research article numerical simulation of rock...

Research ArticleNumerical Simulation of Rock Fragmentation under theImpact Load of Water Jet

Jiang Hongxiang Du Changlong Liu Songyong and Gao Kuidong

School of Mechanical and Electrical Engineering China University of Mining amp Technology Xuzhou 221116 China

Correspondence should be addressed to Du Changlong duclhxjcumteducn

Received 28 October 2013 Accepted 3 December 2013 Published 12 February 2014

Academic Editor Vadim V Silberschmidt

Copyright copy 2014 Jiang Hongxiang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

To investigate the rock fragmentation and its influence factors under the impact load ofwater jet a numericalmethodwhich coupledfinite element method (FEM) with smoothed particle hydrodynamics (SPH) was adopted to simulate the rock fragmentationprocess by water jet Linear and shock equations of state were applied to describe the dynamic characteristics of rock and waterrespectively while the maximum principal stress criterion was used for the rock failure detection The dynamic stresses at theselected element containing points in rock are computed as a function of time under the impact load of water jet The influencesof the factors of boundary condition impact velocity confining pressure and structure plane on rock dynamic fragmentation arediscussed

1 Introduction

Rock fragmentation by the water jet technology has beenwidely used inmining petroleum drilling civil constructiongas drainage and cleaning operations [1ndash4] However therock fragmentation mechanism is still unclear because ofthe opacity and damage instantaneity of the rock Nowadaysthere are extensive arguments on the fragmentation patternsand processes [5] The rock fragmentation by water jetimpact not only is a difficult problem in mechanics but alsopresents a new challenge for physics [6] whereas there arefew investigations of the rock deformation and fragmentationby the water jet impact Bowden and Brunton took noteof the failure pattern on the brittle material surface underthe high speed liquid impact and found that the discretecracks formed regularly surrounding the impact area [7]Meanwhile they studied the fracture of brittle material andnoted that shear and tear occurred on the rock surface dueto the high speed liquid and the so-called water-hammerpressure can cause substantial damage to brittle material[8] Bourne et al studied the fracture of brittle material(PMMA) under liquid jet impact using high-speed photog-raphy and schlieren visualization and they found that the

failure resulted from the interactions of different stress waves[9] Momber investigated the rock erosion due to the liquidjet impact and observed that the failure wasmainly attributedto the lateral jetting [10] Ni et al systematically studied therock fragmentation under high-pressure water jet drillingthrough experimental theoretical and numerical methodrespectively and they considered that the stress wave playedthe leading role in rock fragmentation [11 12] Li and Liaoinvestigated the micromechanism of rock failure by waterjet impact using scanning electron microscope and observedthat the fracturewasmainly caused by two types of transgran-ular fracture and shear dislocations [13] In recent yearsmanyresearchers adopted the numerical method to investigate thefailure of rock or rock-like materials by water jet impact andtheir research works were mainly focused on the influencesof the impact velocity diameter standoff distance andincidence angle of water jet as well as the confining pressureand type of rock on the erosion depth diameter damageand stress distributions of rock [14ndash18] However there is lessinvestigation on the rock fragmentation mechanism by thecompression shear or tensile stress Although many signifi-cant results have been published they are far from complete

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 219489 11 pageshttpdxdoiorg1011552014219489

2 Shock and Vibration

for the numerical study of rock fragmentation by water jetimpact

The present paper is aiming at revealing the rockfragmentation mechanism and explaining the reasons forcrushing zone formation crack initiation and propagationunder the impact load of water jet To achieve this goala numerical method coupled FEM with SPH was adoptedand the following topics were investigated (a) the fractureprocess of rock sample under water jet impact (b) themechanism of crushing zone formation crack initiation andpropagation (c) the effects of boundary condition impactvelocity confining pressure and structure plane on the rockfailure and crack initiation and propagation

2 Numerical Model

21 Coupled SPHFEM Method As we all know the largedeformation of water exists in the rock fragmentation processby water jet impact The calculation would easily terminatedue to the mesh distortion while adopting the conventionalLagrangian FEM The coupled LagrangianEulerian methodcan effectively avoid the mesh distortion but would increasethe computational cost and reduce the computational effi-ciency

SPH is one of the mesh-free particle methods inLagrangian frame which has been widely used in variousfields since it is first invented to solve the astrophysical prob-lems In the SPHmethod a series of particles with some phys-ical quantity such asmass and velocity are used to express thecontinuousmaterialThemesh distortion can be well avoideddue to no structural mesh among these particles Thereforeit is a powerful method for solving the multiphysics flowand large deformation problems Meanwhile the FEM issuitable for the simulation of mechanical characteristics ofsolid materials Nowadays the coupled SPHFEM methodhas been verified to solve the fluid-solid interaction problemcommendably and it could overcome the disadvantagesassociated with the LagrangianEulerian method [19 20]Therefore the coupled SPHFEM method was adopted inthis paper to simulate the rock fragmentation process by thewater jet impact The coupling program is shown in Figure 1the left part shows the computational process of SPH whilethe right part indicates the FEM process Meanwhile theSPH and FEM parts are combined by the node to surfacecontact algorithm [21] where the slave part was defined withSPH particles and the master part was defined with finiteelements

22 Theory of SPH Compared with the finite elementmethod a kernel approximation is used in SPH based onrandomly distributed interpolation points with no assump-tions about which points are neighbors to calculate spatialderivatives Considering a problem domain Ω discretized bya group of particles by collection of particles and assumingkernel function W has a compact supporting domain witha radius of ℎ The approximation 119891(x) and its discretized

differential form nabla119891(119909) at point 119894 can be respectivelyexpressed as [19 20]

119891 (x) asymp intΩ

119891 (x1015840)119882(x minus x1015840 ℎ) dx1015840

nabla119891 (119909119894) asymp minus

119873

sum

119895=1

119898119895

120588119895

119891 (119909119895) nabla119882(119909

119894minus 119909119895 ℎ)

(1)

where 119882 is the smooth kernel function with B-spline typeℎ is the smooth length which defines the supporting domainof the particle x and x1015840 are the location vectors in differentposition and119873 is the total number of the particles includingthe particle 119894 within the supporting domain of the givenparticle 119894 119895 represents those influenced particles nearby theparticle 119894 119898

119895is the mass of particle 119895 120588

119895is the density of

particle 119895The equations of conservation governing the evolution of

mechanical variables can be expressed as followsconservation of mass

d120588119894

d119905=

119873

sum

119895=1

119898119895V120573119894119895

120597119882119894119895

120597119909120573

119894

(2)

conservation of momentum

dV120572119894

d119905=

119873

sum

119895=1

119898119895(120590120572120573

119894

1205882

119894

+

120590120572120573

119895

1205882

119895

)

120597119882119894119895

120597119909120573

119894

(3)

conservation of energy

d119890119894

d119905=1

2

119873

sum

119895=1

119898119895(119901119894

1205882

119894

+

119901119895

1205882

119895

) V120573119894119895

120597119882119894119895

120597119909120573

119894

+120583119894

2120588119894

120576120572120573

119894120576120572120573

119895 (4)

where 119901 is the pressure 119909120573119894is the coordinate of particle 119894 in

120573 direction 120590120572120573119894

and 120576120572120573

119894are the stress and strain tensor of

particle 119894 and 120572 and 120573 are the contravariant indexes 120583 is theviscosity coefficient of fluid V120573

119894119895represents the relative velocity

between two particles in 120573 direction

23 Material Models

231 Equation of State for Water Jet As mentioned abovethere is a large deformation of water in the rock fragmenta-tion processTherefore the shock equation of state is adoptedto describe the mechanical characteristics of water

119875119908=

1198601120574 + 119860

21205742+ 11986031205743+ (1198610+ 1198611120573) 120588119908119890 (120574 ge 0)

1198791120574 + 11987921205742+ 1198610120588119908119890 (120574 le 0)

(5)

where 119875119908is the water pressure 120588

119908is the density of water jet 120574

represents the compression ratio of water jet 120574 ge 0 representsthat water is in compression and on the contrary of it is inexpansion 119890 is the internal energy of water and 119860

1 1198602 1198603

119879111987921198610 and119861

1are thematerial constants of watermedium

In this numerical study 120588119908= 1000 kgm3119860

1= 22GPa119860

2=

954GPa 1198603= 146GPa 119879

1= 22GPa 119879

2= 0 119861

0= 028 and

1198611= 028 [22]

Shock and Vibration 3

of SPH particles

Density strain rate

Pressurestress

Pressurestress

Particle force

Acceleration

Nodes to surfacecontact check

Contact force

Position of finite element

nodes

Displacement

Zone volume

Material model

Conservation of momentum

Direct calculation

Yesintegration

Direct calculation

Direct calculation

Material model

SPH FEM

Velocityposition

and strain rate

ForcemassForcemass

Fmass

Figure 1 Coupled SPHFEMmethod

232 Equation of State and Failure Criterion for Rock Thepressure or deformation of the rock is relative small under thedynamic load by water jet impact and their variations haveless influence on the thermodynamic entropy Therefore thepressure variation could be deemed to be only related to thedensity and volume variations of the rock element Conse-quently the linear equation of state is adopted to describe therockmechanical properties which is very suitable for solvingthe dynamic problem with small deformation and pressureThe equation can be expressed as [21]

119901119903= 119870120583 = 119870(

120588

1205880minus 1

) (6)

where 119901119903is the rock pressure119870 is bulk elastic modulus of the

rock 120588 and 1205880represent real density and reference density of

the rock respectivelyIn order to investigate the mechanism of rock fragmen-

tation under the dynamic load by water jet impact themaximumprincipal stress criterion is adopted to describe therock failure behavior Once the maximum tensile or shearprincipal stresses exceed the rock dynamic tensile or shearstrength the rock element fails The maximum principalstress criterion can be expressed as

1205901(1205902) ge 120590 or 120591

12ge 120591 (7)

where 1205901and 120590

2are the maximum tensile principal stress of

rock element 120590 represents the rock dynamic tensile strength12059112

is the maximum shear principal stress and 120591 is rockdynamic shear strength However it is known that dynamicfracture simulation under high strain rate requires dynamicstrength and this value should increase with increasing strain

WaterRock

Impact velocity

No-reflection boundary

Free boundaryx

y

o

50mm

2mm

20m

m

30

mm

Bottom

Side

Figure 2 Geometric model

rate But the relationship between dynamic strength andstrain rate for the target rock is not available Therefore wehave to select a dynamic strength in this simulation and theimpact target with the following mechanical properties ref-erence density = 2200 kgm3 elasticity modulus = 589GPadynamic tensile strength = 57MPa dynamic shear strength =192MPa and poisson ratio = 022

24 GeometricModel and Boundary Conditions Thegeomet-ricmodel of rock fragmentation under dynamic load bywater


42mm10mm

43mm123

4

5

6

7

8

Crushed zoneRadial crack initiation

(a) 12120583s

123

4

5

6

7

8

Radial crack propagation

Centerline crack

(b) 31 120583s

123

4

5

6

7

8First spall crack

(c) 62 120583s

11mm

23

4

5

6

7

8

Radial tensile cracks

Spall cracksCenterline crack

44mm

(d) 20120583s

Radial crack

Spall crack

Tensile failureShear failure

RockWater

(e) 50120583s

Radial crack

Spall crack


RockWater

(f) Experimental result

Figure 3 Rock status as a function of time by water jet impact

jet impact is shown in Figure 2 The water jet is simplified asa rectangle (2 times 20mm)The smooth particle size is 025mmand there are 640 smooth particles of water jet Similarlythe rock is also simplified as a rectangle (50 times 30mm) Thequadrilateral element is adopted to mesh rock with the sidelength of 025mm and there are 24000 elements of rockTherock bottom is set as the free boundary and its side is setas the no-reflection boundary The numerical model in thispaper all adopts the above geometric model and boundaryconditions unless otherwise stated

3 Simulation Results

31 Rock Fragmentation Process Figure 3 shows the rockstatuses as a function of time by water jet impact In thissimulation the rock bottom and side are set as the freeboundary and no-reflection boundary respectively and theimpact velocity of water jet is 800ms Due to the water jet

impact the rock element containing point 1 (0mm 29mm)sustains an extremely high pressure of 149GPa and thepressure versus time at point 1 is shown in Figure 4 Inthe initial stage the high-density shear stress field formsnearby the impact point under the effect of the extremelyhigh pressure Complex variation of the pressure nearby theimpact point is induced by the intricate stress condition dueto the combined effect of water jet impact load propagatingdisturbance of stress wave compressive energy release and soon The formation of crushing zone nearby the impact pointis conducted by the high-density shear stress field as shownin Figure 3(a) Meanwhile radial crack initiation around thecrushed zone forms the crack initiation zoneThen as shownin Figure 3(b) the crack propagation zone is formed at thetime of 31 120583s It is worth noting that a centerline crack isformed perpendicular to the bottom in rock interior andthe radial crack is generally symmetrical about the impactdirection These phenomena appear as a result of the rockcontinuity and isotropy which have no effect on the rock


0 1 2 3 4 500

05

10

15

20

Pres

sure

(GPa

)

Pressure of point 1

Time (120583s)

Figure 4 Pressure versus time at the element containing point 1

failure behavior andmechanism As shown in Figure 3(c) thespall crack appears nearby the rock bottom because the stresswave reflects at the bottom which is set as the free boundaryand several radial cracks nearly propagate to the rock sideat 62 120583s At the time of 20120583s several spall cracks formedas a result of the stress wave propagation reflection andsuperposition in rock bottom is shown in Figure 3(d) andthe radial and spall cracks initiate and propagate alternatelywithin this zoneWe can notice that the depth of the crushingzone is 42mm at the initial stage of 012 120583s However itincreases by 02mm at 30120583s The phenomenon indicatesthat the formation process of crushing zone is extremelytransitory Although the increase in the depth of crushingzone is inconspicuous the zone width increases obviouslyThe reason is that the walls of crushing zone will suffer theerosion damage under the combined action of compressionand shear due to the return water jet As mentioned abovethe spall crack initiation and propagation are caused by thestress wave reflection at the free boundary However thephenomena are related to the load magnitude of water jetimpact and attenuation characteristics of stress wave as wellas rock volume and mechanical properties For example thespall cracks will not appear under a certain impact load whilethe rock is very big As compared in Figures 3(e) and 3(f) therock failure modes of simulation and experiment (conductedby Lu et al) at the upper and lower part are basicallyconsistent [18] Hence the developed numericalmodel in thispaper can well reappear in the rock fragmentation process bythe water jet impact

32 Formation Mechanism of the Crushed Zone To investi-gate the formation mechanism of the crushing zone by waterjet impact the stresses of the element containing point 2(0mm 27mm) in the crushing zone as a function of timeare recorded as shown in Figure 5 Because of a certaindistance between point 2 and the impact point the stresswavepropagates to point 2 at 015 120583s then the element stresses in

Stre

ss (G

Pa)

Time (120583s)

02

00

minus02

minus04

minus06

minus08

minus10

196MPa 12059112 = (1205901 minus 1205902)2

120591xy

120590x

120590y

00 02 04 06 08 10

Figure 5 Stresses change of the element containing point 2

both 119909 and 119910 directions increase with time at the compressivestatus But the stress in y direction increases faster than thatin 119909 direction because the impact load is perpendicular to therock bottom at the beginning At 036 120583s themaximum shearprincipal stress 120591

12reaches up to 196MPa which is larger

than the rock dynamic shear strength 120591 (120591 = 192MPa) Thenthe element containing point 2 fails in shear and the shearstress declines to 0 in the next time step (0365 120583s) Becausethe maximum principal stress criterion is based on the shearstress equal to 0 the element stresses in x and y directionsare equal to each other once the element fails There are twohighlights of the simulation results (i) the element failuresin the crushing zone are all due to that the maximum shearprincipal stress reaches the rock dynamic shear strengthAlthough element stresses in crushing zone are differentfrom each other before failure the failure behaviors aresimilar Therefore it is unnecessary to present the failurebehaviors of the other elements in this paper (ii) the elementcontaining point 2 fails while the maximum shear principalstress reaches 196MPa but not 192MPa in theory which iscaused by the time step effect in the simulation Of coursethe phenomenon can be avoided if the time step is set tobe small enough In order to improve the computationalefficiency of the developed numerical model the time step isautomatically calculated according to the mesh size and thesimilar phenomena will not be explained anymore below

33 Crack Formation In Figure 3 the element failure is usedto simulate the crack initiation and propagation approxima-tively The element containing point 3 (0mm 25mm) inthe crack initiation zone is selected to investigate the crackinitiation because a radial crack propagation crosses it Thestresses as a function of time at point 3 are recorded shownin Figure 6 The stress in y direction is always compressivestress before failure so the crack through the point 3 isa radial crack The maximum principal stress 120590

1alternates

between compression and tension status twice due to thecoupling effect between the stresses and element strainenergy release then it reaches 603MPa which is larger than


02

01

00

minus01

minus02

minus0306 09 12 15 18

12059112 (max 1779MPa)

1205901 (max 603MPa)120591xy

120590x

120590y

1205902

120591xy = 12059112

120590x = 120590y = 1205901 = 1205902

Time (120583s)

Stre

ss (G

Pa)


1205901 (max 577MPa)

120591xy

120591xy = 12059112 = 0

120590x = 120590y = 1205901 = 1205902

120590x

120590y

1205902

12059112

Time (120583s)

Stre

ss (M

Pa)

3 4 5 6 7 8 9

60

30

0

minus30

minus60

minus90


the rock dynamic tensile strength 120590 (120590 = 57MPa) In thenext time step the element containing point 3 fails resultingin the radial crack After the element fails the shear stressdecreases to 0 and the normal stresses 120590

119909 120590119910 1205901 and 120590

2

are equal Although the element fails because the maximumprincipal stress 120590

1exceeds the rock dynamic tensile strength

its maximum shear principal stress 12059112

(1779MPa) is veryclose to the rock dynamic shear strength at this time Thusit can be seen that the elements nearby point 3 may fail intension or shear in which the radial crack initiated nearbypoint 3 can also be confirmed as shown in Figure 3

The rock bottom is set as the free boundary in thesimulation which results in the appearance of spall crack atrock bottom The element containing point 4 (6mm 7mm)is selected to investigate the mechanism of spall crack andits stresses as a function of time are recorded as shown inFigure 7 The stress wave propagates to point 4 at 35 120583sthen the element stresses in both 119909 and 119910 direction increasewith time at the compressive status Because the distancebetween point 4 and the free boundary is short the element

1205901 = 1205902 = 120590x = 120590y

1205901 of point 4

1205902 asymp 120590x

1205901 asymp 120590y

120591xy

12059112

Time (120583s)

Stre

ss (M

Pa)

0 2 4 6 8 10

60

30

0

minus30

minus60

minus90

minus120


stresses are affected by the reflected wave at the first timeafter 075 120583s Furthermore the maximum principal stress 120590

1

reaches 577MPa at 642 120583s and the element fails because themaximum principal stress exceeds the rock dynamic tensilestrength In particular the stress in 119910 direction is larger thanthat in 119909 direction at tension status before failure whichindicates that the crack through point 4 is a spall crackFurthermore the stress variations in 119909 and 119910 direction bothchange with time which shows that the element stressesaround the point 4 are very complex The crack throughpoint 4 may propagate along x direction or y direction dueto the coupling effect between the dynamic stress wave andreflected stress wave which confirms that the initiation andpropagation of the radial and spall crack are decussate just asshown in Figure 3

As shown in Figure 3 there are transverse cracks in theupper and lower parts of the rock and the transverse crackin the lower part due to the reflected stress wave at thefree boundary has been investigated above To study themechanism of transverse crack in the upper part the stresschange of the element containing point 5 (10mm 28mm)is recorded as shown in Figure 8 The normal stresses arebasically coincident with the maximum principal stressesbecause the shear stress variation is very small and the stressin 119909 direction and the maximum principal stress 120590

2are both

negative before failureThus themaximumprincipal stress1205901

and the stress in 119910 direction reach 5825MPa and 5823MParespectively and they conduce to the crack through point 5to open along y direction and propagate along 119909 directionThe cracks through point 4 and 5 are similar but the failureof point 4 lags behind that of point 5 by about 34 120583s as itcan be seen from Figure 9 Furthermore the reflected stresswave action on point 4 happens before that on point 5 whichindicates that the crack through point 5 is actually a radialcrack but not a spall crack

34 Stress Wave Propagation and Attenuation The elementstresses in different positions (shown in Figure 3) as afunction of time have been recorded as shown in Figure 9


15

12

09

06

03

00

0 2 4 6 8

Time (s)

Pres

sure

(MPa

)

Point 1

Point 3

Point 6 Point 7 Point 8

Point 1 (max 149GPa)Point 3 (max 202MPa)Point 6 (max 653MPa)

Point 7 (max 511MPa)Point 8 (max 342MPa)

Figure 9 Pressure versus time of the elements in different positions

After the water jet starts to impact the rock the pressureof the element containing point 1 achieves the maximumvalue of 149GPa at 0285120583s After 135 120583s the maximumpressure value of point 3 falls down to 202MPa due to energydissipation associated with the stress wave propagation Andthemaximum pressure is only 652MPa when the stress wavepropagates to point 6 which indicates that the stress waveintensity weakens sharply during the propagation in the rockmedium The stress wave is not able to crush rock (shearfailure) when the peak pressure is smaller than the rockdynamic compressive strength At this time they can onlyconduce to the radial and spall cracks The time interval isonly about 135 120583s when the pressure falls from themaximumvalue to the rock dynamic compressive strength and it showsthat action of the stress wave on the rock elements is aloadunload process Therefore the rock fragmentation bywater jet impact can also be regarded as a loadunload pro-cess which can provide a theoretical basis for the formationof the crushing zone at the initial stage

4 Influence Factors on Crack Initiationand Propagation

Many factors influence the rock fragmentation process bywater jet impact such as boundary conditions water jetvelocity and diameter rock confining pressure and structureplane Based the above-mentioned numerical model theinfluence of these factors on rock fragmentation will beinvestigated in the following sections

41 Boundary Condition The rock bottom is set as thefree boundary and no-reflection boundary respectively thenthe rock fragmentation with the two different boundaryconditions is investigated based on the developed numericalmodel Figure 10 shows the rock failure status at the time of

20120583s The free boundary means that the stress wave will bereflected when it reaches the rock bottom however it willgo through with the no-reflection boundary which is usuallyused to simulate thewave propagation in the infinitemediumBased on the above analysis the free boundary can bringabout the spall cracks in the lower part of the rock In contrastto the case with free boundary there is no spall crack nearbythe rock bottom because the stress wave goes through theinterface Due to the effect of the reflected stress wave theextent of rock fragmentation with the free boundary is betterthan that with the no-reflection boundary which indicatesthat the free boundary can improve the rock fragmentationability of water jet

42 Impact Velocity The velocity of water jet determinatesthe impact energy therefore it has a direct effect on rockfragmentationThree different velocities of 300ms 500msand 800ms were selected in the simulation and rock failurebehaviors at 20120583s were obtained as shown in Figures 11(a)11(b) and 3(d) respectively According to the simulationresults the extent of rock fragmentation increases withimpact velocity While the velocity is 300ms the stress wavestrength is too weak so that the radial crack is unable to prop-agate to rock bottom or side Thus the rock fragmentationis mainly concentrated around the impact point When thevelocity is 500ms the radial crack length is longer than thatof 300ms but there is few spall cracks at rock bottom Asfor the velocity of 800ms the reflected stress wave from freeboundary can bring out the formation of spall crack due tothe large amount of impact energy which in return increasesthe extent of rock fragmentation As a consequence it can beseen from the above analysis that the surface erosion of rock isprimary at low impact velocity and the actual failure (such asradial and spall cracks) will occur only when impact velocityreaches up to a certain value

43 Confining Pressure In the deep geotechnical engineer-ing the confining pressure greatly influences the rock frag-mentation and the crack initiation and propagation In orderto avoid the effect of the reflected stresswave on the numericalsimulation results the rock bottom is set as the no-reflectionboundary and both two rock sides were applied with the con-fining pressure Four confining pressures of 10MPa 20MPa40MPa and 60MPa were separately applied to both tworock sides and the rock fragmentation statuses at 20 120583s wereshown in Figure 12The radial crack can propagate to the rockbottomwhen the confining pressure is 10MPa or 20MPa butit cannot propagate to the bottomwith the confining pressureof 40MPa or 60MPa The reason is that the confining pres-sures can effectively suppress the tensile stress perpendicularto the impact direction and the inbibitional effect increaseswith the confining pressure The crack length is 173mm (asshown in Figure 12(c)) with the confining pressure of 40MPabut it decreases to 151mm (as shown in Figure 12(d)) whenthe confining pressure is 60MPaThis phenomenon indicatesthat the confining pressure can not only inhibit the tensilestress perpendicular to the impact direction but also restrainthe cracks propagation rate along the pressure direction


RockTensile failureShear failure

(a) Free boundary


(b) No-reflection boundary

Figure 10 Rock fragmentation status with free and no-reflection boundaries

7mm 19mm


(a) 300ms

83mm38mm


(b) 500ms

Figure 11 Rock fragmentation status at different impact velocities

As a result the stress concentration area is formed nearby theimpact point under the combined effect of the stress wave andthe confining pressure Thus the severe rock fragmentationand high-density tensile crack nearby the impact point arecaused by the stress concentration

44 Structure Plane Therelative position between the impactpoint and the structure plane also influences the rockfragmentation and crack initiation and propagation Thesandstone with a width of 05mm was used to simulate thestructure plane and the distances of 10mm 15mm and20mm as well as the angles of 30∘ and 60∘ were set inthe following simulations The rock bottom and sides wereall set as the no-reflection boundary to avoid the effect ofthe reflected stress wave on the simulation results The rockfragmentation statuses at 20120583s with different structure planeswere obtained as shown in Figure 13 Similar with the freeboundary part of the stress wave was reflected back and theother part would go through the structure plane Thus thestress wave strength will be weakened greatlyThe stress wave

through the structure plane can hardly conduce to the rockfragmentation outside the structure plane But many spallcracks were initiated inside the structure plane because thereflected compressive stress wave changed into the tensilestresswaveMoreover the extent of rock fragmentation insidethe structure plane increases with the decrease in the distancebetween the impact point and the structure plane As shownin Figures 13(e) and 13(f) the spall cracks would propagatealong the structure plane direction due to the stress wavereflection According to the definition of the specific energy(the energy required to crush a unit volume of rock) [23] therock fragmentation effect is better with a larger crushing zoneunder the same impact energyThe rock is overcrushed whenthe distance is only 10mm resulting in a high specific energyThe rock cannot be crushed effectively while the distanceis 20mm whereas it can be mainly crushed to moderatelumpiness with a distance of 15mm As a consequenceanoptimal distance between the impact point and the rockstructural plane can be obtained once the other conditionsare determined


(a) 10MPa (b) 20MPa

173mm


(c) 40MPa

151mm


(d) 60MPa

Figure 12 Rock fragmentation status with different confining pressures

5 Conclusions

In the present study the coupled SPHFEM method isimplemented to simulate the rock fragmentation under theimpact load of water jet The influence factors on therock fragmentation and crack initiation and propagationare extensively investigated From the numerical simulationresults we come to the following conclusions

(1) The rock nearby the impact point is crushed severelydue to the extremely high local pressure at the initialimpact stage Then the pressure attenuates sharplywhen it propagates in the rock medium due to theenergy dissipation Therefore the rock fragmenta-tion by water jet impact can also be regarded as aloadunload process

(2) The rock fragmentationmodes at the upper and lowerparts from the numerical simulation are consistentwith that in the experiment The rock fragmentationby water jet impact is due to the combined action ofshear and tensile failure The crushing zone nearbythe impact point is mainly caused by the shear failureas a result of the high compressive stress while theradial crack initiation and propagation is arousedby the tensile failure However spall crack nearbythe rock bottom is caused by the reflected stresswave At the same time the micromechanism of thecracks initiation and propagation is investigated byanalyzing the element stresses as a function of timein different positions (Figures 5ndash8)

(3) The effect of the free boundary on the rock frag-mentation is very significant and the spall crackcan improve the fragmentation extent The surfaceerosion of rock is primary at low impact velocity andthe actual failure (such as radial and spall cracks)will occur only when impact velocity reaches up toa certain value The confining pressure can restrainthe stress wave and cracks propagation but it willconduce to a severe rock fragmentation as a result ofthe stress concentration closed to the impact pointThe effect of structure plane on rock fragmentationis similar to that of the free boundary An optimaldistance between the impact point and the rockstructural plane can be obtained once the otherconditions are determined Meanwhile the structureplane can lead the spall crack to propagate along theplane direction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to acknowledge the Foundation ofNational 863 Plan of China (2012AA062104) the NationalNatural Science Foundation of China (51375478) the projectfunded by the Priority Academic Program Development


10

5

5

Structure plane30∘ 60∘

(a) Structural plane (b) 10mm

(c) 15mm (d) 20mm


(e) 30∘


(f) 60∘

Figure 13 Rock fragmentation with different structure planes

of Jiangsu Higher Education Institutions (SZBF2011-6-B35)and the Graduate Education Innovation Project of JiangsuProvince (CXLX12 0948)

References

[1] X F Yang X H Li and Y Y Lu ldquoWear characteristics of thecemented carbide blades in drilling limestone with water jetrdquoInternational Journal of Refractory Metals and Hard Materialsvol 29 no 2 pp 320ndash325 2011

[2] C D K Gauert W A Westhhuizen J O Claasen et alldquoA progress report on ultra-high-pressure water-jet cuttingunderground the future of narrow reef gold and PGE miningrdquoThe Journal of the Southern African Institute of Mining andMetallurgy vol 211 no 3 pp 441ndash448 2013

[3] I Karakurt G Aydin and K Aydiner ldquoEffect of the rock grainsize on the cutting performance of abrasive waterjet (AWJ)rdquoMadencilik vol 50 no 1 pp 23ndash32 2011

[4] Y Y Lu J R Tang Z L Ge et al ldquoHard rock drilling techniquewith abrasive water jet assistancerdquo International Journal of RockMechanics and Mining Sciences vol 60 pp 47ndash56 2013

[5] G Li H Liao Z Huang and Z Shen ldquoRock damage mech-anisms under ultra-high pressure water jet impactrdquo Journalof Mechanical Engineering vol 45 no 10 pp 284ndash293 2009(Chinese)

[6] D P Chen H L HeM F Li and F Q Jing ldquoA delayed failure ofinhomogenous brittlematerial under shockwave compressionrdquoActa Physica Sinica vol 56 no 1 pp 423ndash428 2007 (Chinese)

[7] F P Bowden and J H Brunton ldquoDamage to solids by liquidimpact at supersonic speedsrdquoNature vol 181 no 4613 pp 873ndash875 1958

[8] F P Bowden and J H Brunton ldquoThe deformation of solids byliquid impact at supersonic speedsrdquo Proceedings of the RoyalSociety of London A vol 263 pp 433ndash450 1961


[9] N K Bourne T Obara and J E Field ldquoHigh-speed photog-raphy and stress gauge studies of jet impact upon surfacesrdquoPhilosophical Transactions of the Royal Society A vol 355 no1724 pp 607ndash623 1997

[10] AWMomber ldquoDeformation and fracture of rocks due to high-speed liquid impingementrdquo International Journal of Fracturevol 130 no 3 pp 683ndash704 2004

[11] H J Ni and R H Wang ldquoA theoretical study of rock drillingwith high pressure water jetrdquo Petroleum Science vol 11 no 4pp 72ndash76 2004

[12] H J Ni R H Wang and Y Q Zhang ldquoNumerical simulationstudy on rock breaking mechanism and process under highpressure water jetrdquoAppliedMathematics andMechanics vol 26no 12 pp 1595ndash1604 2005

[13] G S Li and H L Liao ldquoMicro-failure mechanism analysisand test study for rock failure surface under water jet impactrdquoChinese Journal of High Pressure Physics vol 19 no 4 pp 337ndash342 2005 (Chinese)

[14] Z Guo andM Ramulu ldquoInvestigation of displacement fields inan abrasivewaterjet drilling process Part 2Numerical analysisrdquoExperimental Mechanics vol 41 no 4 pp 388ndash402 2001

[15] H Si D DWang and X H Li ldquoStress wave effect in numericalsimulation on rock breaking under high pressure water jetrdquoJournal of Chongqing University vol 31 no 8 pp 942ndash945 2008(Chinese)

[16] L Ma R H Bao and Y M Guo ldquoWaterjet penetrationsimulation by hybrid code of SPH and FEArdquo InternationalJournal of Impact Engineering vol 35 no 9 pp 1035ndash1042 2008

[17] J L Liu and H Si ldquoNumerical simulation on damage fieldof high pressure water jet breaking rock under high ambientpressurerdquo Journal of Chongqing University vol 34 no 4 pp 40ndash46 2011 (Chinese)

[18] Y Y Lu S Zhang Y Liu Z H Lu and L Y Jiang ldquoAnalysis onstress wave effect during the process of rock breaking by pulsedjetrdquo Journal of Chongqing University vol 35 no 1 pp 117ndash1242012 (Chinese)

[19] LMaMesh-freemethod and its application on high velocity fluidjet impact and penetration simulation [PhD thesis] ZhejiangUniversity Hangzhou China 2007 (Chinese)

[20] J Limido C Espinosa M Salaun and J L Lacome ldquoSPHmethod applied to high speed cutting modellingrdquo InternationalJournal of Mechanical Sciences vol 49 no 7 pp 898ndash908 2007

[21] S W Attaway M W Heinstein and J W Swegle ldquoCouplingof smooth particle hydrodynamics with the finite elementmethodrdquo Nuclear Engineering and Design vol 150 no 2-3 pp199ndash205 1994

[22] Century Dynamics Inc Autodyn Theory Manual ConcordMass USA 2005

[23] N Sengun and R Altindag ldquoPrediction of specific energy ofcarbonate rock in industrial stones cutting processrdquo ArabianJournal of Geosciences pp 1ndash8 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



for the numerical study of rock fragmentation by water jetimpact

The present paper is aiming at revealing the rockfragmentation mechanism and explaining the reasons forcrushing zone formation crack initiation and propagationunder the impact load of water jet To achieve this goala numerical method coupled FEM with SPH was adoptedand the following topics were investigated (a) the fractureprocess of rock sample under water jet impact (b) themechanism of crushing zone formation crack initiation andpropagation (c) the effects of boundary condition impactvelocity confining pressure and structure plane on the rockfailure and crack initiation and propagation

2 Numerical Model

21 Coupled SPHFEM Method As we all know the largedeformation of water exists in the rock fragmentation processby water jet impact The calculation would easily terminatedue to the mesh distortion while adopting the conventionalLagrangian FEM The coupled LagrangianEulerian methodcan effectively avoid the mesh distortion but would increasethe computational cost and reduce the computational effi-ciency

SPH is one of the mesh-free particle methods inLagrangian frame which has been widely used in variousfields since it is first invented to solve the astrophysical prob-lems In the SPHmethod a series of particles with some phys-ical quantity such asmass and velocity are used to express thecontinuousmaterialThemesh distortion can be well avoideddue to no structural mesh among these particles Thereforeit is a powerful method for solving the multiphysics flowand large deformation problems Meanwhile the FEM issuitable for the simulation of mechanical characteristics ofsolid materials Nowadays the coupled SPHFEM methodhas been verified to solve the fluid-solid interaction problemcommendably and it could overcome the disadvantagesassociated with the LagrangianEulerian method [19 20]Therefore the coupled SPHFEM method was adopted inthis paper to simulate the rock fragmentation process by thewater jet impact The coupling program is shown in Figure 1the left part shows the computational process of SPH whilethe right part indicates the FEM process Meanwhile theSPH and FEM parts are combined by the node to surfacecontact algorithm [21] where the slave part was defined withSPH particles and the master part was defined with finiteelements

22 Theory of SPH Compared with the finite elementmethod a kernel approximation is used in SPH based onrandomly distributed interpolation points with no assump-tions about which points are neighbors to calculate spatialderivatives Considering a problem domain Ω discretized bya group of particles by collection of particles and assumingkernel function W has a compact supporting domain witha radius of ℎ The approximation 119891(x) and its discretized

differential form nabla119891(119909) at point 119894 can be respectivelyexpressed as [19 20]

119891 (x) asymp intΩ

119891 (x1015840)119882(x minus x1015840 ℎ) dx1015840

nabla119891 (119909119894) asymp minus

119873

sum

119895=1

119898119895

120588119895

119891 (119909119895) nabla119882(119909

119894minus 119909119895 ℎ)

(1)

where 119882 is the smooth kernel function with B-spline typeℎ is the smooth length which defines the supporting domainof the particle x and x1015840 are the location vectors in differentposition and119873 is the total number of the particles includingthe particle 119894 within the supporting domain of the givenparticle 119894 119895 represents those influenced particles nearby theparticle 119894 119898

119895is the mass of particle 119895 120588

119895is the density of

particle 119895The equations of conservation governing the evolution of

mechanical variables can be expressed as followsconservation of mass

d120588119894

d119905=

119873

sum

119895=1

119898119895V120573119894119895

120597119882119894119895

120597119909120573

119894

(2)

conservation of momentum

dV120572119894

d119905=

119873

sum

119895=1

119898119895(120590120572120573

119894

1205882

119894

+

120590120572120573

119895

1205882

119895

)

120597119882119894119895

120597119909120573

119894

(3)

conservation of energy

d119890119894

d119905=1

2

119873

sum

119895=1

119898119895(119901119894

1205882

119894

+

119901119895

1205882

119895

) V120573119894119895

120597119882119894119895

120597119909120573

119894

+120583119894

2120588119894

120576120572120573

119894120576120572120573

119895 (4)

where 119901 is the pressure 119909120573119894is the coordinate of particle 119894 in

120573 direction 120590120572120573119894

and 120576120572120573

119894are the stress and strain tensor of

particle 119894 and 120572 and 120573 are the contravariant indexes 120583 is theviscosity coefficient of fluid V120573

119894119895represents the relative velocity

between two particles in 120573 direction

23 Material Models

231 Equation of State for Water Jet As mentioned abovethere is a large deformation of water in the rock fragmenta-tion processTherefore the shock equation of state is adoptedto describe the mechanical characteristics of water

119875119908=

1198601120574 + 119860

21205742+ 11986031205743+ (1198610+ 1198611120573) 120588119908119890 (120574 ge 0)

1198791120574 + 11987921205742+ 1198610120588119908119890 (120574 le 0)

(5)

where 119875119908is the water pressure 120588

119908is the density of water jet 120574

represents the compression ratio of water jet 120574 ge 0 representsthat water is in compression and on the contrary of it is inexpansion 119890 is the internal energy of water and 119860

1 1198602 1198603

119879111987921198610 and119861

1are thematerial constants of watermedium

In this numerical study 120588119908= 1000 kgm3119860

1= 22GPa119860

2=

954GPa 1198603= 146GPa 119879

1= 22GPa 119879

2= 0 119861

0= 028 and

1198611= 028 [22]


of SPH particles

Density strain rate

Pressurestress

Pressurestress

Particle force

Acceleration


Contact force


nodes

Displacement

Zone volume

Material model


Direct calculation

Yesintegration

Direct calculation

Direct calculation

Material model

SPH FEM

Velocityposition

and strain rate

ForcemassForcemass

Fmass



119901119903= 119870120583 = 119870(

120588

1205880minus 1

) (6)





1205901(1205902) ge 120590 or 120591

12ge 120591 (7)

where 1205901and 120590




WaterRock

Impact velocity


Free boundaryx

y

o

50mm

2mm

20m

m

30

mm

Bottom

Side





42mm10mm

43mm123

4

5

6

7

8


(a) 12120583s

123

4

5

6

7

8


Centerline crack

(b) 31 120583s

123

4

5

6

7

8First spall crack

(c) 62 120583s

11mm

23

4

5

6

7

8



44mm

(d) 20120583s

Radial crack

Spall crack


RockWater

(e) 50120583s

Radial crack

Spall crack


RockWater








0 1 2 3 4 500

05

10

15

20

Pres

sure

(GPa

)

Pressure of point 1

Time (120583s)




Stre

ss (G

Pa)

Time (120583s)

02

00

minus02

minus04

minus06

minus08

minus10

196MPa 12059112 = (1205901 minus 1205902)2

120591xy

120590x

120590y

00 02 04 06 08 10






1alternates



02

01

00

minus01

minus02

minus0306 09 12 15 18

12059112 (max 1779MPa)

1205901 (max 603MPa)120591xy

120590x

120590y

1205902

120591xy = 12059112

120590x = 120590y = 1205901 = 1205902

Time (120583s)

Stre

ss (G

Pa)


1205901 (max 577MPa)

120591xy

120591xy = 12059112 = 0

120590x = 120590y = 1205901 = 1205902

120590x

120590y

1205902

12059112

Time (120583s)

Stre

ss (M

Pa)

3 4 5 6 7 8 9

60

30

0

minus30

minus60

minus90



119909 120590119910 1205901 and 120590

2






1205901 = 1205902 = 120590x = 120590y

1205901 of point 4

1205902 asymp 120590x

1205901 asymp 120590y

120591xy

12059112

Time (120583s)

Stre

ss (M

Pa)

0 2 4 6 8 10

60

30

0

minus30

minus60

minus90

minus120



1



2are both





15

12

09

06

03

00

0 2 4 6 8

Time (s)

Pres

sure

(MPa

)

Point 1

Point 3














(a) Free boundary




7mm 19mm


(a) 300ms

83mm38mm


(b) 500ms






(a) 10MPa (b) 20MPa

173mm


(c) 40MPa

151mm


(d) 60MPa


5 Conclusions







Acknowledgment



10

5

5



(c) 15mm (d) 20mm


(e) 30∘


(f) 60∘



References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










of SPH particles

Density strain rate

Pressurestress

Pressurestress

Particle force

Acceleration


Contact force


nodes

Displacement

Zone volume

Material model


Direct calculation

Yesintegration

Direct calculation

Direct calculation

Material model

SPH FEM

Velocityposition

and strain rate

ForcemassForcemass

Fmass



119901119903= 119870120583 = 119870(

120588

1205880minus 1

) (6)





1205901(1205902) ge 120590 or 120591

12ge 120591 (7)

where 1205901and 120590




WaterRock

Impact velocity


Free boundaryx

y

o

50mm

2mm

20m

m

30

mm

Bottom

Side





42mm10mm

43mm123

4

5

6

7

8


(a) 12120583s

123

4

5

6

7

8


Centerline crack

(b) 31 120583s

123

4

5

6

7

8First spall crack

(c) 62 120583s

11mm

23

4

5

6

7

8



44mm

(d) 20120583s

Radial crack

Spall crack


RockWater

(e) 50120583s

Radial crack

Spall crack


RockWater








0 1 2 3 4 500

05

10

15

20

Pres

sure

(GPa

)

Pressure of point 1

Time (120583s)




Stre

ss (G

Pa)

Time (120583s)

02

00

minus02

minus04

minus06

minus08

minus10

196MPa 12059112 = (1205901 minus 1205902)2

120591xy

120590x

120590y

00 02 04 06 08 10






1alternates



02

01

00

minus01

minus02

minus0306 09 12 15 18

12059112 (max 1779MPa)

1205901 (max 603MPa)120591xy

120590x

120590y

1205902

120591xy = 12059112

120590x = 120590y = 1205901 = 1205902

Time (120583s)

Stre

ss (G

Pa)


1205901 (max 577MPa)

120591xy

120591xy = 12059112 = 0

120590x = 120590y = 1205901 = 1205902

120590x

120590y

1205902

12059112

Time (120583s)

Stre

ss (M

Pa)

3 4 5 6 7 8 9

60

30

0

minus30

minus60

minus90



119909 120590119910 1205901 and 120590

2






1205901 = 1205902 = 120590x = 120590y

1205901 of point 4

1205902 asymp 120590x

1205901 asymp 120590y

120591xy

12059112

Time (120583s)

Stre

ss (M

Pa)

0 2 4 6 8 10

60

30

0

minus30

minus60

minus90

minus120



1



2are both





15

12

09

06

03

00

0 2 4 6 8

Time (s)

Pres

sure

(MPa

)

Point 1

Point 3














(a) Free boundary




7mm 19mm


(a) 300ms

83mm38mm


(b) 500ms






(a) 10MPa (b) 20MPa

173mm


(c) 40MPa

151mm


(d) 60MPa


5 Conclusions







Acknowledgment



10

5

5



(c) 15mm (d) 20mm


(e) 30∘


(f) 60∘



References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










42mm10mm

43mm123

4

5

6

7

8


(a) 12120583s

123

4

5

6

7

8


Centerline crack

(b) 31 120583s

123

4

5

6

7

8First spall crack

(c) 62 120583s

11mm

23

4

5

6

7

8



44mm

(d) 20120583s

Radial crack

Spall crack


RockWater

(e) 50120583s

Radial crack

Spall crack


RockWater








0 1 2 3 4 500

05

10

15

20

Pres

sure

(GPa

)

Pressure of point 1

Time (120583s)




Stre

ss (G

Pa)

Time (120583s)

02

00

minus02

minus04

minus06

minus08

minus10

196MPa 12059112 = (1205901 minus 1205902)2

120591xy

120590x

120590y

00 02 04 06 08 10






1alternates



02

01

00

minus01

minus02

minus0306 09 12 15 18

12059112 (max 1779MPa)

1205901 (max 603MPa)120591xy

120590x

120590y

1205902

120591xy = 12059112

120590x = 120590y = 1205901 = 1205902

Time (120583s)

Stre

ss (G

Pa)


1205901 (max 577MPa)

120591xy

120591xy = 12059112 = 0

120590x = 120590y = 1205901 = 1205902

120590x

120590y

1205902

12059112

Time (120583s)

Stre

ss (M

Pa)

3 4 5 6 7 8 9

60

30

0

minus30

minus60

minus90



119909 120590119910 1205901 and 120590

2






1205901 = 1205902 = 120590x = 120590y

1205901 of point 4

1205902 asymp 120590x

1205901 asymp 120590y

120591xy

12059112

Time (120583s)

Stre

ss (M

Pa)

0 2 4 6 8 10

60

30

0

minus30

minus60

minus90

minus120



1



2are both





15

12

09

06

03

00

0 2 4 6 8

Time (s)

Pres

sure

(MPa

)

Point 1

Point 3














(a) Free boundary




7mm 19mm


(a) 300ms

83mm38mm


(b) 500ms






(a) 10MPa (b) 20MPa

173mm


(c) 40MPa

151mm


(d) 60MPa


5 Conclusions







Acknowledgment



10

5

5



(c) 15mm (d) 20mm


(e) 30∘


(f) 60∘



References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1 2 3 4 500

05

10

15

20

Pres

sure

(GPa

)

Pressure of point 1

Time (120583s)




Stre

ss (G

Pa)

Time (120583s)

02

00

minus02

minus04

minus06

minus08

minus10

196MPa 12059112 = (1205901 minus 1205902)2

120591xy

120590x

120590y

00 02 04 06 08 10






1alternates



02

01

00

minus01

minus02

minus0306 09 12 15 18

12059112 (max 1779MPa)

1205901 (max 603MPa)120591xy

120590x

120590y

1205902

120591xy = 12059112

120590x = 120590y = 1205901 = 1205902

Time (120583s)

Stre

ss (G

Pa)


1205901 (max 577MPa)

120591xy

120591xy = 12059112 = 0

120590x = 120590y = 1205901 = 1205902

120590x

120590y

1205902

12059112

Time (120583s)

Stre

ss (M

Pa)

3 4 5 6 7 8 9

60

30

0

minus30

minus60

minus90



119909 120590119910 1205901 and 120590

2






1205901 = 1205902 = 120590x = 120590y

1205901 of point 4

1205902 asymp 120590x

1205901 asymp 120590y

120591xy

12059112

Time (120583s)

Stre

ss (M

Pa)

0 2 4 6 8 10

60

30

0

minus30

minus60

minus90

minus120



1



2are both





15

12

09

06

03

00

0 2 4 6 8

Time (s)

Pres

sure

(MPa

)

Point 1

Point 3














(a) Free boundary




7mm 19mm


(a) 300ms

83mm38mm


(b) 500ms






(a) 10MPa (b) 20MPa

173mm


(c) 40MPa

151mm


(d) 60MPa


5 Conclusions







Acknowledgment



10

5

5



(c) 15mm (d) 20mm


(e) 30∘


(f) 60∘



References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










02

01

00

minus01

minus02

minus0306 09 12 15 18

12059112 (max 1779MPa)

1205901 (max 603MPa)120591xy

120590x

120590y

1205902

120591xy = 12059112

120590x = 120590y = 1205901 = 1205902

Time (120583s)

Stre

ss (G

Pa)


1205901 (max 577MPa)

120591xy

120591xy = 12059112 = 0

120590x = 120590y = 1205901 = 1205902

120590x

120590y

1205902

12059112

Time (120583s)

Stre

ss (M

Pa)

3 4 5 6 7 8 9

60

30

0

minus30

minus60

minus90



119909 120590119910 1205901 and 120590

2






1205901 = 1205902 = 120590x = 120590y

1205901 of point 4

1205902 asymp 120590x

1205901 asymp 120590y

120591xy

12059112

Time (120583s)

Stre

ss (M

Pa)

0 2 4 6 8 10

60

30

0

minus30

minus60

minus90

minus120



1



2are both





15

12

09

06

03

00

0 2 4 6 8

Time (s)

Pres

sure

(MPa

)

Point 1

Point 3














(a) Free boundary




7mm 19mm


(a) 300ms

83mm38mm


(b) 500ms






(a) 10MPa (b) 20MPa

173mm


(c) 40MPa

151mm


(d) 60MPa


5 Conclusions







Acknowledgment



10

5

5



(c) 15mm (d) 20mm


(e) 30∘


(f) 60∘



References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










15

12

09

06

03

00

0 2 4 6 8

Time (s)

Pres

sure

(MPa

)

Point 1

Point 3














(a) Free boundary




7mm 19mm


(a) 300ms

83mm38mm


(b) 500ms






(a) 10MPa (b) 20MPa

173mm


(c) 40MPa

151mm


(d) 60MPa


5 Conclusions







Acknowledgment



10

5

5



(c) 15mm (d) 20mm


(e) 30∘


(f) 60∘



References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











(a) Free boundary




7mm 19mm


(a) 300ms

83mm38mm


(b) 500ms






(a) 10MPa (b) 20MPa

173mm


(c) 40MPa

151mm


(d) 60MPa


5 Conclusions







Acknowledgment



10

5

5



(c) 15mm (d) 20mm


(e) 30∘


(f) 60∘



References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a) 10MPa (b) 20MPa

173mm


(c) 40MPa

151mm


(d) 60MPa


5 Conclusions







Acknowledgment



10

5

5



(c) 15mm (d) 20mm


(e) 30∘


(f) 60∘



References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










10

5

5



(c) 15mm (d) 20mm


(e) 30∘


(f) 60∘



References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article numerical simulation of rock...

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