Research ArticleEnergy Dissipation Characteristic of Red Sandstone in theDynamic Brazilian Disc Test with SHPB Setup
Fengqiang Gong 12 and Jian Hu1
1School of Resources and Safety Engineering Central South University Changsha 410083 China2School of Civil Engineering Southeast University Nanjing 211189 China
Correspondence should be addressed to Fengqiang Gong fengqiangg126com
Received 3 December 2019 Accepted 20 February 2020 Published 22 April 2020
Academic Editor Arnaud Perrot
Copyright copy 2020 Fengqiang Gong and Jian Hu +is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited
In order to quantitatively investigate the energy dissipation characteristic during the dynamic tension failure of rockmaterials thedynamic Brazilian disc tests on red sandstone were conducted using the split Hopkinson pressure bar (SHPB) setup +e states ofthe specimens after different incident energies can be divided into three forms (ie the unruptured state the ruptured state andthe broken state) and the failure processes of the specimens were recorded by using a high-speed camera+e results show that theruptured state of the specimen corresponds to the critical failure strain Taking the critical incident energy as a turning point twopositive linear fitting relations between the dissipated energy and incident energy before and after the point are obtained and thedynamic linear dissipation law is foundWhen the incident energy is less than the critical energy specimens were unruptured afterimpact When the incident energy is greater than the critical energy specimens will be broken after impact According to theobtained linear energy dissipation law the dynamic tensile energy dissipation coefficient (DTEDC) was introduced forquantitatively describing the dynamic energy dissipation capacity of rock materials in the dynamic Brazilian disc test When thespecimen is in the unruptured state the ideal DTEDC is a constant value When the specimen is in a broken state the DTEDCincreases with the increase of incident energy
1 Introduction
+e tensile property is one of the mechanical properties ofrock materials Many rock engineering failures are oftencaused by local or global tension stress Currently theBrazilian disc test is a commonly used indirect tensile testmethod recommended by many experimental procedures[1ndash5] On the other hand rock materials are often subjectedto dynamic loads such as impact blasting and vibrationand the Split Hopkinson pressure bar (SHPB) has become acommon device used for studying the dynamic failure ofrocks under high loading rates or strain rate [3 6ndash9] +estudy of dynamic tensile properties of rock materials byusing the Brazilian disc test has attracted many researchersrsquoattention Gong and Zhao [6] put forward a DIF (dynamicincrease factor) model to describe the dynamic tensileproperties of sandstone in the range of low loading rate to
the high loading rate Asprone et al [10] investigated thedynamic behavior of a Mediterranean natural stone in dy-namic direct tensile tests Wang et al [11] suggested that theFlattened Brazilian Disc specimen could be used in thedynamic tensile test of rock Cadoni [12] reported the dy-namic tensile characteristics of Orthogneiss rock in therange of intermediate to high strain rates Dai and Xia [13]conducted dynamic Brazilian disc tests on Barre granite andconfirmed the dependence between the tensile strength andloading rate Wu et al [14] carried out the dynamic tensiletests of rock under the hydrostatic confinement
+e above research is more focused on the rate effect ofrock tensile strength by numerical simulation or laboratorytest but less from the perspective of energy consumption Infact the absorption and transformation of energy is theessential cause of rock material failure [15ndash18] Especially inthe dynamic test the incident energy carried by the incident
HindawiAdvances in Civil EngineeringVolume 2020 Article ID 7160937 10 pageshttpsdoiorg10115520207160937
wave provides the energy source for the failure of rockspecimens [19ndash21] In this study the dynamic Brazilian disctests of red sandstone under different incident energies werecarried out and the failure processes of rock specimens wererecorded with a high-speed camera According to the ex-perimental results the failure modes and energy relation-ships under different incident energies were analyzed
2 Experimental Methods and Materials
21 Test Apparatus +e test was carried out using a con-ventional SHPB setup [22] As shown in Figure 1 the testapparatus consisted of three main components three bars(the incident bar transmitted bar and absorption bar) adynamic load device and a data acquisition system +ethree bars were constructed of 40 Cr alloy steel with a di-ameter of 50mm a density of 7810 gcm3 and a P-wavevelocity of 5410ms +e lengths of the incident bar and thetransmitted bar were 20m and 15m +e bullet was aspindle-shaped striker which could generate a half-sineincident wave to overcome the shortcoming of high-fre-quency oscillation caused by rectangular waves +e dataacquisition system used a super dynamic strain gauge and aDL-850 oscilloscope +e strain gauges were attached to theincident bar and the transmitted bar at a distance of 100mfrom the specimen and the incident strain reflected strainand transmitted strain could be measured +e dynamicuniaxial and triaxial compression tests can be conducted onthis SHPB device [23]
22 Test Principle According to the principles of dynamicBrazilian disc test and the data collected by the data ac-quisition system the tensile stress at the center of thespecimen can be calculated by the following formula [22]
σ(t) AE
2π DLεI(t) + εR(t) + εT(t)1113858 1113859 (1)
where A and E are the cross-sectional area and Youngrsquosmodulus of the bar D and L are the diameter and length ofthe specimen σ(t) and ε(t) are the tensile stress and strain ofthe specimen at moment t and the subscripts I R and T
denote the incident reflected and transmitted wavesrespectively
+e calculation formulas of the incident energy reflectedenergy and transmitted energy can be expressed as follows
EI(t) ECA 1113946t
0ε2I(t)dt
ER(t) ECA 1113946t
0ε2R(t)dt
ET(t) ECA 1113946t
0ε2T(t)dt
(2)
where EI(t) ER(t) and ET(t) are the cumulative incidentenergy reflected energy and transmitted energy at momentt respectively and C is the one-dimensional longitudinalstress wave velocity of the bar
+e cumulative dissipated energy at moment t can becalculated from the formula
ED(t) EI(t) minus ER(t) minus ET(t) (3)
23 Specimen Preparation A block of red sandstone withgood integrity and homogeneity was selected for preparingthe specimens for the dynamic Brazilian disc testsAccording to the suggested method for rock dynamic tests[3] all the specimens were cored from one piece of rockblock and processed into cylindrical shapes with a diameterof 50mm and a height of 25mm (the length to diameter ratio
DL850 Oscilloscope
Light source High speed camera
Absorption bar
Transmitted barRed sandstone specimen
Strain gauge 2Strain gauge 1
Incident barSpindle-shaped punchAir gun
Firing chamber
Bridge box Bridge box
Super dynamic strain gauge
Computer
Figure 1 Diagram of the SHPB testing apparatus for the dynamic Brazilian disc tests [22]
2 Advances in Civil Engineering
was 1 2) +e ends of the specimens were ground by abuffing machine to control the evenness and non-perpendicularity to be less than 002mm +e averagedensity and wave velocity of red sandstone are 2431 kgm3
and 3386ms
24 Preparation for Formal Test and Dynamic EquilibriumBefore the tests started a nonspecimen test was necessary toverify the stability and operability of the SHPB test system+e incident wave should be basically consistent with thetransmitted wave during a conventional uniaxial test which
IncRe
TraInc + Re
0
50 100 150 200 2500Time (μs)
ndash100
ndash80
ndash60
ndash40
ndash20
20
40
60
80
Stre
ss (M
Pa)
Figure 2 Stress equilibrium check for the dynamic Brazilian disc testing of a red sandstone specimen (lsquoIncrsquo is the incident stress lsquoRersquo is thereflected stress lsquoTrarsquo is the transmitted stress and lsquoInc +Rersquo is the superimposed stress at the end of the incident bar)
Table 1 Energy and mechanical data of red sandstone specimens
indicates that the stability of the test system is up to thestandard of formal tests
+e specimen was sandwiched between the incident barand the transmitted bar in the radial direction and lubricantwas applied at the contact point between the specimen andthe two bars to reduce friction In addition the validity of thetest results should be judged by whether the stress at bothends of the specimen reaches equilibrium before failure
Figure 2 shows the stress history of a typical dynamicBrazilian disc test in which the ends of the rock specimenreached the equilibrium state It is observed that the sum ofthe incident and reflected stress waves lsquoInc +Rersquo is basicallyconsistent with the transmitted stress wave lsquoTrarsquo +is de-notes that the test results are valid and reliable Otherwisethe results will be invalid and should be excluded In thispaper checks of the stress equilibrium were carried out on
each of the specimens to ensure the validity and accuracy ofthe test results
3 Results
+e results of dynamic Brazilian disc tests are listed in Table 1+e states of the specimens after impact can be divided intothree categories namely the unruptured state the rupturedstate and the broken state +e unruptured state refers to thespecimens which remained intact and have the ability to beara load after impact as shown in Figure 3(a)
+e ruptured state refers to the state in which thespecimen produced visible cracks after impact and separatedinto two blocks as shown in Figure 3(b) +e broken staterefers to the state in which the specimen was broken intomany rock blocks after impact as shown in Figure 3(c) It
14 16
(a)
11 11
(b)
4745
(c)
Figure 3 +ree failure states of red sandstone specimens after impact (a) unruptured state (b) ruptured state (c) broken state
4 Advances in Civil Engineering
can be concluded that there is an incident energy whichcorresponds to the state of the specimen that changes fromthe unruptured state to the ruptured state +is incidentenergy was defined as the critical incident energy When theincident energy was less than the critical incident energy thespecimens were in the unruptured state after impact andwhen the incident energy was greater than the critical incidentenergy the specimens were in the broken state after impact
It can be seen from Table 1 and Figure 3 that when theincident energy was 640 J the specimen was split into tworock blocks along the loading direction When the incidentenergy was at a high level and the specimen was brokenthere were triangular broken areas at both ends of thespecimenWith the augmentation of the incident energy thedegree of broken specimens became higher +is is becausebefore the main radial crack penetrated the compressivestresses at both ends of the specimen exceeded the tensilestrength of the rock
31 Mechanical Properties Five typical dynamic load-dis-placement curves are shown in Figure 4 the peak tensilestress of red sandstone specimens under different loadingrates is obtained as shown in Table 1 Figure 5 shows therelationship between the peak tensile stress and the loadingrate It can be found that there is an obvious loading rateeffect with the peak tensile stress of red sandstone and thelogarithmic function can be used to fit this loading effecttrend +is rate effect of peak stress has been confirmed inmany literature studies [4 6 7] and can be considered as oneof the dynamic tensile properties of rock
32 Energy Characteristics Based on the energy calculationprinciple of the SHPB dynamic Brazilian disc test the curvesof incident energy reflected energy transmitted energy anddissipated energy relative to time in the dynamic Braziliandisc testing of red sandstone specimens can be calculated Asshown in Figure 6 there are curves of four kinds of energyfor specimens 1 11 and 26 in three states over time It can beseen that the four kinds of energy all increased with theincrease of time and then no longer increased after ap-proximately 250 μs after which they remained constant
33FailureProcess of theSpecimen In dynamic loading testsfailure of the specimen is a quickly developing process Tostudy the failure processes of specimens effective technicalmeasurements are needed At present a high-speed camerais generally used to record the failure processes of specimensIn this study a high-speed camera was set at 72000 framesper second (fps) to take a photo every 1389 μs Figures 7(a)ndash7(c) shows the typical failure processes of red sandstonespecimens in the three states after impacting specimens 1411 and 47 Specimen 14 was in the unruptured state afterimpact remaining intact after the impact with no visiblecracks on the surface Specimen 11 was in the ruptures stateafter the impact A crack formed on the surface of thespecimen at 25002 μs starting at one end of the incident barand running through the whole specimen along the loading
direction +e specimen was separated into two rock blocksat 79173 μs Specimen 47 was broken under a larger dy-namic load +e specimen was penetrated by a main crackand two additional cracks were produced at the contactpoints between the specimen and the ends of the bars at25002 μs At 79173 μs the specimen was split into severalparts which were flying outward at certain speeds In ad-dition the main cracks of specimens 11 and 47 occurredalong the direction of the loading stress which was becausethe tensile stress of the rock plays a major role in thefracturing of specimens
34 Energy Relationship between the Dissipated Energy andIncident Energy It can be concluded from Table 1 that the
161411
4547
0
5
10
15
20
25
30
Load
(kN
)
01 02 03 04 05 06 07 0800Displacement (mm)
Figure 4 Typical dynamic load-displacement curves
(R2 = 092)
Unrupturted stateRuptured state
Broken stateFitted line of specimens inthree states
Figure 5 Relationship between loading rate and peak tensile stressof red sandstone specimens
Advances in Civil Engineering 5
dissipated energy of the specimens increased with theaugmentation of the incident energy and the states of thered sandstone specimens changed from unruptured state tothe ruptured state and then to the broken state
Figure 8 shows the relationship between the dissipatedenergy and the incident energy in the dynamic Brazilian disctesting of red sandstone specimens It can be seen that thedissipated energy increases approximately linearly with theincrease of incident energy and presenting two differentstages of energy relationship From Table 1 it can be foundthat this feature is related to the state of the specimen afterimpact so the linear fitting formulas were used to fit the datapoints of the unruptured and broken specimens respectively(to correct the fitting curve of the unruptured state the
origin coordinate was also added because when the incidentenergy is 0 J the dissipated energy must be 0 J) +e datapoints for the ruptured state are represented by differentshapes than those for the other two states
+e relationships between the dissipated energy andincident energy are both positively linear for the specimensin the unruptured state and broken state after the impact andcan be fitted by the following
EUD kUEI + cU EI ltECI( 1113857
EBD kBEI + cB EI gtECI( 1113857
⎧⎨
⎩ (4)
where EUD and EB
D are the dissipated energies of the specimenin the unruptured state and in the broken state kU and kB are
Specimen 26Specimen 11Specimen 1
0
5
10
15
20
25
30
35In
cide
nt en
ergy
(J)
50 100 150 200 250 3000Time (μs)
(a)
Specimen 26Specimen 11Specimen 1
0
5
10
15
20
25
Refle
cted
ener
gy (J
)
50 100 150 200 250 3000Time (μs)
(b)
Specimen 26Specimen 11Specimen 1
00
01
02
03
04
05
06
07
08
Tran
smitt
ed en
ergy
(J)
50 100 150 200 250 3000Time (μs)
(c)
Specimen 26Specimen 11Specimen 1
0
2
4
6
8
10D
issip
ated
ener
gy (J
)
50 100 150 200 250 3000Time (μs)
(d)
Figure 6 Energy-time curve of red sandstone specimens in the dynamic Brazilian disc test (a) incident energy-time (b) reflected energy-time (c) transmitted energy-time (d) dissipated energy-time
6 Advances in Civil Engineering
the slopes of the fitted curves of the unruptured state andbroken state EI is the incident energy cU and cB areconstants of the unruptured state and broken state and ECI
is the critical incident energy which corresponds to theruptured state of the specimen after impact For the redsandstone in this study two fitting functions can be ob-tained as follows
EUD
EI
022 minus004EI
EI lt 75 J( 1113857
EBD
EI
034 minus094EI
EI gt 75 J( 1113857
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(5)
For quantitatively describing the dynamic energy dis-sipation capacity of rock materials in the dynamic Braziliandisc test the dynamic tensile energy dissipation coefficient(DTEDC) is introduced as the ratio of dissipated energy toincident energy
In fact the DTEDCs are equal to the values of kU and kBIn formula (5) when the incident energy is 0 J the
dissipated energy must be 0 J However due to the influenceof heterogeneity of rock materials and experimental un-certainty the intercept of energy dissipation function in thestate of unruptured specimen is not 0 (shown in Figure 8)+erefore the ideal energy storage formula should be asfollows
14 14 14000μs 25002μs 79173μs
(a)
11 11 11000μs 25002μs 79173μs
(b)
47 47 47000μs 25002μs 79173μs
(c)
Figure 7 Failure processes of specimens in three states (a) unruptured state (b) ruptured state (c) broken state
Advances in Civil Engineering 7
EUD
EI
022 EI lt 75 J( 1113857
EBD
EI
034 minus094EI
EI gt 75 J( 1113857
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(6)
+e ideal curve obtained according to formula (6) wasshown in red line in Figure 9+eDTEDC of the specimen inthe unruptured state after impact is 022 When the incidentenergy exceeds 75 J the state of the specimen after impactwill develop from a ruptured state to a broken state and theDTEDC begins to increase It can be seen from Figure 9 thatthe DTEDC increases with the increase of incident energyand the velocity decreases gradually +e same conclusioncan be drawn from formula (6) and the DTEDC willgradually increase and approach to 034
4 Discussions
41 Loading Rate Effect As shown in Figure 5 there is noobvious difference among these three failure states +ere-fore from the relationship between the peak tensile stressand loading rate it is impossible to distinguish the differencein mechanical properties of the failure state of the specimenHowever from the point of view of energy as mentionedabove we find that there are two stages of linear energydissipation between these states
42 Failure State It can be seen from Figure 7 that when thespecimen is in the ruptured state that is specimen 11 thereis only one main crack that penetrates the whole specimenwith the specimen divided into two parts along the crack It
is noteworthy that this specimen produced not onlymicrocracks propagation but also macrocracks penetrationand the specimen is at the critical point at which the kineticenergy of the rock fragments is about to be generated
+e data of the ruptured specimen 11 has been fittedtogether with those from the unruptured and broken statesspecimens to create fitting lines and the two previousfitting lines were retained for comparison as shown inFigure 10 +e two previous fitting lines and the other two
Unruptured stateRuptured stateBroken state
Fitted line of unrupturedstateFitted line of broken state
EUD = 022EI ndash 004
(R2 = 091)
EBD = 034EI ndash 094
(R2 = 097)
0
2
4
6
8
10
12
Diss
ipat
ed en
ergy
(J)
5 10 15 20 25 30 350Incident energy (J)
Figure 8 Fitted lines of the dissipated energy-incident energy inthe ldquoUrdquo state and ldquoBrdquo state (ldquoUrdquo state is the unruptured state and ldquoBrdquostate is the broken state)
Ideal curve of DTEDCUnruptured state
Ruptured stateBroken state
01
02
03
04
Dyn
amic
tens
ile en
ergy
diss
ipat
ion
coef
ficie
nt (D
TED
C)
5 10 15 20 25 30 35 400Incident energy (J)
Figure 9 Relation of incident energy versus DTEDC
EprimeDU = 020EI ndash 001
(R2 = 089)
EprimeDB = 034EI ndash 096
(R2 = 097)EDU = 022EI ndash 004(R2 = 091)
EDB = 034EI ndash 094(R2 = 097)
Unruptured stateRuptured stateBroken stateFitted line of ldquoUrdquo state
Fitted line of ldquoBrdquo state
Fitted line of ldquoBrdquo stateamp ldquoRrdquo state
Fitted line of ldquoUrdquo stateamp ldquoRrdquo state
0
2
4
6
8
10
12
Diss
ipat
ed en
ergy
(J)
5 10 15 20 25 30 350Incident energy (J)
Figure 10 Fitted lines of the incident energy-dissipated energy inthe ldquoUrdquo state ldquoBrdquo state ldquoUrdquo state ldquoRrdquo state and ldquoBrdquo state amp ldquoRrdquostate (ldquoUrdquo state is the unruptured state ldquoBrdquo state is the broken stateand ldquoRrdquo state is the ruptured state)
8 Advances in Civil Engineering
fitting lines including the critical-state specimen are listedas follows
EDU 022EI minus 004
EDUprime 020EI minus 001
⎧⎨
⎩
EDB 034EI minus 094
EDBprime 034EI minus 096
⎧⎨
⎩
(7)
It can be seen that the two fitted lines after adding thespecimen in the ruptured state almost coincide with theprevious fitted lines so the data of the specimen in theruptured state can be fitted with the data of other unrupturedand broken specimens As shown in Figure 10 the specimendata point in the ruptured state is approximately at theintersection of the two fitted lines
In addition for the dynamic Brazilian disc test becauseof the relatively small dynamic tensile stress it is difficult toobtain the specimen of the critical state so there is only onecritical-state specimen in this paper
5 Conclusions
In this paper a series of dynamic Brazilian disc tests of redsandstone specimens were carried out using the SHPB setup+e main conclusions can be described as follows
(1) In the dynamic Brazilian disc test the rock specimenhas three states after impact with different incidentenergy the unruptured state ruptured state andbroken state And the ruptured state is corre-sponding to the critical energy When the incidentenergy is less than the critical energy the specimenremains intact and there is no macrocrack on thesurface that is the unruptured state when the in-cident energy is greater than the critical energy thespecimen will be damaged and lose the bearingcapacity along the loading direction that is thebroken state
(2) Under the impact of different incident energy thedynamic peak stress and loading rate of red sand-stone specimens show a unified exponential functionrelationship However the failure states of specimensafter impact cannot be identified from thisrelationship
(3) +ere are linear relationships between the dissipatedenergy and incident energy in both the unrupturedstate and the broken state In the broken state thedissipated energy increases with the increase of in-cident energy and the increasing speed is faster thanthat of the unruptured specimen
(4) According to the loading process recorded with ahigh-speed camera the failure mode of the specimenis splitting along the loading direction When thespecimen is in the critical failure mode after theimpact only one crack is produced along the loadingdirection and it penetrates the whole specimen sothat the specimen is divided into two partsWhen thespecimen is in the broken state except the main
crack along the loading direction additional crackswill be generated at both ends of the specimenconnecting the incident and transmission rein-forcement and these cracks will also expand alongthe loading direction
(5) When the specimen is in the unruptured state afterimpact the ideal DTEDC is a constant value whichis 022 for the red sandstone specimen in this paperHowever when the specimen is in a broken stateafter impact the DTEDC increases with the increaseof incident energy and gradually approaches 034
Data Availability
No data were used to support this study
Conflicts of Interest
+e authors declare that they have no conflicts of interest
Acknowledgments
+is study was supported by the National Natural ScienceFoundation of China (Grant no 41877272)
References
[1] ISRM ldquoSuggested methods for determining tensile strength ofrock materialsrdquo International Journal of Rock Mechanics ampMining Sciences vol 15 no 1 pp 881ndash891 1978
[2] ASTM D3967-08 Standard Test Method for Splitting TensileStrength of Intact Rock Core Specimens ASTM InternationalWest Conshohocken PA USA 2008
[3] 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 pp 105ndash1122012
[4] F Q Gong L Zhang and S Y Wang ldquoLoading rate effect ofrock material with the direct tensile and three Brazilian disctestsrdquo Advances in Civil Engineering vol 2019 Article ID6260351 8 pages 2019
[5] F Q Gong X B Li and J Zhao ldquoAnalytical algorithm toestimate tensile modulus in Brazilian disk splitting testsrdquoChinese Journal of Rock Mechanics and Engineering vol 29no 5 pp 881ndash891 2010
[6] F Q Gong and G F Zhao ldquoDynamic indirect tensile strengthof sandstone under different loading ratesrdquo Rock Mechanicsand Rock Engineering vol 47 no 6 pp 2271ndash2278 2014
[7] Q B Zhang and J Zhao ldquoA review of dynamic experimentaltechniques andmechanical behaviour of rockmaterialsrdquo RockMechanics and Rock Engineering vol 47 no 4 pp 1411ndash14782014
[8] J B Zhu Z Y Liao and C A Tang ldquoNumerical SHPB tests ofrocks under combined static and dynamic loading conditionswith application to dynamic behavior of rocks under in situstressesrdquo Rock Mechanics and Rock Engineering vol 49no 10 pp 3935ndash3946 2016
[9] J C Li L F Rong H B Li and S N Hong ldquoAn SHPB teststudy on stress wave energy attenuation in jointed rockmassesrdquo Rock Mechanics and Rock Engineering vol 52 no 2pp 403ndash420 2019
Advances in Civil Engineering 9
[10] D Asprone E Cadoni A Prota and G Manfredi ldquoDynamicbehavior of a Mediterranean natural stone under tensileloadingrdquo International Journal of Rock Mechanics and MiningSciences vol 46 no 3 pp 514ndash520 2009
[11] Q ZWangW Li and H P Xie ldquoDynamic split tensile test offlattened Brazilian disc of rock with SHPB setuprdquo Mechanicsof Materials vol 41 no 3 pp 252ndash260 2009
[12] 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
[13] F Dai and K Xia ldquoLoading rate dependence of tensilestrength anisotropy of Barre graniterdquo Pure and AppliedGeophysics vol 167 no 11 pp 1419ndash1432 2010
[14] B Wu W Yao and K Xia ldquoAn experimental study of dy-namic tensile failure of rocks subjected to hydrostatic con-finementrdquo Rock Mechanics and Rock Engineering vol 49no 10 pp 3855ndash3864 2016
[15] H Xie L Li R Peng and Y Ju ldquoEnergy analysis and criteriafor structural failure of rocksrdquo Journal of Rock Mechanics andGeotechnical Engineering vol 1 no 1 pp 11ndash20 2009
[16] F Q Gong S Luo and J Y Yan ldquoEnergy storage and dis-sipation evolution process and characteristics of marble inthree tension-type failure testsrdquo Rock Mechanics and RockEngineering vol 51 no 11 pp 3613ndash3624 2018
[17] F Q Gong J Y Yan S Luo and X B Li ldquoInvestigation onthe linear energy storage and dissipation laws of rock ma-terials under uniaxial compressionrdquo RockMechanics and RockEngineering vol 52 no 11 pp 4237ndash4255 2019
[18] Y Chu H Sun and D Zhang ldquoExperimental study onevolution in the characteristics of permeability deformationand energy of coal containing gas under triaxial cyclicloading-unloadingrdquo Energy Science amp Engineering vol 7no 5 pp 2112ndash2123 2019
[19] B Lundberg ldquoA split Hopkinson bar study of energy ab-sorption in dynamic rock fragmentationrdquo InternationalJournal of Rock Mechanics and Mining Sciences amp Geo-mechanics Abstracts vol 13 no 6 pp 187ndash197 1976
[20] A V Mikhalyuk and V V Zakharov ldquoDissipation of dy-namic-loading energy in quasi-elastic deformation processesin rocksrdquo Journal of Applied Mechanics and Technical Physicsvol 38 no 2 pp 312ndash318 1997
[21] X B Li T S Lok and J Zhao ldquoDynamic characteristics ofgranite subjected to intermediate loading raterdquo Rock Me-chanics and Rock Engineering vol 38 no 1 pp 21ndash39 2005
[22] F Q Gong Experimental study of rock mechanical propertiesunder coupled static-dynamic loads and dynamic strengthcriterion PhD thesis Central South University ChangshaChina 2010
[23] 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
10 Advances in Civil Engineering
wave provides the energy source for the failure of rockspecimens [19ndash21] In this study the dynamic Brazilian disctests of red sandstone under different incident energies werecarried out and the failure processes of rock specimens wererecorded with a high-speed camera According to the ex-perimental results the failure modes and energy relation-ships under different incident energies were analyzed
2 Experimental Methods and Materials
21 Test Apparatus +e test was carried out using a con-ventional SHPB setup [22] As shown in Figure 1 the testapparatus consisted of three main components three bars(the incident bar transmitted bar and absorption bar) adynamic load device and a data acquisition system +ethree bars were constructed of 40 Cr alloy steel with a di-ameter of 50mm a density of 7810 gcm3 and a P-wavevelocity of 5410ms +e lengths of the incident bar and thetransmitted bar were 20m and 15m +e bullet was aspindle-shaped striker which could generate a half-sineincident wave to overcome the shortcoming of high-fre-quency oscillation caused by rectangular waves +e dataacquisition system used a super dynamic strain gauge and aDL-850 oscilloscope +e strain gauges were attached to theincident bar and the transmitted bar at a distance of 100mfrom the specimen and the incident strain reflected strainand transmitted strain could be measured +e dynamicuniaxial and triaxial compression tests can be conducted onthis SHPB device [23]
22 Test Principle According to the principles of dynamicBrazilian disc test and the data collected by the data ac-quisition system the tensile stress at the center of thespecimen can be calculated by the following formula [22]
σ(t) AE
2π DLεI(t) + εR(t) + εT(t)1113858 1113859 (1)
where A and E are the cross-sectional area and Youngrsquosmodulus of the bar D and L are the diameter and length ofthe specimen σ(t) and ε(t) are the tensile stress and strain ofthe specimen at moment t and the subscripts I R and T
denote the incident reflected and transmitted wavesrespectively
+e calculation formulas of the incident energy reflectedenergy and transmitted energy can be expressed as follows
EI(t) ECA 1113946t
0ε2I(t)dt
ER(t) ECA 1113946t
0ε2R(t)dt
ET(t) ECA 1113946t
0ε2T(t)dt
(2)
where EI(t) ER(t) and ET(t) are the cumulative incidentenergy reflected energy and transmitted energy at momentt respectively and C is the one-dimensional longitudinalstress wave velocity of the bar
+e cumulative dissipated energy at moment t can becalculated from the formula
ED(t) EI(t) minus ER(t) minus ET(t) (3)
23 Specimen Preparation A block of red sandstone withgood integrity and homogeneity was selected for preparingthe specimens for the dynamic Brazilian disc testsAccording to the suggested method for rock dynamic tests[3] all the specimens were cored from one piece of rockblock and processed into cylindrical shapes with a diameterof 50mm and a height of 25mm (the length to diameter ratio
DL850 Oscilloscope
Light source High speed camera
Absorption bar
Transmitted barRed sandstone specimen
Strain gauge 2Strain gauge 1
Incident barSpindle-shaped punchAir gun
Firing chamber
Bridge box Bridge box
Super dynamic strain gauge
Computer
Figure 1 Diagram of the SHPB testing apparatus for the dynamic Brazilian disc tests [22]
2 Advances in Civil Engineering
was 1 2) +e ends of the specimens were ground by abuffing machine to control the evenness and non-perpendicularity to be less than 002mm +e averagedensity and wave velocity of red sandstone are 2431 kgm3
and 3386ms
24 Preparation for Formal Test and Dynamic EquilibriumBefore the tests started a nonspecimen test was necessary toverify the stability and operability of the SHPB test system+e incident wave should be basically consistent with thetransmitted wave during a conventional uniaxial test which
IncRe
TraInc + Re
0
50 100 150 200 2500Time (μs)
ndash100
ndash80
ndash60
ndash40
ndash20
20
40
60
80
Stre
ss (M
Pa)
Figure 2 Stress equilibrium check for the dynamic Brazilian disc testing of a red sandstone specimen (lsquoIncrsquo is the incident stress lsquoRersquo is thereflected stress lsquoTrarsquo is the transmitted stress and lsquoInc +Rersquo is the superimposed stress at the end of the incident bar)
Table 1 Energy and mechanical data of red sandstone specimens
indicates that the stability of the test system is up to thestandard of formal tests
+e specimen was sandwiched between the incident barand the transmitted bar in the radial direction and lubricantwas applied at the contact point between the specimen andthe two bars to reduce friction In addition the validity of thetest results should be judged by whether the stress at bothends of the specimen reaches equilibrium before failure
Figure 2 shows the stress history of a typical dynamicBrazilian disc test in which the ends of the rock specimenreached the equilibrium state It is observed that the sum ofthe incident and reflected stress waves lsquoInc +Rersquo is basicallyconsistent with the transmitted stress wave lsquoTrarsquo +is de-notes that the test results are valid and reliable Otherwisethe results will be invalid and should be excluded In thispaper checks of the stress equilibrium were carried out on
each of the specimens to ensure the validity and accuracy ofthe test results
3 Results
+e results of dynamic Brazilian disc tests are listed in Table 1+e states of the specimens after impact can be divided intothree categories namely the unruptured state the rupturedstate and the broken state +e unruptured state refers to thespecimens which remained intact and have the ability to beara load after impact as shown in Figure 3(a)
+e ruptured state refers to the state in which thespecimen produced visible cracks after impact and separatedinto two blocks as shown in Figure 3(b) +e broken staterefers to the state in which the specimen was broken intomany rock blocks after impact as shown in Figure 3(c) It
14 16
(a)
11 11
(b)
4745
(c)
Figure 3 +ree failure states of red sandstone specimens after impact (a) unruptured state (b) ruptured state (c) broken state
4 Advances in Civil Engineering
can be concluded that there is an incident energy whichcorresponds to the state of the specimen that changes fromthe unruptured state to the ruptured state +is incidentenergy was defined as the critical incident energy When theincident energy was less than the critical incident energy thespecimens were in the unruptured state after impact andwhen the incident energy was greater than the critical incidentenergy the specimens were in the broken state after impact
It can be seen from Table 1 and Figure 3 that when theincident energy was 640 J the specimen was split into tworock blocks along the loading direction When the incidentenergy was at a high level and the specimen was brokenthere were triangular broken areas at both ends of thespecimenWith the augmentation of the incident energy thedegree of broken specimens became higher +is is becausebefore the main radial crack penetrated the compressivestresses at both ends of the specimen exceeded the tensilestrength of the rock
31 Mechanical Properties Five typical dynamic load-dis-placement curves are shown in Figure 4 the peak tensilestress of red sandstone specimens under different loadingrates is obtained as shown in Table 1 Figure 5 shows therelationship between the peak tensile stress and the loadingrate It can be found that there is an obvious loading rateeffect with the peak tensile stress of red sandstone and thelogarithmic function can be used to fit this loading effecttrend +is rate effect of peak stress has been confirmed inmany literature studies [4 6 7] and can be considered as oneof the dynamic tensile properties of rock
32 Energy Characteristics Based on the energy calculationprinciple of the SHPB dynamic Brazilian disc test the curvesof incident energy reflected energy transmitted energy anddissipated energy relative to time in the dynamic Braziliandisc testing of red sandstone specimens can be calculated Asshown in Figure 6 there are curves of four kinds of energyfor specimens 1 11 and 26 in three states over time It can beseen that the four kinds of energy all increased with theincrease of time and then no longer increased after ap-proximately 250 μs after which they remained constant
33FailureProcess of theSpecimen In dynamic loading testsfailure of the specimen is a quickly developing process Tostudy the failure processes of specimens effective technicalmeasurements are needed At present a high-speed camerais generally used to record the failure processes of specimensIn this study a high-speed camera was set at 72000 framesper second (fps) to take a photo every 1389 μs Figures 7(a)ndash7(c) shows the typical failure processes of red sandstonespecimens in the three states after impacting specimens 1411 and 47 Specimen 14 was in the unruptured state afterimpact remaining intact after the impact with no visiblecracks on the surface Specimen 11 was in the ruptures stateafter the impact A crack formed on the surface of thespecimen at 25002 μs starting at one end of the incident barand running through the whole specimen along the loading
direction +e specimen was separated into two rock blocksat 79173 μs Specimen 47 was broken under a larger dy-namic load +e specimen was penetrated by a main crackand two additional cracks were produced at the contactpoints between the specimen and the ends of the bars at25002 μs At 79173 μs the specimen was split into severalparts which were flying outward at certain speeds In ad-dition the main cracks of specimens 11 and 47 occurredalong the direction of the loading stress which was becausethe tensile stress of the rock plays a major role in thefracturing of specimens
34 Energy Relationship between the Dissipated Energy andIncident Energy It can be concluded from Table 1 that the
161411
4547
0
5
10
15
20
25
30
Load
(kN
)
01 02 03 04 05 06 07 0800Displacement (mm)
Figure 4 Typical dynamic load-displacement curves
(R2 = 092)
Unrupturted stateRuptured state
Broken stateFitted line of specimens inthree states
Figure 5 Relationship between loading rate and peak tensile stressof red sandstone specimens
Advances in Civil Engineering 5
dissipated energy of the specimens increased with theaugmentation of the incident energy and the states of thered sandstone specimens changed from unruptured state tothe ruptured state and then to the broken state
Figure 8 shows the relationship between the dissipatedenergy and the incident energy in the dynamic Brazilian disctesting of red sandstone specimens It can be seen that thedissipated energy increases approximately linearly with theincrease of incident energy and presenting two differentstages of energy relationship From Table 1 it can be foundthat this feature is related to the state of the specimen afterimpact so the linear fitting formulas were used to fit the datapoints of the unruptured and broken specimens respectively(to correct the fitting curve of the unruptured state the
origin coordinate was also added because when the incidentenergy is 0 J the dissipated energy must be 0 J) +e datapoints for the ruptured state are represented by differentshapes than those for the other two states
+e relationships between the dissipated energy andincident energy are both positively linear for the specimensin the unruptured state and broken state after the impact andcan be fitted by the following
EUD kUEI + cU EI ltECI( 1113857
EBD kBEI + cB EI gtECI( 1113857
⎧⎨
⎩ (4)
where EUD and EB
D are the dissipated energies of the specimenin the unruptured state and in the broken state kU and kB are
Specimen 26Specimen 11Specimen 1
0
5
10
15
20
25
30
35In
cide
nt en
ergy
(J)
50 100 150 200 250 3000Time (μs)
(a)
Specimen 26Specimen 11Specimen 1
0
5
10
15
20
25
Refle
cted
ener
gy (J
)
50 100 150 200 250 3000Time (μs)
(b)
Specimen 26Specimen 11Specimen 1
00
01
02
03
04
05
06
07
08
Tran
smitt
ed en
ergy
(J)
50 100 150 200 250 3000Time (μs)
(c)
Specimen 26Specimen 11Specimen 1
0
2
4
6
8
10D
issip
ated
ener
gy (J
)
50 100 150 200 250 3000Time (μs)
(d)
Figure 6 Energy-time curve of red sandstone specimens in the dynamic Brazilian disc test (a) incident energy-time (b) reflected energy-time (c) transmitted energy-time (d) dissipated energy-time
6 Advances in Civil Engineering
the slopes of the fitted curves of the unruptured state andbroken state EI is the incident energy cU and cB areconstants of the unruptured state and broken state and ECI
is the critical incident energy which corresponds to theruptured state of the specimen after impact For the redsandstone in this study two fitting functions can be ob-tained as follows
EUD
EI
022 minus004EI
EI lt 75 J( 1113857
EBD
EI
034 minus094EI
EI gt 75 J( 1113857
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(5)
For quantitatively describing the dynamic energy dis-sipation capacity of rock materials in the dynamic Braziliandisc test the dynamic tensile energy dissipation coefficient(DTEDC) is introduced as the ratio of dissipated energy toincident energy
In fact the DTEDCs are equal to the values of kU and kBIn formula (5) when the incident energy is 0 J the
dissipated energy must be 0 J However due to the influenceof heterogeneity of rock materials and experimental un-certainty the intercept of energy dissipation function in thestate of unruptured specimen is not 0 (shown in Figure 8)+erefore the ideal energy storage formula should be asfollows
14 14 14000μs 25002μs 79173μs
(a)
11 11 11000μs 25002μs 79173μs
(b)
47 47 47000μs 25002μs 79173μs
(c)
Figure 7 Failure processes of specimens in three states (a) unruptured state (b) ruptured state (c) broken state
Advances in Civil Engineering 7
EUD
EI
022 EI lt 75 J( 1113857
EBD
EI
034 minus094EI
EI gt 75 J( 1113857
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(6)
+e ideal curve obtained according to formula (6) wasshown in red line in Figure 9+eDTEDC of the specimen inthe unruptured state after impact is 022 When the incidentenergy exceeds 75 J the state of the specimen after impactwill develop from a ruptured state to a broken state and theDTEDC begins to increase It can be seen from Figure 9 thatthe DTEDC increases with the increase of incident energyand the velocity decreases gradually +e same conclusioncan be drawn from formula (6) and the DTEDC willgradually increase and approach to 034
4 Discussions
41 Loading Rate Effect As shown in Figure 5 there is noobvious difference among these three failure states +ere-fore from the relationship between the peak tensile stressand loading rate it is impossible to distinguish the differencein mechanical properties of the failure state of the specimenHowever from the point of view of energy as mentionedabove we find that there are two stages of linear energydissipation between these states
42 Failure State It can be seen from Figure 7 that when thespecimen is in the ruptured state that is specimen 11 thereis only one main crack that penetrates the whole specimenwith the specimen divided into two parts along the crack It
is noteworthy that this specimen produced not onlymicrocracks propagation but also macrocracks penetrationand the specimen is at the critical point at which the kineticenergy of the rock fragments is about to be generated
+e data of the ruptured specimen 11 has been fittedtogether with those from the unruptured and broken statesspecimens to create fitting lines and the two previousfitting lines were retained for comparison as shown inFigure 10 +e two previous fitting lines and the other two
Unruptured stateRuptured stateBroken state
Fitted line of unrupturedstateFitted line of broken state
EUD = 022EI ndash 004
(R2 = 091)
EBD = 034EI ndash 094
(R2 = 097)
0
2
4
6
8
10
12
Diss
ipat
ed en
ergy
(J)
5 10 15 20 25 30 350Incident energy (J)
Figure 8 Fitted lines of the dissipated energy-incident energy inthe ldquoUrdquo state and ldquoBrdquo state (ldquoUrdquo state is the unruptured state and ldquoBrdquostate is the broken state)
Ideal curve of DTEDCUnruptured state
Ruptured stateBroken state
01
02
03
04
Dyn
amic
tens
ile en
ergy
diss
ipat
ion
coef
ficie
nt (D
TED
C)
5 10 15 20 25 30 35 400Incident energy (J)
Figure 9 Relation of incident energy versus DTEDC
EprimeDU = 020EI ndash 001
(R2 = 089)
EprimeDB = 034EI ndash 096
(R2 = 097)EDU = 022EI ndash 004(R2 = 091)
EDB = 034EI ndash 094(R2 = 097)
Unruptured stateRuptured stateBroken stateFitted line of ldquoUrdquo state
Fitted line of ldquoBrdquo state
Fitted line of ldquoBrdquo stateamp ldquoRrdquo state
Fitted line of ldquoUrdquo stateamp ldquoRrdquo state
0
2
4
6
8
10
12
Diss
ipat
ed en
ergy
(J)
5 10 15 20 25 30 350Incident energy (J)
Figure 10 Fitted lines of the incident energy-dissipated energy inthe ldquoUrdquo state ldquoBrdquo state ldquoUrdquo state ldquoRrdquo state and ldquoBrdquo state amp ldquoRrdquostate (ldquoUrdquo state is the unruptured state ldquoBrdquo state is the broken stateand ldquoRrdquo state is the ruptured state)
8 Advances in Civil Engineering
fitting lines including the critical-state specimen are listedas follows
EDU 022EI minus 004
EDUprime 020EI minus 001
⎧⎨
⎩
EDB 034EI minus 094
EDBprime 034EI minus 096
⎧⎨
⎩
(7)
It can be seen that the two fitted lines after adding thespecimen in the ruptured state almost coincide with theprevious fitted lines so the data of the specimen in theruptured state can be fitted with the data of other unrupturedand broken specimens As shown in Figure 10 the specimendata point in the ruptured state is approximately at theintersection of the two fitted lines
In addition for the dynamic Brazilian disc test becauseof the relatively small dynamic tensile stress it is difficult toobtain the specimen of the critical state so there is only onecritical-state specimen in this paper
5 Conclusions
In this paper a series of dynamic Brazilian disc tests of redsandstone specimens were carried out using the SHPB setup+e main conclusions can be described as follows
(1) In the dynamic Brazilian disc test the rock specimenhas three states after impact with different incidentenergy the unruptured state ruptured state andbroken state And the ruptured state is corre-sponding to the critical energy When the incidentenergy is less than the critical energy the specimenremains intact and there is no macrocrack on thesurface that is the unruptured state when the in-cident energy is greater than the critical energy thespecimen will be damaged and lose the bearingcapacity along the loading direction that is thebroken state
(2) Under the impact of different incident energy thedynamic peak stress and loading rate of red sand-stone specimens show a unified exponential functionrelationship However the failure states of specimensafter impact cannot be identified from thisrelationship
(3) +ere are linear relationships between the dissipatedenergy and incident energy in both the unrupturedstate and the broken state In the broken state thedissipated energy increases with the increase of in-cident energy and the increasing speed is faster thanthat of the unruptured specimen
(4) According to the loading process recorded with ahigh-speed camera the failure mode of the specimenis splitting along the loading direction When thespecimen is in the critical failure mode after theimpact only one crack is produced along the loadingdirection and it penetrates the whole specimen sothat the specimen is divided into two partsWhen thespecimen is in the broken state except the main
crack along the loading direction additional crackswill be generated at both ends of the specimenconnecting the incident and transmission rein-forcement and these cracks will also expand alongthe loading direction
(5) When the specimen is in the unruptured state afterimpact the ideal DTEDC is a constant value whichis 022 for the red sandstone specimen in this paperHowever when the specimen is in a broken stateafter impact the DTEDC increases with the increaseof incident energy and gradually approaches 034
Data Availability
No data were used to support this study
Conflicts of Interest
+e authors declare that they have no conflicts of interest
Acknowledgments
+is study was supported by the National Natural ScienceFoundation of China (Grant no 41877272)
References
[1] ISRM ldquoSuggested methods for determining tensile strength ofrock materialsrdquo International Journal of Rock Mechanics ampMining Sciences vol 15 no 1 pp 881ndash891 1978
[2] ASTM D3967-08 Standard Test Method for Splitting TensileStrength of Intact Rock Core Specimens ASTM InternationalWest Conshohocken PA USA 2008
[3] 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 pp 105ndash1122012
[4] F Q Gong L Zhang and S Y Wang ldquoLoading rate effect ofrock material with the direct tensile and three Brazilian disctestsrdquo Advances in Civil Engineering vol 2019 Article ID6260351 8 pages 2019
[5] F Q Gong X B Li and J Zhao ldquoAnalytical algorithm toestimate tensile modulus in Brazilian disk splitting testsrdquoChinese Journal of Rock Mechanics and Engineering vol 29no 5 pp 881ndash891 2010
[6] F Q Gong and G F Zhao ldquoDynamic indirect tensile strengthof sandstone under different loading ratesrdquo Rock Mechanicsand Rock Engineering vol 47 no 6 pp 2271ndash2278 2014
[7] Q B Zhang and J Zhao ldquoA review of dynamic experimentaltechniques andmechanical behaviour of rockmaterialsrdquo RockMechanics and Rock Engineering vol 47 no 4 pp 1411ndash14782014
[8] J B Zhu Z Y Liao and C A Tang ldquoNumerical SHPB tests ofrocks under combined static and dynamic loading conditionswith application to dynamic behavior of rocks under in situstressesrdquo Rock Mechanics and Rock Engineering vol 49no 10 pp 3935ndash3946 2016
[9] J C Li L F Rong H B Li and S N Hong ldquoAn SHPB teststudy on stress wave energy attenuation in jointed rockmassesrdquo Rock Mechanics and Rock Engineering vol 52 no 2pp 403ndash420 2019
Advances in Civil Engineering 9
[10] D Asprone E Cadoni A Prota and G Manfredi ldquoDynamicbehavior of a Mediterranean natural stone under tensileloadingrdquo International Journal of Rock Mechanics and MiningSciences vol 46 no 3 pp 514ndash520 2009
[11] Q ZWangW Li and H P Xie ldquoDynamic split tensile test offlattened Brazilian disc of rock with SHPB setuprdquo Mechanicsof Materials vol 41 no 3 pp 252ndash260 2009
[12] 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
[13] F Dai and K Xia ldquoLoading rate dependence of tensilestrength anisotropy of Barre graniterdquo Pure and AppliedGeophysics vol 167 no 11 pp 1419ndash1432 2010
[14] B Wu W Yao and K Xia ldquoAn experimental study of dy-namic tensile failure of rocks subjected to hydrostatic con-finementrdquo Rock Mechanics and Rock Engineering vol 49no 10 pp 3855ndash3864 2016
[15] H Xie L Li R Peng and Y Ju ldquoEnergy analysis and criteriafor structural failure of rocksrdquo Journal of Rock Mechanics andGeotechnical Engineering vol 1 no 1 pp 11ndash20 2009
[16] F Q Gong S Luo and J Y Yan ldquoEnergy storage and dis-sipation evolution process and characteristics of marble inthree tension-type failure testsrdquo Rock Mechanics and RockEngineering vol 51 no 11 pp 3613ndash3624 2018
[17] F Q Gong J Y Yan S Luo and X B Li ldquoInvestigation onthe linear energy storage and dissipation laws of rock ma-terials under uniaxial compressionrdquo RockMechanics and RockEngineering vol 52 no 11 pp 4237ndash4255 2019
[18] Y Chu H Sun and D Zhang ldquoExperimental study onevolution in the characteristics of permeability deformationand energy of coal containing gas under triaxial cyclicloading-unloadingrdquo Energy Science amp Engineering vol 7no 5 pp 2112ndash2123 2019
[19] B Lundberg ldquoA split Hopkinson bar study of energy ab-sorption in dynamic rock fragmentationrdquo InternationalJournal of Rock Mechanics and Mining Sciences amp Geo-mechanics Abstracts vol 13 no 6 pp 187ndash197 1976
[20] A V Mikhalyuk and V V Zakharov ldquoDissipation of dy-namic-loading energy in quasi-elastic deformation processesin rocksrdquo Journal of Applied Mechanics and Technical Physicsvol 38 no 2 pp 312ndash318 1997
[21] X B Li T S Lok and J Zhao ldquoDynamic characteristics ofgranite subjected to intermediate loading raterdquo Rock Me-chanics and Rock Engineering vol 38 no 1 pp 21ndash39 2005
[22] F Q Gong Experimental study of rock mechanical propertiesunder coupled static-dynamic loads and dynamic strengthcriterion PhD thesis Central South University ChangshaChina 2010
[23] 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
10 Advances in Civil Engineering
was 1 2) +e ends of the specimens were ground by abuffing machine to control the evenness and non-perpendicularity to be less than 002mm +e averagedensity and wave velocity of red sandstone are 2431 kgm3
and 3386ms
24 Preparation for Formal Test and Dynamic EquilibriumBefore the tests started a nonspecimen test was necessary toverify the stability and operability of the SHPB test system+e incident wave should be basically consistent with thetransmitted wave during a conventional uniaxial test which
IncRe
TraInc + Re
0
50 100 150 200 2500Time (μs)
ndash100
ndash80
ndash60
ndash40
ndash20
20
40
60
80
Stre
ss (M
Pa)
Figure 2 Stress equilibrium check for the dynamic Brazilian disc testing of a red sandstone specimen (lsquoIncrsquo is the incident stress lsquoRersquo is thereflected stress lsquoTrarsquo is the transmitted stress and lsquoInc +Rersquo is the superimposed stress at the end of the incident bar)
Table 1 Energy and mechanical data of red sandstone specimens
indicates that the stability of the test system is up to thestandard of formal tests
+e specimen was sandwiched between the incident barand the transmitted bar in the radial direction and lubricantwas applied at the contact point between the specimen andthe two bars to reduce friction In addition the validity of thetest results should be judged by whether the stress at bothends of the specimen reaches equilibrium before failure
Figure 2 shows the stress history of a typical dynamicBrazilian disc test in which the ends of the rock specimenreached the equilibrium state It is observed that the sum ofthe incident and reflected stress waves lsquoInc +Rersquo is basicallyconsistent with the transmitted stress wave lsquoTrarsquo +is de-notes that the test results are valid and reliable Otherwisethe results will be invalid and should be excluded In thispaper checks of the stress equilibrium were carried out on
each of the specimens to ensure the validity and accuracy ofthe test results
3 Results
+e results of dynamic Brazilian disc tests are listed in Table 1+e states of the specimens after impact can be divided intothree categories namely the unruptured state the rupturedstate and the broken state +e unruptured state refers to thespecimens which remained intact and have the ability to beara load after impact as shown in Figure 3(a)
+e ruptured state refers to the state in which thespecimen produced visible cracks after impact and separatedinto two blocks as shown in Figure 3(b) +e broken staterefers to the state in which the specimen was broken intomany rock blocks after impact as shown in Figure 3(c) It
14 16
(a)
11 11
(b)
4745
(c)
Figure 3 +ree failure states of red sandstone specimens after impact (a) unruptured state (b) ruptured state (c) broken state
4 Advances in Civil Engineering
can be concluded that there is an incident energy whichcorresponds to the state of the specimen that changes fromthe unruptured state to the ruptured state +is incidentenergy was defined as the critical incident energy When theincident energy was less than the critical incident energy thespecimens were in the unruptured state after impact andwhen the incident energy was greater than the critical incidentenergy the specimens were in the broken state after impact
It can be seen from Table 1 and Figure 3 that when theincident energy was 640 J the specimen was split into tworock blocks along the loading direction When the incidentenergy was at a high level and the specimen was brokenthere were triangular broken areas at both ends of thespecimenWith the augmentation of the incident energy thedegree of broken specimens became higher +is is becausebefore the main radial crack penetrated the compressivestresses at both ends of the specimen exceeded the tensilestrength of the rock
31 Mechanical Properties Five typical dynamic load-dis-placement curves are shown in Figure 4 the peak tensilestress of red sandstone specimens under different loadingrates is obtained as shown in Table 1 Figure 5 shows therelationship between the peak tensile stress and the loadingrate It can be found that there is an obvious loading rateeffect with the peak tensile stress of red sandstone and thelogarithmic function can be used to fit this loading effecttrend +is rate effect of peak stress has been confirmed inmany literature studies [4 6 7] and can be considered as oneof the dynamic tensile properties of rock
32 Energy Characteristics Based on the energy calculationprinciple of the SHPB dynamic Brazilian disc test the curvesof incident energy reflected energy transmitted energy anddissipated energy relative to time in the dynamic Braziliandisc testing of red sandstone specimens can be calculated Asshown in Figure 6 there are curves of four kinds of energyfor specimens 1 11 and 26 in three states over time It can beseen that the four kinds of energy all increased with theincrease of time and then no longer increased after ap-proximately 250 μs after which they remained constant
33FailureProcess of theSpecimen In dynamic loading testsfailure of the specimen is a quickly developing process Tostudy the failure processes of specimens effective technicalmeasurements are needed At present a high-speed camerais generally used to record the failure processes of specimensIn this study a high-speed camera was set at 72000 framesper second (fps) to take a photo every 1389 μs Figures 7(a)ndash7(c) shows the typical failure processes of red sandstonespecimens in the three states after impacting specimens 1411 and 47 Specimen 14 was in the unruptured state afterimpact remaining intact after the impact with no visiblecracks on the surface Specimen 11 was in the ruptures stateafter the impact A crack formed on the surface of thespecimen at 25002 μs starting at one end of the incident barand running through the whole specimen along the loading
direction +e specimen was separated into two rock blocksat 79173 μs Specimen 47 was broken under a larger dy-namic load +e specimen was penetrated by a main crackand two additional cracks were produced at the contactpoints between the specimen and the ends of the bars at25002 μs At 79173 μs the specimen was split into severalparts which were flying outward at certain speeds In ad-dition the main cracks of specimens 11 and 47 occurredalong the direction of the loading stress which was becausethe tensile stress of the rock plays a major role in thefracturing of specimens
34 Energy Relationship between the Dissipated Energy andIncident Energy It can be concluded from Table 1 that the
161411
4547
0
5
10
15
20
25
30
Load
(kN
)
01 02 03 04 05 06 07 0800Displacement (mm)
Figure 4 Typical dynamic load-displacement curves
(R2 = 092)
Unrupturted stateRuptured state
Broken stateFitted line of specimens inthree states
Figure 5 Relationship between loading rate and peak tensile stressof red sandstone specimens
Advances in Civil Engineering 5
dissipated energy of the specimens increased with theaugmentation of the incident energy and the states of thered sandstone specimens changed from unruptured state tothe ruptured state and then to the broken state
Figure 8 shows the relationship between the dissipatedenergy and the incident energy in the dynamic Brazilian disctesting of red sandstone specimens It can be seen that thedissipated energy increases approximately linearly with theincrease of incident energy and presenting two differentstages of energy relationship From Table 1 it can be foundthat this feature is related to the state of the specimen afterimpact so the linear fitting formulas were used to fit the datapoints of the unruptured and broken specimens respectively(to correct the fitting curve of the unruptured state the
origin coordinate was also added because when the incidentenergy is 0 J the dissipated energy must be 0 J) +e datapoints for the ruptured state are represented by differentshapes than those for the other two states
+e relationships between the dissipated energy andincident energy are both positively linear for the specimensin the unruptured state and broken state after the impact andcan be fitted by the following
EUD kUEI + cU EI ltECI( 1113857
EBD kBEI + cB EI gtECI( 1113857
⎧⎨
⎩ (4)
where EUD and EB
D are the dissipated energies of the specimenin the unruptured state and in the broken state kU and kB are
Specimen 26Specimen 11Specimen 1
0
5
10
15
20
25
30
35In
cide
nt en
ergy
(J)
50 100 150 200 250 3000Time (μs)
(a)
Specimen 26Specimen 11Specimen 1
0
5
10
15
20
25
Refle
cted
ener
gy (J
)
50 100 150 200 250 3000Time (μs)
(b)
Specimen 26Specimen 11Specimen 1
00
01
02
03
04
05
06
07
08
Tran
smitt
ed en
ergy
(J)
50 100 150 200 250 3000Time (μs)
(c)
Specimen 26Specimen 11Specimen 1
0
2
4
6
8
10D
issip
ated
ener
gy (J
)
50 100 150 200 250 3000Time (μs)
(d)
Figure 6 Energy-time curve of red sandstone specimens in the dynamic Brazilian disc test (a) incident energy-time (b) reflected energy-time (c) transmitted energy-time (d) dissipated energy-time
6 Advances in Civil Engineering
the slopes of the fitted curves of the unruptured state andbroken state EI is the incident energy cU and cB areconstants of the unruptured state and broken state and ECI
is the critical incident energy which corresponds to theruptured state of the specimen after impact For the redsandstone in this study two fitting functions can be ob-tained as follows
EUD
EI
022 minus004EI
EI lt 75 J( 1113857
EBD
EI
034 minus094EI
EI gt 75 J( 1113857
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(5)
For quantitatively describing the dynamic energy dis-sipation capacity of rock materials in the dynamic Braziliandisc test the dynamic tensile energy dissipation coefficient(DTEDC) is introduced as the ratio of dissipated energy toincident energy
In fact the DTEDCs are equal to the values of kU and kBIn formula (5) when the incident energy is 0 J the
dissipated energy must be 0 J However due to the influenceof heterogeneity of rock materials and experimental un-certainty the intercept of energy dissipation function in thestate of unruptured specimen is not 0 (shown in Figure 8)+erefore the ideal energy storage formula should be asfollows
14 14 14000μs 25002μs 79173μs
(a)
11 11 11000μs 25002μs 79173μs
(b)
47 47 47000μs 25002μs 79173μs
(c)
Figure 7 Failure processes of specimens in three states (a) unruptured state (b) ruptured state (c) broken state
Advances in Civil Engineering 7
EUD
EI
022 EI lt 75 J( 1113857
EBD
EI
034 minus094EI
EI gt 75 J( 1113857
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(6)
+e ideal curve obtained according to formula (6) wasshown in red line in Figure 9+eDTEDC of the specimen inthe unruptured state after impact is 022 When the incidentenergy exceeds 75 J the state of the specimen after impactwill develop from a ruptured state to a broken state and theDTEDC begins to increase It can be seen from Figure 9 thatthe DTEDC increases with the increase of incident energyand the velocity decreases gradually +e same conclusioncan be drawn from formula (6) and the DTEDC willgradually increase and approach to 034
4 Discussions
41 Loading Rate Effect As shown in Figure 5 there is noobvious difference among these three failure states +ere-fore from the relationship between the peak tensile stressand loading rate it is impossible to distinguish the differencein mechanical properties of the failure state of the specimenHowever from the point of view of energy as mentionedabove we find that there are two stages of linear energydissipation between these states
42 Failure State It can be seen from Figure 7 that when thespecimen is in the ruptured state that is specimen 11 thereis only one main crack that penetrates the whole specimenwith the specimen divided into two parts along the crack It
is noteworthy that this specimen produced not onlymicrocracks propagation but also macrocracks penetrationand the specimen is at the critical point at which the kineticenergy of the rock fragments is about to be generated
+e data of the ruptured specimen 11 has been fittedtogether with those from the unruptured and broken statesspecimens to create fitting lines and the two previousfitting lines were retained for comparison as shown inFigure 10 +e two previous fitting lines and the other two
Unruptured stateRuptured stateBroken state
Fitted line of unrupturedstateFitted line of broken state
EUD = 022EI ndash 004
(R2 = 091)
EBD = 034EI ndash 094
(R2 = 097)
0
2
4
6
8
10
12
Diss
ipat
ed en
ergy
(J)
5 10 15 20 25 30 350Incident energy (J)
Figure 8 Fitted lines of the dissipated energy-incident energy inthe ldquoUrdquo state and ldquoBrdquo state (ldquoUrdquo state is the unruptured state and ldquoBrdquostate is the broken state)
Ideal curve of DTEDCUnruptured state
Ruptured stateBroken state
01
02
03
04
Dyn
amic
tens
ile en
ergy
diss
ipat
ion
coef
ficie
nt (D
TED
C)
5 10 15 20 25 30 35 400Incident energy (J)
Figure 9 Relation of incident energy versus DTEDC
EprimeDU = 020EI ndash 001
(R2 = 089)
EprimeDB = 034EI ndash 096
(R2 = 097)EDU = 022EI ndash 004(R2 = 091)
EDB = 034EI ndash 094(R2 = 097)
Unruptured stateRuptured stateBroken stateFitted line of ldquoUrdquo state
Fitted line of ldquoBrdquo state
Fitted line of ldquoBrdquo stateamp ldquoRrdquo state
Fitted line of ldquoUrdquo stateamp ldquoRrdquo state
0
2
4
6
8
10
12
Diss
ipat
ed en
ergy
(J)
5 10 15 20 25 30 350Incident energy (J)
Figure 10 Fitted lines of the incident energy-dissipated energy inthe ldquoUrdquo state ldquoBrdquo state ldquoUrdquo state ldquoRrdquo state and ldquoBrdquo state amp ldquoRrdquostate (ldquoUrdquo state is the unruptured state ldquoBrdquo state is the broken stateand ldquoRrdquo state is the ruptured state)
8 Advances in Civil Engineering
fitting lines including the critical-state specimen are listedas follows
EDU 022EI minus 004
EDUprime 020EI minus 001
⎧⎨
⎩
EDB 034EI minus 094
EDBprime 034EI minus 096
⎧⎨
⎩
(7)
It can be seen that the two fitted lines after adding thespecimen in the ruptured state almost coincide with theprevious fitted lines so the data of the specimen in theruptured state can be fitted with the data of other unrupturedand broken specimens As shown in Figure 10 the specimendata point in the ruptured state is approximately at theintersection of the two fitted lines
In addition for the dynamic Brazilian disc test becauseof the relatively small dynamic tensile stress it is difficult toobtain the specimen of the critical state so there is only onecritical-state specimen in this paper
5 Conclusions
In this paper a series of dynamic Brazilian disc tests of redsandstone specimens were carried out using the SHPB setup+e main conclusions can be described as follows
(1) In the dynamic Brazilian disc test the rock specimenhas three states after impact with different incidentenergy the unruptured state ruptured state andbroken state And the ruptured state is corre-sponding to the critical energy When the incidentenergy is less than the critical energy the specimenremains intact and there is no macrocrack on thesurface that is the unruptured state when the in-cident energy is greater than the critical energy thespecimen will be damaged and lose the bearingcapacity along the loading direction that is thebroken state
(2) Under the impact of different incident energy thedynamic peak stress and loading rate of red sand-stone specimens show a unified exponential functionrelationship However the failure states of specimensafter impact cannot be identified from thisrelationship
(3) +ere are linear relationships between the dissipatedenergy and incident energy in both the unrupturedstate and the broken state In the broken state thedissipated energy increases with the increase of in-cident energy and the increasing speed is faster thanthat of the unruptured specimen
(4) According to the loading process recorded with ahigh-speed camera the failure mode of the specimenis splitting along the loading direction When thespecimen is in the critical failure mode after theimpact only one crack is produced along the loadingdirection and it penetrates the whole specimen sothat the specimen is divided into two partsWhen thespecimen is in the broken state except the main
crack along the loading direction additional crackswill be generated at both ends of the specimenconnecting the incident and transmission rein-forcement and these cracks will also expand alongthe loading direction
(5) When the specimen is in the unruptured state afterimpact the ideal DTEDC is a constant value whichis 022 for the red sandstone specimen in this paperHowever when the specimen is in a broken stateafter impact the DTEDC increases with the increaseof incident energy and gradually approaches 034
Data Availability
No data were used to support this study
Conflicts of Interest
+e authors declare that they have no conflicts of interest
Acknowledgments
+is study was supported by the National Natural ScienceFoundation of China (Grant no 41877272)
References
[1] ISRM ldquoSuggested methods for determining tensile strength ofrock materialsrdquo International Journal of Rock Mechanics ampMining Sciences vol 15 no 1 pp 881ndash891 1978
[2] ASTM D3967-08 Standard Test Method for Splitting TensileStrength of Intact Rock Core Specimens ASTM InternationalWest Conshohocken PA USA 2008
[3] 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 pp 105ndash1122012
[4] F Q Gong L Zhang and S Y Wang ldquoLoading rate effect ofrock material with the direct tensile and three Brazilian disctestsrdquo Advances in Civil Engineering vol 2019 Article ID6260351 8 pages 2019
[5] F Q Gong X B Li and J Zhao ldquoAnalytical algorithm toestimate tensile modulus in Brazilian disk splitting testsrdquoChinese Journal of Rock Mechanics and Engineering vol 29no 5 pp 881ndash891 2010
[6] F Q Gong and G F Zhao ldquoDynamic indirect tensile strengthof sandstone under different loading ratesrdquo Rock Mechanicsand Rock Engineering vol 47 no 6 pp 2271ndash2278 2014
[7] Q B Zhang and J Zhao ldquoA review of dynamic experimentaltechniques andmechanical behaviour of rockmaterialsrdquo RockMechanics and Rock Engineering vol 47 no 4 pp 1411ndash14782014
[8] J B Zhu Z Y Liao and C A Tang ldquoNumerical SHPB tests ofrocks under combined static and dynamic loading conditionswith application to dynamic behavior of rocks under in situstressesrdquo Rock Mechanics and Rock Engineering vol 49no 10 pp 3935ndash3946 2016
[9] J C Li L F Rong H B Li and S N Hong ldquoAn SHPB teststudy on stress wave energy attenuation in jointed rockmassesrdquo Rock Mechanics and Rock Engineering vol 52 no 2pp 403ndash420 2019
Advances in Civil Engineering 9
[10] D Asprone E Cadoni A Prota and G Manfredi ldquoDynamicbehavior of a Mediterranean natural stone under tensileloadingrdquo International Journal of Rock Mechanics and MiningSciences vol 46 no 3 pp 514ndash520 2009
[11] Q ZWangW Li and H P Xie ldquoDynamic split tensile test offlattened Brazilian disc of rock with SHPB setuprdquo Mechanicsof Materials vol 41 no 3 pp 252ndash260 2009
[12] 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
[13] F Dai and K Xia ldquoLoading rate dependence of tensilestrength anisotropy of Barre graniterdquo Pure and AppliedGeophysics vol 167 no 11 pp 1419ndash1432 2010
[14] B Wu W Yao and K Xia ldquoAn experimental study of dy-namic tensile failure of rocks subjected to hydrostatic con-finementrdquo Rock Mechanics and Rock Engineering vol 49no 10 pp 3855ndash3864 2016
[15] H Xie L Li R Peng and Y Ju ldquoEnergy analysis and criteriafor structural failure of rocksrdquo Journal of Rock Mechanics andGeotechnical Engineering vol 1 no 1 pp 11ndash20 2009
[16] F Q Gong S Luo and J Y Yan ldquoEnergy storage and dis-sipation evolution process and characteristics of marble inthree tension-type failure testsrdquo Rock Mechanics and RockEngineering vol 51 no 11 pp 3613ndash3624 2018
[17] F Q Gong J Y Yan S Luo and X B Li ldquoInvestigation onthe linear energy storage and dissipation laws of rock ma-terials under uniaxial compressionrdquo RockMechanics and RockEngineering vol 52 no 11 pp 4237ndash4255 2019
[18] Y Chu H Sun and D Zhang ldquoExperimental study onevolution in the characteristics of permeability deformationand energy of coal containing gas under triaxial cyclicloading-unloadingrdquo Energy Science amp Engineering vol 7no 5 pp 2112ndash2123 2019
[19] B Lundberg ldquoA split Hopkinson bar study of energy ab-sorption in dynamic rock fragmentationrdquo InternationalJournal of Rock Mechanics and Mining Sciences amp Geo-mechanics Abstracts vol 13 no 6 pp 187ndash197 1976
[20] A V Mikhalyuk and V V Zakharov ldquoDissipation of dy-namic-loading energy in quasi-elastic deformation processesin rocksrdquo Journal of Applied Mechanics and Technical Physicsvol 38 no 2 pp 312ndash318 1997
[21] X B Li T S Lok and J Zhao ldquoDynamic characteristics ofgranite subjected to intermediate loading raterdquo Rock Me-chanics and Rock Engineering vol 38 no 1 pp 21ndash39 2005
[22] F Q Gong Experimental study of rock mechanical propertiesunder coupled static-dynamic loads and dynamic strengthcriterion PhD thesis Central South University ChangshaChina 2010
[23] 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
10 Advances in Civil Engineering
indicates that the stability of the test system is up to thestandard of formal tests
+e specimen was sandwiched between the incident barand the transmitted bar in the radial direction and lubricantwas applied at the contact point between the specimen andthe two bars to reduce friction In addition the validity of thetest results should be judged by whether the stress at bothends of the specimen reaches equilibrium before failure
Figure 2 shows the stress history of a typical dynamicBrazilian disc test in which the ends of the rock specimenreached the equilibrium state It is observed that the sum ofthe incident and reflected stress waves lsquoInc +Rersquo is basicallyconsistent with the transmitted stress wave lsquoTrarsquo +is de-notes that the test results are valid and reliable Otherwisethe results will be invalid and should be excluded In thispaper checks of the stress equilibrium were carried out on
each of the specimens to ensure the validity and accuracy ofthe test results
3 Results
+e results of dynamic Brazilian disc tests are listed in Table 1+e states of the specimens after impact can be divided intothree categories namely the unruptured state the rupturedstate and the broken state +e unruptured state refers to thespecimens which remained intact and have the ability to beara load after impact as shown in Figure 3(a)
+e ruptured state refers to the state in which thespecimen produced visible cracks after impact and separatedinto two blocks as shown in Figure 3(b) +e broken staterefers to the state in which the specimen was broken intomany rock blocks after impact as shown in Figure 3(c) It
14 16
(a)
11 11
(b)
4745
(c)
Figure 3 +ree failure states of red sandstone specimens after impact (a) unruptured state (b) ruptured state (c) broken state
4 Advances in Civil Engineering
can be concluded that there is an incident energy whichcorresponds to the state of the specimen that changes fromthe unruptured state to the ruptured state +is incidentenergy was defined as the critical incident energy When theincident energy was less than the critical incident energy thespecimens were in the unruptured state after impact andwhen the incident energy was greater than the critical incidentenergy the specimens were in the broken state after impact
It can be seen from Table 1 and Figure 3 that when theincident energy was 640 J the specimen was split into tworock blocks along the loading direction When the incidentenergy was at a high level and the specimen was brokenthere were triangular broken areas at both ends of thespecimenWith the augmentation of the incident energy thedegree of broken specimens became higher +is is becausebefore the main radial crack penetrated the compressivestresses at both ends of the specimen exceeded the tensilestrength of the rock
31 Mechanical Properties Five typical dynamic load-dis-placement curves are shown in Figure 4 the peak tensilestress of red sandstone specimens under different loadingrates is obtained as shown in Table 1 Figure 5 shows therelationship between the peak tensile stress and the loadingrate It can be found that there is an obvious loading rateeffect with the peak tensile stress of red sandstone and thelogarithmic function can be used to fit this loading effecttrend +is rate effect of peak stress has been confirmed inmany literature studies [4 6 7] and can be considered as oneof the dynamic tensile properties of rock
32 Energy Characteristics Based on the energy calculationprinciple of the SHPB dynamic Brazilian disc test the curvesof incident energy reflected energy transmitted energy anddissipated energy relative to time in the dynamic Braziliandisc testing of red sandstone specimens can be calculated Asshown in Figure 6 there are curves of four kinds of energyfor specimens 1 11 and 26 in three states over time It can beseen that the four kinds of energy all increased with theincrease of time and then no longer increased after ap-proximately 250 μs after which they remained constant
33FailureProcess of theSpecimen In dynamic loading testsfailure of the specimen is a quickly developing process Tostudy the failure processes of specimens effective technicalmeasurements are needed At present a high-speed camerais generally used to record the failure processes of specimensIn this study a high-speed camera was set at 72000 framesper second (fps) to take a photo every 1389 μs Figures 7(a)ndash7(c) shows the typical failure processes of red sandstonespecimens in the three states after impacting specimens 1411 and 47 Specimen 14 was in the unruptured state afterimpact remaining intact after the impact with no visiblecracks on the surface Specimen 11 was in the ruptures stateafter the impact A crack formed on the surface of thespecimen at 25002 μs starting at one end of the incident barand running through the whole specimen along the loading
direction +e specimen was separated into two rock blocksat 79173 μs Specimen 47 was broken under a larger dy-namic load +e specimen was penetrated by a main crackand two additional cracks were produced at the contactpoints between the specimen and the ends of the bars at25002 μs At 79173 μs the specimen was split into severalparts which were flying outward at certain speeds In ad-dition the main cracks of specimens 11 and 47 occurredalong the direction of the loading stress which was becausethe tensile stress of the rock plays a major role in thefracturing of specimens
34 Energy Relationship between the Dissipated Energy andIncident Energy It can be concluded from Table 1 that the
161411
4547
0
5
10
15
20
25
30
Load
(kN
)
01 02 03 04 05 06 07 0800Displacement (mm)
Figure 4 Typical dynamic load-displacement curves
(R2 = 092)
Unrupturted stateRuptured state
Broken stateFitted line of specimens inthree states
Figure 5 Relationship between loading rate and peak tensile stressof red sandstone specimens
Advances in Civil Engineering 5
dissipated energy of the specimens increased with theaugmentation of the incident energy and the states of thered sandstone specimens changed from unruptured state tothe ruptured state and then to the broken state
Figure 8 shows the relationship between the dissipatedenergy and the incident energy in the dynamic Brazilian disctesting of red sandstone specimens It can be seen that thedissipated energy increases approximately linearly with theincrease of incident energy and presenting two differentstages of energy relationship From Table 1 it can be foundthat this feature is related to the state of the specimen afterimpact so the linear fitting formulas were used to fit the datapoints of the unruptured and broken specimens respectively(to correct the fitting curve of the unruptured state the
origin coordinate was also added because when the incidentenergy is 0 J the dissipated energy must be 0 J) +e datapoints for the ruptured state are represented by differentshapes than those for the other two states
+e relationships between the dissipated energy andincident energy are both positively linear for the specimensin the unruptured state and broken state after the impact andcan be fitted by the following
EUD kUEI + cU EI ltECI( 1113857
EBD kBEI + cB EI gtECI( 1113857
⎧⎨
⎩ (4)
where EUD and EB
D are the dissipated energies of the specimenin the unruptured state and in the broken state kU and kB are
Specimen 26Specimen 11Specimen 1
0
5
10
15
20
25
30
35In
cide
nt en
ergy
(J)
50 100 150 200 250 3000Time (μs)
(a)
Specimen 26Specimen 11Specimen 1
0
5
10
15
20
25
Refle
cted
ener
gy (J
)
50 100 150 200 250 3000Time (μs)
(b)
Specimen 26Specimen 11Specimen 1
00
01
02
03
04
05
06
07
08
Tran
smitt
ed en
ergy
(J)
50 100 150 200 250 3000Time (μs)
(c)
Specimen 26Specimen 11Specimen 1
0
2
4
6
8
10D
issip
ated
ener
gy (J
)
50 100 150 200 250 3000Time (μs)
(d)
Figure 6 Energy-time curve of red sandstone specimens in the dynamic Brazilian disc test (a) incident energy-time (b) reflected energy-time (c) transmitted energy-time (d) dissipated energy-time
6 Advances in Civil Engineering
the slopes of the fitted curves of the unruptured state andbroken state EI is the incident energy cU and cB areconstants of the unruptured state and broken state and ECI
is the critical incident energy which corresponds to theruptured state of the specimen after impact For the redsandstone in this study two fitting functions can be ob-tained as follows
EUD
EI
022 minus004EI
EI lt 75 J( 1113857
EBD
EI
034 minus094EI
EI gt 75 J( 1113857
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(5)
For quantitatively describing the dynamic energy dis-sipation capacity of rock materials in the dynamic Braziliandisc test the dynamic tensile energy dissipation coefficient(DTEDC) is introduced as the ratio of dissipated energy toincident energy
In fact the DTEDCs are equal to the values of kU and kBIn formula (5) when the incident energy is 0 J the
dissipated energy must be 0 J However due to the influenceof heterogeneity of rock materials and experimental un-certainty the intercept of energy dissipation function in thestate of unruptured specimen is not 0 (shown in Figure 8)+erefore the ideal energy storage formula should be asfollows
14 14 14000μs 25002μs 79173μs
(a)
11 11 11000μs 25002μs 79173μs
(b)
47 47 47000μs 25002μs 79173μs
(c)
Figure 7 Failure processes of specimens in three states (a) unruptured state (b) ruptured state (c) broken state
Advances in Civil Engineering 7
EUD
EI
022 EI lt 75 J( 1113857
EBD
EI
034 minus094EI
EI gt 75 J( 1113857
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(6)
+e ideal curve obtained according to formula (6) wasshown in red line in Figure 9+eDTEDC of the specimen inthe unruptured state after impact is 022 When the incidentenergy exceeds 75 J the state of the specimen after impactwill develop from a ruptured state to a broken state and theDTEDC begins to increase It can be seen from Figure 9 thatthe DTEDC increases with the increase of incident energyand the velocity decreases gradually +e same conclusioncan be drawn from formula (6) and the DTEDC willgradually increase and approach to 034
4 Discussions
41 Loading Rate Effect As shown in Figure 5 there is noobvious difference among these three failure states +ere-fore from the relationship between the peak tensile stressand loading rate it is impossible to distinguish the differencein mechanical properties of the failure state of the specimenHowever from the point of view of energy as mentionedabove we find that there are two stages of linear energydissipation between these states
42 Failure State It can be seen from Figure 7 that when thespecimen is in the ruptured state that is specimen 11 thereis only one main crack that penetrates the whole specimenwith the specimen divided into two parts along the crack It
is noteworthy that this specimen produced not onlymicrocracks propagation but also macrocracks penetrationand the specimen is at the critical point at which the kineticenergy of the rock fragments is about to be generated
+e data of the ruptured specimen 11 has been fittedtogether with those from the unruptured and broken statesspecimens to create fitting lines and the two previousfitting lines were retained for comparison as shown inFigure 10 +e two previous fitting lines and the other two
Unruptured stateRuptured stateBroken state
Fitted line of unrupturedstateFitted line of broken state
EUD = 022EI ndash 004
(R2 = 091)
EBD = 034EI ndash 094
(R2 = 097)
0
2
4
6
8
10
12
Diss
ipat
ed en
ergy
(J)
5 10 15 20 25 30 350Incident energy (J)
Figure 8 Fitted lines of the dissipated energy-incident energy inthe ldquoUrdquo state and ldquoBrdquo state (ldquoUrdquo state is the unruptured state and ldquoBrdquostate is the broken state)
Ideal curve of DTEDCUnruptured state
Ruptured stateBroken state
01
02
03
04
Dyn
amic
tens
ile en
ergy
diss
ipat
ion
coef
ficie
nt (D
TED
C)
5 10 15 20 25 30 35 400Incident energy (J)
Figure 9 Relation of incident energy versus DTEDC
EprimeDU = 020EI ndash 001
(R2 = 089)
EprimeDB = 034EI ndash 096
(R2 = 097)EDU = 022EI ndash 004(R2 = 091)
EDB = 034EI ndash 094(R2 = 097)
Unruptured stateRuptured stateBroken stateFitted line of ldquoUrdquo state
Fitted line of ldquoBrdquo state
Fitted line of ldquoBrdquo stateamp ldquoRrdquo state
Fitted line of ldquoUrdquo stateamp ldquoRrdquo state
0
2
4
6
8
10
12
Diss
ipat
ed en
ergy
(J)
5 10 15 20 25 30 350Incident energy (J)
Figure 10 Fitted lines of the incident energy-dissipated energy inthe ldquoUrdquo state ldquoBrdquo state ldquoUrdquo state ldquoRrdquo state and ldquoBrdquo state amp ldquoRrdquostate (ldquoUrdquo state is the unruptured state ldquoBrdquo state is the broken stateand ldquoRrdquo state is the ruptured state)
8 Advances in Civil Engineering
fitting lines including the critical-state specimen are listedas follows
EDU 022EI minus 004
EDUprime 020EI minus 001
⎧⎨
⎩
EDB 034EI minus 094
EDBprime 034EI minus 096
⎧⎨
⎩
(7)
It can be seen that the two fitted lines after adding thespecimen in the ruptured state almost coincide with theprevious fitted lines so the data of the specimen in theruptured state can be fitted with the data of other unrupturedand broken specimens As shown in Figure 10 the specimendata point in the ruptured state is approximately at theintersection of the two fitted lines
In addition for the dynamic Brazilian disc test becauseof the relatively small dynamic tensile stress it is difficult toobtain the specimen of the critical state so there is only onecritical-state specimen in this paper
5 Conclusions
In this paper a series of dynamic Brazilian disc tests of redsandstone specimens were carried out using the SHPB setup+e main conclusions can be described as follows
(1) In the dynamic Brazilian disc test the rock specimenhas three states after impact with different incidentenergy the unruptured state ruptured state andbroken state And the ruptured state is corre-sponding to the critical energy When the incidentenergy is less than the critical energy the specimenremains intact and there is no macrocrack on thesurface that is the unruptured state when the in-cident energy is greater than the critical energy thespecimen will be damaged and lose the bearingcapacity along the loading direction that is thebroken state
(2) Under the impact of different incident energy thedynamic peak stress and loading rate of red sand-stone specimens show a unified exponential functionrelationship However the failure states of specimensafter impact cannot be identified from thisrelationship
(3) +ere are linear relationships between the dissipatedenergy and incident energy in both the unrupturedstate and the broken state In the broken state thedissipated energy increases with the increase of in-cident energy and the increasing speed is faster thanthat of the unruptured specimen
(4) According to the loading process recorded with ahigh-speed camera the failure mode of the specimenis splitting along the loading direction When thespecimen is in the critical failure mode after theimpact only one crack is produced along the loadingdirection and it penetrates the whole specimen sothat the specimen is divided into two partsWhen thespecimen is in the broken state except the main
crack along the loading direction additional crackswill be generated at both ends of the specimenconnecting the incident and transmission rein-forcement and these cracks will also expand alongthe loading direction
(5) When the specimen is in the unruptured state afterimpact the ideal DTEDC is a constant value whichis 022 for the red sandstone specimen in this paperHowever when the specimen is in a broken stateafter impact the DTEDC increases with the increaseof incident energy and gradually approaches 034
Data Availability
No data were used to support this study
Conflicts of Interest
+e authors declare that they have no conflicts of interest
Acknowledgments
+is study was supported by the National Natural ScienceFoundation of China (Grant no 41877272)
References
[1] ISRM ldquoSuggested methods for determining tensile strength ofrock materialsrdquo International Journal of Rock Mechanics ampMining Sciences vol 15 no 1 pp 881ndash891 1978
[2] ASTM D3967-08 Standard Test Method for Splitting TensileStrength of Intact Rock Core Specimens ASTM InternationalWest Conshohocken PA USA 2008
[3] 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 pp 105ndash1122012
[4] F Q Gong L Zhang and S Y Wang ldquoLoading rate effect ofrock material with the direct tensile and three Brazilian disctestsrdquo Advances in Civil Engineering vol 2019 Article ID6260351 8 pages 2019
[5] F Q Gong X B Li and J Zhao ldquoAnalytical algorithm toestimate tensile modulus in Brazilian disk splitting testsrdquoChinese Journal of Rock Mechanics and Engineering vol 29no 5 pp 881ndash891 2010
[6] F Q Gong and G F Zhao ldquoDynamic indirect tensile strengthof sandstone under different loading ratesrdquo Rock Mechanicsand Rock Engineering vol 47 no 6 pp 2271ndash2278 2014
[7] Q B Zhang and J Zhao ldquoA review of dynamic experimentaltechniques andmechanical behaviour of rockmaterialsrdquo RockMechanics and Rock Engineering vol 47 no 4 pp 1411ndash14782014
[8] J B Zhu Z Y Liao and C A Tang ldquoNumerical SHPB tests ofrocks under combined static and dynamic loading conditionswith application to dynamic behavior of rocks under in situstressesrdquo Rock Mechanics and Rock Engineering vol 49no 10 pp 3935ndash3946 2016
[9] J C Li L F Rong H B Li and S N Hong ldquoAn SHPB teststudy on stress wave energy attenuation in jointed rockmassesrdquo Rock Mechanics and Rock Engineering vol 52 no 2pp 403ndash420 2019
Advances in Civil Engineering 9
[10] D Asprone E Cadoni A Prota and G Manfredi ldquoDynamicbehavior of a Mediterranean natural stone under tensileloadingrdquo International Journal of Rock Mechanics and MiningSciences vol 46 no 3 pp 514ndash520 2009
[11] Q ZWangW Li and H P Xie ldquoDynamic split tensile test offlattened Brazilian disc of rock with SHPB setuprdquo Mechanicsof Materials vol 41 no 3 pp 252ndash260 2009
[12] 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
[13] F Dai and K Xia ldquoLoading rate dependence of tensilestrength anisotropy of Barre graniterdquo Pure and AppliedGeophysics vol 167 no 11 pp 1419ndash1432 2010
[14] B Wu W Yao and K Xia ldquoAn experimental study of dy-namic tensile failure of rocks subjected to hydrostatic con-finementrdquo Rock Mechanics and Rock Engineering vol 49no 10 pp 3855ndash3864 2016
[15] H Xie L Li R Peng and Y Ju ldquoEnergy analysis and criteriafor structural failure of rocksrdquo Journal of Rock Mechanics andGeotechnical Engineering vol 1 no 1 pp 11ndash20 2009
[16] F Q Gong S Luo and J Y Yan ldquoEnergy storage and dis-sipation evolution process and characteristics of marble inthree tension-type failure testsrdquo Rock Mechanics and RockEngineering vol 51 no 11 pp 3613ndash3624 2018
[17] F Q Gong J Y Yan S Luo and X B Li ldquoInvestigation onthe linear energy storage and dissipation laws of rock ma-terials under uniaxial compressionrdquo RockMechanics and RockEngineering vol 52 no 11 pp 4237ndash4255 2019
[18] Y Chu H Sun and D Zhang ldquoExperimental study onevolution in the characteristics of permeability deformationand energy of coal containing gas under triaxial cyclicloading-unloadingrdquo Energy Science amp Engineering vol 7no 5 pp 2112ndash2123 2019
[19] B Lundberg ldquoA split Hopkinson bar study of energy ab-sorption in dynamic rock fragmentationrdquo InternationalJournal of Rock Mechanics and Mining Sciences amp Geo-mechanics Abstracts vol 13 no 6 pp 187ndash197 1976
[20] A V Mikhalyuk and V V Zakharov ldquoDissipation of dy-namic-loading energy in quasi-elastic deformation processesin rocksrdquo Journal of Applied Mechanics and Technical Physicsvol 38 no 2 pp 312ndash318 1997
[21] X B Li T S Lok and J Zhao ldquoDynamic characteristics ofgranite subjected to intermediate loading raterdquo Rock Me-chanics and Rock Engineering vol 38 no 1 pp 21ndash39 2005
[22] F Q Gong Experimental study of rock mechanical propertiesunder coupled static-dynamic loads and dynamic strengthcriterion PhD thesis Central South University ChangshaChina 2010
[23] 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
10 Advances in Civil Engineering
can be concluded that there is an incident energy whichcorresponds to the state of the specimen that changes fromthe unruptured state to the ruptured state +is incidentenergy was defined as the critical incident energy When theincident energy was less than the critical incident energy thespecimens were in the unruptured state after impact andwhen the incident energy was greater than the critical incidentenergy the specimens were in the broken state after impact
It can be seen from Table 1 and Figure 3 that when theincident energy was 640 J the specimen was split into tworock blocks along the loading direction When the incidentenergy was at a high level and the specimen was brokenthere were triangular broken areas at both ends of thespecimenWith the augmentation of the incident energy thedegree of broken specimens became higher +is is becausebefore the main radial crack penetrated the compressivestresses at both ends of the specimen exceeded the tensilestrength of the rock
31 Mechanical Properties Five typical dynamic load-dis-placement curves are shown in Figure 4 the peak tensilestress of red sandstone specimens under different loadingrates is obtained as shown in Table 1 Figure 5 shows therelationship between the peak tensile stress and the loadingrate It can be found that there is an obvious loading rateeffect with the peak tensile stress of red sandstone and thelogarithmic function can be used to fit this loading effecttrend +is rate effect of peak stress has been confirmed inmany literature studies [4 6 7] and can be considered as oneof the dynamic tensile properties of rock
32 Energy Characteristics Based on the energy calculationprinciple of the SHPB dynamic Brazilian disc test the curvesof incident energy reflected energy transmitted energy anddissipated energy relative to time in the dynamic Braziliandisc testing of red sandstone specimens can be calculated Asshown in Figure 6 there are curves of four kinds of energyfor specimens 1 11 and 26 in three states over time It can beseen that the four kinds of energy all increased with theincrease of time and then no longer increased after ap-proximately 250 μs after which they remained constant
33FailureProcess of theSpecimen In dynamic loading testsfailure of the specimen is a quickly developing process Tostudy the failure processes of specimens effective technicalmeasurements are needed At present a high-speed camerais generally used to record the failure processes of specimensIn this study a high-speed camera was set at 72000 framesper second (fps) to take a photo every 1389 μs Figures 7(a)ndash7(c) shows the typical failure processes of red sandstonespecimens in the three states after impacting specimens 1411 and 47 Specimen 14 was in the unruptured state afterimpact remaining intact after the impact with no visiblecracks on the surface Specimen 11 was in the ruptures stateafter the impact A crack formed on the surface of thespecimen at 25002 μs starting at one end of the incident barand running through the whole specimen along the loading
direction +e specimen was separated into two rock blocksat 79173 μs Specimen 47 was broken under a larger dy-namic load +e specimen was penetrated by a main crackand two additional cracks were produced at the contactpoints between the specimen and the ends of the bars at25002 μs At 79173 μs the specimen was split into severalparts which were flying outward at certain speeds In ad-dition the main cracks of specimens 11 and 47 occurredalong the direction of the loading stress which was becausethe tensile stress of the rock plays a major role in thefracturing of specimens
34 Energy Relationship between the Dissipated Energy andIncident Energy It can be concluded from Table 1 that the
161411
4547
0
5
10
15
20
25
30
Load
(kN
)
01 02 03 04 05 06 07 0800Displacement (mm)
Figure 4 Typical dynamic load-displacement curves
(R2 = 092)
Unrupturted stateRuptured state
Broken stateFitted line of specimens inthree states
Figure 5 Relationship between loading rate and peak tensile stressof red sandstone specimens
Advances in Civil Engineering 5
dissipated energy of the specimens increased with theaugmentation of the incident energy and the states of thered sandstone specimens changed from unruptured state tothe ruptured state and then to the broken state
Figure 8 shows the relationship between the dissipatedenergy and the incident energy in the dynamic Brazilian disctesting of red sandstone specimens It can be seen that thedissipated energy increases approximately linearly with theincrease of incident energy and presenting two differentstages of energy relationship From Table 1 it can be foundthat this feature is related to the state of the specimen afterimpact so the linear fitting formulas were used to fit the datapoints of the unruptured and broken specimens respectively(to correct the fitting curve of the unruptured state the
origin coordinate was also added because when the incidentenergy is 0 J the dissipated energy must be 0 J) +e datapoints for the ruptured state are represented by differentshapes than those for the other two states
+e relationships between the dissipated energy andincident energy are both positively linear for the specimensin the unruptured state and broken state after the impact andcan be fitted by the following
EUD kUEI + cU EI ltECI( 1113857
EBD kBEI + cB EI gtECI( 1113857
⎧⎨
⎩ (4)
where EUD and EB
D are the dissipated energies of the specimenin the unruptured state and in the broken state kU and kB are
Specimen 26Specimen 11Specimen 1
0
5
10
15
20
25
30
35In
cide
nt en
ergy
(J)
50 100 150 200 250 3000Time (μs)
(a)
Specimen 26Specimen 11Specimen 1
0
5
10
15
20
25
Refle
cted
ener
gy (J
)
50 100 150 200 250 3000Time (μs)
(b)
Specimen 26Specimen 11Specimen 1
00
01
02
03
04
05
06
07
08
Tran
smitt
ed en
ergy
(J)
50 100 150 200 250 3000Time (μs)
(c)
Specimen 26Specimen 11Specimen 1
0
2
4
6
8
10D
issip
ated
ener
gy (J
)
50 100 150 200 250 3000Time (μs)
(d)
Figure 6 Energy-time curve of red sandstone specimens in the dynamic Brazilian disc test (a) incident energy-time (b) reflected energy-time (c) transmitted energy-time (d) dissipated energy-time
6 Advances in Civil Engineering
the slopes of the fitted curves of the unruptured state andbroken state EI is the incident energy cU and cB areconstants of the unruptured state and broken state and ECI
is the critical incident energy which corresponds to theruptured state of the specimen after impact For the redsandstone in this study two fitting functions can be ob-tained as follows
EUD
EI
022 minus004EI
EI lt 75 J( 1113857
EBD
EI
034 minus094EI
EI gt 75 J( 1113857
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(5)
For quantitatively describing the dynamic energy dis-sipation capacity of rock materials in the dynamic Braziliandisc test the dynamic tensile energy dissipation coefficient(DTEDC) is introduced as the ratio of dissipated energy toincident energy
In fact the DTEDCs are equal to the values of kU and kBIn formula (5) when the incident energy is 0 J the
dissipated energy must be 0 J However due to the influenceof heterogeneity of rock materials and experimental un-certainty the intercept of energy dissipation function in thestate of unruptured specimen is not 0 (shown in Figure 8)+erefore the ideal energy storage formula should be asfollows
14 14 14000μs 25002μs 79173μs
(a)
11 11 11000μs 25002μs 79173μs
(b)
47 47 47000μs 25002μs 79173μs
(c)
Figure 7 Failure processes of specimens in three states (a) unruptured state (b) ruptured state (c) broken state
Advances in Civil Engineering 7
EUD
EI
022 EI lt 75 J( 1113857
EBD
EI
034 minus094EI
EI gt 75 J( 1113857
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(6)
+e ideal curve obtained according to formula (6) wasshown in red line in Figure 9+eDTEDC of the specimen inthe unruptured state after impact is 022 When the incidentenergy exceeds 75 J the state of the specimen after impactwill develop from a ruptured state to a broken state and theDTEDC begins to increase It can be seen from Figure 9 thatthe DTEDC increases with the increase of incident energyand the velocity decreases gradually +e same conclusioncan be drawn from formula (6) and the DTEDC willgradually increase and approach to 034
4 Discussions
41 Loading Rate Effect As shown in Figure 5 there is noobvious difference among these three failure states +ere-fore from the relationship between the peak tensile stressand loading rate it is impossible to distinguish the differencein mechanical properties of the failure state of the specimenHowever from the point of view of energy as mentionedabove we find that there are two stages of linear energydissipation between these states
42 Failure State It can be seen from Figure 7 that when thespecimen is in the ruptured state that is specimen 11 thereis only one main crack that penetrates the whole specimenwith the specimen divided into two parts along the crack It
is noteworthy that this specimen produced not onlymicrocracks propagation but also macrocracks penetrationand the specimen is at the critical point at which the kineticenergy of the rock fragments is about to be generated
+e data of the ruptured specimen 11 has been fittedtogether with those from the unruptured and broken statesspecimens to create fitting lines and the two previousfitting lines were retained for comparison as shown inFigure 10 +e two previous fitting lines and the other two
Unruptured stateRuptured stateBroken state
Fitted line of unrupturedstateFitted line of broken state
EUD = 022EI ndash 004
(R2 = 091)
EBD = 034EI ndash 094
(R2 = 097)
0
2
4
6
8
10
12
Diss
ipat
ed en
ergy
(J)
5 10 15 20 25 30 350Incident energy (J)
Figure 8 Fitted lines of the dissipated energy-incident energy inthe ldquoUrdquo state and ldquoBrdquo state (ldquoUrdquo state is the unruptured state and ldquoBrdquostate is the broken state)
Ideal curve of DTEDCUnruptured state
Ruptured stateBroken state
01
02
03
04
Dyn
amic
tens
ile en
ergy
diss
ipat
ion
coef
ficie
nt (D
TED
C)
5 10 15 20 25 30 35 400Incident energy (J)
Figure 9 Relation of incident energy versus DTEDC
EprimeDU = 020EI ndash 001
(R2 = 089)
EprimeDB = 034EI ndash 096
(R2 = 097)EDU = 022EI ndash 004(R2 = 091)
EDB = 034EI ndash 094(R2 = 097)
Unruptured stateRuptured stateBroken stateFitted line of ldquoUrdquo state
Fitted line of ldquoBrdquo state
Fitted line of ldquoBrdquo stateamp ldquoRrdquo state
Fitted line of ldquoUrdquo stateamp ldquoRrdquo state
0
2
4
6
8
10
12
Diss
ipat
ed en
ergy
(J)
5 10 15 20 25 30 350Incident energy (J)
Figure 10 Fitted lines of the incident energy-dissipated energy inthe ldquoUrdquo state ldquoBrdquo state ldquoUrdquo state ldquoRrdquo state and ldquoBrdquo state amp ldquoRrdquostate (ldquoUrdquo state is the unruptured state ldquoBrdquo state is the broken stateand ldquoRrdquo state is the ruptured state)
8 Advances in Civil Engineering
fitting lines including the critical-state specimen are listedas follows
EDU 022EI minus 004
EDUprime 020EI minus 001
⎧⎨
⎩
EDB 034EI minus 094
EDBprime 034EI minus 096
⎧⎨
⎩
(7)
It can be seen that the two fitted lines after adding thespecimen in the ruptured state almost coincide with theprevious fitted lines so the data of the specimen in theruptured state can be fitted with the data of other unrupturedand broken specimens As shown in Figure 10 the specimendata point in the ruptured state is approximately at theintersection of the two fitted lines
In addition for the dynamic Brazilian disc test becauseof the relatively small dynamic tensile stress it is difficult toobtain the specimen of the critical state so there is only onecritical-state specimen in this paper
5 Conclusions
In this paper a series of dynamic Brazilian disc tests of redsandstone specimens were carried out using the SHPB setup+e main conclusions can be described as follows
(1) In the dynamic Brazilian disc test the rock specimenhas three states after impact with different incidentenergy the unruptured state ruptured state andbroken state And the ruptured state is corre-sponding to the critical energy When the incidentenergy is less than the critical energy the specimenremains intact and there is no macrocrack on thesurface that is the unruptured state when the in-cident energy is greater than the critical energy thespecimen will be damaged and lose the bearingcapacity along the loading direction that is thebroken state
(2) Under the impact of different incident energy thedynamic peak stress and loading rate of red sand-stone specimens show a unified exponential functionrelationship However the failure states of specimensafter impact cannot be identified from thisrelationship
(3) +ere are linear relationships between the dissipatedenergy and incident energy in both the unrupturedstate and the broken state In the broken state thedissipated energy increases with the increase of in-cident energy and the increasing speed is faster thanthat of the unruptured specimen
(4) According to the loading process recorded with ahigh-speed camera the failure mode of the specimenis splitting along the loading direction When thespecimen is in the critical failure mode after theimpact only one crack is produced along the loadingdirection and it penetrates the whole specimen sothat the specimen is divided into two partsWhen thespecimen is in the broken state except the main
crack along the loading direction additional crackswill be generated at both ends of the specimenconnecting the incident and transmission rein-forcement and these cracks will also expand alongthe loading direction
(5) When the specimen is in the unruptured state afterimpact the ideal DTEDC is a constant value whichis 022 for the red sandstone specimen in this paperHowever when the specimen is in a broken stateafter impact the DTEDC increases with the increaseof incident energy and gradually approaches 034
Data Availability
No data were used to support this study
Conflicts of Interest
+e authors declare that they have no conflicts of interest
Acknowledgments
+is study was supported by the National Natural ScienceFoundation of China (Grant no 41877272)
References
[1] ISRM ldquoSuggested methods for determining tensile strength ofrock materialsrdquo International Journal of Rock Mechanics ampMining Sciences vol 15 no 1 pp 881ndash891 1978
[2] ASTM D3967-08 Standard Test Method for Splitting TensileStrength of Intact Rock Core Specimens ASTM InternationalWest Conshohocken PA USA 2008
[3] 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 pp 105ndash1122012
[4] F Q Gong L Zhang and S Y Wang ldquoLoading rate effect ofrock material with the direct tensile and three Brazilian disctestsrdquo Advances in Civil Engineering vol 2019 Article ID6260351 8 pages 2019
[5] F Q Gong X B Li and J Zhao ldquoAnalytical algorithm toestimate tensile modulus in Brazilian disk splitting testsrdquoChinese Journal of Rock Mechanics and Engineering vol 29no 5 pp 881ndash891 2010
[6] F Q Gong and G F Zhao ldquoDynamic indirect tensile strengthof sandstone under different loading ratesrdquo Rock Mechanicsand Rock Engineering vol 47 no 6 pp 2271ndash2278 2014
[7] Q B Zhang and J Zhao ldquoA review of dynamic experimentaltechniques andmechanical behaviour of rockmaterialsrdquo RockMechanics and Rock Engineering vol 47 no 4 pp 1411ndash14782014
[8] J B Zhu Z Y Liao and C A Tang ldquoNumerical SHPB tests ofrocks under combined static and dynamic loading conditionswith application to dynamic behavior of rocks under in situstressesrdquo Rock Mechanics and Rock Engineering vol 49no 10 pp 3935ndash3946 2016
[9] J C Li L F Rong H B Li and S N Hong ldquoAn SHPB teststudy on stress wave energy attenuation in jointed rockmassesrdquo Rock Mechanics and Rock Engineering vol 52 no 2pp 403ndash420 2019
Advances in Civil Engineering 9
[10] D Asprone E Cadoni A Prota and G Manfredi ldquoDynamicbehavior of a Mediterranean natural stone under tensileloadingrdquo International Journal of Rock Mechanics and MiningSciences vol 46 no 3 pp 514ndash520 2009
[11] Q ZWangW Li and H P Xie ldquoDynamic split tensile test offlattened Brazilian disc of rock with SHPB setuprdquo Mechanicsof Materials vol 41 no 3 pp 252ndash260 2009
[12] 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
[13] F Dai and K Xia ldquoLoading rate dependence of tensilestrength anisotropy of Barre graniterdquo Pure and AppliedGeophysics vol 167 no 11 pp 1419ndash1432 2010
[14] B Wu W Yao and K Xia ldquoAn experimental study of dy-namic tensile failure of rocks subjected to hydrostatic con-finementrdquo Rock Mechanics and Rock Engineering vol 49no 10 pp 3855ndash3864 2016
[15] H Xie L Li R Peng and Y Ju ldquoEnergy analysis and criteriafor structural failure of rocksrdquo Journal of Rock Mechanics andGeotechnical Engineering vol 1 no 1 pp 11ndash20 2009
[16] F Q Gong S Luo and J Y Yan ldquoEnergy storage and dis-sipation evolution process and characteristics of marble inthree tension-type failure testsrdquo Rock Mechanics and RockEngineering vol 51 no 11 pp 3613ndash3624 2018
[17] F Q Gong J Y Yan S Luo and X B Li ldquoInvestigation onthe linear energy storage and dissipation laws of rock ma-terials under uniaxial compressionrdquo RockMechanics and RockEngineering vol 52 no 11 pp 4237ndash4255 2019
[18] Y Chu H Sun and D Zhang ldquoExperimental study onevolution in the characteristics of permeability deformationand energy of coal containing gas under triaxial cyclicloading-unloadingrdquo Energy Science amp Engineering vol 7no 5 pp 2112ndash2123 2019
[19] B Lundberg ldquoA split Hopkinson bar study of energy ab-sorption in dynamic rock fragmentationrdquo InternationalJournal of Rock Mechanics and Mining Sciences amp Geo-mechanics Abstracts vol 13 no 6 pp 187ndash197 1976
[20] A V Mikhalyuk and V V Zakharov ldquoDissipation of dy-namic-loading energy in quasi-elastic deformation processesin rocksrdquo Journal of Applied Mechanics and Technical Physicsvol 38 no 2 pp 312ndash318 1997
[21] X B Li T S Lok and J Zhao ldquoDynamic characteristics ofgranite subjected to intermediate loading raterdquo Rock Me-chanics and Rock Engineering vol 38 no 1 pp 21ndash39 2005
[22] F Q Gong Experimental study of rock mechanical propertiesunder coupled static-dynamic loads and dynamic strengthcriterion PhD thesis Central South University ChangshaChina 2010
[23] 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
10 Advances in Civil Engineering
dissipated energy of the specimens increased with theaugmentation of the incident energy and the states of thered sandstone specimens changed from unruptured state tothe ruptured state and then to the broken state
Figure 8 shows the relationship between the dissipatedenergy and the incident energy in the dynamic Brazilian disctesting of red sandstone specimens It can be seen that thedissipated energy increases approximately linearly with theincrease of incident energy and presenting two differentstages of energy relationship From Table 1 it can be foundthat this feature is related to the state of the specimen afterimpact so the linear fitting formulas were used to fit the datapoints of the unruptured and broken specimens respectively(to correct the fitting curve of the unruptured state the
origin coordinate was also added because when the incidentenergy is 0 J the dissipated energy must be 0 J) +e datapoints for the ruptured state are represented by differentshapes than those for the other two states
+e relationships between the dissipated energy andincident energy are both positively linear for the specimensin the unruptured state and broken state after the impact andcan be fitted by the following
EUD kUEI + cU EI ltECI( 1113857
EBD kBEI + cB EI gtECI( 1113857
⎧⎨
⎩ (4)
where EUD and EB
D are the dissipated energies of the specimenin the unruptured state and in the broken state kU and kB are
Specimen 26Specimen 11Specimen 1
0
5
10
15
20
25
30
35In
cide
nt en
ergy
(J)
50 100 150 200 250 3000Time (μs)
(a)
Specimen 26Specimen 11Specimen 1
0
5
10
15
20
25
Refle
cted
ener
gy (J
)
50 100 150 200 250 3000Time (μs)
(b)
Specimen 26Specimen 11Specimen 1
00
01
02
03
04
05
06
07
08
Tran
smitt
ed en
ergy
(J)
50 100 150 200 250 3000Time (μs)
(c)
Specimen 26Specimen 11Specimen 1
0
2
4
6
8
10D
issip
ated
ener
gy (J
)
50 100 150 200 250 3000Time (μs)
(d)
Figure 6 Energy-time curve of red sandstone specimens in the dynamic Brazilian disc test (a) incident energy-time (b) reflected energy-time (c) transmitted energy-time (d) dissipated energy-time
6 Advances in Civil Engineering
the slopes of the fitted curves of the unruptured state andbroken state EI is the incident energy cU and cB areconstants of the unruptured state and broken state and ECI
is the critical incident energy which corresponds to theruptured state of the specimen after impact For the redsandstone in this study two fitting functions can be ob-tained as follows
EUD
EI
022 minus004EI
EI lt 75 J( 1113857
EBD
EI
034 minus094EI
EI gt 75 J( 1113857
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(5)
For quantitatively describing the dynamic energy dis-sipation capacity of rock materials in the dynamic Braziliandisc test the dynamic tensile energy dissipation coefficient(DTEDC) is introduced as the ratio of dissipated energy toincident energy
In fact the DTEDCs are equal to the values of kU and kBIn formula (5) when the incident energy is 0 J the
dissipated energy must be 0 J However due to the influenceof heterogeneity of rock materials and experimental un-certainty the intercept of energy dissipation function in thestate of unruptured specimen is not 0 (shown in Figure 8)+erefore the ideal energy storage formula should be asfollows
14 14 14000μs 25002μs 79173μs
(a)
11 11 11000μs 25002μs 79173μs
(b)
47 47 47000μs 25002μs 79173μs
(c)
Figure 7 Failure processes of specimens in three states (a) unruptured state (b) ruptured state (c) broken state
Advances in Civil Engineering 7
EUD
EI
022 EI lt 75 J( 1113857
EBD
EI
034 minus094EI
EI gt 75 J( 1113857
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(6)
+e ideal curve obtained according to formula (6) wasshown in red line in Figure 9+eDTEDC of the specimen inthe unruptured state after impact is 022 When the incidentenergy exceeds 75 J the state of the specimen after impactwill develop from a ruptured state to a broken state and theDTEDC begins to increase It can be seen from Figure 9 thatthe DTEDC increases with the increase of incident energyand the velocity decreases gradually +e same conclusioncan be drawn from formula (6) and the DTEDC willgradually increase and approach to 034
4 Discussions
41 Loading Rate Effect As shown in Figure 5 there is noobvious difference among these three failure states +ere-fore from the relationship between the peak tensile stressand loading rate it is impossible to distinguish the differencein mechanical properties of the failure state of the specimenHowever from the point of view of energy as mentionedabove we find that there are two stages of linear energydissipation between these states
42 Failure State It can be seen from Figure 7 that when thespecimen is in the ruptured state that is specimen 11 thereis only one main crack that penetrates the whole specimenwith the specimen divided into two parts along the crack It
is noteworthy that this specimen produced not onlymicrocracks propagation but also macrocracks penetrationand the specimen is at the critical point at which the kineticenergy of the rock fragments is about to be generated
+e data of the ruptured specimen 11 has been fittedtogether with those from the unruptured and broken statesspecimens to create fitting lines and the two previousfitting lines were retained for comparison as shown inFigure 10 +e two previous fitting lines and the other two
Unruptured stateRuptured stateBroken state
Fitted line of unrupturedstateFitted line of broken state
EUD = 022EI ndash 004
(R2 = 091)
EBD = 034EI ndash 094
(R2 = 097)
0
2
4
6
8
10
12
Diss
ipat
ed en
ergy
(J)
5 10 15 20 25 30 350Incident energy (J)
Figure 8 Fitted lines of the dissipated energy-incident energy inthe ldquoUrdquo state and ldquoBrdquo state (ldquoUrdquo state is the unruptured state and ldquoBrdquostate is the broken state)
Ideal curve of DTEDCUnruptured state
Ruptured stateBroken state
01
02
03
04
Dyn
amic
tens
ile en
ergy
diss
ipat
ion
coef
ficie
nt (D
TED
C)
5 10 15 20 25 30 35 400Incident energy (J)
Figure 9 Relation of incident energy versus DTEDC
EprimeDU = 020EI ndash 001
(R2 = 089)
EprimeDB = 034EI ndash 096
(R2 = 097)EDU = 022EI ndash 004(R2 = 091)
EDB = 034EI ndash 094(R2 = 097)
Unruptured stateRuptured stateBroken stateFitted line of ldquoUrdquo state
Fitted line of ldquoBrdquo state
Fitted line of ldquoBrdquo stateamp ldquoRrdquo state
Fitted line of ldquoUrdquo stateamp ldquoRrdquo state
0
2
4
6
8
10
12
Diss
ipat
ed en
ergy
(J)
5 10 15 20 25 30 350Incident energy (J)
Figure 10 Fitted lines of the incident energy-dissipated energy inthe ldquoUrdquo state ldquoBrdquo state ldquoUrdquo state ldquoRrdquo state and ldquoBrdquo state amp ldquoRrdquostate (ldquoUrdquo state is the unruptured state ldquoBrdquo state is the broken stateand ldquoRrdquo state is the ruptured state)
8 Advances in Civil Engineering
fitting lines including the critical-state specimen are listedas follows
EDU 022EI minus 004
EDUprime 020EI minus 001
⎧⎨
⎩
EDB 034EI minus 094
EDBprime 034EI minus 096
⎧⎨
⎩
(7)
It can be seen that the two fitted lines after adding thespecimen in the ruptured state almost coincide with theprevious fitted lines so the data of the specimen in theruptured state can be fitted with the data of other unrupturedand broken specimens As shown in Figure 10 the specimendata point in the ruptured state is approximately at theintersection of the two fitted lines
In addition for the dynamic Brazilian disc test becauseof the relatively small dynamic tensile stress it is difficult toobtain the specimen of the critical state so there is only onecritical-state specimen in this paper
5 Conclusions
In this paper a series of dynamic Brazilian disc tests of redsandstone specimens were carried out using the SHPB setup+e main conclusions can be described as follows
(1) In the dynamic Brazilian disc test the rock specimenhas three states after impact with different incidentenergy the unruptured state ruptured state andbroken state And the ruptured state is corre-sponding to the critical energy When the incidentenergy is less than the critical energy the specimenremains intact and there is no macrocrack on thesurface that is the unruptured state when the in-cident energy is greater than the critical energy thespecimen will be damaged and lose the bearingcapacity along the loading direction that is thebroken state
(2) Under the impact of different incident energy thedynamic peak stress and loading rate of red sand-stone specimens show a unified exponential functionrelationship However the failure states of specimensafter impact cannot be identified from thisrelationship
(3) +ere are linear relationships between the dissipatedenergy and incident energy in both the unrupturedstate and the broken state In the broken state thedissipated energy increases with the increase of in-cident energy and the increasing speed is faster thanthat of the unruptured specimen
(4) According to the loading process recorded with ahigh-speed camera the failure mode of the specimenis splitting along the loading direction When thespecimen is in the critical failure mode after theimpact only one crack is produced along the loadingdirection and it penetrates the whole specimen sothat the specimen is divided into two partsWhen thespecimen is in the broken state except the main
crack along the loading direction additional crackswill be generated at both ends of the specimenconnecting the incident and transmission rein-forcement and these cracks will also expand alongthe loading direction
(5) When the specimen is in the unruptured state afterimpact the ideal DTEDC is a constant value whichis 022 for the red sandstone specimen in this paperHowever when the specimen is in a broken stateafter impact the DTEDC increases with the increaseof incident energy and gradually approaches 034
Data Availability
No data were used to support this study
Conflicts of Interest
+e authors declare that they have no conflicts of interest
Acknowledgments
+is study was supported by the National Natural ScienceFoundation of China (Grant no 41877272)
References
[1] ISRM ldquoSuggested methods for determining tensile strength ofrock materialsrdquo International Journal of Rock Mechanics ampMining Sciences vol 15 no 1 pp 881ndash891 1978
[2] ASTM D3967-08 Standard Test Method for Splitting TensileStrength of Intact Rock Core Specimens ASTM InternationalWest Conshohocken PA USA 2008
[3] 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 pp 105ndash1122012
[4] F Q Gong L Zhang and S Y Wang ldquoLoading rate effect ofrock material with the direct tensile and three Brazilian disctestsrdquo Advances in Civil Engineering vol 2019 Article ID6260351 8 pages 2019
[5] F Q Gong X B Li and J Zhao ldquoAnalytical algorithm toestimate tensile modulus in Brazilian disk splitting testsrdquoChinese Journal of Rock Mechanics and Engineering vol 29no 5 pp 881ndash891 2010
[6] F Q Gong and G F Zhao ldquoDynamic indirect tensile strengthof sandstone under different loading ratesrdquo Rock Mechanicsand Rock Engineering vol 47 no 6 pp 2271ndash2278 2014
[7] Q B Zhang and J Zhao ldquoA review of dynamic experimentaltechniques andmechanical behaviour of rockmaterialsrdquo RockMechanics and Rock Engineering vol 47 no 4 pp 1411ndash14782014
[8] J B Zhu Z Y Liao and C A Tang ldquoNumerical SHPB tests ofrocks under combined static and dynamic loading conditionswith application to dynamic behavior of rocks under in situstressesrdquo Rock Mechanics and Rock Engineering vol 49no 10 pp 3935ndash3946 2016
[9] J C Li L F Rong H B Li and S N Hong ldquoAn SHPB teststudy on stress wave energy attenuation in jointed rockmassesrdquo Rock Mechanics and Rock Engineering vol 52 no 2pp 403ndash420 2019
Advances in Civil Engineering 9
[10] D Asprone E Cadoni A Prota and G Manfredi ldquoDynamicbehavior of a Mediterranean natural stone under tensileloadingrdquo International Journal of Rock Mechanics and MiningSciences vol 46 no 3 pp 514ndash520 2009
[11] Q ZWangW Li and H P Xie ldquoDynamic split tensile test offlattened Brazilian disc of rock with SHPB setuprdquo Mechanicsof Materials vol 41 no 3 pp 252ndash260 2009
[12] 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
[13] F Dai and K Xia ldquoLoading rate dependence of tensilestrength anisotropy of Barre graniterdquo Pure and AppliedGeophysics vol 167 no 11 pp 1419ndash1432 2010
[14] B Wu W Yao and K Xia ldquoAn experimental study of dy-namic tensile failure of rocks subjected to hydrostatic con-finementrdquo Rock Mechanics and Rock Engineering vol 49no 10 pp 3855ndash3864 2016
[15] H Xie L Li R Peng and Y Ju ldquoEnergy analysis and criteriafor structural failure of rocksrdquo Journal of Rock Mechanics andGeotechnical Engineering vol 1 no 1 pp 11ndash20 2009
[16] F Q Gong S Luo and J Y Yan ldquoEnergy storage and dis-sipation evolution process and characteristics of marble inthree tension-type failure testsrdquo Rock Mechanics and RockEngineering vol 51 no 11 pp 3613ndash3624 2018
[17] F Q Gong J Y Yan S Luo and X B Li ldquoInvestigation onthe linear energy storage and dissipation laws of rock ma-terials under uniaxial compressionrdquo RockMechanics and RockEngineering vol 52 no 11 pp 4237ndash4255 2019
[18] Y Chu H Sun and D Zhang ldquoExperimental study onevolution in the characteristics of permeability deformationand energy of coal containing gas under triaxial cyclicloading-unloadingrdquo Energy Science amp Engineering vol 7no 5 pp 2112ndash2123 2019
[19] B Lundberg ldquoA split Hopkinson bar study of energy ab-sorption in dynamic rock fragmentationrdquo InternationalJournal of Rock Mechanics and Mining Sciences amp Geo-mechanics Abstracts vol 13 no 6 pp 187ndash197 1976
[20] A V Mikhalyuk and V V Zakharov ldquoDissipation of dy-namic-loading energy in quasi-elastic deformation processesin rocksrdquo Journal of Applied Mechanics and Technical Physicsvol 38 no 2 pp 312ndash318 1997
[21] X B Li T S Lok and J Zhao ldquoDynamic characteristics ofgranite subjected to intermediate loading raterdquo Rock Me-chanics and Rock Engineering vol 38 no 1 pp 21ndash39 2005
[22] F Q Gong Experimental study of rock mechanical propertiesunder coupled static-dynamic loads and dynamic strengthcriterion PhD thesis Central South University ChangshaChina 2010
[23] 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
10 Advances in Civil Engineering
the slopes of the fitted curves of the unruptured state andbroken state EI is the incident energy cU and cB areconstants of the unruptured state and broken state and ECI
is the critical incident energy which corresponds to theruptured state of the specimen after impact For the redsandstone in this study two fitting functions can be ob-tained as follows
EUD
EI
022 minus004EI
EI lt 75 J( 1113857
EBD
EI
034 minus094EI
EI gt 75 J( 1113857
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(5)
For quantitatively describing the dynamic energy dis-sipation capacity of rock materials in the dynamic Braziliandisc test the dynamic tensile energy dissipation coefficient(DTEDC) is introduced as the ratio of dissipated energy toincident energy
In fact the DTEDCs are equal to the values of kU and kBIn formula (5) when the incident energy is 0 J the
dissipated energy must be 0 J However due to the influenceof heterogeneity of rock materials and experimental un-certainty the intercept of energy dissipation function in thestate of unruptured specimen is not 0 (shown in Figure 8)+erefore the ideal energy storage formula should be asfollows
14 14 14000μs 25002μs 79173μs
(a)
11 11 11000μs 25002μs 79173μs
(b)
47 47 47000μs 25002μs 79173μs
(c)
Figure 7 Failure processes of specimens in three states (a) unruptured state (b) ruptured state (c) broken state
Advances in Civil Engineering 7
EUD
EI
022 EI lt 75 J( 1113857
EBD
EI
034 minus094EI
EI gt 75 J( 1113857
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(6)
+e ideal curve obtained according to formula (6) wasshown in red line in Figure 9+eDTEDC of the specimen inthe unruptured state after impact is 022 When the incidentenergy exceeds 75 J the state of the specimen after impactwill develop from a ruptured state to a broken state and theDTEDC begins to increase It can be seen from Figure 9 thatthe DTEDC increases with the increase of incident energyand the velocity decreases gradually +e same conclusioncan be drawn from formula (6) and the DTEDC willgradually increase and approach to 034
4 Discussions
41 Loading Rate Effect As shown in Figure 5 there is noobvious difference among these three failure states +ere-fore from the relationship between the peak tensile stressand loading rate it is impossible to distinguish the differencein mechanical properties of the failure state of the specimenHowever from the point of view of energy as mentionedabove we find that there are two stages of linear energydissipation between these states
42 Failure State It can be seen from Figure 7 that when thespecimen is in the ruptured state that is specimen 11 thereis only one main crack that penetrates the whole specimenwith the specimen divided into two parts along the crack It
is noteworthy that this specimen produced not onlymicrocracks propagation but also macrocracks penetrationand the specimen is at the critical point at which the kineticenergy of the rock fragments is about to be generated
+e data of the ruptured specimen 11 has been fittedtogether with those from the unruptured and broken statesspecimens to create fitting lines and the two previousfitting lines were retained for comparison as shown inFigure 10 +e two previous fitting lines and the other two
Unruptured stateRuptured stateBroken state
Fitted line of unrupturedstateFitted line of broken state
EUD = 022EI ndash 004
(R2 = 091)
EBD = 034EI ndash 094
(R2 = 097)
0
2
4
6
8
10
12
Diss
ipat
ed en
ergy
(J)
5 10 15 20 25 30 350Incident energy (J)
Figure 8 Fitted lines of the dissipated energy-incident energy inthe ldquoUrdquo state and ldquoBrdquo state (ldquoUrdquo state is the unruptured state and ldquoBrdquostate is the broken state)
Ideal curve of DTEDCUnruptured state
Ruptured stateBroken state
01
02
03
04
Dyn
amic
tens
ile en
ergy
diss
ipat
ion
coef
ficie
nt (D
TED
C)
5 10 15 20 25 30 35 400Incident energy (J)
Figure 9 Relation of incident energy versus DTEDC
EprimeDU = 020EI ndash 001
(R2 = 089)
EprimeDB = 034EI ndash 096
(R2 = 097)EDU = 022EI ndash 004(R2 = 091)
EDB = 034EI ndash 094(R2 = 097)
Unruptured stateRuptured stateBroken stateFitted line of ldquoUrdquo state
Fitted line of ldquoBrdquo state
Fitted line of ldquoBrdquo stateamp ldquoRrdquo state
Fitted line of ldquoUrdquo stateamp ldquoRrdquo state
0
2
4
6
8
10
12
Diss
ipat
ed en
ergy
(J)
5 10 15 20 25 30 350Incident energy (J)
Figure 10 Fitted lines of the incident energy-dissipated energy inthe ldquoUrdquo state ldquoBrdquo state ldquoUrdquo state ldquoRrdquo state and ldquoBrdquo state amp ldquoRrdquostate (ldquoUrdquo state is the unruptured state ldquoBrdquo state is the broken stateand ldquoRrdquo state is the ruptured state)
8 Advances in Civil Engineering
fitting lines including the critical-state specimen are listedas follows
EDU 022EI minus 004
EDUprime 020EI minus 001
⎧⎨
⎩
EDB 034EI minus 094
EDBprime 034EI minus 096
⎧⎨
⎩
(7)
It can be seen that the two fitted lines after adding thespecimen in the ruptured state almost coincide with theprevious fitted lines so the data of the specimen in theruptured state can be fitted with the data of other unrupturedand broken specimens As shown in Figure 10 the specimendata point in the ruptured state is approximately at theintersection of the two fitted lines
In addition for the dynamic Brazilian disc test becauseof the relatively small dynamic tensile stress it is difficult toobtain the specimen of the critical state so there is only onecritical-state specimen in this paper
5 Conclusions
In this paper a series of dynamic Brazilian disc tests of redsandstone specimens were carried out using the SHPB setup+e main conclusions can be described as follows
(1) In the dynamic Brazilian disc test the rock specimenhas three states after impact with different incidentenergy the unruptured state ruptured state andbroken state And the ruptured state is corre-sponding to the critical energy When the incidentenergy is less than the critical energy the specimenremains intact and there is no macrocrack on thesurface that is the unruptured state when the in-cident energy is greater than the critical energy thespecimen will be damaged and lose the bearingcapacity along the loading direction that is thebroken state
(2) Under the impact of different incident energy thedynamic peak stress and loading rate of red sand-stone specimens show a unified exponential functionrelationship However the failure states of specimensafter impact cannot be identified from thisrelationship
(3) +ere are linear relationships between the dissipatedenergy and incident energy in both the unrupturedstate and the broken state In the broken state thedissipated energy increases with the increase of in-cident energy and the increasing speed is faster thanthat of the unruptured specimen
(4) According to the loading process recorded with ahigh-speed camera the failure mode of the specimenis splitting along the loading direction When thespecimen is in the critical failure mode after theimpact only one crack is produced along the loadingdirection and it penetrates the whole specimen sothat the specimen is divided into two partsWhen thespecimen is in the broken state except the main
crack along the loading direction additional crackswill be generated at both ends of the specimenconnecting the incident and transmission rein-forcement and these cracks will also expand alongthe loading direction
(5) When the specimen is in the unruptured state afterimpact the ideal DTEDC is a constant value whichis 022 for the red sandstone specimen in this paperHowever when the specimen is in a broken stateafter impact the DTEDC increases with the increaseof incident energy and gradually approaches 034
Data Availability
No data were used to support this study
Conflicts of Interest
+e authors declare that they have no conflicts of interest
Acknowledgments
+is study was supported by the National Natural ScienceFoundation of China (Grant no 41877272)
References
[1] ISRM ldquoSuggested methods for determining tensile strength ofrock materialsrdquo International Journal of Rock Mechanics ampMining Sciences vol 15 no 1 pp 881ndash891 1978
[2] ASTM D3967-08 Standard Test Method for Splitting TensileStrength of Intact Rock Core Specimens ASTM InternationalWest Conshohocken PA USA 2008
[3] 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 pp 105ndash1122012
[4] F Q Gong L Zhang and S Y Wang ldquoLoading rate effect ofrock material with the direct tensile and three Brazilian disctestsrdquo Advances in Civil Engineering vol 2019 Article ID6260351 8 pages 2019
[5] F Q Gong X B Li and J Zhao ldquoAnalytical algorithm toestimate tensile modulus in Brazilian disk splitting testsrdquoChinese Journal of Rock Mechanics and Engineering vol 29no 5 pp 881ndash891 2010
[6] F Q Gong and G F Zhao ldquoDynamic indirect tensile strengthof sandstone under different loading ratesrdquo Rock Mechanicsand Rock Engineering vol 47 no 6 pp 2271ndash2278 2014
[7] Q B Zhang and J Zhao ldquoA review of dynamic experimentaltechniques andmechanical behaviour of rockmaterialsrdquo RockMechanics and Rock Engineering vol 47 no 4 pp 1411ndash14782014
[8] J B Zhu Z Y Liao and C A Tang ldquoNumerical SHPB tests ofrocks under combined static and dynamic loading conditionswith application to dynamic behavior of rocks under in situstressesrdquo Rock Mechanics and Rock Engineering vol 49no 10 pp 3935ndash3946 2016
[9] J C Li L F Rong H B Li and S N Hong ldquoAn SHPB teststudy on stress wave energy attenuation in jointed rockmassesrdquo Rock Mechanics and Rock Engineering vol 52 no 2pp 403ndash420 2019
Advances in Civil Engineering 9
[10] D Asprone E Cadoni A Prota and G Manfredi ldquoDynamicbehavior of a Mediterranean natural stone under tensileloadingrdquo International Journal of Rock Mechanics and MiningSciences vol 46 no 3 pp 514ndash520 2009
[11] Q ZWangW Li and H P Xie ldquoDynamic split tensile test offlattened Brazilian disc of rock with SHPB setuprdquo Mechanicsof Materials vol 41 no 3 pp 252ndash260 2009
[12] 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
[13] F Dai and K Xia ldquoLoading rate dependence of tensilestrength anisotropy of Barre graniterdquo Pure and AppliedGeophysics vol 167 no 11 pp 1419ndash1432 2010
[14] B Wu W Yao and K Xia ldquoAn experimental study of dy-namic tensile failure of rocks subjected to hydrostatic con-finementrdquo Rock Mechanics and Rock Engineering vol 49no 10 pp 3855ndash3864 2016
[15] H Xie L Li R Peng and Y Ju ldquoEnergy analysis and criteriafor structural failure of rocksrdquo Journal of Rock Mechanics andGeotechnical Engineering vol 1 no 1 pp 11ndash20 2009
[16] F Q Gong S Luo and J Y Yan ldquoEnergy storage and dis-sipation evolution process and characteristics of marble inthree tension-type failure testsrdquo Rock Mechanics and RockEngineering vol 51 no 11 pp 3613ndash3624 2018
[17] F Q Gong J Y Yan S Luo and X B Li ldquoInvestigation onthe linear energy storage and dissipation laws of rock ma-terials under uniaxial compressionrdquo RockMechanics and RockEngineering vol 52 no 11 pp 4237ndash4255 2019
[18] Y Chu H Sun and D Zhang ldquoExperimental study onevolution in the characteristics of permeability deformationand energy of coal containing gas under triaxial cyclicloading-unloadingrdquo Energy Science amp Engineering vol 7no 5 pp 2112ndash2123 2019
[19] B Lundberg ldquoA split Hopkinson bar study of energy ab-sorption in dynamic rock fragmentationrdquo InternationalJournal of Rock Mechanics and Mining Sciences amp Geo-mechanics Abstracts vol 13 no 6 pp 187ndash197 1976
[20] A V Mikhalyuk and V V Zakharov ldquoDissipation of dy-namic-loading energy in quasi-elastic deformation processesin rocksrdquo Journal of Applied Mechanics and Technical Physicsvol 38 no 2 pp 312ndash318 1997
[21] X B Li T S Lok and J Zhao ldquoDynamic characteristics ofgranite subjected to intermediate loading raterdquo Rock Me-chanics and Rock Engineering vol 38 no 1 pp 21ndash39 2005
[22] F Q Gong Experimental study of rock mechanical propertiesunder coupled static-dynamic loads and dynamic strengthcriterion PhD thesis Central South University ChangshaChina 2010
[23] 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
10 Advances in Civil Engineering
EUD
EI
022 EI lt 75 J( 1113857
EBD
EI
034 minus094EI
EI gt 75 J( 1113857
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(6)
+e ideal curve obtained according to formula (6) wasshown in red line in Figure 9+eDTEDC of the specimen inthe unruptured state after impact is 022 When the incidentenergy exceeds 75 J the state of the specimen after impactwill develop from a ruptured state to a broken state and theDTEDC begins to increase It can be seen from Figure 9 thatthe DTEDC increases with the increase of incident energyand the velocity decreases gradually +e same conclusioncan be drawn from formula (6) and the DTEDC willgradually increase and approach to 034
4 Discussions
41 Loading Rate Effect As shown in Figure 5 there is noobvious difference among these three failure states +ere-fore from the relationship between the peak tensile stressand loading rate it is impossible to distinguish the differencein mechanical properties of the failure state of the specimenHowever from the point of view of energy as mentionedabove we find that there are two stages of linear energydissipation between these states
42 Failure State It can be seen from Figure 7 that when thespecimen is in the ruptured state that is specimen 11 thereis only one main crack that penetrates the whole specimenwith the specimen divided into two parts along the crack It
is noteworthy that this specimen produced not onlymicrocracks propagation but also macrocracks penetrationand the specimen is at the critical point at which the kineticenergy of the rock fragments is about to be generated
+e data of the ruptured specimen 11 has been fittedtogether with those from the unruptured and broken statesspecimens to create fitting lines and the two previousfitting lines were retained for comparison as shown inFigure 10 +e two previous fitting lines and the other two
Unruptured stateRuptured stateBroken state
Fitted line of unrupturedstateFitted line of broken state
EUD = 022EI ndash 004
(R2 = 091)
EBD = 034EI ndash 094
(R2 = 097)
0
2
4
6
8
10
12
Diss
ipat
ed en
ergy
(J)
5 10 15 20 25 30 350Incident energy (J)
Figure 8 Fitted lines of the dissipated energy-incident energy inthe ldquoUrdquo state and ldquoBrdquo state (ldquoUrdquo state is the unruptured state and ldquoBrdquostate is the broken state)
Ideal curve of DTEDCUnruptured state
Ruptured stateBroken state
01
02
03
04
Dyn
amic
tens
ile en
ergy
diss
ipat
ion
coef
ficie
nt (D
TED
C)
5 10 15 20 25 30 35 400Incident energy (J)
Figure 9 Relation of incident energy versus DTEDC
EprimeDU = 020EI ndash 001
(R2 = 089)
EprimeDB = 034EI ndash 096
(R2 = 097)EDU = 022EI ndash 004(R2 = 091)
EDB = 034EI ndash 094(R2 = 097)
Unruptured stateRuptured stateBroken stateFitted line of ldquoUrdquo state
Fitted line of ldquoBrdquo state
Fitted line of ldquoBrdquo stateamp ldquoRrdquo state
Fitted line of ldquoUrdquo stateamp ldquoRrdquo state
0
2
4
6
8
10
12
Diss
ipat
ed en
ergy
(J)
5 10 15 20 25 30 350Incident energy (J)
Figure 10 Fitted lines of the incident energy-dissipated energy inthe ldquoUrdquo state ldquoBrdquo state ldquoUrdquo state ldquoRrdquo state and ldquoBrdquo state amp ldquoRrdquostate (ldquoUrdquo state is the unruptured state ldquoBrdquo state is the broken stateand ldquoRrdquo state is the ruptured state)
8 Advances in Civil Engineering
fitting lines including the critical-state specimen are listedas follows
EDU 022EI minus 004
EDUprime 020EI minus 001
⎧⎨
⎩
EDB 034EI minus 094
EDBprime 034EI minus 096
⎧⎨
⎩
(7)
It can be seen that the two fitted lines after adding thespecimen in the ruptured state almost coincide with theprevious fitted lines so the data of the specimen in theruptured state can be fitted with the data of other unrupturedand broken specimens As shown in Figure 10 the specimendata point in the ruptured state is approximately at theintersection of the two fitted lines
In addition for the dynamic Brazilian disc test becauseof the relatively small dynamic tensile stress it is difficult toobtain the specimen of the critical state so there is only onecritical-state specimen in this paper
5 Conclusions
In this paper a series of dynamic Brazilian disc tests of redsandstone specimens were carried out using the SHPB setup+e main conclusions can be described as follows
(1) In the dynamic Brazilian disc test the rock specimenhas three states after impact with different incidentenergy the unruptured state ruptured state andbroken state And the ruptured state is corre-sponding to the critical energy When the incidentenergy is less than the critical energy the specimenremains intact and there is no macrocrack on thesurface that is the unruptured state when the in-cident energy is greater than the critical energy thespecimen will be damaged and lose the bearingcapacity along the loading direction that is thebroken state
(2) Under the impact of different incident energy thedynamic peak stress and loading rate of red sand-stone specimens show a unified exponential functionrelationship However the failure states of specimensafter impact cannot be identified from thisrelationship
(3) +ere are linear relationships between the dissipatedenergy and incident energy in both the unrupturedstate and the broken state In the broken state thedissipated energy increases with the increase of in-cident energy and the increasing speed is faster thanthat of the unruptured specimen
(4) According to the loading process recorded with ahigh-speed camera the failure mode of the specimenis splitting along the loading direction When thespecimen is in the critical failure mode after theimpact only one crack is produced along the loadingdirection and it penetrates the whole specimen sothat the specimen is divided into two partsWhen thespecimen is in the broken state except the main
crack along the loading direction additional crackswill be generated at both ends of the specimenconnecting the incident and transmission rein-forcement and these cracks will also expand alongthe loading direction
(5) When the specimen is in the unruptured state afterimpact the ideal DTEDC is a constant value whichis 022 for the red sandstone specimen in this paperHowever when the specimen is in a broken stateafter impact the DTEDC increases with the increaseof incident energy and gradually approaches 034
Data Availability
No data were used to support this study
Conflicts of Interest
+e authors declare that they have no conflicts of interest
Acknowledgments
+is study was supported by the National Natural ScienceFoundation of China (Grant no 41877272)
References
[1] ISRM ldquoSuggested methods for determining tensile strength ofrock materialsrdquo International Journal of Rock Mechanics ampMining Sciences vol 15 no 1 pp 881ndash891 1978
[2] ASTM D3967-08 Standard Test Method for Splitting TensileStrength of Intact Rock Core Specimens ASTM InternationalWest Conshohocken PA USA 2008
[3] 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 pp 105ndash1122012
[4] F Q Gong L Zhang and S Y Wang ldquoLoading rate effect ofrock material with the direct tensile and three Brazilian disctestsrdquo Advances in Civil Engineering vol 2019 Article ID6260351 8 pages 2019
[5] F Q Gong X B Li and J Zhao ldquoAnalytical algorithm toestimate tensile modulus in Brazilian disk splitting testsrdquoChinese Journal of Rock Mechanics and Engineering vol 29no 5 pp 881ndash891 2010
[6] F Q Gong and G F Zhao ldquoDynamic indirect tensile strengthof sandstone under different loading ratesrdquo Rock Mechanicsand Rock Engineering vol 47 no 6 pp 2271ndash2278 2014
[7] Q B Zhang and J Zhao ldquoA review of dynamic experimentaltechniques andmechanical behaviour of rockmaterialsrdquo RockMechanics and Rock Engineering vol 47 no 4 pp 1411ndash14782014
[8] J B Zhu Z Y Liao and C A Tang ldquoNumerical SHPB tests ofrocks under combined static and dynamic loading conditionswith application to dynamic behavior of rocks under in situstressesrdquo Rock Mechanics and Rock Engineering vol 49no 10 pp 3935ndash3946 2016
[9] J C Li L F Rong H B Li and S N Hong ldquoAn SHPB teststudy on stress wave energy attenuation in jointed rockmassesrdquo Rock Mechanics and Rock Engineering vol 52 no 2pp 403ndash420 2019
Advances in Civil Engineering 9
[10] D Asprone E Cadoni A Prota and G Manfredi ldquoDynamicbehavior of a Mediterranean natural stone under tensileloadingrdquo International Journal of Rock Mechanics and MiningSciences vol 46 no 3 pp 514ndash520 2009
[11] Q ZWangW Li and H P Xie ldquoDynamic split tensile test offlattened Brazilian disc of rock with SHPB setuprdquo Mechanicsof Materials vol 41 no 3 pp 252ndash260 2009
[12] 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
[13] F Dai and K Xia ldquoLoading rate dependence of tensilestrength anisotropy of Barre graniterdquo Pure and AppliedGeophysics vol 167 no 11 pp 1419ndash1432 2010
[14] B Wu W Yao and K Xia ldquoAn experimental study of dy-namic tensile failure of rocks subjected to hydrostatic con-finementrdquo Rock Mechanics and Rock Engineering vol 49no 10 pp 3855ndash3864 2016
[15] H Xie L Li R Peng and Y Ju ldquoEnergy analysis and criteriafor structural failure of rocksrdquo Journal of Rock Mechanics andGeotechnical Engineering vol 1 no 1 pp 11ndash20 2009
[16] F Q Gong S Luo and J Y Yan ldquoEnergy storage and dis-sipation evolution process and characteristics of marble inthree tension-type failure testsrdquo Rock Mechanics and RockEngineering vol 51 no 11 pp 3613ndash3624 2018
[17] F Q Gong J Y Yan S Luo and X B Li ldquoInvestigation onthe linear energy storage and dissipation laws of rock ma-terials under uniaxial compressionrdquo RockMechanics and RockEngineering vol 52 no 11 pp 4237ndash4255 2019
[18] Y Chu H Sun and D Zhang ldquoExperimental study onevolution in the characteristics of permeability deformationand energy of coal containing gas under triaxial cyclicloading-unloadingrdquo Energy Science amp Engineering vol 7no 5 pp 2112ndash2123 2019
[19] B Lundberg ldquoA split Hopkinson bar study of energy ab-sorption in dynamic rock fragmentationrdquo InternationalJournal of Rock Mechanics and Mining Sciences amp Geo-mechanics Abstracts vol 13 no 6 pp 187ndash197 1976
[20] A V Mikhalyuk and V V Zakharov ldquoDissipation of dy-namic-loading energy in quasi-elastic deformation processesin rocksrdquo Journal of Applied Mechanics and Technical Physicsvol 38 no 2 pp 312ndash318 1997
[21] X B Li T S Lok and J Zhao ldquoDynamic characteristics ofgranite subjected to intermediate loading raterdquo Rock Me-chanics and Rock Engineering vol 38 no 1 pp 21ndash39 2005
[22] F Q Gong Experimental study of rock mechanical propertiesunder coupled static-dynamic loads and dynamic strengthcriterion PhD thesis Central South University ChangshaChina 2010
[23] 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
10 Advances in Civil Engineering
fitting lines including the critical-state specimen are listedas follows
EDU 022EI minus 004
EDUprime 020EI minus 001
⎧⎨
⎩
EDB 034EI minus 094
EDBprime 034EI minus 096
⎧⎨
⎩
(7)
It can be seen that the two fitted lines after adding thespecimen in the ruptured state almost coincide with theprevious fitted lines so the data of the specimen in theruptured state can be fitted with the data of other unrupturedand broken specimens As shown in Figure 10 the specimendata point in the ruptured state is approximately at theintersection of the two fitted lines
In addition for the dynamic Brazilian disc test becauseof the relatively small dynamic tensile stress it is difficult toobtain the specimen of the critical state so there is only onecritical-state specimen in this paper
5 Conclusions
In this paper a series of dynamic Brazilian disc tests of redsandstone specimens were carried out using the SHPB setup+e main conclusions can be described as follows
(1) In the dynamic Brazilian disc test the rock specimenhas three states after impact with different incidentenergy the unruptured state ruptured state andbroken state And the ruptured state is corre-sponding to the critical energy When the incidentenergy is less than the critical energy the specimenremains intact and there is no macrocrack on thesurface that is the unruptured state when the in-cident energy is greater than the critical energy thespecimen will be damaged and lose the bearingcapacity along the loading direction that is thebroken state
(2) Under the impact of different incident energy thedynamic peak stress and loading rate of red sand-stone specimens show a unified exponential functionrelationship However the failure states of specimensafter impact cannot be identified from thisrelationship
(3) +ere are linear relationships between the dissipatedenergy and incident energy in both the unrupturedstate and the broken state In the broken state thedissipated energy increases with the increase of in-cident energy and the increasing speed is faster thanthat of the unruptured specimen
(4) According to the loading process recorded with ahigh-speed camera the failure mode of the specimenis splitting along the loading direction When thespecimen is in the critical failure mode after theimpact only one crack is produced along the loadingdirection and it penetrates the whole specimen sothat the specimen is divided into two partsWhen thespecimen is in the broken state except the main
crack along the loading direction additional crackswill be generated at both ends of the specimenconnecting the incident and transmission rein-forcement and these cracks will also expand alongthe loading direction
(5) When the specimen is in the unruptured state afterimpact the ideal DTEDC is a constant value whichis 022 for the red sandstone specimen in this paperHowever when the specimen is in a broken stateafter impact the DTEDC increases with the increaseof incident energy and gradually approaches 034
Data Availability
No data were used to support this study
Conflicts of Interest
+e authors declare that they have no conflicts of interest
Acknowledgments
+is study was supported by the National Natural ScienceFoundation of China (Grant no 41877272)
References
[1] ISRM ldquoSuggested methods for determining tensile strength ofrock materialsrdquo International Journal of Rock Mechanics ampMining Sciences vol 15 no 1 pp 881ndash891 1978
[2] ASTM D3967-08 Standard Test Method for Splitting TensileStrength of Intact Rock Core Specimens ASTM InternationalWest Conshohocken PA USA 2008
[3] 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 pp 105ndash1122012
[4] F Q Gong L Zhang and S Y Wang ldquoLoading rate effect ofrock material with the direct tensile and three Brazilian disctestsrdquo Advances in Civil Engineering vol 2019 Article ID6260351 8 pages 2019
[5] F Q Gong X B Li and J Zhao ldquoAnalytical algorithm toestimate tensile modulus in Brazilian disk splitting testsrdquoChinese Journal of Rock Mechanics and Engineering vol 29no 5 pp 881ndash891 2010
[6] F Q Gong and G F Zhao ldquoDynamic indirect tensile strengthof sandstone under different loading ratesrdquo Rock Mechanicsand Rock Engineering vol 47 no 6 pp 2271ndash2278 2014
[7] Q B Zhang and J Zhao ldquoA review of dynamic experimentaltechniques andmechanical behaviour of rockmaterialsrdquo RockMechanics and Rock Engineering vol 47 no 4 pp 1411ndash14782014
[8] J B Zhu Z Y Liao and C A Tang ldquoNumerical SHPB tests ofrocks under combined static and dynamic loading conditionswith application to dynamic behavior of rocks under in situstressesrdquo Rock Mechanics and Rock Engineering vol 49no 10 pp 3935ndash3946 2016
[9] J C Li L F Rong H B Li and S N Hong ldquoAn SHPB teststudy on stress wave energy attenuation in jointed rockmassesrdquo Rock Mechanics and Rock Engineering vol 52 no 2pp 403ndash420 2019
Advances in Civil Engineering 9
[10] D Asprone E Cadoni A Prota and G Manfredi ldquoDynamicbehavior of a Mediterranean natural stone under tensileloadingrdquo International Journal of Rock Mechanics and MiningSciences vol 46 no 3 pp 514ndash520 2009
[11] Q ZWangW Li and H P Xie ldquoDynamic split tensile test offlattened Brazilian disc of rock with SHPB setuprdquo Mechanicsof Materials vol 41 no 3 pp 252ndash260 2009
[12] 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
[13] F Dai and K Xia ldquoLoading rate dependence of tensilestrength anisotropy of Barre graniterdquo Pure and AppliedGeophysics vol 167 no 11 pp 1419ndash1432 2010
[14] B Wu W Yao and K Xia ldquoAn experimental study of dy-namic tensile failure of rocks subjected to hydrostatic con-finementrdquo Rock Mechanics and Rock Engineering vol 49no 10 pp 3855ndash3864 2016
[15] H Xie L Li R Peng and Y Ju ldquoEnergy analysis and criteriafor structural failure of rocksrdquo Journal of Rock Mechanics andGeotechnical Engineering vol 1 no 1 pp 11ndash20 2009
[16] F Q Gong S Luo and J Y Yan ldquoEnergy storage and dis-sipation evolution process and characteristics of marble inthree tension-type failure testsrdquo Rock Mechanics and RockEngineering vol 51 no 11 pp 3613ndash3624 2018
[17] F Q Gong J Y Yan S Luo and X B Li ldquoInvestigation onthe linear energy storage and dissipation laws of rock ma-terials under uniaxial compressionrdquo RockMechanics and RockEngineering vol 52 no 11 pp 4237ndash4255 2019
[18] Y Chu H Sun and D Zhang ldquoExperimental study onevolution in the characteristics of permeability deformationand energy of coal containing gas under triaxial cyclicloading-unloadingrdquo Energy Science amp Engineering vol 7no 5 pp 2112ndash2123 2019
[19] B Lundberg ldquoA split Hopkinson bar study of energy ab-sorption in dynamic rock fragmentationrdquo InternationalJournal of Rock Mechanics and Mining Sciences amp Geo-mechanics Abstracts vol 13 no 6 pp 187ndash197 1976
[20] A V Mikhalyuk and V V Zakharov ldquoDissipation of dy-namic-loading energy in quasi-elastic deformation processesin rocksrdquo Journal of Applied Mechanics and Technical Physicsvol 38 no 2 pp 312ndash318 1997
[21] X B Li T S Lok and J Zhao ldquoDynamic characteristics ofgranite subjected to intermediate loading raterdquo Rock Me-chanics and Rock Engineering vol 38 no 1 pp 21ndash39 2005
[22] F Q Gong Experimental study of rock mechanical propertiesunder coupled static-dynamic loads and dynamic strengthcriterion PhD thesis Central South University ChangshaChina 2010
[23] 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
10 Advances in Civil Engineering
[10] D Asprone E Cadoni A Prota and G Manfredi ldquoDynamicbehavior of a Mediterranean natural stone under tensileloadingrdquo International Journal of Rock Mechanics and MiningSciences vol 46 no 3 pp 514ndash520 2009
[11] Q ZWangW Li and H P Xie ldquoDynamic split tensile test offlattened Brazilian disc of rock with SHPB setuprdquo Mechanicsof Materials vol 41 no 3 pp 252ndash260 2009
[12] 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
[13] F Dai and K Xia ldquoLoading rate dependence of tensilestrength anisotropy of Barre graniterdquo Pure and AppliedGeophysics vol 167 no 11 pp 1419ndash1432 2010
[14] B Wu W Yao and K Xia ldquoAn experimental study of dy-namic tensile failure of rocks subjected to hydrostatic con-finementrdquo Rock Mechanics and Rock Engineering vol 49no 10 pp 3855ndash3864 2016
[15] H Xie L Li R Peng and Y Ju ldquoEnergy analysis and criteriafor structural failure of rocksrdquo Journal of Rock Mechanics andGeotechnical Engineering vol 1 no 1 pp 11ndash20 2009
[16] F Q Gong S Luo and J Y Yan ldquoEnergy storage and dis-sipation evolution process and characteristics of marble inthree tension-type failure testsrdquo Rock Mechanics and RockEngineering vol 51 no 11 pp 3613ndash3624 2018
[17] F Q Gong J Y Yan S Luo and X B Li ldquoInvestigation onthe linear energy storage and dissipation laws of rock ma-terials under uniaxial compressionrdquo RockMechanics and RockEngineering vol 52 no 11 pp 4237ndash4255 2019
[18] Y Chu H Sun and D Zhang ldquoExperimental study onevolution in the characteristics of permeability deformationand energy of coal containing gas under triaxial cyclicloading-unloadingrdquo Energy Science amp Engineering vol 7no 5 pp 2112ndash2123 2019
[19] B Lundberg ldquoA split Hopkinson bar study of energy ab-sorption in dynamic rock fragmentationrdquo InternationalJournal of Rock Mechanics and Mining Sciences amp Geo-mechanics Abstracts vol 13 no 6 pp 187ndash197 1976
[20] A V Mikhalyuk and V V Zakharov ldquoDissipation of dy-namic-loading energy in quasi-elastic deformation processesin rocksrdquo Journal of Applied Mechanics and Technical Physicsvol 38 no 2 pp 312ndash318 1997
[21] X B Li T S Lok and J Zhao ldquoDynamic characteristics ofgranite subjected to intermediate loading raterdquo Rock Me-chanics and Rock Engineering vol 38 no 1 pp 21ndash39 2005
[22] F Q Gong Experimental study of rock mechanical propertiesunder coupled static-dynamic loads and dynamic strengthcriterion PhD thesis Central South University ChangshaChina 2010
[23] 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
