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European Journal of Mechanics A/Solids 29 (2010) 56–67

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European Journal of Mechanics A/Solids

journal homepage: www.elsevier .com/locate/e jmsol

Dynamic compression of metallic sandwich structures during planar impulsiveloading in water

K.P. Dharmasena a,*, D.T. Queheillalt a, H.N.G. Wadley a, P. Dudt b, Y. Chen b, D. Knight b, A.G. Evans c,V.S. Deshpande d

a Department of Materials Science & Engineering, University of Virginia, 395 McCormick Road, Charlottesville, VA 22904, USAb Naval Surface Warfare Center, Carderock, MD 20817, USAc Department of Mechanical Engineering, University of California Santa Barbara, Santa Barbara, CA 93106, USAd Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK

a r t i c l e i n f o

Article history:Received 10 July 2008Accepted 7 May 2009Available online 21 May 2009

Keywords:Sandwich panelsUnderwater impulsive loadingFluid/structure interactionMetal foams, lattices and honeycombs

* Corresponding author.E-mail address: [email protected] (K.P. Dharma

0997-7538/$ – see front matter � 2009 Elsevier Masdoi:10.1016/j.euromechsol.2009.05.003

a b s t r a c t

The compressive response of rigidly supported stainless steel sandwich panels subject to a planarimpulsive load in water is investigated. Five core topologies that spanned a wide range of crush strengthsand strain-dependencies were investigated. They included a (i) square-honeycomb, (ii) triangularhoneycomb, (iii) multi-layer pyramidal truss, (iv) triangular corrugation and (v) diamond corrugation, allwith a core relative density of approximately 5%. Quasi-statically, the honeycombs had the highest peakstrength, but exhibited strong softening beyond the peak strength. The truss and corrugated cores hadsignificantly lower strength, but a post yield plateau that extended to beyond a plastic strain of 60%similar to metal foams. Dynamically, the transmitted pressures scale with the quasi-static strength. Thefinal transmitted momentum increased slowly with core strength (provided the cores were not fullycrushed). It is shown that the essential aspects of the dynamic response, such as the transmittedmomentum and the degree of core compression, are captured with surprising fidelity by modeling thecores as equivalent metal foams having plateau strengths represented by the quasi-static peak strength.The implication is that, despite considerable differences in core topology and dynamic deformationmodes, a simple foam-like model replicates the dynamic response of rigidly supported sandwich panelssubject to planar impulsive loads. It remains to ascertain whether such foam-like models capture morenuanced aspects of sandwich panel behavior when locally loaded in edge clamped configurations.

� 2009 Elsevier Masson SAS. All rights reserved.

1. Introduction

Metallic sandwich panels offer significant advantages overequivalent mass per unit area monolithic plates when exposed tounderwater impulsive loading (Fleck and Deshpande, 2004;Hutchinson and Xue, 2005; McShane et al., 2007). These benefitsarise from two main factors: (i) the enhanced bending strength ofsandwich panels over monolithic plates of equal mass per unit areaand (ii) reduced momentum transfer from the water to the sand-wich structure due to the low inertia of a thin (light), impulsivelyloaded face sheet supported by a crushable core. This studyinvestigates dynamic core compression and the associated fluid–structure interaction (FSI) occurring at the wet face/fluid interface.The assessment is conducted for three classes of cellular topology:honeycombs, prismatics and trusses (Wadley, 2006). These classesdiffer in the following sense. (a) Honeycombs offer high crush

sena).

son SAS. All rights reserved.

resistance and (when square or triangular) also have high in-planestretch strength, but are not amenable to additional functionalities,such as active cooling (Queheillalt et al., 2008; Tian et al., 2007). (b)Trusses have an open architecture suitable for multi-functionality,but have low in-plane strength. (c) Prismatics have intermediatecrush and stretch performance, and benefit from ease of manu-facture. When subjected to impulsive loads, the honeycombs aremost resistant to crushing and thus, transmit large stresses to theback face of the sandwich panel, with deleterious consequences forthe governing metrics: notably, back face deflection, reaction forcesat the supports and face tearing (Liang et al., 2007; Tilbrook et al.,2006; Wadley et al., 2007). The trusses and prismatics, which aremore susceptible to crushing, offer performance benefits in termsof impulsive load resistance (Liang et al., 2007; Tilbrook et al., 2006;Wadley et al., 2008; Dharmasena et al., 2009). However, a unifiedperspective on the role of core topology has yet to emerge.Attaining such a perspective is the primary objective of this article.

The crush dynamics and the associated FSI effects are investigatedusing a dynamic (‘‘Dyno’’) crush test facility (Wadley et al., 2007,

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K.P. Dharmasena et al. / European Journal of Mechanics A/Solids 29 (2010) 56–67 57

2008; Dharmasena et al., 2009; Knight et al., 2005) that employs anexplosive sheet to generate an underwater, planar impulse thatimpinges onto the test structure. In such tests, the impulse in thewater causes the pressure to rise to a peak, p0, almost instanta-neously. The pressure subsequently decreases at a nearly exponentialrate, with a time constant q of order milliseconds such that,

p ¼ p0expð�t=qÞ; (1)

where t is measured from the instant of impulse arrival. Dampedoscillations of the gas bubble containing the explosive productsleads to secondary impulses, but these generate smaller pressures,and are much less damaging.

When the impulse impinges onto a rigid plate at normal inci-dence, it imparts an impulse I0:

I0h2ZN0

p0e�t=qdt ¼ 2p0q: (2)

The factor of two arises due to full reflection of the pressure wave. Ifinstead, the pressure-wave impacts a free-standing plate of arealmass m, it sets the plate in motion, causing the reflected wave to

Fig. 1. Sketches of the (a) square-honeycomb and (b) triangular honeycomb sandwich coreincluded.

become tensile. The ensuing net pressure in the fluid drops tozero and cavitation sets in. The momentum per unit area Itrans

transmitted into the structure is then:

Itrans ¼ xI0 (3a)

where,

xhjj=ð1�jÞ (3b)

and jhrwcwq=m:

For a sandwich panel, these results are modified by the push-backstress exerted on the wet face by the core as it crushes. This effecthas been examined at two levels. (i) Simplified models that endowthe core with a fixed dynamic strength without explicitly modelingthe core topology or the crushing dynamics. One such modelindicates that the transmitted momentum becomes (Hutchinsonand Xue, 2005):

II0¼ jj=ð1�jÞ þ 0:63

�1� jj=ð1�jÞ

�sp

p0(4)

s. The unit cells of the core with all relevant core dimensions marked in mm are also

K.P. Dharmasena et al. / European Journal of Mechanics A/Solids 29 (2010) 56–6758

where sp is the core strength.(ii) Detailed finite element (FE) analysisof the response (Deshpande and Fleck, 2005; Wei et al., 2007; Moriet al., 2007). It has yet to be ascertained whether the simplifiedmodels suffice to capture the response or whether the details areimportant. An additional aim of this paper is to present a detailedexperimental/numerical investigation that addresses this issue.

The outline of the paper is as follows. The manufacture of thevarious cores is described, simple analytical formulae for theirstatic strength are presented and measurements reported. Thedyno-crusher apparatus is briefly described and the dynamicmeasurements detailed. Thereafter, a numerical model is devel-oped based upon the crushing of an equivalent mass and strengthfoam core, and the predictions compared with the measurements.These comparisons are used to draw conclusions regarding theexplicit roles of core topology and strength.

2. Sandwich panel manufacture and core properties

Circular sandwich panels with a diameter D¼ 203 mm and corethickness c z 90 mm, were manufactured from 304 stainless steel.

Fig. 2. Sketch illustrating the manufacturing route of the triangular honeycomb. (a) The 2 sand (c) the assembled triangular honeycomb core.

The panels had two identical stainless steel faces with thicknessh¼ 5 mm. Five different core topologies were produced (Figs.1, 3 and4) by vacuum brazing core/face assemblies using a Ni–Cr–P powder(Wall Colmonoy Nicrobraz 51 alloy) for 1 h at 1050 �C. Estimates of thecore strengths are presented below in terms of the solid material’sYoung’s modulus Es, Poisson’s ratio n and yield strength, sY.

Square-honeycombs (Fig. 1a) were manufactured from steelsheets, with a thickness t¼ 0.76 mm. The sheets were cropped intorectangles with height c¼ 89 mm and length 250 mm. Cross-slots(width Dt¼ 0.76 mm and spacing l¼ 31 mm) were laser cut intothe rectangles to a depth c/2 and assembled as detailed elsewhere(Wadley, 2006; Wadley et al., 2007; Cote et al., 2004). These coreshad a relative density rz0:05. The quasi-static peak strength sp ofsuch cores is elastic buckling governed at low relative densities andyielding dominated at higher densities such that (Cote et al., 2004):

sp ¼( p2

12�1� n2

�Esr3 r <1p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi123Y

�1� n2

�qrsY otherwise

(5)

hapes of slotted sheets employed, (b) the slotting together of the constituent elements


where the yield strain. 3Y hsY=Es and r is the relative density. Fora square-honeycomb (see unit-cell in Fig. 1a),

rz2tl

(6)

Triangular-honeycombs (Fig. 1b) were fabricated from the samesteel sheets, cropped into rectangles with height c¼ 89 mm andlength 250 mm. In one set of the rectangles (labeled shape #1 inFig. 2a) cross-slots having width Dt¼ 0.76 mm and spacingl¼ 51 mm were laser cut to a depth 2c/3, while in another, slots ofdepth c/3, spaced 51 mm apart, were cut from both ends (shape #2in Fig. 2a). The sheets were assembled (Fig. 2b) into equilateraltriangular cells of side l¼ 51 mm forming the core for the samplesused here (Fig. 2c). For the unit-cell shown in Fig. 1b,

rz2ffiffiffi3p

tl

(7)

These cores also have a relative density, rz0:05.Multi-layer pyramidal truss cores were manufactured by laser

cutting a diamond pattern into a flat sheet with a thickness

Fig. 3. Sketch of the multi-layer pyramidal truss sandwich core. The unit-cell of thecore with all relevant core dimensions marked in mm is also included.

t¼ 1.52 mm, leaving a series of intersecting square-section liga-ments each 1.52�1.52 mm in cross-section and length l¼ 17 mm.The perforated sheets were folded along ligament node rows togenerate pyramidal trusses having inclination, u¼ 45�, withrespect to the horizontal plane (Fig. 3). The core comprised 7pyramidal layers, each separated by solid sheets with thickness,ts¼ 0.76 mm. They were assembled to create the unit-cell geometryshown in Fig. 3 and brazed. The total relative density of a multi-layered pyramidal core (including the sheets separating the corelayers) is given by:

rz

ffiffiffi2pð4wt þ tslÞ

l�

lþffiffiffi2p

ts

� (8)

where l is the length of the truss and w� t is the truss cross-section.For the unit-cell shown in Fig. 3 rz0:09 with the contribution ofjust the trusses w5%. The compressive strength is again governedby elastic buckling at low r and by yielding otherwise. In thisconfiguration, to account for the constraints of the interveningsheets in an approximate manner, we regard the trusses as simplysupported at one end and clamped at the other. The strength thenbecomes

sp ¼( ffiffiffi

2p

p2

3

�tl

�3

Estl<

63Y

p2

2ffiffiffi2p �

tl

�2sY otherwise

(9)

Note that the intermediate sheets are assumed not to explicitlycontribute to the uniaxial compressive strength.

Multi-layer triangular corrugated cores were manufactured byfolding the t¼ 0.76 mm sheets into corrugations of sidel¼ 30.5 mm and angle u¼ 45� (Fig. 4a). For a single-layer corru-gation, the relative density in the limit l>> t is given by:

rz2t

lsin2u: (10)

Four such sheets were stacked into a 0–90� layup separated byintermediate sheets of thickness ts¼ 0.76 (Fig. 4a). This multi-layered core had a relative density, rz0:07 including the mass ofthe intermediate sheets. Estimates of the strength are againobtained by regarding the members as built-in at one end andsimply supported at the other, whereupon,

sp ¼(p2

6

�tl

�3

Estl<

1p

ffiffiffiffiffiffiffiffi63Y

p�

tl

�sY otherwise

(11)

Diamond corrugated cores were manufactured in a similarmanner but with a different stacking sequence (Fig. 4b). Twocorrugations were stacked to form diamond-shaped cells. Thediamonds were then stacked in a 0–90� layup separated by anintermediate sheet of thickness ts¼ 0.76 mm as shown in Fig. 4b.Similar to the triangular corrugation, the diamond corrugation alsohad cells of side l¼ 30.5 mm and made from a t¼ 0.76 mm steelsheet with u¼ 45�. The relative density of the core including theintermediate sheets is rz0:06. In this configuration, memberrotation is less constrained such that they are simply supported atboth ends giving the quasi-static strength:

sp ¼( p2

12

�tl

�3

Estl<

1p

ffiffiffiffiffiffiffiffiffiffiffi123Y

p�

tl

�sY otherwise

(12)


3. Quasi-static measurements

The quasi-static compressive responses were measured ina screw driven test machine at a nominal strain rate of 5�10�4 s�1

(Chen and Dudt, 2005). The stress was ascertained from the loadcell, while the average compressive strain was deduced from laserinterferometer measurements of the relative approach of the twofaces. The measured responses are plotted on Fig. 5. The honey-combs (Fig. 5a) display a high initial strength sp followed bya strongly softening response prior to densification as the strainexceeded 60%. Recalling from (5) that cores with rz0:05 resideclose to the transition between elastic buckling and yielding. Weattribute the peak to elastic buckling and the softening to post-buckling plasticity. The reduced strength of the triangular-honey-combs is attributed to manufacturing defects introduced where thethree separate sheets intersect at the nodes. The multi-layer cores(Fig. 5b) crush at almost constant stress with small fluctuations assuccessive layers collapse, reminiscent of the response of metalfoams. The increase in stress at a nominal strain around 60% iscaused by contact of the core members (i.e. densification).Comparisons between the measurements and the estimates(Section 2) are included in Table 1, where we have taken theproperties of 304 stainless steel in its as-brazed state as

Fig. 4. Sketches of the (a) triangular corrugation and (b) diamond corrugation sandwich coincluded.

Es¼ 210 GPa, n¼ 0.3 and sY¼ 220 MPa (Wadley et al., 2007, 2008;Dharmasena et al., 2009; Cote et al., 2006). Reasonableagreement is obtained in all cases: albeit that the predictions(which do not account for imperfections) slightly overestimate themeasurements.

4. Dynamic measurements

4.1. Apparatus

The samples were rigidly back supported by a set of strongpedestals to which strain gages were attached for transmittedpressure measurements. A cardboard cylinder was then placedaround the samples and this was filled with water. Some essentialinformation about the experimental set-up is provided (Fig. 6),with details given elsewhere (Wadley et al., 2007, 2008; Dharma-sena et al., 2009; Knight et al., 2005). The base of the cardboardcylinder containing the water comprises a thick steel plate witha central hole. The circular test panel was aligned such that the wetface was flush with the annular plate. The dry face was supportedon a specimen tray, mass 2.5 kg, and four 2 cm long HY-100steel columns each 3.8 cm in diameter. The four columns wereinstrumented with axial strain gauges to provide temporal

res. The unit cells of the core with all relevant core dimensions marked in mm are also

Fig. 5. Measured quasi-static compressive nominal stress versus nominal strainresponses of the (a) square and triangular honeycomb cores and (b) the prismaticcorrugation and pyramidal truss cores.


measurements of the loads transmitted through the core. Theplanar impulses were generated by detonation of an explosivesheet (Detasheet), diameter 203 mm and thickness 1 mm, placed ata stand-off, H¼ 0.1 m from the wet face.

The temporal dependence of the free field pressure p has beencalculated using the DYSMAS hydrocode (Wei et al., 2007). It isplotted in Fig. 7, where t¼ 0 is the time at which the pressure pulsereaches the front face of the test panel (at a stand-off, H¼ 0.1 m).This pressure is well-approximated by relation (1), with peakpressure p0¼ 260 MPa and decay constant q¼ 0.023 ms as illus-trated in Fig. 7. The impulse per unit area for such a wave incidenton a stationary rigid plate is given from Eq. (2) as, I0 z 12 kPa s.

4.2. Calibration

The fidelity of the apparatus was assessed by first conductinga test using a 6061-T6 solid aluminum cylinder with diameter,

Table 1Comparison between the measured values and predictions of the initial peak quasi-statstainless steel were taken as Es¼ 210 GPa, n¼ 0.3 and sY¼ 220 MPa.

Core topology Measured peak stre

Square-honeycomb 11.5Triangular honeycomb 8.4Multi-layer pyramidal truss 3.9Triangular corrugation 3.4Diamond corrugation 2.4

D¼ 203 mm and height 100 mm. The temporal variation of thetransmitted pressure pt (over the area A h pD2/4 of the cylinder) ofthe sandwich panel and the corresponding areal impulse,

I ¼Zt

0

ptdt (13)

are plotted in Fig. 8a and b, respectively. These figures identify threemetrics: the peak pressure, pmax as well as the time, tss, before theimpulse attains steady-state, Iss. Note that the peak pressure isconsiderably below the value, 2p0¼ 520 MPa, expected from anacoustic analysis of impulse reflection at a rigid structure – waterinterface (2). However, the steady-state transmitted impulse,Iss z 11.5 kPa s, is consistent with the acoustic prediction (2). Thishappens because of signal dispersion and wave reflection withinthe aluminum block and measurement columns reduces themeasured pressures as discussed in (Wei et al., 2007) but does notsignificantly affect the measured transmitted momentum.

4.3. Crush response of sandwich panels

The temporal variation of the transmitted pressure for the fivecores is reported in Fig. 9. While there is some ambiguity regardingthe pressure histories (because of wave dispersion and reflection),the following sequence in peak pressure (from highest to lowest) isclear: honeycomb> truss> triangular corrugation. The diamondcorrugation, which initially transmits a low pressure, undergoessufficient crushing that core fully crushes permitting transmissionof large stresses. This effect is analogous to the case of an air coresandwich, in which the wet face ‘‘slaps’’ into the dry face, resultingin the sudden increase in the pressure at tslap z 1.5 ms as indicatedin Fig. 9e. These dynamic strengths rank in the same order as thequasi-static strengths (Fig. 5).

The impulse waveforms are shown in Fig. 10. The steady-stateimpulse Iss (Fig.10) varies only slightly with core topology. In order toprovide meaningful comparisons with the ensuing simulations, therelationships between the core strength (Table 1) and (i) the corecrushing strain, 3f

chDc=c (where Dc is the final reduction in the corethickness), (ii) the time tss and (iii) the steady-state impulse, Iss, areplotted in Fig.11 (the time tss is defined as the time corresponding tothe knee prior to the plateau in the I versus time curves and Iss asaverage value of I over the range tss�t� 2 ms as indicated in Fig.10).Note that, while 3f

c increases with decreasing sp, the time tss hasa maximum at sp z 4 MPa just prior to the onset of face-sheet slap.Moreover, the measurements plotted in Fig.11c suggest a small dropin Iss/I0 as sp decreases. This indicates that the momentum trans-ferred to sandwich structures supported by a rigid foundation isa weak function of core strength over the practical ranges of corestrengths considered here. The transmitted momentum as predictedby Eq. (4) (with m interpreted as the areal mass of the wet face sheet)is included in Fig. 11c. The simplified model significantly under-predicts the transmitted momentum due to the fact that it does notaccurately account for the re-loading of the sandwich panel afterfirst cavitation in the water column. We note that over this practical

ic strengths of the five core topologies investigated in this study. The properties of

ngth (MPa) Analytical prediction sp (MPa)

10.810.8

4.94.62.3

Fig. 6. Sketch of the dyno-crusher apparatus with key dimensions marked. The inset shows a detailed view of the specimen and four strain gauged columns used to measure theblast loads transmitted through by the sandwich panel.


range of cores strengths, Iss/I0 is about 25% less than the impulsetransmitted into a rigid stationary structure. This result contrastswith the results obtained for free-standing sandwich panels asdiscussed in Section 1 (Fleck and Deshpande, 2004; Hutchinson andXue, 2005; McShane et al., 2007).

To reveal the final deformations, the panels were sectionedalong two perpendicular diametral planes (Fig. 12). For thehoneycombs, the majority of the deformation occurs adjacent tothe wet face while the deformation is more diffuse for the othercores. We surmise that their deformation started adjacent to the

Fig. 7. A DYSMAS prediction of the free field pressure p versus time t history generatedby the denotation of the explosive sheet. The prediction is shown for a point located ata stand-off of H¼ 0.1 m from the explosive sheet and t¼ 0 is defined as the time atwhich the pressure pulse reaches the measurement location. For comparison, theexponential pressure pulse given by Eq. (1) with the choices p0¼ 260 MPa andq¼ 0.023 ms is also included.


wet face and spread through the core. Recall that, for the diamondcores, a sudden increase in the transmitted pressure andmomentum were measured at t z 1.5 ms, attributed to ‘‘slapping’’.This slap mechanism is evident in Fig. 12e where the sheetsbetween adjacent layers of the core are in contact with each otherand the wet face.

Fig. 8. The measured temporal variations of the transmitted (a) pressure s and (b)impulse I for blast loading of a reference solid Al cylinder. Time t¼ 0 is defined as thetime when the blast wave impinges on the wet face of the cylinder.

5. Simulation

A simplified one-dimensional finite element (FE) model ispresented (Fig. 13). It comprises a sandwich panel and a fluidcolumn of height Hw. In order to rationalize and unify theforegoing responses, the crushing dynamics of the cores aresimulated using a simplified (foam) model wherein the quasi-static response is represented by a plateau collapse stress,equated to sp (Fig. 5 and Table 1) and a (logarithmic) densifi-cation strain is taken as 3D ¼ �lnð2rÞ. The sandwich panel hasidentical faces, thickness h, and a core thickness, c. The facesare elastic with Young’s modulus Ef¼ 210 GPa and densityrf¼ 8000 kg m�3, representative of steel. The main simplifica-tion is that the core is modeled as a compressible foam (in a 1Dsetting) with a density rc. Such a model accounts for the shockwave propagation in the compressible sandwich cores (and thusthe differences between the stresses exerted by the core on thewet and dry faces of the sandwich panel). However, the modeldoes not model the enhanced dynamic strengths of the coredue to micro-inertial stabilization of the struts of the coreagainst buckling. The subsequent comparisons with measure-ments shall demonstrate that including this effect is not criticalin modeling the compression of sandwich panels in the Dyno-crusher set-up.

The foam model is analyzed as follows. The total logarithmicstrain rate _3 is written as the sum of an elastic strain rate _3e anda plastic strain rate _3p, resulting in a stress level s.

_3 ¼ _3e þ _3psignðsÞ (14)

The elastic strain rate is related to the stress rate by:

_3e ¼_s

Ec(15)

where Ec is the Young’s modulus of the core. An overstressvisco-plastic model (Radford et al., 2005) governs the plasticstrain rate,

_3p ¼(�

jsj�sp

h

�; if jsj � sp and 3p < 3D

0 otherwise;(16)

where the viscosity h is chosen to give a shock width (Radford et al.,2005):

ls ¼h3D

rcDv¼ c

10(17)

where Dv is the velocity jump across the shock. With ls¼ c/10 weensure that the viscosity h plays no role in the overall structural

Fig. 9. The measured temporal variations of the transmitted pressure pt for the fivesandwich cores investigated here. (a) square-honeycomb, (b) triangular honeycomb,(c) multi-layer pyramidal truss, (d) triangular corrugation and (e) diamond corruga-tion. Time t¼ 0 is defined as the time when the blast wave impinges on the wet face ofthe sandwich panels.

Fig. 10. The measured temporal variations of the transmitted impulse I for the fivesandwich cores investigated here. (a) square-honeycomb, (b) triangular honeycomb,(c) multi-layer pyramidal truss, (d) triangular corrugation and (e) diamond corruga-tion. Time t¼ 0 is defined as the time when the blast wave impinges on the wet face ofthe sandwich panels.


response other than regularizing the numerical problem to obtaina shock of finite width. We employ the estimate Dv¼ p0q/(rfh)based upon a free-standing front face, with fluid–structure inter-action effects neglected. Because large gradients in stress and strainoccur over the shock width a mesh size c/10 is chosen to resolvethese gradients accurately. Calculations are presented for strengthsin the range 1 MPa� s p� 15 MPa and two relative densitiesr ¼ 0:05 and 0.1 corresponding to rchrrf ¼ 400 kg m�3 and800 kg m�3 (i.e. cellular cores constructed from steel withrf¼ 8000 kg m�3). The Young’s modulus of the core material istaken to be Ec ¼ rEf (the response is insensitive to this choice).

The water is modeled as a one-dimensional column of height Hw

(Fig. 13) and thus we neglect the bursting of the cardboard cylindercontaining the water. This is rationalized by noting that the entireevent comprising the transmission of the blast momentum andcore crushing is completed well before the constraint on the watercolumn is lost. Following Bleich and Sandler (1970), the fluidmedium is modeled as a bilinear elastic solid with density

rw¼ 1000 kg m�3 and modulus Ew¼ 1.96 GPa, giving a wave speedcw ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiEw=rw

p¼ 1400 m s�1, representative of water. The stress

s, logarithmic strain 3 relationship is taken to be:

s ¼

Ew3 3 � 00 otherwise

(18)

so that the water is incapable of sustaining a tensile stress.A pressure history,

p ¼ p0e�t0=q (19)

is applied to the top of the fluid column (Fig. 13), with time t0

measured from the instant of application of the pressure. The time tis measured from the instant that the impulse impinges on thestructure. The two are related by t

0¼ tþ (Hw/cw). The height Hw of

the water column is taken to be sufficiently large that the reflectedwave does not reach the top over the duration of the calculations.Namely, the column can be considered semi-infinite. All calcula-tions take p0¼ 260 MPa and q¼ 0.023 ms consistent with theimpulses generated in the experiments.


The calculations were conducted using an updated Lagrangianscheme with the current configuration at time t serving as thereference. The coordinate x denotes the position of a material pointin the current configuration with respect to a fixed Cartesian frame,and u is the displacement of that material point. The principle ofvirtual work (neglecting effects of gravity) for a volume V andsurface S is written in the form:

Fig. 11. The measured (a) final core compression 3fc , (b) time tss corresponding to the

knee prior to the plateau in the impulse versus time curves and (c) the steady-statetransmitted impulse Iss for the five cores tested here. The measured values are plottedagainst the analytical estimates of their respective static core strengths sp from Table 1.Also included are FE predictions for two choices of the core relative densitiesr ¼ 0:05 and 0:1 and the analytical prediction of the transmitted momentum, Eq. (4).

Zsd3dV ¼

ZTdu dS�

Zrv2u

2du dV (20)

V S Vvt

where s is the Cauchy stress, 3hvu=vx is the strain, T is the tractionon the surface S of the current configuration and r is the density ofthe material in the current configuration.

A discretization based on linear, one-dimensional elements isemployed. When this discretization is substituted into the principleof virtual work (20) and the integrations carried out, the discretizedequations of motion are obtained as:

Mv2Uvt2 ¼ R (21)

where U is the vector of nodal displacements, M is the massmatrix and R is the nodal force vector. A lumped mass matrixis used in (21) instead of a consistent mass matrix since this ispreferable for both accuracy and computational efficiency(Krieg and Key, 1973). An explicit time integration scheme,based on the Newmark b-method with b¼ 0, is used to inte-grate equation (21) to obtain the nodal velocities anddisplacements. The sandwich panel geometry in the simula-tions was chosen to match the experiments, i.e. a face-sheetthickness h¼ 5 mm and a core depth c¼ 90 mm. Typically,there are 40,000 one-dimensional elements in the fluid, 2000in the core and 200 in each face sheet. In order to simulate thesupporting structures, the constraint u� 0 was imposed on theoutside of the dry sheet, i.e. the dry face could lose contactwith the support but not move beyond x¼ 0 as sketched inFig. 13.

5.1. Predictions and comparisons with measurements

The predicted temporal variations of the wet face velocity vw forthe r ¼ 0:1 sandwich panels are plotted in Fig. 14a for selectedvalues of the core strength, sp. The velocity rises rapidly and attainsa peak independent of core strength. Thereafter, for an extendedperiod, the wet face decelerates at a rate that increases withincreasing sp. But for the lower sp (sp� 2 MPa) the core undergoesfull densification, resulting in a sudden deceleration, at t z 1.2 ms,followed by elastic vibrations. The temporal variations of the corecompression strains,

3cðtÞh1c

Zt

0

vwdt (22)

and of the impulse I transferred to the rigid supports are plottedin Fig. 14b and c, respectively. The key observations are asfollows: (i) The final core compression 3f

c increases withdecreasing sp, with full densification 3f

c ¼ 1� 2r attained whensp� 2 MPa; (ii) Prior to densification (or arrest of the wet face),the rate of impulse transfer _Izsp; (iii) The final impulse trans-ferred to the supports Iss decreases slowly with core strength forsp> 3 MPa, but rises for lower strengths due to slapping; (iv) Thetime tss at which the steady-state impulse is attained coincideswith the end of core compression (and the time when the wetface comes to rest).

The predicted variations of 3fc, tss and Iss with sp are

superimposed on Fig. 11. Note that there is almost no affect ofthe core relative density between 0.05 and 0.1. The generalfeatures are consistent with the measurements, but with somediscrepancies. In particular, the increase in the transmittedmomentum predicted when sp < 3 MPa is not evident in themeasurements. Nevertheless, given the simplicity of the

Fig. 12. Photographs of the dynamically tested sandwich panels. The panels are sectioned along two orthogonal diametral planes. (a) square-honeycomb, (b) triangular honeycomb,(c) multi-layer pyramidal truss, (d) triangular corrugation and (e) diamond corrugation.


model, it captures the essence of dynamic crushing remark-ably well, even for honeycombs that have extreme strain-dependence of their crushing strength. This is contrasted withthe prediction of the simplified analytical model, Eq. (4) which

Fig. 13. Sketch of the one-dimensional boundary value problem analyzed to investi-gate the response of the sandwich panels in the ‘‘dynocrusher’’ tests.

is non-conservative, and significantly under-predicts thetransmitted momentum.

6. Concluding remarks

The response of rigidly supported sandwich panels to a planarwater borne impulse is investigated using a dyno-crusher appa-ratus. Honeycomb, truss and prismatic core topologies have beenstudied in an attempt to understand the effect of topology. All of thecores had the same overall dimensions and a relative density ofapproximately 5%. The measured quasi-static compressiveresponses revealed that, while the honeycombs had high initialstrength, their post peak response was strongly softening. The trussand corrugated cores had a lower strength, with a long plateau,similar to metal foams. The dyno-crusher measurements revealthat while the transmitted pressures and core compression scalewith the static core strength, the final transmitted momentum isweakly affected. The use of a crushable core in a fully back sup-ported test configuration reduces the transmitted impulse by about25% compared to that transmitted through a rigid, fully supportedblock.

Simulations conducted by modeling the cores as a foam capturethe major experimental trends in the transmitted momentum andcore compression, implying that the static core strength is theprimary variable governing dynamic crushing. It has yet to beascertained whether such foam-like models capture more nuancedaspects of sandwich panel behavior in clamped configurations suchas the soft/strong transition, face tearing and reaction forces.

Fig. 14. FE predictions of the temporal variations of (a) the wet face velocity vw, (b)core compression 3c and (c) areal impulse I transferred to the rigid supports forselected values of the core strength sp of the r ¼ 0:1 sandwich cores.


Acknowledgements

This research was supported by the Office of Naval Researchgrant number N00014-03-1-0281 on Blast and Fragmentation

Protection Sandwich Panel Concepts for Stainless Steel Mono-hull designs monitored by Drs. Edward Johnson and DanielTam.

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