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International Journal of Impact Engineering 38 (2011) 837e848

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International Journal of Impact Engineering

journal homepage: www.elsevier .com/locate/ i j impeng

Wet-sand impulse loading of metallic plates and corrugated core sandwich panels

J.J. Rimoli a, B. Talamini a, J.J. Wetzel b, K.P. Dharmasena b, R. Radovitzky a, H.N.G. Wadley b,*

aDepartment of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139, USAbDepartment of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22904, USA

a r t i c l e i n f o

Article history:Received 11 November 2010Received in revised form17 May 2011Accepted 20 May 2011Available online 30 May 2011

Keywords:Sandwich panelsBlast loadingImpulse mitigation

* Corresponding author.E-mail address: haydn@virginia.edu (H.N.G. Wadle

0734-743X/$ e see front matter � 2011 Elsevier Ltd.doi:10.1016/j.ijimpeng.2011.05.010

a b s t r a c t

We have utilized a combination of experimental and modeling methods to investigate the mechanicalresponse of edge-clamped sandwich panels subject to the impact of explosively driven wet sand.A porthole extrusion process followed by friction stir welding was utilized to fabricate 6061-T6aluminum sandwich panels with corrugated cores. The panels were edge clamped and subjected tolocalized high intensity dynamic loading by the detonation of spherical explosive charges encased bya concentric shell of wet sand placed at different standoff distances. Monolithic plates of the same alloyand mass per unit area were also tested in an identical manner and found to suffer 15e20% largerpermanent deflections. A decoupled wet sand loading model was developed and incorporated intoa parallel finite-element simulation capability. The loading model was calibrated to one of the experi-ments. The model predictions for the remaining tests were found to be in close agreement withexperimental observations for both sandwich panels and monolithic plates. The simulation tool was thenutilized to explore sandwich panel designs with improved performance. It was found that the perfor-mance of the sandwich panel to wet sand blast loading can be varied by redistributing the mass amongthe core webs and the face sheets. Sandwich panel designs that suffer 30% smaller deflections thanequivalent solid plates have been identified.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Metallic sandwich panel structures with cellular cores madefrom ductile, ferritic, and in some cases corrosion resistant stainlesssteels have attracted significant interest for protecting structuresfrom impulsive loading. The performance and advantages ofsandwich panel construction have been thoroughly studied in thecase of water blast, and to a lesser extent for air blast. In each ofthese two cases the sandwich structure performance is controlledby the design of the panels (i.e., core topology and geometry) [1,2],the mechanical properties and density of the material [3], theintegrity of the joints [4], and the nature of the impulse transferfrom the surrounding medium to the structure [5e9]. There hasbeen relatively less literature on impulsive loading with soil as themedium.

In the case of water and air blast, efforts have focused onexploiting the so-called fluid structure interaction (FSI) effect toreduce the impulse imparted to the structure. For explosive loadingin water and air (in the acoustic weak shock limit), the impulsetransferred to a massive plate is twice the incident impulse due to

y).

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complete shock reflection. However, when low inertia (thin andlight) plates are impacted by these shocks, the reflection factor canbe decreased and the impulse acquired by the structure is then lessthan that of the high inertia system [5,3]. As deduced in Taylor’sseminal work [5], the ratio of transferred to incident impulse, I/I0, iscontrolled by a dimensionless parameter b ¼ rwcwt0/mf, where rwand cw are the density and sound speed of the fluid respectively, t0is the characteristic decay time for the pulse, andmf is the mass perunit area of the front face sheet. The acoustic limit for the impulsetransfer ratio derived by Taylor is given by I/I0 ¼ 2bb/(1eb). In water,sandwich panelsmay exploit the FSI effect by using light face sheetssupported by soft cellular structure cores [3]. Reductions of impulseas high as 25% have been experimentally reported for steel facesheets that are several millimeters thick supported by cores witha compressive strength of 5e10 MPa [10]. Improvements to Taylor’sexpression have been proposed by Hutchinson et al. [11] to accountfor the resistance to the motion of the front face sheet offered bythe core in sandwich panels. The level of impulse reductiondepends upon the time and location of cavitation after the shockreflection, which in turn is affected by panel design [12].

Panel designs that seek to exploit the FSI effect in water haveutilized a variety of core topologies and geometries including:Y-truss [2], corrugated [13], triangular honeycomb [14], squarehoneycomb [15], and lattice truss cores [16], among others. These

1 The choice of studying wet sand over dry sand is motivated by the followingconclusions of Deshpande et al. [20]: (i) wet sand blast applies higher averagepressures and impulses to the structure, hence restricting attention to this case isconservative, and (ii) wet sand encapsulates the expanding explosive gases andtherefore avoids the complication of considering separate air and soil blasts.

J.J. Rimoli et al. / International Journal of Impact Engineering 38 (2011) 837e848838

concepts use a tough outer face sheet that is impacted by thewaterborne shock and a relatively low-strength core. The impulsiveloading causes the outer face sheet to accelerate and to crush thecore. At larger deflections, additional resisting forces are activated bymembrane stretching of the front face sheet, and for severe impul-sive loading, by stretching of the back face sheet and core [17,18].

The FSI effect for air blasts has been studied theoretically byKambouchev et al. [8,9,19]. They extended Taylor’s analysis to thenonlinear regime where the compressibility of the fluid is pressuredependent [8]. Their approach allowed a decoupling of the struc-tural loading from the fluids dynamics. It was found that nonlinearcompressibility effects further decrease the transmitted impulse incomparison to the acoustic case. However, the effect is only presentfor very high intensity loading and very low inertia face sheets. Inpractice, the requirement of high intensity and light face sheetsmay limit the practicality of exploiting FSI effects in the design ofair blast-mitigating systems [7,19]. Even so, it is important torecognize that sandwich panels have still demonstrated improvedperformance over equivalent mass per unit area solid plates bytheir increased plastic bending resistance [18].

If an explosive charge is buried in soil, the detonated explosivecreates a shock that propagates through the soil [20]. The shockfront is reflected at the soileair interface causing soil spallation. Themajority of impulse that loads a structure arrives by impact of soilparticles [20]. The success of sandwich panels in blast protectionapplications has naturally extended interest to their use for soilblast. However, engineering design tools for computing the struc-tural loading due to the impacting soil are not well developed.

Soil blasts can be numerically modeled by explicitly consideringthe detonation process, the soil response, and its interaction withthe structure and surrounding air. In general, creating constitutivemodels for the soil response is difficult due towide variations in soilparticle properties and level of water saturation [20,21]. Soilconstitutive models incorporating some of these effects have beenproposed [20e23] and used with coupled codes, e.g. [24e26]. Thismethod has the drawback that it is computationally expensive,requiring significant effort to compute the fluid fields even whenthe primary interest is the structural response. Recent experi-mental and numerical studies [27], indicate that the FSI effectduring soil blast is small (and of secondary importance to thedesign of protective structures), suggesting that decoupled soil andstructural simulation may be a viable and efficient panel designoption.

An alternative to coupled simulation for determining the soilblast loading is to use empirical models, such as that ofWestine el al[28]. This model provides a (spatially varying) specific impulsegiven the problem geometry and the size of the charge. Load isapplied as an initial velocity presuming that the structural responsetime is long compared to impulsive loading from the soil. Theapproach is simple, although its validity is restricted to the range ofthe original experimental conditions. Furthermore, whencompared to other loading models that incorporate a pressure timehistory, the initial velocity approach can significantly overestimatethe core crushing of sandwich panels [7].

The objectives of the present study are to (i) understand themechanical response of edge-clamped sandwich panels subject tothe impact of explosively driven wet sand, (ii) develop a modelingapproach that enables the prediction of such response as a functionof the impulse spatio-temporal distribution, the geometry of thepanels and theirmaterial properties, and (iii) explore amodel-basedoptimization of the panel design. To address these objectives,a combined experimental and modeling approach was adopted.

For the experimental section of the study, a set of 6061-T6aluminum sandwich panels with corrugated cores were fabricated.The panels were subsequently edge clamped and subjected to

localized high intensity dynamic loading by explosively acceleratedwet sand.1 Spherical test charges were detonated at differentstandoff distances from the target panels, and a post-mortemanalysis of the specimens was performed and their permanentdeformation measured. Monolithic plates with same mass per unitarea and made from the same alloy were also tested in an identicalmanner.

A decoupled loading model was developed and incorporatedinto a parallel finite-element simulation capability to analyze theexperiments. The loadingmodel was calibrated tomatch the resultsof one experiment of the test matrix, and the values obtainedthrough the calibration procedure were then adopted for allsubsequent calculations. The simulation tool was then utilized toexplore alternative sandwich panel designs.

The remainder of this article is organized as follows: In Section 2we describe the experimental test setup. In Section 3 we presentthe computational model. In Section 4 experimental results arepresented and compared to simulation results as ameans for modelassessment and experimental interpretation. The model is thenused to investigate the panel design space and potential improve-ments to the panel design are presented. It is shown that significantpanel deflection reductions can be achieved by redistributing panelmass from the core to the back face sheet.

2. Experimental procedures

Experiments were conducted on a simplified system to under-stand the structural response of metallic sandwich panels to wetsand blast (in particular as a function of blast standoff). Theexperimental design, similar to that of Deshpande et al. [20], aimsto produce a controlled loading environment that reproduces therelevant physical interactions. We note that the experimentaldesign is not intended to reproduce the conditions of any particularexplosive device or scenario.

The experiments were conducted on aluminum alloy sandwichpanels with a triangular corrugated and I-core and 1.7 cm thicksolid plates of the same alloy and areal density (mass per unitarea). Spherical test charges were placed at different standoffdistances from the center of the target panels and plates anddetonated to evaluate the effects of the impulsive loading on thestructures. After testing, the plates and panels were water-jetsectioned and their permanent deflections and deformationmodes were evaluated.

A detailed explanation of the panel fabrication processes, thepreparation of the test charge, the panel testing configuration, andthe test matrix adopted for this work is given in the subsectionsbelow.

2.1. Panel fabrication

The sandwich panels were fabricated from a 6061 aluminumalloy using a combination of porthole die extrusion and frictionstir welding. The panels were extruded by Mid East Aluminum(Mountaintop, PA) into 0.143 m wide, w 3 m long stick extrusionswith the corrugated structure shown in Fig. 1, using a 17.8 cmdiameter, 300 ton direct extrusion press operating at 482

�C. The

panels were heat-treated to the T6 (peak strength) condition afterextrusion. The core of the extruded structures had a web

Fig. 1. (a) The aluminum extrusion process. (b) The extruded aluminum panel.

J.J. Rimoli et al. / International Journal of Impact Engineering 38 (2011) 837e848 839

inclination angle of 60�, an inclined web thickness of 3.2 mm,a core height of 19.3 mm, and 5.2 mm thick top and bottom facesheets and (Fig. 1). The relative density of the corrugated core was25.2%. The core’s areal density was 13 kg/m2 while that of each ofthe faces was 14 kg/m2. The edges of the extrusions were termi-nated with vertical solid webs to facilitate the fabrication of widersquare panels. These vertical webs were machined to a thicknessof 4.75 mm.

To create panels of the requiredwidth for testing, five extrusionswere friction stir welded side-by-side at Friction Stir Link, Inc.(Waukesha, WI). The process was performed using a 17 mmdiameter welding tool turning at a speed of 1000 RPM as sche-matically illustrated in Fig. 2. The 4.75 mmwide vertical trusses atthe two sides of the extrusion (Fig.1) were retained tominimize thedistortion of the structure due to the high pressures generatedduring the stir welding process. The vertical trusses also shieldedthe welds from some stress during subsequent testing and

compensated for the local decrease in mechanical propertiesassociated with the welding process.

The resulting test panels were 0.632 m wide and 0.610 m long(Fig. 2). The relative density of the hybrid core including the verticalwebs was 29% and the areal density of the entire sandwich panelwas 46 kg/m2 (equivalent to a solid 6061-T6 aluminum panel of1.7 cm thickness).

Since the sandwich panel face sheets were vulnerable to pull-out at the bolts used to edge clamp the panels during testing,four 10.6 cm wide strips along the panel edges were filled witha Crosslink Epoxy (CLR1061 resin/CLH6930 hardener) to create anouter strengthened panel border.

2.2. Charge preparation

Model test charges were made by packing nominally 200 mmdiameter glass microspheres (Mo-Sci Corp, Rolla, MO) in a 152 mm

Fig. 2. (a) Joining of extruded panels by Friction Stir Welding (b) The test panel.

J.J. Rimoli et al. / International Journal of Impact Engineering 38 (2011) 837e848840

diameter, thin-walled polyester glycol plastic sphere, concentricwithan internal 80mmdiameter, thin-walled acrylic shellfilledwith375gofC-4 explosive. The test charge assembly sequence is shown inFig. 3.

First, a measured mass of explosive was tightly packed insidethe 80 mm diameter plastic sphere. A thin wooden rod was theninserted through the explosive sphere to facilitate subsequentsuspension of the charge. To enable later insertion of a detonator,a thin-walled polymer tube was then inserted into a hole cut in thenorth pole of the plastic sphere and hot glued to the surface(Fig. 3a). This 80 mm plastic sphere was then centered insidea hemispherical, 152 mm diameter plastic shell. The hemisphericalshell was partially filled with 200 mm diameter glass microspheres(artificial “sand”), periodically vibrating the hemisphere so that themicrospheres were tightly packed. The top hemispherical half ofthe outer sphere shell was then positioned on the bottom hemi-sphere, and the seam joint sealed with hot glue (Fig. 3b). Theinternal volume of the outer sphere was then completely filledwith more glass microspheres (Fig. 3c) and the weight of sandmeasured. The mass of sand used in the charges was 2.466 kg.Finally, water was poured into the larger sphere to fill the voidspace between the glass microspheres (Fig. 3d). The mass of wateradded to the charges was 0.617 kg and the total mass of the “watersaturated sand” was 3.083 kg.

2.3. Panel testing configuration

The impulsive loading experiments were conducted at theForce Protection Inc. Explosives Test range located near Edgefield,SC using a test geometry shown schematically in Fig. 4. Approx-imately square (61 cm � 61 cm) test panels were mounted hor-izontally to a 3.8 cm thick steel plate bolted to a welded I-beamframe resting on a concrete support base. A 19.5 mm diameter,32-hole pattern in the steel plate allowed the test panels to besecurely edge clamped in the test configuration. The 3.8 cm thicksteel plate contained a 41 cm � 41 cm square center opening toenable the unrestricted deformation of the test panel whensubjected to a centrally applied impulse load. The sand-encasedspherical charges were suspended above the center of the testpanels at varying standoff distances, defined as the distance fromthe center of the charge to the impact surface of the test panels,Fig. 5.

2.4. Test matrix

A series of 10 experiments were conducted. Five of these wereperformed at standoff distances of 15, 19, 22, 25 and 30 cm(measured from the front face) with an equivalent weight

Fig. 3. Spherical “soil” encapsulated explosive charge preparation.

J.J. Rimoli et al. / International Journal of Impact Engineering 38 (2011) 837e848 841

monolithic 6061-T6 plate of thickness 1.7 cm, Fig. 5a, and theremaining five by replacing the solid plates with five corrugatedcore sandwich panels, at the same standoff distances, Fig. 5b. Thepanels were water-jet sectioned after testing and the paneldeflections and deformation modes were evaluated.

3. Computational modeling

We conducted computer simulations of the plates and sandwichpanels subjected to impulsive wet sand loading (i) to study detailsof the experiments that are difficult to measure; and (ii) to identifypotential improvements to the panel geometry.

In the course of this endeavor, we utilized a simplified, decou-pled method for applying the soil impact loads to the structures.This spatio-temporal distribution of the loading was calibratedagainst one experiment from the testing matrix based upona recently proposed soil model developed by Deshpande et al. [20].In the final step, we compared two strategies for improving thesandwich panels (measured in terms of core crush and maximumback face deflection) with numerical simulations.

3.1. Finite element modeling

The responses of the plates and sandwich panels were simulatedusing a parallel finite-element computational framework [29] basedon a Lagrangian finite deformation formulation of the equations of

motion [30]. The spatial discretization of the plate and panel geom-etries utilized 10-node quadratic tetrahedral elements. The meshused for the analysis of the solid plate comprised approximately160,000 elements while that for the extruded panel containedapproximately 200,000 elements. Time integration was performedwith the explicit, second order accurate, central-difference scheme,which falls within the Newmark family of time stepping algorithms.

3.2. Constitutive modeling

The constitutive response of 6061-T6 aluminum was modeledusing an isotropic, finite deformation variation of the Johnson-Cookviscoplastic model [31], formulated within the framework of vari-ational constitutive updates developed by Radovitzky and Ortiz[32], Ortiz and Stainier [33], Yang et al. [34], and Ortiz and Pandolfi[35]. To avoid complications from fracture, we disregarded theexperimental results that display large-scale failure (the sandwichpanel at 15 cm standoff) sincewe did not attempt tomodel fracture.

We adopted a modified JohnsoneCook flow stress Y defined as

Y ¼ s0

"1þ ep

ep0

#n2641þ Clog

0B@ _ep

2_ep0þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

_ep

2_ep0

vuut !21CA375 (1)

where ep is the equivalent plastic strain, _ep is the equivalent plasticstrain rate,

Fig. 4. Schematic of experimental test setup.

Fig. 5. Variable standoff test arrangement of (a) equivalent weight solid plate (b)extruded and friction stir welded sandwich panel.

Fig. 6. Modified Johnson-Cook material model fit against uniaxial test data.

J.J. Rimoli et al. / International Journal of Impact Engineering 38 (2011) 837e848842

ep0 ¼�s0B

�1n (2)

is a reference plastic strain, and _ep0 is a reference plastic strain rate.The choice of this flow stress avoided singularities in the traditionalJohnson-Cook hardening as _ep/0 and in the hardening modulus asep/0, in accordance with the variational update method require-ments. The constitutive model is supplemented with an isotropicHencky hyperelasticity extension of linear elasticity.

The strain-rate effect parameter C was obtained from the liter-ature [36], fitted at a reference strain rate _ep ¼ 1 s�1. The Young’smodulus E and the strain hardening parameters n, B, and s0 werecalibrated to quasi-static tensile tests, conducted on coupons cutfrom the faces of the finished panels and measured in the directionof extrusion. Fig. 6 compares the measured uniaxial true stress-strain data to the constitutive law prediction. The derived consti-tutive parameter values are shown in Table 1, along with thedensity r.

3.3. Wet sand blast loading model

Traditionally, the modeling of blast loads on solids due toexplosions have been based on empirical formulas, such as theWestine model [28]. This method has proven fruitful in design;however, it is limited in applicability to conditions similar to theoriginal calibration experiments. Another approach has been todevelop constitutive models for soils, e.g. Grujicic et al. [21] andGrujicic and Cheeseman [23]. Such models may then be used incoupledFSI codes to simulate virtuallyany scenario,within the rangeof validity of the constitutive model. In the present context, thisapproach has the disadvantage that significant computational effortis expended on the soilmechanics, which here is of less interest thanthe response of the structure. Since the FSI effect during impulsive

Table 1Material parameters used in numerical simulations.

E n s0 B n C _ep0 r

72 GPa 0.343 250 MPa 365 MPa 0.04 0.002 1 2700 kg/m3

J.J. Rimoli et al. / International Journal of Impact Engineering 38 (2011) 837e848 843

soil loading is small, we created a decoupled loading model witha priori determined tractions that incorporates information froma constitutive model applicable to impulsively loaded soil [20]. Thisapproachmaybeviewedas a compromisebetween the twopreviousmethods, with some of the advantages of both.

Deshpande et al. [20] performed simulations of dry and wetsand impact from test charges similar to those investigated here.These simulations provided values for the time dependence ofthe impulse I and the average pressure p on a target panel orplate as a function of the radial distance r. Our loading consists ofa time-dependent surface traction based upon the impulse andpressure data obtained from this model. The introduced coeffi-cients were calibrated by fitting to one of the sandwich panelexperiments (at 30 cm standoff). A brief description of thisprocedure follows; additional mathematical details are containedin an appendix.

Consider a point P on the target surface located a distance r fromthe center of the charge O as depicted in Fig. 7. As shown in theAppendix, the load at the point P can be reduced to the applicationof a time-dependent pressure pPðtÞ given by

pPðtÞ ¼�pðrÞsin2ðqÞ; if t˛

�tPa ; t

Pa þ tPd

�0 otherwise

(3)

where p(r) is the pressure as a function of distance to the chargecenter, and the arrival time tPa and the duration of the pulse tPd aregiven by

tPa ¼ r0V0

4pað1þ bÞ

"1

r1þb0

� 1r1þb

#(4)

tPd ¼ arb

pðrÞ (5)

In these equations r0 and V0 are the initial average density andvolume of the sand-water mix surrounding the explosive charge. Inaddition, we assumed that the impulse, defined as

IðrÞhZtPd

pðtÞdt (6)

Fig. 7. Sand blast front schematics. The dashed lines represent the position of the wetsand blast front for times t ¼ tP0a and t ¼ tPa .

is a function of the distance r to the charge center, and takesthe form

IðrÞ ¼ arb (7)

The expressions for the impulse I(r) and the pressure p(r) wereobtained by a procedure which utilizes calculated wet sand blastdata for a similar experimental setup (from [20]), followed bycalibration against the current experimental data. The calibrationprocedure is discussed in Section 4.2.

4. Results and validation of the model

4.1. Experimental results

The effect of changing the standoff distance upon the deflectionof a 17 mm thick monolithic aluminum 6061-T6 plate (with a massper unit area identical to the sandwich panels) is shown in Fig. 8. Allof the tests caused significant bending and plastic stretching of thepanels. This increased in severity as the standoff distance wasdecreased. A small crack formed on the loaded surface near themid-point of the gripped edge of the most intensely loaded panel.

The effect of the same impulsive load upon the sandwich panelstructures is shown in Fig. 9. Sandwich panel permanent deflectionwere 15e20% less than those of the solid plates with the exceptionof the most intensely loaded panel, Fig. 9a. In this case, face sheetfracture at the edge and center of the panel occurred. Edge crackingwas also evident in the front and back faces of the panel tested ata standoff of 19 cm (Fig. 9b) and facilitated additional paneldeflection over that due to face sheet stretching and panel bending.Face sheet stretching and deflection between corrugation webnodes in the most intensely loaded regions of the panels is alsoevident (see Fig. 9a and b). At lower impulses, the core was onlymodestly compressed by inelastic buckling of the corrugationwebsand sandwich action was preserved for a significant fraction of theloading event.

4.2. Model analysis

The three loading parameters were obtained by incorporatingresults obtained from Deshpande’s calculations [20] with a cali-bration against one experimental test result. For wet-sand blasts,Deshpande’s model predicts that the pressure is approximately

Fig. 8. Sectioned photos of equivalent weight solid plates tested at standoff distances of,(a) 15 (b) 19 (c) 22 (d) 25 and (e) 30 cm. A small crack formed at the region circled in (a).

Fig. 9. Sectioned photos of sandwich panels tested at standoff distances of, (a) 15 (b)19 (c) 22 (d) 25 and (e) 30 cm.

Fig. 10. Plate displacement vs. time for 30 cm standoff distance.

J.J. Rimoli et al. / International Journal of Impact Engineering 38 (2011) 837e848844

constant for the range of standoff distances considered in this work.Additionally, the impulse decay with distance is well characterizedby a power law:

IðrÞ ¼ arbpðrÞ ¼ p0 (8)

In our calculations, the exponent b is derived from publishedsimulation data in Eq. (20) for a similar experimental setup; theimpulse versus standoff distance data for fully saturated wet sandwerefitted to the power law (Eq. (8)) and the resulting exponentwasadopted for the current simulations. The remaining loading param-eters p0 and a were calibrated through simulation by matching thevalues of the core strain and the permanent deflection at the centerof the back face of the corrugated panel for a standoff of 30 cm.

The loading parameters found through this procedure are listedin Table 2. From Eq. (8), the predicted peak impulses ranged from3.28 kN s/m2 at r¼ 30 cm to 6.00 kN s/m2 at r¼ 19 cm.We note thata b value less than �2 implies a weaker than inverse square lawdependence of impulse upon standoff distance for fully saturatedwet sand. Further assessments of this finding await fully coupledsimulations that account for the direction of the explosives deto-nationwave front (in this case towards the target), impulse transferfrom the explosive to thewet sand and the interaction of the radiallystretching wet sand shell with a dynamically deforming target.

Fig. 10 shows the displacements at the centers of both panel facesheets as a function of time for the calibration case. The applicationof the wet sand blast load to the panel produces an initial rapiddeflection, peaking at tp ¼ 0.36 ms. The initial peak is followed bya series of oscillations with period T z 0.61 ms. The permanent

Table 2Calibrated loading parameters.

a b p0

0.665 kN s/m2 �1.325 90 MPa

deflection of the front and back face sheets is estimated as theaverage of the peak values of the corresponding oscillation:

d ¼ dmax � dmin

2(9)

The last oscillation of the simulationwindow t¼ [0, 0.002] s is usedfor this calculation. From Fig. 10, it can be observed that the frontface exhibits a larger permanent deflection than the back face. Thisdifference, divided by the core height defines the permanent corestrain:

epcore ¼ dfront � dbackhcore

(10)

where hcore ¼ 19.1 mm.After calibrating the loading parameters, the other experiments

were simulated using the same loading parameters.2 In total, thesimulation suite comprises both plates and corrugated panelssubjected to wet sand blast loads at standoffs of 19 cm, 22 cm,25 cm and 30 cm. Fig. 11 shows the deformed shape of sectionedpanels after the simulation concluded.

Fig. 12 shows the numerical results for the corrugated panelstogether with their experimental counterparts. As expected, thedeflection for the 30 cm standoff distance case exactly matched theexperimental observation since this was the calibration case. Theremaining three cases (at standoffs of 25 cm, 22 cm and 19 cm) allare in good agreement.

Fig. 13 compares the numerical deflection results and theexperimental observations for the monolithic plates. Good agree-ment is again observed. The worst case, at 19 cm standoff, exhibitsan 8% difference with the corresponding experiment. It bearsemphasis that the good agreement observed in both the plate andsandwich panel simulations was obtained with identical loadingparameters a, b, and p0 for every simulation, using only one case forcalibration (the sandwich panel at 30 cm standoff).

Comparisons of the center permanent deflections of the sand-wich panel, Fig. 12, and monolithic plate, Fig. 13, show that thesandwich panel suffered 15e20% smaller deflections thanthe monolithic plates. Examination of Fig. 11 shows that the

2 Simulation results for the standoff distance of 15 cm are not reported since thecurrent model neglects failure. The experimental results indicate that extensivefailure occurs in the sandwich panel at this standoff, see Fig. 9.

Fig. 11. Simulation results for panels at standoff distances of, (a) 19 (b) 22 (c) 25 and(d) 30 cm. A quadrant of the panel has been removed (on the left) to show the paneldeflections at the middle of the panel. Fig. 13. Plate center displacement (back face) vs. standoff distance.

J.J. Rimoli et al. / International Journal of Impact Engineering 38 (2011) 837e848 845

simulations predictedmodest core compression, in agreement withexperimental observations, cf. Fig. 9.

4.3. Exploration of the design space

The validated model provides a simple means for exploring thepanel design space by calculating the response of corrugated panelswith variations of the original design. We have restricted ouranalysis to cases where the core compression strain did not exceed50% (whereupon the panels’ sandwich action is lost). Panel modi-fications explored two different strategies:

e Case 1: core mass fraction variation. The overall core area wasreduced by evenly decreasing the thickness of both vertical andinclinedwebs. The total panel mass was kept constant by addingthe subtracted mass to the back face sheet. In addition to theoriginal panel, two cases were considered, with 10% and 20%core density reduction.

Fig. 12. Sandwich panel center displacement (back face) vs. standoff distance. Theincident impulse used in the simulations is also shown.

e Case 2: vertical web variation. The core mass was reducedby decreasing the thickness of the I-core (vertical) webs only.The total panel mass was again kept constant by adding thesubtracted mass to the back face sheet. In addition to the orig-inal panel, three cases were considered, with 20%, 60% and 100%reduction of the vertical web thickness. The latter case corre-sponds to a panel without vertical webs.

Fig. 14 shows the permanent deflection results obtained for case1. A reduction of the total core mass produced a slight decrease ofthe permanent back face deflection for each standoff distance.Larger reductions in core mass result in decreasing back facedeflections but significantly more core crushing, Fig. 17a.

Fig. 15 summarizes the permanent deflection trends for case 2.Here, reduction of the total core mass decreased the permanentback face deflection for all standoff distances. The limiting case of novertical webs exhibited the best performance of any case consideredresulting in 30% reduction in deflection compared to the solid plate.No interpenetration was observed for any of these simulations.

Fig. 14. Panel center displacement (back face) vs. standoff distance for different valuesof core web thickness.

Fig. 15. Panel center displacement (back face) vs. standoff distance for different valuesof vertical web thickness.

Fig. 17. Panel effective plastic strain for 19 cm standoff distance for different designs.

Fig. 18. Center displacement (back face) vs. standoff distance for the monolithic plate,original panel design and optimized panel design.

J.J. Rimoli et al. / International Journal of Impact Engineering 38 (2011) 837e848846

The successful reduction of back face deflection relative to theoriginal design is tempered by increments in core crush strain at allstandoff distances, particularly for case 1. Fig. 16 compares the corecrush strain of the original design to the best performing panelsfrom cases 1 and 2. Fig. 17 illustrates this effect for the 19 cmstandoff distance. The difference in the core crush strains can beexplained by the occurrence of different failure mechanisms.

In case 1, the stubby vertical webs imposed significant constraintupon front facemotion, limiting thebucklingof neighboring inclinedwebs. This constraint resulted in an inhomogeneous buckling(crushing) of the core. As a consequence, failure localized betweenthe innermost vertical webs, and energy was not dissipateduniformly (See Fig. 17a). The effective plastic strain attaineda maximum value ep ¼ 0.48 in the inclined webs, which is signifi-cantlygreater than thevalue reached in theoriginal design, ep¼0.25.This localizedbuckling in turn causedgreater bendingand stretchingin the top face sheet, implying an inefficient use of material.

The response is markedly different when no vertical webs arepresent. In this scenario buckling proceeded more homogeneouslythroughout the core; see Fig. 17b. Consequently, energy wasinitially dissipated though distributed plastic buckling of the core

Fig. 16. Panel core crush strain vs. standoff distance for different designs.

webs across the entire panel, and later through membranestretching of both face sheets. The effective plastic strain in the corewebs attained a maximum value of ep ¼ 0.28, which was similar tothe original design. Strains throughout the face sheets were moreuniform in comparison to cases with vertical webs.

In all cases considered, high effective plastic strains were formedin the face sheets near the clamped edges. Their location corre-sponded to the region where face sheet failure occured in the mostintensely loaded experiments. The deformation in this region waslittle affectedbydesignmodifications. Themaximumeffective plasticstrain was ep ¼ 0.33 in the original design, ep ¼ 0.34 for the designshown in Fig. 17a, and ep ¼ 0.37 for the design depicted in Fig. 17b.

Fig. 18 summarizes the gains realized through the design spaceexploration. The limited space design exploration led to the identifi-cation of panel designs that suffer about 30% smaller centerdeflectionsthan equivalent solid plates under localized wet-sand blast loading.

5. Summary and conclusions

We have used a combined experimental andmodeling approachto investigate wet sand blast loading response of 6061-T6aluminum plates and sandwich panels with corrugated cores of thesame mass per unit area.

The corrugated core panels suffered 15e20% smaller maximumpermanent deflections than equivalent mass per unit area mono-lithic plates of the same alloy. The sandwich panel deflection wasaccommodated by a small (5e10%) crushing of the core and by facesheet stretching. The permanent center deflection of the sandwichpanels increased with load intensity and failed by face-sheet frac-ture at the gripped panel edges and in the panel center for the mostintensely loaded panels.

J.J. Rimoli et al. / International Journal of Impact Engineering 38 (2011) 837e848 847

A parallel finite-element simulation capability based ona Lagrangian large-deformation formulation of the equations ofmotion was used to simulate the response of plates and panelsunder wet sand blast loading. A modified Johnson-Cook constitu-tive model was adopted and its material constants were fitted tomatch the experimentally obtained stress-strain response of thealuminum. We developed a decoupled loading model informedwith Deshpande’s results which predict that the pressure on thetarget is approximately constant for the range of standoff distancesconsidered in this work. By calibrating the loading model for onesingle experiment (corrugated panel, 30 cm standoff distance) wewere able to predict the remaining 7 experiments with excellentaccuracy.

The validated computational model provided the means forperforming a brief exploration of the design parameter space of thestructures. Modifications to the original panel design were studiedby pursuing two different strategies: (i) core mass fraction varia-tion, and (ii) vertical web variation. Different structural failuremechanisms were observed during this exploration: heteroge-neous core crushing for panels with vertical webs, and membranedeformation of the panels as a whole for the case with no verticalwebs. The latter case corresponds to the best design obtained fromour brief exploration of the design space. In this case, the energydissipation is distributed evenly as elasticeplastic buckling of thediagonal webs followed by a subsequent membrane stretch of thepanel front and back sheets as a whole.

Acknowledgments

We are grateful to Vikram Deshpande for helpful discussions ofthis work. The experiments were conducted at the Force ProtectionIndustries Explosives Test Range (Edgefield, South Carolina) withthe considerable assistance of Keith Williams. Research was sup-ported by the Office of Naval Research (ONR grant number N00014-07-1-0764) as part of a Multidisciplinary University ResearchInitiative (David Shifler, Program manager).

Appendix. Derivation of simplified wet sand blast loadingmodel

Let us consider a point P on the target surface at a distance rfrom the explosive charge center O as depicted in Fig. 7. The wetsand velocity, pressure and density at the arrival point arerelated by

rðrÞ ¼ rðrÞvðrÞ2 (11)

In addition, let us consider the mass conservation equationassuming constant density across the sand blast

r0V0 ¼ rðrÞVðrÞ (12)

where r0 and V0 are the initial average density and volume of theexplosive charge respectively and V(r) is the volume of theexpanded wet sand shell, approximated by

VðrÞ ¼ 4pr2wðrÞ (13)

In the previous equation, w(r) is the thickness of the expandedwet sand shell, related to the velocity and duration of the load td by

wðrÞ ¼ tdðrÞvðrÞ ¼ vðrÞIðrÞpðrÞ (14)

Combining the last three equations, the conservation of massreduces to

r0V0 ¼ rðrÞ4pr2vðrÞIðrÞpðrÞ (15)

Solving equations (11) and (15) simultaneously we obtainstandoff-dependent expressions for the arrival velocity v and thedensity r:

rðrÞ ¼�r0V0

4p

2 pðrÞIðrÞ2

1r4

(16)

vðrÞ ¼ 4pr0V0

IðrÞr2 (17)

Since v is not constant, the arrival time tPa must be obtainedthrough integration of equation 17. In order to solve for the arrivaltime, let us assume that the impulse can be fit with a power law:

IðrÞ ¼ arb (18)

This assumption, combined with eq. (17), implies a representa-tion of the velocity of the form

vðrÞ ¼ 4par0V0

rbþ2 (19)

Assuming that t¼0 for r¼ r0 and integrating theprevious equationwe obtain the position of the blast front as a function of time. In thismanner, by simple inversion, the arrival time can be obtained as:

tPa ¼ r0V0

4pað1þ bÞ

"1

r1þb0

� 1r1þb

#(20)

Furthermore, since p(r) is the average pressure, the duration tPdof the pressure pulse is, by definition, given by

tPd ¼ IðrÞpðrÞ (21)

At this point, we know that the point P is loaded by a constantpressure from time tPa to tPa þ tPd , but we must be careful regardingthe definition of such a pressure. The data provided by Deshpandeet al. considers the point where the target surface is perpendicularto the sand velocity, namely P0. Therefore, we must only considerthe normal component of the sand velocity at the arrival point:

vt ¼ v sinðqÞ (22)

In this way, the load at the point P is reduced to the applicationof a time-dependent pressure pPðtÞ given by

pPðtÞ ¼�pðrÞsin2ðqÞ if t˛

�tPa ; t

Pa þ tPd

�0 otherwise

(23)

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