DTIC ADA600479: Concurrent Computational and Dimensional Analyses of Design of Vehicle Floor-Plates for Landmine-Blast Survivability

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Concurrent Computational and
Dimensional Analyses of Design of
Vehicle Floor-Plates for Landmine-Blast
Survivability
M. Grujicic & B. A. Cheeseman
Journal of Materials Engineering and
Performance

ISSN 1059-9495
Volume 23
Number 1

J. of Materi Eng and Perform (2014)
23:1-12
DOI 10.1007/s11665-013-0637-5
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Concurrent Computational and Dimensional Analyses of Design of
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Vehicle Floor-Plates for Landmine-Blast Survivability
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Concurrent Computational and Dimensional Analyses
of Design of Vehicle Floor-Plates for Landmine-Blast
Survivability
M.Grujicic andB.A. Cheeseman
(Submitted January 3,2013;published online August9,2013)
Development of military vehicles capable of surviving landmine blast is seldom done using full-scale pro-
totype testing because of the associated prohibitively-high cost, the destructive nature of testing, and the
requirementsformajorlarge-scaleexperimental-testfacilitiesandalargecrewofengineerscommittedto
thetask.Instead,testsofsmall-scalemodelsaregenerallyemployedandthemodel-basedresultsarescaled
up to the full-size vehicle. In these scale-up efforts, various dimensional analyses are used whose estab-
lishment and validation requires major experimental testing efforts and different-scale models. In the
present work, an approach is proposed within which concurrent and interactive applications of the com-
putationalanalyses(oflandminedetonationandtheinteractionofdetonationproductsandsoilejectawith
the vehicle hull-floor) and the corresponding dimensional analysis are utilized. It is argued that this
approachcanguidethedesignofmilitary-vehiclehull-floorswhichprovidetherequiredlevelofprotection
tothevehicleoccupantsunderlandmineblastattackwithoutintroducingunnecessarilyhighweighttothe
vehicle.Tovalidatethisapproach,acombinedEulerian/Lagrangianformulationforlandminedetonation
and the interaction of detonation products and soil ejecta with the vehicle hull-floor (developed in our
previous work) has been utilized along with the experimental results pertaining to small-scale model and
full-scale vehicletesting.
investigations of whole-vehicle performance under various in-
Keywords dimensional analysis, landmine detonation, vehicle
service (Ref 1) and threat (Ref 2-4) conditions. However, the
hull-floordesign
true utility of these computational analyses remains question-
able considering that critical shortcomings of these analyses,
are either not fully identified or understood. For example,
material models used in these analyses need substantial
1. Introduction improvements to take into account the contribution of various
microstructural effects (at various length scales), the effects of
componentsmanufacturingandassemblyaswellastheeffects
Recent efforts of the U. S. Army have been aimed at
of realistic environmental, thermal, mechanical, and dynamic
becoming more mobile, deployable, and sustainable while
loading conditions on material behavior (Ref 5). In addition,
maintaining or surpassing the current levels of lethality and
many issues associated with the interactions and contacts
survivability. Current battlefield vehicles have reached in
between the vehicle and its surrounding (including fluid/liquid
excess of 70 tons due to ever increasing lethality of ballistic
interactions) are still not fully resolved. This situation requires
threatswhichhinderstheirabilitytobereadilytransportedand
thatextensiveexperimentaltestingandvalidationeffortsstillbe
sustained. Therefore, a number of research and development
employed during development of new or retro fitting of the
programs are under way to engineer light-weight, highly
existing military vehicles. In other words, computational
mobile, transportable, and lethal battlefield vehicles with a
analyses are presently capable of reducing somewhat the
target weight under 20 tons. To attain these goals, significant
new-vehicle development time and of lowering the extent of
advances are needed in the areas of light-weight materials and
(but not fully eliminating a need for) the experimental testing
structuresaswellasintheareasofcomputer-aideddesignand
and validation efforts.
experimental testing/validation techniques for these vehicles.
The main issue with the experimental testing/validation
Overthelasttwodecades,theuseofcomputationalanalyses
efforts is that they should ideally involve full-scale prototypes
in the design of military vehicles has continuously increased.
ofthevehicles,whichisquiteexpensiveandrequiresmajortest
Efficient computational codes and fast, massive, and inexpen-
facilities and a large crew of engineers committed to the task.
sive computational resources enable today computational
Ontheother hand,itisquite advantageous to carry outall the
experimental testing and validation using small-scale models
M.Grujicic,DepartmentofMechanicalEngineering,ClemsonUniversity,
and to then scale-up the model results to the full-scale
241 Engineering Innovation Building, Clemson, SC 29634-0921;
prototype, using the appropriate dimensional analysis (defined
and B.A. Cheeseman, Army Research Laboratory—Survivability
Materials Branch, Aberdeen Proving Ground, MD 21005-5069. later). In trying to attain this goal, computational analyses can
Contacte-mail:[email protected]. play a critical role. That is, the computational analyses can be
Journal of Materials Engineeringand Performance Volume23(1) January 2014—1
Author's personal copy
firstvalidatedexperimentallyusingasinglesmall-scalemodel. cause a sequence of failures throughout the entire vehicle
Then,thecomputationalanalysescanbeemployedtoestablish through components/sub-system connections and interfaces
the basic fundamental parameters and their inter-relations and, (e.g., via fuel lines running along the floor, floor-bolted seats,
in turn to develop the appropriate dimensional analysis. This ammunition storage racks, power-train lines, etc.). Tradition-
could potentially lead to significant reductions in the small ally, the floor-rupture problem is solved through the use of
scale model testing efforts (e.g., different-scale models would thicker floor-plates, stronger, tougher, and usually heavier
not have to be employed to establish/validate the dimensional material and through the utilization of applique armor. This
analysis). Lastly, both the computational and experimental approach is generally associated with unnecessary oversized/
modelscale-uppredictionscanbeused,withmoreconfidence, overweight vehicle-hull floors and, is being abandoned now-
in the design/development of full-scale prototypes. adaysduetoaforementionedmilitary(cid:2)srequirementsforlighter
A review of the public-domain literature carried out as part vehicles. Combined application of the computational analyses
of the present work revealed that the landmine-detonation ofbothsmall-scalemodelsandfull-scaleprototypesofvehicles
related research activities can be broadly divided into three (or their hulls) subjected to landmine blast and a dimensional
maincategories:(a)shockandblastwavemechanics/dynamics analysis for model-results scale-up, proposed in the present
including landmine-detonation phenomena and large-deforma- work, is a way to reduce the weight of the vehicles while
tion/high-deformationrateconstitutivemodelsfortheattendant ensuring the required confidence level for its survivability and
materials (high explosive, air, soil, etc.) (Ref 6); (b) the survivability of its occupants.
kinematic and structural response of the target to blast loading Theorganizationofthepaperisasfollows:Insection 2.1,a
including the role of target design and use of blast attenuation brief overview is provided of the Buckingham P theorem
materials(Ref7);and(c)vulnerabilityofhumanbeingstopost- (Ref 9). The dimensional analysis for military-vehicle floor-
detonation phenomena such as high blast pressures, spall plates subjected to landmine-detonation attack originally pro-
fragments, and large vertical and lateral accelerations (Ref 2). posed by Wenzel and Hennessey (Ref 8) is overviewed in
The present work falls into the categories (a) and (b) of the section 2.2. The combined Eulerian/Lagrangian computational
research listed above since it emphasizes both the role of procedure used in the present work is described in section 2.
material behavior under transient-dynamic loading conditions Themainresultsobtainedinthepresentworkarepresentedand
as well as the kinematic and structural responses of the target discussedinsection 3,whilethekeyconclusionsresultedfrom
structure to detonation of landmines shallow buried in soil. the present study are summarized in section 4.
The main objective of the present work is to extend the
scaling analysis of Wenzel and Hennessey (Ref 8) for military
vehicle-hullfloor-platessubjectedtoimpulseandhigh-pressure 2. Computational Procedure
loads associated with detonation of a landmine buried in soil
under the hull body. The dimensional analysis of Wenzel and
2.1 Buckingham P Theorem (Ref 9)
Hennessey (Ref 8) is based on the so-called Buckingham
Theorem(Ref9),aprocedurecommonlyusedinthesituations As discussed earlier, the Buckingham P theorem (also
inwhichfull-scalestructures/prototypesaredesignedandsized known as the ‘‘similitude theory’’) (Ref 9) is employed in the
basedontheresultsobtainedusingsmall-scalemodeltests.To presentworktoaddresstheproblemofcomputational-analysis
extend the dimensional analysis of Wenzel and Hennessey validation via small-scale model testing and the problem of
(Ref 8), the combined Eulerian/Lagrangian computational scale-up ofthe model results to the full-scale prototype. In the
model and analysis of detonation of landmines shallow-buried remainder of this section, a brief overview is provided of the
in soil and of the interactions between detonation products similitude theory and of its utility and limitations.
and soil ejecta with the target structure developed by Grujicic The similitude theory (also known as the ‘‘dimensional
et al. (Ref 10) is used. analysis’’) is essentially a technique/procedure for describing
Themainroleofthemilitary-vehiclehull-flooristoprotect thebehaviorofasystemintermsofanumberofdimensionless
the vehicle occupants in the case of detonation of a landmine parameters (generally referred to as the P terms). This
buried in soil underneath the vehicle belly. Complex interre- procedure typically involves the following main steps:
lationships generally exist between the type and the extent of
vehicle-occupants injuries and the extent of vehicle damage (a) Identification of all physical parameters which govern
resulting from under-the-vehicle landmine blast. The most the behavior of the system under investigation. This is
serious vehicle-occupants injuries and their fatalities are most the most critical step in the dimensional analysis since it
frequently incurred as a consequence of hull-floor rupture. In requires a good understanding of the underlying physics
addition to generating floor-material fragmentation, floor rup- of the problem at hand (even in the case when the func-
ture enables for the ingress of flame and toxic (detonation- tional forms of the governing equations are not known).
product) gases and can lead to on-board fires and explosions. (b) Specification of the units for the physical parameters
Whilethevehicleunderlandmineattackisalwayssubjectedto defined in (a) in terms of the fundamental physical
high‘‘G’’)forces,thisaspectofthethreatisusuallyconsidered dimensions (typically force, length, and time).
assecondarywhenfloorrupturetakesplace.Ontheotherhand, (c) Identification of a subset of parameters among the physi-
when landmine-detonation results in only hull-floor bulging, cal parameters defined in (a) with the number of param-
the vehicle is then propelled upward and off the ground, to an eters in this subset being equal to the number of
extent which scales directly with size of the landmine and fundamental physical dimensions defined in (b). The
inversely with vehicle weight. The accompanying dynamic parameters in this subset will be used for converting the
shock can cause vehicle occupants and on-board instruments remaining parameters in (a) into a set of dimensionless
andweaponstobethrownaboutwithinthevehicleinterior.In parameters (i.e., the P terms). It should be noted that
addition, large hull-floor deflections and deformations can while the number of the resulting P terms is fully
2—Volume23(1) January 2014 Journal of MaterialsEngineering andPerformance
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defined (equal to the number of physical parameters in identified eleven key physical parameters which govern the
(a) minus the number of fundamental dimensions in (b)), responseoftheplatesubjectedtolandmine-detonationloading.
the functional form of the P terms is not unique. It is Theseparametersarelistedandexplainedandtheirfundamen-
generally advantageous to define the P terms in such a tal dimensions are identified in Table 1.
way that they have a clear physical meaning. However,
regardless of the choice of the P terms, the final impli- P Terms. Next, three of the physical parameters listed in
cations/findings offered by the similitude theory is inde- Table 1 (floor-plate thickness, h, floor-plate material yield
pendent of this choice (provided, all the governing strength, r, and the same material density, q) are used for
physical parameters are correctly identified in (a)). constructing eight P terms (Table 2).
(d) Identification of the functional relationships between It should be noted that the rigid body motion of the floor-
each of the P terms and all other P terms. These rela- plate was neglected due to the fact that it did not modify
tions are generally determined using experimental or landmine-detonation-induced loading and, hence, did not
computational approaches or the combination of the two modify the structural response of the floor-plate. The distrib-
approaches. uted inertial effects, on the other hand, were included through
the selection of the material density and the floor-plate
The power of the similitude theory is that the functional thickness.Itshouldbealsonotedthatnoexplicitconsideration
relationsofferedaregeneralized,i.e.,theeffectofgeometrical, oftheeffectofchangeinthestrainrateswasconsideredwitha
kinematic,ambient,loading,andresponsescalesiseliminated. justification that the strain rates do not appreciably change
Consequently, the similitude theory states that, if all the between the full-scale prototype and its replica-model. How-
pertinent physical parameters were correctly identified in (a) ever,itshouldberecognizedthattheeffectofhighstrainrates
and if all the P terms were kept invariant (i.e. unchanged) was included implicitly, via the use of the dynamic material
between a small-scale model and its full-scale prototype, then yield strength, r.
theresultsobtainedusingasmall-scalemodelcouldbedirectly
used to predict the response of the corresponding full-scale
counterpart.
2.2 Brief Overview of the Wenzel and Hennessey Work
(Ref 8)
As mentioned earlier, one of the main objectives of the
present work is to further advance the dimensional analysis
presented by Wenzel and Hennessey (Ref 8). In the remainder
of the section, a brief overview is provided of this work.
The work of Wenzel and Hennessey (Ref 8) deals with
scalingfloor-platesformilitaryvehiclesexpectedtobetargeted
by landmine blast. The main objective of the work was to
develop and validate a dimensional analysis for the vehicle
Fig.1 Aschematic of theprototype landmine-blasttest set-upused
floor-plate sizing problem, so that the results of small-scale
intheworkofWenzelandHennessey(Ref8)
replica-model can be used to determine the minimal required
floor-plate thickness which will ensure that, under the specific
landmine-detonation scenario, the vehicle floor will not under-
goruptureorextensivedynamicdeflection.Theterm‘‘replica-
Table 1 Key physical parameters controlling structural
model’’ will be defined in greater detail later in this section.
response of the vehicle-floor plates
Untilthen,thistermwillbeusedtodenoteasmall-scalemodel
whichissimilar(geometrically,constitutively,andwithrespect Fundamental
to extent of properly scaled loads and structural responses) to Symbol Parameter dimensions
the corresponding full-scale prototype.
The starting point in the work of Wenzel and Hennessey P Pressureduetolandmine F/L2
(Ref 8) was to take advantage of the fact that floor-plate detonation
deflections do not significantly modify the (landmine-detona- I Specificimpulseduetolandmine FT/L2
detonation
tion-induced)dynamicloading.Towardthatend,theproblemat
h Floor-platethickness L
hand was divided (decoupled) in two separate problems: (a) a
h Arrayofother(non-dimensional) …
floor-plate structural response problem and (b) the problem of i
floor-platedimensions
analyzingpressuresandimpulsesresultingfromdetonationofa r Floor-platematerialyieldstrength F/L2
landmine (shallow buried in soil underneath the vehicle-hull r Arrayofother(non-dimensional) …
i
bottom). strength parameters for floor-plate
2.2.1 Structural Response. Physical Parameters. By material
carrying out a careful and detailed engineering analysis of the e Floor-platematerialequivalent …
structural response of a thick metal plate (a model for the strain
vehicle-hullbottom)supportedonwoodenblocks(toobtainthe q Floor-platematerialdensity FT2/L4
d Floor-platedeflection L
required level of ground clearance), and clamped along its
a Floor-plateacceleration L/T2
edges (to simulate the effect of the surrounding/supported
t Postlandmine-detonationtime T
vehicle-framestructure),Fig. 1,WenzelandHennessey(Ref8)
Journal of Materials Engineering andPerformance Volume23(1) January 2014—3
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Table 2 P terms controlling structural response of the Table 4 Parametersfordefiningloadingfromalandmine
vehicle-floor plates
Fundamental
Similarity Symbol Parameter dimensions
Pterm Physicalmeaning type
P Pressureduetolandminedetona- F/L2
(cid:3) =h Arrayofnon-thickness Geometrical tion
1 i
(non-dimensional) I Specificimpulseduetolandmine FT/L2
floor-platedimensions detonation
(cid:3) =r Arrayofnon-yield-strength Constitutive E Landminedetonation-releaseden- FL
2 i
(non-dimensional) ergy
parametersforfloor-platematerial r Circular-diskshapelandmine …
(cid:3) =e Floor-platematerialequivalent Constitutive, thickness
3
strain response r Arrayofnon-thickness(non- …
i
(cid:3) =d/h Thickness-scaledfloor-plate Response dimensional)landminedimensions
4
deflection d Landminedepth-of-burial L
(cid:3) =aqh/r Non-dimensionalfloor-plate Response R Landmine/floor-platestandoffdis- L
5
acceleration tance
(cid:3) =(tr1/2)/ Non-dimensionaltime Response P Ambientatmospheric-pressure F/L2
6 0
(hq1/2) a Speedofsoundinair L/T
0
(cid:3) =I/ Non-dimensionalimpulse Loading c Ratioofspecificheatsforair …
7 0
(r1/2q1/2h) duetolandmine q Densityofsoil FT2/L4
detonation c Seismicvelocityofsoil L/T
(cid:3) =P/r Non-dimensionalpressure Loading g Gravitationalacceleration L/T2
8
duetolandmine
detonation
Table 3 Replica-to-prototypescalingforstructuralresponse
2.2.2 Landmine-Detonation-InducedImpulsesandPres-
ofthevehicle-floorplates
sures. Wenzel and Hennessey (Ref 8) also carried a detailed
engineering analysis of detonation of landmine shallow-buried
Symbol Parameter Scalefactor
in soil. In order to quantify the resulting time-dependant
h Arrayofnon-thickness k impulses and pressures (input loading parameters in the
i
(non-dimensional)floor-plate structural response analysis), they included the effect of the
dimensions landmine shape and size, type of explosive, soil properties as
q Floor-platematerialdensity 1.0 well as the properties of the surrounding (ambient) air. These
r Floor-platematerialyieldstrength 1.0 parameters along with their fundamental dimensions are listed
e Floor-platematerialequivalent 1.0 in Table 4.
strain
d Floor-platedeflection k
Landmine Related Parameters. The landmine was considered
a Floor-plateacceleration 1/k
to be of a circular-disk shape and to have a size defined by
t Postlandmine-detonationtime k
I Specificimpulseduetolandmine k thickness, r, and an array of non-dimensional parameters, ri
detonation (i = 1, in the present case and r1 represents thickness-formal-
P Pressureduetolandminedetona- 1.0 izedlandminediameter).Thetypeofexplosiveinthelandmine
tion (of a givensize) isthen expressed by thetotal energyreleased
during landmine detonation, E. The location of the landmine
Scaling. As mentioned earlier, in accordance with the simili- with respect to the model/prototype is defined in terms of the
tudetheory,fortheresultsobtainedusingsmall-scalemodelto depth-of-burial, d (a distance between the top surface of the
be applicable at the prototype scale, the P terms must be landmine and the soil/air interface) and standoff distance,
unchanged between the model and the prototype. In order to R (distance between the top surface of the landmine and the
ensure this condition, the model and the prototype must be bottom of the test plate).
similar geometrically, (i.e., P is unchanged), constitutively
1
(i.e.,P andP termsareunchanged),inresponse(i.e.,P -P Ambient-AirParameters. Toaccountfortheeffectsofairblast,
2 3 3 6
terms are unchanged) and with respect to loading (P and P three ambient-air parameters are considered: ambient pressure,
7 8
terms are unchanged). Hence, if the model is obtained by P , sound speed in air, a , and a ratio of the constant-pressure
o 0
scalingdowntheprototypebyafactork(whilemaintainingall and constant-volume specific heats in air, c .
o
theanglesunchanged),themodelandtheprototypearemadeof
the same material (i.e., r and q are unchanged), specific Soil Parameters. To account for the effect of soil (into which
i
impulse due to landmine detonation, I, is scaled by a factor k the landmine is buried) Wenzel and Hennessey (Ref 8),
while pressure due to landmine detonation, P, is kept un- assumed that detonation of shallow-buried landmine can be
changed, then e, d, a, and t responses of the model are scaled relatedtotheproblemofundergroundlandmineexplosionand
withrespecttotheirprototypecounterpartsinaccordancewith theresultingcraterformation.Consequently,theyassumedthat
the values given in Column 3 of Table 3. When the model is two soil-specific parameters should be included: the (initial)
scaled down in accordance with these conditions, then the soildensity,q,anditsseismicvelocity,c.Inaddition,toinclude
model is referred to as the replica-model. the effect of gravitational forces of the soil surrounding the
4—Volume23(1) January 2014 Journal of MaterialsEngineering andPerformance
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Table 5 P terms for determining loading from landmine betweenthesmall-scalemodelandthefull-scaleprototype.The
question then arises as to how to scale the landmine/air/soil-
Similarity related physical parameters listed in Table 4 to achieve this
Pterm Physicalmeaning type condition. In addition, the scaling for the pressure and for the
impulse has already been fixed by the structural-response
(cid:3) =r Arrayofnon-thickness Geometrical
1 i analysis,Table 3andthisscalingcannotbechangedwithinthe
(non-dimensional)
present landmine-detonation-analysis.
landminedimensions
If r, d and, R are scaled by k then P and P terms are
(cid:3) =d/R Scaleddepth-of-burial Geometrical 1 3
2 invariant between the model and the prototype. If the same
(cid:3) =r/R Inverseofscaledstandoffdistance Geometrical
3
(cid:3) =c Ratioofspecificheatsforair Atmospheric ambient air and soil is used in both the model and the
4 0
(cid:3)5=a0/c Airsoundspeedtosoilseismic Kinematic prototype,thenP4andP5areinvariant.Also,theinvarianceof
speedratio the P and P terms requires for E to be scaled as k3. This is
6 7
(cid:3) =P R3/E Sach(cid:2)sNumber Ambient reasonable since all the spatial dimensions are scaled in k and
6 0
(cid:3)7=qc2R3/E ModifiedSach(cid:2)sNumber Soil E depends on the landmine volume.
(cid:3)(cid:3)8==gPRR/3c/2E NFroonu-ddeimNeunmsiboenralpressuredueto LGoraavdiitnagtional Since E scales with k3 and R scales with k, then from P9
9 and P it is seen that P remains fixed while I is scaled as k
landmine 10
betweenthemodelandtheprototype.Asmentionedearlier,this
detonation
is exactly the scaling for specific impulse and pressure which
(cid:3) =IcR2/E Non-dimensionalimpulsedueto Loading
10 was required by the structural-response analysis.
landmine
detonation The invariance of the P8 term, requires that gravitational
accelerationbescaledas1/k.Inotherwords,forexample,fora
one-quarter model, model experiments would have to be done
under conditions corresponding to a four times higher gravi-
Table 6 Replica-to-prototypescalingforblast-loadrelated
tationalaccelerationthantheonenormallypresentattheearth.
parameters
Wenzel and Hennessey (Ref 8) recognized that conducting
Scale small-scale experiments under hyper-gravitational conditions
Symbol Parameter factor would be quite challenging. Consequently, they invoked the
studyofWestine(Ref11)whichshowedthatfailingtoaccount
r,R,d Circular-diskshapelandmine k for the gravitational effects during underground landmine
thickness, explosionformodelinarangebetween1/2and1/8,resultedin
landmine/floor-platestandoffdistance,
relatively small changes in the resulting crater size. Hence,
landminedepth-of-burial
Wenzel and Hennessey (Ref 8) argued that even though the
c,a Seismicvelocityofsoil,speedof 1.0
0 invarianceoftheFroudeNumber(P termwhichisdefinedas
soundinair 8
a ratio of the gravitational and inertial effects) cannot be
c Ratioofspecificheatsforair 1.0
0
q Densityofsoil 1.0 generallysatisfied,thesmall-scalemodelresultscouldbeused,
E Landmine-detonation-released k3 withsignificant confidence,forfull-scale prototypedesignand
energy sizing.
P0,P Ambientatmospheric-pressure, 1.0 2.2.3 Validation of the Dimensional Analysis. To vali-
pressureduetolandminedetonation date their dimensional analysis presented above, Wenzel and
I Specificimpulseduetolandmine k Hennessey (Ref 8) carried out a series of one-half and one-
detonation
quartermodelexperimentsoffloor-plateresponsetolandmine-
detonation loads, and compared the results with available full-
scaleprototypeexperimentaldata.Theyfoundthattheone-half
andone-quartermodelresults(presentedindetailsinsection 3)
landmine, gravitational acceleration, g, is also defined as a
are fully mutually consistent which constituted the proof for
physical parameter affecting the temporal evolution of land-
validity of the similitude theory they developed. On the other
mine-detonation-induced impulses and pressure.
hand, the full-scale results could not be used for dimensional-
analyses validation, due to the lack of knowledge of the
Resulting Landmine-Detonation Loading Parameters. The attendant soil parameters. Nevertheless, the full-scale results
discussion presented above introduced 11 landmine, air and revealed that soil parameter play important role in the
soil-related parameters. The remaining two parameters are landmine-detonation-induced loading.
associated with the landmine-detonation-induced specific
impulse, I, and the associated (peak) pressure, P.
2.3 Computational Problem and Solution Method
P Terms. Three of the parameters listed in Table 4 (i.e., soil Thebasicformulationofthecomputationalproblemdealing
density, q, soil seismic velocity, c, and the total detonation- with interactions between the detonation products, shell frag-
released energy, E) are used for non-dimensionalization of the ments, and soil ejecta (all resulting from the explosion of a
remaining parameters. The resulting 10 P terms are listed in shallow-buried landmine) and the target model/prototype hull-
Table 5. bottomplateispresentedinthissection.Allthecalculationswere
carriedoutusingANSYS/Autodyn(Ref12),ageneralpurpose
Scaling. Asmentionedearlier,inorderfortheresultsobtained transientnonlineardynamicsprogram.Thecomputationalmod-
usingasmall-scalemodeltobeapplicabletothecorresponding elingofthisinteractioninvolvedtwodistinctsteps:(a)geomet-
full-scaleprototype,allPtermsmustbemaintainedunchanged ricalmodelingofthemodel/prototypealongwiththeadjoining
Journal of Materials Engineering andPerformance Volume23(1) January 2014—5
Author's personal copy
mine, air, and soil regions, and (b) the associated transient
nonlineardynamicsanalysisoftheimpulseloading(momentum
transfer)fromthedetonationproducts,shellfragments,andsoil
ejectatothetarget.Thepart(b)ofthisanalysiswasperformed
usingamodifiedversionofthetechniquedevelopedbyFairlie
andBergeron(Ref13)whichcouplesamulti-materialEulerian
meshtotwoLagrangianmeshes.TheEulerianmeshcontained
initiallyaComposition-Bsolid-circular-diskshapeburiedland-
mine (and, after explosion, the resulting high-pressure, high-
internal energy-density gaseous detonation products) and the
(initiallystationary,atmospheric-pressure)air(locatedabovethe
soil).TheEulerianmeshwasconstructedintermsofeightnode
cuboidalcells.OneoftheLagrangianmesheswasusedtomodel
thesoil,whiletheothertorepresentthefloor-plate(alongwiththe
adjoininghull-supportedframestructure).Thesoilandthetarget
werebothdiscretizedusingeight-nodesolidelements(Table 6).
An advantage was taken of the inherent symmetry of the
model.Inotherwords,twomutually-orthogonalverticalplanes
of symmetry were placed along the axis of the target plate as
well as along the axis of the air, landmine, and soil regions
whichenabledonlyaquarterofthecomputationalmodeltobe
analyzed.Representativequartersymmetricmodelsforvarious
computational domainsused inthe present study are shownin
Fig. 2.ItshouldbenotedthatthelowerportionoftheEulerian
domain contains the landmine while the rest of the lower
portion of the Eulerian domain is occupied by the Lagrangian
soil domain. Likewise, the upper portion of the Eulerian
domainwhichextendsabovethesoilcontainsinitiallyairandis
partially occupied by the Lagrangian floor-plate domain.
Air and Composition-B detonation-products are treated as
fluidsandmodeledusingonlytheappropriateequationofstate,
EOS [i.e., the gamma EOS (Ref 12) for air and the JWL EOS
(Ref 12) for Composition-B]. The floor-plate was treated as
being made of Rolled Homogenized Armor (RHA) and
modeled using a linear equation of state, the Johnson Cook
strength and failure models and an instantaneous total equiv-
alenterosionstrainalgorithm.Detailsofthesemodels/relations
and their parameterizations can be found in Ref 10. Soil was
represented using our CU-ARL sand material model
Fig.2 Computational sub-domains used in the present analysis of
(Ref 14-20). The Lagrangian soil elements were allowed to
structural response of vehicle-hull floor-plate subjected to landmine-
erode at an incremental geometric strain of 2.0. Upon the
blast dynamic loading: (a) Euler sub-domain and (b) Lagrange sub-
erosion of an element, the resulting free nodes are allowed to domain
retain their mass/velocity (and, thus, momentum) and to
continue to interact with the soil and the target plate. Several gage points were defined within the landmine, soil,
Interactions between the gaseous fluids (i.e., air and air, and floor-plate which allowed monitoring of the quantities
detonation products) with the target plate were accounted for such as pressure, velocity, and (in the case of the Lagrange
through the use of the appropriate Euler/Lagrange coupling domains) the vertical displacements.
option within ANSYS/Autodyn (Ref 12). Likewise, the soil/ At the beginning of the simulation, all the Lagrange and
target-plate interactions were modeled through the use of the Eulerdomainswereactivatedandthelandminedetonated.The
appropriate Lagrange/Lagrange coupling option. (circular-disk shape) landmine was detonated over its entire
The‘‘flowout’’boundaryconditionswereappliedtoallthe bottom face at the beginning of the simulation. A standard
freefaces(thefaceswhichdonotrepresentinterfacesbetween meshsensitivityanalysiswascarriedout(theresultsnotshown
the different domains) of the Euler domain except for the face for brevity) to ensure that the results obtained are not very
associated with the vertical symmetry planes. To reduce the sensitive to the size of the cells/elements used.
effect of reflection of the shock waves at the outer surfaces of
the Lagrange soil domain, ‘‘transmit’’ boundary conditions
were appliedtoallthefreefaces ofthisdomain except forthe
3. Results and Discussion
faces associated with the vertical symmetry planes and the
upper face which defines the soil/air interface. The transmit
boundary conditions enable propagation of the pressure waves To comply with the experimental procedure of Wenzel and
across the boundaries without reflection mimicking wave Hennessey(Ref8),allthecomputationalanalysescarriedoutin
propagation in an infinitely-large soil domain (Ref 12). the present work involved a single (full-scale) size of the
6—Volume23(1) January 2014 Journal of MaterialsEngineering andPerformance
Author's personal copy
vehicle-hull floor-plate with L9W9H dimensions of 2.13 m (Ref 8) is provided in the subsequent sections. It should be
by 1.52 m by 0.0254 m. Following the present dimensional recalledthattwosetsofPtermswereintroducedinthepresent
analysis,thesedimensionswerereducedbyafactorof2and4 dimensional analysis. One set associated with structural
in the case of one-half and one-quarter models, respectively. response of the floor-plate while the other corresponding to
The initial soil density and its seismic velocity (sound speed) various aspects of detonation of a landmine buried in soil. To
were selected to match their experimental counterparts in the enable clear differentiation between the two sets of P terms
work of Wenzel and Hennessey (Ref 8). Also, the same superscriptsSRandBLareaddedtotherespectivePtermsto
explosive-type(CompositionB)andfloor-platematerial(Rolled denote their structural-response/blast-loading nature.
Homogenized Armor, RHA) were used to match those used in
Ref 8.Initially,twovaluesofthestandoff-distance normalized
3.1 Floor-Plate Deflection Versus Explosive-Charge Energy
ground clearance (G.C./R) of 0.8889 and 0.8235 (correspond-
ing to ground-clearance values of 14¢¢ = 0.3556 m and In this section, the experimental results from Ref 8 and the
24¢¢ = 0.6096 m, respectively and a fixed/single value of the present computational results pertaining to the dependence of
depth-of-burialof3¢¢ = 0.0762 m)wereselected,inagreement thickness-scaled floor-plate average displacement (PSR, in the
4
withthoseusedbyWenzelandHennessey(Ref8).Inorderto structural-responseanalysis)onthescaledchargeenergy(PBL,
7
obtain further insight into the dimensional analysis of land- of the detonation-induced impulse/pressure analysis) are pre-
mine-blastinduceddeformationofthevehicle-hull-floor-plates, sented and discussed.
additional ground-clearance values were also used in the 3.1.1 ExaminationofthePBLTermasDefinedbyWenzel
7
present computational analyses. The explosive-charge sizes and Hennessey (Ref 8). In Fig. 4(a), the PSR versus PBL
4 7
used (at the full scale) at different ground-clearance values are results from the work of Wenzel and Hennessey (Ref 8) are
listed in Table 7. presented for the two aforementioned levels of normalized
Anexampleofthepressurefieldinairduringdetonationof ground clearance. In addition to the measured data points, the
a landmine buried in soil underneath a vehicle-hull floor-plate corresponding power-law best-fit curves are also shown. The
is shown in Fig. 3. A more detail representation and the corresponding computational counterpart results obtained in the
discussionoftheresultsobtainedinthepresentworkandtheir presentworkaredisplayedinFig. 4(b).Furthermore,toassistan
experimental counterparts obtained by Wenzel and Hennessey experiment-with-calculationcomparison,thebest-fitcurvesfrom
Fig. 4(a) are reproduced in Fig.4(b). A simple examination/
comparison of the results displayed in Fig.4(a) and (b)
Table 7 Testmatrixpertainingtotheselectionofexplosive-
reveals that
chargesizes,depth-of-burialandfloor-platedimensionsat
differentfull-scaleground-clearancelevels
(a) At each level of the ground clearance, the corresponding
Full-scalegroundclearance,m full-scale, half-scale, and quarter-scale model computa-
tional results are practically identical. This finding sug-
Parameter 0.356 0.483 0.609 gests that the choice of the two P terms in Fig. 4(b),
was appropriate/justified. Similar conclusion can be
Composition-Bexplosive 1.81 4.54
drawn based on the experimental results of Wenzel and
charge,kg 3.18 9.07
Hennessey (Ref 8), Fig. 4(a). Nevertheless, it should be
4.54 18.14
noted that all the materials (Composition-B, soil, air and
9.07 …
RHA) involved in the present dimensional analysis were
13.61 …
Depth-of-burial,m 0.076 kept unchanged. Also, at a given level of scaled ground
Standoffdistance,m 0.432 0.559 0.686 clearance, P7BL value in Fig. 4(a) and (b) was changed
Floor-plate,L9W9H 2.13491.52490.025 by merely changing the value of explosive-charge
dimensions,m energy, E (via changing the charge thickness, r). Conse-
quently, it is not clear at this point if the PSR versus
4
PBL relations suggested in Fig. 4(a) and (b) will be
7
valid if properties of the attendant materials and explo-
sive-charge shape are changed. This point will be
addressedingreater detail in the remainder of this section.
(b) At each scaled ground-clearance level, the corresponding
best-fit curves for the experimental and computational
data differ by less than 10%. This finding suggests that
there is a reasonably good agreement between the exper-
imental test transient blast-loading conditions used and
the floor-plate response observed in the work of Wenzel
and Hennessey (Ref 8) and the present transient nonlin-
ear dynamic computational modeling and analysis proce-
dure of the same problem. This finding, and the fact
that ground-clearance level is seen to have a significant
effect on the PSR versus PBL relationship, suggested
4 7
the need for additional computational analyses at other
values of ground clearance. Since the two aforemen-
Fig.3 An example of the air-pressure field during detonation of a tioned ground-clearance values of 14 and 24 inches
mineshallow-buriedinsoilunderafloor-platetarget define a fairly realistic range of this quantity in the case
Journal of Materials Engineering andPerformance Volume23(1) January 2014—7