research article the fluid-solid interaction dynamics between underwater explosion ... · 2019. 7....

Research ArticleThe Fluid-Solid Interaction Dynamics between UnderwaterExplosion Bubble and Corrugated Sandwich Plate

Hao Wang,1,2 Yuan Sheng Cheng,1 Jun Liu,1 and Lin Gan3

1School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China2China Institute of Marine Technology & Economic (CIMTEC), Beijing 100081, China3Wuhan Secondary Ship Design & Research Institute, Wuhan 430074, China

Correspondence should be addressed to Yuan Sheng Cheng; [email protected]

Received 12 May 2016; Accepted 25 July 2016

Academic Editor: Tai Thai

Copyright © 2016 Hao Wang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Lightweight sandwich structures with highly porous 2D cores or 3D (three-dimensional) periodic cores can effectively withstandunderwater explosion load. In most of the previous studies of sandwich structure antiblast dynamics, the underwater explosion(UNDEX) bubble phase was neglected. As the UNDEX bubble load is one of the severest damage sources that may lead to structurelarge plastic deformation and crevasses failure, the failure mechanisms of sandwich structures might not be accurate if onlyshock wave is considered. In this paper, detailed 3D finite element (FE) numerical models of UNDEX bubble-LCSP (lightweightcorrugated sandwich plates) interaction are developed by using MSC.Dytran. Upon the validated FE model, the bubble shape,impact pressure, and fluid field velocities for different stand-off distances are studied. Based on numerical results, the failure modesof LCSP and the whole damage process are obtained. It is demonstrated that the UNDEX bubble collapse jet local load plays a moresignificant role than the UNDEX shock wave load especially in near-field underwater explosion.

1. Introduction

Underwater blast is much more destructive than free fieldexplosion in air. Since World War 1 and World War 2, a lotof naval vessels were attacked and destroyed by underwaterexplosion weapons [1]. On March 26, 2010, the CheonanNaval Ship (PCC-772) was attacked and sank into West seaof Korea. The whole ship was cut into two pieces (Figure 1).According to the official Joint Investigation Report (JIR), thisattack was underwater explosion caused by a torpedo with300 kg TNT at a depth of about 6∼9m [2].Thus, theUNDEX-resistant structures that can withstand these extreme loadingconditions and sustain their functionality are critical inmodernmilitary setting.Due to structural efficiency and highenergy absorption capability, sandwich systems have beenextensively used in a variety of applications for many years.They may be the potential structure types of naval vesselhulls. The failure mechanism of this type structure subject toUNDEX is quite different from the ones of traditional marineconstructions of stiffened plates.

To develop and test innovative lightweight structuralconcepts for the US Navy, Wiernicki et al. [3] firstly inves-tigated the elastic and plastic dynamic behaviors of LCSPsubjected to air blast loading. A set of relatively simple closedanalytical expressions were also given to quickly identify theeffects of geometric parameters on dynamic behaviors. Theoptimization problem of LCSP under blast load was firstlydiscussed by Liang et al. [4]. The Feasible Direction Method(FDM) coupled with the Backtrack Program Method (BPM)was used in the optimum design algorithm, and the mainstructural parameters including corrugation angle, face sheetthickness, core thickness, and corrugation pitchwere selectedas design variables in optimum mathematical model. Thecalculations by Wiernicki et al. [3] and Liang et al. [4] arebased on the semianalytical empirical formulations whichcannot capture the details of air blast phenomenon.

By adopting the Taylor plate assumptions, the fluid-solidinteraction (FSI) analysis of LCSP under UNDEX shock waveloading was firstly considered in the investigation proposedby Xue and Hutchinson [5]. If the blast medium is water,

Hindawi Publishing CorporationShock and VibrationVolume 2016, Article ID 6057437, 21 pageshttp://dx.doi.org/10.1155/2016/6057437

2 Shock and Vibration

Damage

Main deck1st platform

2nd platform

Demist stack

Figure 1: Example of Cheonan Naval Ship damage after close-inUNDEX [2].

it was found that FSI effecting can reduce the momentumimparted to a sandwich plate by almost a factor of two relativeto that imparted to a solid plate of the sameweight. Vaziri andHutchinson [6] further complemented the previous studieson the role of FSI by accounting for the nonlinear compress-ibility and finite shock behavior of air medium. The resultsshowed that the FSI enhances the performance of LCSPrelative to monolithic plates under intense air explosion, butnot as significantly as for UNDEX. In practical cases, if theimpulse of impact loading is sufficiently large, the damageof LCSP may occur. Based on the nonlinear finite elementcomputations by adoptingABAQUS/Explicit code, the failuremode maps of LCSP under intense uniform impulsive pres-sure loads were obtained from the investigation of Vaziri et al.[7]. Here, it must be pointed out that the FSI is not consideredin the numerical simulations proposed by Vaziri et al. [7](if the effects of FSI are included, the failure mechanismsof LCSP may be different). To investigate structural designsof vessels against collisions, Rubino et al. [8] measured andanalyzed the dynamic performance of sandwich beams withthe Y-frame and corrugated cores. Both the experimental andfinite element (FE) results revealed that these two topologytype cores sandwich beams with equal mass have similardynamic behavior. Qin et al. [9] studied low velocity impactresistance of a LCSP struck by a heavy mass. In addition,new analytical predictions were obtained and the predictedresults agreed well with FE results. Rimoli et al. [10] utilizedexperimental tests and numerical methods to investigate thedynamic responses of edge-clamped LCSP subject to theshock of explosively driven wet sand. In the modeling, adecoupled wet sand loading curve was incorporated into FEsimulation.

Recently, Wadley et al. [11] further investigated this sand-water-structure coupling phenomenon in mine explosionaccident. The dynamic deformation and fracture processeswere both included in this analysis by employing a particle-basedmethod.The high speed fragment penetration problemof aluminum alloy LCSP with empty and alumina filledcore has been done experimentally by Wadley et al. [12].Zhang et al. [13] also conducted a study on the dynamicresponse of LCSP with unfilled and foam-filled sinusoidalplate cores. A novel analytical procedure was built to eval-uate the dynamic response of LCSP. Very few experimentalinvestigations have been conducted on the blast resistanceof corrugated sandwich panels. Li et al. [14] experimentally

investigated failure mechanisms of two configurations ofthe LCSP specimen by using a ballistic pendulum system.The deflection modes demonstrated that only global defor-mation and small tearing crack occurred. In the previousinvestigation of our group, a three-dimensional fully coupledsimulation is conducted to analyze the dynamic responseof sandwich panels comprising equal thicknesses face sheetssandwiching a corrugated core when subjected to localizedimpulse created by the detonation of cylindrical explosive byZhang et al. [15]. The numerical simulation results showedthat the core configuration has a negligible influence on thepeak reflected pressure, but it has an effect on the deflectionof a panel.

Most of these previous investigations are concernedwith the “pure” shock wave impact loading, while themultidamage sources of load environment such as bubbleare often neglected. Regarding the knowledge of authors, thedetailed failure mechanisms of LCSP subjected to close-inUNDEX bubble loading are not very clear. Thus, the presentpaper is primarily concerned with the detailed dynamicbehavior and the failure mechanism of LCSP subjected toclose-in UNDEX shock wave load and following bubblepulse. An outline of this paper is as follows. In the initial part,the geometry characteristics of LCSP and the theoreticalbackground of UNDEX bubble are introduced. To verifythe numerical simulation model, the 3D near free surfaceUNDEX bubble problem is firstly analyzed and discussedby adopting multimaterial Euler-Lagrange coupling methodin MSC.Dytran code. The detailed integrated response andactual deformation characteristics of LCSP are presented forthree different cases. The bubble shape, fluid field velocity,and bubble collapse jet are investigated. In addition, somenew remarks about failure mode of LCSP are discussedin the last part. The objective of this paper is to study thenonlinear inelastic responses and damage of LCSP underUNDEX shock wave and bubble loading and to developmore accurate predictions of bubble-LCSP interactionbehavior.

2. Theoretical Background of UNDEX Bubble

2.1. Migration of Explosion Bubble. An early investigationof considerable significance in the field of UNDEX bubbleresearch was conducted by Lamb [17] for analyzing thecollapse of a spherical transient bubble in an infinite fluid. Inthat study, it was assumed that the pulsation pressure withinthe explosion bubble varies as

𝑝 = 𝑝

0(

𝑉

0

𝑉

)

𝛾

,(1)

where𝑉 is volume of bubble, 𝛾 is a constant, and the subscriptdenotes initial values. Lamb carried through the analysis forthe cases 𝛾 = 1 and 𝛾 = 4/3.

Based on the work of Lamb, including the buoyancyforces, the influence of viscous damping, and the ocean

Shock and Vibration 3

surface interaction, the complex relation of bubble is givenby Vernon [18]:

�̇� = 𝜎, (2a)

̇

𝜁 = 𝜆, (2b)

�̇� = −

3𝛿

(2𝛿 − 𝛽𝑥)

[

𝜎

2

𝑥

(1 −

2𝛽𝑥

3𝛿

) −

𝜆

2

6𝑥

+

𝜁

𝑥𝜁

0

−

(𝛾 − 1) 𝑘

𝑥

3𝛾+1+

𝛽𝑥

4𝛿

2(𝐶

𝑑

𝜆

2

4𝑥

+

𝜎𝜆

3

−

𝑥

𝜁

0

)] ,

(2c)

̇

𝜆 = −3𝛼 [

1

𝜁

0

+

𝜎𝜆

𝑥

− 𝐶

𝑑

𝜆

2

4𝑥

+

𝛽𝑥

4𝛿

2(3𝜎

2+ 𝑥�̇�)] , (2d)

where

𝑥 =

𝑎

𝐿

;

𝑥

0=

𝑎

0

𝐿

;

𝜁 =

𝑍

𝐿

;

𝜁

0=

𝑍

0

𝐿

;

𝛿 =

𝑑

𝐿

;

𝜏 =

𝑡

𝑇

;

�̇� =

𝑑𝑥

𝑑𝜏

;

̇

𝜁 =

𝑑𝜁

𝑑𝜏

;

�̇� =

𝑑𝜎

𝑑𝜏

;

̇

𝜆 =

𝑑𝜆

𝑑𝜏

;

(3)

and 𝑧 is the water pressure head and the subscript denotesinitial values, d is the depth of charge at initial time, 𝛾 isthe adiabatic gas constant being equal to 1.25, and 𝛼 is themigration control coefficient which can be set to the valueof 1.0 if the migration is considered. 𝛽 is a free surface effectcontrol coefficient which will always be equal to 0 or 1. 𝛽 = 0means that free surface effect is not considered. Here, 𝛼 =1 and 𝛽 = 1 are adopted in the analysis. The other threeunknown parameters in (2a)–(2d) are the length scale factor

L, time scale factor 𝑇, and the nondimensional energy factor𝑘 which can be defined as following relations [18]:

𝐿 = [

3𝐸

0

4𝜋𝜌𝑔𝑍

0

]

1/3

,

𝑇 = [

3

2𝑔𝑍

0

]

1/2

× 𝐿,

𝑘 =

(𝜌𝑔𝑍

0)

𝛾−1

𝛾 − 1

𝑘

1(

𝑊

𝐸

0

)

𝛾−1

,

(4)

where 𝐸0is the total energy of explosion, 𝜌 is the density of

fluid, 𝑔 is acceleration due to gravity,W is the charge weight,and 𝑘

1is the parameter based on the charge type. For the

nondimensional energy charge parameter 𝑘1for TNT, the

simplified expression is specified:

𝑘

1≅ 0.0743𝑍

0.25

0. (5)

Based on the previous bubble dynamics equations (2a)–(5), the characteristics of bubble radius velocity and bubblevertical velocity can be solved using the fourth-order Runge-Kutta method once the initial conditions are given. And theinitial conditions of UNDEX bubble can be defined as 𝑥 = 𝑥

0,

𝜎 = 𝜎

0, 𝜁 = 𝜁

0, and 𝜆 = 𝜆

0and the initial radius of bubble 𝑅

0

can be obtained by solving the following energy conservationequation [19]:

1.39 × 10

5

Δ𝑃

1

(𝛾 − 1)

(

3𝑊

4𝜋𝑅

3

𝑚

)[1 − (

𝑅

0

𝑅

𝑚

)

−3(𝛾−1)

]

= (

𝑅

0

𝑅

𝑚

)

3

− 1,

(6)

where Δ𝑃 = 𝑃∞

− 𝑃

𝑐is the condensing steam pressure of

water and 𝑃∞

is the static water pressure at infinite distance.𝑅

𝑚(inm) is the first maximum radius of UNDEX bubble and

is presented as follows [20]:

𝑅

𝑚= 𝐾

1

𝑊

1/3

(𝑑 + 10.34)

1/3. (7)

The first period of the bubble pulse wave 𝑇1(in s) can be also

expressed using the empirical formula:

𝑇

1= 𝐾

2

𝑊

1/3

(𝑑 + 10.34)

5/6, (8)

where 𝐾1and 𝐾

2are the constants depending on explosive

charge types (for TNT charge, 𝐾1= 3.5, 𝐾

2= 2.11),𝑊 is the

mass of the charge in kilograms, and 𝑑 is the depth of chargein meters.


2.2. Kelvin Impulse and Blake Criteria. The Kelvin impulse isa particularly valuable concept in unsteady fluid dynamics.Benjamin and Ellis [21] seem to be the first to have realizedits value in bubble dynamics.TheKelvin impulsemay be usedto determine aspects of the gross bubble motion, and it isdefined as follows [22]:

I = 𝜌∫𝑆

𝜙n 𝑑𝑆 = ∫𝑡

0

𝐹 (𝑡) 𝑑𝑡, (9)

F (𝑡) = 𝜌𝑔𝑉e + 𝜌∫∑𝑏

{

1

2

(∇𝜙)

2 n − 𝜕𝜙𝜕n

∇𝜙}𝑑𝑆, (10)

where 𝜌 is the fluid density, 𝑉 is the bubble volume, and 𝑒 isthe unit vector of buoyancy force direction. 𝜙 is the velocitypotential, 𝑆 is surface of the bubble,∑ is the boundary, and 𝑛is the outward normal to the fluid.

The first part in (10) indicates the buoyancy force ofbubble, and the second part in (10) is the Bjerknes forcecaused by the variation of fluid field near the boundary.Thus,the sum of these two forces leads to the bubbles migrating.For rigid boundary condition, the second part in (10) can bealso written as [22]

𝜌∫

∑𝑏

{

1

2

(∇𝜙)

2 n − 𝜕𝜙𝜕n

∇𝜙}𝑑𝑆 = −

𝜌𝑚

2(𝑡)

16𝜋ℎ

2(𝑡)

. (11)

According to the Rayleigh spherical bubble model [23](which can be simplified from (2a)–(2d)), one may have

𝑅

𝑑

2𝑅

𝑑𝑡

2+

3

2

𝑑𝑅

𝑑𝑡

+

Δ𝑃

𝜌

= 0. (12)

And suppose that ℎ(𝑡) is constant throughout themotion andequals its initial value ℎ

0. So one may obtain

𝑚(𝑡) = 4𝜋𝑅

2̇

𝑅 = ±4𝜋𝑅

2[

2

3

(

Δ𝑃

𝜌

)(

𝑅

3

𝑚

𝑅

3− 1)]

1/2

.(13)

By using these results and integrating (9) over the lifetime ofbubble, the Kelvin impulse at the end of collapse is obtained[22]:

I =2√6𝜋𝑅

5

𝑚(𝜌Δ𝑃)

1/2

9ℎ

2

0

[2𝜀

2𝛿

2𝐵(

11

6

,

1

2

)

− 𝐵(

7

6

,

3

2

)] ,

(14)

where 𝐵(𝑥, 𝑦) is beta function, and 𝜀 and 𝛿 are defined asfollows:

𝜀 =

ℎ

𝑅

𝑚

,

𝛿 = [

𝜌𝑔𝑅

𝑚

Δ𝑃

]

1/2

.

(15)

Blake and Cerone [22] pointed out that if 𝐼 > 0 the bubblewill migrate away from the rigid boundary. For 𝐼 < 0, thebubble will migrate towards the rigid boundary and for 𝐼 = 0the following relation between 𝜀 and 𝛿 can be observed:

𝜀𝛿 = [

𝐵 (7/6, 3/2)

2𝐵 (11/6, 1/2)

]

1/2

= 0.442.(16)

This is well known as Blake criteria [22].

3. Numerical Modeling and Simulation

Considering the practical application of LCSP in ship build-ing industry, the LCSP with an exposed area 𝑎 ∗ 𝑏 =1200.0mm∗1000.0mm(the totalmass of LCSP𝑀

𝑠= 67.6 kg)

and its geometric description are plotted in Figure 2. Thegeometric parameters of LCSP are 𝜑 = 𝜋/4, 𝑑

𝑐= 120.0mm,

𝑡

𝑓= 2.5mm, 𝑡

𝑏= 2.5mm, 𝑡

𝑐= 1.5mm, and𝐻

𝑐= 38.250mm.

The calculated parameters for all 3 cases are shown in Table 1.

3.1. Lagrange Finite Element Model. The face sheets andcorrugated core are modeled as a plane plate using 52000quadrilateral shell elements (CQUAD4, KEYHOFF formu-lation, hourglass control, five degrees of freedom per node𝑢

𝑥, 𝑢𝑦, 𝑢𝑧, 𝜃𝑥, 𝜃𝑦, and finite membrane strains elements, with

5 integration points). The detailed Lagrange FE model ofLCSP can be seen in Figure 3. In the simulation, the Lagrangematerial of the LCSP is modeled to be Q235 steel. In order toconsider the strain rate effect, the Cowper-Symonds model[24] is adopted in the analysis. And the material propertyconstants are described in previous investigation [24]. Here,the FAILMP Sentry was used to represent the element failuremodel. The value of 0.24 was used from MSC/MVISIONdatabase according to the test results [25].

3.2. Euler Finite Element Model. Boundary integral method(BIM) was widely used in the bubble dynamics early inthe 1960s, which has been validated by many experiments.Gong and Khoo [26] analyzed the transient response ofstiffened composite submersible hull subjected to underwaterexplosion bubble by adopting the coupledBEM-FEMmethodto handle the interaction of the glass-epoxy composite struc-tures and the underwater explosion bubble. And the effect ofbubble locations on the composite submersible hull is alsostudied and analyzed. By using boundary integral method(BIM) and multiple vortex rings model, Zhang et al. foundsome new phenomena such as more splits after the first splitof the toroidal bubble in the splitting of a toroidal bubble neara rigid boundary according to the numerical simulation andtwo experiments [27]. Based on the vortex ring for arbitrary


b

a

dc

Hc

tc

tf

tb 𝜑

Figure 2: Schematic diagram of a lightweight corrugated sandwich plate (LCSP).

Table 1: Various computational cases considered.

Case number 𝑎 (mm) 𝑏 (mm) 𝑊 (kg) 𝑑 (depth of charge, m) 𝑇1(s) 𝑅max (m) 𝑑/𝑅max

1 1200.00 1000.00 0.05 0.6 0.106 0.581 ∼1.02 1200.00 1000.00 0.05 0.3 0.108 0.586 ∼0.53 1200.00 1000.00 5.00 0.6 0.491 2.695 ∼0.2

zy

The LCSP bottom face sheet iscontacting the free surface

Free surface

xb/2a/2

S

sheet ise surface

b/2a/2

Figure 3: Lagrange FE model of LCSP and location of charge.

location in 3D model and a new density potential method(DMP), higher accuracy and stability are obtained to capturedetailed features of bubble deformation especially for thelarge deforming problem and the toroidal bubble phase [28].Though boundary integral method (BIM) is a traditionaltechnique for underwater explosion bubble problem, somevery complex phenomena such as the water splash (especiallywhen the free surface is considered), the fracture of struc-tures, and mesh distortion are hard to overcome. Moreover,some remeshing techniques introduced to decrease the errordue to mesh distortion are studied in [27, 28]. However, theartificial numerical noise cannot be easily avoided during thewhole simulation process. And these complicated numericalalgorithms will increase the complexity of algorithm andthe amount of calculation. Some other different numericaltechniques, for example, finite volume method (FVM) andsmoothed particle hydrodynamics (SPH), are adopted to dealwith underwater explosion bubble problem. Thus, the CFD

Outer Eulerdomain

Free surface

Flow out boundarycondition

Rectangular enclosing

surface

Figure 4: The outer Euler domain.

Inner Euler domain Rectangular enclosing

surface

Figure 5: The inner Euler domain of LCSP.

solver of MSC.Dytran by using an Eulerian approach and afinite volume method is adopted in the present simulation.And the fluid governing equations are the conservation lawsand are integrated in time by a first-order explicit dynamicprocedure [25].

Tomodel the fluid inside and outside the LCSP, two Eulerdomains are used. The outer domain has the LCSP surface(including top face sheet, bottom face sheet, and out-off rigidwall) as part of the fluid boundary. Euler material is outsidethe LCSP surface and there is no material inside the LCSPsurface. The contents inside the LCSP are modeled in theinner domain and this domain is also enclosed by the LCSPsurface. Therefore, both Euler domains use the LCSP surface


Table 2: Material properties (the fresh water and the air).

Material MSC.Dytran model Input parameters

Water Polynomial equation of state 𝑎1 = 2.314 × 109 Pa, 𝑎

2= 6.561 × 109 Pa, 𝑎

3= 1.126 × 1010 Pa, 𝑏

0= 0.4934, 𝑏

1= 1.3937,

𝑏

2= 0.0000, 𝜌

0= 1025 kg/m3, and 𝐸 = 3750.4 J/kg

Air Gamma law𝛾 = 1.40, 𝜌 = 1.185 kg/m3, and 𝐸 = 287 J/kg

ChargeWater

Air

LCSP

Figure 6: Euler domain mesh.

as part of their enclosure (Figures 4 and 5). The outer Eulerdomain and its enclosing surface are shown in Figure 4.

The outer boundary of the outer domain is given by asufficiently large fixed box. Pressure at the outer boundaryis set to the hydrostatic pressure by using HYDRSTATand FLOWDEF keyword cards. This behaves as the openboundary. The Euler mesh contains the water and the air onthe top of the water. The fluid mesh used for this problemconsists of a block of elements, with the dimensions 2.5m ∗2.5m ∗ 4.0m. This fluid block of water and air was meshedwith 100 ∗ 100 ∗ 60 hexahedron elements, and the totalnumber of fluid elements is 900000 (Figure 6).The gird size ofinner Euler domain is 0.02m in this simulation. All boundaryconditions for the fluid mesh shown in Figures 4 and 5 weregiven a “flow” boundary condition by adopting TICEUL andTICVAL keyword cards.

In order to model the fresh water, a polynomial equationof state was conducted. This state equation (EOS) of freshwater relates the pressure in the fluid to the acoustic conden-sation 𝜇 and the specific internal energy by

𝑃 = 𝑎

1𝜇 + 𝑎

2𝜇

2+ 𝑎

3𝜇

3+ (𝑏

0+ 𝑏

1𝜇 + 𝑏

2𝜇

2) 𝜌

0𝐸

(𝜇 > 0) ,

(17a)

𝑃 = 𝑎

1𝜇 + (𝑏

0+ 𝑏

1𝜇

2) 𝜌

0𝐸 (𝜇 < 0) , (17b)

where 𝜇 = (𝜌−𝜌0)/𝜌

0, 𝜌0is the initial density of fresh water.𝐸

is the specific internal energy per unit mass, and 𝑎1, 𝑎2, 𝑎3, 𝑏0,

𝑏

1, and 𝑏

2are the constants of the fluid, respectively. And (17a)

applies to a fluid in a compressed state, while (17b) applies toa fluid in an expanded state. And the constants in (17a) and(17b) are provided in Table 2 [29].

Impact pressure

Lagrange

Euler

Figure 7: General Coupling function and LCSP-UNDEX bubblesimulation model.

The gamma law gas model is adopted for the EOS of air:

𝑃 = (𝛾 − 1) 𝜌𝐸, (18)

where 𝜌 is the density of air, 𝛾 is the heat capacities of the gas,and E is the specific internal energy of air.The initial pressureof air is set to 1.0 × 105 Pa.

The TNT explosive can be modeled by a JWL EOS inMSC.Dytran. However, if the explosive is a spherical ball, theradius of this ball is only 0.04mwhen themass of TNT chargeis set to 0.5 kg. A finer gird has to be modeled to simulatethis small ball. In this analysis, the TNT explosive is definedas a compressed hot gas (𝛾 = 1.25; see (2a)–(2d)). The massand the specific internal energy are those of the TNT charge.The radius of this hot gas ball is calculated using (6) and thedensity of air is adjusted based on the equivalent mass of theTNT explosive.

3.3. Coupled Fluid-Structure Interaction. Multimaterial Eulersolver in MSC.Dytran allows for up to 9 different Eulerianmaterials to be presented in a given investigation. The twodifferent models (General Coupling method and ArbitraryLagrange-Euler method) are available to calculate the FSIproblem between Eulerian and Lagrangian materials. Thedetailed descriptions of these two FSI algorithms can be seenin MSC.Dytran user’s manual [25].

In the present analysis, the “General Coupling” algorithmis used. In this algorithm, the Lagrangian and Eulerianmeshes are geometrically independent and interact via closedcoupling surface attached to LCSP (also see Figures 4 and5). The deformation coupling surface “cuts across” Eulerianelements which contain multimaterial including air andwater, changing their volume and surface areas. As the LCSPFE mesh deforms under the action of the impact pressurefrom the Eulerian mesh, the resulting FE deflection theninfluences subsequentmaterial flow andpressure forces in theEulerianmesh, resulting in automatic and precise coupling ofFSI (Figure 7).


(a) t = 0.01 s (b) t = 0.02 s (c) t = 0.03 s

(d) t = 0.05 s (e) t = 0.07 s (f) t = 0.09 s

(g) t = 0.11 s (h) t = 0.13 s (i) t = 0.15 s

Figure 8: Calculated evolution of the bubble at 0.3 kg detonation at a depth d = 1.0m.

When the pressure force of bubble is sufficiently large,the breach damage may occur in the simulation. Underthis condition, the flow transfer of fluid from the breachbetween outer Euler domain and inner Euler domain willbe important. To simulate the flow transfer of breach, theadaptive multiple Euler domains technology and failure ofcoupling surface technology are adopted in the simulation.The PARAM, FASTCOUP, INPLANE, and FAIL keywordcards are used to consider failure of the coupling surface, andthe PORFLCPL and COUPOR keyword cards are adoptedhere to model the transfer of different Euler domains.

4. Numerical Verification: 3D Near FreeSurface UNDEX Bubble

To verify the reliability of the developed FE model, a three-dimensional (3D) near free surface UNDEX bubble modelis first built for calibration purpose. And the computationalparameters of validation case are as follows: W = 0.3 kg(mass of TNT charge), 𝜌TNT = 1630 kg/m

3 (density of TNTcharge), 𝛾 = 1.25 (for bubble gas), 𝛾 = 1.40 (for air), d =1.0m, 𝑅

0= 0.075m (initial bubble radius, from (6)), 𝜌

0=

168.976 kg/m3 (initial bubble density),𝑃0= 84.68MPa (initial

bubble pressure), 𝑅max = 1.04m (maximum bubble radius),and 𝑇

1= 0.151 s (first period of the bubble pulse).

The entire volume of the bubble in the first bubblecirculation, with use of the developed FE model as simulatedby DYTRAN, is presented in Figure 8. Figure 8(a) shows theinitial conditions of the bubble, at which time the bubblerapidly expands outward with high internal pressure. At t =0.07 s, the bubble expands to its maximum size. At t = 0.09 s,it begins to shrink again. The first bubble circulation timeobtained from the simulated results is 0.145 s, while the resultcalculated from (8) is 0.151 s. The simulated motion of bubbleis in excellent agreement with that in the empirical formula.The relative error is 0.75%. The numerical computational

maximum bubble radius is 1.067m while the result of (7)is 1.04m. The relative error is 2.5%. According to the Blakecriteria (see (16)), the value 𝜀𝛿 of verification case is equalto 0.297 which means the bubble will move from the freesurface. And the bubble collapse jet does not occur. As theeffect of free surface boundary condition, the bubble shape isnot spherical in themigration process.Thus, the Blake criteriain this case will have some error, especially the value 𝜀𝛿whichis near 0.442. In the previous analysis which was given byZhang et al. [30], the similar conclusion is also given.

5. UNDEX Simulation Results and Discussions

5.1. Bubble Shape and Fluid Dynamics

5.1.1. Case 1: 𝑑/𝑅max ∼ 1.0. The bubble dynamics of case 1(W = 0.05 kg and d = 0.6m) are firstly calculated. Figure 9shows the whole process of bubble shape and free surfacecharacteristics at different time instants. As illustrated inFigure 9, the bubble shapes remain spherical in the first t =0.0ms∼0.8ms. The local cavitation behind the LCSP occursdue to the reflection of shock wave front when t = 0.5ms, andthis cavitation region becomes larger at t = 0.8ms. At thistime instant, the bubble shape also changes to nonsphericalsignificantly because of the complex interaction of bubble-LSCP-free surface. The distorted bubble shape is similar to apeach which can be called “peach-bubble” here (t = 0.8ms∼1.0ms). It should be noted that, as shown in Figure 9, the“peach-bubble” changes to the “hill” (here, which is called“hill-bubble”) at about 2.0ms. As a result, the hill-bubblemoves from the LCSP.

To illustrate the bubble impact pressure under case 1,the contour of its characteristics is shown in Figure 10. Thebubble pressure is dissipated rapidly during its propagatingprocess (t = 0.0ms∼0.5ms) in the water. And the bubbleshock wave pressure front arrives at the LCSP at t = 0.5ms


(a) t = 0.0ms (b) t = 0.2ms (c) t = 0.5ms

(d) t = 0.8ms (e) t = 1.0ms (f) t = 1.5ms

(g) t = 2.0ms (h) t = 2.5ms

Figure 9: Bubble shape state of case 1 (W = 0.05 kg, d = 0.6m).

(the reflected pressure wave is clear in Figure 10). Thisreflected wave is tensile in nature, as opposed to othercompressive wave effects. And it is produced from therarefaction of the shock wave from the free surface. Sincewater cannot sustain a significant amount of tension, localcavitation occurs below LCSP at t = 0.8ms. Figures 10(d)–10(h) show the typical bulk cavitation zone. In the figure, thecavitation zone can be seen to be symmetric about the verticalaxis.

Figure 11 depicts the fluid particles velocity behavior ofcase 1. During the initial state, the gas bubble expands out-ward rapidly with the high-pressure and high-temperaturegas in it. At t = 0.5ms, as the reflected wave passes, theparticles are acted on by gravity and atmospheric pressure.Surface effects also occur as a result of an underwaterexplosion bubble at t = 0.8ms. It is obvious that a spray domejet is formed as a result of the bubble pressure pulse, whosevelocity 𝑉max is nearly equal to 80m/s. Later, as presented inFigures 10(e)–10(h), the range of spray dome jet is becominglarger. Furthermore, the jet penetrates the upper surface ofthe bubble at t = 2.5ms as shown in Figure 10(h) because ofthe change of bubble shape. But this jet does not cause thesignificant damage to LCSP.

5.1.2. Case 2: 𝑑/𝑅max ∼ 0.5. Figure 12 depicts the bubbleshape characteristics of case 2 (W = 0.05 kg and d = 0.5m).Like case 1, in the first time period t = 0.0ms∼0.5ms,the bubble generated by the explosion is almost sphericalduring its initial stage of expansion and contraction. Andthe incident shock wave, which is compressive, reflects fromthe free surface and results in a tensile reflected wave. Noticethat the cavitation occurs (see t = 0.8ms) when the absolute

pressure in the water drops below the cavitation pressure,which is about a negative pressure.

The pressure contour characteristic is presented in Fig-ure 13. From the figure, it is obvious that the pressuredistribution characteristic has some differences with that ofcase 1. The first difference is the cavitation area. The one ofcase 2 is larger during the pressure wave front propagationthan that of case 1. And the second is the peak pressure value.The peak pressure value in case 2 is about two times the onein case 1.

It is noted that the bulk cavitation area at time t =0.8ms asshown in Figure 12 is not similar to the one in case 1. As thedistance 𝑑 in case 2 is closer than that of case 1, the bubbleshape is significantly different in the stage of t = 0.8ms∼2.5ms. Particularly, the characteristics of bubble shape forcase 2 are similar to a spindle (which is called “spindlebubble” here). The length of this spindle bubble becomeslonger when the bubble becomes larger. From the figure, it isclear that awater hammer (similar phenomenonwas found inexperiment study [16]) is formed.The average velocity of thiswater hammer is about 40m/s in the initial stage (t = 1.5ms∼2.0ms; see Figure 14), and it increases quickly in the next stage(about 65m/s, t = 2.0ms∼2.5ms; see Figure 14). The shape ofthis water hammer obtained by numerical analysis is similarto that of previous experimental image results as shown inFigures 15 and 16 [16].

5.2. Structural Deformation Patterns. The structural defor-mation of cases 1 and 2 are presented in Figures 17 and 18,respectively. In the initial stage (t = 0.0ms∼1.0ms for case 1,t = 0.0ms∼0.8ms for case 2), the characteristics of deflectionof these two cases are similar. And the time interval (∼0.2ms)is caused by the bubble pressure propagation difference in


8.36 + 0077.80 + 0077.24 + 0076.68 + 0076.13 + 0075.57 + 0075.01 + 0074.46 + 0073.90 + 0073.34 + 0072.79 + 0072.23 + 0071.67 + 0071.11 + 0075.57 + 006

0

(a) t = 0.0ms

3.24 + 0073.03 + 0072.81 + 0072.59 + 0072.38 + 0072.16 + 0071.94 + 0071.73 + 0071.51 + 0071.30 + 0071.08 + 0078.64 + 0066.48 + 0064.32 + 0062.16 + 006

0

(b) t = 0.2ms

1.13 + 0071.05 + 0079.77 + 0069.02 + 0068.26 + 0067.51 + 0066.76 + 0066.01 + 0065.26 + 0064.51 + 0063.76 + 0063.01 + 0062.25 + 0061.50 + 0067.51 + 005

0

(c) t = 0.5ms

7.03 + 0066.56 + 0066.10 + 0065.63 + 0065.16 + 0064.69 + 0064.22 + 0063.75 + 0063.28 + 0062.81 + 0062.34 + 0061.88 + 0061.41 + 0069.38 + 0054.69 + 005

0

(d) t = 0.8ms

5.70 + 0065.32 + 0064.94 + 0064.56 + 0064.18 + 0063.80 + 0063.42 + 0063.04 + 0062.66 + 0062.28 + 0061.90 + 0061.52 + 0061.14 + 0067.60 + 0053.80 + 005

0

(e) t = 1.0ms

4.64 + 0064.33 + 0064.02 + 0063.71 + 0063.41 + 0063.10 + 0062.79 + 0062.48 + 0062.17 + 0061.86 + 0061.55 + 0061.24 + 0069.29 + 0056.19 + 0053.10 + 005

0

(f) t = 1.5ms

3.54 + 0063.31 + 0063.07 + 0062.84 + 0062.60 + 0062.36 + 0062.13 + 0061.89 + 0061.65 + 0061.42 + 0061.18 + 0069.45 + 0057.09 + 0054.73 + 0052.36 + 005

0

(g) t = 2.0ms0

2.93 + 0062.73 + 0062.54 + 0062.34 + 0062.15 + 0061.95 + 0061.76 + 0061.56 + 0061.37 + 0061.17 + 0069.76 + 0057.80 + 0055.85 + 0053.90 + 0051.95 + 005

(h) t = 2.5ms

Figure 10: Bubble impact pressure contour (𝑊 = 0.05 kg, d = 0.6m).


0000000000000000

(a) t = 0.0ms0

2.24 + 0022.09 + 0021.94 + 0021.79 + 0021.64 + 0021.49 + 0021.34 + 0021.19 + 0021.04 + 0028.95 + 0017.46 + 0015.97 + 0014.48 + 0012.98 + 0011.49 + 001

(b) t = 0.2ms

0

2.38 + 0022.22 + 0022.07 + 0021.91 + 0021.75 + 0021.59 + 0021.43 + 0021.27 + 0021.11 + 0029.53 + 0017.94 + 0016.36 + 0014.77 + 0013.18 + 0011.59 + 001

(c) t = 0.5ms

0

1.39 + 0021.30 + 0021.20 + 0021.11 + 0021.02 + 0029.25 + 0018.33 + 0017.40 + 0016.48 + 0015.55 + 0014.63 + 0013.70 + 0012.78 + 0011.85 + 0019.25 + 000

(d) t = 0.8ms

1.14 + 0021.07 + 0029.92 + 0019.16 + 0018.39 + 0017.63 + 0016.87 + 0016.10 + 0015.34 + 0014.58 + 0013.81 + 0013.05 + 0012.29 + 0011.53 + 0017.63 + 0009.09 − 004

(e) t = 1.0ms

8.47 + 0017.91 + 0017.34 + 0016.78 + 0016.21 + 0015.65 + 0015.08 + 0014.52 + 0013.95 + 0013.39 + 0012.82 + 0012.26 + 0011.69 + 0011.13 + 0015.65 + 0001.58 − 003

(f) t = 1.5ms

9.53 + 0018.90 + 0018.26 + 0017.63 + 0016.99 + 0016.35 + 0015.72 + 0015.08 + 0014.45 + 0013.81 + 0013.18 + 0012.54 + 0011.91 + 0011.27 + 0016.36 + 0004.27 − 003

(g) t = 2.0ms

1.47 + 0021.37 + 0021.27 + 0021.18 + 0021.08 + 0029.79 + 0018.81 + 0017.84 + 0016.86 + 0015.88 + 0014.90 + 0013.92 + 0012.94 + 0011.96 + 0019.80 + 0007.70 − 003

(h) t = 2.5ms

Figure 11: Fluid particle velocity contour (W = 0.05 kg, d = 0.6m).


(a) t = 0.0ms (b) t = 0.2ms

(c) t = 0.5ms (d) t = 0.8ms

(e) t = 1.0ms (f) t = 1.5ms

(g) t = 2.0ms (h) t = 2.5ms


the water. During the following stage (t = 1.0ms∼2.0msfor case 1, t = 0.8ms∼1.5ms for case 2), due to the localcavitation phenomenon behind the back face sheet of LCSP,the maximum structural deformation value is much smallerthan that of initial stage. But the closing impact pressureof local cavitation phenomenon does not cause significantdamage to LCSP. It is noted that the local large deformationnear boundary is formed at t = 1.5ms in both cases 1 and2. Compared with the bubble shape evolution contour (seeFigures 9 and 12) and fluid velocity distribution (see Figures 11and 14) at different times, the local high speed of water plumeloaded on the wet face sheet of LCSP is mainly reasonable.At the last stage (t = 1.0ms∼2.0ms for case 1 and t = 0.8ms∼1.5ms for case 2), it is clearly observed that the region of thislocal large deflection is becoming larger as the velocity ofwater plume jet is becoming faster.

5.3. Failure Mode and Structural Damage. As stated in theprevious section, serious damage and failure do not occur forcases 1 and 2.Thus, the mass of TNT charge adopted for case3 is increased in the simulation as shown in Table 1. As thebubble shape is not spherical during the interaction processbetween bubble and LCSP, (6) and (7) do not fit for this case.

But the equation can give a reasonable prediction for bubbleinitial evolution.

The bubble shape evolution process is shown in Figure 19.Like the ones in cases 1 and 2, at the initial stage t =0.0ms∼0.8ms, the bubble shape still remains spherical as thereflection pressure wave does not affect the bubble migration.At the stage of t = 0.5ms∼0.8ms, the bubble shape becomesnonspherical due to the propagation of reflection pressurewave. Unlike the results of cases 1 and 2, the top-half partof UNDEX bubble connects the cavitation area (t = 0.8ms∼1.5ms). So the “peach-bubble” (for case 1) and “spindlebubble” (for case 2) do not occur here. It should be pointedout that the simulation of case 3 is terminated at t = 1.5ms dueto too small time interval of the computational stability step.

To illustrate the damage process of UNDEX bubblecollapse during the time period t = 1.0ms∼1.5ms, the fluidvelocity contour is presented in Figure 20. It is noted that thelocal high velocity region is formed at t = 1.0ms (see the shapeof red dashed point) when the UNDEX bubble connectedwith the cavitation area. And the local high velocity regionchanges to two parts at t = 1.1ms∼1.2ms. In this period, theshape of local high velocity region is similar to a butterflywhich can be called “butterfly bubble.”This “butterfly bubble”


8.25 + 0077.70 + 0077.15 + 0076.60 + 0076.05 + 0075.50 + 0074.95 + 0074.40 + 0073.85 + 0073.30 + 0072.75 + 0072.20 + 0071.65 + 0071.10 + 0075.50 + 006

0

(a) t = 0.0ms

3.22 + 0073.01 + 0072.79 + 0072.58 + 0072.36 + 0072.15 + 0071.93 + 0071.72 + 0071.50 + 0071.29 + 0071.07 + 0078.59 + 0066.44 + 0064.29 + 0062.15 + 006

0

(b) t = 0.2ms

1.24 + 0071.16 + 0071.08 + 0079.94 + 0069.12 + 0068.29 + 0067.46 + 0066.63 + 0065.80 + 0064.97 + 0064.14 + 0063.31 + 0062.49 + 0061.66 + 0068.29 + 005

0

(c) t = 0.5ms

8.53 + 0067.96 + 0067.39 + 0066.82 + 0066.25 + 0065.68 + 0065.12 + 0064.55 + 0063.98 + 0063.41 + 0062.84 + 0062.27 + 0061.71 + 0061.14 + 0065.68 + 005

0

(d) t = 0.8ms

6.97 + 0066.50 + 0066.04 + 0065.58 + 0065.11 + 0064.65 + 0064.18 + 0063.72 + 0063.25 + 0062.79 + 0062.32 + 0061.86 + 0061.39 + 0069.29 + 0054.65 + 005

0

(e) t = 1.0ms

5.77 + 0065.38 + 0065.00 + 0064.61 + 0064.23 + 0063.84 + 0063.46 + 0063.07 + 0062.69 + 0062.31 + 0061.92 + 0061.54 + 0061.15 + 0067.69 + 0053.84 + 005

0

(f) t = 1.5ms

2.12 + 0061.98 + 0061.84 + 0061.70 + 0061.55 + 0061.41 + 0061.27 + 0061.13 + 0069.89 + 0058.48 + 0057.07 + 0055.65 + 0054.24 + 0052.83 + 0051.41 + 005

0

(g) t = 2.0ms

4.32 + 0064.03 + 0063.74 + 0063.46 + 0063.17 + 0062.88 + 0062.59 + 0062.30 + 0062.02 + 0061.73 + 0061.44 + 0061.15 + 0068.64 + 0055.76 + 0052.88 + 005

0

(h) t = 2.5ms

Figure 13: Bubble impact pressure contour (W = 0.05 kg, d = 0.3m).


0000000000000000

(a) t = 0.0ms

2.23 + 0022.08 + 0021.94 + 0021.79 + 0021.64 + 0021.49 + 0021.34 + 0021.19 + 0021.04 + 0028.94 + 0017.45 + 0015.96 + 0014.47 + 0012.98 + 0011.49 + 001

0

(b) t = 0.2ms

2.27 + 0022.12 + 0021.97 + 0021.82 + 0021.67 + 0021.51 + 0021.36 + 0021.21 + 0021.06 + 0029.09 + 0017.57 + 0016.06 + 0014.54 + 0013.03 + 0011.51 + 001

0

(c) t = 0.5ms

1.37 + 0021.28 + 0021.19 + 0021.10 + 0021.00 + 0029.13 + 0018.21 + 0017.30 + 0016.39 + 0015.48 + 0014.56 + 0013.65 + 0012.74 + 0011.83 + 0019.13 + 000

0

(d) t = 0.8ms

1.26 + 0021.17 + 0021.09 + 0021.01 + 0029.22 + 0018.38 + 0017.54 + 0016.71 + 0015.87 + 0015.03 + 0014.19 + 0013.35 + 0012.51 + 0011.68 + 0018.38 + 0009.09 − 004

(e) t = 1.0ms

1.23 + 0021.15 + 0021.07 + 0029.83 + 0019.01 + 0018.19 + 0017.37 + 0016.56 + 0015.74 + 0014.92 + 0014.10 + 0013.28 + 0012.46 + 0011.64 + 0018.20 + 0003.86 − 003

(f) t = 1.5ms

7.78 + 0017.26 + 0016.75 + 0016.23 + 0015.71 + 0015.19 + 0014.67 + 0014.15 + 0013.63 + 0013.11 + 0012.59 + 0012.08 + 0011.56 + 0011.04 + 0015.19 + 0003.70 − 003

(g) t = 2.0ms

1.14 + 0021.06 + 0029.87 + 0019.11 + 0018.35 + 0017.59 + 0016.83 + 0016.07 + 0015.31 + 0014.55 + 0013.79 + 0013.04 + 0012.28 + 0011.52 + 0017.59 + 0004.20 − 003

(h) t = 2.5ms

Figure 14: Fluid particle velocity contour (W = 0.05 kg, d = 0.3m).


Figure 15: UNDEX bubble shape and water hammer [16].

Water hammer

Figure 16: Geometric description of water hammer from present numerical investigation.

has been reported by few studies [31, 32]. And the shape ofthis bubble collapse jet changes during this process due to theinteraction between LCSP and bubble (see the red dashed lineof Figure 20). The fluid velocity of “butterfly bubble” top part(the detailed shape of which can be referred to in Figure 21)is about 150m/s which is very high local impact loading.Under this loading, the significant damage (large deflection)is formulated at about t = 1.3ms∼1.4ms. At about t = 1.4ms,the large deflection reaches its limit where a crevasse can befound as shown in Figures 19 and 20.

In order to compare the damage characteristics of thesethree cases, the curves of equivalent plastic strain of repre-sentative element (central element ofwater contacted surface)are presented in Figure 22. As plotted in Figure 22(a), theplastic occurs at time 𝑡 = 0.8ms simultaneously with the timewhen reflected shock wave passed over LCSP (see Figure 9).Furthermore, the bubble jet is formulated around time𝑡dam = 1.0ms; the plastic damage level increased very quicklyas shown in Figure 22(a). However, the largest plastic valueof case 1 for central element is equal to 6.7 × 10−3, whichmeans that the bubble induced structural damage is minor.The equivalent plastic strain results of case 2, as illustrated inFigure 22(b), give a clear description that the plastic damageoccurs at time 𝑡dam = 0.3ms, which is much earlier thanthat of case 1. And this is mainly caused by the fact thatreflected shock wave front arrives at the LCSP earlier forcase 2 (this phenomenon also exists for case 3, in which

𝑡dam is nearly equal to 0.18ms though the initial damage isvery minor). For both cases 1 and 2, it also shows that theplastic damage increases very fast during the bubble-LCSPinteraction process during which the bubble jet is formulatedand can be proved in Figures 11 and 14. Extremely, the highspeed local bubble jet loading on the LCSP water contactedsurface makes the breach phenomenon of LCSP as depictedin Figure 21 (and this also can be validated in Figure 22(c) asfast increase of plastic strain).

6. Conclusions

In the present investigation, the high-resolution numericalanalysis model of underwater explosion bubble and cor-rugated sandwich plate are built and developed using thefluid-solid interaction nonlinear software MSC.Dytran. Themulti-Euler algorithm is adapted to describe the detaileddynamic behavior between underwater explosion bubble andcorrugated sandwich plate. It is found that the response anddamage of corrugated sandwich plate in thewhole interactionprocess are significantly affected by the bubble responseand such effect should not be neglected. In summary, thefollowing conclusions from the viewpoint of gas bubble pulsedominating failure can be drawn:

(i) The underwater explosion bubble shape changes tononspherical significantly caused by the interaction


2.50 − 0022.33 − 0022.17 − 0022.00 − 0021.83 − 0021.67 − 0021.50 − 0021.33 − 0021.17 − 0021.00 − 0028.33 − 0036.67 − 0035.00 − 0033.33 − 0031.67 − 003

0

(a) t = 0.0ms

2.50 − 0022.33 − 0022.17 − 0022.00 − 0021.83 − 0021.67 − 0021.50 − 0021.33 − 0021.17 − 0021.00 − 0028.33 − 0036.67 − 0035.00 − 0033.33 − 0031.67 − 003

0

(b) t = 0.2ms

4.50 − 0024.20 − 0023.90 − 0023.60 − 0023.30 − 0023.00 − 0022.70 − 0022.40 − 0022.10 − 0021.80 − 0021.50 − 0021.20 − 0029.00 − 0036.00 − 0033.00 − 003

0

(c) t = 0.5ms

4.50 − 0024.20 − 0023.90 − 0023.60 − 0023.30 − 0023.00 − 0022.70 − 0022.40 − 0022.10 − 0021.80 − 0021.50 − 0021.20 − 0029.00 − 0036.00 − 0033.00 − 003

0

(d) t = 0.8ms

4.50 − 0024.20 − 0023.90 − 0023.60 − 0023.30 − 0023.00 − 0022.70 − 0022.40 − 0022.10 − 0021.80 − 0021.50 − 0021.20 − 0029.00 − 0036.00 − 0033.00 − 003

0

(e) t = 1.0ms

2.50 − 0022.33 − 0022.17 − 0022.00 − 0021.83 − 0021.67 − 0021.50 − 0021.33 − 0021.17 − 0021.00 − 0028.33 − 0036.67 − 0035.00 − 0033.33 − 0031.67 − 003

0

(f) t = 1.5ms

2.50 − 0022.33 − 0022.17 − 0022.00 − 0021.83 − 0021.67 − 0021.50 − 0021.33 − 0021.17 − 0021.00 − 0028.33 − 0036.67 − 0035.00 − 0033.33 − 0031.67 − 003

0

(g) t = 2.0ms

2.50 − 0022.33 − 0022.17 − 0022.00 − 0021.83 − 0021.67 − 0021.50 − 0021.33 − 0021.17 − 0021.00 − 0028.33 − 0036.67 − 0035.00 − 0033.33 − 0031.67 − 003

0

(h) t = 2.5ms

Figure 17: Deformation of case 1 (W = 0.05 kg, d = 0.6m).


4.50 − 0024.20 − 0023.90 − 0023.60 − 0023.30 − 0023.00 − 0022.70 − 0022.40 − 0022.10 − 0021.80 − 0021.50 − 0021.20 − 0029.00 − 0036.00 − 0033.00 − 003

0

(a) t = 0.0ms

4.50 − 0024.20 − 0023.90 − 0023.60 − 0023.30 − 0023.00 − 0022.70 − 0022.40 − 0022.10 − 0021.80 − 0021.50 − 0021.20 − 0029.00 − 0036.00 − 0033.00 − 003

0

(b) t = 0.2ms

4.50 − 0024.20 − 0023.90 − 0023.60 − 0023.30 − 0023.00 − 0022.70 − 0022.40 − 0022.10 − 0021.80 − 0021.50 − 0021.20 − 0029.00 − 0036.00 − 0033.00 − 003

0

(c) t = 0.5ms

4.50 − 0024.20 − 0023.90 − 0023.60 − 0023.30 − 0023.00 − 0022.70 − 0022.40 − 0022.10 − 0021.80 − 0021.50 − 0021.20 − 0029.00 − 0036.00 − 0033.00 − 003

0

(d) t = 0.8ms

4.50 − 0024.20 − 0023.90 − 0023.60 − 0023.30 − 0023.00 − 0022.70 − 0022.40 − 0022.10 − 0021.80 − 0021.50 − 0021.20 − 0029.00 − 0036.00 − 0033.00 − 003

0

(e) t = 1.0ms

4.50 − 0024.20 − 0023.90 − 0023.60 − 0023.30 − 0023.00 − 0022.70 − 0022.40 − 0022.10 − 0021.80 − 0021.50 − 0021.20 − 0029.00 − 0036.00 − 0033.00 − 003

0

(f) t = 1.5ms

4.50 − 0024.20 − 0023.90 − 0023.60 − 0023.30 − 0023.00 − 0022.70 − 0022.40 − 0022.10 − 0021.80 − 0021.50 − 0021.20 − 0029.00 − 0036.00 − 0033.00 − 003

0

(g) t = 2.0ms

4.50 − 0024.20 − 0023.90 − 0023.60 − 0023.30 − 0023.00 − 0022.70 − 0022.40 − 0022.10 − 0021.80 − 0021.50 − 0021.20 − 0029.00 − 0036.00 − 0033.00 − 003

0

(h) t = 2.5ms



(a) t = 0.0ms (b) t = 0.2ms

(c) t = 0.3ms (d) t = 0.5ms

(e) t = 0.6ms (f) t = 0.8ms

(g) t = 1.4ms (h) t = 1.5ms



HoleHole

1.21 + 0031.13 + 0031.05 + 0039.69 + 0028.88 + 0028.07 + 0027.27 + 0026.46 + 0025.65 + 0024.84 + 0024.04 + 0023.23 + 0022.42 + 0021.61 + 0028.07 + 0018.62 − 004

1.17 + 0031.09 + 0031.01 + 0039.34 + 0028.56 + 0027.78 + 0027.00 + 0026.23 + 0025.45 + 0024.67 + 0023.89 + 0023.11 + 0022.33 + 0021.56 + 0027.78 + 0014.48 − 003

1.08 + 0031.01 + 0039.36 + 0028.64 + 0027.92 + 0027.20 + 0026.48 + 0025.76 + 0025.04 + 0024.32 + 0023.60 + 0022.88 + 0022.16 + 0021.44 + 0027.20 + 0015.72 − 0.03

1.06 + 0039.85 + 0029.15 + 0028.44 + 0027.74 + 0027.04 + 0026.33 + 0025.63 + 0024.93 + 0024.22 + 0023.52 + 0022.81 + 0022.11 + 0021.41 + 0027.04 + 0015.70 − 003

4.30 + 0034.01 + 0033.72 + 0033.44 + 0033.15 + 0032.86 + 0032.58 + 0032.29 + 0032.00 + 0031.72 + 0031.43 + 0031.15 + 0038.59 + 0025.73 + 0022.86 + 002

0

9.19 + 0038.57 + 0037.96 + 0037.35 + 0036.74 + 0036.12 + 0035.51 + 0034.90 + 0034.29 + 0033.67 + 0033.06 + 0032.45 + 0031.84 + 0031.22 + 0036.12 + 002

0

Large deflection

Figure 20: Bubble jet velocity shape at time = 1.0ms, 1.1ms, 1.2ms, 1.3ms, 1.4ms, and 1.5ms.

of bubble-LSCP-free surface. This distorted bubbleshape is different when the ratio of charge depth tomaximum bubble radius (𝑑/𝑅max) decreases. Here,when d/𝑅max∼1.0, the nonspherical bubble is similarto a peach. But if d/𝑅max∼0.5, the nonspherical bubbleis similar to a spindle. Furthermore, the nonsphericalbubble is similar to a butterfly when 𝑑/𝑅max < 0.2.

(ii) It is found that the bulk cavitation area connectedwith the underwater explosion bubblewhen𝑑/𝑅max ≲0.5. So, the estimation formula of maximum radius

of underwater explosion bubble from previous study(e.g., (7)) cannot be used in these cases.

(iii) Based on the nonlinear finite element analysis results,the failure modes of corrugated sandwich platesubjected to underwater explosion bubble are alsoclarified. The major damage of corrugated sandwichplate is large plastic deformation caused by UNDEXshock wave when d/𝑅max ≳ 0.5. And the crevassedamage caused by butterfly bubble local high velocityjet is major failure type when 𝑑/𝑅max ≲ 0.2. It is


2.80 − 0012.61 − 0012.43 − 0012.24 − 0012.05 − 0011.87 − 0011.68 − 0011.49 − 0011.31 − 0011.12 − 0019.33 − 0027.47 − 0025.60 − 0023.73 − 0021.87 − 002

0

(a) t = 0.0ms

8.34 − 0077.79 − 0077.23 − 0076.68 − 0076.12 − 0075.56 − 0075.01 − 0074.45 − 0073.89 − 0073.34 − 0072.78 − 0072.23 − 0071.67 − 0071.11 − 0075.56 − 008

0

(b) t = 0.2ms

6.03 − 0075.63 − 0075.23 − 0074.83 − 0074.43 − 0074.02 − 0073.62 − 0073.22 − 0072.82 − 0072.41 − 0072.01 − 0071.61 − 0071.21 − 0078.05 − 0084.02 − 008

0

(c) t = 0.5ms

7.97 − 0057.44 − 0056.91 − 0056.37 − 0055.84 − 0055.31 − 0054.78 − 0054.25 − 0053.72 − 0053.19 − 0052.66 − 0052.12 − 0051.59 − 0051.06 − 0055.31 − 006

0

(d) t = 0.8ms

7.97 − 0057.44 − 0056.91 − 0056.37 − 0055.84 − 0055.31 − 0054.78 − 0054.25 − 0053.72 − 0053.19 − 0052.66 − 0052.12 − 0051.59 − 0051.06 − 0055.31 − 006

0

(e) t = 1.0ms

1.27 − 0011.19 − 0011.10 − 0011.02 − 0019.31 − 0028.47 − 0027.62 − 0026.77 − 0025.93 − 0025.08 − 0024.23 − 0023.39 − 0022.54 − 0021.69 − 0028.47 − 003

0

(f) t = 1.3ms

2.41 − 0012.25 − 0012.09 − 0011.93 − 0011.77 − 0011.61 − 0011.45 − 0011.29 − 0011.13 − 0019.65 − 0028.04 − 0026.43 − 0024.82 − 0023.22 − 0021.61 − 002

0

(g) t = 1.4ms

2.41 − 0012.25 − 0012.09 − 0011.93 − 0011.77 − 0011.61 − 0011.45 − 0011.29 − 0011.13 − 0019.65 − 0028.04 − 0026.43 − 0024.82 − 0023.22 − 0021.61 − 002

0

(h) t = 1.5ms



0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0 0.001 0.002 0.003

Equi

vale

nt p

lasti

c str

ain

Time (ms)

(a) Case 1 (W = 0.05 kg, d = 0.6m)

Equi

vale

nt p

lasti

c str

ain

0

0.005

0.01

0.015

0.02

0.025

0.03

0 0.0005 0.001 0.0015 0.002 0.0025 0.003Time (ms)

(b) Case 2 (W = 0.05 kg, d = 0.3m)

Equi

vale

nt p

lasti

c str

ain

0

0.05

0.1

0.15

0.2

0 0.0005 0.001 0.0015 0.002Time (ms)

(c) Case 3 (W = 5.00 kg, d = 0.6m)

Figure 22: Plastic damage of three cases.

demonstrated that the UNDEX bubble collapse jetload plays a more significant role than the UNDEXshock wave load in near-field underwater explosion.

Competing Interests

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

Acknowledgments

This project is supported by the National Natural ScienceFounding of China (under Contract no. 51509096) and theResearch Award Program for Outstanding Doctor Thesis ofHuazhong University of Science and Technology (Contractno. 0109140921). This work was finished atHuazhong Univer-sity of Science and Technology (HUST), Wuhan.

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