Research ArticleOn the Blast Mitigation Ability of Multiple V-Shape Deflectors

Sebastian Stanisławek and Andrzej Morka

Faculty of Mechanical Engineering Military University of Technology Warsaw 00-908 Poland

Correspondence should be addressed to Sebastian Stanisławek sebastianstanislawekwatedupl

Received 23 December 2019 Accepted 3 June 2020 Published 17 June 2020

Academic Editor Yuri S Karinski

Copyright copy 2020 Sebastian Stanisławek and Andrzej Morka -is is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

-is paper presents a 2D numerical study of v-shape deflectors subjected to high explosive charge detonation -e literatureprovides many papers concerning the appropriate geometry of blast protection deflectors Such structures require a relatively highdistance between the ground and the vehicle chassis In most cases the placement of a deflector is not possible or it cannot coverthe whole area under a chassis An interesting question that arises is to what extent a set of small deflectors would be able tomitigate blast effects -e content of this paper constitutes an answer to this question -ree analyses were conducted (1)multiplication of triangle components (2) effect of deflector size and (3) geometry-induced shock dynamics -e problem wassolved with the use of modelling and simulation methods in particular CFD-FEM implemented in the LS-DYNA code It wasconsidered a plain issue in computational fluid dynamics where space discretization for each option was built with two-di-mensional elements to ensure efficient calculations Deflectors were described using a rigid wall boundary condition and anadequate simplified detonation model was assumed -e primary measure of the results was reaction force history with mo-mentum transfer and pressure distribution maps considered supplementary-e studies performed showed that both minimizingthe v-shape deflector size and surrounding it with adjacent structures had a negative impact on its blast mitigation effectivenessHowever for eachmulti-v-shape deflector some improvement was present and therefore in situations where the installation of atypical protector is not possible due to dimensional requirements it may offer a compromise solution

1 Introduction

-e common use of improvised explosive devices (IEDs) andlandmines in conflict zones highlights the need for betterexplosion-resistant vehicles Even if an army dominatestechnologically it can suffer relatively high losses due tovarious types of explosives [1] -e tactics used by guerrillagroups focusing on random attacks make vehiclesrsquo antiblastresistance a primary goal However achieving a suitablespeed for a vehicle and the ability to move through a muddyterrain is difficult when an appropriate level of mine andfragment resistance is also required [2] In order to reconcilemanoeuvrability and protection further development oflight armoured vehicles (LAV) is desirable

In the most common scenario an explosion takes placeunder a vehicle making its chassis responsible for the transferof blast energy to the hull and consequently to the crewmembers Various aspects of blast mitigation problems can be

found in [3 4] where for example the authors of [5ndash8]focused on an optimal material design while the authors inthe following [9ndash11] studied the selection and geometricfeatures of the panels An advanced theoretical and experi-mental study of the oblique shock wave reflection phenomena[12] proved the potential advantages of geometrical barrierslocated on the propagation route of the blast wave in terms ofmomentum and energy transfer -erefore deflectors usingthe shape effect are used to mitigate blast effects with thev-shaped hull (ldquoVrdquo) being the most typical Quantitativeevaluation of a ldquoVrdquo deflector in comparison with a flat panelhas been the subject of many papers [13ndash15]-e action of theprotector aims to preserve a vehiclersquos trajectory and has beenundisputedly shown to reduce injury to occupants Someworks have investigated the hull geometry [16] in order todetermine the effect of deflection for different ldquoVrdquo anglesGenerally sharp angle deflectors disperse more energy buthowever require more space under the chassis In other

HindawiShock and VibrationVolume 2020 Article ID 8708974 8 pageshttpsdoiorg10115520208708974

research studies a u-shaped hull (ldquoUrdquo) has been considered[17ndash19] with deflection patterns usually compared to ldquoVrdquopanels Overall the results show that in general ldquoVrdquo platesbetter disperse blast waves for any type of loading andtherefore can be successfully applied in LAVs

-e conclusions drawn from these scientific investiga-tions resulted in the manufacturing of vehicles with im-proved crew survivability One of the most significantdrawbacks in deflector application is the location of thecentre of gravity which introduces new hazards for soldiersAccording to [20] rollovers were the deadliest and costliestMRAP accident types where the specific hull geometry musthave been a contributory factor Moreover because theinstallation of ldquoVrdquo deflectors requires relatively high groundclearance it often cannot be applied to vehicles which werenot designed for this purpose Another interesting questionwhich arises is how the deflection phenomenon may be usedwhen the size of the protective structure is reduced Un-fortunately few papers [21 22] deal with the problem of howa deflectormdashwhich includes a set of small objects especiallythe ldquoVrdquo typemdashis able to mitigate blast effects -e followingcontent of this paper constitutes an answer to this question

2 Investigation Plan

-emitigation effect depends onmany factors including theexact position of the explosive and the particular design of aprotector A right angle (90deg) v-shaped plate was chosenbecause it was considered to be a good example of an ef-fective blast protective structure In order to eliminate theinfluence of deflector material properties a rigid bodymodelwas utilized in all analysis -e deflector width term wasdefined in the reference case as a v-shape span equal to800mm which is the maximum possible value which sat-isfies the ground clearance requirements for a typical LAV-e charge mass was assumed to be 15 kg of TNT whichprovided a proper relationship between the pressure gra-dient behind the shock wave front and the characteristicdimension of the deflectors studied which guaranteedsufficient differentiation of results

In this paper three groups of analysis results are pre-sented In the first group (Section 41) a single deflector iscompared with multiple v-shape structures where the HEcharge standoff distance is measured from the deflector base-is corresponds to the most realistic situation where aballistic panel is mounted on a vehicle chassis and a charge isburied in the ground A variety of factors such as charge massand type standoff distance deflector material size and ge-ometry vehicle mass and stiffness and ground propertiesinfluence the result of blast loading However two specificphenomena were selected for consideration in this papernamely flow around the v-shape which is dependent onneighbouring structuresrsquo presence and oblique reflectionwhich may be influenced by deflector size In order to dis-tinguish which effect was more significant separate researchwas conducted In Section 42 the influence of deflector size isconsidered and in Section 43 disturbance of air flow isstudied All the tested variants are described in Table 1 in-cluding denotations and schematic configurations

-e problem was solved by the application of a com-putational fluid dynamicmethod In order to provide greaterefficiency and accuracy of simulations it was modelled in a2D domain which implies that transverse dimensions aretreated as infinite -ough it does not fully correspond toreal conditions mutual quantitative relations between re-sults have been conserved for the cases investigated

Table 1 Structure configurations studied

Symbol Deflector description (width (mm)) Scheme

Vf800 Flat (refer)800 (Sections 41 42 and 43)

Vs800 Single800 (Sections 41ndash43)

Vd800 Double800 (Section 41)

Vt800 Triple800 (Section 41)

Vq800 Quadruple800 (Section 41)

Vs560 Single560 (Section 42)

Vs320 Single320 (Section 42)

Vs80 Single80 (Section 42)

Vf800c Flat800 ndash close (Section 43)

Vs560c Single560 ndash close (Section 43)

Vs320c Single320 ndash close (Section 43)

Vs80c Single80 ndash close (Section 43)

Vs200 Single200 (Section 43)

2 Shock and Vibration

Reaction force and momentum transfer on the deflectorsurface were recorded and are considered the primarymeasures of the results Additionally pressure maps wereused for identification of the convergent flow zones

3 Numerical Model Description

-e problem formulated above had to be transformed intothe mathematical language following numerical modeldevelopment -e fundamental laws of mass momentumand energy conservation in terms of continuum mechanicscan be expressed by the system of nonlinear PDEsAccording to the Euler approach and neglecting any externalforce fields they take the following conservative form

zρzt

+ nabla middot (ρv) 0 (1)

z(ρv)

zt+ nabla middot (ρv otimes v minus σ) 0 (2)

z(ρE)

zt+ nabla middot (ρEv minus σ middot v) 0 (3)

where ρ denotes the mass density field and v denotes thefluid velocity field -e total specific energy is defined as thesum of kinetic and internal energies according to

E ≔v2

2+ e (4)

-e system of equations (1)ndash(3) is complemented by theconstitutive relation expressing the Cauchy stress tensor σthrough other problemrsquos variables Finally the mathematicalmodel formulation is completed by adequate initial andboundary conditions (IBC)

-e arbitrary LagrangianndashEulerian (ALE) algorithmimplemented in the LS-DYNA environment a widely usedtool for blast modelling [13 18 21 23] was chosen to providethe numerical solution to the problem -e problem equa-tions could then be rewritten in a more suitable form wherethe convective and advective terms were explicitly presentedas well as taking into account the plain formulation

dρdt

+ ρzvx

zx+

zvy

zy1113888 1113889 0

dρdt≔

zρzt

+ vx

zρzx

+ vy

zρzy

1113888 1113889

(5)

ρdvx

dt+

zp

zx 0

dvx

dt≔

zvx

zt+ vx

zvx

zx+ vy

zvx

zy1113888 1113889

(6)

ρdvy

dt+

zp

zy 0

dvy

dt≔

zvy

zt+ vx

zvy

zx+ vy

zvy

zy1113888 1113889

(7)

ρde

dt+ p

zvx

zx+

zvy

zy1113888 1113889 0

de

dt≔

ze

zt+ vx

ze

zx+ vy

ze

zy1113888 1113889

(8)

and in the case of the perfect fluid model the stress tensorsimplifies to the isotropic part depending only on thepressure which can be described by the polytrophic equa-tion of state (EOS) in the following form

p ρe(c minus 1) (9)

where c denotes the ratio of specific heats -e ALE-basedmethod splits the solution procedure into two steps theLagrangian time step (FEM) followed by the advection step[24] In the first step a solution of (5)ndash(8) is found whichneglects the advection terms (see the right column ofequations terms in brackets) -e advection algorithms arethen used to calculate the transported quantities between thedeformed (Lagrangian) elements and the assumed (in-cluding initial) mesh -e element-centred quantities(density internal energy and stress tensor) are calculatedwith the van Leer MUSCL (Monotone Upwind Scheme forConservation Laws) algorithm while the node-centred(momentumvelocity) uses the HIS (half index shift) algo-rithm Assuming that the advection step is performed afterevery Lagrangian step it is completely equivalent to Eulerianproblem treatment

-e solution for the FEM step is constructed startingfrom the weak formulation (Galerkin approach) of theequations of motion which leads to the integral form of theproblem equations -e semidiscrete form of the variationalequations is then obtained by application of the assumedapproximation formulas with the use of baseshape func-tions in discrete elements for the dependent field variables(trial and test functions) -e system of semidiscrete ODEscan be transformed to the final discrete (matrix) form by theapplication of the central difference scheme for time inte-gration with the time step derived from the CFL (Coura-ntndashFriedrichsndashLewy) stability condition Furthermoreduring the Lagrangian step the mass conservation law isautomatically satisfied by preservation of the mass in thesingle finite element volumes Instead the energy equationneeds additional iteration to gain current pressure andenergy update

-e assumed initial conditions considered the motion-less air at standard conditions for temperature and pressure-e essential boundary conditions of a rigid wall type wererealized by the vanishing of the normal nodal velocitycomponent where the symmetry planes or v-shape struc-tures were localized-e blast loading was modelled througha specific boundary condition where the parameters of theincident shock wave ie pressure and mass flow field actedon the assumed Euler boundary with the application of theso-called ambient zone (elements) ConWep air blast al-gorithms [25 26] were used to generate the incident shockwave where the chosen HE charge was placed in pre-determined locations A description of the infinite domainwas achieved by the nonreflecting boundary condition

Shock and Vibration 3

applied on the outer boundary which resulted in the freeoutflow state

-e geometry of the models was limited to half of thephysical system due to its plane symmetry -e Eulerdomain boundaries were moved away sufficiently to avoidthe impact of the possible boundary effects on the problembeing investigated After the discretization process eachmodel contained between 150 and 200 thousands of 2Dquad elements which provided a high accuracy ofcalculation

-e main result of the simulation ie the reaction forcehistory was acquired by summing up all forces recorded forall boundary nodes which formed the deflector shape -eproper direction (along the symmetry plane) was then se-lected to further study this value as it evolves with time or itstime integral ie impulse Additionally colour pressuremaps were prepared with the nodal averaging technique toimprove readability and interpretation

-e equivalent numerical model was successfully val-idated by a number of authors in the literature For ex-ample Schwer in [27] used a very similar model but in theaxis-symmetric domain studying air blast reflection ratiosand angles of incidence He proved good correspondencebetween the LS-DYNA results and other experimental andquasianalytical predictions In turn Van Dorsselaer et al[28] performed advanced numerical experimental analysisof shock wave propagation around a convex structure-eyconcluded that the LS-DYNA Solver delivered pressureresults within an experimental uncertainty of 20 Asimilar numerical model was investigated by Powell et al in[29] investigating the aspect of the mesh density depen-dence Total impulse characteristics were used as a measureof the results in comparison with different size meshes andconvergence analysis with respect to the discretizationlevel

4 Results and Discussion

41 Analysis of Deflectors with Multiple V-Shapes In thissection deflectors with different number of v-shape com-ponents are compared-ey were situated in such a way thatthe standoff distance (counted from their base) was equal to26m and was fixed for all variants -e assumed distanceallowed a sufficiently flat wave front (below 1) to beformed whose flatness was measured according to thefollowing definition

f l minus s

s (10)

where l is the arc length and s is the chord length for the samecentral angle In such an approach the v-tip positions variedwith respect to HE charge locationWhen they were closer tothe charge the higher intensity blast wave attacked theprotective structures earlier -e deflector location as-sumption refers to the fact that the vehicle chassis is at a fixeddistance from the ground

-e colour map in Figure 1 presents pressure distri-bution in 29ms from the charge detonation moment forflat (a) single (b) double (c) and triple (d) deflectors A

specific time was chosen for the analysis because it is at thismoment that a reflected wave front is formed for a flatsurface -e phenomenon was different for each structurenormal reflection was observed in the case of Vf800 whilein all other cases oblique reflections were registered In thelast two variants multiple reflections were identifiedSurprisingly in the case of Vf800 the pressure level was notthe highest though it covered the whole area of the flatdeflector -e lowest value 05MPa was registered on thereflected shock wave front in the case of Vs800 whichmeans that relatively small amount of energy was trans-ferred to the vehicle body In contrast the highest pressurewas identified for both double and triple deflectors and waslocalized in the limited space where the adjacent trianglesmeet -is effect may be considered to be an advantagebecause of the local action of the pressure but may also havea negative impact if a deflector is perforated due to in-sufficient strength of material

An interesting matter is the interpretation and analysisof wave phenomena associated with the interaction of theincident wave with geometrically complex shapes -e firstpart of the process depicted in Figure 2 was separate re-flections from each segment which created primary wavefronts heading towards each other It should be noted thatthe pressure distribution symmetry was violated in referenceto a single triangular shape because of the presence of ad-jacent objects -e wavesrsquo collision formed a secondaryreflected wave (Figure 2(c)) in the area around the symmetryaxis -e wave propagated towards the point of detonationwith decreasing intensity because of its divergent characterA rapid fall of pressure in points denoted by R was the resultof the rarefaction effect which may have an impact onmeasured parameters -e interference of incident andprimary reflected waves is illustrated as the area denoted byletter I but is not expected to influence the issue beingstudied

-e graph presented in Figure 3 illustrates the reactionforce history for all previously mentioned deflector types-e reference plate Vf800 generated the highest reactionforce while a single deflector Vs800 presented the bestpropertiesmdashin this case it was more than twice smaller Anincreased number of individual structures placed instead of asingle deflector (Vd800 Vt800 and Vq800) caused a steeperreaction force curve with a higher peak A simulation of aquadruple deflector showed results close to the reference flatpanel Nevertheless shaped panels provide some protectionagainst blast with increasing effectiveness in inverse pro-portion to the number of v-shapes Furthermore startingfrom 33ms for multishape variants the time force historydid not differ significantly -e results achieved correspondwell to the fact that deflectors which consist of numerousv-shapes do not have space scale comparable to the reflectionphenomenon [12]

Information about maximum force and momentumtransfer is presented in Table 2 -e study of momentumtransfer showed that the difference between Vf800 andVs800 was not so drastic because the single deflector wassubjected to a stronger wave and load which lasted for alonger period of time -e analogical parameters for others

4 Shock and Vibration

063057052047041036031026020015001

Pressure(MPa)

(a)

Pressure(MPa)048045041037033029025021018014001

(b)Pressure(MPa)14131110

088075062049036023010

(c)

Pressure(MPa)14131110

088075062049036023010

(d)

Figure 1 Pressure map in 29ms for different types of deflectors (a) Vf800 (b) Vs800 (c) Vd800 and (d) Vt800

3028262422201816141201

Pressure(MPa)

(a)

Pressure(MPa)6358524742363126211501

(b)Pressure(MPa)81746659524437302215

008

R

(c)

Pressure(MPa)333027252219161411

008005

I

(d)

Figure 2 Process of shock wave reflection in the case of double deflector (a) 24ms (b) 28ms (c) 31ms and (d) 36ms

Shock and Vibration 5

ranged from 88 to 92 of the reference variant -e resultsfor the assumed scenario investigated in this paragraphproved that it is difficult to isolate a pure geometrical effectand therefore the following research was conducted

42 Analysis of Single Deflector with Different Sizes -issection focuses on the study of the size aspect of singledeflectors whose dimensions varied from 800 to 80mm-estandoff distance was the same as in Section 41 Due to thevariable width of deflectors a normalization procedure wasessential for the comparison of quantitative parameters -eauthors suggested the following formula

Fc Fr

wr

wa

(11)

where Fc is the calculated force Fr is the registered force wr

is the reference width and wa is the actual width-e results shown in Figure 3 were a consequence of two

effects (1) a different deflection process due to the sizereduction of v-shape objects and (2) modification of air flowcaused by neighbouring structures It is important from ascientific point of view to separate these phenomena in theanalysis process -erefore in this section only the size of asingle deflector is investigated regardless of its applicability-emechanics behind energy andmomentum transfer is thesame as the single deflector already discussed so pressuredistributions were not included in this section However the

reaction forces presented in Figure 4 showed that mini-mizing the v-shape deflector had a negative impact on itsblast mitigation effectiveness for both assumed scenariosie fixed position of the base (a) and the tip (b) of thedeflector according to HE charge location Although evenfor a small Vs80 structure the maximum reaction force wasreduced by a third in reference to Vf800 and it was not aseffective as the single deflector Vs800 which was loaded by asignificantly stronger wave Increasing the deflector size ledto a reduction of response force in a monotonic manner Inaddition starting from 39ms the force history was indis-tinguishable from any of the v-shape sizes studied -eresults depicted in Figure 4(b) confirm the superiority of thesingle large deflector where the monotonic character of forcedecay is preserved as above A smaller standoff distanceresulted in a much higher value of force peak for all casesbecause of the more intensive wave

43 Analysis of Boundary Effects All the variants examinedshowed that the application of a smaller deflector resulted indeterioration of the blast mitigation ability However theissue of the rarefaction effect had not yet been investigatedTwo structures with the same position and width weretherefore subjected to a blast load In the first one adjacentobjects were present (Vq800) while in the other there wasonly a single v-shape (Vs200) -e results obtained are

3025 3520Time (ms)

0

100

200

300

400

Forc

e (N

mm

)

Vf800Vs800Vd800

Vt800Vq800

Figure 3 Reaction force history for flat single and multiple v-shape deflectors

Table 2 Maximum force and momentum transfer

VariantMaximum force Momentum transfer

(Nmm) () (Nmmmm) ()Vf800 416 100 272 100Vs800 189 45 194 71Vd800 274 66 239 88Vt800 320 77 247 91Vq800 345 83 251 92

6 Shock and Vibration

depicted in Figure 5 with the normalization procedureapplied -e force history observed for Vq800 was a su-perposition of two contrary events (1) existing rarefactioneffect and (2) appearance of secondary reflecting waves -efact that the variant produced a similar response to thereference case indicated that the first phenomena was not acrucial factor while the second one might play an importantrole In Vs200 rarefaction was not accompanied by otherimportant effects apart from the primary oblique reflection-e force history (Figure 5) shows deterioration of the ef-ficacy of the multiple v-shape deflector

5 Conclusions

-e paper deals with a 2D numerical study of the blastmitigation ability of multi-v-shape deflectors -ree sets of

analysis were conducted (1) multiplication of trianglecomponents (2) influence of deflector size and (3) geom-etry-induced shock dynamics -e primary measure of theresults was reaction force history though momentumtransfer and pressure distribution maps were also taken intoconsideration -e conclusions obtained for plain problemformulation can only be used qualitatively in real conditionsthough a high accuracy of results is guaranteed due to high-dense discretization of space

-e investigation proved that a single deflector ach-ieves a much weaker reaction response Furthermore theforce increased much more slowly in comparison with thereference flat plate even though Vs800 was subjected to astronger wave and load lasted for a longer period of timewhich was confirmed by poorer impulse characteristics-e main mechanism for blast mitigation was obliquereflection and rarefaction effects However the existence ofmultiple v-shapes was responsible for a negative effect inthe form of secondary reflection waves Obviously theimpact of the standoff distance was crucial therefore a fewscenarios were proposed to separately analyse the variousmechanical and geometric effects Studies showed thatdecreasing the size of the single deflector caused themonotonic effectiveness to deteriorate in both assumedscenarios ie fixed position of its base and tip with ref-erence to the HE charge location Analysis of adjacentv-shapesrsquo presence showed that their interaction led toincrease in the force peak in comparison with a singlecomponent (Vs200) where a rarefaction effect wasdominant even though the normalization procedure wasutilized

In summary the results obtained showed that bothdecreasing the v-shape deflector size and surrounding it byneighbouring structures had a negative influence on its blastmitigation effectiveness However for each multishape de-flector some improvement in the protection level was

3025 3520Time (ms)

Vf800Vs800Vs560

Vs320Vs80

0

100

200

300

400

Forc

e (N

mm

)

(a)

3025 3520Time (ms)

Vf800cVs800Vs560c

Vs320cVs80c

0

100

200

300

400

500

600

Forc

e (N

mm

)

(b)

Figure 4 Reaction force for triangle deflectors (a) position of the bottom is fixed and (b) position of the top is fixed

3025 3520Time (ms)

Vf800Vq800Vs200

0

100

200

300

400

Forc

e (N

mm

)

Figure 5 Reaction force for flat quadruple and single deflectors

Shock and Vibration 7

present For situations where installation of a typical pro-tector is not possible mainly due to dimensional require-ments it may offer a compromise solution

Data Availability

-e input data (key file) for the LS-DYNA code used tosupport the findings of this study are available from thecorresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the National Centre for Re-search and Development Poland (Grant no DOBR-BIO4022131492013)

References

[1] Mines Action Canada Landmine Monitor Report 2009 MinesAction Canada Ottawa Canada 2009

[2] NATO Procedures for Evaluating the Protection Level ofArmoured VehiclesndashMine 2reat Vol 2 NATO BrusselsBelgium 2014

[3] A Shukla and D S Rajapakse Blast MitigationndashExperimentaland Numerical Studies Springer Berlin Germany 2010

[4] N Uddin Blast Protection of Civil Infrastructures and VehiclesUsing Composites Woodhead Publishing Sawston UK 2010

[5] G N Nurick and G C Shave ldquo-e deformation and tearingof thin square plates subjected to impulsive loads-An ex-perimental studyrdquo International Journal of Impact Engi-neering vol 18 no 1 pp 99ndash116 1996

[6] A C Jacinto R D Ambrosini and R F Danesi ldquoExperi-mental and computational analysis of plates under air blastloadingrdquo International Journal of Impact Engineering vol 25no 10 pp 927ndash947 2001

[7] S Ambrosini and G Slawinski ldquo-e influence of utilizingdifferent materials and their configurations on ballistic panelsblast resistancerdquo in Advances in Mechanics 2eoreticalComputational and Interdisciplinary Issues CRC Press BocaRaton FL USA 2016

[8] I S Sandhu ldquoMitigation of blast induced acceleration usingopen cell natural rubber and synthetic foamrdquo Defence ScienceJournal vol 69 no 1 pp 53ndash57 2019

[9] C Qi ldquoDynamic response and optimal design of curvedmetallic sandwich panels under blast loadingrdquo 2e ScientificWorld Journal vol 2014 Article ID 853681 14 pages 2014

[10] W Cheng ldquoNumerical analysis of cladding sandwich panelswith tubular cores subjected to uniform blast loadrdquo Inter-national Journal of Impact Engineering vol 133 2019

[11] S Guangyong ldquoDynamic response of sandwich panel withhierarchical honeycomb cores subject to blast loadingrdquo 2in-Walled Structures vol 142 pp 499ndash515 2019

[12] G B Dor Shock Wave Reflection Phenomena Springer-Verlag Berlin Germany 1992

[13] V Denefeld N Heider A Holzwarth A Sattler and M SalkldquoReduction of global effects on vehicles after IED detona-tionsrdquo Defence Technology vol 10 no 2 pp 219ndash225 2014

[14] J Trajkovski J Perenda and R Kunc ldquoBlast response of Lightarmoured vehicles (LAVs) with flat and V-hull floorrdquo 2in-Walled Structures vol 131 pp 238ndash244 2018

[15] R Gieleta W Barnat and T Niezgoda ldquoExperimental in-vestigation of deflectorrsquos angle influence on energy absorp-tionrdquo Journal of KONES Powertrain and Transport vol 19no 4 pp 201ndash205 2015

[16] S Barnat G S Langdon G N Nurick E G Pickering andV H Balden ldquoResponse of V-shape plates to localised blastload experiments and numerical simulationrdquo InternationalJournal of Impact Engineering vol 46 pp 97ndash109 2012

[17] P Kumar J LeBlanc and A Shukla ldquoEffect of curvature onshock loading response of aluminum panelsrdquo Dynamic Be-havior of Materials Volume 1 vol 1 pp 369ndash374 2011

[18] J LeBlanc R Kunc and I Prebil ldquoBlast response of centrallyand eccentrically loaded flat- U- and V-shaped armoredplates comparative studyrdquo Shock Waves vol 27 no 4pp 583ndash591 2017

[19] G Gurumurthy Blast Mitigation Strategies for Vehicles UsingShape Optimization Methods Massachusetts Institute ofTechnology Cambridge MA USA 2008

[20] C Davis Prevention of Injury in Mine Resistant AmbushProtected (MRAP) Vehicle Accidents United States ArmyAeromedical Research Laboratory Fort Rucker AL USA2013

[21] R Hajek M Foglar and J Fladr ldquoInfluence of barrier ma-terial and barrier shape on blast wave mitigationrdquo Con-struction and Building Materials vol 120 pp 54ndash64 2016

[22] K Genson ldquoVehicle shaping for mine blast damage reduc-tionrdquo University of Maryland College Park MA USAMasterrsquos of Science 2006

[23] A Morka ldquoNumerical analysis of momentum transfer in thecase of blast protection structuresrdquo AIP Conference Pro-ceedings vol 2078 no 1 2019

[24] J Hallquist LS-Dyna 2eory Manual LSTC Livermore CAUSA 2018

[25] G Randers-Pehrson and K Bannister AirBlast LoadingModel for DYNA2D and DYNA3D Army Research Labora-tory Adelphi MD USA 1997

[26] C N Kingery and G Bulmash Air-Blast Parameters fromTNT Spherical Airburst and Hemispherical Surface Burst USArmy Ballistic Research Laboratory Aberdeen ProvingGround MD USA 1984

[27] L Schwer ldquoAir blast reflection ratios and angle of incidencerdquoin Proceedings of the 11th European LS-DYNA ConferenceSalzburg Austria 2017

[28] N Van Dorsselaer S Eveillard and S Trelat ldquoExperimentsand simulations of explosives shock wave propagationaround a convex structurerdquo in Proceedings of the 15th In-ternational LS-DYNA Users Conference Detroit MI USAJune 2018

[29] D A Powell D Bogosian and L Schwer ldquoMesh sensitivity ofblast wave propagationrdquo in Proceedings of the 15th Interna-tional LS-DYNA Users Conference Detroit MI USA June2018

8 Shock and Vibration

Reaction force and momentum transfer on the deflectorsurface were recorded and are considered the primarymeasures of the results Additionally pressure maps wereused for identification of the convergent flow zones

3 Numerical Model Description

-e problem formulated above had to be transformed intothe mathematical language following numerical modeldevelopment -e fundamental laws of mass momentumand energy conservation in terms of continuum mechanicscan be expressed by the system of nonlinear PDEsAccording to the Euler approach and neglecting any externalforce fields they take the following conservative form

zρzt

+ nabla middot (ρv) 0 (1)

z(ρv)

zt+ nabla middot (ρv otimes v minus σ) 0 (2)

z(ρE)

zt+ nabla middot (ρEv minus σ middot v) 0 (3)

where ρ denotes the mass density field and v denotes thefluid velocity field -e total specific energy is defined as thesum of kinetic and internal energies according to

E ≔v2

2+ e (4)

-e system of equations (1)ndash(3) is complemented by theconstitutive relation expressing the Cauchy stress tensor σthrough other problemrsquos variables Finally the mathematicalmodel formulation is completed by adequate initial andboundary conditions (IBC)

-e arbitrary LagrangianndashEulerian (ALE) algorithmimplemented in the LS-DYNA environment a widely usedtool for blast modelling [13 18 21 23] was chosen to providethe numerical solution to the problem -e problem equa-tions could then be rewritten in a more suitable form wherethe convective and advective terms were explicitly presentedas well as taking into account the plain formulation

dρdt

+ ρzvx

zx+

zvy

zy1113888 1113889 0

dρdt≔

zρzt

+ vx

zρzx

+ vy

zρzy

1113888 1113889

(5)

ρdvx

dt+

zp

zx 0

dvx

dt≔

zvx

zt+ vx

zvx

zx+ vy

zvx

zy1113888 1113889

(6)

ρdvy

dt+

zp

zy 0

dvy

dt≔

zvy

zt+ vx

zvy

zx+ vy

zvy

zy1113888 1113889

(7)

ρde

dt+ p

zvx

zx+

zvy

zy1113888 1113889 0

de

dt≔

ze

zt+ vx

ze

zx+ vy

ze

zy1113888 1113889

(8)

and in the case of the perfect fluid model the stress tensorsimplifies to the isotropic part depending only on thepressure which can be described by the polytrophic equa-tion of state (EOS) in the following form

p ρe(c minus 1) (9)

where c denotes the ratio of specific heats -e ALE-basedmethod splits the solution procedure into two steps theLagrangian time step (FEM) followed by the advection step[24] In the first step a solution of (5)ndash(8) is found whichneglects the advection terms (see the right column ofequations terms in brackets) -e advection algorithms arethen used to calculate the transported quantities between thedeformed (Lagrangian) elements and the assumed (in-cluding initial) mesh -e element-centred quantities(density internal energy and stress tensor) are calculatedwith the van Leer MUSCL (Monotone Upwind Scheme forConservation Laws) algorithm while the node-centred(momentumvelocity) uses the HIS (half index shift) algo-rithm Assuming that the advection step is performed afterevery Lagrangian step it is completely equivalent to Eulerianproblem treatment

-e solution for the FEM step is constructed startingfrom the weak formulation (Galerkin approach) of theequations of motion which leads to the integral form of theproblem equations -e semidiscrete form of the variationalequations is then obtained by application of the assumedapproximation formulas with the use of baseshape func-tions in discrete elements for the dependent field variables(trial and test functions) -e system of semidiscrete ODEscan be transformed to the final discrete (matrix) form by theapplication of the central difference scheme for time inte-gration with the time step derived from the CFL (Coura-ntndashFriedrichsndashLewy) stability condition Furthermoreduring the Lagrangian step the mass conservation law isautomatically satisfied by preservation of the mass in thesingle finite element volumes Instead the energy equationneeds additional iteration to gain current pressure andenergy update

-e assumed initial conditions considered the motion-less air at standard conditions for temperature and pressure-e essential boundary conditions of a rigid wall type wererealized by the vanishing of the normal nodal velocitycomponent where the symmetry planes or v-shape struc-tures were localized-e blast loading was modelled througha specific boundary condition where the parameters of theincident shock wave ie pressure and mass flow field actedon the assumed Euler boundary with the application of theso-called ambient zone (elements) ConWep air blast al-gorithms [25 26] were used to generate the incident shockwave where the chosen HE charge was placed in pre-determined locations A description of the infinite domainwas achieved by the nonreflecting boundary condition

Shock and Vibration 3

applied on the outer boundary which resulted in the freeoutflow state

-e geometry of the models was limited to half of thephysical system due to its plane symmetry -e Eulerdomain boundaries were moved away sufficiently to avoidthe impact of the possible boundary effects on the problembeing investigated After the discretization process eachmodel contained between 150 and 200 thousands of 2Dquad elements which provided a high accuracy ofcalculation

-e main result of the simulation ie the reaction forcehistory was acquired by summing up all forces recorded forall boundary nodes which formed the deflector shape -eproper direction (along the symmetry plane) was then se-lected to further study this value as it evolves with time or itstime integral ie impulse Additionally colour pressuremaps were prepared with the nodal averaging technique toimprove readability and interpretation

-e equivalent numerical model was successfully val-idated by a number of authors in the literature For ex-ample Schwer in [27] used a very similar model but in theaxis-symmetric domain studying air blast reflection ratiosand angles of incidence He proved good correspondencebetween the LS-DYNA results and other experimental andquasianalytical predictions In turn Van Dorsselaer et al[28] performed advanced numerical experimental analysisof shock wave propagation around a convex structure-eyconcluded that the LS-DYNA Solver delivered pressureresults within an experimental uncertainty of 20 Asimilar numerical model was investigated by Powell et al in[29] investigating the aspect of the mesh density depen-dence Total impulse characteristics were used as a measureof the results in comparison with different size meshes andconvergence analysis with respect to the discretizationlevel

4 Results and Discussion

41 Analysis of Deflectors with Multiple V-Shapes In thissection deflectors with different number of v-shape com-ponents are compared-ey were situated in such a way thatthe standoff distance (counted from their base) was equal to26m and was fixed for all variants -e assumed distanceallowed a sufficiently flat wave front (below 1) to beformed whose flatness was measured according to thefollowing definition

f l minus s

s (10)

where l is the arc length and s is the chord length for the samecentral angle In such an approach the v-tip positions variedwith respect to HE charge locationWhen they were closer tothe charge the higher intensity blast wave attacked theprotective structures earlier -e deflector location as-sumption refers to the fact that the vehicle chassis is at a fixeddistance from the ground

-e colour map in Figure 1 presents pressure distri-bution in 29ms from the charge detonation moment forflat (a) single (b) double (c) and triple (d) deflectors A

specific time was chosen for the analysis because it is at thismoment that a reflected wave front is formed for a flatsurface -e phenomenon was different for each structurenormal reflection was observed in the case of Vf800 whilein all other cases oblique reflections were registered In thelast two variants multiple reflections were identifiedSurprisingly in the case of Vf800 the pressure level was notthe highest though it covered the whole area of the flatdeflector -e lowest value 05MPa was registered on thereflected shock wave front in the case of Vs800 whichmeans that relatively small amount of energy was trans-ferred to the vehicle body In contrast the highest pressurewas identified for both double and triple deflectors and waslocalized in the limited space where the adjacent trianglesmeet -is effect may be considered to be an advantagebecause of the local action of the pressure but may also havea negative impact if a deflector is perforated due to in-sufficient strength of material

An interesting matter is the interpretation and analysisof wave phenomena associated with the interaction of theincident wave with geometrically complex shapes -e firstpart of the process depicted in Figure 2 was separate re-flections from each segment which created primary wavefronts heading towards each other It should be noted thatthe pressure distribution symmetry was violated in referenceto a single triangular shape because of the presence of ad-jacent objects -e wavesrsquo collision formed a secondaryreflected wave (Figure 2(c)) in the area around the symmetryaxis -e wave propagated towards the point of detonationwith decreasing intensity because of its divergent characterA rapid fall of pressure in points denoted by R was the resultof the rarefaction effect which may have an impact onmeasured parameters -e interference of incident andprimary reflected waves is illustrated as the area denoted byletter I but is not expected to influence the issue beingstudied

-e graph presented in Figure 3 illustrates the reactionforce history for all previously mentioned deflector types-e reference plate Vf800 generated the highest reactionforce while a single deflector Vs800 presented the bestpropertiesmdashin this case it was more than twice smaller Anincreased number of individual structures placed instead of asingle deflector (Vd800 Vt800 and Vq800) caused a steeperreaction force curve with a higher peak A simulation of aquadruple deflector showed results close to the reference flatpanel Nevertheless shaped panels provide some protectionagainst blast with increasing effectiveness in inverse pro-portion to the number of v-shapes Furthermore startingfrom 33ms for multishape variants the time force historydid not differ significantly -e results achieved correspondwell to the fact that deflectors which consist of numerousv-shapes do not have space scale comparable to the reflectionphenomenon [12]

Information about maximum force and momentumtransfer is presented in Table 2 -e study of momentumtransfer showed that the difference between Vf800 andVs800 was not so drastic because the single deflector wassubjected to a stronger wave and load which lasted for alonger period of time -e analogical parameters for others

4 Shock and Vibration

063057052047041036031026020015001

Pressure(MPa)

(a)

Pressure(MPa)048045041037033029025021018014001

(b)Pressure(MPa)14131110

088075062049036023010

(c)

Pressure(MPa)14131110

088075062049036023010

(d)

Figure 1 Pressure map in 29ms for different types of deflectors (a) Vf800 (b) Vs800 (c) Vd800 and (d) Vt800

3028262422201816141201

Pressure(MPa)

(a)

Pressure(MPa)6358524742363126211501

(b)Pressure(MPa)81746659524437302215

008

R

(c)

Pressure(MPa)333027252219161411

008005

I

(d)

Figure 2 Process of shock wave reflection in the case of double deflector (a) 24ms (b) 28ms (c) 31ms and (d) 36ms

Shock and Vibration 5

ranged from 88 to 92 of the reference variant -e resultsfor the assumed scenario investigated in this paragraphproved that it is difficult to isolate a pure geometrical effectand therefore the following research was conducted

42 Analysis of Single Deflector with Different Sizes -issection focuses on the study of the size aspect of singledeflectors whose dimensions varied from 800 to 80mm-estandoff distance was the same as in Section 41 Due to thevariable width of deflectors a normalization procedure wasessential for the comparison of quantitative parameters -eauthors suggested the following formula

Fc Fr

wr

wa

(11)

where Fc is the calculated force Fr is the registered force wr

is the reference width and wa is the actual width-e results shown in Figure 3 were a consequence of two

effects (1) a different deflection process due to the sizereduction of v-shape objects and (2) modification of air flowcaused by neighbouring structures It is important from ascientific point of view to separate these phenomena in theanalysis process -erefore in this section only the size of asingle deflector is investigated regardless of its applicability-emechanics behind energy andmomentum transfer is thesame as the single deflector already discussed so pressuredistributions were not included in this section However the

reaction forces presented in Figure 4 showed that mini-mizing the v-shape deflector had a negative impact on itsblast mitigation effectiveness for both assumed scenariosie fixed position of the base (a) and the tip (b) of thedeflector according to HE charge location Although evenfor a small Vs80 structure the maximum reaction force wasreduced by a third in reference to Vf800 and it was not aseffective as the single deflector Vs800 which was loaded by asignificantly stronger wave Increasing the deflector size ledto a reduction of response force in a monotonic manner Inaddition starting from 39ms the force history was indis-tinguishable from any of the v-shape sizes studied -eresults depicted in Figure 4(b) confirm the superiority of thesingle large deflector where the monotonic character of forcedecay is preserved as above A smaller standoff distanceresulted in a much higher value of force peak for all casesbecause of the more intensive wave

43 Analysis of Boundary Effects All the variants examinedshowed that the application of a smaller deflector resulted indeterioration of the blast mitigation ability However theissue of the rarefaction effect had not yet been investigatedTwo structures with the same position and width weretherefore subjected to a blast load In the first one adjacentobjects were present (Vq800) while in the other there wasonly a single v-shape (Vs200) -e results obtained are

3025 3520Time (ms)

0

100

200

300

400

Forc

e (N

mm

)

Vf800Vs800Vd800

Vt800Vq800

Figure 3 Reaction force history for flat single and multiple v-shape deflectors

Table 2 Maximum force and momentum transfer

VariantMaximum force Momentum transfer

(Nmm) () (Nmmmm) ()Vf800 416 100 272 100Vs800 189 45 194 71Vd800 274 66 239 88Vt800 320 77 247 91Vq800 345 83 251 92

6 Shock and Vibration

depicted in Figure 5 with the normalization procedureapplied -e force history observed for Vq800 was a su-perposition of two contrary events (1) existing rarefactioneffect and (2) appearance of secondary reflecting waves -efact that the variant produced a similar response to thereference case indicated that the first phenomena was not acrucial factor while the second one might play an importantrole In Vs200 rarefaction was not accompanied by otherimportant effects apart from the primary oblique reflection-e force history (Figure 5) shows deterioration of the ef-ficacy of the multiple v-shape deflector

5 Conclusions

-e paper deals with a 2D numerical study of the blastmitigation ability of multi-v-shape deflectors -ree sets of

analysis were conducted (1) multiplication of trianglecomponents (2) influence of deflector size and (3) geom-etry-induced shock dynamics -e primary measure of theresults was reaction force history though momentumtransfer and pressure distribution maps were also taken intoconsideration -e conclusions obtained for plain problemformulation can only be used qualitatively in real conditionsthough a high accuracy of results is guaranteed due to high-dense discretization of space

-e investigation proved that a single deflector ach-ieves a much weaker reaction response Furthermore theforce increased much more slowly in comparison with thereference flat plate even though Vs800 was subjected to astronger wave and load lasted for a longer period of timewhich was confirmed by poorer impulse characteristics-e main mechanism for blast mitigation was obliquereflection and rarefaction effects However the existence ofmultiple v-shapes was responsible for a negative effect inthe form of secondary reflection waves Obviously theimpact of the standoff distance was crucial therefore a fewscenarios were proposed to separately analyse the variousmechanical and geometric effects Studies showed thatdecreasing the size of the single deflector caused themonotonic effectiveness to deteriorate in both assumedscenarios ie fixed position of its base and tip with ref-erence to the HE charge location Analysis of adjacentv-shapesrsquo presence showed that their interaction led toincrease in the force peak in comparison with a singlecomponent (Vs200) where a rarefaction effect wasdominant even though the normalization procedure wasutilized

In summary the results obtained showed that bothdecreasing the v-shape deflector size and surrounding it byneighbouring structures had a negative influence on its blastmitigation effectiveness However for each multishape de-flector some improvement in the protection level was

3025 3520Time (ms)

Vf800Vs800Vs560

Vs320Vs80

0

100

200

300

400

Forc

e (N

mm

)

(a)

3025 3520Time (ms)

Vf800cVs800Vs560c

Vs320cVs80c

0

100

200

300

400

500

600

Forc

e (N

mm

)

(b)

Figure 4 Reaction force for triangle deflectors (a) position of the bottom is fixed and (b) position of the top is fixed

3025 3520Time (ms)

Vf800Vq800Vs200

0

100

200

300

400

Forc

e (N

mm

)

Figure 5 Reaction force for flat quadruple and single deflectors

Shock and Vibration 7

present For situations where installation of a typical pro-tector is not possible mainly due to dimensional require-ments it may offer a compromise solution

Data Availability

-e input data (key file) for the LS-DYNA code used tosupport the findings of this study are available from thecorresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the National Centre for Re-search and Development Poland (Grant no DOBR-BIO4022131492013)

References

[1] Mines Action Canada Landmine Monitor Report 2009 MinesAction Canada Ottawa Canada 2009

[2] NATO Procedures for Evaluating the Protection Level ofArmoured VehiclesndashMine 2reat Vol 2 NATO BrusselsBelgium 2014

[3] A Shukla and D S Rajapakse Blast MitigationndashExperimentaland Numerical Studies Springer Berlin Germany 2010

[4] N Uddin Blast Protection of Civil Infrastructures and VehiclesUsing Composites Woodhead Publishing Sawston UK 2010

[5] G N Nurick and G C Shave ldquo-e deformation and tearingof thin square plates subjected to impulsive loads-An ex-perimental studyrdquo International Journal of Impact Engi-neering vol 18 no 1 pp 99ndash116 1996

[6] A C Jacinto R D Ambrosini and R F Danesi ldquoExperi-mental and computational analysis of plates under air blastloadingrdquo International Journal of Impact Engineering vol 25no 10 pp 927ndash947 2001

[7] S Ambrosini and G Slawinski ldquo-e influence of utilizingdifferent materials and their configurations on ballistic panelsblast resistancerdquo in Advances in Mechanics 2eoreticalComputational and Interdisciplinary Issues CRC Press BocaRaton FL USA 2016

[8] I S Sandhu ldquoMitigation of blast induced acceleration usingopen cell natural rubber and synthetic foamrdquo Defence ScienceJournal vol 69 no 1 pp 53ndash57 2019

[9] C Qi ldquoDynamic response and optimal design of curvedmetallic sandwich panels under blast loadingrdquo 2e ScientificWorld Journal vol 2014 Article ID 853681 14 pages 2014

[10] W Cheng ldquoNumerical analysis of cladding sandwich panelswith tubular cores subjected to uniform blast loadrdquo Inter-national Journal of Impact Engineering vol 133 2019

[11] S Guangyong ldquoDynamic response of sandwich panel withhierarchical honeycomb cores subject to blast loadingrdquo 2in-Walled Structures vol 142 pp 499ndash515 2019

[12] G B Dor Shock Wave Reflection Phenomena Springer-Verlag Berlin Germany 1992

[13] V Denefeld N Heider A Holzwarth A Sattler and M SalkldquoReduction of global effects on vehicles after IED detona-tionsrdquo Defence Technology vol 10 no 2 pp 219ndash225 2014

[14] J Trajkovski J Perenda and R Kunc ldquoBlast response of Lightarmoured vehicles (LAVs) with flat and V-hull floorrdquo 2in-Walled Structures vol 131 pp 238ndash244 2018

[15] R Gieleta W Barnat and T Niezgoda ldquoExperimental in-vestigation of deflectorrsquos angle influence on energy absorp-tionrdquo Journal of KONES Powertrain and Transport vol 19no 4 pp 201ndash205 2015

[16] S Barnat G S Langdon G N Nurick E G Pickering andV H Balden ldquoResponse of V-shape plates to localised blastload experiments and numerical simulationrdquo InternationalJournal of Impact Engineering vol 46 pp 97ndash109 2012

[17] P Kumar J LeBlanc and A Shukla ldquoEffect of curvature onshock loading response of aluminum panelsrdquo Dynamic Be-havior of Materials Volume 1 vol 1 pp 369ndash374 2011

[18] J LeBlanc R Kunc and I Prebil ldquoBlast response of centrallyand eccentrically loaded flat- U- and V-shaped armoredplates comparative studyrdquo Shock Waves vol 27 no 4pp 583ndash591 2017

[19] G Gurumurthy Blast Mitigation Strategies for Vehicles UsingShape Optimization Methods Massachusetts Institute ofTechnology Cambridge MA USA 2008

[20] C Davis Prevention of Injury in Mine Resistant AmbushProtected (MRAP) Vehicle Accidents United States ArmyAeromedical Research Laboratory Fort Rucker AL USA2013

[21] R Hajek M Foglar and J Fladr ldquoInfluence of barrier ma-terial and barrier shape on blast wave mitigationrdquo Con-struction and Building Materials vol 120 pp 54ndash64 2016

[22] K Genson ldquoVehicle shaping for mine blast damage reduc-tionrdquo University of Maryland College Park MA USAMasterrsquos of Science 2006

[23] A Morka ldquoNumerical analysis of momentum transfer in thecase of blast protection structuresrdquo AIP Conference Pro-ceedings vol 2078 no 1 2019

[24] J Hallquist LS-Dyna 2eory Manual LSTC Livermore CAUSA 2018

[25] G Randers-Pehrson and K Bannister AirBlast LoadingModel for DYNA2D and DYNA3D Army Research Labora-tory Adelphi MD USA 1997

[26] C N Kingery and G Bulmash Air-Blast Parameters fromTNT Spherical Airburst and Hemispherical Surface Burst USArmy Ballistic Research Laboratory Aberdeen ProvingGround MD USA 1984

[27] L Schwer ldquoAir blast reflection ratios and angle of incidencerdquoin Proceedings of the 11th European LS-DYNA ConferenceSalzburg Austria 2017

[28] N Van Dorsselaer S Eveillard and S Trelat ldquoExperimentsand simulations of explosives shock wave propagationaround a convex structurerdquo in Proceedings of the 15th In-ternational LS-DYNA Users Conference Detroit MI USAJune 2018

[29] D A Powell D Bogosian and L Schwer ldquoMesh sensitivity ofblast wave propagationrdquo in Proceedings of the 15th Interna-tional LS-DYNA Users Conference Detroit MI USA June2018

8 Shock and Vibration

applied on the outer boundary which resulted in the freeoutflow state

-e geometry of the models was limited to half of thephysical system due to its plane symmetry -e Eulerdomain boundaries were moved away sufficiently to avoidthe impact of the possible boundary effects on the problembeing investigated After the discretization process eachmodel contained between 150 and 200 thousands of 2Dquad elements which provided a high accuracy ofcalculation

-e main result of the simulation ie the reaction forcehistory was acquired by summing up all forces recorded forall boundary nodes which formed the deflector shape -eproper direction (along the symmetry plane) was then se-lected to further study this value as it evolves with time or itstime integral ie impulse Additionally colour pressuremaps were prepared with the nodal averaging technique toimprove readability and interpretation

-e equivalent numerical model was successfully val-idated by a number of authors in the literature For ex-ample Schwer in [27] used a very similar model but in theaxis-symmetric domain studying air blast reflection ratiosand angles of incidence He proved good correspondencebetween the LS-DYNA results and other experimental andquasianalytical predictions In turn Van Dorsselaer et al[28] performed advanced numerical experimental analysisof shock wave propagation around a convex structure-eyconcluded that the LS-DYNA Solver delivered pressureresults within an experimental uncertainty of 20 Asimilar numerical model was investigated by Powell et al in[29] investigating the aspect of the mesh density depen-dence Total impulse characteristics were used as a measureof the results in comparison with different size meshes andconvergence analysis with respect to the discretizationlevel

4 Results and Discussion

41 Analysis of Deflectors with Multiple V-Shapes In thissection deflectors with different number of v-shape com-ponents are compared-ey were situated in such a way thatthe standoff distance (counted from their base) was equal to26m and was fixed for all variants -e assumed distanceallowed a sufficiently flat wave front (below 1) to beformed whose flatness was measured according to thefollowing definition

f l minus s

s (10)

where l is the arc length and s is the chord length for the samecentral angle In such an approach the v-tip positions variedwith respect to HE charge locationWhen they were closer tothe charge the higher intensity blast wave attacked theprotective structures earlier -e deflector location as-sumption refers to the fact that the vehicle chassis is at a fixeddistance from the ground

-e colour map in Figure 1 presents pressure distri-bution in 29ms from the charge detonation moment forflat (a) single (b) double (c) and triple (d) deflectors A

specific time was chosen for the analysis because it is at thismoment that a reflected wave front is formed for a flatsurface -e phenomenon was different for each structurenormal reflection was observed in the case of Vf800 whilein all other cases oblique reflections were registered In thelast two variants multiple reflections were identifiedSurprisingly in the case of Vf800 the pressure level was notthe highest though it covered the whole area of the flatdeflector -e lowest value 05MPa was registered on thereflected shock wave front in the case of Vs800 whichmeans that relatively small amount of energy was trans-ferred to the vehicle body In contrast the highest pressurewas identified for both double and triple deflectors and waslocalized in the limited space where the adjacent trianglesmeet -is effect may be considered to be an advantagebecause of the local action of the pressure but may also havea negative impact if a deflector is perforated due to in-sufficient strength of material

An interesting matter is the interpretation and analysisof wave phenomena associated with the interaction of theincident wave with geometrically complex shapes -e firstpart of the process depicted in Figure 2 was separate re-flections from each segment which created primary wavefronts heading towards each other It should be noted thatthe pressure distribution symmetry was violated in referenceto a single triangular shape because of the presence of ad-jacent objects -e wavesrsquo collision formed a secondaryreflected wave (Figure 2(c)) in the area around the symmetryaxis -e wave propagated towards the point of detonationwith decreasing intensity because of its divergent characterA rapid fall of pressure in points denoted by R was the resultof the rarefaction effect which may have an impact onmeasured parameters -e interference of incident andprimary reflected waves is illustrated as the area denoted byletter I but is not expected to influence the issue beingstudied

-e graph presented in Figure 3 illustrates the reactionforce history for all previously mentioned deflector types-e reference plate Vf800 generated the highest reactionforce while a single deflector Vs800 presented the bestpropertiesmdashin this case it was more than twice smaller Anincreased number of individual structures placed instead of asingle deflector (Vd800 Vt800 and Vq800) caused a steeperreaction force curve with a higher peak A simulation of aquadruple deflector showed results close to the reference flatpanel Nevertheless shaped panels provide some protectionagainst blast with increasing effectiveness in inverse pro-portion to the number of v-shapes Furthermore startingfrom 33ms for multishape variants the time force historydid not differ significantly -e results achieved correspondwell to the fact that deflectors which consist of numerousv-shapes do not have space scale comparable to the reflectionphenomenon [12]

Information about maximum force and momentumtransfer is presented in Table 2 -e study of momentumtransfer showed that the difference between Vf800 andVs800 was not so drastic because the single deflector wassubjected to a stronger wave and load which lasted for alonger period of time -e analogical parameters for others

4 Shock and Vibration

063057052047041036031026020015001

Pressure(MPa)

(a)

Pressure(MPa)048045041037033029025021018014001

(b)Pressure(MPa)14131110

088075062049036023010

(c)

Pressure(MPa)14131110

088075062049036023010

(d)

Figure 1 Pressure map in 29ms for different types of deflectors (a) Vf800 (b) Vs800 (c) Vd800 and (d) Vt800

3028262422201816141201

Pressure(MPa)

(a)

Pressure(MPa)6358524742363126211501

(b)Pressure(MPa)81746659524437302215

008

R

(c)

Pressure(MPa)333027252219161411

008005

I

(d)

Figure 2 Process of shock wave reflection in the case of double deflector (a) 24ms (b) 28ms (c) 31ms and (d) 36ms

Shock and Vibration 5

ranged from 88 to 92 of the reference variant -e resultsfor the assumed scenario investigated in this paragraphproved that it is difficult to isolate a pure geometrical effectand therefore the following research was conducted

42 Analysis of Single Deflector with Different Sizes -issection focuses on the study of the size aspect of singledeflectors whose dimensions varied from 800 to 80mm-estandoff distance was the same as in Section 41 Due to thevariable width of deflectors a normalization procedure wasessential for the comparison of quantitative parameters -eauthors suggested the following formula

Fc Fr

wr

wa

(11)

where Fc is the calculated force Fr is the registered force wr

is the reference width and wa is the actual width-e results shown in Figure 3 were a consequence of two

effects (1) a different deflection process due to the sizereduction of v-shape objects and (2) modification of air flowcaused by neighbouring structures It is important from ascientific point of view to separate these phenomena in theanalysis process -erefore in this section only the size of asingle deflector is investigated regardless of its applicability-emechanics behind energy andmomentum transfer is thesame as the single deflector already discussed so pressuredistributions were not included in this section However the

reaction forces presented in Figure 4 showed that mini-mizing the v-shape deflector had a negative impact on itsblast mitigation effectiveness for both assumed scenariosie fixed position of the base (a) and the tip (b) of thedeflector according to HE charge location Although evenfor a small Vs80 structure the maximum reaction force wasreduced by a third in reference to Vf800 and it was not aseffective as the single deflector Vs800 which was loaded by asignificantly stronger wave Increasing the deflector size ledto a reduction of response force in a monotonic manner Inaddition starting from 39ms the force history was indis-tinguishable from any of the v-shape sizes studied -eresults depicted in Figure 4(b) confirm the superiority of thesingle large deflector where the monotonic character of forcedecay is preserved as above A smaller standoff distanceresulted in a much higher value of force peak for all casesbecause of the more intensive wave

43 Analysis of Boundary Effects All the variants examinedshowed that the application of a smaller deflector resulted indeterioration of the blast mitigation ability However theissue of the rarefaction effect had not yet been investigatedTwo structures with the same position and width weretherefore subjected to a blast load In the first one adjacentobjects were present (Vq800) while in the other there wasonly a single v-shape (Vs200) -e results obtained are

3025 3520Time (ms)

0

100

200

300

400

Forc

e (N

mm

)

Vf800Vs800Vd800

Vt800Vq800

Figure 3 Reaction force history for flat single and multiple v-shape deflectors

Table 2 Maximum force and momentum transfer

VariantMaximum force Momentum transfer

(Nmm) () (Nmmmm) ()Vf800 416 100 272 100Vs800 189 45 194 71Vd800 274 66 239 88Vt800 320 77 247 91Vq800 345 83 251 92

6 Shock and Vibration

depicted in Figure 5 with the normalization procedureapplied -e force history observed for Vq800 was a su-perposition of two contrary events (1) existing rarefactioneffect and (2) appearance of secondary reflecting waves -efact that the variant produced a similar response to thereference case indicated that the first phenomena was not acrucial factor while the second one might play an importantrole In Vs200 rarefaction was not accompanied by otherimportant effects apart from the primary oblique reflection-e force history (Figure 5) shows deterioration of the ef-ficacy of the multiple v-shape deflector

5 Conclusions

-e paper deals with a 2D numerical study of the blastmitigation ability of multi-v-shape deflectors -ree sets of

analysis were conducted (1) multiplication of trianglecomponents (2) influence of deflector size and (3) geom-etry-induced shock dynamics -e primary measure of theresults was reaction force history though momentumtransfer and pressure distribution maps were also taken intoconsideration -e conclusions obtained for plain problemformulation can only be used qualitatively in real conditionsthough a high accuracy of results is guaranteed due to high-dense discretization of space

-e investigation proved that a single deflector ach-ieves a much weaker reaction response Furthermore theforce increased much more slowly in comparison with thereference flat plate even though Vs800 was subjected to astronger wave and load lasted for a longer period of timewhich was confirmed by poorer impulse characteristics-e main mechanism for blast mitigation was obliquereflection and rarefaction effects However the existence ofmultiple v-shapes was responsible for a negative effect inthe form of secondary reflection waves Obviously theimpact of the standoff distance was crucial therefore a fewscenarios were proposed to separately analyse the variousmechanical and geometric effects Studies showed thatdecreasing the size of the single deflector caused themonotonic effectiveness to deteriorate in both assumedscenarios ie fixed position of its base and tip with ref-erence to the HE charge location Analysis of adjacentv-shapesrsquo presence showed that their interaction led toincrease in the force peak in comparison with a singlecomponent (Vs200) where a rarefaction effect wasdominant even though the normalization procedure wasutilized

In summary the results obtained showed that bothdecreasing the v-shape deflector size and surrounding it byneighbouring structures had a negative influence on its blastmitigation effectiveness However for each multishape de-flector some improvement in the protection level was

3025 3520Time (ms)

Vf800Vs800Vs560

Vs320Vs80

0

100

200

300

400

Forc

e (N

mm

)

(a)

3025 3520Time (ms)

Vf800cVs800Vs560c

Vs320cVs80c

0

100

200

300

400

500

600

Forc

e (N

mm

)

(b)

Figure 4 Reaction force for triangle deflectors (a) position of the bottom is fixed and (b) position of the top is fixed

3025 3520Time (ms)

Vf800Vq800Vs200

0

100

200

300

400

Forc

e (N

mm

)

Figure 5 Reaction force for flat quadruple and single deflectors

Shock and Vibration 7

present For situations where installation of a typical pro-tector is not possible mainly due to dimensional require-ments it may offer a compromise solution

Data Availability

-e input data (key file) for the LS-DYNA code used tosupport the findings of this study are available from thecorresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the National Centre for Re-search and Development Poland (Grant no DOBR-BIO4022131492013)

References

[1] Mines Action Canada Landmine Monitor Report 2009 MinesAction Canada Ottawa Canada 2009

[2] NATO Procedures for Evaluating the Protection Level ofArmoured VehiclesndashMine 2reat Vol 2 NATO BrusselsBelgium 2014

[3] A Shukla and D S Rajapakse Blast MitigationndashExperimentaland Numerical Studies Springer Berlin Germany 2010

[4] N Uddin Blast Protection of Civil Infrastructures and VehiclesUsing Composites Woodhead Publishing Sawston UK 2010

[5] G N Nurick and G C Shave ldquo-e deformation and tearingof thin square plates subjected to impulsive loads-An ex-perimental studyrdquo International Journal of Impact Engi-neering vol 18 no 1 pp 99ndash116 1996

[6] A C Jacinto R D Ambrosini and R F Danesi ldquoExperi-mental and computational analysis of plates under air blastloadingrdquo International Journal of Impact Engineering vol 25no 10 pp 927ndash947 2001

[7] S Ambrosini and G Slawinski ldquo-e influence of utilizingdifferent materials and their configurations on ballistic panelsblast resistancerdquo in Advances in Mechanics 2eoreticalComputational and Interdisciplinary Issues CRC Press BocaRaton FL USA 2016

[8] I S Sandhu ldquoMitigation of blast induced acceleration usingopen cell natural rubber and synthetic foamrdquo Defence ScienceJournal vol 69 no 1 pp 53ndash57 2019

[9] C Qi ldquoDynamic response and optimal design of curvedmetallic sandwich panels under blast loadingrdquo 2e ScientificWorld Journal vol 2014 Article ID 853681 14 pages 2014

[10] W Cheng ldquoNumerical analysis of cladding sandwich panelswith tubular cores subjected to uniform blast loadrdquo Inter-national Journal of Impact Engineering vol 133 2019

[11] S Guangyong ldquoDynamic response of sandwich panel withhierarchical honeycomb cores subject to blast loadingrdquo 2in-Walled Structures vol 142 pp 499ndash515 2019

[12] G B Dor Shock Wave Reflection Phenomena Springer-Verlag Berlin Germany 1992

[13] V Denefeld N Heider A Holzwarth A Sattler and M SalkldquoReduction of global effects on vehicles after IED detona-tionsrdquo Defence Technology vol 10 no 2 pp 219ndash225 2014

[14] J Trajkovski J Perenda and R Kunc ldquoBlast response of Lightarmoured vehicles (LAVs) with flat and V-hull floorrdquo 2in-Walled Structures vol 131 pp 238ndash244 2018

[15] R Gieleta W Barnat and T Niezgoda ldquoExperimental in-vestigation of deflectorrsquos angle influence on energy absorp-tionrdquo Journal of KONES Powertrain and Transport vol 19no 4 pp 201ndash205 2015

[16] S Barnat G S Langdon G N Nurick E G Pickering andV H Balden ldquoResponse of V-shape plates to localised blastload experiments and numerical simulationrdquo InternationalJournal of Impact Engineering vol 46 pp 97ndash109 2012

[17] P Kumar J LeBlanc and A Shukla ldquoEffect of curvature onshock loading response of aluminum panelsrdquo Dynamic Be-havior of Materials Volume 1 vol 1 pp 369ndash374 2011

[18] J LeBlanc R Kunc and I Prebil ldquoBlast response of centrallyand eccentrically loaded flat- U- and V-shaped armoredplates comparative studyrdquo Shock Waves vol 27 no 4pp 583ndash591 2017

[19] G Gurumurthy Blast Mitigation Strategies for Vehicles UsingShape Optimization Methods Massachusetts Institute ofTechnology Cambridge MA USA 2008

[20] C Davis Prevention of Injury in Mine Resistant AmbushProtected (MRAP) Vehicle Accidents United States ArmyAeromedical Research Laboratory Fort Rucker AL USA2013

[21] R Hajek M Foglar and J Fladr ldquoInfluence of barrier ma-terial and barrier shape on blast wave mitigationrdquo Con-struction and Building Materials vol 120 pp 54ndash64 2016

[22] K Genson ldquoVehicle shaping for mine blast damage reduc-tionrdquo University of Maryland College Park MA USAMasterrsquos of Science 2006

[23] A Morka ldquoNumerical analysis of momentum transfer in thecase of blast protection structuresrdquo AIP Conference Pro-ceedings vol 2078 no 1 2019

[24] J Hallquist LS-Dyna 2eory Manual LSTC Livermore CAUSA 2018

[25] G Randers-Pehrson and K Bannister AirBlast LoadingModel for DYNA2D and DYNA3D Army Research Labora-tory Adelphi MD USA 1997

[26] C N Kingery and G Bulmash Air-Blast Parameters fromTNT Spherical Airburst and Hemispherical Surface Burst USArmy Ballistic Research Laboratory Aberdeen ProvingGround MD USA 1984

[27] L Schwer ldquoAir blast reflection ratios and angle of incidencerdquoin Proceedings of the 11th European LS-DYNA ConferenceSalzburg Austria 2017

[28] N Van Dorsselaer S Eveillard and S Trelat ldquoExperimentsand simulations of explosives shock wave propagationaround a convex structurerdquo in Proceedings of the 15th In-ternational LS-DYNA Users Conference Detroit MI USAJune 2018

[29] D A Powell D Bogosian and L Schwer ldquoMesh sensitivity ofblast wave propagationrdquo in Proceedings of the 15th Interna-tional LS-DYNA Users Conference Detroit MI USA June2018

8 Shock and Vibration

063057052047041036031026020015001

Pressure(MPa)

(a)

Pressure(MPa)048045041037033029025021018014001

(b)Pressure(MPa)14131110

088075062049036023010

(c)

Pressure(MPa)14131110

088075062049036023010

(d)

Figure 1 Pressure map in 29ms for different types of deflectors (a) Vf800 (b) Vs800 (c) Vd800 and (d) Vt800

3028262422201816141201

Pressure(MPa)

(a)

Pressure(MPa)6358524742363126211501

(b)Pressure(MPa)81746659524437302215

008

R

(c)

Pressure(MPa)333027252219161411

008005

I

(d)

Figure 2 Process of shock wave reflection in the case of double deflector (a) 24ms (b) 28ms (c) 31ms and (d) 36ms

Shock and Vibration 5

ranged from 88 to 92 of the reference variant -e resultsfor the assumed scenario investigated in this paragraphproved that it is difficult to isolate a pure geometrical effectand therefore the following research was conducted

42 Analysis of Single Deflector with Different Sizes -issection focuses on the study of the size aspect of singledeflectors whose dimensions varied from 800 to 80mm-estandoff distance was the same as in Section 41 Due to thevariable width of deflectors a normalization procedure wasessential for the comparison of quantitative parameters -eauthors suggested the following formula

Fc Fr

wr

wa

(11)

where Fc is the calculated force Fr is the registered force wr

is the reference width and wa is the actual width-e results shown in Figure 3 were a consequence of two

effects (1) a different deflection process due to the sizereduction of v-shape objects and (2) modification of air flowcaused by neighbouring structures It is important from ascientific point of view to separate these phenomena in theanalysis process -erefore in this section only the size of asingle deflector is investigated regardless of its applicability-emechanics behind energy andmomentum transfer is thesame as the single deflector already discussed so pressuredistributions were not included in this section However the

reaction forces presented in Figure 4 showed that mini-mizing the v-shape deflector had a negative impact on itsblast mitigation effectiveness for both assumed scenariosie fixed position of the base (a) and the tip (b) of thedeflector according to HE charge location Although evenfor a small Vs80 structure the maximum reaction force wasreduced by a third in reference to Vf800 and it was not aseffective as the single deflector Vs800 which was loaded by asignificantly stronger wave Increasing the deflector size ledto a reduction of response force in a monotonic manner Inaddition starting from 39ms the force history was indis-tinguishable from any of the v-shape sizes studied -eresults depicted in Figure 4(b) confirm the superiority of thesingle large deflector where the monotonic character of forcedecay is preserved as above A smaller standoff distanceresulted in a much higher value of force peak for all casesbecause of the more intensive wave

43 Analysis of Boundary Effects All the variants examinedshowed that the application of a smaller deflector resulted indeterioration of the blast mitigation ability However theissue of the rarefaction effect had not yet been investigatedTwo structures with the same position and width weretherefore subjected to a blast load In the first one adjacentobjects were present (Vq800) while in the other there wasonly a single v-shape (Vs200) -e results obtained are

3025 3520Time (ms)

0

100

200

300

400

Forc

e (N

mm

)

Vf800Vs800Vd800

Vt800Vq800

Figure 3 Reaction force history for flat single and multiple v-shape deflectors

Table 2 Maximum force and momentum transfer

VariantMaximum force Momentum transfer

(Nmm) () (Nmmmm) ()Vf800 416 100 272 100Vs800 189 45 194 71Vd800 274 66 239 88Vt800 320 77 247 91Vq800 345 83 251 92

6 Shock and Vibration

depicted in Figure 5 with the normalization procedureapplied -e force history observed for Vq800 was a su-perposition of two contrary events (1) existing rarefactioneffect and (2) appearance of secondary reflecting waves -efact that the variant produced a similar response to thereference case indicated that the first phenomena was not acrucial factor while the second one might play an importantrole In Vs200 rarefaction was not accompanied by otherimportant effects apart from the primary oblique reflection-e force history (Figure 5) shows deterioration of the ef-ficacy of the multiple v-shape deflector

5 Conclusions

-e paper deals with a 2D numerical study of the blastmitigation ability of multi-v-shape deflectors -ree sets of

analysis were conducted (1) multiplication of trianglecomponents (2) influence of deflector size and (3) geom-etry-induced shock dynamics -e primary measure of theresults was reaction force history though momentumtransfer and pressure distribution maps were also taken intoconsideration -e conclusions obtained for plain problemformulation can only be used qualitatively in real conditionsthough a high accuracy of results is guaranteed due to high-dense discretization of space

-e investigation proved that a single deflector ach-ieves a much weaker reaction response Furthermore theforce increased much more slowly in comparison with thereference flat plate even though Vs800 was subjected to astronger wave and load lasted for a longer period of timewhich was confirmed by poorer impulse characteristics-e main mechanism for blast mitigation was obliquereflection and rarefaction effects However the existence ofmultiple v-shapes was responsible for a negative effect inthe form of secondary reflection waves Obviously theimpact of the standoff distance was crucial therefore a fewscenarios were proposed to separately analyse the variousmechanical and geometric effects Studies showed thatdecreasing the size of the single deflector caused themonotonic effectiveness to deteriorate in both assumedscenarios ie fixed position of its base and tip with ref-erence to the HE charge location Analysis of adjacentv-shapesrsquo presence showed that their interaction led toincrease in the force peak in comparison with a singlecomponent (Vs200) where a rarefaction effect wasdominant even though the normalization procedure wasutilized

In summary the results obtained showed that bothdecreasing the v-shape deflector size and surrounding it byneighbouring structures had a negative influence on its blastmitigation effectiveness However for each multishape de-flector some improvement in the protection level was

3025 3520Time (ms)

Vf800Vs800Vs560

Vs320Vs80

0

100

200

300

400

Forc

e (N

mm

)

(a)

3025 3520Time (ms)

Vf800cVs800Vs560c

Vs320cVs80c

0

100

200

300

400

500

600

Forc

e (N

mm

)

(b)

Figure 4 Reaction force for triangle deflectors (a) position of the bottom is fixed and (b) position of the top is fixed

3025 3520Time (ms)

Vf800Vq800Vs200

0

100

200

300

400

Forc

e (N

mm

)

Figure 5 Reaction force for flat quadruple and single deflectors

Shock and Vibration 7

present For situations where installation of a typical pro-tector is not possible mainly due to dimensional require-ments it may offer a compromise solution

Data Availability

-e input data (key file) for the LS-DYNA code used tosupport the findings of this study are available from thecorresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the National Centre for Re-search and Development Poland (Grant no DOBR-BIO4022131492013)

References

[1] Mines Action Canada Landmine Monitor Report 2009 MinesAction Canada Ottawa Canada 2009

[2] NATO Procedures for Evaluating the Protection Level ofArmoured VehiclesndashMine 2reat Vol 2 NATO BrusselsBelgium 2014

[3] A Shukla and D S Rajapakse Blast MitigationndashExperimentaland Numerical Studies Springer Berlin Germany 2010

[4] N Uddin Blast Protection of Civil Infrastructures and VehiclesUsing Composites Woodhead Publishing Sawston UK 2010

[5] G N Nurick and G C Shave ldquo-e deformation and tearingof thin square plates subjected to impulsive loads-An ex-perimental studyrdquo International Journal of Impact Engi-neering vol 18 no 1 pp 99ndash116 1996

[6] A C Jacinto R D Ambrosini and R F Danesi ldquoExperi-mental and computational analysis of plates under air blastloadingrdquo International Journal of Impact Engineering vol 25no 10 pp 927ndash947 2001

[7] S Ambrosini and G Slawinski ldquo-e influence of utilizingdifferent materials and their configurations on ballistic panelsblast resistancerdquo in Advances in Mechanics 2eoreticalComputational and Interdisciplinary Issues CRC Press BocaRaton FL USA 2016

[8] I S Sandhu ldquoMitigation of blast induced acceleration usingopen cell natural rubber and synthetic foamrdquo Defence ScienceJournal vol 69 no 1 pp 53ndash57 2019

[9] C Qi ldquoDynamic response and optimal design of curvedmetallic sandwich panels under blast loadingrdquo 2e ScientificWorld Journal vol 2014 Article ID 853681 14 pages 2014

[10] W Cheng ldquoNumerical analysis of cladding sandwich panelswith tubular cores subjected to uniform blast loadrdquo Inter-national Journal of Impact Engineering vol 133 2019

[11] S Guangyong ldquoDynamic response of sandwich panel withhierarchical honeycomb cores subject to blast loadingrdquo 2in-Walled Structures vol 142 pp 499ndash515 2019

[12] G B Dor Shock Wave Reflection Phenomena Springer-Verlag Berlin Germany 1992

[13] V Denefeld N Heider A Holzwarth A Sattler and M SalkldquoReduction of global effects on vehicles after IED detona-tionsrdquo Defence Technology vol 10 no 2 pp 219ndash225 2014

[14] J Trajkovski J Perenda and R Kunc ldquoBlast response of Lightarmoured vehicles (LAVs) with flat and V-hull floorrdquo 2in-Walled Structures vol 131 pp 238ndash244 2018

[15] R Gieleta W Barnat and T Niezgoda ldquoExperimental in-vestigation of deflectorrsquos angle influence on energy absorp-tionrdquo Journal of KONES Powertrain and Transport vol 19no 4 pp 201ndash205 2015

[16] S Barnat G S Langdon G N Nurick E G Pickering andV H Balden ldquoResponse of V-shape plates to localised blastload experiments and numerical simulationrdquo InternationalJournal of Impact Engineering vol 46 pp 97ndash109 2012

[17] P Kumar J LeBlanc and A Shukla ldquoEffect of curvature onshock loading response of aluminum panelsrdquo Dynamic Be-havior of Materials Volume 1 vol 1 pp 369ndash374 2011

[18] J LeBlanc R Kunc and I Prebil ldquoBlast response of centrallyand eccentrically loaded flat- U- and V-shaped armoredplates comparative studyrdquo Shock Waves vol 27 no 4pp 583ndash591 2017

[19] G Gurumurthy Blast Mitigation Strategies for Vehicles UsingShape Optimization Methods Massachusetts Institute ofTechnology Cambridge MA USA 2008

[20] C Davis Prevention of Injury in Mine Resistant AmbushProtected (MRAP) Vehicle Accidents United States ArmyAeromedical Research Laboratory Fort Rucker AL USA2013

[21] R Hajek M Foglar and J Fladr ldquoInfluence of barrier ma-terial and barrier shape on blast wave mitigationrdquo Con-struction and Building Materials vol 120 pp 54ndash64 2016

[22] K Genson ldquoVehicle shaping for mine blast damage reduc-tionrdquo University of Maryland College Park MA USAMasterrsquos of Science 2006

[23] A Morka ldquoNumerical analysis of momentum transfer in thecase of blast protection structuresrdquo AIP Conference Pro-ceedings vol 2078 no 1 2019

[24] J Hallquist LS-Dyna 2eory Manual LSTC Livermore CAUSA 2018

[25] G Randers-Pehrson and K Bannister AirBlast LoadingModel for DYNA2D and DYNA3D Army Research Labora-tory Adelphi MD USA 1997

[26] C N Kingery and G Bulmash Air-Blast Parameters fromTNT Spherical Airburst and Hemispherical Surface Burst USArmy Ballistic Research Laboratory Aberdeen ProvingGround MD USA 1984

[27] L Schwer ldquoAir blast reflection ratios and angle of incidencerdquoin Proceedings of the 11th European LS-DYNA ConferenceSalzburg Austria 2017

[28] N Van Dorsselaer S Eveillard and S Trelat ldquoExperimentsand simulations of explosives shock wave propagationaround a convex structurerdquo in Proceedings of the 15th In-ternational LS-DYNA Users Conference Detroit MI USAJune 2018

[29] D A Powell D Bogosian and L Schwer ldquoMesh sensitivity ofblast wave propagationrdquo in Proceedings of the 15th Interna-tional LS-DYNA Users Conference Detroit MI USA June2018

8 Shock and Vibration

ranged from 88 to 92 of the reference variant -e resultsfor the assumed scenario investigated in this paragraphproved that it is difficult to isolate a pure geometrical effectand therefore the following research was conducted

42 Analysis of Single Deflector with Different Sizes -issection focuses on the study of the size aspect of singledeflectors whose dimensions varied from 800 to 80mm-estandoff distance was the same as in Section 41 Due to thevariable width of deflectors a normalization procedure wasessential for the comparison of quantitative parameters -eauthors suggested the following formula

Fc Fr

wr

wa

(11)

where Fc is the calculated force Fr is the registered force wr

is the reference width and wa is the actual width-e results shown in Figure 3 were a consequence of two

effects (1) a different deflection process due to the sizereduction of v-shape objects and (2) modification of air flowcaused by neighbouring structures It is important from ascientific point of view to separate these phenomena in theanalysis process -erefore in this section only the size of asingle deflector is investigated regardless of its applicability-emechanics behind energy andmomentum transfer is thesame as the single deflector already discussed so pressuredistributions were not included in this section However the

reaction forces presented in Figure 4 showed that mini-mizing the v-shape deflector had a negative impact on itsblast mitigation effectiveness for both assumed scenariosie fixed position of the base (a) and the tip (b) of thedeflector according to HE charge location Although evenfor a small Vs80 structure the maximum reaction force wasreduced by a third in reference to Vf800 and it was not aseffective as the single deflector Vs800 which was loaded by asignificantly stronger wave Increasing the deflector size ledto a reduction of response force in a monotonic manner Inaddition starting from 39ms the force history was indis-tinguishable from any of the v-shape sizes studied -eresults depicted in Figure 4(b) confirm the superiority of thesingle large deflector where the monotonic character of forcedecay is preserved as above A smaller standoff distanceresulted in a much higher value of force peak for all casesbecause of the more intensive wave

43 Analysis of Boundary Effects All the variants examinedshowed that the application of a smaller deflector resulted indeterioration of the blast mitigation ability However theissue of the rarefaction effect had not yet been investigatedTwo structures with the same position and width weretherefore subjected to a blast load In the first one adjacentobjects were present (Vq800) while in the other there wasonly a single v-shape (Vs200) -e results obtained are

3025 3520Time (ms)

0

100

200

300

400

Forc

e (N

mm

)

Vf800Vs800Vd800

Vt800Vq800

Figure 3 Reaction force history for flat single and multiple v-shape deflectors

Table 2 Maximum force and momentum transfer

VariantMaximum force Momentum transfer

(Nmm) () (Nmmmm) ()Vf800 416 100 272 100Vs800 189 45 194 71Vd800 274 66 239 88Vt800 320 77 247 91Vq800 345 83 251 92

6 Shock and Vibration

depicted in Figure 5 with the normalization procedureapplied -e force history observed for Vq800 was a su-perposition of two contrary events (1) existing rarefactioneffect and (2) appearance of secondary reflecting waves -efact that the variant produced a similar response to thereference case indicated that the first phenomena was not acrucial factor while the second one might play an importantrole In Vs200 rarefaction was not accompanied by otherimportant effects apart from the primary oblique reflection-e force history (Figure 5) shows deterioration of the ef-ficacy of the multiple v-shape deflector

5 Conclusions

-e paper deals with a 2D numerical study of the blastmitigation ability of multi-v-shape deflectors -ree sets of

analysis were conducted (1) multiplication of trianglecomponents (2) influence of deflector size and (3) geom-etry-induced shock dynamics -e primary measure of theresults was reaction force history though momentumtransfer and pressure distribution maps were also taken intoconsideration -e conclusions obtained for plain problemformulation can only be used qualitatively in real conditionsthough a high accuracy of results is guaranteed due to high-dense discretization of space

-e investigation proved that a single deflector ach-ieves a much weaker reaction response Furthermore theforce increased much more slowly in comparison with thereference flat plate even though Vs800 was subjected to astronger wave and load lasted for a longer period of timewhich was confirmed by poorer impulse characteristics-e main mechanism for blast mitigation was obliquereflection and rarefaction effects However the existence ofmultiple v-shapes was responsible for a negative effect inthe form of secondary reflection waves Obviously theimpact of the standoff distance was crucial therefore a fewscenarios were proposed to separately analyse the variousmechanical and geometric effects Studies showed thatdecreasing the size of the single deflector caused themonotonic effectiveness to deteriorate in both assumedscenarios ie fixed position of its base and tip with ref-erence to the HE charge location Analysis of adjacentv-shapesrsquo presence showed that their interaction led toincrease in the force peak in comparison with a singlecomponent (Vs200) where a rarefaction effect wasdominant even though the normalization procedure wasutilized

In summary the results obtained showed that bothdecreasing the v-shape deflector size and surrounding it byneighbouring structures had a negative influence on its blastmitigation effectiveness However for each multishape de-flector some improvement in the protection level was

3025 3520Time (ms)

Vf800Vs800Vs560

Vs320Vs80

0

100

200

300

400

Forc

e (N

mm

)

(a)

3025 3520Time (ms)

Vf800cVs800Vs560c

Vs320cVs80c

0

100

200

300

400

500

600

Forc

e (N

mm

)

(b)

Figure 4 Reaction force for triangle deflectors (a) position of the bottom is fixed and (b) position of the top is fixed

3025 3520Time (ms)

Vf800Vq800Vs200

0

100

200

300

400

Forc

e (N

mm

)

Figure 5 Reaction force for flat quadruple and single deflectors

Shock and Vibration 7

present For situations where installation of a typical pro-tector is not possible mainly due to dimensional require-ments it may offer a compromise solution

Data Availability

-e input data (key file) for the LS-DYNA code used tosupport the findings of this study are available from thecorresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the National Centre for Re-search and Development Poland (Grant no DOBR-BIO4022131492013)

References

[1] Mines Action Canada Landmine Monitor Report 2009 MinesAction Canada Ottawa Canada 2009

[2] NATO Procedures for Evaluating the Protection Level ofArmoured VehiclesndashMine 2reat Vol 2 NATO BrusselsBelgium 2014

[3] A Shukla and D S Rajapakse Blast MitigationndashExperimentaland Numerical Studies Springer Berlin Germany 2010

[4] N Uddin Blast Protection of Civil Infrastructures and VehiclesUsing Composites Woodhead Publishing Sawston UK 2010

[5] G N Nurick and G C Shave ldquo-e deformation and tearingof thin square plates subjected to impulsive loads-An ex-perimental studyrdquo International Journal of Impact Engi-neering vol 18 no 1 pp 99ndash116 1996

[6] A C Jacinto R D Ambrosini and R F Danesi ldquoExperi-mental and computational analysis of plates under air blastloadingrdquo International Journal of Impact Engineering vol 25no 10 pp 927ndash947 2001

[7] S Ambrosini and G Slawinski ldquo-e influence of utilizingdifferent materials and their configurations on ballistic panelsblast resistancerdquo in Advances in Mechanics 2eoreticalComputational and Interdisciplinary Issues CRC Press BocaRaton FL USA 2016

[8] I S Sandhu ldquoMitigation of blast induced acceleration usingopen cell natural rubber and synthetic foamrdquo Defence ScienceJournal vol 69 no 1 pp 53ndash57 2019

[9] C Qi ldquoDynamic response and optimal design of curvedmetallic sandwich panels under blast loadingrdquo 2e ScientificWorld Journal vol 2014 Article ID 853681 14 pages 2014

[10] W Cheng ldquoNumerical analysis of cladding sandwich panelswith tubular cores subjected to uniform blast loadrdquo Inter-national Journal of Impact Engineering vol 133 2019

[11] S Guangyong ldquoDynamic response of sandwich panel withhierarchical honeycomb cores subject to blast loadingrdquo 2in-Walled Structures vol 142 pp 499ndash515 2019

[12] G B Dor Shock Wave Reflection Phenomena Springer-Verlag Berlin Germany 1992

[13] V Denefeld N Heider A Holzwarth A Sattler and M SalkldquoReduction of global effects on vehicles after IED detona-tionsrdquo Defence Technology vol 10 no 2 pp 219ndash225 2014

[14] J Trajkovski J Perenda and R Kunc ldquoBlast response of Lightarmoured vehicles (LAVs) with flat and V-hull floorrdquo 2in-Walled Structures vol 131 pp 238ndash244 2018

[15] R Gieleta W Barnat and T Niezgoda ldquoExperimental in-vestigation of deflectorrsquos angle influence on energy absorp-tionrdquo Journal of KONES Powertrain and Transport vol 19no 4 pp 201ndash205 2015

[16] S Barnat G S Langdon G N Nurick E G Pickering andV H Balden ldquoResponse of V-shape plates to localised blastload experiments and numerical simulationrdquo InternationalJournal of Impact Engineering vol 46 pp 97ndash109 2012

[17] P Kumar J LeBlanc and A Shukla ldquoEffect of curvature onshock loading response of aluminum panelsrdquo Dynamic Be-havior of Materials Volume 1 vol 1 pp 369ndash374 2011

[18] J LeBlanc R Kunc and I Prebil ldquoBlast response of centrallyand eccentrically loaded flat- U- and V-shaped armoredplates comparative studyrdquo Shock Waves vol 27 no 4pp 583ndash591 2017

[19] G Gurumurthy Blast Mitigation Strategies for Vehicles UsingShape Optimization Methods Massachusetts Institute ofTechnology Cambridge MA USA 2008

[20] C Davis Prevention of Injury in Mine Resistant AmbushProtected (MRAP) Vehicle Accidents United States ArmyAeromedical Research Laboratory Fort Rucker AL USA2013

[21] R Hajek M Foglar and J Fladr ldquoInfluence of barrier ma-terial and barrier shape on blast wave mitigationrdquo Con-struction and Building Materials vol 120 pp 54ndash64 2016

[22] K Genson ldquoVehicle shaping for mine blast damage reduc-tionrdquo University of Maryland College Park MA USAMasterrsquos of Science 2006

[23] A Morka ldquoNumerical analysis of momentum transfer in thecase of blast protection structuresrdquo AIP Conference Pro-ceedings vol 2078 no 1 2019

[24] J Hallquist LS-Dyna 2eory Manual LSTC Livermore CAUSA 2018

[25] G Randers-Pehrson and K Bannister AirBlast LoadingModel for DYNA2D and DYNA3D Army Research Labora-tory Adelphi MD USA 1997

[26] C N Kingery and G Bulmash Air-Blast Parameters fromTNT Spherical Airburst and Hemispherical Surface Burst USArmy Ballistic Research Laboratory Aberdeen ProvingGround MD USA 1984

[27] L Schwer ldquoAir blast reflection ratios and angle of incidencerdquoin Proceedings of the 11th European LS-DYNA ConferenceSalzburg Austria 2017

[28] N Van Dorsselaer S Eveillard and S Trelat ldquoExperimentsand simulations of explosives shock wave propagationaround a convex structurerdquo in Proceedings of the 15th In-ternational LS-DYNA Users Conference Detroit MI USAJune 2018

[29] D A Powell D Bogosian and L Schwer ldquoMesh sensitivity ofblast wave propagationrdquo in Proceedings of the 15th Interna-tional LS-DYNA Users Conference Detroit MI USA June2018

8 Shock and Vibration

depicted in Figure 5 with the normalization procedureapplied -e force history observed for Vq800 was a su-perposition of two contrary events (1) existing rarefactioneffect and (2) appearance of secondary reflecting waves -efact that the variant produced a similar response to thereference case indicated that the first phenomena was not acrucial factor while the second one might play an importantrole In Vs200 rarefaction was not accompanied by otherimportant effects apart from the primary oblique reflection-e force history (Figure 5) shows deterioration of the ef-ficacy of the multiple v-shape deflector

5 Conclusions

-e paper deals with a 2D numerical study of the blastmitigation ability of multi-v-shape deflectors -ree sets of

analysis were conducted (1) multiplication of trianglecomponents (2) influence of deflector size and (3) geom-etry-induced shock dynamics -e primary measure of theresults was reaction force history though momentumtransfer and pressure distribution maps were also taken intoconsideration -e conclusions obtained for plain problemformulation can only be used qualitatively in real conditionsthough a high accuracy of results is guaranteed due to high-dense discretization of space

-e investigation proved that a single deflector ach-ieves a much weaker reaction response Furthermore theforce increased much more slowly in comparison with thereference flat plate even though Vs800 was subjected to astronger wave and load lasted for a longer period of timewhich was confirmed by poorer impulse characteristics-e main mechanism for blast mitigation was obliquereflection and rarefaction effects However the existence ofmultiple v-shapes was responsible for a negative effect inthe form of secondary reflection waves Obviously theimpact of the standoff distance was crucial therefore a fewscenarios were proposed to separately analyse the variousmechanical and geometric effects Studies showed thatdecreasing the size of the single deflector caused themonotonic effectiveness to deteriorate in both assumedscenarios ie fixed position of its base and tip with ref-erence to the HE charge location Analysis of adjacentv-shapesrsquo presence showed that their interaction led toincrease in the force peak in comparison with a singlecomponent (Vs200) where a rarefaction effect wasdominant even though the normalization procedure wasutilized

In summary the results obtained showed that bothdecreasing the v-shape deflector size and surrounding it byneighbouring structures had a negative influence on its blastmitigation effectiveness However for each multishape de-flector some improvement in the protection level was

3025 3520Time (ms)

Vf800Vs800Vs560

Vs320Vs80

0

100

200

300

400

Forc

e (N

mm

)

(a)

3025 3520Time (ms)

Vf800cVs800Vs560c

Vs320cVs80c

0

100

200

300

400

500

600

Forc

e (N

mm

)

(b)

Figure 4 Reaction force for triangle deflectors (a) position of the bottom is fixed and (b) position of the top is fixed

3025 3520Time (ms)

Vf800Vq800Vs200

0

100

200

300

400

Forc

e (N

mm

)

Figure 5 Reaction force for flat quadruple and single deflectors

Shock and Vibration 7

present For situations where installation of a typical pro-tector is not possible mainly due to dimensional require-ments it may offer a compromise solution

Data Availability

-e input data (key file) for the LS-DYNA code used tosupport the findings of this study are available from thecorresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the National Centre for Re-search and Development Poland (Grant no DOBR-BIO4022131492013)

References

[1] Mines Action Canada Landmine Monitor Report 2009 MinesAction Canada Ottawa Canada 2009

[2] NATO Procedures for Evaluating the Protection Level ofArmoured VehiclesndashMine 2reat Vol 2 NATO BrusselsBelgium 2014

[3] A Shukla and D S Rajapakse Blast MitigationndashExperimentaland Numerical Studies Springer Berlin Germany 2010

[4] N Uddin Blast Protection of Civil Infrastructures and VehiclesUsing Composites Woodhead Publishing Sawston UK 2010

[5] G N Nurick and G C Shave ldquo-e deformation and tearingof thin square plates subjected to impulsive loads-An ex-perimental studyrdquo International Journal of Impact Engi-neering vol 18 no 1 pp 99ndash116 1996

[6] A C Jacinto R D Ambrosini and R F Danesi ldquoExperi-mental and computational analysis of plates under air blastloadingrdquo International Journal of Impact Engineering vol 25no 10 pp 927ndash947 2001

[7] S Ambrosini and G Slawinski ldquo-e influence of utilizingdifferent materials and their configurations on ballistic panelsblast resistancerdquo in Advances in Mechanics 2eoreticalComputational and Interdisciplinary Issues CRC Press BocaRaton FL USA 2016

[8] I S Sandhu ldquoMitigation of blast induced acceleration usingopen cell natural rubber and synthetic foamrdquo Defence ScienceJournal vol 69 no 1 pp 53ndash57 2019

[9] C Qi ldquoDynamic response and optimal design of curvedmetallic sandwich panels under blast loadingrdquo 2e ScientificWorld Journal vol 2014 Article ID 853681 14 pages 2014

[10] W Cheng ldquoNumerical analysis of cladding sandwich panelswith tubular cores subjected to uniform blast loadrdquo Inter-national Journal of Impact Engineering vol 133 2019

[11] S Guangyong ldquoDynamic response of sandwich panel withhierarchical honeycomb cores subject to blast loadingrdquo 2in-Walled Structures vol 142 pp 499ndash515 2019

[12] G B Dor Shock Wave Reflection Phenomena Springer-Verlag Berlin Germany 1992

[13] V Denefeld N Heider A Holzwarth A Sattler and M SalkldquoReduction of global effects on vehicles after IED detona-tionsrdquo Defence Technology vol 10 no 2 pp 219ndash225 2014

[14] J Trajkovski J Perenda and R Kunc ldquoBlast response of Lightarmoured vehicles (LAVs) with flat and V-hull floorrdquo 2in-Walled Structures vol 131 pp 238ndash244 2018

[15] R Gieleta W Barnat and T Niezgoda ldquoExperimental in-vestigation of deflectorrsquos angle influence on energy absorp-tionrdquo Journal of KONES Powertrain and Transport vol 19no 4 pp 201ndash205 2015

[16] S Barnat G S Langdon G N Nurick E G Pickering andV H Balden ldquoResponse of V-shape plates to localised blastload experiments and numerical simulationrdquo InternationalJournal of Impact Engineering vol 46 pp 97ndash109 2012

[17] P Kumar J LeBlanc and A Shukla ldquoEffect of curvature onshock loading response of aluminum panelsrdquo Dynamic Be-havior of Materials Volume 1 vol 1 pp 369ndash374 2011

[18] J LeBlanc R Kunc and I Prebil ldquoBlast response of centrallyand eccentrically loaded flat- U- and V-shaped armoredplates comparative studyrdquo Shock Waves vol 27 no 4pp 583ndash591 2017

[19] G Gurumurthy Blast Mitigation Strategies for Vehicles UsingShape Optimization Methods Massachusetts Institute ofTechnology Cambridge MA USA 2008

[20] C Davis Prevention of Injury in Mine Resistant AmbushProtected (MRAP) Vehicle Accidents United States ArmyAeromedical Research Laboratory Fort Rucker AL USA2013

[21] R Hajek M Foglar and J Fladr ldquoInfluence of barrier ma-terial and barrier shape on blast wave mitigationrdquo Con-struction and Building Materials vol 120 pp 54ndash64 2016

[22] K Genson ldquoVehicle shaping for mine blast damage reduc-tionrdquo University of Maryland College Park MA USAMasterrsquos of Science 2006

[23] A Morka ldquoNumerical analysis of momentum transfer in thecase of blast protection structuresrdquo AIP Conference Pro-ceedings vol 2078 no 1 2019

[24] J Hallquist LS-Dyna 2eory Manual LSTC Livermore CAUSA 2018

[25] G Randers-Pehrson and K Bannister AirBlast LoadingModel for DYNA2D and DYNA3D Army Research Labora-tory Adelphi MD USA 1997

[26] C N Kingery and G Bulmash Air-Blast Parameters fromTNT Spherical Airburst and Hemispherical Surface Burst USArmy Ballistic Research Laboratory Aberdeen ProvingGround MD USA 1984

[27] L Schwer ldquoAir blast reflection ratios and angle of incidencerdquoin Proceedings of the 11th European LS-DYNA ConferenceSalzburg Austria 2017

[28] N Van Dorsselaer S Eveillard and S Trelat ldquoExperimentsand simulations of explosives shock wave propagationaround a convex structurerdquo in Proceedings of the 15th In-ternational LS-DYNA Users Conference Detroit MI USAJune 2018

[29] D A Powell D Bogosian and L Schwer ldquoMesh sensitivity ofblast wave propagationrdquo in Proceedings of the 15th Interna-tional LS-DYNA Users Conference Detroit MI USA June2018

8 Shock and Vibration

present For situations where installation of a typical pro-tector is not possible mainly due to dimensional require-ments it may offer a compromise solution

Data Availability

-e input data (key file) for the LS-DYNA code used tosupport the findings of this study are available from thecorresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the National Centre for Re-search and Development Poland (Grant no DOBR-BIO4022131492013)

References

[1] Mines Action Canada Landmine Monitor Report 2009 MinesAction Canada Ottawa Canada 2009

[2] NATO Procedures for Evaluating the Protection Level ofArmoured VehiclesndashMine 2reat Vol 2 NATO BrusselsBelgium 2014

[3] A Shukla and D S Rajapakse Blast MitigationndashExperimentaland Numerical Studies Springer Berlin Germany 2010

[4] N Uddin Blast Protection of Civil Infrastructures and VehiclesUsing Composites Woodhead Publishing Sawston UK 2010

[5] G N Nurick and G C Shave ldquo-e deformation and tearingof thin square plates subjected to impulsive loads-An ex-perimental studyrdquo International Journal of Impact Engi-neering vol 18 no 1 pp 99ndash116 1996

[6] A C Jacinto R D Ambrosini and R F Danesi ldquoExperi-mental and computational analysis of plates under air blastloadingrdquo International Journal of Impact Engineering vol 25no 10 pp 927ndash947 2001

[7] S Ambrosini and G Slawinski ldquo-e influence of utilizingdifferent materials and their configurations on ballistic panelsblast resistancerdquo in Advances in Mechanics 2eoreticalComputational and Interdisciplinary Issues CRC Press BocaRaton FL USA 2016

[8] I S Sandhu ldquoMitigation of blast induced acceleration usingopen cell natural rubber and synthetic foamrdquo Defence ScienceJournal vol 69 no 1 pp 53ndash57 2019

[9] C Qi ldquoDynamic response and optimal design of curvedmetallic sandwich panels under blast loadingrdquo 2e ScientificWorld Journal vol 2014 Article ID 853681 14 pages 2014

[10] W Cheng ldquoNumerical analysis of cladding sandwich panelswith tubular cores subjected to uniform blast loadrdquo Inter-national Journal of Impact Engineering vol 133 2019

[11] S Guangyong ldquoDynamic response of sandwich panel withhierarchical honeycomb cores subject to blast loadingrdquo 2in-Walled Structures vol 142 pp 499ndash515 2019

[12] G B Dor Shock Wave Reflection Phenomena Springer-Verlag Berlin Germany 1992

[13] V Denefeld N Heider A Holzwarth A Sattler and M SalkldquoReduction of global effects on vehicles after IED detona-tionsrdquo Defence Technology vol 10 no 2 pp 219ndash225 2014

[14] J Trajkovski J Perenda and R Kunc ldquoBlast response of Lightarmoured vehicles (LAVs) with flat and V-hull floorrdquo 2in-Walled Structures vol 131 pp 238ndash244 2018

[15] R Gieleta W Barnat and T Niezgoda ldquoExperimental in-vestigation of deflectorrsquos angle influence on energy absorp-tionrdquo Journal of KONES Powertrain and Transport vol 19no 4 pp 201ndash205 2015

[16] S Barnat G S Langdon G N Nurick E G Pickering andV H Balden ldquoResponse of V-shape plates to localised blastload experiments and numerical simulationrdquo InternationalJournal of Impact Engineering vol 46 pp 97ndash109 2012

[17] P Kumar J LeBlanc and A Shukla ldquoEffect of curvature onshock loading response of aluminum panelsrdquo Dynamic Be-havior of Materials Volume 1 vol 1 pp 369ndash374 2011

[18] J LeBlanc R Kunc and I Prebil ldquoBlast response of centrallyand eccentrically loaded flat- U- and V-shaped armoredplates comparative studyrdquo Shock Waves vol 27 no 4pp 583ndash591 2017

[19] G Gurumurthy Blast Mitigation Strategies for Vehicles UsingShape Optimization Methods Massachusetts Institute ofTechnology Cambridge MA USA 2008

[20] C Davis Prevention of Injury in Mine Resistant AmbushProtected (MRAP) Vehicle Accidents United States ArmyAeromedical Research Laboratory Fort Rucker AL USA2013

[21] R Hajek M Foglar and J Fladr ldquoInfluence of barrier ma-terial and barrier shape on blast wave mitigationrdquo Con-struction and Building Materials vol 120 pp 54ndash64 2016

[22] K Genson ldquoVehicle shaping for mine blast damage reduc-tionrdquo University of Maryland College Park MA USAMasterrsquos of Science 2006

[23] A Morka ldquoNumerical analysis of momentum transfer in thecase of blast protection structuresrdquo AIP Conference Pro-ceedings vol 2078 no 1 2019

[24] J Hallquist LS-Dyna 2eory Manual LSTC Livermore CAUSA 2018

[25] G Randers-Pehrson and K Bannister AirBlast LoadingModel for DYNA2D and DYNA3D Army Research Labora-tory Adelphi MD USA 1997

[26] C N Kingery and G Bulmash Air-Blast Parameters fromTNT Spherical Airburst and Hemispherical Surface Burst USArmy Ballistic Research Laboratory Aberdeen ProvingGround MD USA 1984

[27] L Schwer ldquoAir blast reflection ratios and angle of incidencerdquoin Proceedings of the 11th European LS-DYNA ConferenceSalzburg Austria 2017

[28] N Van Dorsselaer S Eveillard and S Trelat ldquoExperimentsand simulations of explosives shock wave propagationaround a convex structurerdquo in Proceedings of the 15th In-ternational LS-DYNA Users Conference Detroit MI USAJune 2018

[29] D A Powell D Bogosian and L Schwer ldquoMesh sensitivity ofblast wave propagationrdquo in Proceedings of the 15th Interna-tional LS-DYNA Users Conference Detroit MI USA June2018

8 Shock and Vibration