StudyofTestingMethodforDynamicInitiationToughnessof...

Research ArticleStudy of Testing Method for Dynamic Initiation Toughness ofSandstone under Blasting Loading

Dingjun Xiao 12 Zheming Zhu 1 Rong Hu1 and Lin Lang1

1MOE Key Laboratory of Deep Underground Science and Engineering College of Architecture and EnvironmentSichuan University Chengdu 610065 China2Shock and Vibration of Engineering Materials and Structures Key Laboratory of Sichuan ProvinceSouthwest University Science and Technology Mianyang Sichuan 621010 China

Correspondence should be addressed to Zheming Zhu zhemingzhuhotmailcom

Received 24 September 2018 Revised 15 November 2018 Accepted 25 November 2018 Published 24 December 2018

Academic Editor Alvaro Cunha

Copyright copy 2018 Dingjun Xiao et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper an internal central single-cracked disk (ICSCD) specimen was proposed for the study of dynamic fracture initiationtoughness of sandstone under blasting loadinge ICSCD specimen had a diameter of 400mm sandstone disc with a 60mm longcrack Blasting tests were conducted by using the ICSCD specimens e blasting strain-time curve was obtained from the radialstrain gauges placed around the blast hole e fracture initiation time was determined by circumferential strain gauges placedaround the crack tip e stress history on the blast hole of the sandstone specimen was then derived from measured strain curvethrough the Laplace transform e numerical solutions were further obtained by the numerical inversion method A numericalmodel was established using the finite element software ANSYS e type I dynamic stress intensity factor curves of sandstoneunder blasting loading were derived by the mutual interaction integration method e results showed that (1) the ICSCDspecimen can be used to measure dynamic initiation fracture toughness of rocks (2) the stress on the blast hole wall can beobtained by the Laplace numerical inversion method (3) the dynamic initiation fracture toughness of the ICSCD sandstonespecimen can be calculated by the experimental-numerical method with a maximum error of only 7

1 Introduction

Drilling and blasting is one of the most common eco-nomical and efficient technologies and it has been widelyutilized in rock mass excavation and engineering con-struction applications However the use of such techniquesin engineering practices has also raised considerable engi-neering stability and safety issues for researchers all over theworld [1ndash3] Understanding the dynamic fracturing be-havior of rock as a heterogeneous material under blastingloading has important implications for both realizing moreefficient rock breakage and preservation of rock masseserefore exploring the behavior of rock under dynamicloading conditions has drawn special attention [4 5] Inparticular how rocks break under blasting loading the sizedistribution of the rock fragments and the level of damage tothe remaining rock body are some of the key issues exploredin many research studies

As natural materials rocks contain large numbers ofmicrocracks ese microcracks can affect the sizes of thefragments after blasting and in certain cases result in in-stability issues in rock engineering e dynamic fracturetoughness is an important rock parameter that measures itsresistance to dynamic crack initialization and growth as wellas its ability to arrest cracks Exploring the dynamic fracturetoughness of rock samples allows researchers understand thecharacteristics of crack initialization growth and arrestthus enabling them to predict and control the fracturingbehavior of rocks In summary measuring the dynamicfracture toughness of rocks is the basis for investigating theirdynamic fracturing behavior which requires a strong the-oretical background and advanced experimental techniques

A disc specimen and developed different configura-tions for loading the test sample were redesigned based onthe Hopkinson pressure bar [6 7] Zhou et al [8] and Zhanget al [9] redesigned a disc specimen and developed different

HindawiShock and VibrationVolume 2018 Article ID 1043298 13 pageshttpsdoiorg10115520181043298

configurations for loading the test sample based on theHopkinson pressure bar He further loaded the experimentalmeasurement data onto commercial finite element softwareto perform extensive calculations and proposed a combinedexperimental-numerical approach to test the dynamicfracture toughness of rock samples Zhou et al [10] proposedan NSCB method to test the dynamic initiation fracturetoughness of rock samples using the Hopkinson pressure baras the loading platform His method was further recom-mended by ISRM as a standard dynamic rock testingmethod However many achievements have been made indynamic rock studies using the Hopkinson pressure bar asthe loading platform Blasting load is more complicated thanthe impacting load First the wave types are different eimpact-induced stress wave is one-dimensional wavewhereas the blast-induced stress wave can be a cylindricalspherical or plane wave Second the loading rates aredifferent e loading rate of blasts is generally larger thanthat of impacts Both the explosive stress wave and explosivegases in the drilling and blasting processes affect rockfracture [11 12] erefore exploring the fracture toughnessof rock samples under blasting loading is a more appropriateapproach to resolving engineering blasting issues

Rocks are brittle material with natural cracks eirinertia and size can often affect test results [13 14] emaximum allowable diameter of the Hopkinson pressure baris usually around 100mm which imposes a severe limitationon the size of the rock specimen It is generally morechallenging to analyze the dynamic fracture behavior of rockspecimens with smaller sizes erefore design of testingplatforms that allows the study of fracture toughness of largerock specimens is a necessary task which can providevaluable information on rock blasting excavation engi-neering and stability analysis of rock mass after excavation

In this study we designed and rationalized a test con-figuration for studying the dynamic fracture toughnessunder blasting loading by drilling and blasting in rocks Anew testing method was also proposed for obtaining theinitiation fracture toughness of rocks Using the proposedmethod we obtained the dynamic initiation fracturetoughness of the rock specimen under blasting loadingconditions Such methods can contribute to the enrichmentof testing approaches for evaluating the dynamic fracturetoughness of rocks e flow chart of experimental nu-merical method is shown in Figure 1

2 Design of Blasting Loading and TestSpecimen Configurations

21 Blasting Loading Device e use of high-pressureloading devices is a popular approach for studying thedynamic performance of materials at high strain rates Withincreasing in-depth studies of the dynamic properties ofmaterials high-pressure loading techniques have been re-ceiving more attention from researchers Techniques such asdropping hammer Hopkinson pressure bar light gas gunand explosive blasting [15 16] have been widely used forstudying the dynamic properties of materials Compared toother dynamic loading methods explosive blasting is a more

advantageous approach for being simple convenient lowcost and free of size limitations However the explosiveblasting method also entails complex loading mechanismand has poor repeatability and stability erefore manyresearchers have developed alternative explosive blastingmethods such as the explosive plane-wave generator and theexplosive expansion ring [17] for analyzing the dynamicproperties of materials under blasting loading conditions

In this study the structure of an explosive loading devicewas designed from the engineering drilling and blastingperspective An industrial detonating cord was used togenerate stable and reliable blasting loads e detonatingcord was filled with 12 g of powder at every meter e outerdiameter and detonation velocity of the detonating cordwere 5mm and 6690ms respectively In order to minimizethe radius of the explosive pack in the rock-pulverizing zone(3ndash7) r and to strengthen the stress wave the pulverizingzone was filled with water so that the detonating cord and theblasting wall were coupled by water which enabled trans-mission of the explosive stress wave In order to preventearly release of the explosive gases and to restrict the dis-placement in the Z direction for achieving a quasi-planestrain condition the specimen was covered with a platemade from the same materials on both the top and bottomsurfaces e blasting holes were coupled with the plate-covering by high-strength explosion-proof tubes for re-ducing the damage on the plate-covering from the deto-nating cord e configuration of the loading device isshown in Figure 2

According to the literature the testing specimen can beconsidered to be under plane stress if the strain of thespecimen in the Z direction under dynamic loading is lessthan 15 of the strain in the X or Y direction [18] In order tovalidate the quasi-plane stress model strain gaugesdeforming in theX and Z directions were placed at a distanceof 80mm away from the center of the blasting hole and thestrain tests were subsequently performed e loadingconfiguration and the testing results are shown in Figure 3 Itcan be seen that the peak strain in the Z direction is only 16of that in the X direction erefore the requirements fortreating the testing specimen as under plane stress aresatisfied

22 Design and Rationalization of the SpecimenConfiguration As early as 1955 ISRM proposed to use alambdoidal slotted Brazilian disc for testing the staticfracture toughness of rocks However it was not until 2012that the first dynamic testing methods for rocks were

Strain measurementnear the blast hole

Backcalculatedpressure curve

Calculation of dynamicstress intensity factor

Fracture initiationtime measurement

Dynamic initiationfracture toughness

Figure 1 Flow chart of experimental numerical method

2 Shock and Vibration

developed ese included the dynamic compression dy-namic stretching and the dynamic fracturing of rocks [19]e dynamic test of rocks is a complex process where almostall the available test designs are developed using the Hop-kinson pressure bar as the loading approach Such loadingconfiguration only considers the effects of the stress wavewhich is different from the stress wave and explosive gasgenerated in an actual blasting engineering project [12]Furthermore when using the Hopkinson pressure bar as theloading tool the size of the testing specimen is limited by themaximum available size of the Hopkinson pressure bar

erefore considering the typical characteristics ofdrilling and blasting loading in common field practices alarge testing specimen was selected for this studye testingspecimen was a disc containing a center hall with a diameterof 44mm is configuration allows the blasting shock waveto transmit to the coupling water medium which can po-tentially increase the range of the cracking zone and min-imize the pulverizing area e diameter of the disc was

designed to be 400mm in order to ensure that the ratiobetween the size of the center hole and the disc diameter fellwithin 01ndash03 is value was also used in a previous studyby Zhang and Li [20] Figure 3 displays the configuration ofthe testing specimen e blasting tests were performed infour internal center single crack discs (ICSCD) with thesame dimensions all of which were made from the samesandstone blocks as shown in Figure 4

A minimum thickness of the specimen is required toensure a plane-stress condition is value can be calculatedbased on equation (1) which yields d ge 10mm Consideringthe convenience in obtaining the target specimen and thelarge number of dynamic fracturing tests required in thisstudy the final thickness was selected to be 20mm whichsatisfies the requirement for a plane-stress condition

dge 25KIC

σs1113888 1113889

2

(1)

where d and σs are the thickness and yield strength of thespecimen respectively Since the yield strength does notapply to rock-type materials we substitute this value withthe dynamic strength of rock [21] which yieldsσs 100 Mpa Based on a prior study by Yang et al [22] whomeasured the fracture toughness of sandstone using theHopkinson pressure bar we have KIC 653 MPam12

To avoid the effect of the reflection wave a small crackwith a length of 60mm and a width of 05mm was manuallycarved into the specimen 80mm away from the center of theblast hole e tip of the crack was treated with a finemachining process to ensure that the width of the crack tipwas less than 01mm e minimum distance required forthe reflection wave to reach G2 is 320mm Considering thatthe longitudinal wave velocity is around 2339ms it takes1368 μs for the reflection wave to reach the crack top at G2We experimentally determined the maximum fracture ini-tiation time to be 864 μs is value is smaller than the timerequired for the reflection wave to reach G2 erefore bythe time the reflection wave hits G2 an initial fracture hasalready formed at the crack tip us the initiation fracturetoughness test is free from the impact of the reflection wave

3 Dynamic Fracture Test of Sandstone

31 Dynamic Strain Test of Sandstone under Blasting LoadingBlasting tests are usually destructive tests in which the hightemperature and high pressure environment near the testingcenter make it difficult to perform measurements e costof tests can also be quite prohibitive in most casesereforein our tests strain gauges were used as low-cost and highlyefficient tools for measuring the dynamic strain During theexperiment the DH5939 high speed data acquisition systemwith a maximum sampling frequency of 10MHz was used tocollect the testing data e response frequency of the strainamplifier was DC sim 1MHz e entire strain test system isshown in Figure 5

e choice of the right strain gauge has a huge impact onthe test results Specifically the response rate of the straingauge is determined by its gate length where a smaller gate

6000

4000

2000

0

0 20 40 60Time (μs)

Stra

in (μ

ε)

G1G2G3

G1

40mm

G2 G3

80mm

Figure 3 Contrast curve of strain

Detonator

PMMADetonating cordWaterTube

AirCovering plateSpecimen

Figure 2 Sketch of specimen under explosive loading

Shock and Vibration 3

length indicates a higher response ratee BA120-1AA foil-type strain gauge was used to measure the strain responsecurve near the blasting center while the BA120-10AA foil-type strain gauge was used to measure the fracturing time faraway from the blasting center e parameters of the straingauges are listed in Table 1

As the detonation wave generated by the explosionpropagates to the interface between the explosive pack and therock a shock wave with extremely high peak pressure will begenerated inside the rock at 3ndash7r (75ndash175mm r is the radiusof the explosive pack) away from the center of blast hole epower of such shock wave can often exceed the dynamiccompressive strength of the rock and therefore results inplastic deformation or pulverization of the rock Most of theexplosion energy is consumed during this process erefore

the shock wave decays into a stress wave beyond the crushingzone and the parameters of the wave vibration surface tend tochange slowly afterwards is region covered by the stresswave can extend to 120ndash150r (300ndash375mm) Based on adetonating cord diameter of 5mm the maximum distance ofthe area under the influence of the shock wave influence wasfound to be 175mm erefore the stress waves measuring

Premade crack

Strain gauge

Fracture initiationstrain gauge

Explosive

Water

Sandstone

(a)

60 mm

60 m

m40

mm

40 m

m

40 mm 40 mm 80 mm

G3 G2 G1 G7

G4

G5

G6

(b)

Figure 4 Sketch map of specimens (a) Actual image of the specimen (b) Schematic of the specimen

Trigger system Data acquisition instrument

Data analysis systemStrain amplifier

Specimen

Figure 5 Strain test system

Table 1 Parameters of the strain gauges

Model Gate sensitivity(mm)

Resistance(Ω)

Sensitivity()

BA120-10AA 98 times 30 120 plusmn 02 221 plusmn 1



points were set at 40mm away from the center of the blasthole to mitigate the impact from the shock wave as well as toobtain accurate measurements of the strain curve duringblasting Gauges 4 5 and 6 were used to characterize theattenuation of the explosive strain wave Gauges 1 and 7 wereused as backups for Gauges 4 and 5 in case the measurementdata were lost and Gauges 2 and 3 were used to measure thefracture initiation time

32 Basic Dynamic Mechanical Parameters of SandstoneRock is a nonlinear elastic material composed of mineralparticles with many different internal structures such asweak surfaces and cracks e high nonuniformity of rock isa primary difficulty in dynamic testing Ultrasonic testing isone of the prevalent approaches for obtaining the basicdynamic parameters of rock samples and has been widelyused in the field of rock mechanics [23] In this study thelongitudinal wave and shear wave velocities of the testspecimen were measured using RSM-SY5(T) nonmetallicacoustic wave detector which yielded Cp 2339ms and Cs 1430ms e density of the rock was measured to be ρ

2163 kgm3e dynamic parameters can be calculated usingequations (2)ndash(5) based on the elastic wave theory [24] efinal dynamic parameters of the test specimen were υd 02Ed 1063 GPa Gd 442 GPa and Kd 444 GPa

υd C2p minus 2C2

s

2 C2p minusC2

s1113872 1113873 (2)

Ed ρC2

p 1 + υd( 1113857 1minus 2υd( 1113857

1minus υd (3)

Gd Ed

2 1 + υd( 1113857 (4)

Kd Ed

3 1minus 2υd( 1113857 (5)

where υd is dynamic Poissonrsquos ratio Ed is dynamic elasticmodulus Gd is dynamic shear modulus and Kd is anddynamic bulk modulus

33 Analysis of Dynamic Strain Test Results of Sandstonee ratio between the distance to the center of the explosivepack and the radius of the explosive pack is defined as r rr0e initial time is defined to be the instance where the slope ofthe blasting test strain curve is maximized e ending time isdefined to be the time when the peak value of the strain curvedecays to 20 of the maximum value e loading time isdefined as the difference between the strain peak time and thestrain start time e unloading time is defined as the dif-ference between the strain end time and strain peak time

Blasting loading is a dynamic loading whose magnitudechanges significantly over time is change becomes morepronounced as the distance ratio increases e strain starttime is proportional to the velocity of the longitudinal waveIncreasing the distance ratio can result in a longer loading

time of the strain wave With respect to the unloading timeof the strain wave it will first increase and approach a criticalvalue near the crack tip with the increasing distance ratio Asthe crack starts to fracture during the test the internalpressure drops radically which leads to a sharp reduction inthe unloading time afterwards

When the distance ratio was set as 16 the maximumdifference in the strain start time between the four specimenswas only 04 μs which indicated a good match in the testingsystem between the different samples When the distanceratio varied between 16 and 32 the loading time and theunloading time of the blasting strain wave ranged between13 to 46 μs and 191 to 1397 μs respectively e testingtime parameters and the corresponding peak strains arelisted in Table 2

e peak radial strain during the blasting εr decreasedexponentially with increasing the distance ratio r e peakstrain decayed at a much faster rate with increasing thedistance ratio with an average decay coefficient of 078However once the peak strain was over it took a substantiallylong time before the strain reduced completely to zeroTypical strain curves during the test are shown in Figure 6

e dynamic fracturing time of the crack was a keyparameter in our test Two types of strain gauges were usedin the tests e gate lengths of the first and second types ofthe strain gauges were 98mm and 1mm respectively estrain gauge with the smaller gate size exhibited a higherfrequency response and could be used to record the elasticstrain wave near the blasting hole e strain gauge with thelarger gate size could be used to measure the fracture ini-tiation time of the crack e large strain gauge was pre-processed prior to use as shown in Figure 4 A smalltriangular notch was cut on the basis of the strain gaugeusing an art blade e notch extended all the wayapproaching the sensitive gate of the strain gauge such thatthe fracture initiation time could be recorded at the instancewhen the crack fractures startede fracturing time and theblasting loading signals were collected at the same startingtime In other words the fracturing signal and the blastingloading signals shared the same initial time t 0

Due to the stress concentration effect on the crack tipfracture would only occur if the accumulation of the strainenergy has reached a certain level at the crack tip as shown inFigure 7 Due to the impact of the crack tip it was difficult todetermine the start time of the fracture signal at Gauge 2

Ideally Gauges 2 5 and 7 should have shared the samestrain start time since they were distributed around the centerof the blast hole with the same radial distance erefore theaverage start time measured from Gauge 5 and Gauge 7 wasused as the starting time for Gauge 2 e fracture time as-sociated with Gauge 2 was obtained by differentiating thefracture signal measured by Gauge 2 [25] e specimenfailure patterns after blasting are presented in Figure 8

e detailed values are listed in Table 3 e strain starttime for specimens 1ndash4 were 380 μs 375 μs 395 μs and387 μs respectively e average start time was 384 μs efracture time for specimens 1ndash4 was 799 μs 844 μs 864 μsand 788 μs respectively e average fracture time was824 μs e fracture accumulation time for specimens 1ndash4


Table 2 Location of strain gauge point and test value

Specimen number Strain gauge number rStrain time parameters (μs)

Peak strain (με)Start time Peak time End time Loading time Unloading time

1

Gauge 4 16 205 228 614 23 386 11055Gauge 5 32 386 403 180 17 1397 4795Gauge 6 56 672 693 1247 33 554 2496Gauge 7 32 374 391 1561 17 117 5559

2


3


4


ndash8000

ndash4000

0

4000

8000

12000

Stra

in (μ

ε)

Gauge 2Gauge 4Gauge 5

Gauge 6Gauge 7

0 20 40 60 80 100Time (μs)

(a)


Gauge 6Gauge 7

ndash8000

ndash4000

0

4000

8000

12000St

rain

(με)

0 20 40 60 80 100Time (μs)

(b)


Gauge 6Gauge 7

0 15 30 45 60 75 10590

ndash5000

0

5000

10000

15000

Stra

in (μ

ε)

Time (μs)

(c)


Gauge 6Gauge 7

ndash5000

0

5000

10000

15000

Stra

in (μ

ε)

0 20 40 60 80 100Time (μs)

(d)

Figure 6 Typical curves of strain near the blast hole (a) Specimen 1 (b) Specimen 2 (c) Specimen 3 (d) Specimen 4


was 419 μs 469 μs 47 μs and 40 μs respectively e av-erage fracture accumulation time was 440 μs e bestconsistency between the four specimens was found in thestrain start time where the maximum difference was only2 μs However a time difference of a few microseconds wasfound in both the fracture time and the fracture accumu-lation time due to the presence of the premade crack and theinhomogeneity of the rock specimen

4 Counter Radial Stress at Blast Hole

In this paper the strain curve and the dynamic fracture timeof the specimen in the elastic deformation region wereobtained through dynamic strain tests e relationshipbetween the pressure on the wall of the blast hole and themeasured strain curve was further derived using the elasticwave theory Based on the dynamic mechanical parametersof the specimen obtained from the acoustic wave detectorwe obtained an expression of the counter stress curve of theblast hole wall using the Laplace transform and the nu-merical inversion method

41 9eoretical Derivation of the Stress on the Wall of BlastHole To simplify the problem the drilling and blastingprocesses on the sandstone were assumed to be equivalent tothose for a cylindrical cavity with a uniform and elasticmedium experiencing radial displacement under a suddenload p(t) is allowed us to simplify the complex blastingevent to a linear and elastic strain problem on an axisym-metric plane According to the theory of elastic dynamicsthe wave equations associated with such processes are givenby the following equation [26]

z2Φ(r t)

zr2+1r

zΦ(r t)

zr

1c2p

z2Φ(r t)

zt2

Φ(r 0) _Φ(r 0) 0 rge r0

limr⟶infinΦ(r t) 0 (tgt 0)

σr r0 t( 1113857 p(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

ndash6000

ndash4000

ndash2000

0

2000St

rain

(με)

Time (μs)

Specimen 1Specimen 2


0 15 30 45 60 75 90 105

(a)



78 80 82 84 86 88 90

ndash250000

ndash200000

ndash150000

ndash100000

ndash50000

0

Diff

eren

tiatio

n of

the f

ract

ure s

igna

l

Time (micros)

(b)

Figure 7 Fracture signal and initiation time (a) Crack fracture signal (b) Differentiation of the crack fracture signal

(a) (b) (c) (d)

Figure 8 Specimen failure patterns after blasting (a) Specimen 1 (b) Specimen 2 (c) Specimen 3 (d) Specimen 4


where Φ(r t) is the potential function Cp is the velocity ofthe longitudinal wave Cs is the velocity of the shear waveσr(r t) is the radial stress σθ(r t) is the tangential stressp(t) is the load on the blast hole and r0 is the radius of theblast hole Taking the Laplace transformation with respect tot on equation (6) yields

d2Φ(r s)

dr2+1r

dΦ(r s)

dr k

2dΦ(r s) (7)

where kd scp and s are the variables used in Laplacetransformation A generation solution to equation (7) isgiven by

Φ(r s) A(s)I0 kdr( 1113857 + B(s)K0 kdr( 1113857 (8)

As r approaches infinity we have limxsiminfinΦ(r s) 0 In

order to satisfy this condition wemust have A(s) 0 I0 andK0 are the zero-order Bessel functions of the first and secondkind after Laplace transformation respectively Based onequation (8) and the relationship between the displacementand the potential function given by u ur zΦzr we have

Φ(r s) B(s)K0 kdr( 1113857 (9)

σr λnabla2Φ + 2μz2Φzr2

σθ λnabla2Φ +(2μr)zΦzr

σz υ σr + σθ( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)

In the above equations nabla2Φ is given by the left term inequation (6) Taking the Laplace equation of σr in equation(10) and substituting the results into equation (9) yields

σ(r s) (λ + 2μ)k2dK0 kdr( 1113857 +

λrkdK1 kdr( 11138571113890 1113891B(s) (11)

At the blast hole the following condition must be sat-isfied σ(r0 s) L(p(t)) p(s) erefore we have

B(s) p(s)

(λ + 2μ)Flowast(s) (12)

where Flowast(s) k2dK0(kdr0) + (2r0D2)kdK1(kdr0) and

D2 (λ + 2μμ) c2pc2s Here λ and μ are the Lame con-stants r0 is the radius of the blast hole cs is the velocity of theshear wave and K1 is the first order Bessel function of thesecond kind Combining equations (9) and (12) we have

Φ(r s) p(s)

(λ + 2μ)Flowast(s)K0 kdr( 1113857 (13)

ε(r s) z2Φ(r s)

zr2 (14)

e rock on the wall of the blast hole can take a pressurewith orders of magnitude of up to GPa under the blastingloading erefore these rocks will be pulverized intoparticles and it becomes challenging to measure the pressureat the blast hole directly Even if we can obtain the pressureon the wall of blast hole predicting the stress at other lo-cations in the specimen accurately still requires the stress-strain relationship of the rocks near the blast hole Howeversuch correlations can be very complex with high discrete-ness and can thus be difficult to obtain accurately fromdirect measurements In order to resolve this issue wemeasured the radial strain-time curve ε(rb t) at an arbitraryradial location in the elastic region of the specimen in thestudy We further obtained the Laplace transform of thestrain curve as ε(rb s) and substituted it into equations (13)and (14) After rearrangement of the final equation weobtained an expression of the pressure in the Laplace do-main as

p(s) ε rb s( 1113857(λ + 2μ)Flowast(s)

k2d K0 kdrb( 1113857 + K1 kdrb( 1113857kdrb( 11138571113858 1113859

(15)

where rb is the distance between the testing point and thecenter of the blast hole Taking the inverse Laplace transformof equation (15) yields the pressure-time expression asexpressed in equation (16) is equation allows us tobackcalculate the pressure curve p(t) on the blast hole wallfrom the strain curve measurements

p(t) 12πi

1113946c+iinfin

cminusiinfinp(s)e

stds (16)

42 Derivation of Radial Stress at Blast Hole Using the Nu-merical InversionMethod ere are a number of numericalinversion methods for performing the Laplace transformincluding the Stehfest algorithm the Dubner and Abatealgorithm and the Crump algorithm In this study we usedthe Stehfest algorithm which was developed by Stehfest in1970 for performing Laplace transforms We performed thenumerical inversion on equation (16) using MATLAB andobtained the stress field at the blast hole Since we did notmeasure the stress in the specimen the accuracy of the

Table 3 Time of the crack fracture

Specimen number Strain gauge number Distance ratioStrain time parameters (μs)

Start time Fracture time Fracture accumulation time1 Gauge 2 4 380 799 4192 Gauge 2 4 375 844 4693 Gauge 2 4 395 864 4704 Gauge 2 4 387 788 401Average 4 384 824 440


numerical inversion method was validated by comparing thestrain curve obtained from the numerical inversion withthe actual measurements 80mm from the center of the blasthole as shown in Figure 9 e comparisons show that thegeneral trend of the curve obtained from the numericalinversion agrees well with that obtained from experimentalmeasurements despite certain fluctuations in the numericalresults

e strain curve measured by Gauges 5 and 7 wereused to backcalculate the pressure data in our study esetwo strain gauges were placed at the same radial distancefrom the center of the blast hole e dynamic parametersof the rock used in the backcalculation were obtained fromacoustic measurements e backcalculated pressurecurve of the blast hole wall is shown in Figure 10 epresence of premade crack led to a high level of disparitybetween the latter sections of the pressure data back-calculated from the strain curve measured by Gauge 5Instead the pressure data derived from the strain curvemeasured by Gauge 7 showed good consistency betweenthe different specimens

5 Calculating the Dynamic Stress IntensityFactor of Sandstone

51 Calculating Stress Intensity Factor Using the Mutual In-teraction Integration Method e numerical methods usedfor calculating the stress intensity factor can be divided intodirect and indirect methods e commonly used directmethods include the displacement method and the stressmethod e commonly used indirect methods include theJ-integration method and the mutual interaction in-tegration method When calculating the stress intensityfactor using the mutual interaction integration method anauxiliary field at the crack tip is first constructed to helpisolate and obtain the stress intensity factor in the real fielde auxiliary field at the crack tip is an arbitrary dis-placement field and the stress field satisfies the equilibriumconditions the physical equations and the geometric re-lationships e real field at the crack tip is the actualdisplacement field and the stress field was explored in thestudy

According to the basic principles of fracture mechanicsand RICErsquos definition of J-integration expressed in equation(17) the integral loop Γ as shown in Figure 11 is an arbitrarysmooth curve that starts from a single point on the bottomsurface of the crack bypasses the end point of the crack inthe counterclockwise direction and finally ends on a singlepoint on the top surface of the crack e mutual interactionintegral I is defined in equation (18) σij εij and ui are thereal stress strain and displacement σauxij εauxij and uaux

ij arethe auxiliary stress strain and displacement and qi is theexpansion vector of the crack Both the auxiliary stress-displacement fields and the real stress-displacement fieldsare introduced in the solving process when calculating thestress intensity factor by the mutual interaction integrationmethod e collective integral Js is taken as the sum of theintegral J from real stress fields the integral Jaux from the

auxiliary stress field and the mutual interaction integral I asexpressed in equation (19)

J 1113946Γ

Wdx2 minusTi

zui

zx1dx1113888 1113889 (17)

I 1113938

Vqij σkiεauxki δij minus σauxkj uki minus σkju

auxki1113874 1113875 dv

1113938sδqn ds

(18)

Js

J + Jaux

+ I (19)

Neglecting the nonlinearity of rock the relationshipbetween the integral J energy releasing rate G and the stressintensity factor K in a plane-strain model with the samematerial is given by J G (K2

I + K2II)E Based on this

correlation we can further obtain the expression shown inequation (20) Separating the integrals J associated with theauxiliary and real fields from equation (20) yields the ex-pression of the mutual interaction integral shown inequation (21)

Js

1E

KI + KauxI( 1113857

2+ KII + K

auxI( 1113857

21113960 1113961 (20)

I 2E

KIKauxI + KIIK

auxII( 1113857 (21)

By assigning the intensity factor of the auxiliary fieldwith reasonable values ie Kaux

I 1 and KauxII 0 we can

obtain KI EII2 For KauxI 0 and Kaux

II 1 these con-ditions lead to KII EIII2 erefore KI and KII can beobtained through two mutual interaction integrations

52 Numerical Calculation of the Dynamic Stress IntensityFactor A highly condensed meshing is required to obtainaccurate results of stress variation at the crack tip due to thesudden changes in the stress field at the crack tip However

20 40 60 80 100 120

0

1500

3000

4500

6000

Stra

in (μ

ε)

Time (μs)

Strain curve obtained from back-calculationStrain curve obtained from measurement

Figure 9 Contrast curve of strain for inversion and testing


using a large number of meshes reduces the computationefficiency significantly erefore we used singular ele-ments in the Ansys finite element simulation to model thestress singularity at the crack tip e simulation model isshown in Figure 12 e PLANE183 quadrilateral meshingelements with eight nodes were used in the simulation eangle of singular element the length of the singular ele-ment and the number of the integration segments at thecrack tip can all affect the calculation of the stress intensityfactor [27] In our study the angle of the singular elementthe length of singular element and the number of in-tegration segments at the crack tip were set as 30deg 01mmand 8 respectively e total number of meshing elementswas 140000

53 9e Initiation Fracture Toughness of Sandstone A finiteelement model was first constructed in Ansys as shown inFigure 12 By applying the mutual interaction integrationmethod on the four sandstone specimens and loading thetwo sets of pressure curve data of blast hole walls obtainedfrom backcalculations we could derive the stress intensity

curve as shown in Figure 13 Using the fracture initiationtime measured at the crack (located at 80mm distance) wewere able to obtain the initiation fracture toughness ofthe sandstone as presented in Table 4 e results showthat using the stress curves backcalculated at Gauges 5 and7 yields different values of the stress intensity factore initiation fracture toughness calculated along the di-rection vertical to the crack is generally smaller than thatcalculated along the direction parallel to the crack einitiation fracture toughness of the four sandstone speci-mens was 495MPam12 548MPam12 515MPam12and 479MPam12 respectively e average initiationfracture toughness was 509MPam12

6 Conclusion

A loading method for studying the drilling and blasting ofrock was designed and developed in this study Validationtests based on the proposed loading method were performedon four identical sandstone specimens Expressions of thestress curves at the blast hole were derived using the strain ofthe blast hole in the elastic region as the parameter edynamic initiation fracture toughness of the sandstonespecimen was also obtained through the mutual interactionintegration method e following conclusions are drawnfrom the study

(1) is study provided a numerical backcalculationmethod for obtaining the counter stress-time curveon the wall of the blast hole Due to the presence ofexplosive gases the unloading time of the straincurve measured at locations with distance ratiosbetween 16 and 56 is 14 to 54 times the loading timee loading time ranges from 19 to 3 μs and in-creases slightly with increasing distance ratio e

Crack

Y X2

X X1

ds

Γ

Figure 11 Definition of the J integral

0

50

100

150

200

250

300

350Pr

essu

re (M

Pa)

Time (micros)0 15 30 45 60 75 90 105



(a)

0 15 30 45 60 75 90 1050

50

100

150

200

250

300

350

Pres

sure

(MPa

)

Time (micros)



(b)

Figure 10 Curve of pressure on the wall of the borehole (a) Backcalculated pressure curve from Gauge 5 (b) Backcalculated pressure curvefrom Gauge 7


peak strain decreases exponentially with increasingdistance ratio

(2) e fracture initiation time of the different spec-imens varies slightly due to the inhomogeneity ofthe rock itself Specifically the fracture initiationtime for the four test specimens was 799 μs

844 μs 864 μs and 788 μs respectively e av-erage fracture initiation time was 824 μs andthe average fracture accumulation time was440 μs

(3) A stable and reasonable initiation fracture toughnesscan be obtained through numerical simulations

350

300

250

200

150

100

50

00 15 30 45 60 75 90 105

Pres

sure

(MPa

)

Time (micros)

K L OP

N

PLANE183JM



Figure 12 Finite element calculation loading model

20 40 60 80 100ndash15

00

15

30

45

60

75

Stre

ss in

tens

ity fa

ctor

(MPa

times m

12 )

Time (micros)



(a)

15 30 45 60 75 90 105ndash15

00

15

30

45

60

75

Stre

ss in

tens

ity fa

ctor

(MPa

times m

12 )

Time (micros)



(b)

Figure 13 Curve of the stress intensity factor (a) Stress intensity factor curve backcalculated from Gauge 5 (b) Stress intensity factor curvebackcalculated from Gauge 7


without using a very high mesh density at the cracktip by the mutual interaction integration methode initiation fracture toughness of the four sand-stone specimens was 495MPam12 548MPam12515MPam12 and 479MPam12 respectively eaverage initiation fracture toughness was found to be509MPam12

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was financially supported by the National NaturalScience Foundation of China (Grant Nos 1167219411702181 and 11802255) Sichuan Administration of WorkSafety (aj20170515161307) Natural Science Foundation ofSouthwest University of Science and Technology (17zx7138)and Southwest University of Science and Technology Lab-oratory Experimental Technology Research Project (14syjs-60)

References

[1] E Trigueros M Canovas J M Muntildeoz and J Cospedal ldquoAmethodology based on geomechanical and geophysicaltechniques to avoid ornamental stone damage caused by blast-induced ground vibrationsrdquo International Journal of RockMechanics and Mining Sciences vol 93 no 3 pp 196ndash2002017

[2] M Hasanipanah H B Amnieh H Arab and M S ZamzamldquoFeasibility of PSO-ANFIS model to estimate rock frag-mentation produced by mine blastingrdquo Neural Computingand Applications vol 30 no 4 pp 1015ndash1024 2016

[3] W Li and J S Zhang ldquoStudy on rock mass bedding slopestability under blast seismrdquo Explosion and Shock Wavesvol 27 no 05 pp 426ndash430 2007

[4] X P Li J H Huang Y Luo and P P Chen ldquoA study ofsmooth wall blasting fracture mechanisms using the timingsequence control methodrdquo International Journal of RockMechanics and Mining Sciences vol 92 no 2 pp 1ndash8 2017

[5] Z Yue P Qiu R Yang S Zhang K Yuan and Z Li ldquoStressanalysis of the interaction of a running crack and blastingwaves by caustics methodrdquo Engineering Fracture Mechanicsvol 184 pp 339ndash351 2017

[6] F Dai M D Wei N W Xu T Zhao and Y Xu ldquoNumericalinvestigation of the progressive fracture mechanisms of fourISRM-suggested specimens for determining the mode ifracture toughness of rocksrdquo Computers and Geotechnicsvol 69 pp 424ndash441 2015

[7] M Li Z Zhu R Liu B Liu L Zhou and Y Dong ldquoStudy ofthe effect of empty holes on propagating cracks under blastingloadsrdquo International Journal of Rock Mechanics and MiningSciences vol 103 pp 186ndash194 2018

[8] Y Zhou C Zhang J Yang and Q Wang ldquoComprehensivecalibration of the stress intensity factor for the holed flattenedbrazilian disc with an inner single crack or double cracksrdquoApplied Mathematics and Mechanics vol 36 no 1 pp 16ndash302015

[9] C Zhang F Cao L Li Y Zhou R Huang and Q WangldquoDetermination of dynamic fracture initiation propagationand arrest toughness of rock using SCDC specimenrdquo ChineseJournal of 9eoretical and Applied Mechanics vol 03pp 624ndash635 2016

[10] 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

[11] X Zhao D Liu G Cheng and D Li ldquoAnalysis of blasting gasmechanism and rock crack growthrdquo Journal of ChongqingUniversity vol 34 no 6 pp 75ndash80 2011

[12] R Yang C Ding Y Wang and C Chen ldquoAction-effect studyof medium under loading of explosion stress wave and ex-plosion gasrdquo Chinese Journal of Rock Mechanics and Engi-neering vol 35 no 2 pp 3501ndash3506 2016

[13] Z Zhu Y Li J Xie and B Liu ldquoe effect of principal stressorientation on tunnel stabilityrdquo Tunnelling and UndergroundSpace Technology vol 49 pp 279ndash286 2015

[14] J Xie Z Zhu R Hu and J Liu ldquoA calculation method ofoptimal water injection pressures in natural fractured res-ervoirsrdquo Journal of Petroleum Science and Engineeringvol 133 pp 705ndash712 2015

[15] P Navas R C Yu B Li and G Ruiz ldquoModeling the dynamicfracture in concrete an eigensoftening meshfree approachrdquoInternational Journal of Impact Engineering vol 113 pp 9ndash20 2018

[16] H Du F Dai K Xia N Xu and Y Xu ldquoNumerical in-vestigation on the dynamic progressive fracture mechanism ofcracked chevron notched semi-circular bend specimens insplit hopkinson pressure bar testsrdquo Engineering FractureMechanics vol 184 pp 202ndash217 2017

[17] M-t Liu T-g Tang Z-l Guo Z-t Zhang and C-l LiuldquoExpanding ring experimental platform and its applicationin material dynamic mechanical behavior investigationrdquoJournal of Experimental Mechanics vol 31 no 1 pp 47ndash562016

[18] F F Shi R Merle B Hou J G Liu Y L Li and H Zhao ldquoAcritical analysis of plane shear tests under quasi-static andimpact loadingrdquo International Journal of Impact Engineeringvol 74 pp 107ndash119 2014

[19] K Xia S Huang and F Dai ldquoEvaluation of the frictionaleffect in dynamic notched semi-circular bend testsrdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 62 pp 148ndash151 2013

Table 4 Fracture toughness value for different specimens

Specimennumber

Location associatedwith the

backcalculation

Initiationfracturetoughness(MPamiddotm12)

Averagevalue

(MPamiddotm12)

1 Gauge 5 473 495Gauge 7 516

2 Gauge 5 506 548Gauge 7 589

3 Gauge 5 483 515Gauge 7 5464 Gauge 5 482 479


[20] S Zhang and X Li ldquoInfluence of diameter of center holes onmeasured values of dynamic fracture toughness of rockrdquoChinese Journal of Rock Mechanics and Engineering vol 34no 8 pp 1660ndash1666 2015

[21] L Hong X Li C Ma T Yin Z Ye and G Liao ldquoStudy onsize effect of rock dynamic strength and strain rate sensi-tivityrdquo Chinese Journal of Rock Mechanics and Engineeringvol 27 no 03 pp 526ndash533 2008

[22] J Yang C Zhang Y Zhou Z Zhu and Q Wang ldquoA newmethod for determining dynamic fracture toughness of rockusing SCDC specimensrdquo Chinese Journal of Rock Mechanicsand Engineering vol 34 no 2 pp 279ndash292 2015

[23] P Zhang X Zhang and T Wang ldquoRelationship betweenelastic moduli and wave velocities in rockrdquo Chinese Journal ofRock Mechanics and Engineering vol 20 no 06 pp 785ndash7882001

[24] G Hu H Wang J Jia Z Wu K Li and C Pang ldquoRe-lationship between dynamic and static value of elasticmodulus in rockrdquo Journal of Chongqing University (NaturalScience Edition) vol 28 no 03 pp 102ndash105 2005

[25] W Xu Z Zhu and L Zeng ldquoTesting method study of mode-Idynamic fracture toughness under blasting loadsrdquo ChineseJournal of Rock Mechanics and Engineering vol 34 no 1pp 2767ndash2772 2015

[26] C Yi D Johansson and J Greberg ldquoEffects of in-situ stresseson the fracturing of rock by blastingrdquo Computers and Geo-technics vol 104 pp 321ndash330 2017

[27] J Gong S Zhang H Li and C Zhang ldquoComputation of thestress intensity factor based on the interaction integralmethodrdquo Journal of Nanchang Hangkong University (NaturalSciences) vol 29 no 01 pp 42ndash48 2015


International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018


Active and Passive Electronic Components

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Shock and Vibration


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Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

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

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Multimedia

Submit your manuscripts atwwwhindawicom

configurations for loading the test sample based on theHopkinson pressure bar He further loaded the experimentalmeasurement data onto commercial finite element softwareto perform extensive calculations and proposed a combinedexperimental-numerical approach to test the dynamicfracture toughness of rock samples Zhou et al [10] proposedan NSCB method to test the dynamic initiation fracturetoughness of rock samples using the Hopkinson pressure baras the loading platform His method was further recom-mended by ISRM as a standard dynamic rock testingmethod However many achievements have been made indynamic rock studies using the Hopkinson pressure bar asthe loading platform Blasting load is more complicated thanthe impacting load First the wave types are different eimpact-induced stress wave is one-dimensional wavewhereas the blast-induced stress wave can be a cylindricalspherical or plane wave Second the loading rates aredifferent e loading rate of blasts is generally larger thanthat of impacts Both the explosive stress wave and explosivegases in the drilling and blasting processes affect rockfracture [11 12] erefore exploring the fracture toughnessof rock samples under blasting loading is a more appropriateapproach to resolving engineering blasting issues

Rocks are brittle material with natural cracks eirinertia and size can often affect test results [13 14] emaximum allowable diameter of the Hopkinson pressure baris usually around 100mm which imposes a severe limitationon the size of the rock specimen It is generally morechallenging to analyze the dynamic fracture behavior of rockspecimens with smaller sizes erefore design of testingplatforms that allows the study of fracture toughness of largerock specimens is a necessary task which can providevaluable information on rock blasting excavation engi-neering and stability analysis of rock mass after excavation

In this study we designed and rationalized a test con-figuration for studying the dynamic fracture toughnessunder blasting loading by drilling and blasting in rocks Anew testing method was also proposed for obtaining theinitiation fracture toughness of rocks Using the proposedmethod we obtained the dynamic initiation fracturetoughness of the rock specimen under blasting loadingconditions Such methods can contribute to the enrichmentof testing approaches for evaluating the dynamic fracturetoughness of rocks e flow chart of experimental nu-merical method is shown in Figure 1

2 Design of Blasting Loading and TestSpecimen Configurations

21 Blasting Loading Device e use of high-pressureloading devices is a popular approach for studying thedynamic performance of materials at high strain rates Withincreasing in-depth studies of the dynamic properties ofmaterials high-pressure loading techniques have been re-ceiving more attention from researchers Techniques such asdropping hammer Hopkinson pressure bar light gas gunand explosive blasting [15 16] have been widely used forstudying the dynamic properties of materials Compared toother dynamic loading methods explosive blasting is a more

advantageous approach for being simple convenient lowcost and free of size limitations However the explosiveblasting method also entails complex loading mechanismand has poor repeatability and stability erefore manyresearchers have developed alternative explosive blastingmethods such as the explosive plane-wave generator and theexplosive expansion ring [17] for analyzing the dynamicproperties of materials under blasting loading conditions

In this study the structure of an explosive loading devicewas designed from the engineering drilling and blastingperspective An industrial detonating cord was used togenerate stable and reliable blasting loads e detonatingcord was filled with 12 g of powder at every meter e outerdiameter and detonation velocity of the detonating cordwere 5mm and 6690ms respectively In order to minimizethe radius of the explosive pack in the rock-pulverizing zone(3ndash7) r and to strengthen the stress wave the pulverizingzone was filled with water so that the detonating cord and theblasting wall were coupled by water which enabled trans-mission of the explosive stress wave In order to preventearly release of the explosive gases and to restrict the dis-placement in the Z direction for achieving a quasi-planestrain condition the specimen was covered with a platemade from the same materials on both the top and bottomsurfaces e blasting holes were coupled with the plate-covering by high-strength explosion-proof tubes for re-ducing the damage on the plate-covering from the deto-nating cord e configuration of the loading device isshown in Figure 2

According to the literature the testing specimen can beconsidered to be under plane stress if the strain of thespecimen in the Z direction under dynamic loading is lessthan 15 of the strain in the X or Y direction [18] In order tovalidate the quasi-plane stress model strain gaugesdeforming in theX and Z directions were placed at a distanceof 80mm away from the center of the blasting hole and thestrain tests were subsequently performed e loadingconfiguration and the testing results are shown in Figure 3 Itcan be seen that the peak strain in the Z direction is only 16of that in the X direction erefore the requirements fortreating the testing specimen as under plane stress aresatisfied

22 Design and Rationalization of the SpecimenConfiguration As early as 1955 ISRM proposed to use alambdoidal slotted Brazilian disc for testing the staticfracture toughness of rocks However it was not until 2012that the first dynamic testing methods for rocks were

Strain measurementnear the blast hole

Backcalculatedpressure curve

Calculation of dynamicstress intensity factor

Fracture initiationtime measurement

Dynamic initiationfracture toughness

Figure 1 Flow chart of experimental numerical method






dge 25KIC

σs1113888 1113889

2

(1)






6000

4000

2000

0

0 20 40 60Time (μs)

Stra

in (μ

ε)

G1G2G3

G1

40mm

G2 G3

80mm


Detonator








Premade crack

Strain gauge


Explosive

Water

Sandstone

(a)

60 mm

60 m

m40

mm

40 m

m

40 mm 40 mm 80 mm

G3 G2 G1 G7

G4

G5

G6

(b)




Specimen




Resistance(Ω)

Sensitivity()







υd C2p minus 2C2

s

2 C2p minusC2

s1113872 1113873 (2)

Ed ρC2

p 1 + υd( 1113857 1minus 2υd( 1113857

1minus υd (3)

Gd Ed

2 1 + υd( 1113857 (4)

Kd Ed

3 1minus 2υd( 1113857 (5)















1


2


3


4


ndash8000

ndash4000

0

4000

8000

12000

Stra

in (μ

ε)


Gauge 6Gauge 7

0 20 40 60 80 100Time (μs)

(a)


Gauge 6Gauge 7

ndash8000

ndash4000

0

4000

8000

12000St

rain

(με)

0 20 40 60 80 100Time (μs)

(b)


Gauge 6Gauge 7

0 15 30 45 60 75 10590

ndash5000

0

5000

10000

15000

Stra

in (μ

ε)

Time (μs)

(c)


Gauge 6Gauge 7

ndash5000

0

5000

10000

15000

Stra

in (μ

ε)

0 20 40 60 80 100Time (μs)

(d)







z2Φ(r t)

zr2+1r

zΦ(r t)

zr

1c2p

z2Φ(r t)

zt2

Φ(r 0) _Φ(r 0) 0 rge r0


σr r0 t( 1113857 p(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

ndash6000

ndash4000

ndash2000

0

2000St

rain

(με)

Time (μs)



0 15 30 45 60 75 90 105

(a)



78 80 82 84 86 88 90

ndash250000

ndash200000

ndash150000

ndash100000

ndash50000

0

Diff

eren

tiatio

n of

the f

ract

ure s

igna

l

Time (micros)

(b)


(a) (b) (c) (d)




d2Φ(r s)

dr2+1r

dΦ(r s)

dr k

2dΦ(r s) (7)





Φ(r s) B(s)K0 kdr( 1113857 (9)



σz υ σr + σθ( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)


σ(r s) (λ + 2μ)k2dK0 kdr( 1113857 +

λrkdK1 kdr( 11138571113890 1113891B(s) (11)


B(s) p(s)




Φ(r s) p(s)

(λ + 2μ)Flowast(s)K0 kdr( 1113857 (13)

ε(r s) z2Φ(r s)

zr2 (14)




(15)


p(t) 12πi

1113946c+iinfin

cminusiinfinp(s)e

stds (16)













J 1113946Γ

Wdx2 minusTi

zui

zx1dx1113888 1113889 (17)

I 1113938


auxki1113874 1113875 dv

1113938sδqn ds

(18)

Js

J + Jaux

+ I (19)




Js

1E

KI + KauxI( 1113857

2+ KII + K

auxI( 1113857

21113960 1113961 (20)

I 2E

KIKauxI + KIIK

auxII( 1113857 (21)






20 40 60 80 100 120

0

1500

3000

4500

6000

Stra

in (μ

ε)

Time (μs)







6 Conclusion



Crack

Y X2

X X1

ds

Γ


0

50

100

150

200

250

300

350Pr

essu

re (M

Pa)

Time (micros)0 15 30 45 60 75 90 105



(a)

0 15 30 45 60 75 90 1050

50

100

150

200

250

300

350

Pres

sure

(MPa

)

Time (micros)



(b)







350

300

250

200

150

100

50

00 15 30 45 60 75 90 105

Pres

sure

(MPa

)

Time (micros)

K L OP

N

PLANE183JM




20 40 60 80 100ndash15

00

15

30

45

60

75

Stre

ss in

tens

ity fa

ctor

(MPa

times m

12 )

Time (micros)



(a)

15 30 45 60 75 90 105ndash15

00

15

30

45

60

75

Stre

ss in

tens

ity fa

ctor

(MPa

times m

12 )

Time (micros)



(b)




Data Availability




Acknowledgments


References





















Specimennumber


backcalculation


Averagevalue

(MPamiddotm12)

1 Gauge 5 473 495Gauge 7 516

2 Gauge 5 506 548Gauge 7 589














RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






dge 25KIC

σs1113888 1113889

2

(1)






6000

4000

2000

0

0 20 40 60Time (μs)

Stra

in (μ

ε)

G1G2G3

G1

40mm

G2 G3

80mm


Detonator








Premade crack

Strain gauge


Explosive

Water

Sandstone

(a)

60 mm

60 m

m40

mm

40 m

m

40 mm 40 mm 80 mm

G3 G2 G1 G7

G4

G5

G6

(b)




Specimen




Resistance(Ω)

Sensitivity()







υd C2p minus 2C2

s

2 C2p minusC2

s1113872 1113873 (2)

Ed ρC2

p 1 + υd( 1113857 1minus 2υd( 1113857

1minus υd (3)

Gd Ed

2 1 + υd( 1113857 (4)

Kd Ed

3 1minus 2υd( 1113857 (5)















1


2


3


4


ndash8000

ndash4000

0

4000

8000

12000

Stra

in (μ

ε)


Gauge 6Gauge 7

0 20 40 60 80 100Time (μs)

(a)


Gauge 6Gauge 7

ndash8000

ndash4000

0

4000

8000

12000St

rain

(με)

0 20 40 60 80 100Time (μs)

(b)


Gauge 6Gauge 7

0 15 30 45 60 75 10590

ndash5000

0

5000

10000

15000

Stra

in (μ

ε)

Time (μs)

(c)


Gauge 6Gauge 7

ndash5000

0

5000

10000

15000

Stra

in (μ

ε)

0 20 40 60 80 100Time (μs)

(d)







z2Φ(r t)

zr2+1r

zΦ(r t)

zr

1c2p

z2Φ(r t)

zt2

Φ(r 0) _Φ(r 0) 0 rge r0


σr r0 t( 1113857 p(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

ndash6000

ndash4000

ndash2000

0

2000St

rain

(με)

Time (μs)



0 15 30 45 60 75 90 105

(a)



78 80 82 84 86 88 90

ndash250000

ndash200000

ndash150000

ndash100000

ndash50000

0

Diff

eren

tiatio

n of

the f

ract

ure s

igna

l

Time (micros)

(b)


(a) (b) (c) (d)




d2Φ(r s)

dr2+1r

dΦ(r s)

dr k

2dΦ(r s) (7)





Φ(r s) B(s)K0 kdr( 1113857 (9)



σz υ σr + σθ( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)


σ(r s) (λ + 2μ)k2dK0 kdr( 1113857 +

λrkdK1 kdr( 11138571113890 1113891B(s) (11)


B(s) p(s)




Φ(r s) p(s)

(λ + 2μ)Flowast(s)K0 kdr( 1113857 (13)

ε(r s) z2Φ(r s)

zr2 (14)




(15)


p(t) 12πi

1113946c+iinfin

cminusiinfinp(s)e

stds (16)













J 1113946Γ

Wdx2 minusTi

zui

zx1dx1113888 1113889 (17)

I 1113938


auxki1113874 1113875 dv

1113938sδqn ds

(18)

Js

J + Jaux

+ I (19)




Js

1E

KI + KauxI( 1113857

2+ KII + K

auxI( 1113857

21113960 1113961 (20)

I 2E

KIKauxI + KIIK

auxII( 1113857 (21)






20 40 60 80 100 120

0

1500

3000

4500

6000

Stra

in (μ

ε)

Time (μs)







6 Conclusion



Crack

Y X2

X X1

ds

Γ


0

50

100

150

200

250

300

350Pr

essu

re (M

Pa)

Time (micros)0 15 30 45 60 75 90 105



(a)

0 15 30 45 60 75 90 1050

50

100

150

200

250

300

350

Pres

sure

(MPa

)

Time (micros)



(b)







350

300

250

200

150

100

50

00 15 30 45 60 75 90 105

Pres

sure

(MPa

)

Time (micros)

K L OP

N

PLANE183JM




20 40 60 80 100ndash15

00

15

30

45

60

75

Stre

ss in

tens

ity fa

ctor

(MPa

times m

12 )

Time (micros)



(a)

15 30 45 60 75 90 105ndash15

00

15

30

45

60

75

Stre

ss in

tens

ity fa

ctor

(MPa

times m

12 )

Time (micros)



(b)




Data Availability




Acknowledgments


References





















Specimennumber


backcalculation


Averagevalue

(MPamiddotm12)

1 Gauge 5 473 495Gauge 7 516

2 Gauge 5 506 548Gauge 7 589














RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





Premade crack

Strain gauge


Explosive

Water

Sandstone

(a)

60 mm

60 m

m40

mm

40 m

m

40 mm 40 mm 80 mm

G3 G2 G1 G7

G4

G5

G6

(b)




Specimen




Resistance(Ω)

Sensitivity()







υd C2p minus 2C2

s

2 C2p minusC2

s1113872 1113873 (2)

Ed ρC2

p 1 + υd( 1113857 1minus 2υd( 1113857

1minus υd (3)

Gd Ed

2 1 + υd( 1113857 (4)

Kd Ed

3 1minus 2υd( 1113857 (5)















1


2


3


4


ndash8000

ndash4000

0

4000

8000

12000

Stra

in (μ

ε)


Gauge 6Gauge 7

0 20 40 60 80 100Time (μs)

(a)


Gauge 6Gauge 7

ndash8000

ndash4000

0

4000

8000

12000St

rain

(με)

0 20 40 60 80 100Time (μs)

(b)


Gauge 6Gauge 7

0 15 30 45 60 75 10590

ndash5000

0

5000

10000

15000

Stra

in (μ

ε)

Time (μs)

(c)


Gauge 6Gauge 7

ndash5000

0

5000

10000

15000

Stra

in (μ

ε)

0 20 40 60 80 100Time (μs)

(d)







z2Φ(r t)

zr2+1r

zΦ(r t)

zr

1c2p

z2Φ(r t)

zt2

Φ(r 0) _Φ(r 0) 0 rge r0


σr r0 t( 1113857 p(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

ndash6000

ndash4000

ndash2000

0

2000St

rain

(με)

Time (μs)



0 15 30 45 60 75 90 105

(a)



78 80 82 84 86 88 90

ndash250000

ndash200000

ndash150000

ndash100000

ndash50000

0

Diff

eren

tiatio

n of

the f

ract

ure s

igna

l

Time (micros)

(b)


(a) (b) (c) (d)




d2Φ(r s)

dr2+1r

dΦ(r s)

dr k

2dΦ(r s) (7)





Φ(r s) B(s)K0 kdr( 1113857 (9)



σz υ σr + σθ( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)


σ(r s) (λ + 2μ)k2dK0 kdr( 1113857 +

λrkdK1 kdr( 11138571113890 1113891B(s) (11)


B(s) p(s)




Φ(r s) p(s)

(λ + 2μ)Flowast(s)K0 kdr( 1113857 (13)

ε(r s) z2Φ(r s)

zr2 (14)




(15)


p(t) 12πi

1113946c+iinfin

cminusiinfinp(s)e

stds (16)













J 1113946Γ

Wdx2 minusTi

zui

zx1dx1113888 1113889 (17)

I 1113938


auxki1113874 1113875 dv

1113938sδqn ds

(18)

Js

J + Jaux

+ I (19)




Js

1E

KI + KauxI( 1113857

2+ KII + K

auxI( 1113857

21113960 1113961 (20)

I 2E

KIKauxI + KIIK

auxII( 1113857 (21)






20 40 60 80 100 120

0

1500

3000

4500

6000

Stra

in (μ

ε)

Time (μs)







6 Conclusion



Crack

Y X2

X X1

ds

Γ


0

50

100

150

200

250

300

350Pr

essu

re (M

Pa)

Time (micros)0 15 30 45 60 75 90 105



(a)

0 15 30 45 60 75 90 1050

50

100

150

200

250

300

350

Pres

sure

(MPa

)

Time (micros)



(b)







350

300

250

200

150

100

50

00 15 30 45 60 75 90 105

Pres

sure

(MPa

)

Time (micros)

K L OP

N

PLANE183JM




20 40 60 80 100ndash15

00

15

30

45

60

75

Stre

ss in

tens

ity fa

ctor

(MPa

times m

12 )

Time (micros)



(a)

15 30 45 60 75 90 105ndash15

00

15

30

45

60

75

Stre

ss in

tens

ity fa

ctor

(MPa

times m

12 )

Time (micros)



(b)




Data Availability




Acknowledgments


References





















Specimennumber


backcalculation


Averagevalue

(MPamiddotm12)

1 Gauge 5 473 495Gauge 7 516

2 Gauge 5 506 548Gauge 7 589














RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





υd C2p minus 2C2

s

2 C2p minusC2

s1113872 1113873 (2)

Ed ρC2

p 1 + υd( 1113857 1minus 2υd( 1113857

1minus υd (3)

Gd Ed

2 1 + υd( 1113857 (4)

Kd Ed

3 1minus 2υd( 1113857 (5)















1


2


3


4


ndash8000

ndash4000

0

4000

8000

12000

Stra

in (μ

ε)


Gauge 6Gauge 7

0 20 40 60 80 100Time (μs)

(a)


Gauge 6Gauge 7

ndash8000

ndash4000

0

4000

8000

12000St

rain

(με)

0 20 40 60 80 100Time (μs)

(b)


Gauge 6Gauge 7

0 15 30 45 60 75 10590

ndash5000

0

5000

10000

15000

Stra

in (μ

ε)

Time (μs)

(c)


Gauge 6Gauge 7

ndash5000

0

5000

10000

15000

Stra

in (μ

ε)

0 20 40 60 80 100Time (μs)

(d)







z2Φ(r t)

zr2+1r

zΦ(r t)

zr

1c2p

z2Φ(r t)

zt2

Φ(r 0) _Φ(r 0) 0 rge r0


σr r0 t( 1113857 p(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

ndash6000

ndash4000

ndash2000

0

2000St

rain

(με)

Time (μs)



0 15 30 45 60 75 90 105

(a)



78 80 82 84 86 88 90

ndash250000

ndash200000

ndash150000

ndash100000

ndash50000

0

Diff

eren

tiatio

n of

the f

ract

ure s

igna

l

Time (micros)

(b)


(a) (b) (c) (d)




d2Φ(r s)

dr2+1r

dΦ(r s)

dr k

2dΦ(r s) (7)





Φ(r s) B(s)K0 kdr( 1113857 (9)



σz υ σr + σθ( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)


σ(r s) (λ + 2μ)k2dK0 kdr( 1113857 +

λrkdK1 kdr( 11138571113890 1113891B(s) (11)


B(s) p(s)




Φ(r s) p(s)

(λ + 2μ)Flowast(s)K0 kdr( 1113857 (13)

ε(r s) z2Φ(r s)

zr2 (14)




(15)


p(t) 12πi

1113946c+iinfin

cminusiinfinp(s)e

stds (16)













J 1113946Γ

Wdx2 minusTi

zui

zx1dx1113888 1113889 (17)

I 1113938


auxki1113874 1113875 dv

1113938sδqn ds

(18)

Js

J + Jaux

+ I (19)




Js

1E

KI + KauxI( 1113857

2+ KII + K

auxI( 1113857

21113960 1113961 (20)

I 2E

KIKauxI + KIIK

auxII( 1113857 (21)






20 40 60 80 100 120

0

1500

3000

4500

6000

Stra

in (μ

ε)

Time (μs)







6 Conclusion



Crack

Y X2

X X1

ds

Γ


0

50

100

150

200

250

300

350Pr

essu

re (M

Pa)

Time (micros)0 15 30 45 60 75 90 105



(a)

0 15 30 45 60 75 90 1050

50

100

150

200

250

300

350

Pres

sure

(MPa

)

Time (micros)



(b)







350

300

250

200

150

100

50

00 15 30 45 60 75 90 105

Pres

sure

(MPa

)

Time (micros)

K L OP

N

PLANE183JM




20 40 60 80 100ndash15

00

15

30

45

60

75

Stre

ss in

tens

ity fa

ctor

(MPa

times m

12 )

Time (micros)



(a)

15 30 45 60 75 90 105ndash15

00

15

30

45

60

75

Stre

ss in

tens

ity fa

ctor

(MPa

times m

12 )

Time (micros)



(b)




Data Availability




Acknowledgments


References





















Specimennumber


backcalculation


Averagevalue

(MPamiddotm12)

1 Gauge 5 473 495Gauge 7 516

2 Gauge 5 506 548Gauge 7 589














RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





1


2


3


4


ndash8000

ndash4000

0

4000

8000

12000

Stra

in (μ

ε)


Gauge 6Gauge 7

0 20 40 60 80 100Time (μs)

(a)


Gauge 6Gauge 7

ndash8000

ndash4000

0

4000

8000

12000St

rain

(με)

0 20 40 60 80 100Time (μs)

(b)


Gauge 6Gauge 7

0 15 30 45 60 75 10590

ndash5000

0

5000

10000

15000

Stra

in (μ

ε)

Time (μs)

(c)


Gauge 6Gauge 7

ndash5000

0

5000

10000

15000

Stra

in (μ

ε)

0 20 40 60 80 100Time (μs)

(d)







z2Φ(r t)

zr2+1r

zΦ(r t)

zr

1c2p

z2Φ(r t)

zt2

Φ(r 0) _Φ(r 0) 0 rge r0


σr r0 t( 1113857 p(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

ndash6000

ndash4000

ndash2000

0

2000St

rain

(με)

Time (μs)



0 15 30 45 60 75 90 105

(a)



78 80 82 84 86 88 90

ndash250000

ndash200000

ndash150000

ndash100000

ndash50000

0

Diff

eren

tiatio

n of

the f

ract

ure s

igna

l

Time (micros)

(b)


(a) (b) (c) (d)




d2Φ(r s)

dr2+1r

dΦ(r s)

dr k

2dΦ(r s) (7)





Φ(r s) B(s)K0 kdr( 1113857 (9)



σz υ σr + σθ( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)


σ(r s) (λ + 2μ)k2dK0 kdr( 1113857 +

λrkdK1 kdr( 11138571113890 1113891B(s) (11)


B(s) p(s)




Φ(r s) p(s)

(λ + 2μ)Flowast(s)K0 kdr( 1113857 (13)

ε(r s) z2Φ(r s)

zr2 (14)




(15)


p(t) 12πi

1113946c+iinfin

cminusiinfinp(s)e

stds (16)













J 1113946Γ

Wdx2 minusTi

zui

zx1dx1113888 1113889 (17)

I 1113938


auxki1113874 1113875 dv

1113938sδqn ds

(18)

Js

J + Jaux

+ I (19)




Js

1E

KI + KauxI( 1113857

2+ KII + K

auxI( 1113857

21113960 1113961 (20)

I 2E

KIKauxI + KIIK

auxII( 1113857 (21)






20 40 60 80 100 120

0

1500

3000

4500

6000

Stra

in (μ

ε)

Time (μs)







6 Conclusion



Crack

Y X2

X X1

ds

Γ


0

50

100

150

200

250

300

350Pr

essu

re (M

Pa)

Time (micros)0 15 30 45 60 75 90 105



(a)

0 15 30 45 60 75 90 1050

50

100

150

200

250

300

350

Pres

sure

(MPa

)

Time (micros)



(b)







350

300

250

200

150

100

50

00 15 30 45 60 75 90 105

Pres

sure

(MPa

)

Time (micros)

K L OP

N

PLANE183JM




20 40 60 80 100ndash15

00

15

30

45

60

75

Stre

ss in

tens

ity fa

ctor

(MPa

times m

12 )

Time (micros)



(a)

15 30 45 60 75 90 105ndash15

00

15

30

45

60

75

Stre

ss in

tens

ity fa

ctor

(MPa

times m

12 )

Time (micros)



(b)




Data Availability




Acknowledgments


References





















Specimennumber


backcalculation


Averagevalue

(MPamiddotm12)

1 Gauge 5 473 495Gauge 7 516

2 Gauge 5 506 548Gauge 7 589














RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






z2Φ(r t)

zr2+1r

zΦ(r t)

zr

1c2p

z2Φ(r t)

zt2

Φ(r 0) _Φ(r 0) 0 rge r0


σr r0 t( 1113857 p(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

ndash6000

ndash4000

ndash2000

0

2000St

rain

(με)

Time (μs)



0 15 30 45 60 75 90 105

(a)



78 80 82 84 86 88 90

ndash250000

ndash200000

ndash150000

ndash100000

ndash50000

0

Diff

eren

tiatio

n of

the f

ract

ure s

igna

l

Time (micros)

(b)


(a) (b) (c) (d)




d2Φ(r s)

dr2+1r

dΦ(r s)

dr k

2dΦ(r s) (7)





Φ(r s) B(s)K0 kdr( 1113857 (9)



σz υ σr + σθ( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)


σ(r s) (λ + 2μ)k2dK0 kdr( 1113857 +

λrkdK1 kdr( 11138571113890 1113891B(s) (11)


B(s) p(s)




Φ(r s) p(s)

(λ + 2μ)Flowast(s)K0 kdr( 1113857 (13)

ε(r s) z2Φ(r s)

zr2 (14)




(15)


p(t) 12πi

1113946c+iinfin

cminusiinfinp(s)e

stds (16)













J 1113946Γ

Wdx2 minusTi

zui

zx1dx1113888 1113889 (17)

I 1113938


auxki1113874 1113875 dv

1113938sδqn ds

(18)

Js

J + Jaux

+ I (19)




Js

1E

KI + KauxI( 1113857

2+ KII + K

auxI( 1113857

21113960 1113961 (20)

I 2E

KIKauxI + KIIK

auxII( 1113857 (21)






20 40 60 80 100 120

0

1500

3000

4500

6000

Stra

in (μ

ε)

Time (μs)







6 Conclusion



Crack

Y X2

X X1

ds

Γ


0

50

100

150

200

250

300

350Pr

essu

re (M

Pa)

Time (micros)0 15 30 45 60 75 90 105



(a)

0 15 30 45 60 75 90 1050

50

100

150

200

250

300

350

Pres

sure

(MPa

)

Time (micros)



(b)







350

300

250

200

150

100

50

00 15 30 45 60 75 90 105

Pres

sure

(MPa

)

Time (micros)

K L OP

N

PLANE183JM




20 40 60 80 100ndash15

00

15

30

45

60

75

Stre

ss in

tens

ity fa

ctor

(MPa

times m

12 )

Time (micros)



(a)

15 30 45 60 75 90 105ndash15

00

15

30

45

60

75

Stre

ss in

tens

ity fa

ctor

(MPa

times m

12 )

Time (micros)



(b)




Data Availability




Acknowledgments


References





















Specimennumber


backcalculation


Averagevalue

(MPamiddotm12)

1 Gauge 5 473 495Gauge 7 516

2 Gauge 5 506 548Gauge 7 589














RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



d2Φ(r s)

dr2+1r

dΦ(r s)

dr k

2dΦ(r s) (7)





Φ(r s) B(s)K0 kdr( 1113857 (9)



σz υ σr + σθ( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)


σ(r s) (λ + 2μ)k2dK0 kdr( 1113857 +

λrkdK1 kdr( 11138571113890 1113891B(s) (11)


B(s) p(s)




Φ(r s) p(s)

(λ + 2μ)Flowast(s)K0 kdr( 1113857 (13)

ε(r s) z2Φ(r s)

zr2 (14)




(15)


p(t) 12πi

1113946c+iinfin

cminusiinfinp(s)e

stds (16)













J 1113946Γ

Wdx2 minusTi

zui

zx1dx1113888 1113889 (17)

I 1113938


auxki1113874 1113875 dv

1113938sδqn ds

(18)

Js

J + Jaux

+ I (19)




Js

1E

KI + KauxI( 1113857

2+ KII + K

auxI( 1113857

21113960 1113961 (20)

I 2E

KIKauxI + KIIK

auxII( 1113857 (21)






20 40 60 80 100 120

0

1500

3000

4500

6000

Stra

in (μ

ε)

Time (μs)







6 Conclusion



Crack

Y X2

X X1

ds

Γ


0

50

100

150

200

250

300

350Pr

essu

re (M

Pa)

Time (micros)0 15 30 45 60 75 90 105



(a)

0 15 30 45 60 75 90 1050

50

100

150

200

250

300

350

Pres

sure

(MPa

)

Time (micros)



(b)







350

300

250

200

150

100

50

00 15 30 45 60 75 90 105

Pres

sure

(MPa

)

Time (micros)

K L OP

N

PLANE183JM




20 40 60 80 100ndash15

00

15

30

45

60

75

Stre

ss in

tens

ity fa

ctor

(MPa

times m

12 )

Time (micros)



(a)

15 30 45 60 75 90 105ndash15

00

15

30

45

60

75

Stre

ss in

tens

ity fa

ctor

(MPa

times m

12 )

Time (micros)



(b)




Data Availability




Acknowledgments


References





















Specimennumber


backcalculation


Averagevalue

(MPamiddotm12)

1 Gauge 5 473 495Gauge 7 516

2 Gauge 5 506 548Gauge 7 589














RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia









J 1113946Γ

Wdx2 minusTi

zui

zx1dx1113888 1113889 (17)

I 1113938


auxki1113874 1113875 dv

1113938sδqn ds

(18)

Js

J + Jaux

+ I (19)




Js

1E

KI + KauxI( 1113857

2+ KII + K

auxI( 1113857

21113960 1113961 (20)

I 2E

KIKauxI + KIIK

auxII( 1113857 (21)






20 40 60 80 100 120

0

1500

3000

4500

6000

Stra

in (μ

ε)

Time (μs)







6 Conclusion



Crack

Y X2

X X1

ds

Γ


0

50

100

150

200

250

300

350Pr

essu

re (M

Pa)

Time (micros)0 15 30 45 60 75 90 105



(a)

0 15 30 45 60 75 90 1050

50

100

150

200

250

300

350

Pres

sure

(MPa

)

Time (micros)



(b)







350

300

250

200

150

100

50

00 15 30 45 60 75 90 105

Pres

sure

(MPa

)

Time (micros)

K L OP

N

PLANE183JM




20 40 60 80 100ndash15

00

15

30

45

60

75

Stre

ss in

tens

ity fa

ctor

(MPa

times m

12 )

Time (micros)



(a)

15 30 45 60 75 90 105ndash15

00

15

30

45

60

75

Stre

ss in

tens

ity fa

ctor

(MPa

times m

12 )

Time (micros)



(b)




Data Availability




Acknowledgments


References





















Specimennumber


backcalculation


Averagevalue

(MPamiddotm12)

1 Gauge 5 473 495Gauge 7 516

2 Gauge 5 506 548Gauge 7 589














RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





6 Conclusion



Crack

Y X2

X X1

ds

Γ


0

50

100

150

200

250

300

350Pr

essu

re (M

Pa)

Time (micros)0 15 30 45 60 75 90 105



(a)

0 15 30 45 60 75 90 1050

50

100

150

200

250

300

350

Pres

sure

(MPa

)

Time (micros)



(b)







350

300

250

200

150

100

50

00 15 30 45 60 75 90 105

Pres

sure

(MPa

)

Time (micros)

K L OP

N

PLANE183JM




20 40 60 80 100ndash15

00

15

30

45

60

75

Stre

ss in

tens

ity fa

ctor

(MPa

times m

12 )

Time (micros)



(a)

15 30 45 60 75 90 105ndash15

00

15

30

45

60

75

Stre

ss in

tens

ity fa

ctor

(MPa

times m

12 )

Time (micros)



(b)




Data Availability




Acknowledgments


References





















Specimennumber


backcalculation


Averagevalue

(MPamiddotm12)

1 Gauge 5 473 495Gauge 7 516

2 Gauge 5 506 548Gauge 7 589














RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






350

300

250

200

150

100

50

00 15 30 45 60 75 90 105

Pres

sure

(MPa

)

Time (micros)

K L OP

N

PLANE183JM




20 40 60 80 100ndash15

00

15

30

45

60

75

Stre

ss in

tens

ity fa

ctor

(MPa

times m

12 )

Time (micros)



(a)

15 30 45 60 75 90 105ndash15

00

15

30

45

60

75

Stre

ss in

tens

ity fa

ctor

(MPa

times m

12 )

Time (micros)



(b)




Data Availability




Acknowledgments


References





















Specimennumber


backcalculation


Averagevalue

(MPamiddotm12)

1 Gauge 5 473 495Gauge 7 516

2 Gauge 5 506 548Gauge 7 589














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


References





















Specimennumber


backcalculation


Averagevalue

(MPamiddotm12)

1 Gauge 5 473 495Gauge 7 516

2 Gauge 5 506 548Gauge 7 589














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


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