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International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng

Underwater blast loading of water-backed sandwich plates with elastic cores: Theoretical modelling and simulations TagedPD1X XA. SchifferD2Xa,X *, D3X XV.L. TagarielliD4XbX a

TagedP Department of Mechanical Engineering, Khalifa University, Abu Dhabi, UAE b Department of Aeronautics, Imperial College London, South Kensington Campus, SW7 2AZ, United Kingdom

TAGEDPA R T I C L E

I N F O

Article History: Received 27 June 2016 Accepted 24 November 2016 Available online 1 December 2016 TagedPKeywords: Underwater blast Cavitation Fluidstructure interaction Wave propagation

TAGEDPA B S T R A C T

Analytical predictions and ﬁnite element (FE) calculations are performed to predict the 1D response to underwater blast loading of sandwich plates with elastic cores, in contact with water on both sides and loaded by an exponentially decaying shock wave on one side. The theoretical models explicitly account for cavitation processes and effects of deep water, and their formulation helps identifying the governing parameters of the problem. Three characteristic regimes of behaviour are identiﬁed and regime maps are constructed. The analytical models are validated by FE simulations and used to explore the sensitivity of the predictions to the governing non-dimensional parameters. It is shown that, in the absence of plastic core deformation, sandwich plates with stiff cores are imparted higher blast impulses compared to those with softer cores and equivalent areal mass. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction TagedPA major consideration in the design of naval components is their resistance to withstand blast loading in water. In the last decades, sandwich panels have found increasing engineering use in commercial and military marine vehicles because they combine high stiffness, low mass and the capability to absorb kinetic energy by plastic core crushing. However, if the intensity of blast loading is relatively low (e.g. in case of a large stand-off distance from the detonation point) and the collapse strength of the core is sufﬁciently high (e.g. metal honeycomb cores), core crushing does not occur and the sandwich undergoes a linear elastic response. A detailed examination of this latter case is necessary if sandwich plates are to be designed to sustain blast loading with negligible plastic deformations. TagedPThe loading applied to a structure in an underwater blast event is highly sensitive to the details of the ensuing ﬂuidstructure interaction (FSI) processes taking place during the early stage of the loading phase. A profound understanding of these processes is crucial to achieve optimal design against underwater blast. Early studies on FSI in underwater blast date back to World War II andD5X X were published in the 1950s by the Ofﬁce of Naval Research [1]. The pioneering work of Taylor [2] examined the response of a free-standing rigid plate loaded by an underwater shock wave in water and found that the transmitted momentum is highly sensitive to the plate's mass. He showed theoretically that the reductions in momentum *

Corresponding author. Fax C971 2 4472442. E-mail address: [email protected] (A. Schiffer).

http://dx.doi.org/10.1016/j.ijimpeng.2016.11.014 0734-743X/© 2016 Elsevier Ltd. All rights reserved.

TagedP re a consequence of the occurrence of cavitation spreading at the a ﬂuid-structure interface. TagedPThe underlying physical phenomena of shock-wave induced cavitation in water were ﬁrst studied by Kennard [3]. He found that the cavitated liquid expands by propagation of two breaking fronts (BF) emerging from a single nucleation point and propagating in opposite directions at supersonic speed. Subject to the conditions in the water, a propagating BF may turn into a closing front (CF) forcing contraction of the cavitation zone. TagedPLater, it was shown that Kennard's theory can be used to predict the underwater blast response of submerged structures [4], concluding that the propagation of BFs and CFs (and hence, the loading on the structure) depend on the problem geometry, material properties, the characteristics of the shock wave and on the hydrostatic pressure in the ﬂuid prior to the blast. TagedPRecent literature in underwater blast loading focused on the beneﬁts of replacing monolithic structures with crushable sandwich plates. Several studies have shown that sandwich panels widely outperform monolithic designs of equal mass if the core material and geometry are adequately selected [59]. It was found that sandwich plates exhibit a different FSI mechanism compared to that of monolithic plates [5,10]: for a sandwich plate with crushable core, cavitation initiates at a ﬁnite distance from the ﬂuid-structure interface due to the resistance offered by the core. On the other hand, if the core response is purely elastic, two cavitation zones may initially €kinen [11]. Experimental eviappear in the water, as reported by Ma dence for these ﬁndings was provided by other researchers [1214] who measured the propagation of cavitation fronts using a

A. Schiffer and V.L. Tagarielli / International Journal of Impact Engineering 102 (2017) 6273

TagedPtransparent shock tube. Other authors used steel shock tubes or explosive devices to measure the underwater blast resistance of sandwich plates [1518] but did not visualise the cavitation process. TagedPThe underwater blast response of sandwich plates does not only entail complex FSI processes, it may also result in structural damage and failure if the loading intensity is sufﬁciently high. Wei et al. [19] developed detailed 3D FE models to examine deformation and failure of monolithic composite plates and sandwich panels, concluding that core crushing and face sheet failure are highly sensitive to the details of the cavitation process. Avachat and Zhou [20] followed a similar approach to examine damage and failure mechanisms in both air-backed and water-backed sandwich plates, concluding that low-density cores consistently outperform those with higher densities in terms of face sheet deﬂection and sustained blast impulse. TagedPOther authors developed simpliﬁed analytical models to study the details of the cavitation process, and its effect on the structural response. For example, Deshpande and Fleck [10] and Hutchinson and Xue [21] developed analytical models for the 1D underwater blast response of sandwich plates with crushable cores and accounted for the FSI effect by assigning an attached water layer to the front face sheet, as a consequence of cavitation occurring at a ﬁnite distance. McMeeking et al. [22] modelled the cavitation process in more detail, assuming that the cavitated ﬂuid spreads by propagation of two BFs, and considering the possibility of the emergence of a reconstitution wave (equivalent to the notion of a closing front). On the other hand, the analysis of McMeeking et al. [22] did not explicitly account for the reﬂection of pressure waves at the propagating cavitation front and the effect of a non-vanishing initial hydrostatic ﬂuid pressure. Theoretical work by Schiffer et al. [4] analysed such effects for the case of underwater blast loading of a rigid plate supported by a linear spring, concluding that FSI is extremely sensitive to initial pressure in the ﬂuid. These models capture propagation of breaking fronts and closing fronts as well as their interactions with the structure in a blast event. Later, the latter approach [4] was used by other authors to assess the performance of different types of claddings in terms of underwater blast mitigation [23,24]. TagedPIn this study we examine the 1D response to underwater blast loading of a sandwich plate with an elastic core and rigid face sheets; both face sheets are considered to be in contact with water. Following the approach of Schiffer et al.D6X X[4], we construct analytical models capable of replicating the details of the cavitation process, including propagation of breaking fronts and closing fronts in the ﬂuid, and its effect on the response of the sandwich plate. Theoretical predictions are compared to results obtained from fully-coupled dynamic FE calculations, and the effect of face sheet mass, spring stiffness and initial ﬂuid pressure on the impulse imparted to the sandwich are explored. Non-dimensional regime maps and performance charts are constructed to provide guidelines for blast resistant design. TagedPThe outline of the paper is as follows: in Section 2 we derive the governing equations of the analytical model, identify characteristic regimes of behaviour and present non-dimensional regime maps; a description of the FE scheme is presented in Section 3; we compare analytical and FE predictions in Section 4; in Section 5, we explore the sensitivity of the structural response to the governing nondimensional parameters, and ﬁnally, in Section 6, we summarise the main ﬁndings of this study.

63

sTagedP ufﬁciently far away from the structure, resulting in a nearly planar shock front travelling at approximately sonic speed in water. According to Cole [25], the primary shock wave can be expressed as an exponentially decaying pressure versus time pulse ppos ðx; tÞ D p0 e¡ðt¡x=cw Þ=u ;

ð1Þ

for a wave travelling in the positive x direction at an arbitrary time t. The peak pressure p0 and the decay time u are set by the characteristics of the blast [25]. At time t D 0, the shock-wave (1) reaches the ﬂuid-structure interface located at x D 0, and reﬂects back into the ﬂuid column. The reﬂected wave, travelling in the negative x direction, is given by pneg ðx; tÞ D p0 e¡ðt C x=cw Þ=u :

ð2Þ

TagedPIn case of a rigid, stationary interface the total interface pressure is given by ppos(0, t) C pneg(0, t) and results in an imparted impulse Z1 I0 D 2

p0 e¡t=u dt D 2p0 u:

ð3Þ

0

TagedPNow, instead of a stationary interface, assume that the loaded structure is an unsupported rigid plate free to move in the x direction. Then, upon arrival of the pressure wave (1) at the ﬂuid-structure interface, the plate is set in motion and compatibility dictates that the plate and the ﬂuid at the interface must have equal velocity vf(t); plate motion in the positive x direction gives rise to a rarefaction wave of magnitude prare ðx; tÞ D ¡rw cw vf ðt C x=cw Þ

ð4Þ

emanating from the ﬂuid-structure interface and travelling in the negative x direction, away from the plate. The total ﬂuid pressure at an arbitrary point x in the front water column is then given by superposition of Eqs. (1), (2), (4) and the initial hydrostatic ﬂuid pressure pst: pðx; tÞ D pst C ppos C pneg C prare D pst C p0 e¡ðt¡x=cw Þ=u ¡ðt C x=cw Þ=u

C p0 e

ð5Þ

¡rw cw vf ðt C x=cw Þ:

T imilarly, the particle velocity ﬁeld in the water vw(x, t) is agedPS obtained by superimposing the particle velocity ﬁelds associated with incident, reﬂected and rarefaction waves. This gives vw ðx; tÞ D

ppos

rw cw

C

pneg

rw cw

C rw cw vf ðx; t C x=cw Þ :

ð6Þ

TagedPThe tensile rarefaction wave can cause the pressure to drop to the value of the cavitation pressure of the ﬂuid pc, at location xc and time tc. In typical underwater blast events (p0 100 - 200 MPa), the value of the cavitation pressure can be neglected; hence, we assume pc D 0 in all calculations, consistent with assumptions of previous studies in this ﬁeld [1216]. TagedPThe cavitation process is manifested by an expanding zone of cavitated water bounded by propagating cavitation fronts, acting as reﬂecting interfaces and affecting the pressure ﬁelds in the ﬂuid. Hence, Eq. (5) needs to be modiﬁed to account for these effects. Accordingly, we deﬁne as ‘Stage-I’ the response prior to the onset of cavitation, while we denote as ‘Stage-II’ the response subsequent to this event.

2.2. Response prior to cavitation (Stage-I) 2. Analytical modelling 2.1. Wave propagation and ﬂuidstructure interaction TagedPIn this section we present the governing equations for the propagation of blast waves in water and their interaction with surrounding structural interfaces. We assume that the explosive charge is

TagedPWith reference to Fig. 1, we proceed to derive the governing equations for the Stage-I response of a sandwich plate comprising of an elastic core (stiffness k per unit area) and two rigid face sheets of equal mass per unit area m; both face sheets are in contact with water (density rw, speed of sound cw) at uniform initial pressure pst. The sandwich is loaded at the front face by

64

A. Schiffer and V.L. Tagarielli / International Journal of Impact Engineering 102 (2017) 6273

Fig. 1. Schematic of problem geometry, loading condition and reference system.

TagedP TagedPan exponentially decaying shock wave (Eq. (1)) propagating in the water towards the structure. TagedPIn a coordinate system, where x D 0 denotes the equilibrium position of the front face sheet prior to blast loading, the motion of front and back face sheet is dictated by t ð7Þ m€u f C k uf ¡ub D 2p0 exp ¡ ¡rw cw u_ f

u

and

m€u b C k ub ¡uf D ¡rw cw u_ b ;

ð8Þ

respectively, where uf and ub are the displacements of front and back face sheets, respectively, and the over-dots represent derivatives with respect to time. TagedPWe now deﬁne the non-dimensional parameters t D t=u ; x D x=ðcw uÞ ; uf D uf =ðcw uÞ ; ub D ub =ðcw uÞ ; vw D vw m=I0 vf D vf m=I0 ; vb D vb m=I0 ; ; It D It =I0 ; p D p=p0 ; pst D pst =p0 ð9Þ representing non-dimensional time, position, front face displacement, back face displacement, ﬂuid particle velocity, front face velocity, back face velocity, imparted impulse, ﬂuid pressure and initial static pressure, respectively (vw denotes the velocity of water particles). TagedPConsidering the initial conditions uf D 0, u_ f D 0, ub D 0 and u_ b D 0, Eqs. (9) and (10) can be solved simultaneously to obtain time histories of the front and back face sheet displacements, uf(t) and ub(t), respectively, in closed form. In terms of the parameters (9), the solutions for uf ðtÞand ub ðtÞ are 2 3 pﬃﬃﬃ t pﬃﬃﬃ pﬃﬃﬃ t A C 2c¡4 e2ð A¡cÞ 7 61 2ðk¡c C 1Þe¡t e¡2ð A C cÞ b pﬃﬃﬃ 5 uf t D I 4 ¡ C ¡ c ð2k¡c C 1Þðc¡1Þ 2ð2k¡c C 1Þ 2ð2k¡c C 1Þ A

ð10Þ and 2 61 ub t D bI 4 C

c

3 pﬃﬃﬃ t pﬃﬃﬃ pﬃﬃﬃ t A C 2c¡4 e2ð A¡cÞ 7 e¡ct 2ke¡t e¡2ð A C cÞ pﬃﬃﬃ 5; ¡ ¡ C cðc¡1Þ ð2k¡c C 1Þðc¡1Þ 2ð2k¡c C 1Þ 2ð2k¡c C 1Þ A

ð11Þ respectively. Here, the parameters

kD

ku I0 u r cw u ; ^I D ; cD w m m m 2

ð12Þ

denote non-dimensional core stiffness, impulse and face sheet mass, respectively, and A D c2 ¡ 8k. The non-dimensional face sheet velocities follow as

vf t D

2ðk¡c C 1Þe¡t ¡ ð2k¡c C 1Þðc¡1Þ

pﬃﬃﬃ t pﬃﬃﬃ A C c e¡2ð A C cÞ 4ð2k¡c C 1Þ

t pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ A¡c e2ð A¡cÞ A C 2c¡4 pﬃﬃﬃ ¡ 2ð2k¡c C 1Þ A

ð13Þ and vb t D

pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ ¡t ACc ACc e 2 A C 2c¡4

2ke¡t e¡ct ¡ C ð2k¡c C 1Þðc¡1Þ ðc¡1Þ pﬃﬃﬃ t pﬃﬃﬃ pﬃﬃﬃ A¡c A¡c e2 A C 2c¡4 pﬃﬃﬃ C : 4ð2k¡c C 1Þ A

4ð2k¡c C 1Þ

ð14Þ TagedPNow combine D7X X Eq. (5) with Eq. (13) to obtain the ﬂuid pressure ﬁeld in the front water column in non-dimensional form: p x; t D pst C e¡ðt¡xÞ C e¡ðt C xÞ 2 3 6 7 6 7 pﬃﬃﬃ 6 7 t 6 7 pﬃﬃﬃ ¡ ACc 6 7 6 2ðk¡c C 1Þe¡t 7 ACc e 2 6 7 ¡ ¡c6 7: 6ð2k¡c C 1Þðc¡1Þ 7 4ð2k¡c C 1Þ 6 7 pﬃﬃﬃ 6 7 t pﬃﬃﬃ pﬃﬃﬃ 6 7 c A ¡ 6 7 A C 2c¡4 A¡c e2 4 5 pﬃﬃﬃ ¡ 2ð2k¡c C 1Þ A

ð15Þ T inally, using Eqs. (6), (9), (13) and (14), we write the non-dimenagedPF sional particle velocity ﬁeld in the front water as pﬃﬃﬃ pﬃﬃﬃ ¡ tCx ACc 2 ACc e

e¡ðt¡xÞ e¡ðt C xÞ 2ðk¡c C 1Þe¡ðt C xÞ ¡ vw x; t D C C ð2k¡c C 1Þðc¡1Þ 2c 2c pﬃﬃﬃ t C x pﬃﬃﬃ pﬃﬃﬃ A¡c A¡c e 2 A C 2c¡4 pﬃﬃﬃ ¡ 2ð2k¡c C 1Þ A

4ð2k¡c C 1Þ

ð16Þ

2.3. Response affected by cavitation (Stage-II) TagedPIn this Section we describe the solution scheme and the governing equations for the response subsequent to the occurrence of ﬁrst cavitation, as referred to here as the Stage-II response. Our analysis rests on the assumption that the ﬂuid pressure p varies linearly with the compressive volumetric strain ec when p > 0; in contrast, we assume that the ﬂuid is unable to sustain any pressure when p < 0;

A. Schiffer and V.L. Tagarielli / International Journal of Impact Engineering 102 (2017) 6273

TagedPthis gives the following constitutive relation for water ( 2 Kw ɛc D rw cw ɛvc for p > 0 p D 0 for p 0

65

ð17Þ

where Kw denotes the bulk modulus of water. TagedPOnce the ﬂuid pressure drops to zero at time tc and distance xc, two breaking fronts (BFs) start propagating from this point, travelling into opposite directions with supersonic speed and opening a zone of cavitated water [3]; such BFs can be seen as propagating phase fronts separating regions of cavitated and uncavitated water. In the cavitated zone, the pressure vanishes and the ﬂuid offers no resistance to continued tensile loading, according to Eq. (17). The resistance offered by the core forces the front sheet to decelerate and this may cause the BF to arrest and become a closing frontD8X X (CF), propagating into the cavitated water and forcing collapse of the cavitation zone; such CF always propagates at subsonic speed according to Kennard [3]. The propagation of the CF is coupled to the structural motion and depends on the ﬂuid properties, core stiffness, face sheet mass, initial static pressure and details of the blast event. In our analysis, the effect of the latter quantities is described through the three non-dimensional parameters k, c and pst (Eqs. (9) and (12)), governing the response of the sandwich. TagedPIt is important to note that the cavitation process may be fully suppressed if the system is subject to a sufﬁciently high static pressure pst . In this latter case, the absolute pressure in the ﬂuid does not drop to zero at any point in the ﬂuid during the blast event, and the Stage-I analysis (see Section 2.2) is sufﬁcient to predict the structural response. On the other hand, both Stage-I and Stage-II analyses are needed in the presence of cavitation. TagedPThe coupling of front and back face sheets through the elastic core gives rise to more complex cavitation phenomena compared to those of monolithic plates [4]. For the problem investigated here, two cavitation mechanisms are identiﬁed, namely a double cavitation mechanism (i) where two subsequent cavitation events occur in the €kinen [11]), and a single cavitation mechwater (also predicted by Ma anism (ii) where only one cavitation event takes place. TagedPIn the following we develop the solution scheme for the Stage-II response, addressing both cavitation mechanisms outlined above. We will make use of Kennard's equations [3] to predict the propagation of BFs and CFs, and to determine the face sheet velocities, interface pressures as well as the impulse imparted to the sandwich during the Stage-II response. TagedP2.3.1. The primary cavitation process TagedPWith reference to Fig. 2a, the cavitation process ﬁrst occurs at a ﬁnite distance from the ﬂuid-structure interface, xc1 < 0, at time tc1. The location and time of ﬁrst cavitation, xc1 and tc1, respectively, can be found by numerically solving the two equations pðx; tÞ D 0 D pst C p0 e¡ðt¡x=cw Þ=u C p0 e¡ðt C x=cw Þ=u ¡rw cw vf ðt C x=cw Þ; ð18Þ @pðx; tÞ D 0; @x

ð19Þ

deﬁning the onset of cavitation in water [3,5]. As shown in Fig. 2a, two breaking fronts (denoted here as BF1) originate from the point x D xc1 and travel into opposite directions with supersonic speed. Kennard [3] described the criteria for propagation of a BF as @p @vw < r cw @vw >0 ðpÞx D xBF D 0 w @x x D xBF @x x D xBF @x x D xBF @p 0 ðpropagation in the positive x directionÞ @x x D xBF @p 0 ðpropagation in the negative x directionÞ @x x D xBF ð20Þ

Fig. 2. Schematic illustration of the double cavitation process: (a) onset of ﬁrst cavitation, (b) propagation and arrest of breaking fronts (BF1), (c) emergence of a closing front (CF1) and onset of secondary cavitation, (d) propagation and arrest of secondary breaking fronts (BF2), and (e) emergence of a secondary closing front (CF2).

TagedPwhere xBF denotes the actual position of the BF. Here, the time derivatives of velocity and pressure, @vw/@t and @p/@t, respectively, are to be evaluated in the uncavitated region ahead of the BF. For the BF propagating away from the ﬂuid-structure interface, the above conditions (20) are always met for t tc1; hence this front will continue propagating and will not affect the structural response. On the other hand, for the BF travelling towards the front face sheet, Kennard's criteria (20) may not hold for t tc1. Hence, BF1 may arrest at time tr1 at a point closer to the front face sheet, xr1 > xc1, as sketched in Fig. 2b. We note that since BF1 propagates at supersonic speed, the pressure ﬁeld ahead of this front is not affected by its propagation. Therefore, xr1 and tr1 can be found by an iterative algorithm where the above conditions (20) are checked on the implicit curve p(x,t) D 0 (Eq. (18)) at discrete points in time ti D tc1 C iDt (i D 0,1,. . .), typically with Dt10¡8 s, until one of the criteria fails. Then tr1 D ti and xr1 is found by back-substitution into Eq. (18). TagedPOnce BF1 has arrested, a closing front may emerge at x D xr1 (denoted as CF1 in Fig. 2b), and propagate in the negative x direction

66

A. Schiffer and V.L. Tagarielli / International Journal of Impact Engineering 102 (2017) 6273

TagedP(away from the plate) at a velocity cCF1, forcing collapse of the cavitated ﬂuid. It can be shown mathematically [3] that cCF1 < cw, hence CF1 will never reach BF1. Kennard [3] describes the criteria for propagation of a CF as 2pCF;in > rw cw vc

;

hðx; tÞ D

Zt tBF ðxÞ

@vc dt > 0 @x

ð21Þ

where vc represents the particle velocity of cavitated water ahead of the propagating CF and h(x, t) denotes the strain ﬁeld in the cavitated water. We note that in all calculations performed here, Kennard's criteria (Eq. (20)) were met immediately after the BF1 had arrested. Hence, propagation of CF1 always commenced at t D tr1. TagedPSince p D 0 and @p/@x D 0 in the cavitated water (Eqs. (18) and (19)), it follows from the acoustic equations [3] that the particle velocity ﬁeld is time-independent, i.e. vc D vc(x). Therefore, vc(x) can be obtained from vw(x, t) in the uncavitated water at the time of arrival of the BF. To do this, we ﬁrst determine the time of ﬁrst cavitation, tBF(xi), at discrete points xi < xr1 by using Eq. (18). Then, we substitute tBF(xi) into Eq. (16), yielding the velocity ﬁeld of discrete ﬂuid particles at the time of ﬁrst cavitation, vc(xi). Finally, the function h(x, t) (Eq. (21)) is obtained via time integration of the numerical velocity gradients [vc(xi C 1) ¡ vc(xi)]/(xi C 1 ¡ xi) for xi < xr1 with respect to the time interval [tBF(xi), t], resulting in a spatially discrete ﬂuid strain function h(xi, t). TagedPThe quantity pCF, in in Eq. (21) represents the pressure at x D xCF1, associated with the wave train reﬂected from the ﬂuidstructure interface and approaching the CF from the uncavitated side. With reference to Eq. (5), this wave train can be obtained as the sum of the rarefaction wave and the shock wave reﬂected from the front face sheet. Hence, pCF;in ðtÞ D p0 e¡ðt C xCF1 =cw Þ=u ¡rw cw vf ðt C xCF1 =cw Þ:

ð22Þ

TagedPNote that the above equations are valid for a CF travelling into the negative x direction. Applying the principle of superposition, the total ﬂuid pressure at the propagating CF located at x D xCF1 can be written as pCF D pst C pCF;in ðtÞ C pCF;out ðtÞ:

ð23Þ

T he pressure pCF, out depends on the hydrostatic ﬂuid pressure, agedPT the ﬂuid strain in the cavitated liquid and on the propagation speed of the closing front, cCF. According to Kennard [3] and Schiffer et al. [4], cCF is given by cCF D

λcw λ C h cw ¡λ

ð24Þ

where the parameter λ is deﬁned as

λD

2pCF;in C pst ¡vc : rw cw ð1¡hÞλ rw cw : 2λ C h cw ¡2λ

€u b ðtÞ D ¡rw cw u_ b ðtÞ¡kub ðtÞ C kuf ðtÞ =m

ð29Þ

with respect to the initial conditions calculated from the Stage-I solutions of front and back face sheet velocities and displacements (Eqs. (10), (11), (13) and (14)) evaluated at the transition time t D tI. Once a solution is obtained, the total interface pressure 0 pf ðtÞ D 2ppos ð0; tÞ¡rw cw vf ðtÞ

ð30Þ

is calculated and used to determine the outgoing wave train given by 0 0 pneg ðx; tÞ D ppos ðt C x=cw Þ¡rw cw vf ðt C x=cw Þ:

ð31Þ

TagedPDue to the fact that cCF1 < cw, the latter wave train (31) will eventually reach CF1 and affect its propagation; this interaction is accounted for in our analysis. TagedP2.3.2. The secondary cavitation process TagedPAfter the primary closing front (CF1) has started propagating (t > tr1), oscillations of the front face sheet may force the pressure in the front water column to drop to zero again at distance xc2 and time tc2, as sketched in Fig. 2c; consequently, a secondary cavitation process may initiate. We note that secondary cavitation may not necessarily occur if the ﬂuid pressure is sufﬁciently high; in this latter case, FSI is controlled by a single cavitation mechanism. TagedPThe position and time of secondary cavitation, xc2 and tc2, respectively, can be found by evaluating, in each time step of the numerical integration, the pressure ﬁeld at discrete points xi D ¡ixCF1/n (iD 1, 2, .., nD 400) within the range xCF1(t) x 0. If pD 0 is detected at any calculation point, the point of secondary cavitation is found, xi D xc2. The secondary cavitation zone evolves in the same manner as the primary cavitation zone described above and can be modelled by the same technique: propagation of two BFs (denoted as BF2, Fig. 2c); arrest of the BF2 approaching the plate at point xr2 and time tr2 (see Fig. 2d); transition of the latter BF to a CF (referred to as CF2 here, Fig. 2e); propagation of CF2 at speed cCF2, resulting in a reﬂected wave train 00 ðx; tÞ D pCF;out ðt¡x=cw C xCF2 ðtÞ=cw Þ ; ppos

ð32Þ

approaching the sandwich plate and impinging on the front face sheet at time t2 D tr2 ¡ xr2/cw (see Fig. 2e); hence, for t > t2, Eq. (28) is modiﬁed to h i

00 ðx D 0; tÞ¡rw cw u_ f ðtÞ¡kuf ðtÞ C kub ðtÞ m ð33Þ €u f ðtÞ D 2ppos

ð25Þ

2

ð26Þ

TagedPThe pressure associated with the wave train reﬂected from the propagating CF, pCF, out(t), is then obtained by rearranging Eq. (23) and combining with Eqs. (22) and (26). The latter wave train can be described as 0 ppos ðx; tÞ D pCF;out ðt¡x=cw C xCF1 ðtÞ=cw Þ

and

and solved together with Eq. (29) to obtain the time histories of front and back face sheet velocities via numerical time integration.

TagedPThen, the absolute pressure pCF at the propagating CF follows as pCF D

TagedPHence, for t > t1, Eq. (27) is employed to solve numerically the equations of motion h i

0 m ð28Þ ðx D 0; tÞ¡rw cw u_ f ðtÞ¡kuf ðtÞ C kub ðtÞ €u f ðtÞ D 2ppos

ð27Þ

and propagates towards the ﬂuid-structure interface, impinging on the front face sheet at t D tr1 ¡ xr1/cw · t1, and invalidating the pressure loading associated with the incident blast wave (1).

TagedP2.3.3. Collapse of the secondary cavitation zone TagedPFor t tr2, the secondary cavitation zone is bounded at one end by CF2 and at the opposite end by BF2, travelling into the negative x direction and approaching the propagating CF1 associated with the primary cavitation zone due to cBF2 > cw > cCF1 (see Fig. 2d). However, BF2 may not reach CF1, if one of the criteria given in Eq. (20) fails. In the latter case, BF2 will invert its direction of motion (provided that the criteria given in Eq. (21) are met) and become a closing front (CF3), travelling towards the ﬂuid-structure interface and in opposite direction to CF2. Consequently, the secondary cavitation zone will completely collapse when CF3 and CF2 coalesce at distance xcoll and time tcoll. Note that the motion of CF3 and its interaction with the surrounding ﬂuid is modelled in the same way as explained above for the primary cavitation zone, and is not reiterated here for the sake of brevity.

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TagedPThe collapse of the secondary cavitation zone has an effect on the structural response for t > (tcoll ¡ xcoll/cw) · t3, owing to the fact that the wave reﬂected from CF2 (Eq. (32)) no longer impinges on the structure; instead, the front face sheet is loaded by the wave train emanated from CF1 (Eq. (27)) and Eqs. (28) and (29) become valid again during this ﬁnal stage of the response. TagedP2.3.4. Cavitation at the back face sheet TagedPIt is important to recognise that the Stage-II response may also be affected by cavitation processes in the back water column. Initially, blast loading of the sandwich causes the back face sheet to move into the positive x direction, resulting in an over-pressure of rwcwvb(t) at its ﬂuid-structure interface. As a consequence of the spring action, the back face sheet may invert its direction of motion, causing the absolute interface pressure pb ðtÞ D pst C rw cw vb ðtÞ

ð34Þ

to drop below zero, giving rise to cavitation. Recalling that pressure waves incident on the back face sheet are not considered in this study, it can be inferred from Eq. (34) that the cavitation process in the back water column always initiates at the ﬂuid-structure interface. This is accounted for in our analysis by replacing Eq. (29) with €u b ðtÞ D ¡kub ðtÞ C kuf ðtÞ m ð35Þ as soon as pb(t) D 0 is detected in Eq. (37). The cavitation zone in the back water column will eventually collapse once the structural response has ceased; however, this typically occurs very late during blast loading (after t > 30 u) and does not appreciably affect the response of the sandwich plate. 2.4. Regimes of behaviour TagedPThe response of the sandwich plate in a blast event is highly sensitive to the sequence of cavitation processes, as described in Section 2.3, and can be classiﬁed by deﬁning three characteristic regimes of behaviour. We refer to Regime 1 in the event of a double cavitation while we denote as Regime 2 all responses characterised by a single cavitation mechanism; ﬁnally, in the absence of cavitation, we refer to Regime 3. TagedPThe regime charts in Fig. 3 can be used to determine the active regime of behaviour for a given set of non-dimensional parameters c, k and pst . The contours included in Fig. 3a represent the regime transitions for the case pst D 0, showing that when the initial static pressure is vanishingly small, Regime 3 does not occur. In addition, for c > 8, Regime 2 is predominantly active, concluding that a low face sheet mass causes the single cavitation mechanism to dominate. The effect of an increased static pressure, pst D 0:13, on the regime transitions is presented in Fig. 3b. For this choice of pst , all three regimes coexist within the plotted k ¡ c space. We also observe that the dominance of Regime 2 increases by increasing pst . Finally, Fig. 3c shows that, if the initial static pressure is further increased to pst D 0:4, Regime 1 is fully suppressed and Regime 3 is predominantly active, indicating that cavitation does not occur for a large range of parameters. We note that if pst D 0:82, cavitation is fully suppressed for any choice of k and c within the plotted range of parameters. 3. Finite element calculations TagedPWe compare our analytical predictions to fully-coupled dynamic FE calculations performed in ABAQUS/Explicit [17] to validate the accuracy of our analytical models (see Section 2). In the FE calculations, both face sheets are modelled as rigid bodies (of unit area) while two linear springs (SpringA in ABAQUS/Explicit), each of stiffness k/2, are used to simulate the action of the elastic core, as illustrated in Fig. 4a. The sandwich plate is tied to two water columns, each of unit width and length L, at the front and back face sheets.

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TagedP he length L of each column was chosen sufﬁciently long to prevent T spurious loading by pressure waves reﬂected at their free ends. Four-noded plane strain elements with reduced integration (CPE4R in ABAQUS/Explicit) were used to discretise the ﬂuid columns, with the element size set to 0.1 mm in the direction of wave propagation, while only one element was used along the height of each column, to mimic 1D loading conditions. TagedPThe constitutive response of water is modelled using the MieGruneisen equation of state together with a linear Hugoniot relation between particle and the shock velocity, with the parameters adjusted such to give a linear relationship between ﬂuid pressure and volumetric strain [4], in accordance with Eq. (17). The shear modulus was assumed to be zero and cavitation was modelled by implementing a tensile failure criterion (available in ABQAUS/ Explicit) to achieve unrestricted straining of the ﬂuid when p D 0, as in Eq. (17). The density and speed of sound of water was chosen to be rw D 1000 kgm-3and cw D 1498 ms-1, respectively. TagedPThe effect of a nonzero initial hydrostatic pressure is modelled by imposing an initial hydrostatic stress ﬁeld of magnitude ¡ pst on all ﬂuid elements. To equilibrate the ﬂuid-structure system prior to blast loading, a D9X X traction of ¡ pst was applied at the free ends of the water columns, see Fig. 4a. TagedPIn order to simulate 1D blast loading, a pressure boundary condition according to Eq. (1) was imposed at the free end of the front water column, while the transverse displacements of the sandwich plate and the ﬂuid elements were constrained to zero, as shown in Fig. 4a. To minimise artiﬁcial energy dissipation by volumetric straining of the ﬂuid elements, we set the bulk viscosity coefﬁcients to 20% of the default values in ABAQUS, as in previous studies [13,14]. TagedPTo examine the effect of stress wave propagation in the throughthickness direction of the sandwich, an alternate model was developed where both face sheets and the core were modelled using fournoded plane strain elements with reduced integration (CPE4R in ABAQUS/Explicit) as sketched in Fig. 4b. The number of elements along the direction of wave propagation was approximately 12 in the face sheets and 100 in the core. The constitutive response of the face sheets was taken to be linear-elastic with Young's modulus Ef and Poisson's ratio nf. The core material was also assumed to be linear elastic (modulus Ec, Poisson's ration nc). Note that plastic deformation in the face sheets and the core was not considered. The constitutive response of the ﬂuid columns as well as the boundary and loading conditions were modelled as described above. 4. Comparison of analytical and FE predictions TagedPWe proceed to validate the accuracy of our analytical models by comparing their predictions to those obtained from the dynamic FE calculations. In doing so, we shall explore all three regimes of behaviour, and provide time histories of the position of cavitation fronts, face sheet velocities and interface pressures. 4.1. Propagation of cavitation fronts TagedPIn Fig. 5a, we present a non-dimensional cavitation map in the x ¡t space, showing analytical and FE predictions of breaking and closing front positions as a function of time for the case k D 15 and c D 2; contours are included for two choices of non-dimensional static pressure, pst D 0 and pst D 0:08. Both choices give rise to a Regime 1 response (see Fig. 3) and should therefore involve doublecavitation. Fig. 5a clearly shows that this is the case, with two subsequent cavitation zones spreading in the water for both choices pst D 0 and pst D 0:08. TagedPFor the case pst D 0, the minimum on the plotted contour represents the location and time of ﬁrst cavitation, xc1 and tc1 , respectively, as indicated in Fig. 5a. The two BFs emerging from this point

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Fig. 3. Non-dimensional regime charts in the k c space showing the regime transitions for pst D 0 (a), pst D 0.13 (b) and pst D 0.4 (c).

Fig. 4. Schematics of the FE models used to simulate 1D underwater blast loading of water-backed sandwich plates with elastic cores: spring model (a) and continuum model (b).

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cTagedP ontrast to what presented in Fig. 5a, the leftgoing BF, triggered by the secondary event, arrests at t 2:7 before reaching the CF associated with primary cavitation. Subsequently, the arrested BF inverts its direction of motion, becomes a CF and coaelesces with the opposite CF, forcing complete collapse of the secondary cavitation zone at time t 3:05. Also for this type of response, analytical and FE predictions are found in good agreement. TagedPFig. 6 presents FE and analytical predictions of BF and CF trajectories for the choice k D 5 and c D 18 with two contours included for the choices pst D 0 and pst D 0:13, respectively. For both cases, the regime map presented in Fig. 3 predicts Regime 2 to be dominant, suggesting that a single-cavitation mechanism is active. The trajectories of Fig. 6 conﬁrm this, showing that the secondary cavitation process does not occur. It can also be seen that the onset of ﬁrst cavitation occurs very close to the ﬂuid-structure interface, and at an earlier time, tc1 0:25, than predicted for the case shown in Fig. 5a. This can be justiﬁed by the lower face sheet mass used here, c D 18, resulting in rapid acceleration of the front face sheet and early cavitation. Fig. 6 also shows that an increase in pst causes the location of ﬁrst cavitation to occur at a larger distances from the front face sheet, and in faster propagation of the ensuing CF, in line with what presented in Fig. 5a. TagedPThe FE and analytical predictions presented in Figs. 5 and 6 are in excellent agreement, providing evidence that our analytical models are accurate and able to predict the details of the cavitation process. In the following section we proceed to probe the accuracy of our analytical predictions, focusing on structural motion and applied loading. 4.2. Structural motion and applied pressure

Fig. 5. (a) Non-dimensional time versus distance chart showing trajectories of breaking fronts (BF) and closing fronts (CF) for the case of a Regime 1 response (kD 5, c D 2): analytical and FE predictions are compared for two different choices of static pressure pst D 0 andpst D 0.08 and (b) similar information for the case k D 15, c D 2 and pst D 0.

TagedPare represented by a leftgoing and a rightgoing branch; the leftgoing branch is clearly visible and shows that the corresponding BF propagates indeﬁnitely, as predicted theoretically. Noting that the speed of sound in water cw corresponds to a slope of ¡1 in this chart, it is clear that the propagation speed of the latter BF is initially greater than cw, but approaches this value as time elapses. The opposite branch, however, cannot be clearly identiﬁed, as the BF approaching the structure arrests immediately after tc1 , and transitions into a CF, propagating away from the ﬂuid-structure interface at a velocity cCF < cw. Subsequently, at t 2:3, secondary cavitation is triggered very close to the ﬂuid-structure interface. Again, the BF approaching the structure immediately inverts its direction of motion and turns into a CF while the BF propagating away from the structure approaches the CF associated with the primary cavitation event. The two cavitation zones will eventually join up due to cBF > cCF; however, for illustration purposes, this is not shown in Fig. 5a. TagedPA similar sequence of cavitation events is reported for the case of a non-negligible static pressure, pst D 0:08. It can be seen that both the distance and time of ﬁrst cavitation increase D10X X compared to the what predicted for the pst D 0 case, and the speed of CF propagation increases as well. Excellent agreement between analytical and FE predictionsD1X iX s reported for both cases shown in Fig. 5a. TagedPIn Fig. 5b, we present similar information for the case k D 15, c D 2 and pst D 0, giving rise to a Regime 1 response (see Fig. 3a). In

TagedPIn Fig. 7 we plot analytical and FE predictions of non-dimensional interface pressure and face sheet velocities as functions of nondimensional time for the case k D 5, c D 2 and pst D 0: Note that this response corresponds to the cavitation front trajectories presented in Fig. 5a. Fig. 7a, and shows that the pressure applied on the front face sheet instantaneously rises to pf D 2 at t D 0, as expected, and subsequently decays to zero after going through oscillations without dropping below the cavitation limit (p D 0), indicating that cavitation at the ﬂuid-structure interface does not occur (in line with Fig. 5a). The pressure at the back face sheet, pb , rises more moderately from t D 0 but attenuates in a similar manner as pf ; it is also clear from

Fig. 6. Non-dimensional time versus distance chart showing trajectories of breaking fronts (BF) and closing fronts (CF) for the case of a Regime 2 response (kD 5, c D 18): analytical and FE predictions are compared for two different choices of static pressure pst D 0 and pst D 0.13.

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Fig. 7. Regime 1 (kD 5, c D 2, pst D 0): Time histories of non-dimensional interface pressure (a) and face sheet velocities (b); analytical and FE predictions are compared.

Fig. 8. Regime 2 (kD 5, c D 18, pst D 0): Time histories of non-dimensional interface pressure (a) and face sheet velocities (b); analytical and FE predictions are compared.

TagedPFig. 7a that cavitation at the back face sheet does not initiate in this case. TagedPThe corresponding non-dimensional face sheet velocities, vf and vb , respectively, are presented in Fig. 7bD12X X and show similar trends. We note that the analytical predictions of pressure and velocity histories are in excellent agreement with those obtained from the detailed FE calculations. TagedPFig. 8 illustrates the predicted non-dimensional time histories of interface pressure and face sheet velocities for a Regime 2 response with k D 5, c D 18 and pst D 0; the corresponding CF and BF positions are shown in Fig. 6. For the relatively low face sheet mass chosen here (c D 18), the front face sheet rapidly accelerates (see Fig. 8b) causing the applied pressure, pf , to rapidly drop to zero (see Fig. 8a), giving rise to cavitation at the ﬂuid-structure interface. Subsequently, the pressure pf , quickly rises again due to the rapid collapse of the adjacent cavitation zone and this results in additional loading on the structure. After t 6:5, both pf and pb drop to zero (see Fig. 8a), almost simultaneously, resulting in cavitation at both interfaces which is followed by face sheet oscillations, as shown in Fig. 8b. Also for this Regime 2 response, FE and analytical predictions are found to be in excellent agreement. TagedPFor the sake of brevity, we omit showing detailed time histories for Regime 3 where cavitation is fully suppressed by the initial hydrostatic pressure. However, good correlation between analytical and FE predictions was also achieved for this latter regime of response.

4.3. Effects of elastic wave propagation in the sandwich plate TagedPOur analytical models do not consider propagation of stress waves in the through thickness direction of the face sheets and core, which may affect the details of the cavitation process as well as the response of the sandwich to the blast. To examine this sensitivity, additional FE calculations were performed using the continuum model described in Section 3 and sketched in Fig. 4b. In these calculations, a sandwich plate, consisting of a 200 mm thick Rohacell 200 core (Ec D 250 MPa, nc D 0.2, rc D 200 kg m¡3, [26]) encased between two steel face sheets (Ef D 210 GPa, nf D 0.3, rf D 8000 kg m¡3) of thickness h D 20 mm, was loaded by a blast wave (p0 D 10 MPa, u D 0.15 ms) in shallow water, pst D 0. TagedPThe corresponding non-dimensional parameters were calculated (k D 0.18, c D 1.4, pst D 0) and used to perform predictions using the developed analtyical model. TagedPIn Fig. 9 we present the non-dimensional time histories of interface pressure and face sheet velocities as predicted by the FE and analytical model, respectively. The FE calculations predict signiﬁcant oscillations in the pressure and velocity time histories, resulting from stress waves propagting in the core between the two face sheets. Although the analytical predictions don't capture these effects, they are sufﬁciently accurate for practical purposes and capable of replicating major trends associated with the structural and ﬂuid response. In terms of the impulse imparted to the structure

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Fig. 10. Analytical and FE predictions of non-dimensional impulse imparted to the front face sheet as a function of the spring stiffness parameter k for the case pst D 0; contours of non-dimensional face sheet mass c are included for three different values; the markers represent FE predictions.

Fig. 9. Time histories of non-dimensional interface pressure and plate velocities for both front and back face sheets for the case k D 0.18, c D 1.4 and pst D 0; analytical predictions are compared to those obtained with the continuum FE model.

rTagedP ise to double cavitation and resulting in additional impulse imparted to the sandwich. Such trend can also be observed for the choice c D 5; however, the effect of the regime transition on If is less pronounced here. It is also interesting to note that when c is large (e.g. c D 12), If converges to a value of 0.5 with increasing k, in line with the ﬁndings of SchifferD13X X et al. [4] for the case of a rigid water-backed plate. TagedPIn Fig. 11, analytical predictions of If are plotted as functions of c for the case pst D 0 with contours of k included. Here, the trends are similar to what presented in Fig. 10. When Regime 1 is dominant, If decreases rapidly with increasing c. On the other hand, upon transition to a Regime 1 response (Fig. 3a), If rises with increasing values of c due to the additional impulse imparted to the plate in the event of double cavitation. We note that if c > 5, the effect of increasing k is

TagedP(to be introduced with Eq. (36)), the discrepancy between the two types of predictions is less than 5%. 5. Sensitivity of the response to the non-dimensional parameters TagedPThe severity of blast loading is often described by measuring the maximum impulse imparted to the structure during the blast event [4,10,12]. For a sandwich plate, such impulse can be deﬁned as Z t If D max0 t < 1 ðpf ¡pst Þdt ð36Þ 0

or, If D If =I0 in non-dimensional terms. In the following we make use of the validated analytical models to explore the sensitivity of If to the non-dimensional parameters k, c and pst , as presented in Figs. 1012. We also include in Figs. 1012 FE predictions for selected sets of parameters, as represented by full and empty markers. TagedPThe sensitivity of If to variations of non-dimensional core stiffness k is presented in Fig. 10 for the case of vanishing initial hydrostatic pressure, pst D 0; contours of non-dimensional face sheet mass c are included for three different values. It can be seen that If decreases monotonically with increasing k for all choices of c, if k is sufﬁciently small. However, for the case c D 2, If increases with increasing values of k when k > 2.5, due to the fact that Regime 2 becomes active at k > 2.5 for this choice of c (see Fig. 3a), giving

Fig. 11. Analytical and FE predictions of non-dimensional impulse imparted to the front face sheet as a function of the parameter c for the case pst D 0; contours of the stiffness parameter k are included for three different values; the markers represent FE predictions.

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TagedPThe validated analytical models were used to examine the effect of the governing non-dimensional parameters on the impulse imparted to the sandwich in the blast. The results of this parametric study are summarised as follows: TagedP The transition from single to double cavitation results in an increase of the imparted impulse to the sandwich. TagedP A decrease in core stiffness generally leads to an increase of the imparted impulse, concluding that in absence of core crushing, stiff cores widely outperform soft cores in terms of blast resistance. TagedP The imparted impulse decreases with increasing initial static pressure, suggesting that blast loading is less severe in deep water than in shallow water.

Fig. 12. Analytical and FE predictions of non-dimensional impulse imparted to the front face sheet as a function of non-dimensional static pressure pst for the case c D 12; contours of the stiffness parameter k are included for three different values; the markers represent FE predictions.

TagedPto reduce the imparted impulse If ; hence, sandwich plates with elastic cores beneﬁt from a stiff core response in terms of blast resistance. TagedPFinally, Fig. 12 presents analytical predictions of If as functions of non-dimensional static pressure pst for the choice c D 12 with contours of k included. It is clear from Fig. 12 that the effect of increasing pst is to reduce the imparted impulse If for any choice of k. This effect is more pronounced for small values of k and becomes insigniﬁcant if k 15. For all choices of k shown here, If converges to a value of 0.5 if pst ! 1, in line with the results obtained by SchifferD14X X et al. [4] for the case of a rigid water-backed plate.

6. Conclusions TagedPAnalytical predictions and dynamic FE calculations were performed to investigate the 1D underwater blast response of a waterbacked sandwich plate with an elastic core. The developed analytical models account for ﬂuidstructure interaction by explicitly modelling the propagation of breaking fronts and closing fronts, as well as the effect of an initially applied hydrostatic ﬂuid pressure, as encountered in deep water blast events. TagedPThe response of the sandwich plate was found to be controlled by three non-dimensional parameters, namely non-dimensional core stiffness k, face sheet mass c and initial static pressureD15X X pst . Three regimes of behaviour were identiﬁed based on the details of the cavitation process, and non-dimensional regime maps were constructed. In Regime 1, the response of the sandwich results in double cavitation, characterised by two subsequent cavitation events triggered in the water. In Regime 2, the ensuing cavitation process is similar to that of a rigid monolithic plate, with only one cavitation zone occurring in the water. On the other hand, it was found that cavitation may be fully suppressed if the initial static pressure is sufﬁciently high. The latter type of response is referred to as Regime 3. TagedPThe accuracy of the analytical models was probed by comparing their predictions to fully-coupled dynamic FE calculations and excellent agreement was reported for a wide range of input parameters. In addition, it was shown that the analytical model can also be used to predict the response of relatively thick sandwich plates whose response is affected by propagation of stress waves in the throughthickness direction.

TagedPThe developed analytical models provide useful guidelines for blast resistant design of sandwich plates as well as clear physical insight into the ﬂuidstructure interaction phenomena triggered by an elastic core response. However, the analytical calculations are based on linear elasticity and do not account for dissipative plastic deformation processes in the core, which severe blast loading typically entails. Therefore, more reﬁned elasto-plastic models are needed to accurately predict the blast response of sandwich plates under large deformations. This is left as a topic for future studies. Acknowledgements TagedPThe authors are grateful to EPSRC and Dstl for ﬁnancial support [Grant number EP/G042586/1]. References TagedP [1] ONR. Underwater explosion research: a compendium of British and American reports. Washington, DC, USA; 1950. TagedP [2] Taylor GI. The pressure and impulse of submarine explosion waves on plates. In: Batchelor GK, editor. The scientiﬁc papers of G.I. Taylor, Vol III. Cambridge, UK: Cambridge University Press; 1963. p. 287–303. TagedP [3] Kennard EH. Cavitation in an elastic liquid. Phys Rev 1943;63:172–81. TagedP [4] Schiffer A, Tagarielli VL, Petrinic N, Cocks AFC. The response of rigid plates to deep water blast: analytical models and ﬁnite element predictions. J Appl Mech 2012;79. TagedP [5] Liang Y, Spuskanyuk AV, Flores SE, Hayhurst DR, Hutchinson JW, McMeeking RM, et al. The response of metallic sandwich panels to water blast. J Appl Mech 2007;71:81–99. TagedP [6] McShane GJ, Deshpande VS, Fleck NA. The underwater blast resistance of metallic sandwich beams with prismatic lattice cores. J Appl Mech. 2007;74:352–64. TagedP [7] Qiu X, Deshpande VS, Fleck NA. Finite element analysis of the dynamic response of clamped sandwich beams subject to shock loading. Eur J Mech A 2003;22:801–14. TagedP [8] Xue Z, Hutchinson JW. A comparative study of impulse-resistant metal sandwich plates. Int J Impact Eng 2004;30:1283–305. TagedP [9] Fleck NA, Deshpande VS. The resistance of clamped sandwich beams to shock loading. J Appl Mech 2004;71:386–401. TagedP[10] Deshpande VS, Fleck NA. One-dimensional response of sandwich plates to underwater shock loading. J Mech Phys Solids 2005;53:2347–83. €kinen K. The transverse response of sandwich panels to an underwater shock TagedP[11] Ma wave. J Fluid Struct 1999;13:631–46. TagedP[12] Schiffer A, Tagarielli VL. The response of rigid plates to blast in deep water: ﬂuidstructure interaction experiments. Proc R Soc Lond A 2012;468:2807–28. TagedP[13] Schiffer A, Tagarielli VL. One-dimensional response of sandwich plates to underwater blast: ﬂuidstructure interaction experiments and simulations. Int J Impact Eng 2014;71:34–49. TagedP[14] Schiffer A, Tagarielli VL. The one-dimensional response of a water-ﬁlled double hull to underwater blast: experiments and simulations. Int J Impact Eng 2014;63:177–87. TagedP[15] Deshpande VS, Heaver A, Fleck NA. An underwater shock simulator. Proc R Soc Lond A 2006;462:1021–41. TagedP[16] McShane GJ, Deshpande VS, Fleck NA. Underwater blast response of free-standing sandwich plates with metallic lattice cores. Int J Impact Eng 2010;37:1138– 49. TagedP[17] Latourte F, Gregoire D, Zenkert D, Wei X, Espinosa HD. Failure mechanisms in composite panels subjected to underwater impulsive loads. J Mech Phys Solids 2011;59:1623–46.

A. Schiffer and V.L. Tagarielli / International Journal of Impact Engineering 102 (2017) 6273 TagedP[18] Wadley H, Dharmasena K, Chen Y, Dudt P, Knight D, Charette R, et al. Compressive response of multilayered pyramidal lattices during underwater shock loading. Int J Impact Eng 2008;35:1102–14. TagedP[19] Wei X, Tran P, de Vaucorbail A, Ramaswamy RB, Latourte F, Espinosa HD. Threedimensional numerical modeling of composite panels subjected to underwater blast. J Mech Phys Solids 2013;61:1319–36. TagedP[20] Avachat S, Zhou M. High-speed digital imaging and computational modeling of dynamic failure in composite structures subjected to underwater impulsive loads. Int J Impact Eng 2015;77:147–65. TagedP[21] Hutchinson JW, Xue Z. Metal sandwich plates optimized for pressure impulses. Int J Mech Sci 2005;47:545–69.

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TagedP[22] McMeeking RM, Spuskanyuk AV, He MY, Deshpande VS, Fleck NA, Evans AG. An analytical model for the response to water blast of unsupported metallic sandwich panels. Int J Solids Struct 2007;45:478–96. TagedP[23] Jin Z, Yin C, Chen Y, Hua H. One-dimensional analytical model for the response of elastic coatings to water blast. J Fluid Struct 2015;59:37–56. TagedP[24] Yin C, Jin Z, Chen Y, Hua H. One-dimensional response of single/double-layer cellular cladding to water blast. Int J Impact Eng 2016;88:125–38. TagedP[25] Cole RH. Underwater explosions. Princeton, NJ, USA: Princeton University Press; 1948. TagedP[26] Arezoo S, Tagarielli VL, Petrinic N, Reed JM. The mechanical response of Rohacell foams at different length scales. J Mater Sci 2011;46:6863–70.

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