Design of Metal Foam Cladding Subjectedto Close-Range Blast

Hongyuan Zhou1; Guowei Ma2; Jingde Li3; and Zhiye Zhao, M.ASCE4

Abstract: The response of a blast mitigation cladding consisting of a face sheet and metal foam core subjected to a close-range blastis predicted. Whereas the cladding is sufficiently wide compared to the standoff distance between the explosion center and the cladding,the boundary of the blast induced bulge is released. The face sheet is considered as a rigid perfectly plastic membrane as the deformationof the sheet always exceeds half its thickness. A procedure predicting the depth and extent of the bulge is proposed with energy method.Subsequently, the minimum thickness of the foam layer is calculated based on the bulge depth. This design-oriented approach, in aready-to-use manner, can be straightforwardly applied, facilitating the preliminary design of blast mitigation claddings with metal foamcore. DOI: 10.1061/(ASCE)CF.1943-5509.0000606. © 2014 American Society of Civil Engineers.

Author keywords: Blast loads; Protective structures; Foam; Cladding; Cellular structures; Blast mitigation; Close-range; Energyabsorption; Metal foam.

Introduction

To protect key buildings against accidental and terrorist explosions,a conventional method is to strengthen structural members andmake them thicker in dimension and higher in strength, which isexpensive, labor intensive, and time consuming in construction andretrofit. Besides, with the same high-energy product and standoffdistance, the intensity of blast load exerted on a strengthenedstructural member of greater rigidity is higher than that of theunstrengthened one. Alternatively, attaching blast mitigation clad-dings with metal foam cores to protect structures as a blast allevia-tion philosophy emerged (Ashby et al. 2000). Low-density foamundergoes a large plastic deformation at a controllable relativelylow stress level when compressed, showing great potential in blastmitigation. In addition, the cladding is made of metal and can berecycled after use. In contrast to concrete for strengthening andretrofitting the existing structures, the application of an energy ab-sorption cladding is advantageous in terms of flexibility and sus-tainability. Therefore, in recent years, the dynamic properties offoam and response of structures with foam components subjectedto blast loads have been intensively studied experimentally, analyti-cally, and numerically (e.g., Lu and Yu 2003; Ma and Ye 2007; Zhuet al. 2009; Wu et al. 2011; Shen et al. 2011; Goel et al. 2013).Various models delineating dynamic behaviors of such cellular

material have been proposed. Among which, the rigid-perfectly-plastic-locking model based on a shock theory (Reid and Peng1997) which predicts the response of low-density foam under highvelocity crushing with adequate accuracy in explicit formulationsthus has been widely applied (e.g., Xue and Hutchinson 2003; Yeand Ma 2007).

Analyses of metal foam cladding response under blast load canbe categorized into two major approaches. In the former, the ex-plosion is assumed occurring at a sufficient distance from the struc-ture of concern and the blast load is considered to be uniformlyapplied on the face sheet, where the structure, load, and responseare essentially one dimensional (Reid and Peng 1997; Ma and Ye2007; Ye and Ma 2007). As a consequence, there is no deformationof the face sheet and it does not dissipate energy during claddingcrushing. The main merit of this idealization is that the time historyof the cladding response can be readily obtained in close form,allowing for convenient parametric study and key parameter iden-tification, thus the response mechanism is revealed explicitly. Thedrawbacks of these models are obvious: for a spherical airburst, theuniformly distributed load assumption implies that the event shouldbe sufficiently distant; thus, the model cannot be used for a close-range detonation; further, the boundary condition of the claddingis not considered; lastly, the energy dissipation by the face sheet iscompletely neglected.

The latter approach adopts, in contrast, two-dimensional modelsfor sandwich beams and plates with metal foam cores subjected toblasts. By assuming deflection fields, response time history can bederived in one or more phases, or by adopting energy method, thefinal deformation state can be directly determined without analyz-ing response time histories. In fact, through assuming deformedshapes of the claddings, the two-dimensional problems are reducedto essentially one-dimensional problems. The main advantage ofthese models is that a close-range blast can be studied and the en-ergy dissipation by the face sheet can be taken into consideration.However, in most tests and models of foam cladding subjected toblasts, the cladding is clamped or simply supported (or assumed soin theoretical work) over a span or diameter not significantly largerthan the standoff distance between the explosion center and thecladding (Jones 1989). With these boundaries, the extent of theface sheet bulge evolving from the original flat plate is inevitably

1Research Fellow, School of Civil and Environmental Engineering,Nanyang Technological Univ., Singapore 639798. E-mail: hyzhou@ntu.edu.sg

2Professor, School of Civil and Resource Engineering, Univ. of WesternAustralia, Crawley WA 6009, Australia. E-mail: ma@civil.uwa.edu.au

3Ph.D. Candidate, School of Civil and Resource Engineering, Univ. ofWestern Australia, Crawley WA 6009, Australia. E-mail: jaden_li@hotmail.com

4Associate Professor, School of Civil and Environmental Engineering,Nanyang Technological Univ., Singapore 639798 (corresponding author).E-mail: czzhao@ntu.edu.sg

Note. This manuscript was submitted on October 31, 2013; approved onMarch 3, 2014; published online on March 5, 2014. Discussion period openuntil January 28, 2015; separate discussions must be submitted for indivi-dual papers. This paper is part of the Journal of Performance of Con-structed Facilities, © ASCE, ISSN 0887-3828/04014110(11)/$25.00.

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prescribed as the whole beam span or plate diameter. Whether theloads are applied on the whole face sheet or only on a patch, thecladding always exhibits global deformation.

In engineering practice, the boundary condition of a blast mit-igation cladding is totally different. When a metal foam claddingattached to a structure is subjected to a close-range explosion, theinduced bulge extent can no longer be prescribed. Instead, it de-pends on the properties of both the load and cladding. In the presentstudy, the boundary is released (neither fixed nor simply supportedwith a fixed span) to reflect the real situation of a typical full-sizecladding subjected to a close-range spherical airburst. The metalfoam cladding is directly attached to the protected structure whichis assumed rigid and unmovable, justified by small-scale test(“Justification and Validation of the Prediction”). Compared tothe localized area on which the blast load effectively exerts, thecladding is considered sufficiently wide. More importantly, the ex-tent of the bulge is no longer predefined; it depends on the blastload distribution and the cladding parameters such as the foam pla-teau stress. It is evident that a spherical airburst imposes an axisym-metrical load on the face sheet, resulting in an axisymmetricalbulge with the largest transverse deflection in the center. By assum-ing a deflection profile and adopting energy method, the bulge ex-tent and depth can be determined. Subsequently, the minimumfoam thickness corresponding to a certain blast load is calculated,which can be used in preliminary design for protective claddings.A reasonable design must ensure that for a blast of a specified pre-defined intensity, the metal foam core should not be fully densified.Otherwise, the protected structure may experience undesirabletransmitted load, even stronger than that without applying the clad-ding, which should be avoided. Therefore, for a specific blast sce-nario in practical design, the depth of the bulge is more critical thanthe extent. The present study provides a simplified and straightfor-ward procedure for preliminary design of blast mitigation claddingwith metal foam core.

Deformation Profile of the Face Sheet

Debris penetration and perforation of the face sheet are not consid-ered in the present study. When the maximum deflection of a plateis smaller than half its thickness, the deformation is dominated bybending and plastic hinges, whereas the membrane force within itcan be neglected (Jones 1989). Under this circumstance, the bulgeshape can be approximated by (Jones 1989)

wðrÞ ¼ W0

�1 −

�rη

�α�; r ≤ η ð1Þ

whereW0 and η are the depth and extent of the bulge, respectively;r is the radial coordinate measured from the bulge center; and α is aparameter controlling the bulge shape. Profiles with α ¼ 1 or 2 areoften used in small deformation analysis of plates.

When the maximum deflection is greater than half the platethickness, the bulging process is dominated by membrane forceand the contribution from bending becomes relatively trivial andneglected (Jones 1989), where the deformation profiles in Eq. (1)are no longer appropriate. Based on the observation of the bulgeprofile of a metal foam cladding subjected to close-range explosion(Liu et al. 2013), as expected by symmetry, the rotation at the bulgecenter is zero. Thus, for a reasonable assumed bulge shape inducedby a blast, the rotation at the bulge center should be zero. Further,for large deflection of a plate (maximum deformation greater thanhalf its thickness) induced by a close-range blast, the plate defor-mation is dominated by in-plane membrane stretching, leadingto the transition region around the bulge periphery to be smooth

without sharp corner and the rotation at the bulge periphery shouldbe nearly zero provided that the cladding is sufficiently wide com-pared to the standoff distance. From Fig. 1, it is evident that theshapes delineated by Eq. (1) are not suitable for approximatingthe bulge in the present study because even if the zero center ro-tation is satisfied, there are still inevitable sharp corners around thebulge periphery. Observation of the small-scale test of metal foamcladding subjected to close-range blast (Liu et al. 2013), shown inFig. 2, suggests that the bulge profile of the face sheet is similar tothe normal distribution function. Therefore, generalized Gaussianfunction is used to approximate the bulge

wðrÞ ¼ W0e−r2=ðkη2Þ; r ≤ η ð2Þ

where k is a constant determining the bulge extent. Obviously, thegeneralized Gaussian function satisfies the requirement of zerobulge center rotation. Moreover, it has the capacity to representthe nearly zero rotation around the bulge periphery. Comparisonof the generalized Gaussian function in Fig. 1 with the bulge profileobserved in test in Fig. 2 confirms that the generalized Gaussianfunction is a reasonable choice.

Determination of the Bulge Extent

The face sheet of a typical protective cladding is generally thin formass efficiency, with thickness ranging from less than 1 mm to sev-eral millimeters (Zhu et al. 2009). Because of the small standoffdistance between the detonation center and the cladding as wellas the small face sheet thickness, the bulge depth is always greaterthan half the face sheet thickness. For sufficiently large transversedisplacement of a circular plate greater than half its thickness, theinfluence of bending moment can be entirely neglected and theresponse is solely controlled by membrane force (Jones 1989).In addition, for large plastic deflection of a thin plate, the contri-bution from elasticity, either in the initial preyield stage or the un-loading spring-back, is negligibly small compared to that from theplastic deformation (Jones 1989). Therefore, in the current study,the face sheet is idealized as a rigid perfectly plastic membrane.

When a blast mitigation cladding consisting of a thin metal facesheet and a layer of metal foam is subjected to a blast load, thebulge extent can be estimated before the bulge depth is determined,

Fig. 1. Various idealized face sheet deformation profiles

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whose prediction accuracy is justified by small-scale test (“Justifi-cation and Validation of the Prediction”). The blast load induced bya close-range airburst lasting for an extremely short duration,typically from several microseconds to 1 ms, is often simplifiedas a triangular load with zero rise time (Smith and Hetherington1994). The bulge size can thus be determined by analyzing theload applied on, and resistance from, the cladding, as shownin Fig. 3.

Before deformation, the nearest point in the face sheet to theexplosion center as the coordinate origin and downward directionis positive was defined. It is evident that both the blast load distri-bution and bulge profile are independent of radius direction; thus, itis essentially an axisymmetrical problem. Observations on a num-ber of tests (U.S. Army 1986, 2008) show that when the incidenceangle (defined as the sharp angle between the wave front and thetarget plane) of a blast load exceeds 70°, the loading intensity nor-mal to the cladding becomes negligibly small compared to its peakload at the coordinate origin, where the wave front impinges theface sheet perpendicularly. In fact, there is no exact analytical ex-pression of the blast loading for a spherical explosion applied ontoa nearby plane, whose intensity distribution perpendicular to theplane resembles a bell and is sometimes called a bell function.Gaussian function has a variation similar to the bell function, thusreasonably approximating the blast loading spatial distribution onthe cladding

pðr; tÞ ¼ pce−r2=ð2b2Þ

�1 − t

Td

�ð3Þ

where pc is the peak load applied on the cladding and b is the char-acteristic size of the blast loading area in a sense that at r ¼ 3b; theloading intensity is 1% of that at the origin. This condition can beperfectly satisfied when b is taken as the standoff distance betweenthe explosion center and the cladding (when r ¼ 3b, e−r2=ð2b2Þ ¼e−4.5 ¼ 0.011, Arc tan½3b=b� ¼ Arc tan½3� ¼ 71.6∘). Td is an effec-tive blast duration (different from real positive blast duration wherethe blast undergoes an exponential decay, thereafter omit effective)

of the idealized triangular load preserving the same peak load andspecific impulse (impulse per unit area).

At the arrival instant of the blast, the deformation and velocity ofthe face sheet are zero; thus, the dynamic enhancement of the metalfoam is zero as there is no high velocity foam crushing. Accordingto D’Alembert’s principle, taking the face sheet to analyze, the blastload applied on the face sheet should be equal to the sum of theinertia force on the face sheet and the resistance from the metalfoam, as indicated in Fig. 3. At the bulge periphery, the deflection,velocity and acceleration are constantly zero, resulting in a zeroinertia force. Then, by equating the blast load intensity to the foamplateau stress, the extent of the bulge is estimated

pce−r2=ð2b2Þ ¼ σpl; r ¼ η ð4Þ

η is the bulge radius and it is derived to be

η ¼ffiffiffi2

pb

ffiffiffiffiffiffiffiffiffiffiffiln

pc

σpl

rð5Þ

where σpl is the foam plateau stress, dependent on foam relativedensity and configuration (open cell or close cell). Based on rigidplastic assumption of the metal foam (Ashby et al. 2000; Lu and Yu2003), the cladding region with radius greater than η remainsundeformed, where the load does not contribute to the bulging.In other words, the load on cladding is truncated at the extentperiphery and only the part within the extent contributes to thebulging process.

The boundary is assumed not moving. In fact, the central partof the cladding is loaded slightly earlier and other part slightly later,which implies that there is a coupling between the space and timeof the blast load, and the bulge is evolving in size. In the presentstudy, by applying the energy method, only the initial and finalstates are concerned. The accuracy of the approximate approachis validated by a small-scale test (“Justification and Validationof the Prediction”).

Determination of the Bulge Depth

Once the bulge extent of the cladding is estimated, the bulge depthcan be determined by applying energy method. The region with theradius greater than the bulge periphery will not be examined be-cause as stated in the preceding section, although blast load applies,it does not deform. The duration of a close-range blast is extremelyshort, generally significantly shorter than the structural responsetime of the cladding. It is reasonable to assume that the impulseis imparted to the face sheet impulsively before any deformationoccurs (Qiu et al. 2004; Zhu et al. 2009). The response of the metalfoam cladding is divided into two sequential phases (similarphase division can be found: Qiu et al. 2004; Zhu et al. 2009),

Fig. 2. Deformation of a metal foam cladding subjected to a close-range blast (the length scale should be 5 mm rather than 1 cm) [reprinted fromLiu et al. (2013), with permission from Elsevier]

Fig. 3. Determination of the bulge extent

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i.e., Phase I: blast–structure interaction, resulting in an initialvelocity field of the face sheet before any deformation occurs;Phase II: retardation of the cladding to rest by stretching of the facesheet and crushing. The approach is justified by Shen et al. (2011).

Energy Input to the Cladding

Taking the cladding within the bulge extent to analyze, the impulseimparted into the face sheet is

I ¼ZA

ZTd

0

pce−r2=ð2b2Þ

�1 − t

Td

�dtdA ¼ πTdpcb2

�1 − σpl

pc

�

ð6Þ

The face sheet is assumed to gain a velocity field with gener-alized Gaussian distribution (Ramu and Iyengar 1976; Kennedyand Iyengar 1981)

v0ðrÞ ¼ V0e−r2=ð2c2Þ; r ≤ η ð7Þ

where V0 is the initial velocity at the bulge center and c is the char-acteristic size of the bulge. At η ¼ 3c, the velocity is 1% of V0

(based on the characteristics of Gaussian function) and consideredzero. The initial momentum gained by the face sheet is

ΔMV ¼ZAV0e−r

2=ð2c2ÞρbhbdA ¼ 2πV0ρbhbc2ð1 − e−4.5Þ

≈ 2πV0ρbhbc2 ð8Þ

The item e−4.5 is significantly small compared to unity, thusneglected without affecting the accuracy. Thereafter, the approxi-mately equal is directly written as equal in equations. According tothe impulse-momentum relation, the initial velocity at the bulgecenter is determined by equating the imparted impulse to the initialmomentum of the face sheet

V0 ¼Tdpcb2

2ρbhbc2

�1 − σpl

pc

�ð9Þ

Then, the kinetic energy imparted to the face sheet is

Ek ¼ZA

1

2½v0ðrÞ�2ρbhbdA ¼ π

2V20ρbhbc

2ð1 − e−9Þ

¼ πT2dp

2cb4

8ρbhbc2

�1 − σpl

pc

�2

ð10Þ

This kinetic energy of the face sheet will be dissipated by foamcore crushing and face sheet stretching.

Energy Dissipated by Foam and Face Sheet

Based on the observed deformed shape of metal foam cladding sub-jected to close-range blast and the discussion in “Deformation Pro-file of the Face Sheet,” the bulge shape is approximated by ageneralized Gaussian function

wðrÞ ¼ W0e−r2=ð2c2Þ ð11Þ

When the metal foam is crushed at sufficiently high velocity(velocity higher than certain threshold, which is observed in testsbut not yet fully understood in mechanism), the densification stress,in fact the crushing resistance, at the shock front is (Ashby et al.2000; Lu and Yu 2003)

σD ¼ σpl þρfεD

v2 ð12Þ

where ρf is the foam density before crushing; εD is the foam den-sification strain; and v is the crushing velocity. The first term is thefoam plateau stress (the velocity-independent part of the crushingresistance) and the second term is dynamic enhancement (thevelocity-dependent part).

During the cladding crushing, the resistance to the face sheetand densified foam varies because even though the foam plateaustress remains constant, the dynamic enhancement related to thecrushing velocity decreases from the initial peak value to zeromonotonically. Therefore, the work done by the foam to resistthe crushing of the face sheet and densified foam can be dividedinto two parts: one from the foam plateau stress and the other fromthe dynamic enhancement. Among which, the part of energy asso-ciated with the ever-changing dynamic enhancement is difficult toquantify exactly. To estimate the energy absorbed by the foam core,a one-dimensional cladding consisting of a face sheet and a foamcore subjected to high velocity crushing with an initial face sheetvelocity V0 was first considered. Then, another one-dimensionalcrushing process with the same initial velocity and final crushedstroke was considered, except that the dynamic enhancement isassumed constant throughout the densification, illustrated as theshaded area in Fig. 4. A constant β is introduced to describe theaverage dynamic enhancement in the sense that the energy dissi-pations by these two dynamic enhancements are the same

σD ¼ σpl þρfεD

βV20 ð13Þ

where β is a constant introduced by assuming two densificationprocesses [the resistance in Eqs. (12) and (13), respectively] to havethe same energy absorption and final crushed distance umax

β ¼R umax0 v2duV20umax

ð14Þ

From Eq. (12), it is evident that the resistance at the initial stageis relatively greater (higher velocity implies higher dynamic en-hancement), resulting in a relatively greater deceleration, thus ashorter duration with high velocity crushing. Similarly, the smallerresistance (lower velocity implies lower dynamic enhancement) inthe later stage with smaller deceleration causes the duration of lowvelocity crushing to last relatively longer. Curve 1 in Fig. 4 sche-matically shows the trend. Therefore, the motion of the systemconsisting of the crushed foam and face sheet is a retardationwith decreasing deceleration. It is always lower than the linearlydecreasing line (Line 2 in Fig. 4), which has a β equal to 0.5.Then, 0.5 can be considered as the upper bound of β.

Fig. 4. Dynamic crushing of a metal foam cladding and its equivalency(one-dimensional)

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The aforementioned parameter β is for one-dimensional crush-ing. When it comes to two-dimensional cladding as in the presentstudy, the initially gained kinetic energy of the face sheet is dissi-pated not only by foam crushing but also by face sheet stretching.However, the velocity trends of every point within the bulge extentare similar because the resistance decreases with the crusheddistance.

On one hand, the upper bound of β ¼ 0.5 in one-dimensionalcrushing is used for the two-dimensional foam crushing. Duringthe entire densification process, the crushing resistance consistingof the quasi-static and dynamic parts at any point of the two-dimensional system monotonically decreases with crushed dis-tance. According to Fig. 4, the value of β must be less than 0.5,as the curve is concave up decreasing. Therefore, applying β equalto 0.5 leads to overly estimated energy dissipation by dynamicenhancement and subsequently a shallower bulge than in the realsituation. This predicted bulge depth can be used as the lowerbound of the depth.

In contrast, assuming that β equal to 0 implies that there is nodynamic enhancement throughout the crushing process, the crush-ing is essentially quasi-static and velocity independent, resulting inunderestimated energy absorption by foam and subsequently adeeper bulge than in the real situation. The predicted bulge canbe used as the upper bound of the depth. Until now, both the upperand lower bounds of the bulge depth are predicted. The real depthmust be in between. For a specific event, two simple calculationswith β equal to 0 and 0.5 establish the range of the bulge depth.

In fact, when the crushing velocity is lower than certain value(dependent on foam characteristics), no shock front forms and theentire crushing is in a uniform manner. Obviously, in the initial andlater stage of a densification process, whether the loading is strongor weak, the crushing velocity falls below the threshold forprogressive collapse. With energy method and the upper and lowerbounds of energy absorption by varying β from 0 to 0.5, the den-sification mode is no longer a concern; subsequently, the velocitycheck for crushing mode becomes not necessary. In particular, if thecrushing velocity is relatively low, e.g., less than 30 m=s for a typ-ical aluminum foam, there is no dynamic enhancement, and theupper bound of the bulge depth is the true prediction.

With the dynamic resistance from foam crushing estimatedwith Eq. (13), the total energy dissipated by the foam during crush-ing is

Ef ¼ZA

1

2

�σpl þ σpl þ

ρfεD

βv20ðrÞ�εD

�wðrÞεD

�dA

¼ 2πc2W0

�σpl þ

β6

ρfεD

V20

�ð15Þ

The energy dissipated by the constant foam plateau stress, infact the energy dissipated by quasi-static crushing, can be calcu-lated as the work done by the quasi-static resistance (the foamplateau stress)

Efqs ¼ σpl

ZAwðrÞdA ¼ 2πW0c2σpl ð16Þ

which is exactly the same as the first term of Eq. (15). According toTeeling-Smith and Nurick (1991), the plastic strain energy for largedeformation of a rigid plastic membrane with an axisymmetricalbulging is

Eb ¼ hb

Z2π

0

Zη

0

1

2

γσysffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − ν þ ν2

p�dwdr

�2

rdrdθ ¼ πhbγσysW20

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − ν þ ν2

p

ð17Þ

where σys and ν are the quasi-static yield strength and Poisson’sratio of the face sheet material, respectively; γ is a dynamic en-hancement factor accounting for strain rate effect of the face sheetmade from rate-sensitive material.

Determination of the Bulge Depth

Equating the energy dissipated by the face sheet stretching and dy-namic foam crushing to the kinetic energy input to the face sheet

Ek ¼ Eb þ Ef ð18Þyields the governing equation for the bulge depth

1

2V20ρbhbc

2 ¼ 2c2W0

�σpl þ

β6

ρfεD

V20

�þ hbγσysW2

0

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − ν þ ν2

p ð19Þ

In a short form, it becomes

AW20 þ BW0 þ C ¼ 0 ð20Þ

where

A ¼ hbγσys

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − ν þ ν2

p ; B ¼ 2c2�σpl þ

β6

ρfεD

V20

�;

C ¼ −Ek ¼ − 1

2V20ρbhbc

2; V0 ¼Tdpcb2

2ρbhbc2

�1 − σpl

pc

�

where V0 is given in Eq. (9), rewritten in this section for convenientreference. The bulge depth is [the other solution of Eq. (20) is neg-ative and discarded]

W0 ¼−Bþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB2 − 4AC

p

2Að21Þ

with β equal to 0 and 0.5; the upper and lower bounds of the bulgedepth can be established. The real depth is in between.

Justification and Validation of the Prediction

Before the theoretical prediction can be applied in engineeringpractice for protective cladding design, it must be justified andvalidated.

Preliminary Justification with Limit Conditions

A basic requirement for a reasonable prediction of structural re-sponse is that it should be able to demonstrate some special casesof the structural behavior even without detailed calculation. In thepresent study, if the blast load is not strong, i.e., with intensity onlyslightly higher than the foam plateau stress, from Eq. (5), withpc=σpl approaching unity, the bulge extent is decreasing to zero.Subsequently, the constants B and C in Eq. (20) will approachzero. Then, according to Eq. (21), the bulge depth will be deter-mined as very small, approaching zero, which is reasonable. Thesuccessful representation of the limit condition case justifies themodel to some extent. The prediction is not valid for weak blastswith intensity lower than the foam plateau stress, where applying ametal foam cladding is not necessary.

Small-Scale Test

The most effective and direct method to validate and verify themodel is experiment, either a full-scale test or a small-scale test.In this section, the prediction is validated with the published resultof a small-scale test (Liu et al. 2013). The side view schematic of

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the test setup is shown in Fig. 5, a steel square 300 × 300 mm in-plane and 30-mm thick. In the middle of the setup, a hollow square(100 × 100 mm) is created. A metal foam cladding, consisting of aface sheet, a foam core, and a back plate is mounted in the middle ofthe support frame as shown in Fig. 5. The back plate is made of twomild steel sheets, each is 4.2 mm, thus 8.4 mm in total. Similarly,the face sheet is also made of two mild steel sheets (one is 2 mm,the other is 4.2 mm), which is 6.2 mm in total. Sandwiched be-tween the face sheet and the back plate is a layer of close-cell alu-minum foam, with a relative density of 0.16.

A small amount of pentaerythritol tetranitrate (PETN), 1.14 g, isused as detonator, whereas 10 g of research department explosive(RDX) (Cyclonite, 98% RDX and 2% paraffin) is used as mainexplosive charge, made into a cylinder 20 mm in diameter and20 mm in height. The density of the main charge is 1.5 g=cm3.The standoff distance between the charge center and the face sheetis 20 mm. With the protection from the foam layer, in this test, theback plate experiences no deformation (Liu et al. 2013) after theshot, as shown in Fig. 2. Therefore, the back plate can be consid-ered to represent the protected structural member, assumed rigidand unmovable in the present study.

Validation of the Prediction

The deformation of the metal foam cladding subjected to theprescribed blast is shown in Fig. 2, where both the 2-mm part ofthe face sheet and the 4.2-mm sheet of the back plate are removed.The length scale (1 cm) in Fig. 2 should read 5 mm, to be consistentwith the description in the literature (Liu et al. 2013).

With a trinitrotoluene (TNT) equivalent factor of 1.185 for RDX(Smith and Hetherington 1994), the equivalent TNT charge weightis 0.01185 kg. Recalling the standoff distance of 0.02 m, the scaleddistance is calculated as

Z ¼ R

W1=3 ¼0.02

0.011851=3¼ 0.0877 m=kg1=3 ð22Þ

To utilize the latest chart for blast load parameters, shown inFig. 6, the scaled distance is converted to U.S. customary unitas 0.221 ft=lb1=3. The blast duration is not available in the chart(U.S. Army 2008), possibly because of the extremely short standoffdistance. Then, the blast duration (effective) is obtained indirectlyfrom the same chart (U.S. Army 2008) with the directly read specificreflected impulse and the peak reflected pressure. With the scaled dis-tance of 0.0877 m=kg1=3 (0.221 ft=lb1=3), the peak reflected pressureis read as 379.2 MPa (55,000 psi). The scaled specific reflected im-pulse (ir=W1=3) is read as 13; 244 Pa-s=kg1=3 ð2,500 psi-ms=lb1=3Þ,leading to specific reflected impulse of 5,115 Pa-s (741.8 psi-ms), re-calling the charge weight of 0.01185 kg (0.02612 lb). Subsequently,

the duration is calculated as 0.02697 ms (741.8 × 2=55,000 ¼0.02697).

The plateau stress and densification strain of the foam usedin the test are not specified in the test. With the well-establishedempirical formulas, the plateau stress for close-cell foam can bepredicted as (Ashby et al. 2000)

σpl ¼ C1σys

�0.5

�ρfρs

�3=1

þ 0.3

�ρfρs

��ð23Þ

and the densification strain (Ashby et al. 2000) as

εD ¼ 1 − 1.4ρfρs

þ 0.4

�ρfρs

�3

ð24Þ

where C1 is a constant equal to 0.3 and σys is the quasi-static yieldstrength of the material from which the foam is made, 200 MPa for

Fig. 5. Side view of the test setup [reprinted from Liu et al. (2013), with permission from Elsevier]

Fig. 6. Parameters for a spherical TNT explosion in free air at sea level(reproduced from U.S. Army 2008)

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aluminum; ρs is the density of the base material, 2,700 kg=m3

for aluminum. Knowing the relative density of 0.16 (Liu et al.2013), the foam plateau stress and densification strain are calcu-lated as 4.8 MPa and 0.778, respectively.

Subsequently, with Eqs. (5) and (21), the bulge extent diameteris predicted as 59 mm, whereas the bulge depth is predicted as 9.03to 9.22 mm in the center. From Fig. 2 and Liu et al. (2013), thecircled number 1 denotes the periphery of the visible deformation,therefore the bulge extent diameter in the test can be taken as ap-proximately 50 mm. Further, the maximum deformation of the facesheet is measured as 7.44 mm (Liu et al. 2013).

It is shown from the comparison that the prediction reasonablyagrees with the test results, which justifies the prediction. Carefulexamination reveals that both the extent and depth of the bulge areoverestimated. The major reason is in the present study, the facesheet is assumed as a membrane, where bending and shear are to-tally neglected. In fact, the bending and shear of the face sheet doexist, which will absorb additional energy than analyzed in themodel, resulting in smaller bulge than predicted. However, the in-fluence of the bending and shear will become relatively insignifi-cant in full-sized blast events, because in real situation, the facesheet is typically thin and the strength of the close-range explosionis high (compared to the relatively thick face sheet and the smallcharge weight in this small-scale test).

There is no doubt that a field test with real structure/structuralmember protected by foam cladding is more convincing. To furthervalidate and calibrate the proposed model, a series of full-scale fieldtests will be conducted in the future for the foam cladding protectedstructural member subjected to localized blast.

Discussions

Practical Concern of Applying Foam Claddingfor Blast Mitigation

Before discussing the cladding response subjected to a close-rangeairburst, the selection of metal foam core, which is a practical prob-lem in design, is addressed first. The product of foam plateau stressand densification strain is equal to the energy absorbed per unitvolume or the specific strain energy. On one hand, for a specificblast, to minimize the foam thickness, the foam specific energyshould be as high as possible. Take open-cell aluminum foam asexample. With the empirical formulas for plateau stress and den-sification strain (Ashby et al. 2000), the specific strain energy is

Ws ¼ σplεD ¼ C2σys

�ρfρs

�3=2

�1 − 1.4

ρfρs

�ð25Þ

where C2 is a constant and 0.3 is for aluminum foam. According toEq. (25), the specific energy of foam depends only on the relativedensity, indicated in Fig. 7. Generally, only metal foam with rela-tive density lower than 0.3 exhibits exceptional capacity in energyabsorption (Gibson and Ashby 1997). Thus, in real applicationrange, the specific energy increases monotonically with increasingrelative density. To maximize volume efficiency, the foam shouldbe as dense as possible provided that the relative density is lessthan 0.3. In contrast, for a structure protected by a metal foam clad-ding, the transmitted load from the foam, which is roughly equal totwice the foam plateau stress, should not overload the protectedstructure. Therefore, the foam should be judiciously selected in abalanced approach: it should have a plateau stress just below halfthe structure tolerance so that the cladding thickness is minimized,and at the same time, the protected structure is not overloaded.

Parametric Study

With Eqs. (5) and (21), when the parameters of the close-range air-burst and cladding are known, the procedures for determining thebulge extent and depth are straightforward. After defining the blastload with empirical formulations and/or charts (U.S. Army 1986,2008), the bulge extent is obtained with Eq. (5). Subsequently, thebulge depth can be predicted with Eq. (21). Finally, the bulge depthdivided by the foam densification strain yields the minimum foamthickness for mitigating this specific close-range airburst. Whencertain foam part in the crushing direction is completely densified,the load transmitted to the protected structure will increasedramatically, even higher than that without the cladding (Ye andMa 2007) which should be avoided. It is emphasized again thatfor a specific blast, the required thickness of the metal foam coreshould be larger than the densified foam thickness rather than thebulge depth.

Fig. 7. Relationship between the specific energy of open-cell alumi-num foam under quasi-static compression and relative density

Fig. 8. Relationship between the bulge extent and peak load

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From Eq. (5), the nondimensional bulge extent is

ηR¼

ffiffiffi2

p ffiffiffiffiffiffiffiffiffiffiffiln

pc

σpl

rð26Þ

where R is the standoff distance between the explosion center andthe face sheet. From Fig. 8, for a specific foam core, the bulge ex-tent increases with the peak load in a decreasing rate. The reason isthat both the square root function and the logarithmic function in-crease with independent variable in a decreasing rate, and the non-dimensional bulge extent in Eq. (26) is a compound of these twofunctions. To create a larger bulge extent, a remarkably higher peakload is required.

Eqs. (5) and (26) for bulge extent prediction are valid when thepeak load pc is equal to or greater than the foam plateau stress σpl.If pc is smaller than σpl, these two equations predict complex val-ues not meaningful for application. In fact, it is obvious that therewill be no bulge in this situation. Further, if the peak load is exactlyequal to the foam plateau stress, based on Eqs. (5) and (21), boththe bulge extent and depth are zero. The prediction is valid.

Parametric study is conducted through analyzing a typical clad-ding subjected to a close-range airburst, where open-cell aluminumfoam of relative density 5% with a 0.67-MPa plateau stress is used.The standoff distance is fixed as 0.3 m and the charge weight variesto yield different loading intensities. The face sheets is 2 mm thick,made of aluminum, and are rate insensitive. Certain parametervaries whereas others remain unchanged to identify its influence.

From Fig. 9, the bulge depth increases with increasing peak loadon cladding. Recall that when the peak load increases, with thesame standoff distance, the bulge extent also increases. Therefore,a stronger blast results in not only a larger extent but also a deeperdepth. Similarly, as expected, the bulge depth increases with in-creasing impulse (per unit area) as shown in Fig. 10.

When a material is loaded dynamically, its yield strength isno longer a constant and becomes a function of the strain rate,delineated by Cowper-Symonds equation (Jones 1989)

σyd

σys¼ 1þ

�ε̇D

�1=q

ð27Þ

where ε̇ is the strain rate; σyd the dynamic yield strength; and Dand q are two material constants determined by experiments.

For mild steel, D is 40 and q is 5, whereas for aluminum, D is6,500 and q is 4 (Jones 1989), implying that mild steel is rate sen-sitive whereas the aluminum is not. On one hand, the strain rateeffect of aluminum foam, used as core of almost all the metal foamcladding core, is negligible (Ashby et al. 2000). In contrast, a dy-namic factor γ is added into the formulation to account for thedynamic effect of face sheet under large strain rate deformation.The value of γ can be estimated from Cowper-Symonds relationwith well-established material constants D and q for a specificmaterial and the estimated strain rate ε̇. The strain rate in the re-sponse process in the present study is a function of time, not aconstant.

Based on Eq. (9), the initial velocity of the bulge center, andEq. (21), the final bulge depth (obviously, final velocity is zero),the nominal average strain rate for the center point can be roughlyestimated. However, it may deviate remarkably from the strain ratetime history in the real situation because the estimated is not fromthe response time history (recall energy method is used in thecurrent study). Fig. 11 shows the relationship between increasingface sheet yield strength and bulge depth in the case study. In fact,in preliminary design, this kind of figure for a specific event can beused to account for the strain rate effect of face sheet if it is ratesensitive. When the face sheet undergoes dynamic stretching, itsyield strength increases and becomes higher than that under quasi-static load, resulting in more energy absorption during the plasticstretching and consequently a smaller bulge depth. Therefore, in thepresent study, neglecting the strain rate effect of face sheet (assumeγ ¼ 1) leads to a slightly conservative prediction of the bulgedepth.

Fig. 12 indicates that the bulge depth decreases with increasingface sheet density and thickness. The mechanism is that subjectedto the same blast, the initial velocity field (integration over thebulge extent) imparted by the impulse is inversely proportionalto the mass of the face sheet within the bulge periphery—a productof density, thickness, and area. The kinetic energy gained, or theenergy to be dissipated, is proportional to the initial velocity fieldsquared (integration over the bulge extent). Therefore, the greaterthe face sheet density and thickness, the less energy to be dissi-pated. For this reason, the face sheet should not be too thin; other-wise, a larger amount of energy is imparted. However, a claddingwith heavy face sheet is not practical in terms of mass efficiency.

Fig. 9. Relationship between the bulge depth and peak load: upper andlower bounds

Fig. 10. Relationship between the bulge depth and impulse (per unitarea): upper and lower bounds

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Then, the face sheet density and thickness should be optimized toachieve a balance between acceptable protection effect and massefficiency.

Now, when the blast load is fixed and the foam relative density isvaried on which the plateau stress and densification strain depend,the bulge depth decreases, as expected, with increasing foam rel-ative density, indicated in Fig. 13. According to the principle offoam selection discussed previously, the foam relative density can-not be too high, or the protected structure is overloaded. As a ruleof thumb, the foam relative density should be judiciously selectedso that the plateau stress of the chosen foam is slightly smallerthan half the damage tolerance of the protected structure. The foamprotected structural member should be typically slabs (both two-way and one-way) and wide beams, whose surface is flat and thesmallest in-plane dimension is remarkably larger than the diameterof bulge induced by close-range blast shown in Fig. 14. The presentmodel cannot be applied to typical columns protected by foam clad-ding, even columns with rectangular cross-section and flat surface.The reason is that the typical column width may be not remarkablylarger than the bulge, thus the face sheet outside the bulge cannotprovide tension for the bulge (if a flat-surfaced column is hugecompared to the bulge, the current model can be applied; however,such scenarios are quite few). For the same reason, obviously, thecurrent model is not applicable to columns with round cross-section.

Limitations and Future Study

The blast mitigation philosophy by attaching cladding with foamcore to the exterior of structure is that when subjected to a blastload, the cladding sacrifices: it undergoes large deformation whichabsorbs a large amount of energy and at the same time controls thetransferred load to the protected structure to an acceptable level (in-cident load roughly equal to the foam crushing strength). After theblast event, a new cladding will replace the sacrificed one. To pre-vent the protected structure from permanent damage, the thicknessand crushing strength of the foam core should be judiciously de-signed to ensure the protected structure to deform within the elasticlimit. The elastic deformation of typical structure members is gen-erally significantly small compared to the cladding crushing, whichis intentionally designed large to lower down the transferred load

Fig. 13. Relationship between the bulge depth and foam relativedensity: upper and lower bounds

Fig. 11. Relationship between the bulge depth and face sheet yieldstrength: upper and lower bounds

Fig. 12. Relationship between the bulge depth and: (a) face sheetdensity; (b) face sheet thickness, both upper and lower bounds

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intensity to the protected structure. In fact, the current study isbased on an assumption that the cladding is properly designedwith sufficient thickness so that under a specific blast event, theprotected structural member deforms elastically. Otherwise, if themember experiences large inelastic deformation, major damage, oreven collapse, the large structural deformation of the member willinvalidate the proposed model, which is out of the scope of thecurrent study.

Typically, the thickness of the face sheet of a protective claddingis small, always not greater than several millimeters. When sub-jected to a close-range blast, the deflection is always greater thanhalf the face sheet thickness. Therefore, the bending and shear re-sistance is relatively small and neglected. However, if the face sheetis relatively thick, the bulge extent will be overestimated because ofthe negligence of bending and shear, subsequently unreasonablepredicted bulge depth. In this case, the face sheet cannot be treatedas a rigid plastic membrane and a refined analysis with bending andshear incorporated is required.

It is evident that the bulge profile in the present study is an en-gineering approximation of the real situation. The prediction accu-racy may be improved if more complicated approaches are used.For instance, the blast load applied on the face sheet and the bulgeprofile can be modeled by refined functions. However, the diffi-culty of mathematical manipulation increases. Besides, after theinitial velocity field of the face sheet is assumed, the final bulgeprofile can be determined by integrating its velocity field with time,with retardation from the dynamic foam crushing and face sheetstretching taken into account. Nevertheless, it is difficult or evenimpossible to get a close-form solution because generally, there isno explicit integral expression if function e−kr2 (k is a real constantand r is variable) is included in the integrand (Thomas et al. 2010).Then, the numerical integration scheme must be used, which maybe more complex and expensive than conducting a finite elementanalysis. In fact, it is neither meaningful nor necessary for a pre-diction; the accuracy of the analysis procedure is higher than thatof the input data (Biggs 1964). Recalling the scattering of the testdata and the lack of knowledge for close-range blasts (Smith andHetherington 1994; U.S. Army 1986, 2008), the present studyestablishes a reasonable approximate procedure that balances the

prediction accuracy and solution expense in an explicit andready-to-use manner.

With formulas in the present study, the bulge extent and depthof a metal foam cladding subjected to a close-range blast can bequickly estimated, although the accuracy is not very high and onlydepth upper and lower bounds are established. A calculator, ratherthan a computer, is sufficient for such preliminary design.

Conclusions

The response of a blast mitigation cladding with metal foam coresubjected to a close-range airburst is estimated. Whereas the stand-off distance between the explosion center and the cladding is rel-atively small compared to the cladding size, the in-plane dimensioncan be considered sufficiently wide. To reflect the real situation,the boundary of the blast-induced bulge is released. With theenergy method, the extent and depth of the bulge are predictedexplicitly. When the parameters of a specific blast event and clad-ding are known, the bulge can be straightforwardly calculated.Subsequently, with the estimated bulge depth, the minimum thick-ness of the foam core can be readily obtained, which is of sig-nificance in engineering design. The prediction can be used inpreliminary design of blast mitigation claddings with metalfoam core.

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Fig. 14. Illustration of a metal foam cladding protected structural member subjected to a close-range blast: (a) side view; (b) plan view; note: the blastload outside the bulge, not causing cladding deformation, is not illustrated

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