimulation of rock fragmentation mechanisms subject towedge penetration for TBMsrdquo Tunnelling and UndergroundSpace Technology vol 53 pp 96ndash108 2016

[3] J C Li H B Li and J Zhao ldquoAn improved equivalentviscoelastic medium method for wave propagation acrosslayered rock massesrdquo International Journal of Rock Mechanicsand Mining Sciences vol 73 no 1 pp 62ndash69 2015

[4] F Gong W Wuxing Wu T Tianbin Li and X Xuefeng SildquoExperimental simulation and investigation of spalling failureof rectangular tunnel under different three-dimensional stressstatesrdquo International Journal of Rock Mechanics and MiningSciences vol 122 p 104081 2019

[5] D A Shockey D R Curran L Seaman J T Rosenberg andC F Petersen ldquoFragmentation of rock under dynamic loadsrdquoInternational Journal of Rock Mechanics and Mining Sciencesamp Geomechanics Abstracts vol 11 no 8 pp 303ndash317 1974

[6] F Q Gong S Luo G Lin and X B Li ldquoEvaluation of shearstrength parameters of rocks by preset angle shear directshear and triaxial compression testsrdquo Rock Mechanics andRock Engineering vol 53 no 5 pp 2505ndash2519 2020

[7] X F Li H B Li Q B Zhang J L Jiang and J ZhaoldquoDynamic fragmentation of rock material characte-ristic sizefragment distribution and pulverization lawrdquo EngineeringFracture Mechanics vol 199 pp 739ndash759 2018

[8] J D Hogan R J Rogers J G Spray and S Boonsue ldquoDynamicfragmentation of granite for impact energies of 6ndash28 Jrdquo En-gineering Fracture Mechanics vol 79 pp 103ndash125 2012

[9] P Rosin and E Rammler ldquo+e laws governing the fineness ofpowdered coalrdquo Journal of Institute of Fuel vol 7 pp 29ndash36 1933

[10] N F Mott and E H Linfoot ldquoMinistry of supply +e rawmaterial guiderdquo Ae Analyst vol 68 no 806 1943

[11] D E Grady ldquoLocal inertial effects in dynamic fragmentationrdquoJournal of Applied Physics vol 53 no 1 pp 322ndash325 1982

[12] F Zhou and J F Molinari ldquoOn the rate-dependency ofdynamic tensile strength of a model ceramic systemrdquo Com-puter Methods in Applied Mechanics and Engineering vol 194no 12-16 pp 1693ndash1709 2005

[13] X F Li X Li H B Li Q B Zhang and J Zhao ldquoDynamictensile behaviours of heterogeneous rocks the grain scalefracturing characteristics on strength and fragmentationrdquoInternational Journal of Impact Engineering vol 118pp 98ndash118 2018

[14] F Q Gong X B Li and X L Liu ldquoPreliminary experimentalstudy of characteristics of rock subjected to 3D coupled staticand dynamic loadsrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 6 pp 1179ndash1190 2011

[15] S Levy and J F Molinari ldquoDynamic fragmentation of ce-ramics signature of defects and scaling of fragment sizesrdquoJournal of the Mechanics and Physics of Solids vol 58 no 1pp 12ndash26 2010

[16] B Paliwal and K T Ramesh ldquoAn interacting micro-crackdamage model for failure of brittle materials under com-pressionrdquo Journal of the Mechanics and Physics of Solidsvol 56 no 3 pp 896ndash923 2008

[17] R J Wu H B Li X F Li X Xia and L W Liu ldquoExperimentalstudy and numerical simulation of the dynamic behavior oftransversely isotropic phylliterdquo International Journal of Geo-mechanics vol 20 no 8 Article ID 04020105 2020

[18] X Ou X Zhang H Feng C Zhang and J Yang ldquoEffect of theconfining pressure on the dynamic compression properties of

20 Advances in Civil Engineering

transversely isotropic rocksrdquo Advances in Civil Engineeringvol 6 p 11 2019

[19] H-B Du F Dai Y Xu Y Liu and H-n Xu ldquoNumericalinvestigation on the dynamic strength and failure behavior ofrocks under hydrostatic confinement in SHPB testingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 108 pp 43ndash57 2018

[20] X F Li Q B Zhang H B Li and J Zhao ldquoGrain-baseddiscrete element method (GB-DEM) modelling of multi-scalefracturing in rocks under dynamic loadingrdquo Rock Mechanicsand Rock Engineering vol 51 no 12 pp 3785ndash3817 2018

[21] F Dai M D Wei N W Xu Y Ma and D S Yang ldquoNu-merical assessment of the progressive rock fracture mecha-nism of cracked chevron notched Brazilian disc specimensrdquoRock Mechanics and Rock Engineering vol 48 no 2pp 463ndash479 2015

[22] B Park and K-B Min ldquoBonded-particle discrete elementmodeling of mechanical behavior of transversely isotropicrockrdquo International Journal of Rock Mechanics and MiningSciences vol 76 pp 243ndash255 2015

[23] E Cadoni ldquoDynamic characterization of orthogneiss rocksubjected to intermediate and high strain rates in tensionrdquoRock Mechanics and Rock Engineering vol 43 no 6pp 667ndash676 2010

[24] Y X Zhou K Xia X B Li et al ldquoSuggested methods fordetermining the dynamic strength parameters and Mode-Ifracture toughness of rock materialsrdquo International Journal ofRock Mechanics and Mining Sciences vol 49 no 1pp 105ndash112 2011

[25] F Dai K Xia J P Zuo R Zhang and N W Xu ldquoStatic anddynamic flexural strength anisotropy of Barre graniterdquo Rock Me-chanics and Rock Engineering vol 46 no 6 pp 1589ndash1602 2013

[26] F-Q Gong X-F Si X-B Li and S-Y Wang ldquoDynamictriaxial compression tests on sandstone at high strain ratesand low confining pressures with split Hopkinson pressurebarrdquo International Journal of Rock Mechanics and MiningSciences vol 113 pp 211ndash219 2019

[27] X F Li H B Li L W Liu et al ldquoInvestigating the crackinitiation and propagation mechanism in brittle rocks usinggrain-based finite-discrete element methodrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 127p 104219 2020

[28] F Gong and J Hu ldquoEnergy dissipation characteristic of redsandstone in the dynamic Brazilian disc test with SHPBsetuprdquo Advances in Civil Engineering vol 2020 Article ID7160937 10 pages 2020

[29] J C Li H B Li G W Ma and J Zhao ldquoA time-domainrecursive method to analyse transient wave propagationacross rock jointsrdquo Geophysical Journal Internationalvol 188 no 2 pp 631ndash644 2012

[30] P A Cundall ldquoA computer model for simulating progressivelarge-scale movement in blocky rock systemrdquo in Proceedingsof the International Symposium on Rock Mechanics NancyFrance October 1971

[31] B Park K-B Min N +ompson and P Horsrud ldquo+ree-dimensional bonded-particle discrete element modeling ofmechanical behavior of transversely isotropic rockrdquo Inter-national Journal of Rock Mechanics and Mining Sciencesvol 110 pp 120ndash132 2018

[32] Y M Tien M C Kuo and C H Juang ldquoAn experimentalinvestigation of the failure mechanism of simulated transverselyisotropic rocksrdquo International Journal of Rock Mechanics andMining Sciences vol 43 no 8 pp 1163ndash1181 2006

Advances in Civil Engineering 21

deformation having no longer an effect on the crackpropagation

44Effect ofWeakBeddingAickness As shown in Figure 12weak plane models with thicknesses of 08mm 14mm20mm 36mm and 50mm are constructed +e beddingstrength is half of the original matrix rather than using SJmodel to avoid the distortion of the simulation results led byall the SJ model damage during loading +e smallestthickness represents the existence of a weak surface In thecase of 50mm half sample is strong matrix and the other isweak It could be seen from Figure 12 that when there existweak planes in the 45deg specimen the central bedding planedivides the specimen into two parts bearing the load sep-arately As a result the bearing capacity of the 45deg specimenschanged little under different conditions which lead tosimilar failure patterns for 45deg specimens

However the bearing capacity and failure mode ofspecimens with other bedding angles change greatly when theproportion of weak plane increases +e simulation revealsthat more defects lead to greater fragmentation and the sizehas a significant downward trend for all angles except 45degwhich means most angle rocks are sensitive to it At the sametime the distribution curves for 45deg specimen have littledifferences and others move left as a whole +e weak planealways makes the specimen with high angle easier to developmore cracks regardless of its width compared to 225degspecimen being the most difficult to be further damaged

45 Effect of Weak Bedding Space A smaller spacing meansthat the number of laminations increases which changesfrom 10 to 1 It could be analyzed from Figure 13 that whenthe bedding angle is greater than 45deg the failure of thespecimens under any spacing is influenced by the beddingplane And more split cracks begin to appear due to thebarrier effect of weak plane being weakened when there isonly one weak plane +e smaller fragment would emerge asthe split crack intersects with the weak plane +e localfailure of specimens is controlled by the bedding planeswhen the spacing is less than 12mm for the specimens withlow bedding angle However the effect of bedding planes for0deg specimens is no longer significant with the space in-creasing while that of 225 specimens still affects localfracture As a result the fragment distribution of otherbedding dip angle samples except 225deg correspondinglyshifts to the left when the bedding space decreases Nev-ertheless the average fragment size rises slightly when thespace exceeds a quarter

5 Conclusions

+e effect of the microscopic properties of bedding andstrain rate on the fragment size distributions of layered rockwith different bedding dip angles was investigated by usingthe 2D discrete element model Several conclusions weredrawn from the experiment

(1) A SHPBnumerical model was established and validatedA new method to obtain reasonable layered rock

Stre

ngth

(MPa

)90

180

270

β = 675degβ = 90deg

β = 45degβ = 225degβ = 0deg

10 15 20 255Space (mm)

(g)

Figure 13+e cumulative distribution curve and average fragment size of layered rocks with different bedding space (andashe)+e distributioncurve of 0deg 225deg 45deg 675deg and 90deg samples red space 5mm blue space 65mm yellow space 10mm green space 125mm andgray space 25mm (f) Average fragment size vs bedding space under different bedding angles (g) Dynamic strength vs bedding spaceunder different bedding angles

Advances in Civil Engineering 19

dynamic simulation result is proposed +e quasistaticand dynamic test results are combined to calibrate theelastic modulus and strength of layered rock to repro-duce the stressndashstrain curve and ultimate fragment state

(2) +e average size of samples with different beddingangles decreases as the strain rate increases while thestrength presents an opposite trend +e samplestend to become pulverized under high strain rateespecially for the low bedding angle samples as morecracks initiate along the loading direction andpropagate to intersect with the cracks

(3) It is harder for the samples with low dip angle to bedamaged if the bedding shear strength is added whilethe fragmentation of high angle samples does notchange significantly +e average fragment size of 45degsamples varies greatly with the shear strength becausethe failure mode of the specimens changed And thefailure of layered specimens with all dip angles is notaffected by the tensile strength and stiffness

(4) +e wider bedding may promote the crack initiationand propagation under the same impact load whichmeans large fragments develop into small ones It wasinitially considered that the effect of bedding plane onlowdip angle samples is not considerableHowever thebedding has a dramatic impact on the final fragment

(5) Because the effect of hindering splitting crack hasalways existed for the bedding plane of 225deg sampleand more cracks initiate along the loading directionthe fragment size distribution curve of all beddingangle samples gradually moves to the right when thebedding space increases except for the 225deg samplesNevertheless the average fragment size rises slightlywhen the space exceeds a range

Data Availability

Some data models or codes generated or used during thestudy are available from the corresponding author uponrequest static mechanical properties of layered phyllite withdifferent bedding angles and all results of the dynamiccompression test

Conflicts of Interest

+e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

+e constructive comments of three anonymous reviewersare appreciated +is work received funding from the Na-tional Natural Science Foundation of China (Nos 51439008and 51679231) and China Scholarship Council (to the firstauthor No 202004910629)

References

[1] C V B Cunningham ldquoFragmentation estimations and theKuz-Ram modelndashfour years onrdquo in Proceedings of the 2nd

International Symposium on Rock Fragmentation by Blastingpp 475ndash487 Keystone CO USA August 1987

[2] X F Li H B Li Y Q Liu Q C Zhou and X Xia ldquoNumericalsimulation of rock fragmentation mechanisms subject towedge penetration for TBMsrdquo Tunnelling and UndergroundSpace Technology vol 53 pp 96ndash108 2016

[3] J C Li H B Li and J Zhao ldquoAn improved equivalentviscoelastic medium method for wave propagation acrosslayered rock massesrdquo International Journal of Rock Mechanicsand Mining Sciences vol 73 no 1 pp 62ndash69 2015

[4] F Gong W Wuxing Wu T Tianbin Li and X Xuefeng SildquoExperimental simulation and investigation of spalling failureof rectangular tunnel under different three-dimensional stressstatesrdquo International Journal of Rock Mechanics and MiningSciences vol 122 p 104081 2019

[5] D A Shockey D R Curran L Seaman J T Rosenberg andC F Petersen ldquoFragmentation of rock under dynamic loadsrdquoInternational Journal of Rock Mechanics and Mining Sciencesamp Geomechanics Abstracts vol 11 no 8 pp 303ndash317 1974

[6] F Q Gong S Luo G Lin and X B Li ldquoEvaluation of shearstrength parameters of rocks by preset angle shear directshear and triaxial compression testsrdquo Rock Mechanics andRock Engineering vol 53 no 5 pp 2505ndash2519 2020

[7] X F Li H B Li Q B Zhang J L Jiang and J ZhaoldquoDynamic fragmentation of rock material characte-ristic sizefragment distribution and pulverization lawrdquo EngineeringFracture Mechanics vol 199 pp 739ndash759 2018

[8] J D Hogan R J Rogers J G Spray and S Boonsue ldquoDynamicfragmentation of granite for impact energies of 6ndash28 Jrdquo En-gineering Fracture Mechanics vol 79 pp 103ndash125 2012

[9] P Rosin and E Rammler ldquo+e laws governing the fineness ofpowdered coalrdquo Journal of Institute of Fuel vol 7 pp 29ndash36 1933

[10] N F Mott and E H Linfoot ldquoMinistry of supply +e rawmaterial guiderdquo Ae Analyst vol 68 no 806 1943

[11] D E Grady ldquoLocal inertial effects in dynamic fragmentationrdquoJournal of Applied Physics vol 53 no 1 pp 322ndash325 1982

[12] F Zhou and J F Molinari ldquoOn the rate-dependency ofdynamic tensile strength of a model ceramic systemrdquo Com-puter Methods in Applied Mechanics and Engineering vol 194no 12-16 pp 1693ndash1709 2005

[13] X F Li X Li H B Li Q B Zhang and J Zhao ldquoDynamictensile behaviours of heterogeneous rocks the grain scalefracturing characteristics on strength and fragmentationrdquoInternational Journal of Impact Engineering vol 118pp 98ndash118 2018

[14] F Q Gong X B Li and X L Liu ldquoPreliminary experimentalstudy of characteristics of rock subjected to 3D coupled staticand dynamic loadsrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 6 pp 1179ndash1190 2011

[15] S Levy and J F Molinari ldquoDynamic fragmentation of ce-ramics signature of defects and scaling of fragment sizesrdquoJournal of the Mechanics and Physics of Solids vol 58 no 1pp 12ndash26 2010

[16] B Paliwal and K T Ramesh ldquoAn interacting micro-crackdamage model for failure of brittle materials under com-pressionrdquo Journal of the Mechanics and Physics of Solidsvol 56 no 3 pp 896ndash923 2008

[17] R J Wu H B Li X F Li X Xia and L W Liu ldquoExperimentalstudy and numerical simulation of the dynamic behavior oftransversely isotropic phylliterdquo International Journal of Geo-mechanics vol 20 no 8 Article ID 04020105 2020

[18] X Ou X Zhang H Feng C Zhang and J Yang ldquoEffect of theconfining pressure on the dynamic compression properties of

20 Advances in Civil Engineering

transversely isotropic rocksrdquo Advances in Civil Engineeringvol 6 p 11 2019

[19] H-B Du F Dai Y Xu Y Liu and H-n Xu ldquoNumericalinvestigation on the dynamic strength and failure behavior ofrocks under hydrostatic confinement in SHPB testingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 108 pp 43ndash57 2018

[20] X F Li Q B Zhang H B Li and J Zhao ldquoGrain-baseddiscrete element method (GB-DEM) modelling of multi-scalefracturing in rocks under dynamic loadingrdquo Rock Mechanicsand Rock Engineering vol 51 no 12 pp 3785ndash3817 2018

[21] F Dai M D Wei N W Xu Y Ma and D S Yang ldquoNu-merical assessment of the progressive rock fracture mecha-nism of cracked chevron notched Brazilian disc specimensrdquoRock Mechanics and Rock Engineering vol 48 no 2pp 463ndash479 2015

[22] B Park and K-B Min ldquoBonded-particle discrete elementmodeling of mechanical behavior of transversely isotropicrockrdquo International Journal of Rock Mechanics and MiningSciences vol 76 pp 243ndash255 2015

[23] E Cadoni ldquoDynamic characterization of orthogneiss rocksubjected to intermediate and high strain rates in tensionrdquoRock Mechanics and Rock Engineering vol 43 no 6pp 667ndash676 2010

[24] Y X Zhou K Xia X B Li et al ldquoSuggested methods fordetermining the dynamic strength parameters and Mode-Ifracture toughness of rock materialsrdquo International Journal ofRock Mechanics and Mining Sciences vol 49 no 1pp 105ndash112 2011

[25] F Dai K Xia J P Zuo R Zhang and N W Xu ldquoStatic anddynamic flexural strength anisotropy of Barre graniterdquo Rock Me-chanics and Rock Engineering vol 46 no 6 pp 1589ndash1602 2013

[26] F-Q Gong X-F Si X-B Li and S-Y Wang ldquoDynamictriaxial compression tests on sandstone at high strain ratesand low confining pressures with split Hopkinson pressurebarrdquo International Journal of Rock Mechanics and MiningSciences vol 113 pp 211ndash219 2019

[27] X F Li H B Li L W Liu et al ldquoInvestigating the crackinitiation and propagation mechanism in brittle rocks usinggrain-based finite-discrete element methodrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 127p 104219 2020

[28] F Gong and J Hu ldquoEnergy dissipation characteristic of redsandstone in the dynamic Brazilian disc test with SHPBsetuprdquo Advances in Civil Engineering vol 2020 Article ID7160937 10 pages 2020

[29] J C Li H B Li G W Ma and J Zhao ldquoA time-domainrecursive method to analyse transient wave propagationacross rock jointsrdquo Geophysical Journal Internationalvol 188 no 2 pp 631ndash644 2012

[30] P A Cundall ldquoA computer model for simulating progressivelarge-scale movement in blocky rock systemrdquo in Proceedingsof the International Symposium on Rock Mechanics NancyFrance October 1971

[31] B Park K-B Min N +ompson and P Horsrud ldquo+ree-dimensional bonded-particle discrete element modeling ofmechanical behavior of transversely isotropic rockrdquo Inter-national Journal of Rock Mechanics and Mining Sciencesvol 110 pp 120ndash132 2018

[32] Y M Tien M C Kuo and C H Juang ldquoAn experimentalinvestigation of the failure mechanism of simulated transverselyisotropic rocksrdquo International Journal of Rock Mechanics andMining Sciences vol 43 no 8 pp 1163ndash1181 2006

Advances in Civil Engineering 21

dynamic simulation result is proposed +e quasistaticand dynamic test results are combined to calibrate theelastic modulus and strength of layered rock to repro-duce the stressndashstrain curve and ultimate fragment state

(2) +e average size of samples with different beddingangles decreases as the strain rate increases while thestrength presents an opposite trend +e samplestend to become pulverized under high strain rateespecially for the low bedding angle samples as morecracks initiate along the loading direction andpropagate to intersect with the cracks

(3) It is harder for the samples with low dip angle to bedamaged if the bedding shear strength is added whilethe fragmentation of high angle samples does notchange significantly +e average fragment size of 45degsamples varies greatly with the shear strength becausethe failure mode of the specimens changed And thefailure of layered specimens with all dip angles is notaffected by the tensile strength and stiffness

(4) +e wider bedding may promote the crack initiationand propagation under the same impact load whichmeans large fragments develop into small ones It wasinitially considered that the effect of bedding plane onlowdip angle samples is not considerableHowever thebedding has a dramatic impact on the final fragment

(5) Because the effect of hindering splitting crack hasalways existed for the bedding plane of 225deg sampleand more cracks initiate along the loading directionthe fragment size distribution curve of all beddingangle samples gradually moves to the right when thebedding space increases except for the 225deg samplesNevertheless the average fragment size rises slightlywhen the space exceeds a range

Data Availability

Some data models or codes generated or used during thestudy are available from the corresponding author uponrequest static mechanical properties of layered phyllite withdifferent bedding angles and all results of the dynamiccompression test

Conflicts of Interest

+e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

+e constructive comments of three anonymous reviewersare appreciated +is work received funding from the Na-tional Natural Science Foundation of China (Nos 51439008and 51679231) and China Scholarship Council (to the firstauthor No 202004910629)

References

[1] C V B Cunningham ldquoFragmentation estimations and theKuz-Ram modelndashfour years onrdquo in Proceedings of the 2nd

International Symposium on Rock Fragmentation by Blastingpp 475ndash487 Keystone CO USA August 1987

[2] X F Li H B Li Y Q Liu Q C Zhou and X Xia ldquoNumericalsimulation of rock fragmentation mechanisms subject towedge penetration for TBMsrdquo Tunnelling and UndergroundSpace Technology vol 53 pp 96ndash108 2016

[3] J C Li H B Li and J Zhao ldquoAn improved equivalentviscoelastic medium method for wave propagation acrosslayered rock massesrdquo International Journal of Rock Mechanicsand Mining Sciences vol 73 no 1 pp 62ndash69 2015

[4] F Gong W Wuxing Wu T Tianbin Li and X Xuefeng SildquoExperimental simulation and investigation of spalling failureof rectangular tunnel under different three-dimensional stressstatesrdquo International Journal of Rock Mechanics and MiningSciences vol 122 p 104081 2019

[5] D A Shockey D R Curran L Seaman J T Rosenberg andC F Petersen ldquoFragmentation of rock under dynamic loadsrdquoInternational Journal of Rock Mechanics and Mining Sciencesamp Geomechanics Abstracts vol 11 no 8 pp 303ndash317 1974

[6] F Q Gong S Luo G Lin and X B Li ldquoEvaluation of shearstrength parameters of rocks by preset angle shear directshear and triaxial compression testsrdquo Rock Mechanics andRock Engineering vol 53 no 5 pp 2505ndash2519 2020

[7] X F Li H B Li Q B Zhang J L Jiang and J ZhaoldquoDynamic fragmentation of rock material characte-ristic sizefragment distribution and pulverization lawrdquo EngineeringFracture Mechanics vol 199 pp 739ndash759 2018

[8] J D Hogan R J Rogers J G Spray and S Boonsue ldquoDynamicfragmentation of granite for impact energies of 6ndash28 Jrdquo En-gineering Fracture Mechanics vol 79 pp 103ndash125 2012

[9] P Rosin and E Rammler ldquo+e laws governing the fineness ofpowdered coalrdquo Journal of Institute of Fuel vol 7 pp 29ndash36 1933

[10] N F Mott and E H Linfoot ldquoMinistry of supply +e rawmaterial guiderdquo Ae Analyst vol 68 no 806 1943

[11] D E Grady ldquoLocal inertial effects in dynamic fragmentationrdquoJournal of Applied Physics vol 53 no 1 pp 322ndash325 1982

[12] F Zhou and J F Molinari ldquoOn the rate-dependency ofdynamic tensile strength of a model ceramic systemrdquo Com-puter Methods in Applied Mechanics and Engineering vol 194no 12-16 pp 1693ndash1709 2005

[13] X F Li X Li H B Li Q B Zhang and J Zhao ldquoDynamictensile behaviours of heterogeneous rocks the grain scalefracturing characteristics on strength and fragmentationrdquoInternational Journal of Impact Engineering vol 118pp 98ndash118 2018

[14] F Q Gong X B Li and X L Liu ldquoPreliminary experimentalstudy of characteristics of rock subjected to 3D coupled staticand dynamic loadsrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 6 pp 1179ndash1190 2011

[15] S Levy and J F Molinari ldquoDynamic fragmentation of ce-ramics signature of defects and scaling of fragment sizesrdquoJournal of the Mechanics and Physics of Solids vol 58 no 1pp 12ndash26 2010

[16] B Paliwal and K T Ramesh ldquoAn interacting micro-crackdamage model for failure of brittle materials under com-pressionrdquo Journal of the Mechanics and Physics of Solidsvol 56 no 3 pp 896ndash923 2008

[17] R J Wu H B Li X F Li X Xia and L W Liu ldquoExperimentalstudy and numerical simulation of the dynamic behavior oftransversely isotropic phylliterdquo International Journal of Geo-mechanics vol 20 no 8 Article ID 04020105 2020

[18] X Ou X Zhang H Feng C Zhang and J Yang ldquoEffect of theconfining pressure on the dynamic compression properties of

20 Advances in Civil Engineering

transversely isotropic rocksrdquo Advances in Civil Engineeringvol 6 p 11 2019

[19] H-B Du F Dai Y Xu Y Liu and H-n Xu ldquoNumericalinvestigation on the dynamic strength and failure behavior ofrocks under hydrostatic confinement in SHPB testingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 108 pp 43ndash57 2018

[20] X F Li Q B Zhang H B Li and J Zhao ldquoGrain-baseddiscrete element method (GB-DEM) modelling of multi-scalefracturing in rocks under dynamic loadingrdquo Rock Mechanicsand Rock Engineering vol 51 no 12 pp 3785ndash3817 2018

[21] F Dai M D Wei N W Xu Y Ma and D S Yang ldquoNu-merical assessment of the progressive rock fracture mecha-nism of cracked chevron notched Brazilian disc specimensrdquoRock Mechanics and Rock Engineering vol 48 no 2pp 463ndash479 2015

[22] B Park and K-B Min ldquoBonded-particle discrete elementmodeling of mechanical behavior of transversely isotropicrockrdquo International Journal of Rock Mechanics and MiningSciences vol 76 pp 243ndash255 2015

[23] E Cadoni ldquoDynamic characterization of orthogneiss rocksubjected to intermediate and high strain rates in tensionrdquoRock Mechanics and Rock Engineering vol 43 no 6pp 667ndash676 2010

[24] Y X Zhou K Xia X B Li et al ldquoSuggested methods fordetermining the dynamic strength parameters and Mode-Ifracture toughness of rock materialsrdquo International Journal ofRock Mechanics and Mining Sciences vol 49 no 1pp 105ndash112 2011

[25] F Dai K Xia J P Zuo R Zhang and N W Xu ldquoStatic anddynamic flexural strength anisotropy of Barre graniterdquo Rock Me-chanics and Rock Engineering vol 46 no 6 pp 1589ndash1602 2013

[26] F-Q Gong X-F Si X-B Li and S-Y Wang ldquoDynamictriaxial compression tests on sandstone at high strain ratesand low confining pressures with split Hopkinson pressurebarrdquo International Journal of Rock Mechanics and MiningSciences vol 113 pp 211ndash219 2019

[27] X F Li H B Li L W Liu et al ldquoInvestigating the crackinitiation and propagation mechanism in brittle rocks usinggrain-based finite-discrete element methodrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 127p 104219 2020

[28] F Gong and J Hu ldquoEnergy dissipation characteristic of redsandstone in the dynamic Brazilian disc test with SHPBsetuprdquo Advances in Civil Engineering vol 2020 Article ID7160937 10 pages 2020

[29] J C Li H B Li G W Ma and J Zhao ldquoA time-domainrecursive method to analyse transient wave propagationacross rock jointsrdquo Geophysical Journal Internationalvol 188 no 2 pp 631ndash644 2012

[30] P A Cundall ldquoA computer model for simulating progressivelarge-scale movement in blocky rock systemrdquo in Proceedingsof the International Symposium on Rock Mechanics NancyFrance October 1971

[31] B Park K-B Min N +ompson and P Horsrud ldquo+ree-dimensional bonded-particle discrete element modeling ofmechanical behavior of transversely isotropic rockrdquo Inter-national Journal of Rock Mechanics and Mining Sciencesvol 110 pp 120ndash132 2018

[32] Y M Tien M C Kuo and C H Juang ldquoAn experimentalinvestigation of the failure mechanism of simulated transverselyisotropic rocksrdquo International Journal of Rock Mechanics andMining Sciences vol 43 no 8 pp 1163ndash1181 2006

Advances in Civil Engineering 21

transversely isotropic rocksrdquo Advances in Civil Engineeringvol 6 p 11 2019

[19] H-B Du F Dai Y Xu Y Liu and H-n Xu ldquoNumericalinvestigation on the dynamic strength and failure behavior ofrocks under hydrostatic confinement in SHPB testingrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 108 pp 43ndash57 2018

[20] X F Li Q B Zhang H B Li and J Zhao ldquoGrain-baseddiscrete element method (GB-DEM) modelling of multi-scalefracturing in rocks under dynamic loadingrdquo Rock Mechanicsand Rock Engineering vol 51 no 12 pp 3785ndash3817 2018

[21] F Dai M D Wei N W Xu Y Ma and D S Yang ldquoNu-merical assessment of the progressive rock fracture mecha-nism of cracked chevron notched Brazilian disc specimensrdquoRock Mechanics and Rock Engineering vol 48 no 2pp 463ndash479 2015

[22] B Park and K-B Min ldquoBonded-particle discrete elementmodeling of mechanical behavior of transversely isotropicrockrdquo International Journal of Rock Mechanics and MiningSciences vol 76 pp 243ndash255 2015

[23] E Cadoni ldquoDynamic characterization of orthogneiss rocksubjected to intermediate and high strain rates in tensionrdquoRock Mechanics and Rock Engineering vol 43 no 6pp 667ndash676 2010

[24] Y X Zhou K Xia X B Li et al ldquoSuggested methods fordetermining the dynamic strength parameters and Mode-Ifracture toughness of rock materialsrdquo International Journal ofRock Mechanics and Mining Sciences vol 49 no 1pp 105ndash112 2011

[25] F Dai K Xia J P Zuo R Zhang and N W Xu ldquoStatic anddynamic flexural strength anisotropy of Barre graniterdquo Rock Me-chanics and Rock Engineering vol 46 no 6 pp 1589ndash1602 2013

[26] F-Q Gong X-F Si X-B Li and S-Y Wang ldquoDynamictriaxial compression tests on sandstone at high strain ratesand low confining pressures with split Hopkinson pressurebarrdquo International Journal of Rock Mechanics and MiningSciences vol 113 pp 211ndash219 2019

[27] X F Li H B Li L W Liu et al ldquoInvestigating the crackinitiation and propagation mechanism in brittle rocks usinggrain-based finite-discrete element methodrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 127p 104219 2020

[28] F Gong and J Hu ldquoEnergy dissipation characteristic of redsandstone in the dynamic Brazilian disc test with SHPBsetuprdquo Advances in Civil Engineering vol 2020 Article ID7160937 10 pages 2020

[29] J C Li H B Li G W Ma and J Zhao ldquoA time-domainrecursive method to analyse transient wave propagationacross rock jointsrdquo Geophysical Journal Internationalvol 188 no 2 pp 631ndash644 2012

[30] P A Cundall ldquoA computer model for simulating progressivelarge-scale movement in blocky rock systemrdquo in Proceedingsof the International Symposium on Rock Mechanics NancyFrance October 1971

[31] B Park K-B Min N +ompson and P Horsrud ldquo+ree-dimensional bonded-particle discrete element modeling ofmechanical behavior of transversely isotropic rockrdquo Inter-national Journal of Rock Mechanics and Mining Sciencesvol 110 pp 120ndash132 2018

[32] Y M Tien M C Kuo and C H Juang ldquoAn experimentalinvestigation of the failure mechanism of simulated transverselyisotropic rocksrdquo International Journal of Rock Mechanics andMining Sciences vol 43 no 8 pp 1163ndash1181 2006

Advances in Civil Engineering 21