Study on the Pressure Characteristics of Shock Wave...

Research ArticleStudy on the Pressure Characteristics of Shock WavePropagating across Multilayer Structures duringUnderwater Explosion

Zi-Fei Meng 1 Xue-Yan Cao 1 Fu-Ren Ming 1 A-Man Zhang1 and Bin Wang2

1College of Shipbuilding Engineering Harbin Engineering University Harbin 150001 China2National Key Laboratory of Shock Wave and Detonation Physics Institute of Fluid PhysicsChina Academy of Engineering Physics Mianyang Sichuan 621900 China

Correspondence should be addressed to Xue-Yan Cao caoxueyanhrbeueducn and Fu-Ren Ming mingfurengmailcom

Received 29 July 2018 Revised 8 November 2018 Accepted 21 November 2018 Published 13 January 2019

Academic Editor M I Herreros

Copyright copy 2019 Zi-Fei Meng 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

e propagation of the shock wave across multilayer structures during underwater explosion is a very complex physicalphenomenon involving violent fluid-structure interaction (FSI) problems In this paper the coupled EulerianndashLagrangian (CEL)method in AUTODYN is used to simulate the process of shock wave propagation and solve FSI problems Firstly the governingequation and the treatment of fluid and structure interface of the CEL method are briefly reviewed Afterwards two underwaterexplosion numerical models are established and the results are compared with the empirical formula and experimental datarespectively to verify the reliability of numerical solutions e results obtained by this method show good agreements with thoseof the empirical formula and experiment Furthermore the model of the multilayer structures composed of two hemisphericalshells and the fluid filled between the shells subjected to underwater explosion is established and the pressure characteristics of theshock wave propagating across the multilayer structures are analyzed regarding the wave reflection and transmission Finally theeffects of the shell thickness and the filled fluid type among the multilayer structures on the wave reflection and transmissionare studied

1 Introduction

e shock wave produced by underwater explosion can causeserious damage to submarines and threaten survivability ofsubmarines [1] erefore knowledge on the pressure char-acteristics of shock waves is of particular importance to anoverall understanding of the shock resistance of submarinesAlthough numerous studies have been conducted [2ndash5] thereare still many tough problems to solve due to the violentinteractions between the structure and the shock wave es-pecially for the understanding of the wave propagation inmultilayer structures Among them the double-layer hemi-spherical shell is often used as the protective structure ofsubmarines and other underwater vehicles Hence in thispaper the pressure propagation characteristics of the un-derwater explosion shock wave in two hemispherical shellswith air or water among them are investigated

Regarding shock wave propagation in structures a greatdeal of researches have been performed Taylor [6] assumedthat the underwater explosion shock wave is a one-dimensional planar exponentially decaying pressure waveand adopted this planar wave as shock load to calculate thepressure and impulse on the wetted surface (the interface ofwater and structure) Although Taylorrsquos solution can giveaccurate pressure time histories for a simple plate it isdifficult to capture the reflected wave from the multilayerstructures On this basis Jin et al [7] presented an analyticalmethod to obtain interfacial pressure of coated plates underunderwater weak shock waves e method can take intoaccount multiple reflections and transmissions in multilayerstructures But the major limitation to this method is that theshock wave should be weak enough to guarantee someassumptions they adopted that rigid body motion is negli-gible and the variation of the wave propagation time in each

HindawiShock and VibrationVolume 2019 Article ID 9026214 19 pageshttpsdoiorg10115520199026214

medium caused by the medium deformation is negligibleRecently Chen et al [8] applied one-dimensional cavitationtheory and section-varying bar theory to discuss the propa-gation characteristics of stress waves in simple plates andstiffened plates (T profiles and I profiles) Considering thecavitation effect the pressure and impulse on the wettedsurface are predicted accurately by analytical solution eyfound that obviously pressure drops and oscillations occur onthe wetted surface in the stiffened plates model due to thesuperposition of the reflected wave from the structure and theincident wave compared to the simple plate model Howeverthis analytical model is limited to the one-dimensional system

Valuable researches on pressure characteristics of theshock wave in structures using numerical simulation havealso been conducted Otsuka et al [9] utilized LS-DYNA3D an analysis code using the arbitraryLagrangianndashEulerian approach to simulate the propaga-tion process of the explosion shock waves produced by thetwo modes of detonation (a detonation cord and anemulsion explosive) in the three layers of air water andstructure ey found that as the air layer becomes thickthe peak pressure decreases in the water However theimpulse remains unchanged since the duration of pressureis increased by the effect of a reflected wave Yet theirsimulation lacks a comparative study between the differentmaterial layers Desceliers et al [10] presented a newnumerical hybrid method dedicated to the simulation ofthe transient elastic wave propagation in multilayer un-bounded media which can be fluid or solid subjected togiven transient loadsis method can capture clearly someprofiles of the elastic waves including reflected wavestransmitted waves and refracted surface waves However itis restricted to simple geometrical structures Wang et al[11] used the coupled EulerianndashLagrangian (CEL) methodin AUTODYN to investigate the shock wave propagationcharacteristics near the air-backed plate and water-backedplate ey found that two pressure peaks occurred in thesetwo numerical models where the first peak is generated bythe incident wave and the second peak is induced by thereflected wave from the plate due to the impedance of thewater being lower than that of the plate But in that workthey did not apply the CELmethod to multilayer structuresWu et al [12] investigated the protection effects of amitigation layer on the structure exerted by an underwaterexplosion with the help of the Mie-Greurouneisen mixturemodel and discussed the affections of the layer material(steel neoprene and polyethylene) layer thickness andexplosive-structure distance on protection effects eyconcluded that the acoustic impedance of the mitigationlayer plays an important role in determining not only theproperty of shock loads but also the occurrence of pro-tection effects of the mitigation layer while the layerthickness and explosive-structure distance have a littleeffect on the main shock loads Nevertheless the loads ofthe inner shock wave in the mitigation layer are their mainconcern In addition other scholars mainly focus onstructural response and damage characteristics [13ndash21] Ina word theoretical research based on certain assumptionsis suitable for a structure with a simple geometry while

numerical simulation is easier to implement to deal withshock wave propagation problems in complex structures

Owing to the presence of complex superposition ofmultiple waves geometry with curvature moving interfaceand large plastic deformation traditional theoretical re-search comes across many difficulties in solving shock wavepropagation characteristics in multilayer structures In orderto tackle these problems the CEL method in AUTODYN isadopted to study the propagation characteristics of the shockwave in the multilayer structures during underwater ex-plosion in the present worke shell thickness and the filledmedium between the double shells are focused and theirinfluences on the propagation of the shock wave are ana-lyzed e relationships among the incident the reflectedand the transmitted shock waves are discussed

2 Theoretical Background

21 Coupled Eulerian and Lagrangian Method Actually theprocess of underwater explosion near structures is a com-plicated fluid-structure interaction (FSI) problem [22 23]Two sorts of methods are usually used to solve this problemnamely the Lagrangianmethod and Eulerianmethod [11 24]Both of them have their own advantages and limitations eLagrangian method is suitable for describing the stress strainand deformation of the structure However when encoun-tering large deformation problems the Lagrangian mesh isseverely distorted which leads to adverse effects on calcu-lation Conversely the Eulerian method can deal with largedeformation problems and some flow problems But itcannot track accurately material interfaces erefore acoupled EulerianndashLagrangian (CEL) analysis method whichcombines the advantages of these two methods is introducedto tackle FSI problems e structure part is solved by theLagrangian method while the fluid part is solved by theEulerian methode FSI between the two parts is carried outat the interface [11 25]is approach has beenmade possiblein some hydrocodes eg AUTODYN [25] and ABAQUS[26] Specifically in the Lagrangian method based on thefinite difference scheme the material is attached to the meshelement which deforms with the flow of the material and themass of the element remains constant [25] In the Lagrangiansolver given the node position x (x y z) the node velocityu the element density ρ the element volume V the internalenergy e and the governing equations are as follows[25 27 28]

ρ ρ0V0

Vm

V

ρdudt nabla middot σ

de

dtσρnabla middot u

(1)

where ρ0 V0 m and σ are the initial density initial volumemass and total stress tensor respectivelye stress tensor isseparated into a hydrostatic component and a deviatoriccomponent which can be referred from the study in [29]

2 Shock and Vibration

A series of calculations in a Lagrangian subgrid are shownschematically in Figure 1 (the left column) A detailed de-scription about this process can be referred from the study in[25 30ndash32]

In the Eulerian solver assuming that the fluid is inviscidthe explosion process is adiabatic and the body force isneglected [27 28 33 34] e governing equations can bewritten as follows [35 36]

zρzt+ nabla middot (ρu) 0

z(ρu)zt

+ nabla middot (ρuu) minusnablaP

z(ρE)zt

+ nabla middot (ρEu) minusnabla middot (Pu)

(2)

where ρ is the density u is the velocity E is the total energyP is the pressure ρu is the mass flux ρuu is the momentumflux and ρEu is the energy flux

In fact the Eulerian method contains two steps to ac-count for both the changes in the element solution caused bythe source and the transport of the material through themesh More details can be found in [32 37ndash39]

Besides the calculation process of the Eulerian solver isshown in Figure 1 (the right column) ere are also manycomplicated works that need to be accomplished includingthe calculation of intersection points intersecting linescommon areas surface area and three-dimensional volumeese detail descriptions can be found in [40ndash43]

22 Treatments of Fluid-Structure Interface e interactionbetween the structure and fluid is performed at the couplingsurface At the interface the structure described by theLagrangian view is regarded as the geometric and velocityboundary for the fluid Conversely the fluid is regarded asthe pressure boundary for the structure [25 32 37]

For the Lagrangian structure element the position xvelocity u force F and massm are defined at the node whilethe stress σ strain ε pressure P energy e and density ρ aredefined at the center of the element For the Eulerian fluidelement all variables are defined at the center of the elementexcept the position x (Figures 2(a) and 2(b)) is definitionmode facilitates the coupling between the Lagrangianstructure and the Eulerian fluid [25]

23 Numerical Validation Firstly the free-field underwaterexplosion model is established to validate the feasibility of theCELmethod inAUTODYN [25] for simulating the shockwavepressure e spherical TNTcharge with a massmr of 0853 kgand a radiusR0 of 005m is detonated in the cylindrical domainwith a radius of 20R0 and a height of 50R0 as shown inFigure 3(a) A nondimensional distance r hereinafter is definedto describe the distance from the measuring points to the TNTcenter namely r RR0 where R is the detonation distanceDifferent distances of r 10minus 16 are chosen e transmittingboundary condition is set around the computational domain toreduce the nonphysical reflection In the present work we

adopt a two-dimensional axisymmetric model to simulate theprocess of free-field underwater explosion

e density of TNT is 1630 kgm3 and the detonationvelocity is 6930ms Besides the state equation of JonesndashWilkinsndashLee (JWL) hereinafter is used to simulate the TNTdetonation process and it can be written as equation (3)where P is the pressure produced by the explosive chargesVis the specific volume of detonation products over thespecific volume of undetonated explosives E is the specificinternal energy and A B R1 R2 and ω are material con-stants listed in Table 1 [25 44] e equation of state forwater hereinafter can be defined as equation (4) whereμ ρρ0 minus 1 in which ρ0 is the initial density of waterie ρ0 1 000 kgm3 e term e is the specific internalenergy for water A1 A2 A3 B0 B1 T1 and T2 are constantsdefined in the AUTODYN material library ese param-eters are listed in Table 2 [25 44]

P A 1minusωR1V

1113888 1113889eminusR1V + B 1minus

ωR2V

1113888 1113889eminusR2V +

ωEV (3)

P A1μ + A2μ2 + A3μ3 + B0 + B1μ 1113857ρ0e μgt 0T1μ + T2μ2 + B0ρ0e μlt 0

1113896 (4)

e water and explosive are modeled by the Euleriansubgrid In order to test the convergence of the present CELmethod three mesh resolutions are used in the Eulerian part25mm 5mm and 10mm respectively An evenly parti-tioned grid is used e shock wave pressure versus time atr 10 is measured as shown in Figure 4 Clearly the resultof the mesh size 25mm shows a little difference comparedwith the result of the mesh size 5mm e convergence rateof three resolutions is calculated to be 195 using their peakpressure Generally if the convergence rate is over 1 then itmeans this numerical method can give better results [45] Inorder to reduce the computational cost the element size forthis free-field underwater explosion model is 5mm and thenumber of elements is about 100000

e attenuation of shock wave pressure varying withtime p(t) is given by the Zamyshlyayev empirical formula[3]

p(t)

Pmeminustθ 0le tle θ

0368Pmθt

1 minust

tp1113888 1113889

15⎡⎣ ⎤⎦ θlt tle tp

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(5)

Pm

441 times 107 W13

R1113872 1113873

15 6le

R

R0lt 12

524 times 107 W13

R1113872 1113873

113 12le

R

R0lt 240

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(6)

tp R0

C⎡⎣850

Ph

Patm1113888 1113889

081

minus 20Ph

Patm1113888 1113889

13

+ 114minus 106 timesR0

R1113874 1113875

013+ 151

R0

R1113874 1113875

126⎤⎦

(7)

Shock and Vibration 3

θ

045R0RR0

1113874 1113875045

times 10minus3R

R0lt 30

35R0

C

lgR

R01113888 1113889minus 09

1113971

R

R0ge 30

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(8)

Ph Patm + ρgh (9)

where Pm is the peak pressure of the shock wave (Pa) θ isthe time decay constant of the shock wave (s) W is theweight of the explosive charge (kg) R is the distance

between the explosion center and measuring point (m) R0is the initial radius of the explosive (m) tp is the time whenthe bubble reaches its maximum radius (s) Ph is thehydrostatic pressure near the explosion center (Pa) Patm isthe atmospheric pressure (Pa) C is the sound speed ofwater (ms) and h is the initial depth of the explosioncenter (m)

e numerical model of free-field underwater explo-sion and the pressure contours at two typical instants areshown in Figure 3 e shock wave propagating in water isspherical (Figure 3(b)) When the shock wave arrives atthe boundary it is not reflected due to the enforcement of

Lagrangian solver Eulerian solver

Nodal displacement

Strain rate and density

Internal energyand pressure

Deviatoric stressand nodal force

Nodal acceleration

Nodal velocity

Inte

grat

ion

Element mass momentumand energy

Strain rate andelement density

Internal energyand element pressure

Element deviatoric stressand face impulse

Nodal acceleration

Momentummass

Mass conservationequation

Energy conservationequation and state equation

Material model

Momentumconservation equation

Coupling interaction

Nodal velocity

Transport

Forcemass

Initial condition

Figure 1 Calculation frame of the CEL method [25 32]

x u F m

σ ε P e ρ

(a)

σ ε P eρ u m

x

(b)

Lagrangiansubgrid

Euleriansubgrid

(c)

Figure 2 (a) Lagrangian elements x u F andm in the node and σ ε P e and ρ in the element center (b) Eulerian elements x in the nodeand other variables in the element center (c) fluid-structure interaction interface Euler as the pressure condition for Lagrange and Lagrangeas the velocity condition for Euler Blue represents the material in the Eulerian element [25 32 37] σ ε P e ρ x u F andm are the stressstrain pressure energy density coordinate velocity force and mass respectively


the transmitting boundary condition as shown inFigure 3(c)e time histories of the pressure at r 10 andr 13 obtained from CEL and Zamyshlyayev empiricalformulas [3] are compared in Figure 5 It reveals that theyaccord with each other If one defines the relative error ofthe peak pressure between the numerical P0 and theZamyshlyayev results P as ε |P0 minusP|P the values can belisted in Table 3 We can find that the relative errors arewithin 10 Although there are some oscillations causedby the strong discontinuity of the shock wave in thenumerical results the whole attenuation process and peakpressure values are in good agreement with the empiricalformula

In fact during the shock wave propagating in multilayermedia the FSI process is involved In order to further test thereliability of the CEL method in dealing with FSI problemsan experiment (this experimental case and relevant data areprovided by the Institute of Fluid Physics China Academy ofEngineering Physics) is performed in a water tank with thesize of 2m times 2m times 2m e depth of water is 16m Asshown in Figure 6 the square test plate made from the mildsteel Q235 is fixed on the wall of the tank It has a length of08m and a thickness of 0003me cylindrical charge has asize of Φ 20mm times 18mm and thus the mass is 10 g echarge is placed in the water and its depth isD 08m eaxis of the charge is parallel with the plate and the distance R

r

1

3

654

2

R0

Water

TNT

7

Measuringpoints

20R0

50R0

Transmitting boundary

(a)

1659P (MPa)

1494132811629968306644983321660

(b)

333P (MPa)

2972602241881511157842560

(c)

Figure 3 e numerical model and pressure distributions of free-field underwater explosion (a) Numerical model (b) t 015ms(c) t 072ms

Table 1 e parameters of the JWL equation [25 44]

A B R1 R2 W E3738GPa 375GPa 415 09 035 60 times 109 Jmiddotmminus3

Reprinted by permission from Springer Nature Customer Service Center GmbH Springer Nature Transactions of Tianjin University Numerical simulation ofunderwater explosion loads Chunliang Xin Gengguang XU Kezhong Liu 2008 advance online publication 15 November 2008 (doi httpsdoiorg101007s12209-008-0089-4)

Table 2 e parameters of polynomial EOS for water [25 44]

A1 A2 A3 B0 B1 T1 T2 e22GPa 954GPa 1457GPa 028 028 22GPa 0 361875 JmiddotkgReprinted by permission from Springer Nature Customer Service Center GmbH Springer Nature Transactions of Tianjin University Numerical simulation ofunderwater explosion loads Chunliang Xin Gengguang XU Kezhong Liu 2008 advance online publication 15 November 2008 (doi httpsdoiorg101007s12209-008-0089-4)


P (M

Pa)

120

100

80

60

40

20

010 02 04

t (ms)06 08

Element size = 25mmElement size = 5mmElement size = 10mm

Figure 4 Shock wave pressure versus time from three different element sizes at r 10

120

100

80

60

40

20

0

P (M

Pa)

0 02 04t (ms)

06 08 1

Empirical formula [3]CEL method

(a)

t (ms)

80

60

40

20

0

P (M

Pa)

0 02 04 06 08 1


(b)

Figure 5 Comparison of shock wave pressure time histories from numerical results and Zamyshlyayev empirical formula [3] (a) r 10(b) r 13

Table 3 Comparison of peak pressure

r P0 (MPa) P (MPa) ε ()

10 1143 1152 0811 993 998 0612 873 876 0413 775 803 3514 698 738 5515 631 683 7616 574 635 96

TNT

Water surface

Water tank

RD

Plate

R0

Figure 6 e experimental layout


from the axis to the plate is about four times the charge radiusR0 namely R 4R0 e experimental layout is illustrated inFigure 6 Two measuring points (D1 andD2) are placed on theside of the plate to record the displacement and velocity(Figure 7)

e three-dimensional numerical model is establishedaccording to the layout of the experiment e water andTNTare also modeled by the Eulerian subgrid Note that theair in this numerical model can be ignored e steel ismodeled by the Lagrangian subgrid and its density is7830 kgm3 e JohnsonndashCook strength and fracture pa-rameters for the steel are listed in Tables 4 and 5 (found in[46]) e smallest element size of the Eulerian mesh is5mm and it is 15mm for the Lagrangian mesh e Eulerdomain consists of about 38 million cells and the La-grangian domain contains 570000 cells

e displacement and velocity time histories calculated bythe CEL method are compared with the experimental results inFigure 8 It is not hard to see from Figure 8 that they are in goodaccordance especially the displacement time histories ere-fore the CEL method can simulate the FSI process well and itcan predict accurately the shock wave propagation in thecomplex model

3 Results and Discussion

31 Numerical Model e two-dimensional axisymmetricnumerical model of a double-layer hemispherical shellsubjected to underwater explosion is established to in-vestigate the propagation characteristics of the shock wave inmultilayer media as plotted in Figure 9e outer shell has aradius of R1 and a thickness of d1 while the inner shell has aradius of R2 and a thickness of d2 e parameters related todistance are normalized by the charge radius R0 namelyR1 R1R0 20 d1 d1R0 R2 R2R0 17 d2 d2R0and r RR0 10minus16 e thickness of the outer shell ischanged to test the effects so d1 01minus08 are selected with aconstant of d2 04 e mass of the spherical TNT chargethe charge radius R0 and the size of the Eulerian domain arethe same as those of the free-field model in Section 23 emeasuring points 1 and 3 are located near the outside surfaceof the outer shell and the inner shell respectively emeasuring point 2 is located near the inside of the outershell and the distance from point 2 to the outer shell isd 02R0 e shell structure is fixed around the boundaryand the transmitting condition is enforced around the fluiddomain so as to overcome the nonphysical reflection eparameters related to time hereinafter are normalized by thetime decay constant θ when r 10 in equation (8)

Medium 1 and medium 2 are water or air e materialparameters of water and charge are the same as those in thefree-field model Air is modeled by the ideal gas equation ofstate which can be written as p ρρ0((cminus 1))e where thespecific heat ratio c is 14 the initial density of air ρ0 is1225 kgm3 and the specific energy e is 2534 kJm3 [25]e density of steel is 7830 kgm3 [25] Note that theJohnsonndashCook strength parameters listed in Table 6 [25] forthe steel are different from those of the experiment in Section23 e element size for the Eulerian part is also 5mm the

same as that of the free-field model and more than fourelements in the thickness direction are discretized for theLagrangian part us the Eulerian domain contains about100000 cells and the number of Lagrangian cells rangesfrom 4230 to 8940 due to different outer shell thicknesses

32 Shock Wave Propagation Process Figure 10 shows theshock wave propagation process for the case of media 1 and2 filled with water and air respectively at the detonationdistance r 10 Several typical instants during the processare presented Obviously they show a similar process afterthe charge detonation and a spherical shock wave prop-agates to the outer shell surface (Figure 10(a)) when thewave arrives at the outer shell whose other side is backed towater it is reflected and transmitted by the shell as plottedin Figure 10(b) (the left column) Accordingly when thewave arrives at the outer shell whose other side is backed toair there exists a rarefaction wave reflected which travelsbackwards in water Due to the much lower impedance ofthe air the transmitted wave is weaker than the reflectedwave erefore Figure 10(b) (the right column) cannotrecognize the transmitted wave in the air-backed casewhen the transmitted shock wave reaches the inner shellwhose other side is backed to water the shock wave isreflected and transmitted over the shell again inFigure 10(c) (the left column) Cavitation occurs near theouter shell in the air-backed case as shown in Figure 10(c)(the right column) Also the pressure of the reflected andtransmitted waves on the inner shell in the water-backedcase is obviously larger than that in the air-backed caseLater the wave reflected from the inner shell propagates tothe outer shell and then it is reflected and transmittedagain Due to the complex superposition of the incidentwave the reflected wave and the transmitted wave thepressure becomes very complicated (Figure 10(d) (the leftcolumn)) At the same time the cavitation region in the air-backed case is further expanding

On the whole because of the severe mismatch of theimpedance between different media the intensity of thetransmitted wave of the shells backed to water is larger thanthat of the shells backed to air and the cavitation region nearthe outer shell occurs in the air-backed case

33 ShockWaveReflection Figure 11 shows the time historyof the shock wave pressure at point 1 e outer shell isbacked to water in case a and to air in case b Meanwhiledifferent thicknesses of the outer shell are also taken intoaccount and the detonation distance r is 10 Note thatmedium 2 behind the inner shell has a little effect on thereflected pressure near the outer shell As the results show ifd1 gt 04 with the increase of the outer shell thickness thepressure of the reflected wave tends to be steady in bothcases By comparing the above two cases it can be seen thatthe peak pressure of the reflected wave caused by the water-backed shell is larger than that of the air-backed case but thelatter tends to be consistent with the former in value ifd1 ge 04 What is more the pressure fluctuates more vio-lently for the air-backed case


is can be explained from the perspective of wavepropagations e first interface for the wave arrival is thesame for the two cases namely the interface between thewater and the outside of the outer shell erefore the re-flected wave pressure at this interface should be equal in bothcases However at the next interface the shock wave trans-mits from the inside of the outer shell to the water in case awhile from the inside to the air in case b Because the acousticimpedances of water and air are much lower than that of steelthere is a rarefaction wave reflected at this interface [47ndash51]erefore the intensity of the incident shock wave isunloaded Moreover the acoustic impedance of air is lowerthan that of water the unloading effect in case a is less seriousthan that in case b so the peak pressure of the reflected wavein case a is larger than that in case b e stiffness of the outershell however is improved with the increase of shell thick-ness Hence the intensity of the reflected wave in these twocases is gradually enhanced To sum up if the outer shellthickness d1 ge 04 the medium behind the outer shell willaffect the peak pressure of the reflected wave slightly

e reflection coefficient of shock wave pressure is de-fined as λr P1P0 where P1 is the peak pressure of thereflected wave at point 1 and P0 is the peak pressure at thesame distance in the free-field underwater explosion modelIn the present work if the detonation distance is far enoughthe numerical model can be regarded as a one-dimensionalmodel Based on Taylorrsquos assumptions [6] we can obtain

P1 P0 + Pr minus ρcv where Pr is the pressure of the reflectedwave and ρcv is the pressure of the rarefaction wave pro-duced by the motion of the structure ρ is the density ofwater c is the sound speed of water and v is the velocity ofthe outer shell If the perfect reflection of the incident waveoccurs [52 53] the pressure of the reflected wave would readPr P0 erefore P1 should satisfy P1 2P0 minus ρcv theo-retically And then the reflection coefficient holdsλr P1P0 2minus ρcvP0

Figure 12 illustrates the reflection coefficients of theshock wave varying with the thickness of the outer shell atthe distance r 10 Obviously the reflection coefficientsincrease with the increase of shell thickness but the trendgradually slows down for the two cases Moreover thecoefficients of the air-backed case are almost the same asthose of the water-backed case if d1 ge 04 e reflectioncoefficients exceed 1 no matter what the medium is filledwith at the back of the outer shell It means that thepresence of the outer shell enhances the peak pressure atpoint 1

If the thickness d1 04 the shock wave reflection co-efficients varying with the distance r in these two cases areplotted in Figure 13 e coefficients decrease with the in-crease of detonation distance as shown in Figure 13 erelationship between the reflection coefficients and thedetonation distance is approximately linear for these twocases In the present work we can recast the Zamyshlyayevempirical formula ie equation (6) as P0 k(1r)

α wherek and α are the ratio parameter and exponent parameterrespectively So a relation between the reflection coefficientλr and the distance r can be obtained asλr P1P0 2minus (ρcvk)rα As described in the literature[54] the rarefaction wave (ie ρcv) can be neglected at theearly stage of shock wave impinging on the absolutely rigidstructure For the elastic shell if we extract the velocities ofthe outer shell near point 1 from numerical results listed inTable 7 we can find these velocities are positive for thesecases So the rarefaction wave cannot be neglected In thiscase the reflection coefficient of peak pressure varying withthe distance r can be obtained from the numerical simu-lation as shown in Figure 13 One should notice that thedecreasing trend of the coefficient is only suitable for thecases of close distances (ie 10le rle 16)

e impulse reflection coefficient can be defined asIr I1I0 where I1 is the pressure impulse at point 1 and I0is the pressure impulse at the same distance in the free-fieldunderwater explosion model Figure 14 shows the impulsereflection coefficients varying with shell thickness in thewater-backed case and air-backed case at the detonationdistance r 10 e impulse reflection coefficients increasewith the increase of shell thickness and the tendency be-comes gradually slow If d1 lt 04 the impulse reflectioncoefficients of the air-backed case will be smaller than thoseof the water-backed case because the deformation of theouter shell in the air-backed case is more serious than that inthe water-backed case If the thickness d1 ge 04 the co-efficients of the air-backed case are larger than those of thewater-backed case Under this circumstance the de-formation of the outer shell is not the principal factor

D2D1

07R

Figure 7 e position of measuring points

Table 4 e relevant parameters for material strength [46]

A B C n m2492MPa 8890MPa 0058 0746 094Reprinted from Ocean Engineering 117 FR Ming AM Zhang YZ XueSP Wang Damage characteristics of ship structures subjected to shockwavesof underwater contact explosions Pages No359ndash382 2016 with permissionfrom Elsevier (doi httpsdoiorg101016joceaneng201603040)

Table 5 e relevant parameters for the fracture model [46]

D1 D2 D3 D4 D5

038 147 258 minus00015 807Reprinted from Ocean Engineering 117 FR Ming AM Zhang YZ XueSP Wang Damage characteristics of ship structures subjected to shockwavesof underwater contact explosions Pages No359ndash382 2016 with permissionfrom Elsevier (doi httpsdoiorg101016joceaneng201603040)


Disp

lace

men

t (m

m)

40

30

20

10

00 10050 150 200 250

t (μs)

ExperimentCEL method

(a)

Disp

lace

men

t (m

m)

40

30

20

10

00 10050 150 200 250

t (μs)


(b)

Velo

city

(ms

)

250

200

150

50

100

00 100 200 300 400

t (μs)


(c)

Velo

city

(ms

)

200

150

100

50

00 200100 300 400

t (μs)


(d)

Figure 8 Displacement and velocity time histories at D1 and D2 points (these experimental data are provided by the Institute of FluidPhysics China Academy of Engineering Physics) (a) the displacement at D1 point (b) the displacement at D2 point (c) the velocity at D1point (d) the velocity at D2 point

Structure

Transmittingboundary

Medium 1

Medium 2

Water

TNT

(a)

R1 (outer shell)

R2 (innershell)

r

d1d2

TNT

321

(b)

Figure 9 Numerical model and the arrangement of measuring points (a) Numerical model (b) Model size and measuring points



A B C n m7920MPa 5100MPa 0014 026 103

1200P (MPa)

108096084072060048036024012000

1200P (MPa)

108096084072060048036024012000

120010809608407206004803602401200 0

120010809608407206004803602401200 0

(a)

800P (MPa)

7206405604804003202401608000

8007206405604804003202401608000

P (MPa)800720640560480400320240160800 0

800720640560480400320240160800 0

(b)

600P (MPa)

5404804203603002401801206000

6005404804203603002401801206000

P (MPa)

(c)

Figure 10 Continued


influencing impulse since the stiffness of the outer shell ishigh enough e reason for this phenomenon is that theincident wave transmits over the outer shell in the water-backed case while the incident wave is mainly reflected inthe air-backed case because of the impedance differencebetween water and air

If the thickness d1 04 the impulse reflection co-efficients varying with different detonation distances in thesetwo cases are plotted in Figure 15 e coefficients ap-proximately decrease linearly with the increase of detonationdistance whatever the outer shell is exposed to water or air Itshould be noticed that Ir of the air-backed case changesfaster than that of the water-backed case and varies in a widerange which may be due to the fact that complex super-position of the wave system between the outer shell and the

inner shell affects seriously the pressure of point 1 aftert 709θ found in Figure 10

34 Shock Wave Transmission When the shock wavetransmits over the outer shell to the inner shell the peakpressure and the energy of the shock wave gradually at-tenuate In Figure 16 the pressure histories of the trans-mitted shock wave for different shell thicknesses arepresented Medium 1 between the two shells and medium 2behind the inner shell are water or air respectively edetonation distance r is 10 In Figure 16(a) medium 1 andmedium 2 are both water (noted as water-water) whilemedium 1 is water and medium 2 is air (noted as water-air)in Figure 16(b) Obviously when the wave transmits over the

500P (MPa)

4504003503002502001501005000

5004504003503002502001501005000

P (MPa)

(d)

Figure 10 Comparison of the shock wave propagation when the double-layer shell was filled with water (the left column) and air (the rightcolumn) (a) t 363θ (b) t 520θ (c) t 630θ (d) t 709θ

315θ 473θ 630θ 788θt

1

378θ 394θ

0

50

100

150

200

250

P (M

Pa)

d1 = 02d1 = 04

d1 = 06d1 = 08

200

180

220

(a)

d1 = 02d1 = 04

d1 = 06d1 = 08

378θ 394θ

220

200

180

160

315θ 473θ 630θ 788θt

0

50

100

150

200

250

P (M

Pa)

1

(b)

Figure 11e pressure at gauging point 1 for the cases of different thicknesses of the outer shell (dark color represents water and light colorrepresents air) (a) Media 1 and 2 filled with water (b) Media 1 and 2 filled with air


outer shell a peak pressure called the transmitted peakoccurs in these two cases and then it attenuates quicklyWith the attenuation of the pressure the waves reflected bythe outer and inner surfaces of the inner shell reach point 2at about t 630θ as shown in Figure 17 Owing to thesuperposition of the transmitted wave and the reflectedwave it causes the pressure to rise and generates a secondpeak pressure called the reflected peak which might have asecond impact on the shell (Figures 16(a) and 16(b)) In

01 02 03 04 05 06 07 0812

14

16

18

2

λr

Water-backedAir-backed

d1

Figure 12 e reflection coefficients of peak pressure for differentouter shell thicknesses

10 11 12 13 14 15 16174

176

178

18

182

184

186

λr


r

Figure 13 e reflection coefficients of peak pressure for differentdetonation distances

Table 7 e time of the peak pressure occurring the corre-sponding velocity of the outer shell and the reflection coefficientwith different detonation distances

r t (θ) v (ms) λr10 390 121 18411 438 111 18212 487 100 18113 538 97 18014 588 92 17915 639 87 17816 689 82 177

01 02 03 04 05 06 07 0804

08

12

16

Ir


d1

Figure 14 e reflection coefficients of impulse for different outershell thicknesses

10 11 12 13 14 15 1612

13

14

15

16

Ir


r

Figure 15 e reflection coefficients of impulse for differentdetonation distances


addition both the peaks decrease with the increase of theouter shell thickness It is worth noting that the reflectedpeak can be larger than the transmitted peak if d1 ge 06 Itindicates that the reflected wave produced by the innersurface of the outer shell plays a leading role under thiscircumstance

Media 1 and 2 are both air (noted as air-air) inFigure 16(c) while medium 1 is air and medium 2 is water(noted as air-water) in Figure 16(d) e peak pressures ofthe transmitted shock wave at point 2 are almost the same atthe same shell thickness for both the cases while they are

about 1330 times that of the water-water case It may be dueto the lower impedance of air which is about 15000 timesthat of the water If the shell thickness d1 02 in these twocases there are no values in pressure history curves when thetime exceeds 946θ which is due to the fact that the outershell after deformation is over the position of the measuringpoint 2 With the increase of the shell thickness the pressureattenuates gradually Different from the water-water case orwater-air case the second peak arrives later (at aboutt 1577θ) which may be due to the strong impedancemismatch Also the pressure time histories show no

2

d1 = 04d1 = 02 d1 = 06

d1 = 08

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60Transmitted peak

Reflected peak

(a)

2

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60

Transmitted peak

Reflected peak

d1 = 02d1 = 04

d1 = 06d1 = 08

(b)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(c)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(d)

Figure 16 Transmitted pressure at gauging point 2 for different media filled between the shells (dark color represents water and light colorrepresents air) (a) Water-water (b) Water-air (c) Air-air (d) Air-water


differences for medium 2 filled with water or air behind theinner shell

As seen in Figures 16(a)ndash16(d) it can be found thatmedium 1 has a great influence on the pressure of thetransmitted wave On the contrary medium 2 behind theinner shell has few effects on the transmitted pressure

e transmission coefficient λt P2P0 can be defined asthe ratio of the peak pressure P2 of the transmitted wave atpoint 2 ie the first peak in Figure 16 to the peak pressureP0 of the incident shock wave in free field e transmissioncoefficients of the shock wave varying with different shellthicknesses are plotted in Figure 18 As described in theanalysis above medium 2 behind the inner shell has littleeffect on the peak pressure of the transmitted shock waveerefore the cases in which medium 2 is water whilemedium 1 is water or air are hereinafter taken for examplesWith the increase of the shell thickness the transmissioncoefficients of the shock wave decrease exponentially for thetwo cases Although the peak pressure P2 of the air-backedcase is much smaller than that of the water-backed case thetrend of transmission coefficients varying with different shellthicknesses is similar to that of the water-backed case

Taking the thickness d1 04 for example the curves ofthe transmission coefficients varying with different deto-nation distances can be plotted in Figure 19 Unlike

Figure 13 with the increase of the detonation distance thetransmission coefficients have a trend of increase by degreese difference may be because the peak pressure P2 is af-fected by the inner shell and the outer shell together whileP1 is mainly affected by the outer shell although the peakpressure P1 and P2 decrease as the detonation distanceincreases

e total energy released by the TNT is 2703 kJ att 75θ and the proportion of the energy absorbed by theouter shell accounting for the total energy is plotted inFigure 20 Because the TNTmass is quite small and the shellhas not been destroyed the energy absorbed by the outershell is less With the increase of shell thickness the energyabsorption proportion increases generally Another featureis that the proportion of the energy absorbed by the outershell can be up to 2 for the air-backed case while about 1for the water-backed case when the shell thickness d1 rangesfrom 01 to 08 For the same shell thickness the energyabsorption of the air-backed case is two to three times that ofthe water-backed case is may attribute to the largerdeformation of the shell in the air-backed case

Figure 21 shows the proportion of the energy transmittedover the outer shell versus different thicknesses etransmission energy gradually decreases with the increase ofshell thickness for the two cases For the same shell

1000 Pressure (MPa)

Water

04

(a)

1000

Water

04Pressure (MPa)

(b)

1000

Air

04Pressure (MPa)

(c)

1000

Air

04Pressure (MPa)

(d)

Figure 17e pressure contours if themedium is water-water and water-air at the same time (a)Water-water at t 630θ (b)Water-waterat t 709θ (c) Water-air at t 630θ (d) Water-air at t 709θ


thickness the transmission energy of the air-backed shell ismuch smaller than that of the water-backed shell whichcoincides with the above results in Figure 18 at is thetransmission coefficient λt of the air-backed case is muchsmaller than that of the water-backed case

Figure 22 shows the Mises stress contours of the outerand inner shells for the case of the outer shell thicknessd1 04 It can be seen that if medium 1 between the two

shells is air the maximum stress of the shell will be about 15times that of the water-backed shell and there are almost nostresses in the inner shell erefore it can be concluded thatif the medium is filled with air between two shells the innershell is prevented from serious impact

35 Comparison of Shock Wave Reflection and TransmissionActually the shock wave pressure measured on the outersurface of the shell is from the superposition of the incidentshock wave and the reflected shock wave In order to study

01 02 03 04 05 06 07 080

05

1

15

2

25

3

()


d1

Figure 20 e absorbed energy for different outer shellthicknesses

06 07 080

05

1

15

2

25

3

()

01 02 03 04 050

001

002

003

004


d1

Figure 21 e total transmission energy for different outer shellthicknessesewater-backed case is plotted in the left axis and theair-backed case is plotted in the right axis

01 02 03 04 05 0802

03

04

05

06

07

λt

06 0708

09

1

11

12

13times 10ndash3


d1

Figure 18 e transmission coefficients of peak pressure fordifferent outer shell thicknessese water-backed case is plotted inthe left axis and the air-backed case is plotted in the right axis

10 11 12 13 14 15 1603

035

04

045

λt

0

1

2

3times 10ndash3


r

Figure 19 e transmission coefficients of peak pressure fordifferent detonation distances e water-backed case is plotted inthe left axis and the air-backed case is plotted in the right axis


the relationship between the reflection and transmission ofthe shock wave the ratios λ1 (P1 minusP0)P2 of the peakpressure of the reflected shock wave to that of the trans-mitted shock wave are plotted in Figure 23 e peakpressure of the reflected shock wave is the difference betweenthe measured pressure P1 near the surface of the outer shelland the incident pressure P0 in free field

Obviously with the increase of the thickness no matterwhat the outer shell is backed to the ratios increase and tendto be slow when the thickness exceeds to some extents for theair-backed case e reason for this phenomenon is that thepeak pressure P2 of the transmission wave decreases dras-tically with the increase of shell thickness (Figures 16 and18) Another feature is that the ratios of the shell backed toair are much larger than those of the shell backed to waterat is the reflection proportion is dominant for the air-backed case is can be explained by the fact which can befound in Section 34 that the peak pressure P2 of the air-backed case is smaller than that of the water-backed case fortwo orders of magnitude

Figure 24 shows the peak pressure ratios λ2 P3P1 atthe measuring points 1 and 3 versus different outer shellthicknesses Generally the ratios decrease with the increaseof the shell thickness no matter what media 1 and 2 are filledwith It indicates the reflection proportion increase as theshell thickens In addition the ratios of the water-backedcases (water-water case and water-air case) are obviouslylarger than those of the air-backed cases (air-air case and air-water case) which almost coincide with each other eseaccord well with the analysis in Sections 33 and 34

After the directive wave propagating over the outer shellthe transmitted wave arrives at the inner shell which can beregarded as a directive wave producing a new charge ie theequivalent charge mass me at is to say the effect of thetransmitted wave on the inner shell is to some extentequivalent to that of a new charge on the inner shell Withthe aid of the analysis in Section 33 me can be transformedfrom a free-field component of incident peak pressure on theinner shell namely P4 P3λr by Zamyshlyayev empiricalformula (6)

Table 8 presents the ratios of the equivalent charge massme to the actual charge mass mr e cases of the mediumbetween the two shells filled with water are considered andair-backed cases are absent because P3 is too small As thetable shows the equivalent charge mass for the inner shellcan be decreased about 75 if the outer shell thickness isover 02 times the charge radius With the increasingthickness the equivalent charge mass decreases sharply

4 Conclusions

e shock wave propagation between two hemisphericalshells which are filled with different media is studied based onthe coupled EulerianndashLagrangian method in AUTODYN

Water

4263 00Misstress (MPa)

(a)

Air


(b)

Figure 22eMises stress distribution of the double-layer shell structure (a) Water filled between the two shells (b) Air filled between thetwo shells

0

1

2

3

4

λ1

01 02 03 04 05 06 07 080

200

400

600

800

1000


d1

Figure 23 e reflection and transmission ratios for differentouter shell thicknesses e water-backed case is plotted in the leftaxis and the air-backed case is plotted in the right axis


e relationships among the incident reflected and trans-mitted waves are discussed

emedium between two hemispherical shells will affectthe peak pressure of the reflected wave generated by theouter shell slightly if the outer shell thickness is thicker thana certain level (here it is over 04 times the charge radius)e reflection coefficients of peak pressure and impulseincrease with the increase of shell thickness and decreaselinearly with the increase of detonation distance Besides thecavitation area caused by the reflected rarefaction wave nearthe outer shell occurs in the air-backed case

When the wave transmits over the outer shell a peakpressure called the transmitted peak occurs whatever themedium behind the outer shell is water or air It is worthmentioning that a second peak which is relatively comparedto the transmitted peak can be produced by the reflections ofthe inner shell and it will be larger than the transmitted peakif the outer shell thickness reaches a certain extent (here it isover 06 times the charge radius) for the water-water andwater-air casesemedium between the two shells has great

influences on the transmitted wave while the medium be-hind the inner shell has slight effects on the transmittedwave With the increase of the shell thickness the trans-mission coefficients of the shock wave decrease exponen-tially and increase linearly with the detonation distancewhatever the outer shell is exposed to water or air

For the same outer shell thickness the energy absorptionof the air-backed case is about two to three times that of thewater-backed case With the increasing thickness theequivalent charge mass decreases drastically e equivalentcharge mass for the inner shell can be decreased about 75 ifthe outer shell thickness is over 02 times the charge radius

Data Availability

All data included in this study are available upon request bycontacting the corresponding author

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is paper was supported by the National Natural ScienceFoundation of China (51609049 and U1430236) and theNatural Science Foundation of Heilongjiang Province(QC2016061)

References

[1] F R Ming AM Zhang H Cheng and P N Sun ldquoNumericalsimulation of a damaged ship cabin flooding in transversalwaves with smoothed particle hydrodynamics methodrdquoOcean Engineering vol 165 pp 336ndash352 2018

[2] R H Cole Underwater Explosions Princeton UniversityPress Princeton NJ USA 1948

[3] B V Zamyshlyaev and Y S Yakovlev ldquoDynamic loads inunderwater explosionrdquo Tech Rep AD-757183 IntelligenceSupport Center Washington DC USA 1973

[4] J M Brett ldquoNumerical modelling of shock wave and pressurepulse generation by underwater explosionsrdquo Tech Rep AR-010-558 DSTO Aeronautical and Maritime Research Labo-ratory Canberra Australia 1998

[5] F R Ming P N Sun and A M Zhang ldquoInvestigation oncharge parameters of underwater contact explosion based onaxisymmetric sph methodrdquo Applied Mathematics and Me-chanics vol 35 no 4 pp 453ndash468 2014

[6] G I Taylor ldquoe pressure and impulse of submarine ex-plosion waves on platesrdquo in Underwater Explosion Researchpp 1155ndash1173 Office of Naval Research Arlington VA USA1950

[7] Z Y Jin C Y Yin Y Chen X C Huang and H X Hua ldquoAnanalytical method for the response of coated plates subjectedto one-dimensional underwater weak shock waverdquo Shock andVibration vol 2014 no 2 Article ID 803751 13 pages 2014

[8] Y Chen X Yao and W Xiao ldquoAnalytical models for theresponse of the double-bottom structure to underwater ex-plosion based on the wave motion theoryrdquo Shock and Vi-bration vol 2016 no 9 Article ID 7368624 21 pages 2016

0

01

02

03

04

05

06

07

λ2

01 02 03 04 05 06 07 084

6

8

10

12

14times 10ndash4

Water-waterWater-air

Air-airAir-water

d1

Figure 24 e peak pressures of P3P1 for different outer shellthicknessese water-backed case is plotted in the left axis and theair-backed case is plotted in the right axis

Table 8 e equivalent charge mass for the inner shell withdifferent outer shell thicknesses

Case Water-water (memr ) Water-air (memr )

d1 01 814 788d1 02 261 252d1 03 119 116d1 04 68 65d1 05 43 41d1 06 28 27d1 07 21 19d1 08 16 14


[9] M Otsuka S Tanaka and S Itoh Research for Explosion ofHigh Explosive in Complex Media Springer Netherlands NewYork City NY USA 2006

[10] C Desceliers C Soize Q Grimal G Haiat and S Naili ldquoAtime-domain method to solve transient elastic wave propa-gation in a multilayer medium with a hybrid spectral-finiteelement space approximationrdquo Wave Motion vol 45 no 4pp 383ndash399 2008

[11] G Wang S Zhang M Yu H Li and Y Kong ldquoInvestigationof the shock wave propagation characteristics and cavitationeffects of underwater explosion near boundariesrdquo AppliedOcean Research vol 46 no 2 pp 40ndash53 2014

[12] Z D Wu L Sun and Z Zong ldquoComputational investigationof the mitigation of an underwater explosionrdquo ActaMechanica vol 224 no 12 pp 3159ndash3175 2013

[13] L Hammond and R Grzebieta ldquoStructural response ofsubmerged air-backed plates by experimental and numericalanalysesrdquo Shock and Vibration vol 7 no 6 pp 333ndash3412000

[14] R Rajendran and J M Lee ldquoA comparative damage study ofair- and water-backed plates subjected to non-contact un-derwater explosionrdquo International Journal of Modern PhysicsB vol 22 pp 1311ndash1318 2008

[15] X L Yao J Guo L H Feng and A M Zhang ldquoCompa-rability research on impulsive response of double stiffenedcylindrical shells subjected to underwater explosionrdquo In-ternational Journal of Impact Engineering vol 36 no 5pp 754ndash762 2009

[16] Z F Zhang F R Ming and A M Zhang ldquoDamage char-acteristics of coated cylindrical shells subjected to underwatercontact explosionrdquo Shock and Vibration vol 2014 Article ID763607 15 pages 2014

[17] Z Wang X Liang and G Liu ldquoAn Analytical Method forEvaluating the Dynamic Response of Plates Subjected toUnderwater Shock Employing Mindlin Plate eory andLaplace Transformsrdquo Mathematical Problems in Engineeringvol 2013 Article ID 803609 11 pages 2013

[18] G Z Liu J H Liu J Wang J Q Pan and H B Mao ldquoAnumerical method for double-plated structure completelyfilled with liquid subjected to underwater explosionrdquoMarineStructures vol 53 pp 164ndash180 2017

[19] Z Zhang L Wang and V V Silberschmidt ldquoDamage re-sponse of steel plate to underwater explosion effect of shapedcharge linerrdquo International Journal of Impact Engineeringvol 103 pp 38ndash49 2017

[20] D Su Y Kang X Wang et al ldquoAnalysis and numericalsimulation for tunnelling through coal seam assisted by waterjetrdquo Computer Modeling in Engineering and Sciences vol 111no 5 pp 375ndash393 2016

[21] Y S Ferezghi M Sohrabi and S Nezhad ldquoDynamic analysisof non-symmetric functionally graded (FG) cylindricalstructure under shock loading by radial shape function usingmeshless local Petrov-Galerkin (MLPG) method with non-linear grading patternsrdquo Computer Modeling in Engineeringand Sciences vol 113 no 4 pp 497ndash520 2017

[22] N N Liu F R Ming L T Liu and S F Ren ldquoe dynamicbehaviors of a bubble in a confined domainrdquo Ocean Engi-neering vol 144 pp 175ndash190 2017

[23] N N Liu W B Wu A M Zhang and Y L Liu ldquoExperi-mental and numerical investigation on bubble dynamics neara free surface and a circular opening of platerdquo Physics ofFluids vol 29 no 10 article 107102 2017

[24] F R Ming P N Sun and A M Zhang ldquoNumerical in-vestigation of rising bubbles bursting at a free surface through

a multiphase sph modelrdquo Meccanica vol 52 no 11-12pp 1ndash20 2017

[25] Autodyn explicit software for nonlinear dynamics eorymanual version 43 Century Dynamics Inc Concord CAUSA 2003

[26] D Simulia Abaqus Version 2016 Documentation USA Das-sault Systems Simulia Corp Johnston RI USA 2016

[27] B G R Liu and M B Liu Smoothed Particle Hydrodynamicsa Meshfree Particle Method World Scientific Singapore 2004

[28] A M Zhang W S Yang and X L Yao ldquoNumerical sim-ulation of underwater contact explosionrdquo Applied OceanResearch vol 34 pp 10ndash20 2012

[29] G Luttwak and M S Cowler ldquoAdvanced eulerian techniquesfor the numerical simulation of impact and penetration usingautodyn-3drdquo Tech rep 1999

[30] D J Steinberg ldquoSpherical explosions and the equation of stateof waterrdquo UCID-20974 Lawrence Livermore National Lab-oratory Livermore CA USA 1987

[31] M L Wilkins ldquoCalculation of elastic-plastic flowrdquo Journal ofBiological Chemistry vol 280 no 13 pp 12833ndash12839 1969

[32] D J Benson ldquoComputational methods in Lagrangian andeulerian hydrocodesrdquo Computer Methods in Applied Me-chanics and Engineering vol 99 no 2-3 pp 235ndash394 1992

[33] J H Kim and H C Shin ldquoApplication of the ale technique forunderwater explosion analysis of a submarine liquefied ox-ygen tankrdquo Ocean Engineering vol 35 no 8-9 pp 812ndash8222008

[34] J S Peery and D E Carroll ldquoMulti-material ale methods inunstructured gridsrdquo Computer Methods in Applied Mechanicsand Engineering vol 187 no 3 pp 591ndash619 2000

[35] F L Addessio D E Carroll J K Dukowicz et al ldquoCaveat acomputer code for fluid dynamics problems with large dis-tortion and internal sliprdquo vol 46 no 3 pp 1019ndash1022 1992

[36] W E Johnson ldquoOil a continuous two-dimensional eulerianhydrodynamic coderdquo Tech rep Defense Technical In-formation Center Fort Belvoir VA USA 1965

[37] D J Benson and S Okazawa ldquoContact in a multi-materialeulerian finite element formulationrdquo Computer Methods inApplied Mechanics and Engineering vol 193 no 39-41pp 4277ndash4298 2004

[38] B V Leer ldquoTowards the ultimate conservative differencescheme Iv A new approach to numerical convectionrdquoJournal of Computational Physics vol 23 no 3 pp 276ndash2991977

[39] B V Leer ldquoTowards the ultimate conservative differencescheme V - a second-order sequel to godunovrsquos methodrdquoJournal of Computational Physics vol 32 no 1 pp 101ndash1361997

[40] D J Benson ldquoA mixture theory for contact in multi-materialeulerian formulationsrdquo Computer Methods in Applied Me-chanics and Engineering vol 140 no 1-2 pp 59ndash86 1997

[41] CW Hirt and B D Nichols ldquoVolume of fluid (VOF) methodfor the dynamics of free boundariesrdquo Journal of Computa-tional Physics vol 39 no 1 pp 201ndash225 1981

[42] L Olovsson ldquoEulerian-lagrangian mapping for finite elementanalysisrdquo US 7167816 B1 2007

[43] D L Youngs ldquoAn interface tracking method for a 3D eulerianhydrodynamics coderdquo Scientific Research PublishingWuhan China 1984

[44] C L Xin G G Xu and K Z Liu ldquoNumerical simulation ofunderwater explosion loadsrdquo Transactions of Tianjin Uni-versity vol 14 no B10 pp 519ndash522 2008


[45] S Marrone ldquoEnhanced SPH modeling of free-surface flowswith large deformationsrdquo PhD thesis University of Rome LaSapienza Rome Italy 2012

[46] F R Ming A M Zhang Y Z Xue and S P Wang ldquoDamagecharacteristics of ship structures subjected to shockwaves ofunderwater contact explosionsrdquo Ocean Engineering vol 117pp 359ndash382 2016

[47] A Chernishev N Petrov and A Schmidt Numerical In-vestigation of Processes Accompanying Energy Release inWaterNeare Free Surface 28th International Symposium on ShockWaves Springer Berlin Heidelberg 2012

[48] M Kamegai C E Rosenkilde and L S Klein ldquoComputersimulation studies on free surface reflection of underwatershock wavesrdquo UCRL-96960 UCRL Berkeley CA USA 1987

[49] N V Petrov and A A Schmidt ldquoMultiphase phenomena inunderwater explosionrdquo Experimental ermal and FluidScience vol 60 pp 367ndash373 2015

[50] M Dubesset and M Lavergne ldquoCalculation of the cavitationdue to underwater explosions at small depthsrdquo Acta AcusticaUnited with Acustica vol 20 no 5 pp 289ndash298 1968

[51] A R Pishevar and R Amirifar ldquoAn adaptive ale method forunderwater explosion simulations including cavitationrdquoShock Waves vol 20 no 5 pp 425ndash439 2010

[52] N A Fleck and V Deshpande ldquoe resistance of clampedsandwich beams to shock loadingrdquo Journal of Applied Me-chanics vol 71 no 3 pp 386ndash401 2004

[53] Z K Liu and Y L Young ldquoTransient response of submergedplates subject to underwater shock loading an analyticalperspectiverdquo Journal of Applied Mechanics vol 75 no 4article 044504 2008

[54] Z Zong and K Y Lam ldquoViscoplastic response of a circularplate to an underwater explosion shockrdquo Acta Mechanicavol 148 no 1-4 pp 93ndash104 2001


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medium caused by the medium deformation is negligibleRecently Chen et al [8] applied one-dimensional cavitationtheory and section-varying bar theory to discuss the propa-gation characteristics of stress waves in simple plates andstiffened plates (T profiles and I profiles) Considering thecavitation effect the pressure and impulse on the wettedsurface are predicted accurately by analytical solution eyfound that obviously pressure drops and oscillations occur onthe wetted surface in the stiffened plates model due to thesuperposition of the reflected wave from the structure and theincident wave compared to the simple plate model Howeverthis analytical model is limited to the one-dimensional system

Valuable researches on pressure characteristics of theshock wave in structures using numerical simulation havealso been conducted Otsuka et al [9] utilized LS-DYNA3D an analysis code using the arbitraryLagrangianndashEulerian approach to simulate the propaga-tion process of the explosion shock waves produced by thetwo modes of detonation (a detonation cord and anemulsion explosive) in the three layers of air water andstructure ey found that as the air layer becomes thickthe peak pressure decreases in the water However theimpulse remains unchanged since the duration of pressureis increased by the effect of a reflected wave Yet theirsimulation lacks a comparative study between the differentmaterial layers Desceliers et al [10] presented a newnumerical hybrid method dedicated to the simulation ofthe transient elastic wave propagation in multilayer un-bounded media which can be fluid or solid subjected togiven transient loadsis method can capture clearly someprofiles of the elastic waves including reflected wavestransmitted waves and refracted surface waves However itis restricted to simple geometrical structures Wang et al[11] used the coupled EulerianndashLagrangian (CEL) methodin AUTODYN to investigate the shock wave propagationcharacteristics near the air-backed plate and water-backedplate ey found that two pressure peaks occurred in thesetwo numerical models where the first peak is generated bythe incident wave and the second peak is induced by thereflected wave from the plate due to the impedance of thewater being lower than that of the plate But in that workthey did not apply the CELmethod to multilayer structuresWu et al [12] investigated the protection effects of amitigation layer on the structure exerted by an underwaterexplosion with the help of the Mie-Greurouneisen mixturemodel and discussed the affections of the layer material(steel neoprene and polyethylene) layer thickness andexplosive-structure distance on protection effects eyconcluded that the acoustic impedance of the mitigationlayer plays an important role in determining not only theproperty of shock loads but also the occurrence of pro-tection effects of the mitigation layer while the layerthickness and explosive-structure distance have a littleeffect on the main shock loads Nevertheless the loads ofthe inner shock wave in the mitigation layer are their mainconcern In addition other scholars mainly focus onstructural response and damage characteristics [13ndash21] Ina word theoretical research based on certain assumptionsis suitable for a structure with a simple geometry while

numerical simulation is easier to implement to deal withshock wave propagation problems in complex structures

Owing to the presence of complex superposition ofmultiple waves geometry with curvature moving interfaceand large plastic deformation traditional theoretical re-search comes across many difficulties in solving shock wavepropagation characteristics in multilayer structures In orderto tackle these problems the CEL method in AUTODYN isadopted to study the propagation characteristics of the shockwave in the multilayer structures during underwater ex-plosion in the present worke shell thickness and the filledmedium between the double shells are focused and theirinfluences on the propagation of the shock wave are ana-lyzed e relationships among the incident the reflectedand the transmitted shock waves are discussed

2 Theoretical Background

21 Coupled Eulerian and Lagrangian Method Actually theprocess of underwater explosion near structures is a com-plicated fluid-structure interaction (FSI) problem [22 23]Two sorts of methods are usually used to solve this problemnamely the Lagrangianmethod and Eulerianmethod [11 24]Both of them have their own advantages and limitations eLagrangian method is suitable for describing the stress strainand deformation of the structure However when encoun-tering large deformation problems the Lagrangian mesh isseverely distorted which leads to adverse effects on calcu-lation Conversely the Eulerian method can deal with largedeformation problems and some flow problems But itcannot track accurately material interfaces erefore acoupled EulerianndashLagrangian (CEL) analysis method whichcombines the advantages of these two methods is introducedto tackle FSI problems e structure part is solved by theLagrangian method while the fluid part is solved by theEulerian methode FSI between the two parts is carried outat the interface [11 25]is approach has beenmade possiblein some hydrocodes eg AUTODYN [25] and ABAQUS[26] Specifically in the Lagrangian method based on thefinite difference scheme the material is attached to the meshelement which deforms with the flow of the material and themass of the element remains constant [25] In the Lagrangiansolver given the node position x (x y z) the node velocityu the element density ρ the element volume V the internalenergy e and the governing equations are as follows[25 27 28]

ρ ρ0V0

Vm

V

ρdudt nabla middot σ

de

dtσρnabla middot u

(1)

where ρ0 V0 m and σ are the initial density initial volumemass and total stress tensor respectivelye stress tensor isseparated into a hydrostatic component and a deviatoriccomponent which can be referred from the study in [29]





z(ρu)zt


z(ρE)zt


(2)









P A 1minusωR1V


ωR2V

1113888 1113889eminusR2V +

ωEV (3)


1113896 (4)



p(t)


0368Pmθt

1 minust

tp1113888 1113889


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(5)

Pm

441 times 107 W13

R1113872 1113873

15 6le

R

R0lt 12

524 times 107 W13

R1113872 1113873

113 12le

R

R0lt 240

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(6)

tp R0

C⎡⎣850

Ph

Patm1113888 1113889

081

minus 20Ph

Patm1113888 1113889

13


R1113874 1113875

013+ 151

R0

R1113874 1113875

126⎤⎦

(7)


θ

045R0RR0

1113874 1113875045

times 10minus3R

R0lt 30

35R0

C

lgR

R01113888 1113889minus 09

1113971

R

R0ge 30

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(8)

Ph Patm + ρgh (9)





Nodal displacement




Nodal acceleration

Nodal velocity

Inte

grat

ion





Nodal acceleration

Momentummass



Material model



Nodal velocity

Transport

Forcemass

Initial condition


x u F m

σ ε P e ρ

(a)

σ ε P eρ u m

x

(b)

Lagrangiansubgrid

Euleriansubgrid

(c)





r

1

3

654

2

R0

Water

TNT

7

Measuringpoints

20R0

50R0


(a)

1659P (MPa)

1494132811629968306644983321660

(b)

333P (MPa)

2972602241881511157842560

(c)








P (M

Pa)

120

100

80

60

40

20

010 02 04

t (ms)06 08



120

100

80

60

40

20

0

P (M

Pa)

0 02 04t (ms)

06 08 1


(a)

t (ms)

80

60

40

20

0

P (M

Pa)

0 02 04 06 08 1


(b)




10 1143 1152 0811 993 998 0612 873 876 0413 775 803 3514 698 738 5515 631 683 7616 574 635 96

TNT

Water surface

Water tank

RD

Plate

R0





















D2D1

07R





D1 D2 D3 D4 D5



Disp

lace

men

t (m

m)

40

30

20

10

00 10050 150 200 250

t (μs)


(a)

Disp

lace

men

t (m

m)

40

30

20

10

00 10050 150 200 250

t (μs)


(b)

Velo

city

(ms

)

250

200

150

50

100

00 100 200 300 400

t (μs)


(c)

Velo

city

(ms

)

200

150

100

50

00 200100 300 400

t (μs)


(d)


Structure


Medium 1

Medium 2

Water

TNT

(a)

R1 (outer shell)

R2 (innershell)

r

d1d2

TNT

321

(b)




A B C n m7920MPa 5100MPa 0014 026 103

1200P (MPa)

108096084072060048036024012000

1200P (MPa)

108096084072060048036024012000

120010809608407206004803602401200 0

120010809608407206004803602401200 0

(a)

800P (MPa)

7206405604804003202401608000

8007206405604804003202401608000

P (MPa)800720640560480400320240160800 0

800720640560480400320240160800 0

(b)

600P (MPa)

5404804203603002401801206000

6005404804203603002401801206000

P (MPa)

(c)

Figure 10 Continued






500P (MPa)

4504003503002502001501005000

5004504003503002502001501005000

P (MPa)

(d)


315θ 473θ 630θ 788θt

1

378θ 394θ

0

50

100

150

200

250

P (M

Pa)

d1 = 02d1 = 04

d1 = 06d1 = 08

200

180

220

(a)

d1 = 02d1 = 04

d1 = 06d1 = 08

378θ 394θ

220

200

180

160

315θ 473θ 630θ 788θt

0

50

100

150

200

250

P (M

Pa)

1

(b)




01 02 03 04 05 06 07 0812

14

16

18

2

λr


d1


10 11 12 13 14 15 16174

176

178

18

182

184

186

λr


r



r t (θ) v (ms) λr10 390 121 18411 438 111 18212 487 100 18113 538 97 18014 588 92 17915 639 87 17816 689 82 177

01 02 03 04 05 06 07 0804

08

12

16

Ir


d1


10 11 12 13 14 15 1612

13

14

15

16

Ir


r






2

d1 = 04d1 = 02 d1 = 06

d1 = 08

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60Transmitted peak

Reflected peak

(a)

2

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60

Transmitted peak

Reflected peak

d1 = 02d1 = 04

d1 = 06d1 = 08

(b)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(c)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(d)










1000 Pressure (MPa)

Water

04

(a)

1000

Water

04Pressure (MPa)

(b)

1000

Air

04Pressure (MPa)

(c)

1000

Air

04Pressure (MPa)

(d)







01 02 03 04 05 06 07 080

05

1

15

2

25

3

()


d1


06 07 080

05

1

15

2

25

3

()

01 02 03 04 050

001

002

003

004


d1


01 02 03 04 05 0802

03

04

05

06

07

λt

06 0708

09

1

11

12

13times 10ndash3


d1


10 11 12 13 14 15 1603

035

04

045

λt

0

1

2

3times 10ndash3


r








4 Conclusions


Water


(a)

Air


(b)


0

1

2

3

4

λ1

01 02 03 04 05 06 07 080

200

400

600

800

1000


d1








Data Availability




Acknowledgments


References









0

01

02

03

04

05

06

07

λ2

01 02 03 04 05 06 07 084

6

8

10

12

14times 10ndash4


Air-airAir-water

d1




d1 01 814 788d1 02 261 252d1 03 119 116d1 04 68 65d1 05 43 41d1 06 28 27d1 07 21 19d1 08 16 14





















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





z(ρu)zt


z(ρE)zt


(2)









P A 1minusωR1V


ωR2V

1113888 1113889eminusR2V +

ωEV (3)


1113896 (4)



p(t)


0368Pmθt

1 minust

tp1113888 1113889


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(5)

Pm

441 times 107 W13

R1113872 1113873

15 6le

R

R0lt 12

524 times 107 W13

R1113872 1113873

113 12le

R

R0lt 240

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(6)

tp R0

C⎡⎣850

Ph

Patm1113888 1113889

081

minus 20Ph

Patm1113888 1113889

13


R1113874 1113875

013+ 151

R0

R1113874 1113875

126⎤⎦

(7)


θ

045R0RR0

1113874 1113875045

times 10minus3R

R0lt 30

35R0

C

lgR

R01113888 1113889minus 09

1113971

R

R0ge 30

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(8)

Ph Patm + ρgh (9)





Nodal displacement




Nodal acceleration

Nodal velocity

Inte

grat

ion





Nodal acceleration

Momentummass



Material model



Nodal velocity

Transport

Forcemass

Initial condition


x u F m

σ ε P e ρ

(a)

σ ε P eρ u m

x

(b)

Lagrangiansubgrid

Euleriansubgrid

(c)





r

1

3

654

2

R0

Water

TNT

7

Measuringpoints

20R0

50R0


(a)

1659P (MPa)

1494132811629968306644983321660

(b)

333P (MPa)

2972602241881511157842560

(c)








P (M

Pa)

120

100

80

60

40

20

010 02 04

t (ms)06 08



120

100

80

60

40

20

0

P (M

Pa)

0 02 04t (ms)

06 08 1


(a)

t (ms)

80

60

40

20

0

P (M

Pa)

0 02 04 06 08 1


(b)




10 1143 1152 0811 993 998 0612 873 876 0413 775 803 3514 698 738 5515 631 683 7616 574 635 96

TNT

Water surface

Water tank

RD

Plate

R0





















D2D1

07R





D1 D2 D3 D4 D5



Disp

lace

men

t (m

m)

40

30

20

10

00 10050 150 200 250

t (μs)


(a)

Disp

lace

men

t (m

m)

40

30

20

10

00 10050 150 200 250

t (μs)


(b)

Velo

city

(ms

)

250

200

150

50

100

00 100 200 300 400

t (μs)


(c)

Velo

city

(ms

)

200

150

100

50

00 200100 300 400

t (μs)


(d)


Structure


Medium 1

Medium 2

Water

TNT

(a)

R1 (outer shell)

R2 (innershell)

r

d1d2

TNT

321

(b)




A B C n m7920MPa 5100MPa 0014 026 103

1200P (MPa)

108096084072060048036024012000

1200P (MPa)

108096084072060048036024012000

120010809608407206004803602401200 0

120010809608407206004803602401200 0

(a)

800P (MPa)

7206405604804003202401608000

8007206405604804003202401608000

P (MPa)800720640560480400320240160800 0

800720640560480400320240160800 0

(b)

600P (MPa)

5404804203603002401801206000

6005404804203603002401801206000

P (MPa)

(c)

Figure 10 Continued






500P (MPa)

4504003503002502001501005000

5004504003503002502001501005000

P (MPa)

(d)


315θ 473θ 630θ 788θt

1

378θ 394θ

0

50

100

150

200

250

P (M

Pa)

d1 = 02d1 = 04

d1 = 06d1 = 08

200

180

220

(a)

d1 = 02d1 = 04

d1 = 06d1 = 08

378θ 394θ

220

200

180

160

315θ 473θ 630θ 788θt

0

50

100

150

200

250

P (M

Pa)

1

(b)




01 02 03 04 05 06 07 0812

14

16

18

2

λr


d1


10 11 12 13 14 15 16174

176

178

18

182

184

186

λr


r



r t (θ) v (ms) λr10 390 121 18411 438 111 18212 487 100 18113 538 97 18014 588 92 17915 639 87 17816 689 82 177

01 02 03 04 05 06 07 0804

08

12

16

Ir


d1


10 11 12 13 14 15 1612

13

14

15

16

Ir


r






2

d1 = 04d1 = 02 d1 = 06

d1 = 08

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60Transmitted peak

Reflected peak

(a)

2

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60

Transmitted peak

Reflected peak

d1 = 02d1 = 04

d1 = 06d1 = 08

(b)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(c)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(d)










1000 Pressure (MPa)

Water

04

(a)

1000

Water

04Pressure (MPa)

(b)

1000

Air

04Pressure (MPa)

(c)

1000

Air

04Pressure (MPa)

(d)







01 02 03 04 05 06 07 080

05

1

15

2

25

3

()


d1


06 07 080

05

1

15

2

25

3

()

01 02 03 04 050

001

002

003

004


d1


01 02 03 04 05 0802

03

04

05

06

07

λt

06 0708

09

1

11

12

13times 10ndash3


d1


10 11 12 13 14 15 1603

035

04

045

λt

0

1

2

3times 10ndash3


r








4 Conclusions


Water


(a)

Air


(b)


0

1

2

3

4

λ1

01 02 03 04 05 06 07 080

200

400

600

800

1000


d1








Data Availability




Acknowledgments


References









0

01

02

03

04

05

06

07

λ2

01 02 03 04 05 06 07 084

6

8

10

12

14times 10ndash4


Air-airAir-water

d1




d1 01 814 788d1 02 261 252d1 03 119 116d1 04 68 65d1 05 43 41d1 06 28 27d1 07 21 19d1 08 16 14





















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


θ

045R0RR0

1113874 1113875045

times 10minus3R

R0lt 30

35R0

C

lgR

R01113888 1113889minus 09

1113971

R

R0ge 30

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(8)

Ph Patm + ρgh (9)





Nodal displacement




Nodal acceleration

Nodal velocity

Inte

grat

ion





Nodal acceleration

Momentummass



Material model



Nodal velocity

Transport

Forcemass

Initial condition


x u F m

σ ε P e ρ

(a)

σ ε P eρ u m

x

(b)

Lagrangiansubgrid

Euleriansubgrid

(c)





r

1

3

654

2

R0

Water

TNT

7

Measuringpoints

20R0

50R0


(a)

1659P (MPa)

1494132811629968306644983321660

(b)

333P (MPa)

2972602241881511157842560

(c)








P (M

Pa)

120

100

80

60

40

20

010 02 04

t (ms)06 08



120

100

80

60

40

20

0

P (M

Pa)

0 02 04t (ms)

06 08 1


(a)

t (ms)

80

60

40

20

0

P (M

Pa)

0 02 04 06 08 1


(b)




10 1143 1152 0811 993 998 0612 873 876 0413 775 803 3514 698 738 5515 631 683 7616 574 635 96

TNT

Water surface

Water tank

RD

Plate

R0





















D2D1

07R





D1 D2 D3 D4 D5



Disp

lace

men

t (m

m)

40

30

20

10

00 10050 150 200 250

t (μs)


(a)

Disp

lace

men

t (m

m)

40

30

20

10

00 10050 150 200 250

t (μs)


(b)

Velo

city

(ms

)

250

200

150

50

100

00 100 200 300 400

t (μs)


(c)

Velo

city

(ms

)

200

150

100

50

00 200100 300 400

t (μs)


(d)


Structure


Medium 1

Medium 2

Water

TNT

(a)

R1 (outer shell)

R2 (innershell)

r

d1d2

TNT

321

(b)




A B C n m7920MPa 5100MPa 0014 026 103

1200P (MPa)

108096084072060048036024012000

1200P (MPa)

108096084072060048036024012000

120010809608407206004803602401200 0

120010809608407206004803602401200 0

(a)

800P (MPa)

7206405604804003202401608000

8007206405604804003202401608000

P (MPa)800720640560480400320240160800 0

800720640560480400320240160800 0

(b)

600P (MPa)

5404804203603002401801206000

6005404804203603002401801206000

P (MPa)

(c)

Figure 10 Continued






500P (MPa)

4504003503002502001501005000

5004504003503002502001501005000

P (MPa)

(d)


315θ 473θ 630θ 788θt

1

378θ 394θ

0

50

100

150

200

250

P (M

Pa)

d1 = 02d1 = 04

d1 = 06d1 = 08

200

180

220

(a)

d1 = 02d1 = 04

d1 = 06d1 = 08

378θ 394θ

220

200

180

160

315θ 473θ 630θ 788θt

0

50

100

150

200

250

P (M

Pa)

1

(b)




01 02 03 04 05 06 07 0812

14

16

18

2

λr


d1


10 11 12 13 14 15 16174

176

178

18

182

184

186

λr


r



r t (θ) v (ms) λr10 390 121 18411 438 111 18212 487 100 18113 538 97 18014 588 92 17915 639 87 17816 689 82 177

01 02 03 04 05 06 07 0804

08

12

16

Ir


d1


10 11 12 13 14 15 1612

13

14

15

16

Ir


r






2

d1 = 04d1 = 02 d1 = 06

d1 = 08

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60Transmitted peak

Reflected peak

(a)

2

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60

Transmitted peak

Reflected peak

d1 = 02d1 = 04

d1 = 06d1 = 08

(b)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(c)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(d)










1000 Pressure (MPa)

Water

04

(a)

1000

Water

04Pressure (MPa)

(b)

1000

Air

04Pressure (MPa)

(c)

1000

Air

04Pressure (MPa)

(d)







01 02 03 04 05 06 07 080

05

1

15

2

25

3

()


d1


06 07 080

05

1

15

2

25

3

()

01 02 03 04 050

001

002

003

004


d1


01 02 03 04 05 0802

03

04

05

06

07

λt

06 0708

09

1

11

12

13times 10ndash3


d1


10 11 12 13 14 15 1603

035

04

045

λt

0

1

2

3times 10ndash3


r








4 Conclusions


Water


(a)

Air


(b)


0

1

2

3

4

λ1

01 02 03 04 05 06 07 080

200

400

600

800

1000


d1








Data Availability




Acknowledgments


References









0

01

02

03

04

05

06

07

λ2

01 02 03 04 05 06 07 084

6

8

10

12

14times 10ndash4


Air-airAir-water

d1




d1 01 814 788d1 02 261 252d1 03 119 116d1 04 68 65d1 05 43 41d1 06 28 27d1 07 21 19d1 08 16 14





















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




r

1

3

654

2

R0

Water

TNT

7

Measuringpoints

20R0

50R0


(a)

1659P (MPa)

1494132811629968306644983321660

(b)

333P (MPa)

2972602241881511157842560

(c)








P (M

Pa)

120

100

80

60

40

20

010 02 04

t (ms)06 08



120

100

80

60

40

20

0

P (M

Pa)

0 02 04t (ms)

06 08 1


(a)

t (ms)

80

60

40

20

0

P (M

Pa)

0 02 04 06 08 1


(b)




10 1143 1152 0811 993 998 0612 873 876 0413 775 803 3514 698 738 5515 631 683 7616 574 635 96

TNT

Water surface

Water tank

RD

Plate

R0





















D2D1

07R





D1 D2 D3 D4 D5



Disp

lace

men

t (m

m)

40

30

20

10

00 10050 150 200 250

t (μs)


(a)

Disp

lace

men

t (m

m)

40

30

20

10

00 10050 150 200 250

t (μs)


(b)

Velo

city

(ms

)

250

200

150

50

100

00 100 200 300 400

t (μs)


(c)

Velo

city

(ms

)

200

150

100

50

00 200100 300 400

t (μs)


(d)


Structure


Medium 1

Medium 2

Water

TNT

(a)

R1 (outer shell)

R2 (innershell)

r

d1d2

TNT

321

(b)




A B C n m7920MPa 5100MPa 0014 026 103

1200P (MPa)

108096084072060048036024012000

1200P (MPa)

108096084072060048036024012000

120010809608407206004803602401200 0

120010809608407206004803602401200 0

(a)

800P (MPa)

7206405604804003202401608000

8007206405604804003202401608000

P (MPa)800720640560480400320240160800 0

800720640560480400320240160800 0

(b)

600P (MPa)

5404804203603002401801206000

6005404804203603002401801206000

P (MPa)

(c)

Figure 10 Continued






500P (MPa)

4504003503002502001501005000

5004504003503002502001501005000

P (MPa)

(d)


315θ 473θ 630θ 788θt

1

378θ 394θ

0

50

100

150

200

250

P (M

Pa)

d1 = 02d1 = 04

d1 = 06d1 = 08

200

180

220

(a)

d1 = 02d1 = 04

d1 = 06d1 = 08

378θ 394θ

220

200

180

160

315θ 473θ 630θ 788θt

0

50

100

150

200

250

P (M

Pa)

1

(b)




01 02 03 04 05 06 07 0812

14

16

18

2

λr


d1


10 11 12 13 14 15 16174

176

178

18

182

184

186

λr


r



r t (θ) v (ms) λr10 390 121 18411 438 111 18212 487 100 18113 538 97 18014 588 92 17915 639 87 17816 689 82 177

01 02 03 04 05 06 07 0804

08

12

16

Ir


d1


10 11 12 13 14 15 1612

13

14

15

16

Ir


r






2

d1 = 04d1 = 02 d1 = 06

d1 = 08

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60Transmitted peak

Reflected peak

(a)

2

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60

Transmitted peak

Reflected peak

d1 = 02d1 = 04

d1 = 06d1 = 08

(b)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(c)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(d)










1000 Pressure (MPa)

Water

04

(a)

1000

Water

04Pressure (MPa)

(b)

1000

Air

04Pressure (MPa)

(c)

1000

Air

04Pressure (MPa)

(d)







01 02 03 04 05 06 07 080

05

1

15

2

25

3

()


d1


06 07 080

05

1

15

2

25

3

()

01 02 03 04 050

001

002

003

004


d1


01 02 03 04 05 0802

03

04

05

06

07

λt

06 0708

09

1

11

12

13times 10ndash3


d1


10 11 12 13 14 15 1603

035

04

045

λt

0

1

2

3times 10ndash3


r








4 Conclusions


Water


(a)

Air


(b)


0

1

2

3

4

λ1

01 02 03 04 05 06 07 080

200

400

600

800

1000


d1








Data Availability




Acknowledgments


References









0

01

02

03

04

05

06

07

λ2

01 02 03 04 05 06 07 084

6

8

10

12

14times 10ndash4


Air-airAir-water

d1




d1 01 814 788d1 02 261 252d1 03 119 116d1 04 68 65d1 05 43 41d1 06 28 27d1 07 21 19d1 08 16 14





















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


P (M

Pa)

120

100

80

60

40

20

010 02 04

t (ms)06 08



120

100

80

60

40

20

0

P (M

Pa)

0 02 04t (ms)

06 08 1


(a)

t (ms)

80

60

40

20

0

P (M

Pa)

0 02 04 06 08 1


(b)




10 1143 1152 0811 993 998 0612 873 876 0413 775 803 3514 698 738 5515 631 683 7616 574 635 96

TNT

Water surface

Water tank

RD

Plate

R0





















D2D1

07R





D1 D2 D3 D4 D5



Disp

lace

men

t (m

m)

40

30

20

10

00 10050 150 200 250

t (μs)


(a)

Disp

lace

men

t (m

m)

40

30

20

10

00 10050 150 200 250

t (μs)


(b)

Velo

city

(ms

)

250

200

150

50

100

00 100 200 300 400

t (μs)


(c)

Velo

city

(ms

)

200

150

100

50

00 200100 300 400

t (μs)


(d)


Structure


Medium 1

Medium 2

Water

TNT

(a)

R1 (outer shell)

R2 (innershell)

r

d1d2

TNT

321

(b)




A B C n m7920MPa 5100MPa 0014 026 103

1200P (MPa)

108096084072060048036024012000

1200P (MPa)

108096084072060048036024012000

120010809608407206004803602401200 0

120010809608407206004803602401200 0

(a)

800P (MPa)

7206405604804003202401608000

8007206405604804003202401608000

P (MPa)800720640560480400320240160800 0

800720640560480400320240160800 0

(b)

600P (MPa)

5404804203603002401801206000

6005404804203603002401801206000

P (MPa)

(c)

Figure 10 Continued






500P (MPa)

4504003503002502001501005000

5004504003503002502001501005000

P (MPa)

(d)


315θ 473θ 630θ 788θt

1

378θ 394θ

0

50

100

150

200

250

P (M

Pa)

d1 = 02d1 = 04

d1 = 06d1 = 08

200

180

220

(a)

d1 = 02d1 = 04

d1 = 06d1 = 08

378θ 394θ

220

200

180

160

315θ 473θ 630θ 788θt

0

50

100

150

200

250

P (M

Pa)

1

(b)




01 02 03 04 05 06 07 0812

14

16

18

2

λr


d1


10 11 12 13 14 15 16174

176

178

18

182

184

186

λr


r



r t (θ) v (ms) λr10 390 121 18411 438 111 18212 487 100 18113 538 97 18014 588 92 17915 639 87 17816 689 82 177

01 02 03 04 05 06 07 0804

08

12

16

Ir


d1


10 11 12 13 14 15 1612

13

14

15

16

Ir


r






2

d1 = 04d1 = 02 d1 = 06

d1 = 08

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60Transmitted peak

Reflected peak

(a)

2

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60

Transmitted peak

Reflected peak

d1 = 02d1 = 04

d1 = 06d1 = 08

(b)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(c)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(d)










1000 Pressure (MPa)

Water

04

(a)

1000

Water

04Pressure (MPa)

(b)

1000

Air

04Pressure (MPa)

(c)

1000

Air

04Pressure (MPa)

(d)







01 02 03 04 05 06 07 080

05

1

15

2

25

3

()


d1


06 07 080

05

1

15

2

25

3

()

01 02 03 04 050

001

002

003

004


d1


01 02 03 04 05 0802

03

04

05

06

07

λt

06 0708

09

1

11

12

13times 10ndash3


d1


10 11 12 13 14 15 1603

035

04

045

λt

0

1

2

3times 10ndash3


r








4 Conclusions


Water


(a)

Air


(b)


0

1

2

3

4

λ1

01 02 03 04 05 06 07 080

200

400

600

800

1000


d1








Data Availability




Acknowledgments


References









0

01

02

03

04

05

06

07

λ2

01 02 03 04 05 06 07 084

6

8

10

12

14times 10ndash4


Air-airAir-water

d1




d1 01 814 788d1 02 261 252d1 03 119 116d1 04 68 65d1 05 43 41d1 06 28 27d1 07 21 19d1 08 16 14





















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




















D2D1

07R





D1 D2 D3 D4 D5



Disp

lace

men

t (m

m)

40

30

20

10

00 10050 150 200 250

t (μs)


(a)

Disp

lace

men

t (m

m)

40

30

20

10

00 10050 150 200 250

t (μs)


(b)

Velo

city

(ms

)

250

200

150

50

100

00 100 200 300 400

t (μs)


(c)

Velo

city

(ms

)

200

150

100

50

00 200100 300 400

t (μs)


(d)


Structure


Medium 1

Medium 2

Water

TNT

(a)

R1 (outer shell)

R2 (innershell)

r

d1d2

TNT

321

(b)




A B C n m7920MPa 5100MPa 0014 026 103

1200P (MPa)

108096084072060048036024012000

1200P (MPa)

108096084072060048036024012000

120010809608407206004803602401200 0

120010809608407206004803602401200 0

(a)

800P (MPa)

7206405604804003202401608000

8007206405604804003202401608000

P (MPa)800720640560480400320240160800 0

800720640560480400320240160800 0

(b)

600P (MPa)

5404804203603002401801206000

6005404804203603002401801206000

P (MPa)

(c)

Figure 10 Continued






500P (MPa)

4504003503002502001501005000

5004504003503002502001501005000

P (MPa)

(d)


315θ 473θ 630θ 788θt

1

378θ 394θ

0

50

100

150

200

250

P (M

Pa)

d1 = 02d1 = 04

d1 = 06d1 = 08

200

180

220

(a)

d1 = 02d1 = 04

d1 = 06d1 = 08

378θ 394θ

220

200

180

160

315θ 473θ 630θ 788θt

0

50

100

150

200

250

P (M

Pa)

1

(b)




01 02 03 04 05 06 07 0812

14

16

18

2

λr


d1


10 11 12 13 14 15 16174

176

178

18

182

184

186

λr


r



r t (θ) v (ms) λr10 390 121 18411 438 111 18212 487 100 18113 538 97 18014 588 92 17915 639 87 17816 689 82 177

01 02 03 04 05 06 07 0804

08

12

16

Ir


d1


10 11 12 13 14 15 1612

13

14

15

16

Ir


r






2

d1 = 04d1 = 02 d1 = 06

d1 = 08

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60Transmitted peak

Reflected peak

(a)

2

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60

Transmitted peak

Reflected peak

d1 = 02d1 = 04

d1 = 06d1 = 08

(b)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(c)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(d)










1000 Pressure (MPa)

Water

04

(a)

1000

Water

04Pressure (MPa)

(b)

1000

Air

04Pressure (MPa)

(c)

1000

Air

04Pressure (MPa)

(d)







01 02 03 04 05 06 07 080

05

1

15

2

25

3

()


d1


06 07 080

05

1

15

2

25

3

()

01 02 03 04 050

001

002

003

004


d1


01 02 03 04 05 0802

03

04

05

06

07

λt

06 0708

09

1

11

12

13times 10ndash3


d1


10 11 12 13 14 15 1603

035

04

045

λt

0

1

2

3times 10ndash3


r








4 Conclusions


Water


(a)

Air


(b)


0

1

2

3

4

λ1

01 02 03 04 05 06 07 080

200

400

600

800

1000


d1








Data Availability




Acknowledgments


References









0

01

02

03

04

05

06

07

λ2

01 02 03 04 05 06 07 084

6

8

10

12

14times 10ndash4


Air-airAir-water

d1




d1 01 814 788d1 02 261 252d1 03 119 116d1 04 68 65d1 05 43 41d1 06 28 27d1 07 21 19d1 08 16 14





















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Disp

lace

men

t (m

m)

40

30

20

10

00 10050 150 200 250

t (μs)


(a)

Disp

lace

men

t (m

m)

40

30

20

10

00 10050 150 200 250

t (μs)


(b)

Velo

city

(ms

)

250

200

150

50

100

00 100 200 300 400

t (μs)


(c)

Velo

city

(ms

)

200

150

100

50

00 200100 300 400

t (μs)


(d)


Structure


Medium 1

Medium 2

Water

TNT

(a)

R1 (outer shell)

R2 (innershell)

r

d1d2

TNT

321

(b)




A B C n m7920MPa 5100MPa 0014 026 103

1200P (MPa)

108096084072060048036024012000

1200P (MPa)

108096084072060048036024012000

120010809608407206004803602401200 0

120010809608407206004803602401200 0

(a)

800P (MPa)

7206405604804003202401608000

8007206405604804003202401608000

P (MPa)800720640560480400320240160800 0

800720640560480400320240160800 0

(b)

600P (MPa)

5404804203603002401801206000

6005404804203603002401801206000

P (MPa)

(c)

Figure 10 Continued






500P (MPa)

4504003503002502001501005000

5004504003503002502001501005000

P (MPa)

(d)


315θ 473θ 630θ 788θt

1

378θ 394θ

0

50

100

150

200

250

P (M

Pa)

d1 = 02d1 = 04

d1 = 06d1 = 08

200

180

220

(a)

d1 = 02d1 = 04

d1 = 06d1 = 08

378θ 394θ

220

200

180

160

315θ 473θ 630θ 788θt

0

50

100

150

200

250

P (M

Pa)

1

(b)




01 02 03 04 05 06 07 0812

14

16

18

2

λr


d1


10 11 12 13 14 15 16174

176

178

18

182

184

186

λr


r



r t (θ) v (ms) λr10 390 121 18411 438 111 18212 487 100 18113 538 97 18014 588 92 17915 639 87 17816 689 82 177

01 02 03 04 05 06 07 0804

08

12

16

Ir


d1


10 11 12 13 14 15 1612

13

14

15

16

Ir


r






2

d1 = 04d1 = 02 d1 = 06

d1 = 08

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60Transmitted peak

Reflected peak

(a)

2

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60

Transmitted peak

Reflected peak

d1 = 02d1 = 04

d1 = 06d1 = 08

(b)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(c)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(d)










1000 Pressure (MPa)

Water

04

(a)

1000

Water

04Pressure (MPa)

(b)

1000

Air

04Pressure (MPa)

(c)

1000

Air

04Pressure (MPa)

(d)







01 02 03 04 05 06 07 080

05

1

15

2

25

3

()


d1


06 07 080

05

1

15

2

25

3

()

01 02 03 04 050

001

002

003

004


d1


01 02 03 04 05 0802

03

04

05

06

07

λt

06 0708

09

1

11

12

13times 10ndash3


d1


10 11 12 13 14 15 1603

035

04

045

λt

0

1

2

3times 10ndash3


r








4 Conclusions


Water


(a)

Air


(b)


0

1

2

3

4

λ1

01 02 03 04 05 06 07 080

200

400

600

800

1000


d1








Data Availability




Acknowledgments


References









0

01

02

03

04

05

06

07

λ2

01 02 03 04 05 06 07 084

6

8

10

12

14times 10ndash4


Air-airAir-water

d1




d1 01 814 788d1 02 261 252d1 03 119 116d1 04 68 65d1 05 43 41d1 06 28 27d1 07 21 19d1 08 16 14





















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



A B C n m7920MPa 5100MPa 0014 026 103

1200P (MPa)

108096084072060048036024012000

1200P (MPa)

108096084072060048036024012000

120010809608407206004803602401200 0

120010809608407206004803602401200 0

(a)

800P (MPa)

7206405604804003202401608000

8007206405604804003202401608000

P (MPa)800720640560480400320240160800 0

800720640560480400320240160800 0

(b)

600P (MPa)

5404804203603002401801206000

6005404804203603002401801206000

P (MPa)

(c)

Figure 10 Continued






500P (MPa)

4504003503002502001501005000

5004504003503002502001501005000

P (MPa)

(d)


315θ 473θ 630θ 788θt

1

378θ 394θ

0

50

100

150

200

250

P (M

Pa)

d1 = 02d1 = 04

d1 = 06d1 = 08

200

180

220

(a)

d1 = 02d1 = 04

d1 = 06d1 = 08

378θ 394θ

220

200

180

160

315θ 473θ 630θ 788θt

0

50

100

150

200

250

P (M

Pa)

1

(b)




01 02 03 04 05 06 07 0812

14

16

18

2

λr


d1


10 11 12 13 14 15 16174

176

178

18

182

184

186

λr


r



r t (θ) v (ms) λr10 390 121 18411 438 111 18212 487 100 18113 538 97 18014 588 92 17915 639 87 17816 689 82 177

01 02 03 04 05 06 07 0804

08

12

16

Ir


d1


10 11 12 13 14 15 1612

13

14

15

16

Ir


r






2

d1 = 04d1 = 02 d1 = 06

d1 = 08

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60Transmitted peak

Reflected peak

(a)

2

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60

Transmitted peak

Reflected peak

d1 = 02d1 = 04

d1 = 06d1 = 08

(b)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(c)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(d)










1000 Pressure (MPa)

Water

04

(a)

1000

Water

04Pressure (MPa)

(b)

1000

Air

04Pressure (MPa)

(c)

1000

Air

04Pressure (MPa)

(d)







01 02 03 04 05 06 07 080

05

1

15

2

25

3

()


d1


06 07 080

05

1

15

2

25

3

()

01 02 03 04 050

001

002

003

004


d1


01 02 03 04 05 0802

03

04

05

06

07

λt

06 0708

09

1

11

12

13times 10ndash3


d1


10 11 12 13 14 15 1603

035

04

045

λt

0

1

2

3times 10ndash3


r








4 Conclusions


Water


(a)

Air


(b)


0

1

2

3

4

λ1

01 02 03 04 05 06 07 080

200

400

600

800

1000


d1








Data Availability




Acknowledgments


References









0

01

02

03

04

05

06

07

λ2

01 02 03 04 05 06 07 084

6

8

10

12

14times 10ndash4


Air-airAir-water

d1




d1 01 814 788d1 02 261 252d1 03 119 116d1 04 68 65d1 05 43 41d1 06 28 27d1 07 21 19d1 08 16 14





















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






500P (MPa)

4504003503002502001501005000

5004504003503002502001501005000

P (MPa)

(d)


315θ 473θ 630θ 788θt

1

378θ 394θ

0

50

100

150

200

250

P (M

Pa)

d1 = 02d1 = 04

d1 = 06d1 = 08

200

180

220

(a)

d1 = 02d1 = 04

d1 = 06d1 = 08

378θ 394θ

220

200

180

160

315θ 473θ 630θ 788θt

0

50

100

150

200

250

P (M

Pa)

1

(b)




01 02 03 04 05 06 07 0812

14

16

18

2

λr


d1


10 11 12 13 14 15 16174

176

178

18

182

184

186

λr


r



r t (θ) v (ms) λr10 390 121 18411 438 111 18212 487 100 18113 538 97 18014 588 92 17915 639 87 17816 689 82 177

01 02 03 04 05 06 07 0804

08

12

16

Ir


d1


10 11 12 13 14 15 1612

13

14

15

16

Ir


r






2

d1 = 04d1 = 02 d1 = 06

d1 = 08

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60Transmitted peak

Reflected peak

(a)

2

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60

Transmitted peak

Reflected peak

d1 = 02d1 = 04

d1 = 06d1 = 08

(b)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(c)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(d)










1000 Pressure (MPa)

Water

04

(a)

1000

Water

04Pressure (MPa)

(b)

1000

Air

04Pressure (MPa)

(c)

1000

Air

04Pressure (MPa)

(d)







01 02 03 04 05 06 07 080

05

1

15

2

25

3

()


d1


06 07 080

05

1

15

2

25

3

()

01 02 03 04 050

001

002

003

004


d1


01 02 03 04 05 0802

03

04

05

06

07

λt

06 0708

09

1

11

12

13times 10ndash3


d1


10 11 12 13 14 15 1603

035

04

045

λt

0

1

2

3times 10ndash3


r








4 Conclusions


Water


(a)

Air


(b)


0

1

2

3

4

λ1

01 02 03 04 05 06 07 080

200

400

600

800

1000


d1








Data Availability




Acknowledgments


References









0

01

02

03

04

05

06

07

λ2

01 02 03 04 05 06 07 084

6

8

10

12

14times 10ndash4


Air-airAir-water

d1




d1 01 814 788d1 02 261 252d1 03 119 116d1 04 68 65d1 05 43 41d1 06 28 27d1 07 21 19d1 08 16 14





















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



01 02 03 04 05 06 07 0812

14

16

18

2

λr


d1


10 11 12 13 14 15 16174

176

178

18

182

184

186

λr


r



r t (θ) v (ms) λr10 390 121 18411 438 111 18212 487 100 18113 538 97 18014 588 92 17915 639 87 17816 689 82 177

01 02 03 04 05 06 07 0804

08

12

16

Ir


d1


10 11 12 13 14 15 1612

13

14

15

16

Ir


r






2

d1 = 04d1 = 02 d1 = 06

d1 = 08

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60Transmitted peak

Reflected peak

(a)

2

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60

Transmitted peak

Reflected peak

d1 = 02d1 = 04

d1 = 06d1 = 08

(b)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(c)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(d)










1000 Pressure (MPa)

Water

04

(a)

1000

Water

04Pressure (MPa)

(b)

1000

Air

04Pressure (MPa)

(c)

1000

Air

04Pressure (MPa)

(d)







01 02 03 04 05 06 07 080

05

1

15

2

25

3

()


d1


06 07 080

05

1

15

2

25

3

()

01 02 03 04 050

001

002

003

004


d1


01 02 03 04 05 0802

03

04

05

06

07

λt

06 0708

09

1

11

12

13times 10ndash3


d1


10 11 12 13 14 15 1603

035

04

045

λt

0

1

2

3times 10ndash3


r








4 Conclusions


Water


(a)

Air


(b)


0

1

2

3

4

λ1

01 02 03 04 05 06 07 080

200

400

600

800

1000


d1








Data Availability




Acknowledgments


References









0

01

02

03

04

05

06

07

λ2

01 02 03 04 05 06 07 084

6

8

10

12

14times 10ndash4


Air-airAir-water

d1




d1 01 814 788d1 02 261 252d1 03 119 116d1 04 68 65d1 05 43 41d1 06 28 27d1 07 21 19d1 08 16 14





















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





2

d1 = 04d1 = 02 d1 = 06

d1 = 08

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60Transmitted peak

Reflected peak

(a)

2

P (M

Pa)

315θ 473θ 630θ 788θt

946θndash10

0

10

20

30

40

50

60

Transmitted peak

Reflected peak

d1 = 02d1 = 04

d1 = 06d1 = 08

(b)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(c)

Transmitted peak

2

P (M

Pa)

315θ 630θ 946θ 1262θt

1577θ009

01

011

012

013

014

015

d1 = 04d1 = 02 d1 = 06

d1 = 08

(d)










1000 Pressure (MPa)

Water

04

(a)

1000

Water

04Pressure (MPa)

(b)

1000

Air

04Pressure (MPa)

(c)

1000

Air

04Pressure (MPa)

(d)







01 02 03 04 05 06 07 080

05

1

15

2

25

3

()


d1


06 07 080

05

1

15

2

25

3

()

01 02 03 04 050

001

002

003

004


d1


01 02 03 04 05 0802

03

04

05

06

07

λt

06 0708

09

1

11

12

13times 10ndash3


d1


10 11 12 13 14 15 1603

035

04

045

λt

0

1

2

3times 10ndash3


r








4 Conclusions


Water


(a)

Air


(b)


0

1

2

3

4

λ1

01 02 03 04 05 06 07 080

200

400

600

800

1000


d1








Data Availability




Acknowledgments


References









0

01

02

03

04

05

06

07

λ2

01 02 03 04 05 06 07 084

6

8

10

12

14times 10ndash4


Air-airAir-water

d1




d1 01 814 788d1 02 261 252d1 03 119 116d1 04 68 65d1 05 43 41d1 06 28 27d1 07 21 19d1 08 16 14





















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia









1000 Pressure (MPa)

Water

04

(a)

1000

Water

04Pressure (MPa)

(b)

1000

Air

04Pressure (MPa)

(c)

1000

Air

04Pressure (MPa)

(d)







01 02 03 04 05 06 07 080

05

1

15

2

25

3

()


d1


06 07 080

05

1

15

2

25

3

()

01 02 03 04 050

001

002

003

004


d1


01 02 03 04 05 0802

03

04

05

06

07

λt

06 0708

09

1

11

12

13times 10ndash3


d1


10 11 12 13 14 15 1603

035

04

045

λt

0

1

2

3times 10ndash3


r








4 Conclusions


Water


(a)

Air


(b)


0

1

2

3

4

λ1

01 02 03 04 05 06 07 080

200

400

600

800

1000


d1








Data Availability




Acknowledgments


References









0

01

02

03

04

05

06

07

λ2

01 02 03 04 05 06 07 084

6

8

10

12

14times 10ndash4


Air-airAir-water

d1




d1 01 814 788d1 02 261 252d1 03 119 116d1 04 68 65d1 05 43 41d1 06 28 27d1 07 21 19d1 08 16 14





















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






01 02 03 04 05 06 07 080

05

1

15

2

25

3

()


d1


06 07 080

05

1

15

2

25

3

()

01 02 03 04 050

001

002

003

004


d1


01 02 03 04 05 0802

03

04

05

06

07

λt

06 0708

09

1

11

12

13times 10ndash3


d1


10 11 12 13 14 15 1603

035

04

045

λt

0

1

2

3times 10ndash3


r








4 Conclusions


Water


(a)

Air


(b)


0

1

2

3

4

λ1

01 02 03 04 05 06 07 080

200

400

600

800

1000


d1








Data Availability




Acknowledgments


References









0

01

02

03

04

05

06

07

λ2

01 02 03 04 05 06 07 084

6

8

10

12

14times 10ndash4


Air-airAir-water

d1




d1 01 814 788d1 02 261 252d1 03 119 116d1 04 68 65d1 05 43 41d1 06 28 27d1 07 21 19d1 08 16 14





















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







4 Conclusions


Water


(a)

Air


(b)


0

1

2

3

4

λ1

01 02 03 04 05 06 07 080

200

400

600

800

1000


d1








Data Availability




Acknowledgments


References









0

01

02

03

04

05

06

07

λ2

01 02 03 04 05 06 07 084

6

8

10

12

14times 10ndash4


Air-airAir-water

d1




d1 01 814 788d1 02 261 252d1 03 119 116d1 04 68 65d1 05 43 41d1 06 28 27d1 07 21 19d1 08 16 14





















































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









0

01

02

03

04

05

06

07

λ2

01 02 03 04 05 06 07 084

6

8

10

12

14times 10ndash4


Air-airAir-water

d1




d1 01 814 788d1 02 261 252d1 03 119 116d1 04 68 65d1 05 43 41d1 06 28 27d1 07 21 19d1 08 16 14





















































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


Study on the Pressure Characteristics of Shock Wave...

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Transcript of Study on the Pressure Characteristics of Shock Wave...