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Research ArticleEffect of Nonparallel End Face on Energy Dissipation Analyses ofRocklike Materials Based on SHPB Tests

Pu Yuan 123 Ning-Ning Wei 1 and Qin-Yong Ma 123

1School of Civil Engineering and Architecture Anhui University of Science and Technology Huainan 232001 China2State Key Laboratory of Mining Response and Disaster Prevention and Control in Deep Coal MinesAnhui University of Science and Technology Huainan 232001 China3Engineering Research Center of Underground Mine Construction Ministry of EducationAnhui University of Science and Technology Huainan 232001 China

Correspondence should be addressed to Pu Yuan puy2012126com

Received 9 May 2019 Accepted 2 July 2019 Published 18 July 2019

Academic Editor Itzhak Green

Copyright copy 2019 PuYuan et alis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

To evaluate the effect of nonparallel end face of rocklike specimens in SHPB tests the characteristics of energy dissipation areanalyzed based on numerical simulations for end-face nonparallelism from 0 to 040 and Youngrsquos modulus from 14GPa to42GPa With the increment of end-face nonparallelism both energy consumption density and dissipated energy density show aslight increase trend while releasable elastic strain energy density presents a slight decrease trend Existence of elastic unloading inthe damaged rocklike specimen leads to a reduction of energy consumption density and a constant dissipated energy densityduring total strain shrinkage At peak dynamic stress dissipated energy density presents a linear upward trend with the incrementof end-face nonparallelism and Youngrsquos modulus while releasable elastic strain energy density shows an inverse trend A binarylinear regression equation is deduced to estimate the energy dissipation ratio Mechanical damage evolution of the rocklikespecimen is divided into two regions in line with the two regions in dynamic stress-strain curves and the transition between theslow-growth region and rapid-growth region is shifted to the right with the increment of end-face nonparallelism Due to thepresence of nonparallel end face fluctuation presents in energy density evolution and mechanical damage evolution efluctuation is enhanced with the increment of end-face nonparallelism and weakened with the increase of Youngrsquos modulus Basedon energy density evolution and mechanical damage evolution analyses the maximum end-face nonparallelism should becontrolled within 020 twice the value in ISRM suggested methods which reduces the cost and time for processingrocklike specimens

1 Introduction

e split Hopkinson pressure bar (SHPB) also known asKolsky bar is an extensive convenient and reliable tech-nique to characterize the behaviors of solid materials at ahigh strain rate mainly in the range of 102ndash104sminus1 [1] It iswidely used to quantify the dynamic properties of metallicmaterials [2 3] en it is generalized to brittle materialsuch as concrete [4] ceramics [5] frozen soil [6] and rocks[7 8] To overcome the major limitations of SHPB apparatusfor rocklike materials pulse-shaping technique or changingthe shape of the striker is employed to generate a rampedincident stress wave [9] and some fundamental issues in the

SHPB test are also analyzed for rocklike materials [10] Testmethods based on the SHPB apparatus are suggested byISRM to determine the dynamic properties of the rockmaterial [11]

With the development of SHPB technique the accuracyand precision of dynamic mechanical characteristics are thekey issues for SHPB tests Based on numerical simulationssix types of incident bar misalignment in SHPB apparatusare investigated and the distorted signal generated by barmisalignment is mainly induced by the presence of flexuralmodes of vibration and affects the SHPB test results ad-versely [12] To evaluate the effects of imperfect condition onincident stress waves both experiments and numerical

HindawiShock and VibrationVolume 2019 Article ID 2040947 11 pageshttpsdoiorg10115520192040947

simulations are carried out and the inclination and in-dentation of impact end-surface show a great impact onincident stress waves [13] For a physical SHPB apparatusthe errors induced by imperfect impact interface and barmisalignment can be minimized and eliminated by aligningand manufacturing the bars precisely In SHPB tests thepending tested specimen is sandwiched between the incidentbar and transmitted bar Considering the processing accu-racy of the specimen the specimen geometry imperfectionson SHPB tests for ductile materials are analyzed from thereflected wave transmitted wave and dynamic stress-straincurve and the error induced by specimen geometry im-perfections is small for ductile materials and can beneglected when imperfection angles no larger than 03deg [14]While for rocklike materials the specimen geometry im-perfections in SHPB test is more adverse than ductilemartials When end-face nonparallelism is within 040 thenonparallel end face shows a small and negligible influenceon dynamic stress while it shows a great impact on dynamicstrain and strain rate [15] According to ISRM suggested testmethods [11] cylinder rock specimens with diameter of50mm and length-to-diameter ratio of 05 are widely used inSHPB tests Each rock specimen is processed throughdrilling cutting and grinding processes [16] For shortcylinder rock specimen the processing accuracy and pre-cision are very difficult to control especially the perpen-dicularity of two ends to the axis Besides high processingaccuracy and precision of rock specimens are always relatedwith expensive manufacturing technologies and long pro-cessing time By analyzing the end-face nonparallel effect inthe SHPB test the allowable processing deviation is in-vestigated to reduce the cost and save the processing timewithout affecting the reliability of SHPB tests

e deformation and failure of rock can be considered asan irreversible process of energy dissipation [17] ereforeenergy dissipation of the rock under dynamic loads can bestudied based on SHPB tests [18] and is becoming a majorissue in rock mechanics and rock engineering [19 20]

Considering the processing deviation of the rocklikespecimen numerical simulations of SHPB tests are con-ducted for nonparallel end-face rocklike specimens withvarious Youngrsquos moduli by LS-DYNA During numericalsimulation end-face nonparallelism ranges from 00 to040 and Youngrsquos modulus ranges from 14GPa to 42GPaen energy density dissipation is analyzed to reveal theeffect of end-face nonparallellism As mechanical damageevolution is closely related with energy dissipation the in-fluence of nonparallel end face on mechanical damageevolution is also studied

2 Setup of 3D Numerical Model for SHPB Testand Verification

21 Setup of 3D Numerical Model Based on the physicalΦ50mm SHPB apparatus a series of 3D finite elementmodels without a striker are set up to conduct SHPB tests forrocklike materials Basically a typical SHPB apparatusconsists of a striker an incident bar and a transmitted barAs shown in Figure 1 an incident bar a rocklike specimen

and a transmitted bar are considered and built in a 3D finiteelement model and the nonparallel end face of the rocklikespecimen is contacted with the transmitted bar e lengthand diameter for both incident and transmitted bars are2000mm and 50mm respectively In numerical simulationof SHPB tests an automatic single surface contact isemployed for the contact between two elastic bars androcklike specimen Automatic single surface contact is thesimplest type of contact with no definition of contact ortarget surface and LS-DYNA automatically determineswhich surfaces within a model may come into contact Aslubricant such as Vaseline is applied on both ends ofrocklike specimen in physical SHPB tests the friction effectbetween elastic bars and rocklike specimen is eliminatederefore the interfacial friction effect is negligible in nu-merical simulations

ANSYS is used to prepare 3D finite element models andthe SOLID164 element with one integration point is employedto save the computer time [15 21] e SOLID164 element isan eight-node solid hexahedron element in explicit dynamicanalyses After establishing 3D finite element model a key-word file is output fromANSYS and is modified for LS-DYNAby applying the HolmquistndashJohnsonndashCook (HJC) model tothe rocklike specimen In numerical simulations the meshsensitivity is evaluated by a dimensionless mesh parameter theratio of smallest model size to largest element size Accordingto the research of Kariem et al [12] numerical simulationresults is insensitive to the mesh smaller than 15mm Con-sidering the smallest model size is the diameter 127mm thecritical dimensionless mesh parameter is about 85 for SHPBnumerical simulation en a dimensionless mesh parameterof 10 is chosen and both incident bar and transmitted barconsist of 60000 hexahedron elements Considering smallnonparallel end face a finer mesh is employed for the rocklikespecimen and rocklike specimen consists of 60000 hexahedronelements Hence a total of 180000 hexahedron elements areinvolved in a 3D finite element model

In physical SHPB tests a compressive loading stresswave is generated by launching a striker impacting on theincident bar For traditional rectangular compressive stresswave premature failure of the rocklike specimen beforestress equilibrium makes test results unreliable Moreoverhigh signal oscillation presents in rectangular compressivestress wave due to the wave dispersion [22] erefore thetraditional rectangular compressive loading stress waveshould be modified Half-sine loading stress wave generatedby a cone-shape striker proves to be a suitable rational andeffective waveform for rocklike materials with good im-munity to premature failure before stress equilibriumgeometric dispersion effect and PochhammerndashChree os-cillation [11 22ndash24] Moreover half-sine loading stress wavegives the possibility to an approximate constant strain ratecondition According to the typical example of dynamicstress balance analysis in ISRM suggested test methods [11]the amplitude and duration of half-sine loading stress fornumerical simulation is assumed to be 260MPa and 240 μsrespectively As no striker is in the 3D finite element modelthe half-sine loading stress wave is straightly loaded on thefront-end face of the incident bar

2 Shock and Vibration

e rocklike specimen with a length to diameter ratio of05 is modeled and sandwiched between the incident bar andtransmitted bar [11] During specimen processing the endfaces of the rocklike specimen can be easily grinded to besmooth and flat As the length to diameter ratio is just 05 itis very difficult to control the parallelism between two endfaces End-face nonparallelism c which is a measure ofnonparallel deviation is defined as the ratio of maximumheight deviation δ to average height h and it varies from 0to 040 with an increment of 005 [15 21] Hence thecorresponding maximum height deviation ranges from0mm to 010mm

22 ConstitutiveModels andMaterial Parameters As for thedynamic characteristics test of rocklike materials the in-cident and transmitted bars in physical SHPB apparatus areall made of a homogenous and isotropic alloy steel and theykeep in a linear elastic deformation state during SHPB testserefore the elastic constitutive model for an isotropicelastic material in LS-DYNA is selected for both incident andtransmitted bars According to the alloy steel properties inphysical SHPB apparatus the density Youngrsquos modulus andPoissonrsquos ratio are set as 785 gcm3 210GPa and 030respectively

Considering the high strain rate in SHPB tests the HJCconstitutive model for materials subjected to large strainhigh strain rate and high pressure in LS-DYNA is employedfor the rocklike specimen [15 25ndash27] Youngrsquos modulusvaries from 14GPa to 42GPa with an increment of 7GPaWhen Youngrsquos modulus of rocklike material is 28GPamaterial parameters of HJC constitutive model are shown inTable 1

As five kinds of Youngrsquos moduli is considered relatedmaterial parameters of the HJC constitutive model should bemodified with Youngrsquos modulus In the HJC constitutivemodel shear modulus G and crushing volumetric strain μCchange with Youngrsquos modulus and they can be expressed asfollows

G E

2(1 + ])

μC pC

K

pC(1minus 2])

E

(1)

23 Verification of 3D Numerical Model In line with thephysical SHPB test four hexahedron elements at the same

cross section referring to pair strain gages symmetricallymounted on the surface of the bars are chosen to export Z-direction stress-time histories for stress uniformity analyses[21] When Youngrsquos modulus is 28GPa acquired incidentstress σ(t)I reflected stress σ(t)R and transmitted stress σ(t)Tfor parallel rocklike specimen are illustrated in Figure 2

As shown in Figure 2 the acquired incident stress wave isconsistent with applied half-sine incident loading stresswave Compared with the SHPB test results in ISRM sug-gested test methods [11] and literature [23] it can be foundthat the waveforms of acquired stress waves are similar withSHPB test results and a clear flat region is also presented inreflected stress wave Slight difference is the result of dif-ferent incident stress amplitudes and HJC material pa-rameters When failure occurs deleting the failure rocklikespecimen in numerical simulation leads to stress wavereflecting to incident bar which causes a second peak in thereflected stress wave

e fundamental assumptions of SHPB technique areone-dimensional stress wave propagation and stress uni-formity [1 7 10] To verify the numerical simulation resultsof SHPB tests stress-time histories at two ends of rocklikespecimens are compared and the unbalance stress is alsochecked [21 28 29] Numerical model verifications areperformed for parallel end-face rocklike specimen withYoungrsquos modulus of 14GPa 21GPa 28GPa 35GPa and42GPa According to acquired incident stress σ(t)I reflectedstress σ(t)R and transmitted stress σ(t)T exporting fromnumerical simulations the stress-time histories on two endsof parallel end-face rocklike specimens are illustrated inFigure 3 In Figure 3 the unbalance stress defined asσ(t)I + σ(t)R minus σ(t)T is also presented

As shown in Figure 3 stress-time histories on two ends ofparallel end-face rocklike specimens are basically the sameand the unbalanced stress is very small and can be omittederefore the stress uniformity state is achieved duringnumerical simulation and the 3D finite element model isvalid for the following energy dissipation analyses Besidesthe duration of transmitted stress wave extends with theincrease of Youngrsquos modulus of the HJC constitutive model

3 Energy Dissipation Analyses duringSHPB Tests

31 Energy Evolution Analyses When half-sine loadingstress wave propagates in an elastic steel bar both elasticdeformation and motion are generated in elastic steel bars

Incident bar Transmitted bar

δ

h

Rocklike specimen

0 60 120 180 2400

100

200

300

σ (M

Pa)

t (μs)

YX Z

Figure 1 A schematic diagram of SHPB setup in the 3D finite element model

Shock and Vibration 3

e energy carried by stress wave is composed of elasticstrain energy and kinetic energy and the elastic strain energyis basically equal to the kinetic energy for the elastic stresswave [30] e energy carried by incident stress wave re-flected stress wave and transmitted stress wave can becalculated as follows [18 19 30]

Wi AC

E1113946 σ(t)

2i dt i I R T (2)

where σ(t) and ε(t) denote the stress-time history and strain-time history in elastic bars and the subscript i can be I Rand T which refer to incident stress wave reflected stresswave and transmitted stress wave respectively E A and Care the Youngrsquos modulus cross-sectional area and longi-tudinal wave velocity of elastic bars respectively

An isothermal process is assumed in SHPB tests andthere is no heat exchange with the external environmentAccording to the first law of thermodynamics also known asthe energy conservation law the absorption energy ofrocklike specimen can be expressed as follows by neglectingthe energy loss in SHPB tests [31 32]

WL WI minusWR minusWT AC

E1113946σ(t)

2Idt

minusAC

E1113946σ(t)

2Rdtminus

AC

E1113946σ(t)

2Tdt

(3)

According to the fundamental assumption of SHPBtechnique stress uniformity equation (3) can be rewritten asfollows

WL AC

E11139462σ(t)Rσ(t)Tdt (4)

Based on equations (2) and (4) time histories of incidentenergy reflected energy transmitted energy and absorptionenergy for parallel end-face rocklike specimen with Youngrsquosmodulus of 28GPa are illustrated in Figure 4

As illustrated in Figure 4 the foregoing energies increasewith the increment of loading time and the transmittedenergy accounts for most of incident energy

32 Effect of Nonparallel End Face on Reflected Transmittedand Absorption Energies As only one type of half-sineloading stress wave is considered the incident energynamely the whole input energy is a constant value 38878 JBased on SHPB numerical simulations the variation ofreflected energy and transmitted energy with the incrementof end-face nonparallelism c is illustrated in Figure 5

Table 1 Material parameters of HJC constitutive model for rocklike material [15 25]

ρ (gcm3) FC (GPa) A B C SFMAX G (GPa) D1 D2 N247 013 079 160 0007 40 1167 0045 10 061EFMIN T (MPa) pC (MPa) μC pL μL K1 (GPa) K2 (GPa) K3 (GPa) FS0005 707 4333 000278 1 01 85 minus171 208 0004

0 50 100 150

σ(t)I

σ(t)R

σ(t)T

σ(t)I + σ(t)R

200 250ndash200

ndash100

0

100

200

300

σ (M

Pa)

t (micros)

Figure 2 Acquired incident reflected and transmitted stresseswhen Youngrsquos modulus is 28GPa

0

100

200

300

0

100

200

300

0

100

200

300

0

100

200

300

0 50 100 150 200 250 300

0

100

200

30042GPa

35GPa

28GPa

21GPa

σ (M

Pa)

σ (M

Pa)

σ (M

Pa)

σ (M

Pa)

σ (M

Pa)

14GPa

σ(t)I + σ(t)R

σ(t)I + σ(t)R ndash σ(t)T

t (μs)

σ(t)T

Figure 3 Stress-time histories on two ends of parallel end-facerocklike specimens


As shown in Figure 5 with the increment of end-facenonparallelism reflected energy shows a slight increasetrend while transmitted energy presents a slight decreasetrend Moreover for a constant end-face nonparallelism thereflected energy decreases with the increase of Youngrsquosmodulus while the transmitted energy shows an inversetrend ese results are consistent with the characteristics ofstress wave propagation As described in the research ofYuan et al [21] the amplitude of reflected stress wavegradually rises with the increment of end-face non-parallelism and decreases with the increase of Youngrsquosmodulus while the amplitude of transmitted stress waveshows an opposite variation trend

Figure 6 shows the variation of total absorption energywith the increment of end-face nonparallelism

As illustrated in Figure 6 total absorption energy showsa general upward trend with the increment of end-facenonparallelism and an overall downward trend with theincrease of Youngrsquos modulus of the HJC constitutive modelWhen Youngrsquos modulus is within 28GPa there is almost alinear relation between total absorption energy and end-facenonparallelism and the slope of the linear trend for Youngrsquosmodulus of 14GPa 21GPa and 28GPa is small and roughlythe same While for Youngrsquos modulus of 35GPa and 42GPaa rapid increase of total absorption energy is followed by aslow linear variation According to the dynamic stress-straincurve [15] there is an elastic unloading and total strainshrinkage due to slight broken or damage of the rocklikespecimen As a result of the presence of elastic unloading theabsorption energy releases during the unloading phase erapid increase of total absorption energy indicates a quickdecrease of released absorption energy As energy dissipa-tion is closely related with damage evolution the totaldamage of the rocklike specimen after SHPB tests also in-creases rapidly with end-face nonparallelism increasingfrom 020 to 035 for Youngrsquos modulus of 35GPa andfrom 030 to 040 for Youngrsquos modulus of 42GPa

33 Characteristics of Energy Density Evolution To illustratethe effect of energy dissipation per unit volume energy

consumption density also known as specific energy ab-sorption is defined as the energy consumed for breaking therocklike specimen per unit volume erefore the energyconsumption density presents a similar variation trend toabsorption energy In one-dimensional loading conditionenergy consumption density is defined as the area of dy-namic stress-strain curve and can by calculated as follows

U WL

Vs 1113946σ dε 1113946σ _ε dt 1113946σ(t)T

2C

ls

σ(t)RE

dt

C

Els11139462σ(t)Tσ(t)Rdt

(5)

where Vs and ls are the volume and length of the rocklikespecimen and U is the energy consumption density

According to the research of Wang et al [20] the energyconsumption of a rocklike material consists of dissipatedenergy and releasable elastic strain energy e releasableelastic strain energy density and dissipated energy densitycan be calculated as follows

005 010 015 020 025 030 035 040000

50

100

150

200

250

300

350

Ener

gy (J

)

γ ()

WR 14GPaWT 14GPaWR 28GPa

WT 28GPaWR 42GPaWT 42GPa

0

Figure 5 Variation of reflected energy and transmitted energy withend-face nonparallelism

000 005 010 015 020 025 030 035 04030

60

90

120

150

14GPa21GPa28GPa

35GPa42GPa

WL (

J)

γ ()

Figure 6 Variation of total absorption energy with end-facenonparallelism

0 50 100 150 200 250 300

0

100

200

300

400

500

Ener

gy (J

)

t (micros)

WIWR

WTWL

Figure 4 Energy-time histories during SHPB numericalsimulation


Ue

12σεe

12σ(t)2T

Ed (6)

Ud

UminusUe (7)

where Ed is dynamic Youngrsquos modulus of rocklike specimenwhich can be obtained from the dynamic stress-strain curveandUe andUd are the releasable elastic strain energy densityand dissipated energy density respectively

Based on equations (5)ndash(7) the energy density evolutioncurves of the rocklike specimens are illustrated in Figure 7

Obviously from Figure 7 both Youngrsquos modulus of theHJC constitutive model and end-face nonparallelism affectenergy density evolution On the one hand both energyconsumption density and dissipated energy density increasewith the increment of end-face nonparallelism while re-leasable elastic strain energy density reduces slightly On theother hand all three energy densities decrease with theincrease of Youngrsquos modulus of the HJC constitutive modelIt is worth mentioning that the presence of elastic unloadingin slight broken or damaged rocklike specimen leads to anobvious total strain shrinkage after peak dynamic stresswhich causes release of elastic strain energy density andreduction of energy consumption density erefore thedissipated energy density is approximately a constant valuewith the shrinkage of total strain and the final dissipatedenergy density increases with the increment of end-facenonparallelism

Due to the nonparallel end face fluctuation presents inreflected stresses and transmitted stresses [21] Hencefluctuation also presents in the evolution of both releasableelastic strain energy density and dissipated energy densitye larger the end-face nonparallelism is the greater thefluctuation appears e larger the Youngrsquos modulus is theweaker the fluctuation is

34 Energy Dissipation and Elastic Energy Release at PeakDynamic Stress Under uniaxial compression a typicalcomplete stress-strain curve consists of five stages crackclosure elastic cracking postfailure and residual and theidealized stress-strain curve can be basically divided into tworegions prefailure region and postfailure region [33] Innumerical simulation both the crack closure and residualstages cannot be presented for no crack in the finite elementmodel and elements deleted after failure erefore at theend of postfailure the dissipated energy density is basicallyequal to the energy consumption density due to residualstress approaching zero e critical point between elasticand cracking stages is difficult to determine while the criticalpoint between cracking or prefailure and postfailure is easyto determine which is the peak dynamic stress Consideringthe critical point between prefailure and postfailure regionsthe detail energy densities at the peak dynamic stress arecalculated and listed in Table 2 e releasable elastic strainenergy density and dissipated energy density at the peakdynamic stress are marked as Ue

p and Udp respectively

As shown in Table 2 dissipated energy density at thepeak dynamic stress presents a linear upward trend with the

increment of end-face nonparallelism and Youngrsquos moduluswhile releasable elastic strain energy density at the peakdynamic stress shows a linear downward trend e smallerthe Youngrsquos modulus is the more obvious the effect of end-face nonparallelism is

In order to study the influence of nonparallel end face onenergy dissipation characteristics the energy dissipationratio Kp is defined as dividing dissipated energy density byenergy consumption density at the peak dynamic stress andcan be calculated as follows

Kp Ud

p

Up (8)

With the increment of end-face nonparallelism energydissipation ratios for various Youngrsquos moduli are drawn inFigure 8

As clearly illustrated in Figure 8 the energy dissipationratio increases linearly with the increment of end-facenonparallelism and the growth trends for various Youngrsquosmoduli are approximate parallel to each other e smallerYoungrsquos modulus is the less the energy dissipation ratio isLinear regression equations with constant slope and dif-ferent intercepts are employed Linear regression equationscan be expressed as follows

Kp 023c + C (9)

where C is the intercept of linear regression equationIntercept C is closely related with Youngrsquos modulus of

the HJC constitutive model ere is also a linear relationbetween the intercept C and Youngrsquos modulus A linearregression equation is also employed by a dimensionlessYoungrsquos modulus Eprime and can be expressed as follows

C 022Eprime + 0129 022EminusEmin

Emax minusEmin+ 0129 (10)

where Emax and Emin are maximum and minimum value ofconsidered Youngrsquos modulus which are 42GPa and 14GParespectively

Hence with end-face nonparallelism and dimensionlessYoungrsquos modulus as variables a binary linear regressionequation is deduced for energy dissipation ratio and isexpressed as follows

Kp 023c + 022Eprime + 0129 (11)

4 Mechanical Damage Evolution Based onEnergy Density Analyses

As deformation and failure progress of the rocklike materialis also the progress of energy dissipation mechanicaldamage can be defined as the ratio of dissipated energydensity to total energy consumption density which can becalculated as follows [20]

D Ud

U (12)


For a certain dynamic stress-strain curve the total en-ergy consumption density U is a definite value For rocklikespecimens with total strain shrinkage in the dynamic stress-strain curve total energy consumption density is reducedduring the release of elastic strain energy in total strain

shrinkage If total energy consumption density is used forthe rocklike specimen with strain shrinkage the mechanicaldamage is approximate to 1 which is inconsistent with theslight broken or damaged rocklike specimen in numericalsimulations Hence energy consumption density at the peak

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

Ener

gy d

ensit

y (J

middotcmndash3

)

28GPaγ = 040

28GPaγ = 020

28GPaγ = 0

14GPaγ = 040

14GPaγ = 020

14GPaγ = 0

42GPaγ = 020

42GPaγ = 0

42GPaγ = 040

Ener

gy d

ensit

y (J

middotcmndash3

)

UUe

Ud

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

ε ε ε

ε ε

ε ε ε

Figure 7 Energy density evolution curves of the rocklike specimen

Table 2 Releasable elastic strain energy density and dissipated energy density at peak dynamic stress (unit Jmiddotcmminus3)

c ()14GPa 21GPa 28GPa 35GPa 42GPa

Uep Ud

p Uep Ud

p Uep Ud

p Uep Ud

p Uep Ud

p

0 1808 0260 1372 0344 1101 0416 0947 0409 0818 0422005 1802 0254 1368 0378 1100 0431 0941 0453 0809 0442010 1803 0319 1356 0394 1098 0420 0937 0445 0804 0459015 1791 0280 1350 0393 1092 0447 0937 0450 0806 0462020 1790 0303 1343 0407 1092 0468 0937 0471 0809 0476025 1786 0348 1342 0415 1093 0473 0936 0492 0808 0511030 1773 0384 1330 0448 1081 0507 0938 0526 0805 0548035 1742 0439 1316 0483 1070 0539 0935 0565 0808 0581040 1708 0471 1286 0534 1051 0582 0923 0599 0799 0629


value of the evolution curve is employed to calculate themechanical damage e energy consumption densities formechanical damage calculation are listed in Table 3

Figure 9 shows the mechanical damage evolution forvarious end-face nonparallelism and Youngrsquos moduli eshape of mechanical damage evolution curve is similar tothat of dissipated energy density

As clearly seen from Figure 9 the mechanical damageevolution is influenced by both end-face nonparallelism andYoungrsquos modulus of the HJC constitutive model In line withtwo regions in the dynamic stress-strain curve prefailure andpostfailure the mechanical damage evolution of the rocklikespecimen in the dynamic loading condition can also be dividedinto to two regions slow-growth region and rapid-growthregion especially in small Youngrsquos modulus In slow-growthregion fluctuation presents in mechanical damage evolutiondue to the existence of nonparallel end face e smaller theYoungrsquos modulus is the more remarkable the fluctuation is Inthe rapid-growth region the end-face nonparallelism showslittle influence on mechanical damage development whileYoungrsquos modulus demonstrates an impact on mechanicaldamage developmente smaller the Youngrsquos modulus is themore rapidly the mechanical damage develops

Transition between two regions of damage evolution isobvious and easy to determine in small Youngrsquos moduluswhile it becomes difficult to determine in large Youngrsquosmodulus as illustrated in Figure 9 With the increment ofend-face nonparallelism the transition is shifted to the rightwhich indicates an increase of both strain and damagethreshold value at the transition While with the increase ofYoungrsquos modulus the strain at the transition decreases andthe damage threshold value at the transition increases and thismay be the result of decreasing crushing volumetric strain μCwith Youngrsquos modulus of the HJC constitutive model

5 Discussion on AllowableProcessing Deviation

For parallel end face rocklike specimens the dynamic stress-strain curve for various Youngrsquos moduli is illustrated inFigure 10

As obvious in Figure 10 Youngrsquos modulus has a greatimpact on the shape of the dynamic stress-strain curveWhen Youngrsquos modulus is 35GPa or 42GPa an elasticunloading with total strain shrinkage is presented whichindicates a slight breakage of the rocklike specimen In theelastic unloading phase the energy consumption decreaseswith total strain shrinkage due to the release of elastic strainenergy With the increment of end-face nonparallelism theelastic unloading phase diminishes gradually thereforereleasable elastic strain energy in elastic unloading alsodecreases Hence the reduction of energy consumptiondensity for Youngrsquos modulus of 42GPa in Figure 7 decreaseswith the increment of end-face nonparallelism With thecontinuous increases of end-face nonparallelism for Youngrsquosmodulus of 35GPa and 42GPa the curve shape of bothdynamic stress-strain and energy evolution is changed andthen the mechanical damage evolution is also changedwhich makes the SHPB test results unreliable

Dynamic characteristics energy density evolution andmechanical damage evolution are desired by conductingSHPB tests Both nonparallel end face and Youngrsquos modulushave a great impact on SHPB test results of rocklike speci-mens Youngrsquos modulus of the rocklike material is an intrinsiccharacteristic of pending tested rocklike materials and it isunknown before the test In order to make the SHPB testresults reliable the errors induced by the rocklike specimenprocessing deviation should be controlled within an ac-ceptable level It is infeasible to give an allowable processingdeviation for various Youngrsquos moduli of rocklike materialserefore a common practice is given an allowable pro-cessing deviation without regard to Youngrsquos modulus

When end-face nonparallelism is 020 the curve shapeof both energy density evolution and mechanical damageevolution remain unchanged and the error induced bynonparallel end face is small According to above analysesmaximum end-face nonparallelism can be controlled within020 namely the allowable processing deviation is005mm for 25mm height rocklike specimen which is twicethe value in ISRM suggested methods [11] Hence the costand time for processing rocklike specimens can be reduced

6 Conclusions

Regarding nonparallel end face of rocklike specimens inSHPB tests numerical simulations have been performed

000 005 010 015 020 025 030 035 04000

01

02

03

04

05

14GPa21GPa28GPa

35GPa42GPa

Kp

γ ()

Figure 8 Energy dissipation ratio versus end-face nonparallelism

Table 3 Energy consumption density for mechanical damagecalculation (unit Jmiddotcmminus3)

c ()U

14GPa 21GPa 28GPa 35GPa 42GPa0 2658 2439 2142 1630 1546005 2627 2427 2190 1661 1576010 2728 2436 2246 1691 1607015 2731 2462 2273 1725 1641020 2838 2482 2296 1769 1673025 2846 2526 2337 1904 1729030 2886 2576 2407 2012 1754035 2930 2646 2407 2272 1895040 2969 2689 2456 2345 2229


with end-face nonparallelism varying from 0 to 040 andYoungrsquos modulus ranging from 14GPa to 42GPa en thecharacteristics of energy dissipation and mechanical damageare analyzed to evaluate the effects of nonparallel end facee main conclusions are summarized as follows

(1) With the increment of end-face nonparallelism bothabsorption energy and reflected energy show a slightincrease trend while transmitted energy presents aslight decrease trend

(2) Both energy consumption density and dissipatedenergy density increase with the increment of end-face nonparallelism while releasable elastic strainenergy density reduces slightly Due to the presenceof nonparallel end face fluctuation presents in theevolution of both releasable elastic strain energydensity and dissipated energy density e fluctua-tion is enhanced with the increment of end-facenonparallelism and weakened with the increase ofYoungrsquos modulus

(3) At the peak dynamic stress dissipated energy densitypresents a linear upward trend with the increment ofend-face nonparallelism and Youngrsquos modulus

while releasable elastic strain energy density shows alinear downward trend A binary linear regressionequation is deduced to estimate energy dissipationratio with end-face nonparallelism and Youngrsquosmodulus

(4) In line with two regions in the dynamic stress-straincurve mechanical damage evolution of the rocklikespecimen is also divided into to two regions slow-growth region and rapid-growth region In theslow-growth region fluctuation presents due to thepresence of nonparallel end face and it weakenswith the increase of Youngrsquos modulus Transitionbetween two regions is shifted to the right with theincrement of end-face nonparallelism which in-dicates an increase of both strain and damagethreshold values

(5) Based on energy density evolution and mechanicaldamage evolution analyses maximum end-facenonparallelism can be controlled within 020namely the allowable processing deviation is005mm for 25mm height rocklike specimen esuggested allowable processing deviation is twice thevalue in ISRM suggested methods which reduces thecost and time for processing rocklike specimens

Data Availability

e datasets generated and analyzed during the currentstudy are available from the corresponding author on rea-sonable request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is research was funded by the National Natural ScienceFoundation of China (no 51774011) Anhui ProvincialNatural Science Foundation (no 1808085QE148) projectfunded by China Postdoctoral Science Foundation (no2018M642504) Natural Science Research Project of Colleges

000 001ε ε

002 003000

025

050

075

100

000 001 002 003000

025

050

075

100

000 001 002 003000

025

050

075

10042GPa28GPa14GPa

D

0010020

030040

ε

D D

Figure 9 Mechanical damage evolution curves of the rocklike specimen

0000 0005 0010 0015 0020 0025 00300

50

100

150

200

250

300

σ (M

Pa)

14GPa21GPa28GPa

35GPa42GPa

ε

Figure 10 Dynamic stress-strain curve for various Youngrsquosmoduli


and Universities in Anhui Province (no KJ2017A097)Young Teacher Scientific Research Project of Anhui Uni-versity of Science and Technology (no QN201607) DoctoralFund Project of Anhui University of Science and Technology(no 11674) Science and Technology Project of Departmentof Housing and Urban-Rural Development of AnhuiProvince (no 2017YF-08) National Innovation and En-trepreneurship Training Program for College Students (no201810361029) and Anhui Provincial Innovation and En-trepreneurship Training Program for College Students (no201810361174)

References

[1] H Kolsky ldquoAn investigation of the mechanical properties ofmaterials at very high rates of loadingrdquo Proceedings of thePhysical Society Section B vol 62 no 11 pp 676ndash700 1949

[2] S Yadav D R Chichili and K T Ramesh ldquoe mechanicalresponse of a 6061-T6 A1A12O3 metal matrix composite athigh rates of deformationrdquo Acta Metallurgica et Materialiavol 43 no 12 pp 4453ndash4464 1995

[3] R L Woodward and R H Brown ldquoDynamic stress-strainproperties of a steel and a brass at strain rates up to 104 persecondrdquo Proceedings of the Institution of Mechanical Engi-neers vol 189 no 1 pp 107ndash115 1975

[4] C A Ross P Y ompson and J W Tedesco ldquoSplit-Hopkinson pressure-bar tests on concrete and mortar intension and compressionrdquo ACI Materials Journal vol 86no 5 pp 475ndash481 1989

[5] G Ravichandran and G Subhash ldquoCritical appraisal oflimiting strain rates for compression testing of ceramics in asplit Hopkinson pressure barrdquo Journal of the American Ce-ramic Society vol 77 no 1 pp 263ndash267 1994

[6] D Ma Q Ma and P Yuan ldquoSHPB tests and dynamicconstitutive model of artificial frozen sandy clay underconfining pressure and temperature staterdquo Cold RegionsScience and Technology vol 136 pp 37ndash43 2017

[7] K Xia and W Yao ldquoDynamic rock tests using split Hop-kinson (Kolsky) bar systemmdasha reviewrdquo Journal of RockMechanics and Geotechnical Engineering vol 7 no 1pp 27ndash59 2015

[8] B Xie D Ai and Y Yang ldquoCrack detection and evolution lawfor rock mass under SHPB impact testsrdquo Shock and Vibrationvol 2019 Article ID 3956749 12 pages 2019

[9] P Baranowski J Malachowski R Gieleta K DamaziakL Mazurkiewicz and D Kolodziejczyk ldquoNumerical study fordetermination of pulse shaping design variables in SHPBapparatusrdquo Bulletin of the Polish Academy of Sciences Tech-nical Sciences vol 61 no 2 pp 459ndash466 2013

[10] F Dai S Huang K Xia and Z Tan ldquoSome fundamentalissues in dynamic compression and tension tests of rocksusing split Hopkinson pressure barrdquo Rock Mechanics andRock Engineering vol 43 no 6 pp 657ndash666 2010

[11] 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

[12] M A Kariem J H Beynon and D Ruan ldquoMisalignmenteffect in the split Hopkinson pressure bar techniquerdquo In-ternational Journal of Impact Engineering vol 47 pp 60ndash702012

[13] X Wu Q Yin Y Wei and C Huang ldquoEffects of imperfectexperimental conditions on stress waves in SHPB experi-mentsrdquo Acta Mechanica Sinica vol 31 no 6 pp 827ndash8362015

[14] R Panowicz J Janiszewski and K Kochanowski ldquoEffects ofsample geometry imperfections on the results of split Hop-kinson pressure bar experimentsrdquo Experimental Techniquespp 1ndash7 2018

[15] P Yuan and Q Ma ldquoCorrection of non-parallel end-faces ofrock specimens in SHPB testsrdquo Explosion and Shock Wavesvol 37 no 5 pp 929ndash936 2017

[16] P Yuan and Q Y Ma ldquoSplit Hopkinson pressure bar tests onsandstone in coalmine under cyclic wetting and dryingrdquo Rockand Soil Mechanics vol 34 no 9 pp 2557ndash2562 2013

[17] H P Xie R D Peng Y Ju and H W Zhou ldquoEnergy analysisof rock failurerdquo Chinese Journal of Rock Mechanics and En-gineering vol 24 no 15 pp 2603ndash2608 2005

[18] 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

[19] J Feng E Wang R Shen L Chen X Li and Z Xu ldquoIn-vestigation on energy dissipation and its mechanism of coalunder dynamic loadsrdquoGeomechanics and Engineering vol 11no 5 pp 657ndash670 2016

[20] PWang J Xu X Fang and PWang ldquoEnergy dissipation anddamage evolution analyses for the dynamic compressionfailure process of red-sandstone after freeze-thaw cyclesrdquoEngineering Geology vol 221 pp 104ndash113 2017

[21] P Yuan Q Y Ma and D D Ma ldquoStress uniformity analyseson nonparallel end-surface rock specimen during loadingprocess in SHPB testsrdquo Advances in Civil Engineeringvol 2018 Article ID 5406931 12 pages 2018

[22] Z Zhou X Li A Liu and Y Zou ldquoStress uniformity of splitHopkinson pressure bar under half-sine wave loadsrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 48 no 4 pp 697ndash701 2011

[23] Z Y Liao J B Zhu K W Xia and C A Tang ldquoDe-termination of dynamic compressive and tensile behavior ofrocks from numerical tests of split Hopkinson pressure andtension barsrdquo Rock Mechanics and Rock Engineering vol 49no 10 pp 3917ndash3934 2016

[24] G P Zou X H Shen Z L Chang Y WWang and PWangldquoA method of restraining the geometric dispersion effect onsplit-Hopkinson pressure bar by the modified striker barrdquoExperimental Techniques vol 40 no 4 pp 1249ndash1261 2016

[25] G M Zhao W W Ma and X R Meng ldquoDamage modes andenergy characteristics of rock-like materials under dynamicloadrdquo Rock and Soil Mechanics vol 36 no 12 pp 3598ndash36052015

[26] G-M Ren H Wu Q Fang and X-Z Kong ldquoParameters ofHolmquist-Johnson-Cook model for high-strength concrete-like materials under projectile impactrdquo International Journalof Protective Structures vol 8 no 3 pp 352ndash367 2017

[27] T J Holmquist G R Johnson and W H Cook ldquoA com-putational constitutive model for concrete subjected to largestrains high strain rates and high pressuresrdquo in Proceedings ofthe 14th International Symposium on Ballistics Quebec CityCanada September 1993

[28] H Zhao ldquoMaterial behaviour characterisation using SHPBtechniques tests and simulationsrdquo Computers amp Structuresvol 81 no 12 pp 1301ndash1310 2003

[29] D J Frew M J Forrestal and W Chen ldquoA split Hopkinsonpressure bar technique to determine compressive stress-strain


data for rock materialsrdquo Experimental Mechanics vol 41no 1 pp 40ndash46 2001

[30] B Song and W Chen ldquoEnergy for specimen deformation in asplit Hopkinson pressure bar experimentrdquo ExperimentalMechanics vol 46 no 3 pp 407ndash410 2006

[31] Y Deng M Chen Y Jin and D Zou ldquoeoretical analysisand experimental research on the energy dissipation of rockcrushing based on fractal theoryrdquo Journal of Natural GasScience and Engineering vol 33 pp 231ndash239 2016

[32] Y Ju H Wang Y Yang Q Hu and R Peng ldquoNumericalsimulation of mechanisms of deformation failure and energydissipation in porous rock media subjected to wave stressesrdquoScience China Technological Sciences vol 53 no 4pp 1098ndash1113 2010

[33] L Tutluoglu I F Oge and C Karpuz ldquoRelationship betweenpre-failure and post-failure mechanical properties of rockmaterial of different originrdquo Rock Mechanics and Rock En-gineering vol 48 no 1 pp 121ndash141 2015


International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

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Active and Passive Electronic Components

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Volume 2018

Control Scienceand Engineering

Journal of



Journal ofEngineeringVolume 2018

SensorsJournal of



RotatingMachinery


Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation


Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

simulations are carried out and the inclination and in-dentation of impact end-surface show a great impact onincident stress waves [13] For a physical SHPB apparatusthe errors induced by imperfect impact interface and barmisalignment can be minimized and eliminated by aligningand manufacturing the bars precisely In SHPB tests thepending tested specimen is sandwiched between the incidentbar and transmitted bar Considering the processing accu-racy of the specimen the specimen geometry imperfectionson SHPB tests for ductile materials are analyzed from thereflected wave transmitted wave and dynamic stress-straincurve and the error induced by specimen geometry im-perfections is small for ductile materials and can beneglected when imperfection angles no larger than 03deg [14]While for rocklike materials the specimen geometry im-perfections in SHPB test is more adverse than ductilemartials When end-face nonparallelism is within 040 thenonparallel end face shows a small and negligible influenceon dynamic stress while it shows a great impact on dynamicstrain and strain rate [15] According to ISRM suggested testmethods [11] cylinder rock specimens with diameter of50mm and length-to-diameter ratio of 05 are widely used inSHPB tests Each rock specimen is processed throughdrilling cutting and grinding processes [16] For shortcylinder rock specimen the processing accuracy and pre-cision are very difficult to control especially the perpen-dicularity of two ends to the axis Besides high processingaccuracy and precision of rock specimens are always relatedwith expensive manufacturing technologies and long pro-cessing time By analyzing the end-face nonparallel effect inthe SHPB test the allowable processing deviation is in-vestigated to reduce the cost and save the processing timewithout affecting the reliability of SHPB tests

e deformation and failure of rock can be considered asan irreversible process of energy dissipation [17] ereforeenergy dissipation of the rock under dynamic loads can bestudied based on SHPB tests [18] and is becoming a majorissue in rock mechanics and rock engineering [19 20]

Considering the processing deviation of the rocklikespecimen numerical simulations of SHPB tests are con-ducted for nonparallel end-face rocklike specimens withvarious Youngrsquos moduli by LS-DYNA During numericalsimulation end-face nonparallelism ranges from 00 to040 and Youngrsquos modulus ranges from 14GPa to 42GPaen energy density dissipation is analyzed to reveal theeffect of end-face nonparallellism As mechanical damageevolution is closely related with energy dissipation the in-fluence of nonparallel end face on mechanical damageevolution is also studied

2 Setup of 3D Numerical Model for SHPB Testand Verification

21 Setup of 3D Numerical Model Based on the physicalΦ50mm SHPB apparatus a series of 3D finite elementmodels without a striker are set up to conduct SHPB tests forrocklike materials Basically a typical SHPB apparatusconsists of a striker an incident bar and a transmitted barAs shown in Figure 1 an incident bar a rocklike specimen

and a transmitted bar are considered and built in a 3D finiteelement model and the nonparallel end face of the rocklikespecimen is contacted with the transmitted bar e lengthand diameter for both incident and transmitted bars are2000mm and 50mm respectively In numerical simulationof SHPB tests an automatic single surface contact isemployed for the contact between two elastic bars androcklike specimen Automatic single surface contact is thesimplest type of contact with no definition of contact ortarget surface and LS-DYNA automatically determineswhich surfaces within a model may come into contact Aslubricant such as Vaseline is applied on both ends ofrocklike specimen in physical SHPB tests the friction effectbetween elastic bars and rocklike specimen is eliminatederefore the interfacial friction effect is negligible in nu-merical simulations

ANSYS is used to prepare 3D finite element models andthe SOLID164 element with one integration point is employedto save the computer time [15 21] e SOLID164 element isan eight-node solid hexahedron element in explicit dynamicanalyses After establishing 3D finite element model a key-word file is output fromANSYS and is modified for LS-DYNAby applying the HolmquistndashJohnsonndashCook (HJC) model tothe rocklike specimen In numerical simulations the meshsensitivity is evaluated by a dimensionless mesh parameter theratio of smallest model size to largest element size Accordingto the research of Kariem et al [12] numerical simulationresults is insensitive to the mesh smaller than 15mm Con-sidering the smallest model size is the diameter 127mm thecritical dimensionless mesh parameter is about 85 for SHPBnumerical simulation en a dimensionless mesh parameterof 10 is chosen and both incident bar and transmitted barconsist of 60000 hexahedron elements Considering smallnonparallel end face a finer mesh is employed for the rocklikespecimen and rocklike specimen consists of 60000 hexahedronelements Hence a total of 180000 hexahedron elements areinvolved in a 3D finite element model

In physical SHPB tests a compressive loading stresswave is generated by launching a striker impacting on theincident bar For traditional rectangular compressive stresswave premature failure of the rocklike specimen beforestress equilibrium makes test results unreliable Moreoverhigh signal oscillation presents in rectangular compressivestress wave due to the wave dispersion [22] erefore thetraditional rectangular compressive loading stress waveshould be modified Half-sine loading stress wave generatedby a cone-shape striker proves to be a suitable rational andeffective waveform for rocklike materials with good im-munity to premature failure before stress equilibriumgeometric dispersion effect and PochhammerndashChree os-cillation [11 22ndash24] Moreover half-sine loading stress wavegives the possibility to an approximate constant strain ratecondition According to the typical example of dynamicstress balance analysis in ISRM suggested test methods [11]the amplitude and duration of half-sine loading stress fornumerical simulation is assumed to be 260MPa and 240 μsrespectively As no striker is in the 3D finite element modelthe half-sine loading stress wave is straightly loaded on thefront-end face of the incident bar






G E

2(1 + ])

μC pC

K

pC(1minus 2])

E

(1)









δ

h

Rocklike specimen

0 60 120 180 2400

100

200

300

σ (M

Pa)

t (μs)

YX Z




Wi AC

E1113946 σ(t)

2i dt i I R T (2)




E1113946σ(t)

2Idt

minusAC

E1113946σ(t)

2Rdtminus

AC

E1113946σ(t)

2Tdt

(3)


WL AC

E11139462σ(t)Rσ(t)Tdt (4)






0 50 100 150

σ(t)I

σ(t)R

σ(t)T

σ(t)I + σ(t)R

200 250ndash200

ndash100

0

100

200

300

σ (M

Pa)

t (micros)


0

100

200

300

0

100

200

300

0

100

200

300

0

100

200

300

0 50 100 150 200 250 300

0

100

200

30042GPa

35GPa

28GPa

21GPa

σ (M

Pa)

σ (M

Pa)

σ (M

Pa)

σ (M

Pa)

σ (M

Pa)

14GPa

σ(t)I + σ(t)R


t (μs)

σ(t)T








U WL

Vs 1113946σ dε 1113946σ _ε dt 1113946σ(t)T

2C

ls

σ(t)RE

dt

C


(5)



005 010 015 020 025 030 035 040000

50

100

150

200

250

300

350

Ener

gy (J

)

γ ()



0


000 005 010 015 020 025 030 035 04030

60

90

120

150

14GPa21GPa28GPa

35GPa42GPa

WL (

J)

γ ()


0 50 100 150 200 250 300

0

100

200

300

400

500

Ener

gy (J

)

t (micros)

WIWR

WTWL



Ue

12σεe

12σ(t)2T

Ed (6)

Ud

UminusUe (7)










Kp Ud

p

Up (8)



Kp 023c + C (9)







Kp 023c + 022Eprime + 0129 (11)



D Ud

U (12)




000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

Ener

gy d

ensit

y (J

middotcmndash3

)

28GPaγ = 040

28GPaγ = 020

28GPaγ = 0

14GPaγ = 040

14GPaγ = 020

14GPaγ = 0

42GPaγ = 020

42GPaγ = 0

42GPaγ = 040

Ener

gy d

ensit

y (J

middotcmndash3

)

UUe

Ud

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

ε ε ε

ε ε

ε ε ε




Uep Ud

p Uep Ud

p Uep Ud

p Uep Ud

p Uep Ud

p

0 1808 0260 1372 0344 1101 0416 0947 0409 0818 0422005 1802 0254 1368 0378 1100 0431 0941 0453 0809 0442010 1803 0319 1356 0394 1098 0420 0937 0445 0804 0459015 1791 0280 1350 0393 1092 0447 0937 0450 0806 0462020 1790 0303 1343 0407 1092 0468 0937 0471 0809 0476025 1786 0348 1342 0415 1093 0473 0936 0492 0808 0511030 1773 0384 1330 0448 1081 0507 0938 0526 0805 0548035 1742 0439 1316 0483 1070 0539 0935 0565 0808 0581040 1708 0471 1286 0534 1051 0582 0923 0599 0799 0629











6 Conclusions


000 005 010 015 020 025 030 035 04000

01

02

03

04

05

14GPa21GPa28GPa

35GPa42GPa

Kp

γ ()



c ()U










Data Availability




Acknowledgments


000 001ε ε

002 003000

025

050

075

100

000 001 002 003000

025

050

075

100

000 001 002 003000

025

050

075

10042GPa28GPa14GPa

D

0010020

030040

ε

D D


0000 0005 0010 0015 0020 0025 00300

50

100

150

200

250

300

σ (M

Pa)

14GPa21GPa28GPa

35GPa42GPa

ε




References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






G E

2(1 + ])

μC pC

K

pC(1minus 2])

E

(1)









δ

h

Rocklike specimen

0 60 120 180 2400

100

200

300

σ (M

Pa)

t (μs)

YX Z




Wi AC

E1113946 σ(t)

2i dt i I R T (2)




E1113946σ(t)

2Idt

minusAC

E1113946σ(t)

2Rdtminus

AC

E1113946σ(t)

2Tdt

(3)


WL AC

E11139462σ(t)Rσ(t)Tdt (4)






0 50 100 150

σ(t)I

σ(t)R

σ(t)T

σ(t)I + σ(t)R

200 250ndash200

ndash100

0

100

200

300

σ (M

Pa)

t (micros)


0

100

200

300

0

100

200

300

0

100

200

300

0

100

200

300

0 50 100 150 200 250 300

0

100

200

30042GPa

35GPa

28GPa

21GPa

σ (M

Pa)

σ (M

Pa)

σ (M

Pa)

σ (M

Pa)

σ (M

Pa)

14GPa

σ(t)I + σ(t)R


t (μs)

σ(t)T








U WL

Vs 1113946σ dε 1113946σ _ε dt 1113946σ(t)T

2C

ls

σ(t)RE

dt

C


(5)



005 010 015 020 025 030 035 040000

50

100

150

200

250

300

350

Ener

gy (J

)

γ ()



0


000 005 010 015 020 025 030 035 04030

60

90

120

150

14GPa21GPa28GPa

35GPa42GPa

WL (

J)

γ ()


0 50 100 150 200 250 300

0

100

200

300

400

500

Ener

gy (J

)

t (micros)

WIWR

WTWL



Ue

12σεe

12σ(t)2T

Ed (6)

Ud

UminusUe (7)










Kp Ud

p

Up (8)



Kp 023c + C (9)







Kp 023c + 022Eprime + 0129 (11)



D Ud

U (12)




000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

Ener

gy d

ensit

y (J

middotcmndash3

)

28GPaγ = 040

28GPaγ = 020

28GPaγ = 0

14GPaγ = 040

14GPaγ = 020

14GPaγ = 0

42GPaγ = 020

42GPaγ = 0

42GPaγ = 040

Ener

gy d

ensit

y (J

middotcmndash3

)

UUe

Ud

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

ε ε ε

ε ε

ε ε ε




Uep Ud

p Uep Ud

p Uep Ud

p Uep Ud

p Uep Ud

p

0 1808 0260 1372 0344 1101 0416 0947 0409 0818 0422005 1802 0254 1368 0378 1100 0431 0941 0453 0809 0442010 1803 0319 1356 0394 1098 0420 0937 0445 0804 0459015 1791 0280 1350 0393 1092 0447 0937 0450 0806 0462020 1790 0303 1343 0407 1092 0468 0937 0471 0809 0476025 1786 0348 1342 0415 1093 0473 0936 0492 0808 0511030 1773 0384 1330 0448 1081 0507 0938 0526 0805 0548035 1742 0439 1316 0483 1070 0539 0935 0565 0808 0581040 1708 0471 1286 0534 1051 0582 0923 0599 0799 0629











6 Conclusions


000 005 010 015 020 025 030 035 04000

01

02

03

04

05

14GPa21GPa28GPa

35GPa42GPa

Kp

γ ()



c ()U










Data Availability




Acknowledgments


000 001ε ε

002 003000

025

050

075

100

000 001 002 003000

025

050

075

100

000 001 002 003000

025

050

075

10042GPa28GPa14GPa

D

0010020

030040

ε

D D


0000 0005 0010 0015 0020 0025 00300

50

100

150

200

250

300

σ (M

Pa)

14GPa21GPa28GPa

35GPa42GPa

ε




References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Wi AC

E1113946 σ(t)

2i dt i I R T (2)




E1113946σ(t)

2Idt

minusAC

E1113946σ(t)

2Rdtminus

AC

E1113946σ(t)

2Tdt

(3)


WL AC

E11139462σ(t)Rσ(t)Tdt (4)






0 50 100 150

σ(t)I

σ(t)R

σ(t)T

σ(t)I + σ(t)R

200 250ndash200

ndash100

0

100

200

300

σ (M

Pa)

t (micros)


0

100

200

300

0

100

200

300

0

100

200

300

0

100

200

300

0 50 100 150 200 250 300

0

100

200

30042GPa

35GPa

28GPa

21GPa

σ (M

Pa)

σ (M

Pa)

σ (M

Pa)

σ (M

Pa)

σ (M

Pa)

14GPa

σ(t)I + σ(t)R


t (μs)

σ(t)T








U WL

Vs 1113946σ dε 1113946σ _ε dt 1113946σ(t)T

2C

ls

σ(t)RE

dt

C


(5)



005 010 015 020 025 030 035 040000

50

100

150

200

250

300

350

Ener

gy (J

)

γ ()



0


000 005 010 015 020 025 030 035 04030

60

90

120

150

14GPa21GPa28GPa

35GPa42GPa

WL (

J)

γ ()


0 50 100 150 200 250 300

0

100

200

300

400

500

Ener

gy (J

)

t (micros)

WIWR

WTWL



Ue

12σεe

12σ(t)2T

Ed (6)

Ud

UminusUe (7)










Kp Ud

p

Up (8)



Kp 023c + C (9)







Kp 023c + 022Eprime + 0129 (11)



D Ud

U (12)




000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

Ener

gy d

ensit

y (J

middotcmndash3

)

28GPaγ = 040

28GPaγ = 020

28GPaγ = 0

14GPaγ = 040

14GPaγ = 020

14GPaγ = 0

42GPaγ = 020

42GPaγ = 0

42GPaγ = 040

Ener

gy d

ensit

y (J

middotcmndash3

)

UUe

Ud

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

ε ε ε

ε ε

ε ε ε




Uep Ud

p Uep Ud

p Uep Ud

p Uep Ud

p Uep Ud

p

0 1808 0260 1372 0344 1101 0416 0947 0409 0818 0422005 1802 0254 1368 0378 1100 0431 0941 0453 0809 0442010 1803 0319 1356 0394 1098 0420 0937 0445 0804 0459015 1791 0280 1350 0393 1092 0447 0937 0450 0806 0462020 1790 0303 1343 0407 1092 0468 0937 0471 0809 0476025 1786 0348 1342 0415 1093 0473 0936 0492 0808 0511030 1773 0384 1330 0448 1081 0507 0938 0526 0805 0548035 1742 0439 1316 0483 1070 0539 0935 0565 0808 0581040 1708 0471 1286 0534 1051 0582 0923 0599 0799 0629











6 Conclusions


000 005 010 015 020 025 030 035 04000

01

02

03

04

05

14GPa21GPa28GPa

35GPa42GPa

Kp

γ ()



c ()U










Data Availability




Acknowledgments


000 001ε ε

002 003000

025

050

075

100

000 001 002 003000

025

050

075

100

000 001 002 003000

025

050

075

10042GPa28GPa14GPa

D

0010020

030040

ε

D D


0000 0005 0010 0015 0020 0025 00300

50

100

150

200

250

300

σ (M

Pa)

14GPa21GPa28GPa

35GPa42GPa

ε




References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







U WL

Vs 1113946σ dε 1113946σ _ε dt 1113946σ(t)T

2C

ls

σ(t)RE

dt

C


(5)



005 010 015 020 025 030 035 040000

50

100

150

200

250

300

350

Ener

gy (J

)

γ ()



0


000 005 010 015 020 025 030 035 04030

60

90

120

150

14GPa21GPa28GPa

35GPa42GPa

WL (

J)

γ ()


0 50 100 150 200 250 300

0

100

200

300

400

500

Ener

gy (J

)

t (micros)

WIWR

WTWL



Ue

12σεe

12σ(t)2T

Ed (6)

Ud

UminusUe (7)










Kp Ud

p

Up (8)



Kp 023c + C (9)







Kp 023c + 022Eprime + 0129 (11)



D Ud

U (12)




000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

Ener

gy d

ensit

y (J

middotcmndash3

)

28GPaγ = 040

28GPaγ = 020

28GPaγ = 0

14GPaγ = 040

14GPaγ = 020

14GPaγ = 0

42GPaγ = 020

42GPaγ = 0

42GPaγ = 040

Ener

gy d

ensit

y (J

middotcmndash3

)

UUe

Ud

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

ε ε ε

ε ε

ε ε ε




Uep Ud

p Uep Ud

p Uep Ud

p Uep Ud

p Uep Ud

p

0 1808 0260 1372 0344 1101 0416 0947 0409 0818 0422005 1802 0254 1368 0378 1100 0431 0941 0453 0809 0442010 1803 0319 1356 0394 1098 0420 0937 0445 0804 0459015 1791 0280 1350 0393 1092 0447 0937 0450 0806 0462020 1790 0303 1343 0407 1092 0468 0937 0471 0809 0476025 1786 0348 1342 0415 1093 0473 0936 0492 0808 0511030 1773 0384 1330 0448 1081 0507 0938 0526 0805 0548035 1742 0439 1316 0483 1070 0539 0935 0565 0808 0581040 1708 0471 1286 0534 1051 0582 0923 0599 0799 0629











6 Conclusions


000 005 010 015 020 025 030 035 04000

01

02

03

04

05

14GPa21GPa28GPa

35GPa42GPa

Kp

γ ()



c ()U










Data Availability




Acknowledgments


000 001ε ε

002 003000

025

050

075

100

000 001 002 003000

025

050

075

100

000 001 002 003000

025

050

075

10042GPa28GPa14GPa

D

0010020

030040

ε

D D


0000 0005 0010 0015 0020 0025 00300

50

100

150

200

250

300

σ (M

Pa)

14GPa21GPa28GPa

35GPa42GPa

ε




References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Ue

12σεe

12σ(t)2T

Ed (6)

Ud

UminusUe (7)










Kp Ud

p

Up (8)



Kp 023c + C (9)







Kp 023c + 022Eprime + 0129 (11)



D Ud

U (12)




000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

Ener

gy d

ensit

y (J

middotcmndash3

)

28GPaγ = 040

28GPaγ = 020

28GPaγ = 0

14GPaγ = 040

14GPaγ = 020

14GPaγ = 0

42GPaγ = 020

42GPaγ = 0

42GPaγ = 040

Ener

gy d

ensit

y (J

middotcmndash3

)

UUe

Ud

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

ε ε ε

ε ε

ε ε ε




Uep Ud

p Uep Ud

p Uep Ud

p Uep Ud

p Uep Ud

p

0 1808 0260 1372 0344 1101 0416 0947 0409 0818 0422005 1802 0254 1368 0378 1100 0431 0941 0453 0809 0442010 1803 0319 1356 0394 1098 0420 0937 0445 0804 0459015 1791 0280 1350 0393 1092 0447 0937 0450 0806 0462020 1790 0303 1343 0407 1092 0468 0937 0471 0809 0476025 1786 0348 1342 0415 1093 0473 0936 0492 0808 0511030 1773 0384 1330 0448 1081 0507 0938 0526 0805 0548035 1742 0439 1316 0483 1070 0539 0935 0565 0808 0581040 1708 0471 1286 0534 1051 0582 0923 0599 0799 0629











6 Conclusions


000 005 010 015 020 025 030 035 04000

01

02

03

04

05

14GPa21GPa28GPa

35GPa42GPa

Kp

γ ()



c ()U










Data Availability




Acknowledgments


000 001ε ε

002 003000

025

050

075

100

000 001 002 003000

025

050

075

100

000 001 002 003000

025

050

075

10042GPa28GPa14GPa

D

0010020

030040

ε

D D


0000 0005 0010 0015 0020 0025 00300

50

100

150

200

250

300

σ (M

Pa)

14GPa21GPa28GPa

35GPa42GPa

ε




References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

000 001 002 0030

1

2

3

Ener

gy d

ensit

y (J

middotcmndash3

)

28GPaγ = 040

28GPaγ = 020

28GPaγ = 0

14GPaγ = 040

14GPaγ = 020

14GPaγ = 0

42GPaγ = 020

42GPaγ = 0

42GPaγ = 040

Ener

gy d

ensit

y (J

middotcmndash3

)

UUe

Ud

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

Ener

gy d

ensit

y (J

middotcmndash3

)

ε ε ε

ε ε

ε ε ε




Uep Ud

p Uep Ud

p Uep Ud

p Uep Ud

p Uep Ud

p

0 1808 0260 1372 0344 1101 0416 0947 0409 0818 0422005 1802 0254 1368 0378 1100 0431 0941 0453 0809 0442010 1803 0319 1356 0394 1098 0420 0937 0445 0804 0459015 1791 0280 1350 0393 1092 0447 0937 0450 0806 0462020 1790 0303 1343 0407 1092 0468 0937 0471 0809 0476025 1786 0348 1342 0415 1093 0473 0936 0492 0808 0511030 1773 0384 1330 0448 1081 0507 0938 0526 0805 0548035 1742 0439 1316 0483 1070 0539 0935 0565 0808 0581040 1708 0471 1286 0534 1051 0582 0923 0599 0799 0629











6 Conclusions


000 005 010 015 020 025 030 035 04000

01

02

03

04

05

14GPa21GPa28GPa

35GPa42GPa

Kp

γ ()



c ()U










Data Availability




Acknowledgments


000 001ε ε

002 003000

025

050

075

100

000 001 002 003000

025

050

075

100

000 001 002 003000

025

050

075

10042GPa28GPa14GPa

D

0010020

030040

ε

D D


0000 0005 0010 0015 0020 0025 00300

50

100

150

200

250

300

σ (M

Pa)

14GPa21GPa28GPa

35GPa42GPa

ε




References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia











6 Conclusions


000 005 010 015 020 025 030 035 04000

01

02

03

04

05

14GPa21GPa28GPa

35GPa42GPa

Kp

γ ()



c ()U










Data Availability




Acknowledgments


000 001ε ε

002 003000

025

050

075

100

000 001 002 003000

025

050

075

100

000 001 002 003000

025

050

075

10042GPa28GPa14GPa

D

0010020

030040

ε

D D


0000 0005 0010 0015 0020 0025 00300

50

100

150

200

250

300

σ (M

Pa)

14GPa21GPa28GPa

35GPa42GPa

ε




References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia









Data Availability




Acknowledgments


000 001ε ε

002 003000

025

050

075

100

000 001 002 003000

025

050

075

100

000 001 002 003000

025

050

075

10042GPa28GPa14GPa

D

0010020

030040

ε

D D


0000 0005 0010 0015 0020 0025 00300

50

100

150

200

250

300

σ (M

Pa)

14GPa21GPa28GPa

35GPa42GPa

ε




References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia










RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Documents

EffectofNonparallelEndFaceonEnergyDissipationAnalysesof ...downloads.hindawi.com/journals/sv/2019/2040947.pdffTff fTfS fTfH fTfB f S H B f f S B fTff fTfS fTfH fTfB f S H B f S H B