peerin

Impulse transferABAQUSAUTODYNSimulation validation

loadtheexpstict al

ling, asure loiameteressur

charge radius as a function of plate radius/width. In a further var-iation of this method, Balden and Nurick [6] used the programAUTODYN [7] to obtain a pressure prole for a localised blast load.The pressure prole was based on the prole of maximum pressureobtained from the simulations. The pressure magnitude for theprole was adjusted so that the applied impulse corresponded withthe measured experimental ballistic pendulum impulse.

AUTODYN [7] which uses an Eulerianmesh for the explosive and airand the JonesWilkinsLee (JWL) equation of state [8] for theexplosive. If the experimental set-up including the clampingarrangement is modelled (Fig. 1), then a priori knowledge of howthe impulse is imparted to the plate is not required. The geometryin the simulation will allow any physical phenomena to be repro-duced and the impulse will be imparted as in the experiment. Axi-symmetric or 3-dimensional models can be used but 3-dimensionalmodels greatly increase the computational expense. Experiments

Contents lists availab

o

ls

International Journal of Impact Engineering 36 (2009) 4052* Corresponding author. Tel.: 27 21 6503234; fax: 27 21 6503240.duration which is usually taken as the time it takes a detonationwave to travel the length of the charge. This simple method hasbeen used with various levels of success to predict the deformationand failure of locally (and uniformly) blast loaded plates.

A variation of this method was presented by Bimha [5] (seeSection 3.4) where the blast load was represented by a constantpressure acting over the charge area with the pressure decaying tothe edge of the plate. Based on comparisonwith experimental platedeection proles the decay constant was found for variations in

localised load. The focus of this paper is to investigate this as-sumption in order to determine the proportion of the measuredballistic pendulum impulse responsible for plate deformation. Theresults of this investigation are used to develop a localised blastload model. The loading model is used in a nite element model ofclamped rectangular plates subjected to localised blast loading. Acomparison of the nite element results with an independent set ofexperiments on clamped rectangular plates provides a validation ofthe loading model.

An alternative is to model the explosive using a program such as\1. Introduction

Traditionally, in numerical modelbeen represented by a constant presequivalent to the charge or burn dsumption of impulsive loading, the pE-mail address: gerald.nurick@uct.ac.za (G.N. Nuri

0734-743X/$ see front matter 2008 Elsevier Ltd.doi:10.1016/j.ijimpeng.2008.03.003impulse from the ballistic pendulum should not simply be applied as a centrally localised pressure load.Numerical simulations of localised blast loading in combination with the aforementioned experimentalresults are used to develop a localised blast loading model. The loading model is a simplied pressureloading model which only imparts the deformation causing impulse as opposed to the total ballisticpendulum impulse. The model is validated using an independent set of localised blast load experimentson clamped mild steel plates. Results obtained using a published localised blast load model are alsocompared.

2008 Elsevier Ltd. All rights reserved.

localised blast load hasad acting over an arear [14]. Under the as-e is applied over a short

The preceding localised blast load models use the measuredballistic pendulum impulse to determine the magnitude of thepressure prole. This ensures that the total measured experimentalimpulse is applied impulsively to the plate in the form of a pressureload. These methods assume that the total impulse measured bythe ballistic pendulum is transferred to the plate as a centrallyKeywords:Localised blast loadthe plate deformation is unchanged. This suggests that not all of the impulse measured by the ballisticpendulum resulted in plate deformation. Therefore, in numerical and analytical modelling, the totalThe inuence of boundary conditions onsubjected to localised blast loading Im

D. Bonorchis, G.N. Nurick*

Blast Impact and Survivability Research Unit (BISRU), Department of Mechanical Engin

a r t i c l e i n f o

Article history:Received 14 November 2007Received in revised form 3 March 2008Accepted 3 March 2008Available online 15 May 2008

a b s t r a c t

A series of localised blastmental set-up inuencesa ballistic pendulum. Thepulse measured by a balliSignicantly, it is found tha

International Journal

journal homepage: www.eck).

All rights reserved.the loading of rectangular platesortance in numerical simulations

g, University of Cape Town, Private Bag, Rondebosch 7701, South Africa

ing experiments are performed in order to understand how the experi-impulse imparted to a plate. The imparted impulse is measured usingerimental results show that for both rigid and deformable plates the im-pendulum increases as the height of the boundary (clamps) increases.though the measured impulse varies as a function of the boundary height,

le at ScienceDirect

f Impact Engineering

evier .com/locate/ i j impengsuch as rectangular plate experiments cannot be reduced to

axi-symmetric models and therefore, for practical reasons, de-veloping a simple validated pressure loadingmodel is advantageous.

The paper begins by describing an experimental investigationto determine the effect of the presence of clamps (boundaries)and clamp height on both rigid and deformable plates forlocalised blast loading. The rigid plate tests are used to de-termine the effect of clamping arrangement on impulse transferas measured by a ballistic pendulum. The effect of the varyingimpulse transfer is subsequently investigated in terms of platedeformation. Axi-symmetric numerical simulation results fromAUTODYN are used to develop a pressure loading model whichis applied to rectangular plates and compared with an in-dependent set of experiments. The results of the investigationinto the effect of the clamping arrangement are incorporated inthe loading model. The results of the proposed loading modelare compared with the results obtained using a published [5]localised loading model.

2. Blast loading prole

2.1. Localised blast load experimental investigation

A series of experiments are performed in order to gain furtherinsight into localised blast loading of rectangular plates. The ex-periments are as follows:

Fig. 1. Photograph showing the experimental set-up of a typical clamped platesubjected to localised blast loading.

Nomenclature

_3P equivalent plastic strain rates material stress3p equivalent plastic strainI*(r,m) impulse density prole as

function of radius and charge massm charge mass including 1 g leaderP pressureT temperaturetb burn time

D. Bonorchis, G.N. Nurick / International Journal of Impact Engineering 36 (2009) 4052 41Fig. 2. Quarter-symmetry diagrams showing the rigid and deformable plate layouts used tothe plate is the explosive at a stand-off of 13 mm.determine the effect of clamp height for a constant charge mass. The quarter disc above

Rigid plate tests To determine the effect of the boundary height (clampheight) on the impulse transferred for a given charge mass.

To determine the impulse transferred to the exposed area ofthe plate only with no boundary effects.

Deformable plate tests To determine the effect of the boundary height (clamp

Fig. 3. Photograph showing the ground side and the post-

D. Bonorchis, G.N. Nurick / International Journa42height) on the deformation of the plate for a given chargemass.

Fig. 2 shows the layouts used to determine the effect of clampheight (for a constant charge mass) on impulse transfer and de-formation. In all cases with an exposed area the breadth (B) is120 mm and the length (L) is 200 mm. The total plate dimensionsare 260 mm 300 mm. Tests with charge masses of 8 g, 12 g and15 g (excluding the 1 g leader) are performed with either noclamps, one clamp or two clamps on the loaded side of the plate(additional charge masses are tested with two clamps). Each clampis 16 mm in height. The charge diameter is kept constant at 40 mmwhich in this case results in a charge diameter to plate breadth ratioof one-third. As a result of the diameter being held constant thecharge height increases with increasing charge mass. For the 8 g,12 g and 15 g charge masses the calculated nominal charge heightsare 4.0 mm, 6.0 mm and 7.5 mm, respectively. The clamp assemblyis attached to a ballistic pendulum which measures the totaltransferred impulse (see [1] for details of the experimental set-up).

The deformable plates are machined from 10 mm thick mildsteel plates. The plates are rst ground on one side to provide a atsurface and baseline fromwhich to machine the thickness from theFig. 4. Photograph of the rigid exposed area plate. A stand-off is used to ensure nodirect blast loading of the pendulum.opposite side. The plates were machined with a CNC milling ma-chine to an average thickness of 3.74 mm 0.02 mm. The materialproperties are not required for these plates since they are only usedfor comparison with each other. Photographs of the ground sideand the post-blast deformed machined side of the deformableplates are shown in Fig. 3.

The rigid exposed area plate tests are performed with a chargemass of 15 g (excluding the 1 g leader). The rigid exposed area plateis of the same size as the exposed area of the deformable plates andthe plate is sufciently offset from the pendulum attachments so asto ensure no direct blast loading of the pendulum, as shown inFig. 4.

2.2. Localised blast load experimental results

2.2.1. Rigid platesThe experimental impulse versus mass of explosive results of

the localised blast load tests performed on rigid plates with varyingclamp heights is shown in Fig. 5. The rigid plate results are com-pared with previous deformable plate results to show that themeasured ballistic pendulum impulse is independent of plate de-formation. The single clamp rigid plate results () correspond to theprevious single clamp deformable plate (A) trends. Another im-portant point is that the various trend lines clearly show that for thesame charge mass the impulse recorded on the pendulum varies asa function of clamp height. Increasing clamp height results in in-creased transfer of impulse to the ballistic pendulum and thereforethe effect of clamp height (experimental conguration) needs to betaken into account.

For the cases where there is no clamp height, both sets of datashow a reduced impulse as compared with clamped plate tests. Theresults of the exposed area rigid plate tests (,) are also shown inFig. 5 and the trend is below the rigid plate tests (:) with no clamp.This is due to the reduced area of the exposed area plate testscompared to the plate tests with no boundary. These tests give anindication of the impulse transferred to the plate with no clamp

blast deformed machined side of a deformable plate.

l of Impact Engineering 36 (2009) 4052effects and no direct loading on top of the clamps. This informationis necessary to determine howmuch extra impulse is transferred tothe plates as a result of the clamps.

To get a quantitative idea of the effect of the clamps and clampheight, the trends of the 151 g case are examined. Taking the oneclamp height case as a baseline the increase (decrease) in impulsefor the other cases is given as

Two clamps: 8.5%. No clamps: 9.6%. Exposed area only: 10.0%.

It is clear that when using the measured ballistic pendulumimpulse to apply a load either analytically or numerically, the ex-perimental conguration used to obtain the impulse value needs tobe considered with great care.

2.2.2. Deformable platesIt has been established that for the same charge mass signi-

cantly different impulse measurements are possible depending on

Fig. 5. Graph showing the rigid and previous deformable (one clamp) plate impulse

D. Bonorchis, G.N. Nurick / International Journal of Impact Engineering 36 (2009) 4052 43Fig. 6. Graphs showing the effect of varying clamp heights on deformable plateresponse. (a) Graph comparing impulse measured using deformable plates versus rigidplates. (b) Graph of midpoint deection for a constant charge mass of 151 g.the boundary conditions used in the experiment. The next step is todetermine whether for the same charge mass, the variation inimpulse results in a variation in midpoint deections for deform-able plates.

The deformable plate impulses as a result of varying clampheights are compared with the rigid plate results at a charge massof 151 g as shown in Fig. 6(a). The grouping of the deformableplate results corresponds to the grouping of the rigid plate resultsfor varying clamp heights. There is no trend of the deformableplates transferring more impulse than the rigid plates and viceversa. For the current deformable plate experiments it is againshown that, within experimental variation, deformable plates donot produce different impulse transfers when compared with rigidplates.

The plate midpoint deections versus the measured impulse forthe various clamp heights at a constant charge mass of 151 g areshown in Fig. 6(b). The experimental results indicate that eventhough there is a large variation in impulse transferred to theballistic pendulum due to the varying clamp heights, the platemidpoint deections are unaffected. Therefore for a given charge

s as a function of clamp height. The 1 g leader is included in the charge mass.mass the clamp height greatly affects the measured impulsetransfer but has no effect on the plate midpoint deection. Thissuggests that not all the impulse measured by the ballistic pen-dulum results in plate deformation. This result is extremely im-portant when representing the blast load with a simpliedimpulsive pressure load for analytical and numerical work. Eventhough the impulse measured by the ballistic pendulum is a truemeasure of the impulse being transferred through the plate, the

Fig. 7. Schematic showing the rigid plate model set-up in AUTODYN for the 151 gcase.

total measured impulse cannot simply be applied as a centrallylocalised impulsive pressure load. As a result of the clamps, theapplied impulse must be reduced to take account of the measured

rigid plate test is shown in Fig. 7. The rigid plate is modelled by

Table 1Material properties used in the rigid plate simulations obtained from the AUTODYN mat

PE4 (Equation of state JWL)r0 (g/cm

3) A (kPa) B (kPa) R1 R2 u

1.601 6.0977 108 1.295 107 4.5 1.4 0.25Air (Equation of state Ideal gas)r0 (g/cm

3) g T0 (K)

1.225 103 1.4 288.2

D. Bonorchis, G.N. Nurick / International Journa44not specifying a boundary condition at the edge of the air zone. Thisimpulse which does not cause deformation.

2.3. Localised blast load numerical model

Localised blast loading simulations are performed using AUTO-DYN-2D v6.1, a state of-the-art nonlinear dynamics modelling andsimulation software package [7]. AUTODYN and similar codes areoften referred to as hydrocodes and are particularly suited tomodelling blast, impact and penetration events. Their power comesfrom their ability to handle complex problems where a Lagrangeprocessor and an Eulerian processor can work side by side on thesame problem. The Lagrange processor uses a mesh which deformswith the material it contains while the Eulerian processor hasa xedmesh in space which allows the material to move through it.The Lagrange processor is typically used for solid, continuumstructures while the Eulerian processor is used for modelling gases,liquids or solids where large deformations are likely to occur [9].

The localised blast loading simulations are performed axi-symmetrically in AUTODYN-2D. The reason for using an axi-sym-metric model as opposed to a 3-dimensional model is to reducecomputational costs. The axi-symmetric model is used to charac-terise how the impulse density varies as a function of distance fromthe blast centre and to gain insight into the effect of the interactionbetween the blast wave and the boundary (clamps). A simpliedpressure loading model is sought to replace the 3-dimensionalmodelling of the explosive and plate interaction within reasonablelimits of accuracy.

2.3.1. Rigid plate modelThe AUTODYN model used to obtain pressure histories from theFig. 8. Graph of impulse versus charge mass for axi-symmetric rigid plate simulations.allows no transmission of material and acts as a rigid boundary. Thereected pressure on the rigid plate is measured using gaugepoints in the air cell on the edge of the air zone. Transmissionboundaries are included at the outer edges of the mesh. Theseboundaries are sufciently far away from the plate so that themajority of the loading on the plate is complete before any materialis lost through the transmission boundaries. The majority of thecentral load transfer is assumed to be complete within approxi-mately 40 ms (Fig. 10(d)) and the explosive products have not yetreached the boundary (Fig. 9).

The size of the Eulerian cells is approximately 0.5 mm 0.5 mm.The air is modelled using the ideal gas law and the explosive(PE4 C4) is modelled using the JWL equation of state [8]. Thematerial properties of the air and the explosive are given in Table 1and are obtained from the material library in AUTODYN [7]. Whenlling the air Eulerian mesh the specic internal energy (E0) of theair is required and details of this calculation can be found in Refs.[9,10].

The detonation is modelled using a programmed burn wherethe detonation front travels at a constant detonation velocity of8193 m/s for PE4 [7, 11]. The programmed burn assumes that det-onation takes place instantaneously. As the detonation front rea-ches a material point within the un-reacted explosive material, theexplosive material is instantaneously transformed into gaseousdetonation products together with the release of energy associatedwith the chemical reaction [9]. The 1 g leader used with the deto-nator is included in the model as seen in Fig. 7. Detonation is ini-tiated at the end of the leader furthest from the plate and thedetonation front travels as a planar wave along the leader. Afterreaching the end of the leader the detonation progresses into theexplosive disc automatically using the programmed burnalgorithm.

2.3.2. Loading simulation resultsThe rigid plate simulation results for charge masses of 8 1 g,

121 g and 151 g are shown graphically in Fig. 8 for variousclamp heights. The results follow the same trend as the experi-mental results with an increase in impulse recorded for an in-creasing clamp height at a constant charge mass. To get

erial library [7]

CJ detonationvelocity (m/s)

CJ energy/volume (kJ/m3)

CJ pressure (kPa) Auto-convert toideal gas

8193 9.0 106 28.0 106 Yes

cp (J/kg K) E0 (J/kg)

7.176 102 2.068 105

l of Impact Engineering 36 (2009) 4052a quantitative idea of the effect of the clamps and clamp height, thetrends of the 151 g case are examined for the numerical model.Taking the one clamp height case as a baseline, the increase (de-crease) in impulse for the other cases is given as (rectangular plateexperimental values are given in parenthesis)

Two clamps: 9.9% (8.5%) No clamps: 21.6% (9.6%) Exposed area only:33.3% (10.0%) (trend not shown in Fig. 8)

The exposed area only and the no clamps case have slightlydifferent impulses because the no clamps case has a larger areathan the exposed area only case. Localised blast loading is notuniform and therefore the ratio of impulse of the exposed area onlyand the no clamps cases is not directly proportional to their

urnaD. Bonorchis, G.N. Nurick / International Jorespective areas. It is, however, useful to compare the no clampscase with the clamped cases because there is some loading on thetop surfaces of the clamps and this area is included in the no clampscase.

The increase in impulse for an increase in clamp height is moresevere for the numerical results due to the fact that they are axi-symmetric and that the radius is taken as the half-width of therectangular plate. The entire boundary of the axi-symmetric plate is

Fig. 9. Diagrams showing the pressure contours of the rigid plate simulations with varying cThe bottom edge of each contour plot is the axis of symmetry.l of Impact Engineering 36 (2009) 4052 45close to the centre of the localised blast load as opposed to therectangular plate where only the half-width boundaries are close.The axi-symmetric plate using the half-width as a radius gives anupper bound of the effect of boundary conditions on impulsetransfer. Since only a trend of the effect of clamp height is soughtthe upper bound is sufcient. Direct comparisons between rect-angular and axi-symmetric plates should not be made from thischoice.

lamp heights at different times. Only the regions above the plate and clamp are shown.

urnaD. Bonorchis, G.N. Nurick / International Jo46For the axi-symmetric plates with clamps, as the explosiveproducts expand radially they are met in all directions bya clamp face perpendicular to the radial expansion. This con-straint is equal in all directions and does not allow a relief ofpressure as is the case with the rectangular plates used in theexperiments. For the rectangular plates, only the explosiveproducts interacting with the closest point on the shortest andlongest boundaries do so at right angles. Explosive productsinteracting with other points along the boundaries do so at

Fig. 10. Graphs showing the effect of clamp height on thel of Impact Engineering 36 (2009) 4052acute angles which will result in lower reected pressures andtherefore lower impulse transfer.

Fig. 9 shows the pressure build-up due to the clamped bound-ary. The pressure build-up acts on the plate near the boundary aftermost of the pressure has receded above the plate with no clamp.This lingering pressure, although of relatively low magnitude, actsover a long duration and therefore increases the total impulsetransferred to the plate. Graphs of the intermediate impulse densityproles are shown in Fig. 10. The impulse density is dened as the

impulse density prole for a charge mass of 151 g.

Fig. 11. Contour of the mapped impulse density prole onto a quarter-symmetric

Fig. 12. Graph of experimental versus numerical impulses for the exposed area rigidplate tests.

D. Bonorchis, G.N. Nurick / International Journal of Impact Engineering 36 (2009) 4052 47impulse per unit area. The impulse density at the centre of the platehas almost reached a maximum by the time the impulse densitybegins to increase at the boundary as shown in Fig. 10(b) and (c).The reected pressure from the clamp travels back and forth acrossthe plate and gradually adds more impulse. This is seen in the nalimpulse density prole in Fig. 10(f) where the plate with twoclamps has a slightly higher impulse density than the plate witha single clamp which in turn has a higher impulse density than theplate with no clamp.

2.4. Experimental versus numerical impulse results

In order to compare the rectangular experimental impulseresults with the axi-symmetric numerical impulse results, theaxi-symmetric impulse density results are mapped onto the quar-ter-rectangular layout as shown in Fig. 11. The impulse is calculatedby creating a rectangular grid of 1 mm 1 mm square elementsand multiplying the area of each element by its impulse density.The impulse density of each element is determined by calculatingits distance from the plate centre and using the distance to nd theimpulse density from the AUTODYN impulse density prole withno clamps. An example of the impulse density prole obtainedfrom AUTODYN is shown in Fig. 13(a).

Once the impulse acting on each element is known the impulses

rectangular grid.are summed over the rectangle to obtain the total impulse (which ismultiplied by 4). Fig. 12 shows a graph comparing the

Fig. 13. Graphs showing how the AUTODYN impulse density distribution is matched to the edensity distribution. (b) The AUTODYN t is shifted up to matched the experimental impulexperimentally obtained impulses with the numerically obtainedimpulses. The trend of the numerical impulses obtained fromAUTODYN is below the experimental impulses. Given the necessarysimplications and approximations required to model this com-plicated event the numerical trend is nevertheless acceptable.Halving the element size in the region adjacent to the plate andusing double precision as opposed to the usual single precisionAUTODYN processor resulted in no change in the simulation results.

2.5. Final form of simplied pressure loading

Axi-symmetric impulse density proles obtained from the8 1 g, 121 g and 151 g AUTODYN simulations with no clampsare each t (using a least-squares minimisation method) witha function of the form given in Eq. (1). The function is purely a curve-tandnophysicalmeaning is attached.Any functionwhichcant theimpulse density prole could be used. The 151 g case is shown inFig.13(a). TheAUTODYNts are shiftedupslightly so that the impulsethey produce over the quarter-rectangular layout (Fig. 11) matchestheir respective experimental impulses (exposedareaonly). The shiftto match the experimental impulse is shown in Fig. 13(b) for the151 g case (the shift is barely noticeable but nevertheless increasesthe impulse by approximately 7.7%). The 151 g case is chosen asa baseline impulse density function. The baseline function is multi-plied by a scaling factor which is varied until the difference betweenthe scaled baseline function and the 121 g experimentally

matched function is minimised. The process is repeated for the

xperimental impulse for the 151 g case. (a) A function is t to the AUTODYN impulsese.

8 1 g case. The scale factors are plotted as a function of chargemass(including the 1 g leader) in Fig. 14 together with a linear trend.

The linear trend allows the scale factor to be determined for any

scaling factor, the impulse density distribution from the experi-mentally corrected AUTODYN simulations can be matched.

3. Loading model validation

3.1. Material model

In order to validate the scaled impulse density function, an in-dependent set of experiments are performed on 3 mm thick,monolithic single-clamped mild steel plates with an exposed areaof 120 mm 200 mm. The chemical composition of the commer-

Fig. 14. Graph of the optimised specic impulse scaling factor and its trend.

Table 2Constants for baseline 151 g impulse density function

Ih* (N s/m2) Il*(N s/m

2) I0* (N s/m2) I1* (N s/m

2) I2* (N s/m2)

14500 5657.6 68.17 5743.1 153.67

s1 (mm1) s2 (mm

1) s3 s4 rinI (mm)

0.10288 0.16348 1.22759 0.91143 20.25

D. Bonorchis, G.N. Nurick / International Journal of Impact Engineering 36 (2009) 405248charge mass in the range 8 1 g to 151 g. It is not known how farthe linear trend can be extrapolated on either side of the range; thisneeds to be further investigated. The scale factor trend and thebaseline function allow the impulse density distribution to bedetermined for any charge mass in the specied range. This meansthat by simply specifying the charge mass, the impulse densitydistributionwhich corresponds to the correct experimental impulsecan be found. The baseline function (the 151 g case) is given as

I*r I*h I*l

1 exps1rinI rs3 I*1 for r rinI

I*r I*l I*0

exp

s2r rinIs4

I*2 for r > rinI 1where r (mm) is the distance from the centre and the constantsused to t the function are given in Table 2. The scaled impulsedensity function, using the trend from Fig. 14, is given as

I*r;m 0:059m 0:068I*r (2)The results of scaling the baseline impulse density function areshown in Fig. 15 for the 8 1 g and 121 g cases, respectively. Theresults show that by multiplying the baseline function by a simpleFig. 15. Graph showing how the scaled 8 1 g and 1cial quality mild steel plates is given in Table 3. The plate materialmechanical properties are characterised based on tensile tests andSplit Hopkinson Pressure Bar (SHPB) tests. The behaviour at quasi-static (8 104 s1), intermediate (8 102 s1) and dynamic(2.6103 s1) strain rates is characterised.

The quasi-static and intermediate results are obtained using aniterative experimental/numerical method in which the tensile testsare simulated and the material properties adjusted until the ex-perimental forcedeection curves are matched. A more detaileddescription of this method is given by Bao and Wierzbicki [12]. Theresult of the quasi-static strain hardening characterisation is givenin Fig. 16(a). The high temperature response is based on high strainrate, high temperature results of a similar mild steel as presentedby Gilat and Wu [13]. The JohnsonCook [14] temperature model ischaracterised based on these results as shown in Fig. 16(b).

The strain rate effects are characterised based on a strain of 0.1.The strain rate effects are additive as opposed to multiplicative andtherefore the strain rate effect causes a parallel shift of the owstress curve. With a multiplicative strain rate model, such asJohnsonCook [14], the work-hardening rate increases as the strainrate increases which is not always true for BCC metals as discussedby Liang and Khan [15]. The temperature effects are taken intoaccount when characterising the high strain rate response because21 g curves match the shifted AUTODYN ts.

the experimental results implicitly include temperature for strainsup to 0.1. The effect of including temperature in the strain ratecharacterisation is shown in Fig. 16(c).

The nal form of the material model is similar to that of Zhao[16] except that the coupling term between strain hardening andstrain rate is omitted. The material model as a function of strainhardening, strain rate and temperature is given as

s hs3p s

_3pi

f T (3)

where

s3p A ssat A1 expb3pg

aB3pn

s_3p sref

"C ln

_3p

_3p0

! C1

_3p

_3p0

!1=C2#

f T 1

T TrefTmelt Tref

!M

The coefcients used in the material model are summarised inTable 4. The coefcients cp and h are the specic heat and inelasticheat fraction (TaylorQuinney coefcient), respectively. 100% of theplastic work is assumed to be converted into heat.

A graphical summary of the material response at various strainrates is given in Fig. 17. The tensile test results at quasi-static andintermediate strain rates are only presented up to the UltimateTensile Stress (UTS).

3.2. Finite element model

The nite element program ABAQUS/Explicit v6.56 [17] isused for the numerical model of the monolithic plates. Thenumerical model consists of a 1/4 scale model of the clampedplate as shown in Fig. 18(a). Appropriate symmetry boundaryconditions are included on the symmetry edges. A rigid clamp isplaced at the boundary and a clamping force is applied. Theplate is meshed using 3-dimensional continuum 8-node linearbrick elements with reduced integration and hourglass control(C3D8R). The clamps are meshed using 4-node 3-dimensionaldiscrete rigid brick elements (R3D4). Hard contact with separa-tion allowed is dened between the clamp and the plate. Tan-gential behaviour is included with a friction coefcient of 0.5.Simulating an actual clamp as opposed to simply constrainingthe nodes at the boundary allows slight pull-in of the platewhich is more realistic [18]. The scaled impulse density prole isapplied to the exposed area of the plate by converting it toa pressure prole. The pressure is calculated as

Table 3Plate material chemical composition specication (ladle analysis, percent) [19]

Thickness, t (mm) C (max) Mn (max) P (max) S (max) Si (max)

t< 4.5 0.15 1.00 0.035 0.035 0.30t 4.5 0.25 1.60 0.040 0.035 0.50

D. Bonorchis, G.N. Nurick / International Journal of Impact Engineering 36 (2009) 4052 49Fig. 16. Graphs showing the strain hardening, temperature and strain rate response of the phigh strain rate from Gilat et al [13]). (c) High strain rate response including temperature elate material. (a) Quasi-static strain hardening response. (b) Temperature response (atffects.

P I*

tb(4)

where the pressure prole is applied for a duration equal to theburn time, tb, which is calculated as the charge radius divided bythe detonation velocity. The burn time equals 2.44 ms in this case.The pressure load is applied using a Fortran coded VDLOAD user-subroutine in ABAQUS/Explicit v6.56. The extent of the de-formation for a 151 g charge mass is shown in Fig. 18(b) wherethe nal midpoint deection is more than 10 times the platethickness.

validate the pressure loading prole and the scaled appliedimpulse.

3.4. Comparison with published prole

A localised blast load pressure model was developed by Bimha[5]. The model consists of a constant pressure acting over thecharge radius with a decaying pressure prole to the plateboundary. The pressure prole is given as

P P0; r a

P P0ekra; a < r R 5

where P0 central pressure, a charge radius, r position alongradius and R plate outer radius. The decay constant, k, obtainedusing a combined experimental/numerical method is given by

k 130 261aR

948

aR

2; 0:15