The impact of sand slugs against beams and plates: coupled discrete/continuum calculations

Transcription

1 The impat f sand slugs against beams and plates: upled disrete/ntinuum alulatins T. Liu, N.A. Flek, V.S. Deshpande * 1 Department f Engineering, University f Cambridge, Trumpingtn Street, Cambridge CB 1PZ, UK. and H.N.G. Wadley 1 Department f Material Siene & Engineering, Shl f Engineering and Applied Siene, University f Virginia Charlttesville, VA 903, USA. Abstrat Cupled finite element (FE) and disrete partile simulatins f sand lumns (slugs) mprising idential partiles impating end-lamped mnlithi and sandwih beams as well as plates are used t eluidate the sand-struture interatin. The majrity f the alulatins are perfrmed using slugs mprising idential irular ylindrial partiles impating beams. These alulatins demnstrate that the lading due t the sand was primarily inertial with little fluid-struture interatin effets, i.e. the mmentum transmitted t the strutures was apprximately equal t the mmentum f the inming sand slug. The lading and respnse f the strutures is well haraterised by aunting fr just tw parameters: the initial mmentum f the sand lumn and the lading time given by the rati f the height f the lumn and the initial velity f the sand partiles. Sandwih beams with thik and strng res were shwn t signifiantly utperfrm mnlithi beams f equal areal mass. This perfrmane enhanement is shwn t result frm the sandwih effet whereby the high bending strength f sandwih beams gives superir perfrmane when the beam defletins are less than the re thikness. The main sand-struture interatin mehanisms eluidated by the beam alulatins are shwn t hld in a threedimensinal setting where slugs mprising spherial sand partiles are impated against edge-lamped irular mnlithi and sandwih plates. The fluid-struture interatin effets were again fund t be negligible and sandwih plates with thik strng res ut-perfrmed mnlithi plates f equal areal mass. Finally, we present a fam analgy where impat f the sand partiles is shwn t be equivalent t the impat f an equivalent rushable fam prjetile. The alulatins n the equivalent prjetile are signifiantly less intensive mputatinally and yet give preditins t within 5% f the full disrete partile alulatins fr the mnlithi and sandwih beams and plates. These fam prjetile alulatins suggest that "mdel" materials an be used t simulate the lading by sand partiles within a labratry setting. Keywrds: fluid-struture interatin, disrete/ntinuum upling, dynami lading. Submitted t Jurnal f the Mehanis and Physis f Slids, May 011 * Crrespnding authr, (vsd@eng.am.a.uk)

2 1. Intrdutin The air and water blast resistane f strutures has reently reeived nsiderable attentin with the aim f designing lightweight blast resistant strutures. Several reent theretial studies have shwn that sandwih strutures subjeted t water blast lading utperfrm mnlithi strutures f equal mass, see fr example Flek and Deshpande (004) and Xue and Huthinsn (004). These preditins have been nfirmed experimentally by Wadley et al (008) and Wei et al (009). This enhaned perfrmane is mainly due t fluid-struture interatin effets with a smaller fratin f the impulse being transmitted int sandwih strutures mpared t their mnlithi unterparts. On the ther hand, under air blast lading, sandwih strutures prvide nly marginal benefits ver mnlithi strutures as the fluidstruture interatin effets are signifiantly less imprtant (Kambuhev et al., 007). The main aim f this study is t investigate the relative perfrmane f mnlithi and sandwih strutures subjeted t lading by the impat f high velity granular media that are aelerated by the explsin f shallw buried explsives (landmines). The prttypial prblem is skethed in Fig. 1 where the sandwih struture represents the under-bdy f a vehile. The phenmena in play are mplex and an typially be separated int three sequential phases: (i) The transfer f impulse frm the explsive t surrunding sil, leading t the frmatin f a dispersin f high velity partiles, (ii) the prpagatin (with spreading) f the expanding sil ejeta and (iii) impat f the sil ejeta with the struture leading t mmentum transfer t the prtetin system (i.e. the sil-struture interatin). The fus f this study is n the third phase; the interatin f high velity granular media with a struture and the subsequent respnse f the struture. Diret experimental haraterizatins f buried explsive events (Bergern et al., 1998; Braid, 001; Wekert and Andersen, 006; Neuberger et al., 007) have been the main fus f researh t-date and have led t the develpment f empirial mdels t quantify impulsive lads impsed by the sil ejeta (Westine et al., 1985) as well as strutural design des suh as thse prpsed by Mrris (1993). The preditive apability f these empirial mdels is limited and annt be extraplated t new strutural nepts suh as the sandwih struture skethed in Fig. 1. Mst researh at develping a preditive apability has fussed n develping apprpriate nstitutive mdels fr the sil that an be implemented within Eulerian numerial des. These Eulerian des an then be upled t Lagrangian finite element (FE) alulatins t simulate the strutural respnse. Readers are referred t Grujii et al. (006) and Grujii et al. (008) fr a detailed analysis f sil mdels used t simulate landmine explsins. Ntable amng these are the s-alled three phase mdel f Wang et al. (003) and Wang et al. (004) whih is a mdified Druker and Prager (195) mdel and the prus-material/mpatin mdel develped by Laine and Sandvik (001). The sil mdels listed abve are restrited t a regime where the paking density f the sil is suffiiently high that the partile-partile ntats are semi-permanent. While these mdels are apprpriate during the initial stages f a buried explsin r during an avalanhe r land slide, their appliability when widely dispersed partiles impat a struture is questinable. Deshpande et al. (009) mdified a nstitutive mdel f Bagnld (1954) t develp a ntinuum sil mdel appliable t sils in bth their densely paked and dispersed states. Hwever, the suessful implementatin f this mdel within a upled Eulerian-Lagrangian mputatinal framewrk has been elusive beause f well knwn mputatinal

3 prblems assiated with the analysis f lw density partile sprays; see Wang et al. (004) fr a disussin f these numerial issues. An alternative mdelling strategy has reently been emplyed by Brvik et al. (011) and Pingle et al. (011) wherein the lw density sil is treated as an aggregate f partiles with the ntat law between the partiles ditating the verall aggregate behaviur. This apprah has several advantages: (i) there is n need t make a-priri assumptins abut the nstitutive respnse f the aggregate (this bemes an utme f the simulatins), (ii) it prvides a fundamental tl t study the essential physis f the sand-struture interatin and (iii) given that the sand is represented by a disrete set f partiles we d nt fae the usual numerial prblems assiated with slving the equatins assiated with the equivalent ntinuum desriptins. Pingle et al. (011) have reently investigated the respnse f rigid targets impated by lumns f partiles. This rather idealised but fundamental fluid-struture interatin (FSI) prblem is the sand-blast analgue t the lassial water-blast FSI prblem studied by Taylr (1963). On the ther hand, Brvik et al. (011) analysed the respnse f mnlithi plates laded by spherially expanding sand shells and mpared their preditins with measurements. Their study demnstrated the apabilities f this methdlgy in mdelling mplex sand-struture interatin events. Hwever, given the mplexity f the bundary value prblem analysed, it was hard t extrat a mre fundamental understanding f the key phenmena at play. 1.1 Apprah and spe Our aim in this study is t nsider a relatively simple lading senari invlving a spatially unifrm lumn f disrete, idential partiles impating mnlithi and sandwih strutures. This lumn mprising either idential irular ylindrial r spherial partiles will subsequently be referred t as a sand slug. While this slug is nt diretly representative f the ejeta reated during a landmine explsin, it enables us t develp a physial understanding f the phenmena at play and mpare the relative perfrmane f mnlithi and sandwih strutures fr a well haraterised lading senari. The utline f the paper is as fllws. First, the methdlgy emplyed t uple the disrete and ntinuum Lagrangian FE alulatins is desribed. Next, the respnse f lamped mnlithi and sandwih beams impated by sand slugs mprising idential irular ylindrial partiles is disussed. Finally we extend the analysis t three-dimensins (3D) and present the respnse f lamped irular plates laded by ylindrial sand slugs mprising idential spherial partiles.. Summary f the plane strain upled disrete element/finite element methdlgy The plane strain bundary value prblems are skethed in Fig. and mprise a slug f a lw density granular medium (subsequently referred t as sand) whih impats a lamped beam. The granular medium was mdelled as a set f tw-dimensinal (D) disrete idential irular ylindrial partiles using the GRANULAR pakage in the multi-purpse mleular dynamis de LAMMPS (Plimptn, 1995) while the beams were mdelled using a D finite strain Lagrangian finite element (FE) framewrk. We shall first briefly desribe the disrete partile and FE numerial methds and then prvide details f the upling f these tw tehniques. 3

4 .1 Disrete element alulatins The granular medium was mdelled as D ylindrial partiles f diameter D lying in the x 1 x plane (thikness b in the x3 -diretin). The granular pakage in LAMMPS is based n the sft-partile ntat mdel intrdued by Cundall and Strak (1979) and extended t large sale simulatins by Campbell and -wrkers (Campbell and Brennen, 1985; Campbell, 00). A shemati f the sft-partile ntat mdel is shwn in Fig. 3 and mprises a linear spring K n and linear dashpt with damping nstant n, gverning the nrmal mtin and a linear spring K s and Culmb fritin effiient, gverning the tangential mtin during the ntat and defrmatin f tw partiles f mass m p. Define unit vetrs e ˆn and e ˆs suh that e ˆn is in the diretin f the utward nrmal t the partiles alng the line nneting the entres f the tw partiles and eˆ ˆ n es is a unit vetr in the x 3 diretin. The fre ating n eah partile is then given as Feˆ ˆ n nfe s s if n0 F (.1) 0 therwise where r is the distane between the partile entres n r D is the interpenetratin. The nrmal fre is given as Fn Kn n meff n n (.) where m eff is the effetive r redued mass f the tw ntating bdies. In the alulatins presented here impats were either between partiles r between partiles and the beams with meff mp / and mp in these tw ases, respetively. It nw remains t speify the tangential fre F s. Define s as the tangential displaement rate between the ntating partiles. We then stipulate F s via an elasti-plasti relatin s as t simulate Culmb fritin, i.e. K ss if Fs Fn r F ss 0 F s (.3) 0 therwise As disussed by Bathurst and Rthenburg (1988) the Pissn's rati f the slid is related t the rati Ks / K n, while K n is a funtin f the Yung's mdulus. Furthermre, it is lear that the damping nstant n, will determine the lss f energy during nrmal llisins and will therefre be diretly related t the effiient f restitutin e fr nrmal llisins. Cnsequently, apprpriate values f n an be determined frm the knwn r measured effiient f restitutin via the relatin e exp. (.4) 8 K ( ) 1 1/ n nm p Nte that this partile interatin mdel leads t a llisin time fr individual binary llisins t e given by ln( e) te. (.5) n Thus, in the limit f plasti llisins with e 0, the ntat time t e. This simplest ntat mdel has the feature that the ntat prperties an be readily related t the effiient f restitutin e used in mst analytial treatments. 4

5 The alulatins were perfrmed using the GRANULAR pakage in the mleular dynamis de LAMMPS. The Newtn equatins fr bth the translatinal and rtatinal mtins f the partiles were integrated using a Verlet time-integratin sheme (i.e. Newmark-Beta with 0.5 ). The time-step was typially taken t be abut t e /10 in rder t ensure aurate integratin f the ntat relatins, Eqs. (.1)- (.3) and equal t that used in the finite element alulatins desribed belw.. Tw-dimensinal finite element alulatins The D plane strain alulatins were perfrmed using an updated Lagrangian finite element (FE) sheme with the urrent nfiguratin at time t serving as the referene. The rdinate x i dentes the psitin f a material pint in the urrent nfiguratin with respet t a fixed Cartesian frame, and v i is the velity f that material pint. Fr the plane strain prblem under nsideratin, the priniple f virtual pwer (negleting effets f gravity) fr a vlume V and surfae S is written in the frm dv Tv ds v v dv (.6) where ij is the Cauhy stress, ij 0.5vi, j vj, i n the surfae S T V ij ij S i i i i T V is the strain rate, T i the tratins S due t the impats f the partiles while is the material density in the urrent nfiguratin. The symbl dentes arbitrary virtual variatins in the respetively quantities. A finite element disretisatin based n linear, plane strain three nde triangular elements (i.e. nstant strain triangles) is emplyed. When the finite element disretisatin f the displaement field is substituted int the priniple f virtual pwer (.6) and the integratins are arried ut, the disretised equatins f mtin are btained as U M F (.7) t where U is the vetr f ndal displaements, M is the mass matrix and F is the ndal fre vetr. An expliit time integratin sheme based n the Newmark β- methd with 0 was used t integrate Eq. (.7) t btain the ndal velities and the ndal displaements. A lumped mass matrix is used in (.7) instead f a nsistent mass matrix, sine this is preferable fr expliit time integratin predures, fr bth auray and mputatinal effiieny..3 Cupling f the disrete element and finite element alulatins The upling between the disrete and finite element alulatins was arried ut as fllws. At time t, ntat between the partiles and the beam in its urrent nfiguratin was deteted. The displaement n is defined as the n r D/, where r is the distane between the partile entre and the ntat pint n the beam while s is the tangential displaement rate f the partile relative t the beam measured. Nte that the rate n is the relative velity f the partile and the pint f ntat n the beam surfae alng the nrmal t the beam surfae at the pint f ntat n the beam. The nrmal and tangential ntat fres are then alulated using Eqs. (.1) - (.3). These nrmal and tangential fres are then transfrmed t The velity f any pint n the beam is alulated by interplating the ndal velities using the shape funtins f the nstant strain triangle elements. 5

6 the glbal rdinate system and a vetr f equivalent ndal fres determined fr eah element n the surfae f the beam. These vetrs f elemental ndal fres are then inserted int the vetr f glbal ndal fres F in Eq. (.7). Subsequently, the disrete and finite element equatins were integrated as desribed in Setins.1 and. and the new psitins and velities f the partiles in the disrete alulatins and material pints in the finite element alulatins determined at time t t. 3. The mnlithi and sandwih beam bundary value prblems The tw plane strain bundary value prblems investigated in this study are skethed in Figs. a and b. In bth prblems a slug f height H and width a impats the lamped beams entrally and nrmally at time t 0. The slug mprises disrete idential ylindrial partiles f diameter D made frm f slid material f density s and randmly paked with unifrm spatial relative density. Eah partile has an initial velity v in the negative x diretin. In Fig. a shws a sketh f a mnlithi beam f span L and made frm a material with a density m. The beam has a thikness h and is fully lamped at its tw ends. The equivalent lamped sandwih beam f span L skethed in Fig. b mprises tw idential fae sheets f thikness h f als made frm the same material and separated by a mpressible fam re f density and thikness. The areal mass m b f the mnlithi and sandwih beam are equal suh that mb mh h f m. (3.1) In rder t redue the variables needed t speify the sandwih gemetry we restrit attentin in this study t the situatin where mass f the sandwih is divided equally between the tw faes and the re, i.e. mb hfm (3.) 3 This mass distributin was shwn t be near ptimal fr sandwih beams under impulsive lading (Xue and Huthinsn, 004). Thus, given a mnlithi beam f areal mass M there remain nly tw free gemetri variables fr the sandwih beam (, ) that are nstrained via Eq. (3.). In all the alulatins presented here, the sandwih and mnlithi beams have a halfspan L 0.5 m. The mnlithi beam is made f DH 36 steel and thikness - h 50mm rrespnding t an areal mass mb 400 kgm fr an assumed steel -3 density m 8000 kgm. Tw sandwih gemetries with aspet rati / L 0.11 and 0.33 are investigated. Fr eah fr these gemetries the thikness f the sandwih beam fae sheets and re density fllws frm Eq. (3.). Only half the beam spans were mdelled with symmetry bundary nditins impsed alng x 1 0 and all degrees f freedm nstrained alng x 1 L in rder t simulate the lamped bundary nditins. The beams were disretised using quadrilaterals mprising fur nstant strain triangles (Peire et al., 1984) with elements f size h /15 and h /8 in the mnlithi and sandwih beams, respetively. f 6

7 3.1 Material prperties The mnlithi beam and sandwih beam fae sheets were made frm DH36 steel. This steel was mdelled as an elasti-plasti J flw thery slid with a Yung s -3 mdulus E 00 GPa, Pissn s rati 0.3 and density m 8000 kgm. The rate dependent uniaxial yield strength f the steel was speified via the relatin (Vaziri et al., 007) 0.4 eff Y Y 1 1.5( eff) ln,(3.3) 0-1 where Y 470 MPa, 0 1s and eff is the vn-mises equivalent plasti strain rate. The re in the sandwih beams was mdelled a hmgeneus mpressible visplasti rthtrpi fam-like material fllwing Tilbrk et al. (006). Assume the rthtrpi axes x i f the re are aligned with the axes f the beam as skethed in Fig., i.e x 1 and x are aligned with the lngitudinal and transverse diretins, respetively. Intrdue the stress and plasti strain matries in the usual way as T T σ ( 1,, 3, 4, 5, 6 ) ( 11,, 33, 13, 3, 1), (3.4) and p p p p p p p T p p p p p p T ε ( 1,, 3, 4, 5, 6 ) ( 11,, 33, 13, 3, 1), (3.5) respetively. Cmplete deupling f material respnse between the rthgnal material diretins is assumed and we define the plasti strain rate p i via an verstress relatin as p i Yi ( i ) p p if i Yi ( i ) i (3.6) 0 therwise, where the yield strength ( p p Yi i ) is a funtin nly f the plasti strain i and is the material vissity. The ttal strain rate i is btained by supplementing the abve anistrpi plastiity mdel with istrpi elastiity suh that p i L ij j i sign( i ) (summatin ver j). (3.7) In the ase f istrpi elastiity, the mpliane matrix L ij f the re material is speified in terms f the Yung s mdulus E and Pissn s rati as 1/ E / E / E / E / E / E L (3.8) (1 ) / E 0 0 sym (1 ) / E 0 (1 ) / E We emply an istrpi elasti respnse fr simpliity; this suffies as the re respnse is ditated by the plasti branh. The abve elasti-plasti nstitutive relatin is expeted t be adequate t mdel sandwih res suh as the squarehneymb re r the rrugated re; see fr example Xue et al. (005). 7

8 In all the alulatins reprted here we assume an istrpi plasti respnse in the sense that the all the strengths Y i are assumed t be equal and strain hardening is negleted in all diretins ther than the transverse diretin. The transverse strength p Y is assumed t be independent f the plasti strain up t a nminal densifiatin strain D : beynd densifiatin a linear hardening behaviur is assumed with a very large tangent mdulus E t 0. 1E. The re has a Yung s mdulus E E where the relative density f the fam re / m (assuming that the fam is made f the same slid material as the fae sheets) and a Pissn s rati 0.5 ; numerial experiments nfirmed that the preditins are insensitive t the elasti stiffness f the re. The nminal densifiatin strain D is related t the re relative density via the relatin (Ashby et al., 000) D 11. 5, (3.9) while the transverse strength Y is varied in the parametri studies reprted subsequently. Finally, we nte that the vissity f the re was hsen suh that the shk width (Radfrd et al., 005) l D, (3.10) v 10 with v interpreted as the initial impat velity f the sand partiles. This presriptin ensures that the shk width is always muh less than the re depth yet is larger than the mesh size. Nte that large gradients in stress and strain ur ver the shk width and thus a mesh size smaller than l is required t reslve these gradients aurately. 3. Prperties f the impating partiles The partiles prperties were hsen s as t mimi silia sand partiles. Unless, therwise speified the fllwing parameters were emplyed in all the alulatins presented here. The idential ylindrial partiles f diameter D 00 μm and -3 thikness b 1mm (in the x 3 diretin) had a slid material density s 700 kg m. -1 The nrmal stiffness between the partiles was taken t be Kn 7300 knm and the -effiient f restitutin fr bth impats between the partiles and the partiles and the beam assumed t be e 0.7 (i.e. the value f n fr inter-partile ntats was twie that fr impats between the partiles and the beam). Fllwing Silbert et al. (001) we fixed the rati Ks / Kn / 7 and the referene value f the fritin effiient was assumed t be Impat against the mnlithi beam Cnsider the mnlithi end-lamped beam impated by a sand slug at time t 0 as skethed in Fig. a. Define the nn-dimensinal lading parameters a sh I a, m,, and I (4.1) L m m / b b Y m where I shv is the areal mmentum f the impating slug, mb mh is the areal mass f the beam. Dimensinal analysis ditates that fr a given set f beam 8

9 material and sand prperties, the defletin w f the mid-span f the beam has the funtinal frm w h m w f t,,, a, I, m,. (4.) L L s t t/ L / is the time nrmalised by the strutural respnse time f a where m Y plasti string f length L and made frm a material f density m and yield strength Y. Reall that all alulatins are presented fr a beam f aspet rati h/ L 0.05 and a parametri study is reprted fr the sensitivity f the respnse t the lading parameters a, I, m and. Unless therwise speified these parameters had the fllwing referene values: a 0., 0., m (sand slug height H 300 mm ). We nte in passing that nsistent with the findings f Pingle et al. (011) fr sand impating a rigid target, the ntat prperties f sand partiles have a very minr effet n the respnse f the beam; varying K n by tw rders f magnitude and hanging e and frm by fatrs f five had less than a % effet n the predited beam defletins. 4.1 Effet f sand mmentum Preditins f the nrmalised mid-span defletin w versus nrmalised time t respnse f the beam are pltted in Fig. 4a fr three seleted values f the nrmalised areal mmentum f the sand I (the mmentum was varied by keeping all parameters fixed at their referene values and inreasing the sand impat velity frm -1-1 v 100 ms t 400 ms ). The imum defletins inrease with inreasing I and all ur at t 1. The imum nrmalised defletins w w / L are summarised in Fig. 4b as a funtin f I while snapshts shwing the defrmatin f the sand slug and the beam at seleted times are inluded in Fig. 5 fr the I 0.67 ase. In the initial stages f the defrmatin, the sand slug mpats against the beam with a near planar densifiatin frnt prpagating thrugh the slug frm the impated t the distal end f the slug. With inreasing time, there is signifiant lateral spreading f the slug and the densifiatin frnt n lnger remains planar. This densifiatin frnt reahes the distal end f the slug at tv / H 1 and subsequently the beam defletin ntinues t inrease due t its aquired mmentum while the sand flws laterally. Elasti vibratins f the beam ntinue after the beam has attained its peak defletin. We preed t quantify the lading f the beam due t the impat f the sand slug. Fr this purpse we alulate the pressure p exerted by the sand partiles n the beam as fllws. At any time t, there are M sand partiles in ntat with ne f the finite elements n the impated surfae f the beam. In the undefrmed state, the mid-pint f that element n the surfae f the beam had a x 1 rdinate X m. The pressure at latin X m at time t is then defined as px (, t) m M i1 l e F i (4.3) 9

10 i where F is the ntat fre in the x diretin between the i th sand partile and the beam and l e is the undefrmed length f the finite element n the surfae f the beam. The nrmalised pressure p /( sv ) is pltted in Fig. 6 fr the I 0.67 ase as a funtin f the spatial rdinate X, where X dentes the x 1 rdinate a material pint n the beam surfae in its underfrmed nfiguratin, i.e. X 0 is at the beam mid-span and X L is at the supprts. Results are shwn in Fig. 6 fr three seleted times suh that at t 0.1 and 0.5, the sand slug has nt fully mpated while at t 1, the beam has reahed its peak defletin with the sand flwing laterally as seen in Fig. 5. It is lear frm Fig. 6 that while the sand slug is mpating, the spatial pressure prfile ver the beam surfae is apprximately retangular with the pressure mainly ating ver the width f undefrmed slug, i.e. ver a width a at the mid-span. The magnitude f this pressure is initially apprximately sv but dereases with inreasing time. After the sand slug has fully mpated, it mainly flws laterally, and then exerts a negligible pressure n the beam. Given that the pressure n the beam is apprximately spatially unifrm ver a entral path f width a and zer utside this path we define an average pressure L 1 p() t pdx a. (4.4) 0 The tempral variatin f the nrmalised pressure p /( sv ) is shwn in Fig. 7a fr tw values f I. Here we hse t plt the pressure as funtin f the nrmalised time tˆ tv / H fr reasns that will be larified subsequently. We make tw main bservatins: (i) The pressure p/( sv) 1 at t 0 in all ases and then dereases slwly with inreasing time. (ii) The pressure drps t nearly zer at tˆ 1 in all ases. We ratinalise these bservatins as fllws. First, as disussed in Pingle et al. (011), the lading due t the sand is mainly inertial and hene sales as s v, where v is the relative velity between the sand and the beam. At time t 0, the beam is statinary and hene v v resulting in a pressure p/( sv) 1. With inreasing time, the velity f the beam inreases whih implies that v dereases. Hene the pressure p gradually redues. Send, the distal end f the sand slug reahes the beam at time t H / v and the subsequent mtin f the sand is mainly in the lateral diretin. Thus, the pressure drps dramatially at tˆ 1. The rrespnding preditins f the nrmalised mmentum It / I transmitted int the beam are pltted in Fig. 7b, where I () t pdt. (4.5) t The tempral variatin f the mmentum I t displays tw knees. One knee urs at tˆ 1 where after the rate I t dereases signifiantly. The transmitted mmentum I t ntinues t rise until It / I 1 at whih pint the send knee urs and I t remains nstant thereafter. It is lear that the first knee rrespnds t the instant when the distal end f the sand slug reahes the beam and there is a signifiant drp in the t 0 10

11 pressure exerted by the sand n the beam. Subsequently, the lateral flw f the sand alng the defrmed beam exerts a small pressure as the sand aquires a small velity in the psitive x due t the defrmed prfile f the beam (see snapshts in Fig. 5). This small pressure results in the gradual inrease in I t. After the beam reahes its imum defletin and rebunds bak elastially, the sand lses ntat with the beam (see Fig. 5) and hene I t plateaus ut. Reall that the beam attains its peak defletin at t 1, irrespetive f the value f I. Thus, the send knee in Fig. 7b urs values f ˆt that inrease with inreasing I (in these simulatins v inreased with inreasing I ). 4. Parametri study We investigate here the sensitivity f the imum defletin w t the parameters m, and a. First nsider the effet f m and. The effet f these tw parameters is best understd by examining the rati f the lading time due t the sand slug and the respnse time f the beam. Reall that the respnse time f the beam is apprximately L m / Y while the lading time is H / v. Thus, is given by m h m (4.6) I L s Qiu et al. (003) as well as Xue and Huthinsn (003) have demnstrated that the dynami respnse f the beam is nly dependent n I fr values f 0.01, i.e. the lading is impulsive and the respnse f the beam is independent f the applied pressure. Fr higher values f, imum defletins derease with inreasing as the respnse nw depends n bth the applied pressure and the impulse. We shall shw subsequently that effet f bth m and is adequately aptured via the single parameter. Calulatins were nduted by varying m ver the range 0.05 t 1. fr tw seleted values f I with and a bth held fixed at the referene values f 0.. The preditins f w are summarised in Fig. 8 (the filled squares) in terms f : the imum defletins are insensitive t m fr 0.01 and then derease with inreasing m r. Next alulatins were perfrmed with m and a held fixed at their referene values and the effet f ( ) investigated again fr I 0.67 and 1.0. The preditins f w are inluded in Fig. 8 (pen irles). It is evident that the dependene f w n bth m and is adequately aptured via the single parameter. The dependene f the imum defletins w n the nrmalised width a f the sand slug is summarised in Fig. 9a fr tw seleted values f I and all ther parameters held fixed at their referene values. Similar t the impulsive lading results f Qiu et al. (005), the defletins inrease with inreasing a fr a given value f I. Snapshts shwing the defrmatin f the beam and sand slug are 11

12 inluded in Fig. 9b fr the ase with I 0.67 and a 0.8. The defrmatin mde is very similar t that seen in Fig. 5 fr the a 0. ase. Thus, the inrease in the defletin is nt due t a hange in the defrmatin mde but rather nly beause the ttal mmentum f the sand slug inreases with inreasing a with all ther parameters held fixed. 4.3 The effet f the lateral spreading f the sand In rder t investigate the effet f the spreading f the sand, we perfrm alulatins by inluding a nstraint in the simulatins suh that n sand partiles an mve int the regin x 1 a, i.e. the width f the sand slug remains nstant during the entire defrmatin histry. Time snapshts shwing the defrmatin f the beam and sand slug frm a simulatin emplying this nstraint are inluded in Fig. 10a fr I 0.67 and the ther parameters held fixed at their referene values. The defrmatin f the sand slug in Fig. 10a is reminisent f the impat f a fam n a target with a planar densifiatin frnt travelling frm the impated end f the slug twards the distal end (Radfrd et al., 005). After the densifiatin frnt reahes the distal end it reflets as a tensile wave and the sand spalls and travels in the negative x diretin. Preditins f the variatin f w with I are inluded in Fig. 10b fr tw values and all ther parameters held fixed at their referene values. The figure inludes tw types f simulatins: in the unnstrained simulatins lateral spreading f the sand is permitted while in the nstrained simulatins n spreading f the sand int the regin x1 a is permitted. The preditins f w fr the 0. ase are nearly idential in the nstrained and unnstrained ases while fr 0.45, w is abut 5% higher in the nstrained ase mpared t the unnstrained ase. These results indiate (i) the lateral spreading des nt have a signifiant effet n the defrmatin f the beam and (ii) the additinal defletin in the nstrained 0.45 ase is due t the strnger bune-bak f the sand at higher values f as disussed by Pingle et al. (011). This strnger bune-bak results in a higher transmitted mmentum int the beam and a slightly enhaned defletin. 4.4 The fam analgue Reall that nstraining the sand against lateral flw des nt signifiantly affet the respnse f the beam. Mrever, the impat press in this nstrained ase as seen in Fig. 10a is very reminisent f a fam impating a rigid statinary target. Pingle et al. (011) develped an analgy between a nstrained sand slug and a fam prjetile with all prperties f the fam derived frm the prperties f the sand partile. Here, we investigate whether suh a fam mdel an als be used t predit the respnse f beams impated by sand slugs. Plane strain finite defrmatin FE alulatins f fam prjetiles impating beams were nduted using the mmerial FE pakage ABAQUS. Similar t the alulatins desribed abve, nly half the beams were mdelled in the alulatins with symmetry impsed at x 1 0 and all degrees f freedm nstrained at x 1 L t simulate the lamped bundary nditins. The beam and the fam prjetile were disretised using 4-nded plane strain elements with redued integratin (CPE4R in 1

13 the ABAQUS ntatin). Typially, the elements were square and had a size h /15 in the beam and H /100 in the fam prjetile. The prjetile was given an initial velity v and brught int ntat with the beam at mid-span at time t 0 with ntat between the beam and prjetile mdelled using the general ntat ptin in ABAQUS. The beam material was mdelled as elasti-plasti using J flw thery and prperties as speified in Setin 3.1. The prperties endwed t the fam prjetile are thse speified by Pingle et al. (011) and detailed belw. The fam f initial density s is mdelled as a mpressible ntinuum using the metal fam nstitutive mdel f Deshpande and Flek (000). Write s ij as the usual deviatri stress and the vn Mises effetive stress as e 3sijs ij /. Then, the istrpi yield surfae fr the metal fam is speified by ˆ Y 0, (4.7) where the equivalent stress ˆ is a hmgeneus funtin f e and mean stress m kk / 3 arding t 1 ˆ e m. 1 / 3 (4.8) The material parameter dentes the rati f deviatri strength t hydrstati strength, and the nrmalisatin fatr n the right hand side f relatin (4.8) is hsen suh that ˆ dentes the stress in a uniaxial tensin r mpressin test. An verstress mdel is emplyed with the yield stress Y speified by p Y ˆ, (4.9) p in terms f the vissity and the plasti strain-rate ˆ (wrk njugate t ˆ ). The p harateristi ( ) is the stati uniaxial stress versus plasti strain relatin f the fam. Nrmality f plasti flw is assumed, and this implies that the plasti p p Pissn s rati p / 11 fr uniaxial mpressin in the 1-diretin is given by 1/ / 3 p. 1 / 3 (4.10) The mdulus f the fam is related t the sand partile prperties via the relatin 3 3Kn E f 4b (4.11) and bth elasti Pissn s rati and a plasti Pissn s rati p 0. Sine the fam is used t mdel a lse aggregate f partiles whih have n stati strength up t densifiatin we assume a stati strength f the frm p ˆ pl -ln(1 D) therwise (4.1) where D is the nminal densifiatin strain given by D 1. (4.13) and pl sv / 100 (i.e. signifiantly less than the dynami strength but suffiient t give numerial stability). The disrete alulatins suggest that 0.9 and this 13

14 is the value used in Eq. (4.13). It nw remains t speify the material vissity. Radfrd et al. (005) have shwn that the linear vissity results in a shk within the fam f width l given by D l (4.14) sv Using (4.14), we hse a vissity s that w H in all the alulatins nsistent with the disrete alulatins f Pingle et al. (011) that suggested that the shk width was n larger than abut 10 partile diameters. Cmparisns between the full disrete alulatins and the preditins using the fam prjetile are shwn in Fig. 4. The fam prjetile lading aptures the defrmatin f the beam with remarkable auray. The slight ver-preditin f the defletin is due t the fat that the fam prjetile des nt apture the redutin in the applied pressure due t the lateral spreading f the sand. We nlude that the fam prjetile represents a gd mprmise between auray and mputatinal st t simulate the lading f beams by sand slugs. 5. Impat against sandwih beams We preed t analyse the respnse f sandwih beams subjeted t the lading by a high velity sand slug. These sandwih beams have the same areal mass m b 400 kgm and half-span L 0.5 m as the mnlithi beams analysed in Setin 4. Als reall that we partitin the mass f the sandwih beam as per Eq. (3.) s that a third f the mass is in the re and eah f the fae sheets. These nstraints imply that the defletin w f the bak fae f the sandwih beam at midspan may be written in nn-dimensinal terms as w m w f t,,,, a, I, m,. (5.1) L s where / L is the aspet rati f the beam and /( Y) is the nrmalised re strength: fr a given tplgy, a strething-dminated ellular fam re will have a unique value f independent f its density (Wadley et al., 003). Nte that the the nstraint, Eq. (3.), implies that the re relative density is nt an independent parameter, but is related t the beam aspet rati via the relatin mb m 3mL. (5.) The effet f the lading parameters ( ai,, m, ) was larified in Setin 4. Our aim here is t ntrast the mnlithi and sandwih beam respnses and hene we fus attentin n the sandwih beam parameter and present results fr tw beam aspet ratis 0.11 and 0.33 rrespnding t re relative densities 0.3 and 0.1, respetively. Unless therwise speified the fllwing referene parameters will be emplyed in all the alulatins reprted here: a 0., 0., m and -1 I 0.67 rrespnding t a impat velity v 400 ms. The sandwih parameters and are varied in this study and will be speified in all the alulatins reprted belw. 14

15 The simulated tempral variatin f the nrmalised defletin w is shwn in Figs. 11a and 11b fr sandwih beams with aspet ratis 0.11 and 0.33 respetively, and seleted values f the re strength. The respnse f the equivalent mnlithi beam is als inluded in Fig. 11. The defletins typially derease with inreasing fr bth values f. Hwever, there is a key differene between the tw sandwih beam aspet ratis. While the peak defletins w f the 0.11 sandwih beams with weak res (i.e. lw values) are mparable t their mnlithi unterparts, the 0.33 sandwih beams have a signifiantly lwer peak defletins mpared t the equal mass mnlithi beams ver the entire range f nsidered here. Snapshts frm the simulatins shwing the defrmatin f the sand slug and the 0.11 and 0.33 sandwih beams ( 0.11 ) are shwn in Figs. 1a and 1b, respetively. The verall defrmatin f the sand slug is similar t that seen in Fig. 5 fr the mnlithi beams. This press mprises first the transmissin f a densifiatin frnt thrugh the slug, then the lateral spreading f the sand and fllwed by lss f ntat between the beam and the sand as the beam bunes bak after attaining its peak defletin. Hwever, the defrmatin f the sandwih beams is markedly different frm the mnlithi beams. In the initial phase, the defrmatin mprises mainly f re mpressin with negligible defletin f the bak fae f the sandwih beam. This is fllwed by the bending f the entire beam. Fr the ases shwn in Figs. 1a and 1b, the re mpressin phase typially ends befre the peak defletin has been attained. Preditins f the re mpressin strain /, where is the redutin in re thikness at the mid-span are inluded in Figs. 13a and 13b fr the 0.11 and 0.33 ases, respetively (rrespnding t the w preditins in Fig. 11). The re mpressin inreases with dereasing but the time t required fr the imum re mpressin t be attained is reasnably independent f. The effets f re strength n the nrmalised imum defletins w w / L, imum re mpressin and the nrmalised time t t / L m / Y, are inluded in Figs. 14a, 14b and 14, respetively. Reall that the imum defletin f the equivalent mnlithi beam subjeted t the same lading is w Thus, ver the entire range f values nsidered, the 0.33 sandwih beam utperfrm their mnlithi unterparts in terms f having f lwer w while the 0.11 sandwih beams have a superir perfrmane nly fr values greater than 0.. The re mpressin and the rrespnding re mpressin time t were higher fr the 0.33 sandwih beam mpared t the 0.11 beam fr all values f nsidered here. This bservatin is ratinalised as fllws. Reall that the 0.33 re has a relative density 0.1 while 0.3 fr the 0.11 beam. Thus, the atual re strength f the 0.33 beam is lwer mpared t the 0.11 beam fr any given values f nrmalized strength,. This lwer atual 15

16 re strength results in a large re mpressin and a lnger re mpressin time t. 5.1 Analysis f the superir perfrmane f sandwih beams The superir perfrmane f apprpriately designed sandwih beams an in general be a result f tw effets: (i) A fluid-struture interatin effet whereby the pressure exerted by the sand n the sandwih beam is less than that exerted n the mnlithi beam (and nsequently the mmentum transmitted t the sandwih panel is less than that transmitted int the mnlithi beam). This was the dminant phenmenn respnsible fr the superir perfrmane f sandwih panels subjeted t underwater blast lading (Flek and Deshpande, 004). (ii) The strutural perfrmane f sandwih beams is superir t mnlithi beams. In rder t understand the relative ntributins frm these tw effets we first plt in Fig. 15 the tempral variatin f the nrmalised pressure p /( sv ) fr the 0.33 sandwih beam and tw extreme values f ( p is alulated in the manner desribed in Setin 4 fr the mnlithi beams). The rrespnding preditin fr the mnlithi beam is als inluded in Fig. 15. It is lear that the lading due t the sand is nearly idential fr the sandwih and mnlithi beams and thus the superir perfrmane f the sandwih beams must be due t differenes in the strutural perfrmane f the sandwih and mnlithi beams. We disuss the tw fatrs that result in differene between the dynami perfrmane f sandwih and mnlithi beams. (a) The sandwih effet: The stati bending strength f sandwih beams is typially greater than that f mnlithi beams f equal mass. Thus, as lng as the defletin f the sandwih beams is less than the beam thikness (i.e. respnse is dminated by the bending strength rather than streth resistane), the sandwih beams deflet less than mnlithi beams fr a given applied lad. If the lad is suffiiently large that the defletins are greater than the beam thikness, the respnse bemes dminated by beam strething and sandwih beams then lse their advantage ver mnlithi beams. It is lear frm Fig. 14a that w / 1 fr the 0.33 beams ver the entire range f nsidered, i.e. the beams are in a regime where sandwih beams are expeted t utperfrm mnlithi beams. On the ther hand, the smaller re thikness f the 0.11 beams implies that fr the ladings nsidered here w typially exeeds and these beams thus have a perfrmane mparable t their mnlithi unterparts. We nte in passing that re mpressin redues with inreasing. This means that the separatin f the tw fae sheets and the sandwih effet is maintained fr high re strengths resulting in a redutin in w with inreasing. (b) Cupling between the re mpressin and beam bending phases: MShane et al. (007) and Liang et al. (007) identified the sl-alled sft-re effet in whih the sandwih beams utperfrmed mnlithi beams under dynami lading nditins when there was a strng tempral upling between the re mpressin and beam bending phases f the defrmatin, i.e. the re mpressin time t was mparable t the time t required fr the beams t attain their imum defletin. 16

17 Preditins f t inluded in Fig. 14 indiate that while there is signifiant upling f the re mpressin and beam bending phases at lw values f, the tw phases are deupled at the high values f, i.e. in the range where the perfrmane benefits f using sandwih nstrutin are imum the re mpressin and beam bending phases are deupled. We thus nlude that sandwih effet is the largest ntributr t the perfrmane benefits bserved in the sandwih simulatins reprted here. These perfrmane benefits are the imum fr high values f and ur ver a wide range f sand slug impulses as seen in Fig. 16 fr the 0.33 sandwih beams. 5. Fam analgue The fam prjetile was shwn t aurately represent sand slug lading and apture the defrmatin f mnlithi beams subjeted t sand slug lading. Here we demnstrate that this fam prjetile analgy als wrks fr sandwih beams laded by sand slugs. The prperties f the fam prjetile as hsen as desribed in Setin 4.4 and alulatins f fam prjetile lading f the sandwih beams arried ut in the mmerial FE pakage ABAQUS. Preditins f w, and t / t f the 0.11 and 0.33 sandwih beams are inluded in Fig. 14 while preditins f w fr the 0.33 sandwih beam are inluded in Fig. 16 as a funtin f I fr tw seleted values f (labelled fam impat in these figures). In all ases exellent agreement is btained between the preditins f the f the sand slug impat and the fam prjetile equivalent. We thus nlude that fam prjetiles an als be used t simulate sand slug lading f sandwih beams. 6. Cirular plates laded by ylindrial sand slugs We preed t briefly illustrate that the main mehanisms desribed abve are nt restrited t beams impated by sand slug mprising ylindrial sand partiles but als hld fr irular plates impated by sand slugs mprising spherial sand partiles. Here we analyse the respnse f irular mnlithi and sandwih plates (Fig. 17) impated entrally and nrmally by a ylindrial sand slug f radius a and height H, mprising spherial sand partiles f diameter D and with a spatially unifrm relative density. Befre impat, the sand partiles all have a velity v in the negative z diretin as shwn in Fig. 17. The sand slug impats lamped irular plates f radius R and areal mass m at time t 0. b 6.1 Pseud three-dimensinal alulatins The bundary value prblems skethed in Fig. 17 are inherently three-dimensinal (3D) as we need t mdel spherial sand partiles. Hwever, in rder t redue the mputatinal st, we analysed these prblems using a pseud 3D tehnique wherein the sand slug is mdelled in a 3D setting but the plate is assumed t underg axisymmetri defrmatin. We briefly explain this methdlgy. A slie f the ylindrial sand slug subtending an angle abut the entral axis as shwn in Fig. 18 was mdelled using the GRANULAR pakage f the disrete de 17

18 LAMMPS. The irumferential expansin f this slie was nstrained, i.e. the sand partiles were nstrained by radial rigid statinary walls alng 0 and as shwn in Fig. 18. This ensures that nsistent with the plate defrmatin, the defrmatin f the ylindrial sand slug is als apprximately axi-symmetri. Cntat between the spherial sand partiles f diameter D was haraterised by the mdel desribed in Setin.1. These disrete partile alulatins are upled t the axi-symmetri FE alulatin as fllws. At any time instant t, the FE alulatin prvides the radial and axial rdinates ( rp, z p ) and radial and axial velities ( vr, v z) f every material pint n the plate surfae. Realling that the FE alulatin is axi-symmetri the rdinate z p is independent f and the tangential velity v 0. The ntat fres between the partiles and the plate surfae ( F r and F z in the radial and axial diretins, respetively) are then alulated as desribed in Setin.3 and a vetr f ndal fres determined (the fre F due t any mtin f the sand partiles in the diretin is negleted as it annt be inluded in the axi-symmetri FE alulatin). These fres are then saled by the fatr / and inserted int the vetr f glbal ndal fres F in Eq. (.5). Subsequently, the disrete and finite element equatins are integrated as desribed in Setins.1 and. and the new psitins and velities f the partiles in the disrete alulatins and material pints in the finite element alulatins determined at time t t. 6. Gemetry and material prperties Clamped plates f radius R 0.5 m and areal mass m b 160 kgm were analysed. The plates were taken t be made frm the same materials as the beams desribed in Setins 4 and 5, i.e. mnlithi plates made frm DH36 steel and sandwih plates with DH36 steel fae sheets and a fam re als made frm DH36 parent material. The mnlithi plates had a thikness h 0mm and the sandwih plates f equal areal mass were designed in a manner similar t the sandwih beams suh that the mass was equally distributed between the tw fae-sheets eah f thikness hf 6.67 mm and re f thikness that is related t relative density f the re via Eq. (5.). Similar t the sandwih beams, we shall present results fr tw re relative densities 0.1 and 0.3 rrespnding t / R and The material prperties f the mnlithi plate material, the sandwih plate fae sheets and the fam re are idential t thse desribed in Setin 3.1. The nly free material parameter is the re strength : results are presented fr a range f values f in rder t parameterise the effet f re strength. In all the alulatins, the sand slug has a radius a 100 mm s that a/ R 0., initial height H 150 mm with a relative density 0. f the sand partiles f diameter D 00μm. The sand partiles were assumed t be made frm silia with density and ntat prperties listed in Setin 3.. We nte that the mid-span defletins f the mnlithi and sandwih plates made frm DH36 steel may be written in nn-dimensinal frm as (fr a given set f sand prperties) 18

19 w h m w f t,,, a, I, m, (6.1) R R s and w m w gt,,,, a, I, m,, (6.) R s respetively. The terms are as defined in Setins 4 and 5 with L replaed by R, i.e. t t / R /, / and / m Y R a a R. (6.3) Based n the material prperties desribed abve, the fllwing nn-dimensinal grups are fixed in the alulatins: m 0.405, h/ R 0.04, a 0., 0. and m / s.96. Results are presented fr the effet f the sand mmentum I, sandwih beam aspet rati and re strength n the plate defletins. All alulatins reprted subsequent were defrmed using a ylindrial slie f the sand slug with 10 : spt alulatins demnstrated that inreasing the value f had n appreiable effet n the preditins. 6.3 Respnse f mnlithi and sandwih plates Preditins f the nrmalised imum bak fae defletins w w / R at the mid-span f the mnlithi and the sandwih plates are pltted in Fig. 19a as a funtin f the nrmalised sand slug impulse I. Results fr the sandwih plate are inluded fr tw values f the re strength. Over the entire range f impulses nsidered, the sandwih plates have a lwer defletin mpared t the mnlithi plates. Further, similar t the sandwih beams, the defletins are lwer fr the sandwih plates with the higher re strength. Snap shts shwing the defrmatin f 10 slies f the mnlithi and sandwih ( 0.133, 0.1 ) plates are inluded in Figs. 0a and 0b, respetively. The defrmatin mdes are qualitatively very similar t the mnlithi and sandwih beams; see Figs. 5 and 1, respetively. Thus, we antiipate that all the findings fr the beams will qualitatively extraplate t the plates. The nrmalised imum bak fae defletins, w, are summarised in Fig. 19b fr and sandwih plates as a funtin f the re strength. The rrespnding defletin f the mnlithi plate f equal area mass is als inluded. Cnsistent the findings fr the sandwih beams, the sandwih plate with the thiker re (i.e ) utperfrms bth the mnlithi plate and the sandwih plate ver the entire range f values investigated here. By ntrast, the sandwih plates have a smaller defletin mpared t the mnlithi plates nly fr 0., i.e. strng res. Similar t the beams, the superir perfrmane f these sandwih plates is due t the sandwih effet disussed in Setin 5.1: the sandwih effet is greater fr thiker res and hene the plates utperfrm the thinner plates. 6.4 Fam analgue In Setins 4.4 and 5. we demnstrated that the respnse f mnlithi and sandwih beams laded by sand slugs was adequately mdelled by replaing the sand slug by an 19

20 equivalent fam prjetile. We nw extend this analgy t the plates analysed in this setin. Here we replae the ylindrial sand slug by an equivalent ylindrial fam prjetile f the same verall dimensins and density as the un-defrmed sand slug and travelling with an initial velity equal t that f the sand partiles. The material prperties f the fam are idential t thse detailed in Setin 4.4 with tw exeptins. The sand slug nw mprises f spherial rather than ylindrial sand partiles. These partiles an pak t a imum relative density 0.74 (rather than the 0.9 in the D ase). Assuming that rrespnds t a ubi lse paking, the mdulus E f f the fam under uniaxial straining is then determined by using the Otet truss prperties given by Deshpande et al. (001). This analysis gives E in terms f the sand prperties as f 4 Kn E f. (6.4) 3D Thus, the nly mdifiatins t the presriptin presented in Setin 4.4 are: (i) Eq. (4.11) is replaed by Eq. (6.4) and (ii) use 0.74 in Eq. (4.13) rather than 0.9. Axisymmetri FE alulatins f the fam prjetiles impating the mnlithi and sandwih plates were perfrmed using ABAQUS. Preditins f w are inluded in Figs. 19a and 19b and learly shw the exellent agreement between the full disrete alulatins and the alulatins using the equivalent fam prjetiles. 7. Cnluding remarks We have presented a upled disrete/ntinuum mputatinal framewrk t analyse the lading f mnlithi and sandwih strutures by sand lumns (slugs) travelling at high velities. The sand is mdelled as either disrete idential irular ylindrial r spherial partiles while the strutures (beams r irular plates) are mdelled using a Lagrangian finite element (FE) framewrk. The lamped beams and plates are laded by sand slugs impating the strutures nrmally and entrally: this is a mdel prblem devised t develp a fundamental understanding f the sand-struture interatin that urs in mre mplex lading events suh as the lading f strutures by landmine explsins. The alulatins f the mnlithi beams have demnstrated that the lading due t the sand is primarily inertial with nly a small fluid-struture interatin effet, i.e. the mmentum transmitted t the strutures was apprximately equal t the mmentum f the inming sand slug and relatively insensitive t the rati f the areal masses f the sand slug and plate. The lading and the strutural respnse was well haraterised by aunting fr just tw parameters: (i) the initial mmentum f the sand slug and (ii) the lading time given by the rati f the height f the slug and the initial velity f the sand partiles. The effet f parameters suh as the areal mass rati f the sand slug and struture, relative density f the sand partiles in the sand slug and beam aspet rati are all aptured within a single nn-dimensinal parameter, viz, the rati f the lading time t the strutural respnse time. Sandwih 0

21 beams with thik strng res were shwn t signifiantly utperfrm mnlithi beams f equal areal mass. This enhaned perfrmed f the sandwih beams was due t the sandwih effet whereby the high bending strength gives the sandwih beams superir perfrmane. The main sand/struture interatins mehanisms eluidated by the beam alulatins are shwn t hld in a three-dimensinal setting where slugs mprising spherial sand partiles impat lamped irular mnlithi and sandwih plates. Again, fluid-struture interatin effets were fund t be negligible with sandwih plates with thik and strng res ut-perfrming mnlithi plates f equal areal mass. The impat f a sand slug against the strutures is shwn t the analgus t a fam impat. We shw that replaing the sand slug by an equivalent fam prjetile aptures the respnse f the mnlithi and sandwih beams and plates with remarkable auray. This fam analgy thus nt nly presents a numerially effiient methd t slve the sand-struture interatin prblem but als suggests that fam prjetile lading might be a nvenient physial analgue fr studying this interatin in a labratry setting. Referenes Ashby, M.F., Evans, A.G., Flek, N.A., Gibsn, L.J., Huthinsn, J.W. and Wadley, H.N.G. (000). Metal fams: a design guide, Butterwrth-Heinemann, Bstn MA. Bagnld, R.A. (1954). Experiments n a gravity-free dispersin f large slid partiles in a Newtnian fluid under shear, Preedings f the Ryal Siety, Lndn, A5, Bathurst, R.J. and Rthenburg, L. (1988). Mirmehanial aspets f istrpi granular assemblies with linear ntat interatins, Jurnal f Applied Mehanis, ASME, 55, Bergern, D., Walker, R. and Cffey, C. (1998). Detnatin f 100-gram antipersnnel mine surrgate harges in sand-a test ase fr mputer de validatin. Suffield Reprt N. 668, Defene Researh Establishment Suffield, Ralstn, Alberta, Canada. Brvik, T., Olvssn, L., Hanssen, A.G., Dharmasena, K., Hanssn, H. and Wadley, H.N.G. (011). A disrete partile apprah t simulate the mbined effet f blast and sand impat lading f steel plates, Submitted t Jurnal f the Mehanis and Physis f Slids. Braid, M.P. (001). Experimental investigatin and analysis f the effets f antipersnnel landmine blasts. Defene Researh Establishment Suffield, Canada. Speial Publiatin, DRES SSSP Campbell, C.S., and Brennen, C.E. (1985). Chute flws f granular material: sme mputer simulatins. Jurnal f Applied Mehanis, ASME, 5, Campbell, C.S. (00). Granular shear flws at the elasti limit. Jurnal f Fluid Mehanis, 465,

22 Cundall, P.A. and Strak, O.D.L. (1979). A disrete numerial mdel fr granular assemblies. Getehnique, 9, Deshpande, V.S. and Flek, N.A. (000). Istrpi nstitutive mdels fr metalli fams, Jurnal f the Mehanis and Physis f Slids, 48, Deshpande, V.S., Flek, N.A. and Ashby, M.F. (001). Effetive prperties f an tet truss lattie material, Jurnal f the Mehanis and Physis f Slids, 49, Deshpande, V.S., MMeeking, R.M., Wadley, H.N.G. and Evans, A.G. (009). Cnstitutive mdel fr prediting dynami interatins between sil ejeta and strutural panels. Jurnal f Mehanis and Physis f Slids, 57, Druker, D.C. and Prager, W. (195). Sil mehanis and plasti analysis r limit design, Quarterly f Applied Mathematis, 10, Flek, N.A. and Deshpande, V.S. (004). The resistane f lamped beams t shk lading, Jurnal f Applied Mehanis, 71, Kambuhev, N., Nels, L., and Radvitzky, R. (007). Fluid-struture interatin effets in the lading f free-standing plates by unifrm shks, Jurnal f Applied Mehanis, 74, Grujii, M., Pandurangan, B. and Cheeseman, B.A. (006). The effet f degree f saturatin f sand n detnatin phenmena assiated with shallw-buried and grund-laid mines. Jurnal f Shk and Vibratin, 13, Grujii, M., Pandurangan, B., Qia, R., Cheeseman, B.A., Ry, W.N., Skaggs, R.R. and Gupta, R. (008). Parameterizatin f the prus-material mdel fr sand with different levels f water saturatin. Sil dynamis and Earthquake Engineering, 8, Laine, P. and Sandvik, A. (001). Derivatin f Mehanial Prperties fr Sand, Preedings f the 4th Asia-Paifi Cnferene n Shk and Impat Lads n Strutures, CI-Premier PTE LTD, Singapre, pp Liang, Y.M., Spuskanyuk, A.V., Flres, S.E., Hayhurst, D.R., Huthinsn, J.W., MMeeking, R. M. and Evans, A.G. (006) The respnse f metalli sandwih panels f water blast, ASME Jurnal f Applied Mehanis, 74, MShane, G.J., Deshpande, V.S. and Flek, N.A. (007). The underwater blast resistane f metalli sandwih beams with prismati lattie res, ASME Jurnal f Applied Mehanis, 74, Mrris, B.L. (1993). Analysis f Imprved Crew Survivability in Light Vehiles Subjeted t Mine Blast. Final reprt fr ntrat n. DAAK70-9-C-0058 fr the U.S. Army Belvir RDEC, Ft. Belvir, VA.

23 Neuberger, A., Peles, S. and Rittel, D. (007). Saling the respnse f irular plates subjeted t large and lse-range spherial explsins. Part II: Buried harges, Internatinal Jurnal f Impat Engineering, 34, Peire, D., Shih, C.F. and Needleman, A. (1984). A tangent mdulus methd fr rate dependent slids, Cmputers and Strutures, 18, Pingle, S.M., Flek, N.A., Wadley, H.N.G. and Deshpande, V.S. (011). Disrete element alulatins f the impat f a sand lumn against rigid strutures, submitted t Internatinal Jurnal f Impat Engineering. Plimptn, S. (1995). Fast parallel algrithms fr shrt-range mleular dynamis. Jurnal f Cmputatinal Physis 117, Radfrd, D.D., Deshpande, V.S. and Flek, N.A. (005). The use f metal fam prjetiles t simulate shk lading n a struture, Internatinal Jurnal f Impat Engineering, 31, Silbert L.E., Ertaş D., Grest G.S., Halsey, T.C., Levine, D., Plimptn, S.J. (001). Granular flw dwn an inlined plane: Bagnld saling and rhelgy. Physis review E, 64, Qiu, X., Deshpande, V.S. and Flek, N.A. (003). Finite element analysis f the dynami respnse f lamped sandwih beams subjet t shk lading, Eurpean Jurnal f Mehanis A/Slids,, Qui, X., Deshpande, V.S. and Flek, N.A. (005). Impulsive lading f sandwih and mnlithi beams ver a entral path, Jurnal f the Mehanis and Physis f Slids, 53, Taylr, G.I. (1963). The pressure and impulse f submarine explsin waves n plates. The Sientifi Papers f Sir Geffrey Ingram Taylr, Vl. III: Aerdynamis and the Mehanis f Prjetiles and Explsins, G. Bathelr, ed., Cambridge University Press, Cambridge, UK, Tilbrk, M.T., Deshpande, V.S. and Flek, N.A. (006). The impulsive respnse f sandwih beams: Analytial and numerial investigatin f the regimes f behaviur, Jurnal f the Mehanis and Physis f Slids, 54, Vaziri, A., Xue, Z.Y. and Huthinsn, J.W. (007). Perfrmane and failure f metal sandwih plates subjeted t shk lading, Jurnal f Mehanis f Materials and Strutures,, Wadley, H.N.G., Flek, N.A. and Evans, A.G. (003). Fabriatin and strutural perfrmane f peridi ellular metal sandwih strutures, Cmpsites Siene and Tehnlgy, 63, Wadley, H.N.G., Dharmasena, K.P., Chen, Y., Dudt P., Knight, D., Charette, R., and Kiddy, K. (008) Cmpressive Respnse f Multilayered Pyramidal Lattie During 3

24 Underwater Shk Lading, Internatinal Jurnal f Impat Engineering, 35, Wang, Z. and Lu, Y. (003). Numerial Analysis n Dynami Defrmatin Mehanisms f Sils under Blast Lading, Sil Dynamis and Earthquake Engineering, 3, Wang, Z., Ha, H. and Lu, Y. (004). A three-phase sil mdel fr simulating stress wave prpagatin due t blast lading, Internatinal Jurnal fr Numerial and Analytial Methds in Gemehanis, 8, Wekert, S. and Andersen, C. (006). A Preliminary Cmparisn Between TNT and PE4 Landmines, Reprt n. DSTO-TN-073, Defene Siene and Tehnlgy Organizatin, Edinburgh, Suth Australia. Wei, Z., Deshpande, V.S., Evans, A.G., Dharmasena, K.P., Queheillalt, D.T., Wadley, H.N.G., Murty, Y., Elzey, R.K., Dudt, P., Chen, Y., Knight, D. and Kiddy, K (008). The resistane f metalli plates t lalized impulse. Jurnal f Mehanis and Physis f Slids, 56, Westine, P.S., Mrris, B.L., Cx, P.A. and Plh, E. (1985). Develpment f mputer prgram fr flr plate respnse frm land mine explsins, Cntrat Reprt N fr U.S. Army TACOM Researh and Develpment Centre. Xue, Z. and Huthinsn, J.W. (003). Preliminary assessment f sandwih plates subjeted t blast lads, Internatinal Jurnal f Mehanial Sienes 45(4): Xue, Z. and Huthinsn, J.W. (004). A mparative study f blast-resistant metal sandwih plates, Internatinal Jurnal f Impat Engineering 30(10): Xue, Z., Vaziri, A. and Huthinsn, J.W. (005). Nn-unifrm hardening nstitutive mdel fr mpressible rthtrpi materials with appliatin t sandwih plate res, Cmputer Mdelling and Simulatin in Engineering, 10, Aknwledgements This researh was supprted by the Offie f Naval Researh (ONR NICOP grant number N ) as part f a Multidisiplinary University Researh Initiative n Cellular materials nepts fr fre prtetin, Prime Award N. N is gratefully aknwledged. The prgram manager was Dr David Shifler. Figure Captins Figure 1: Sketh f a prttypial prblem f a lamped sandwih struture laded by a shallw mine explsin. 4

25 Figure : The plane strain bundary value prblem f a slug f sand mprising irular ylindrial partiles impating a lamped (a) mnlithi and (b) sandwih beam. Figure 3: Sketh illustrating the ntat law between tw partiles in the disrete alulatins. Figure 4: (a) The nrmalised mid-span defletin w versus nrmalised time t histries fr three seleted values f the nrmalised sand slug impulse I. (b) The rrespnding preditins f the nrmalised imum mid-span defletin w as a funtin f I. All ther parameters are held fixed at their referene values. Preditins using the equivalent fam prjetile are als inluded in (a) and (b). Figure 5: Snapshts frm the upled disrete/ntinuum alulatins f the sand slug impating the mnlithi beam at six seleted nrmalised times t. The results are shwn fr a nrmalised impulse I 0.67 with all ther parameters fixed at their referene values. Figure 6: The distributin f the nrmalised pressure p /( sv ) ver the span f the beam at three times t after impat f the sand slug against the mnlithi beam with a nrmalised impulse I 0.67 (all ther parameters fixed at their referene values). The rdinate X dentes the x 1 rdinate a material pint n the beam surfae in its undefrmed nfiguratin, i.e. X 0 is at the beam mid-span and X L is at the supprts. Figure 7: Preditins f (a) the nrmalised average pressure p /( sv ) and (b) nrmalised transmitted mmentum It / I a funtin f the nrmalised time tˆ tv / H fr the sand slug impat against the mnlithi beam. Results are shwn fr tw values f the nrmalised mmentum I (all ther parameters fixed at their referene values). Figure 8: Preditins f the nrmalised imum defletin w f the mnlithi beams as a funtin f the nrmalised lading time. Tw sets f alulatins are reprted: (i) vary m with 0. and (ii) vary with m In eah ase results are presented fr tw values f the nrmalised impulse I 0.67 and 1.0. Figure 9: (a) Preditins f the nrmalised imum defletin w f the mnlithi beam as a funtin f the nrmalised width a f the sand slug fr tw seleted values f the sand slug impulse I. (b) Snapshts shwing the defrmatin f the mnlithi beam and sand slug at six seleted times t fr the a 0.8 slug with an impulse I All parameters nt listed here have their referene values. 5

26 Figure 10: (a) Snapshts shwing the defrmatin f the mnlithi beam and nstrained sand slug with an impulse I 0.67 at six seleted times t. (b) Preditins f the nrmalised imum defletins w as a funtin f the sand slug impulse I. Results are inluded fr tw relative densities f bth the nstrained and unnstrained sand slugs. All parameters nt listed here have their referene values. Figure 11: Preditins f the nrmalised defletin w versus nrmalised time t histries f the (a) 0.11 and (b) 0.33 sandwih beams impated by sand slugs with a nrmalised impulse I Results are shwn fr three values f the nrmalised re strength f the sandwih beams. Preditins f the defletin f the mnlithi beam f equal areal mass are als inluded. All parameters nt listed here have their referene values. Figure 1: Snapshts at six seleted times t shwing the defrmatin f the (a) 0.11 and (b) 0.33 sandwih beams with re strength 0.11 impated by sand slugs with a nrmalised impulse I All parameters nt listed here have their referene values. Figure 13: Preditins f the variatin f the re mpressin with nrmalised time t fr the (a) 0.11 and (b) 0.33 sandwih beams impated by sand slugs with a nrmalised impulse I Results are shwn fr the three seleted values f re strength rrespnding t Fig. 11 with all parameters fixed at their referene values. Figure 14: Preditins f (a) nrmalised imum defletin w, (b) imum re mpressin and () nrmalised re mpressin time t as a funtin f the nrmalised re strength fr the 0.11 and 0.33 sandwih beams impated by sand slugs with a nrmalised impulse I All parameters nt listed here have their referene values. Preditins using the equivalent fam prjetile are inluded in (a) and (b). Figure 15: Predited nrmalised average pressure p /( sv ) versus nrmalised time t histry fr the 0.33 sandwih beams impated by the sand slug with impulse I Results are shwn fr tw extreme values f the nrmalised re strength with all ther parameters equal t their referene values. The rrespnding preditins fr the equal areal mass mnlithi beam are als inluded. Figure 16: Preditins f the nrmalised imum mid-span defletin f the bak fae f the 0.33 sandwih beams as a funtin f the sand slug nrmalised impulse I. Results are shwn fr tw values f the nrmalised re strength with all ther parameters equal t their referene values. Preditins using the equivalent fam prjetile are als inluded. 6

27 Figure 17: Skethes illustrating the gemetries f the lamped irular (a) mnlithi and (b) sandwih plates impated by a ylindrial sand slug mprising idential spherial sand partiles. Figure 18: Sketh illustrating the slie f the irular ylindrial sand slug analysed using disrete alulatins. The applied bundary nditins n the sides f the slie are als illustrated. Figure 19: (a) Preditins f the nrmalised imum defletin w f the mnlithi and the sandwih plate as a funtin f the nrmalised sand slug impulse I. Results fr the sandwih plate are shwn fr tw values f the nrmalised re strength. (b) The defletins w f the and sandwih plates as a funtin f fr an impulse I The rrespnding defletin f the mnlithi plate f equal areal mass is inluded in (b). Preditins using the equivalent fam prjetile are als inluded in (a) and (b). Figure 0: Snapshts at six seleted times t shwing the defrmatin f the (a) mnlithi and (b) sandwih plate with re strength 0.1 impated by sand slugs with a nrmalised impulse I All parameters nt listed here have their referene values. The figures shw the 10 slie analysed in the disrete alulatins. 7

28 Figure 1: Sketh f a prttypial prblem f a lamped sandwih struture laded by a shallw mine explsin.

29 Figure : The plane strain bundary value prblem f a slug f sand mprising irular ylindrial partiles impating a lamped (a) mnlithi and (b) sandwih beam.

30 Figure 3: Sketh illustrating the ntat law between tw partiles in the disrete alulatins.

31 Figure 4: (a) The nrmalised mid-span defletin w versus nrmalised time t histries fr three seleted values f the nrmalised sand slug impulse I. (b) The rrespnding preditins f the nrmalised imum mid-span defletin w as a funtin f I. All ther parameters are held fixed at their referene values. Preditins using the equivalent fam prjetile are als inluded in (a) and (b).

32 Figure 5: Snapshts frm the upled disrete/ntinuum alulatins f the sand slug impating the mnlithi beam at six seleted nrmalised times t. The results are shwn fr a nrmalised impulse I = 0.67 with all ther parameters fixed at their referene values.

33 Figure 6: The distributin f the nrmalised pressure p / ( ρρ sv ) ver the span f the beam at three times t after impat f the sand slug against the mnlithi beam with a nrmalised impulse I = 0.67 (all ther parameters fixed at their referene values). The rdinate X dentes the x 1 rdinate a material pint n the beam surfae in its undefrmed nfiguratin, i.e. X = 0 is at the beam mid-span and X = L is at the supprts.

34 Figure 7: Preditins f (a) the nrmalised average pressure p / ( sv ) ρρ and (b) nrmalised transmitted mmentum It / I a funtin f the nrmalised time tˆ tv / H fr the sand slug impat against the mnlithi beam. Results are shwn fr tw values f the nrmalised mmentum I (all ther parameters fixed at their referene values).

35 Figure 8: Preditins f the nrmalised imum defletin w f the mnlithi beams as a funtin f the nrmalised lading time τ. Tw sets f alulatins are reprted: (i) vary m with ρ = 0. and (ii) vary ρ with m = In eah ase results are presented fr tw values f the nrmalised impulse I = 0.67 and 1.0.

36 Figure 9: (a) Preditins f the nrmalised imum defletin w f the mnlithi beam as a funtin f the nrmalised width a f the sand slug fr tw seleted values f the sand slug impulse I. (b) Snapshts shwing the defrmatin f the mnlithi beam and sand slug at six seleted times t fr the a = 0.8 slug with an impulse I = All parameters nt listed here have their referene values.

37 Figure 10: (a) Snapshts shwing the defrmatin f the mnlithi beam and nstrained sand slug with an impulse I = 0.67 at six seleted times t. (b) Preditins f the nrmalised imum defletins w as a funtin f the sand slug impulse I. Results are inluded fr tw relative densities ρ f bth the nstrained and unnstrained sand slugs. All parameters nt listed here have their referene values.

38 Figure 11: Preditins f the nrmalised defletin w versus nrmalised time t histries f the (a) = 0.11 and (b) = 0.33 sandwih beams impated by sand slugs with a nrmalised impulse I = Results are shwn fr three values f the nrmalised re strength σ f the sandwih beams. Preditins f the defletin f the mnlithi beam f equal areal mass are als inluded. All parameters nt listed here have their referene values.

39 Figure 1: Snapshts at six seleted times t shwing the defrmatin f the (a) = 0.11 and (b) = 0.33 sandwih beams with re strength σ = 0.11 impated by sand slugs with a nrmalised impulse I = 0.67 their referene values.. All parameters nt listed here have

40 Figure 13: Preditins f the variatin f the re mpressin ε with nrmalised time t fr the (a) = 0.11 and (b) = 0.33 sandwih beams impated by sand slugs with a nrmalised impulse I = Results are shwn fr the three seleted values f re strength σ rrespnding t Fig. 11 with all parameters fixed at their referene values.

41 Figure 14: Preditins f (a) nrmalised imum defletin w, (b) imum re mpressin ε and () nrmalised re mpressin time t as a funtin f the nrmalised re strength σ fr the = 0.11 and = 0.33 sandwih beams impated by sand slugs with a nrmalised impulse I = All parameters nt listed here have their referene values. Preditins using the equivalent fam prjetile are inluded in (a) and (b).

42 Figure 15: Predited nrmalised average pressure p / ( ρρ sv ) versus nrmalised time t histry fr the = 0.33 sandwih beams impated by the sand slug with impulse I = Results are shwn fr tw extreme values f the nrmalised re strength σ with all ther parameters equal t their referene values. The rrespnding preditins fr the equal areal mass mnlithi beam are als inluded.

43 Figure 16: Preditins f the nrmalised imum mid-span defletin f the bak fae f the = 0.33 sandwih beams as a funtin f the sand slug nrmalised impulse I. Results are shwn fr tw values f the nrmalised re strength σ with all ther parameters equal t their referene values. Preditins using the equivalent fam prjetile are als inluded.

44 Figure 17: Skethes illustrating the gemetries f the lamped irular (a) mnlithi and (b) sandwih plates impated by a ylindrial sand slug mprising idential spherial sand partiles.

45 Figure 18: Sketh illustrating the slie f the irular ylindrial sand slug analysed using disrete alulatins. The applied bundary nditins n the sides f the slie are als illustrated.