models for projectile impact into hybrid multi-layer armor systems with axisymmetric or biaxial ...

233 Pages·2011·13.21 MB·English

MODELS FOR PROJECTILE IMPACT INTO
HYBRID MULTI-LAYER ARMOR SYSTEMS WITH
AXISYMMETRIC OR BIAXIAL LAYERS WITH
GAPS
ADissertation
PresentedtotheFacultyoftheGraduateSchool
ofCornellUniversity
inPartialFulfillmentoftheRequirementsfortheDegreeof
DoctorofPhilosophy
by
WeiLi
May2011
⃝c 2011WeiLi
ALLRIGHTSRESERVED
MODELSFORPROJECTILEIMPACTINTOHYBRIDMULTI-LAYERARMOR
SYSTEMSWITHAXISYMMETRICORBIAXIALLAYERSWITHGAPS
WeiLi,Ph.D.
CornellUniversity2011
Analytical and numerical modeling of fibrous material resistance to penetra-
tion under impact by high-velocity projectiles has been of great interest not
only for personnel protection reasons but also because trial-and-error testing is
costly and time-consuming. In this thesis, two PC-based models are developed
for projectile impact into a multi-layer system of membrane layers with non-
zero spacings between them. The projectile is a standard right circular cylinder
(RCC) often used in laboratory experiments to compare material systems, and
the models blend theoretical analysis and numerical simulation to characterize
theinteractionbetweentheprojectileandthevariouslayers.
We first consider a system of axisymmetric layers under impact by an RCC
projectile. In particular, we consider such performance measures as the critical
strains in layers resulting in their failure, the strains in unfailed layers, criti-
cal layer gaps, the number of layers penetrated, and the residual velocities in
cases where all layers have been penetrated. The model allows variation of
mechanical properties from layer to layer as well as variations in spacings be-
tween layers, in order to study their combined effects on the ballistic perfor-
mance of the system. Case studies are performed on the ballistic impact re-
sponse of fibrous material systems of particular interest in body armor. These
areultra-highmolecularweightpolyethylene(UHMWPE)fiberssuchasDSM’s
DyneemaSK76,aswellasaramidfiberssuchasduPont’sKevlar-29.
We also develop a semi-analytical model for a multi-layered biaxial, elas-
tic membrane system impacted by an RCC projectile. The model builds on a
single-layer membrane model, which has been under development by collabo-
rators in the overall body armor work at Cornell University. Key assumptions
and parameter values in the single layer model were guided by simulation re-
sults using a code based on the finite difference method (FDM) incorporating
an algorithm frequently used in molecular dynamics simulations. The code
was originally developed by researchers at DSM (makers of Dyneema) and has
been modified by the author and several collaborators at Cornell University to
suppress local strain concentrations and dynamic artifacts resulting from the
descretizationofthestructure,andtobetterhandlethecurrentgeometry.
Numerical simulations of impact into a flexible panel are performed where
the main emphasis is on a comprehensive understanding of the strain and dis-
placement fields, as well as on the velocity fields versus time. The panel is
treated as a single biaxial membrane with negligible shear stiffness compared
with the tensile stiffness, and is assigned the properties of Dyneema SK76 or
Kevlar 29 biaxial fabrics and flexible composites having about 15 to 20 percent
matrixcontent. Numericalresultsareobtainedthroughincrementalintegration
ofdifferentialequationsusingsmalltimesteps. Comparedtosimulationsusing
themodifiedDSMcode,thereareseveralimportantimprovements: (1)Thecal-
culation time has been accelerated by at least a factor of 1000; (2) Results under
differentparametercombinationscanbeobtainedforlargergeometricsizesand
much longer times; and (3) Modeling multi-layer systems is now possible, and
wepresentresultsforseveralcases.
BIOGRAPHICALSKETCH
The author was born in Dalian, China in June, 1980. After completing his high
school education in Dalian No. 24 High School in 1999, the author was admit-
tedintoDepartmentofMechanicalEngineeringinTsinghuaUniversity,Beijing,
wherehereceivedhisBachelor’sdegreeinJuly,2003andhisMaster’sdegreein
July, 2006. The author was admitted into the Sibley School of Mechanical and
Aerospace Engineering, Cornell University in August, 2006, to pursue his PhD
degree.
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ThisthesisisdedicatedtomyfatherLi,YongqingandmymotherWang,Liyan
fortheirconsistentsupportandencouragementonmyacademicpursuits
duringschoolyears.
iv
ACKNOWLEDGEMENTS
I would like to express my appreciation to my thesis advisor, Professor Stu-
art Leigh Phoenix, for his patient instructions and guidance during my Ph.D.
study. I would also like to thank Professors Subrata Mukherjee and Gennady
Samorodnitsky for serving on my special committee and for their encourage-
ment and advice during my studies at Cornell. Dr. Abdulkadir Yavuz has also
contributed much to the models developed in this thesis, for which I am eter-
nallygrateful.
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TABLEOFCONTENTS
BiographicalSketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
TableofContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
ListofTables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
ListofFigures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1 Introduction 1
2 Single-andmulti-layermodelsforisotropicmaterials 12
2.1 Single-layermodel . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.2 Unitsolution . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.3 Conewavebehavior . . . . . . . . . . . . . . . . . . . . . . 16
2.1.4 Velocityprofilesforvaryingstrainattheprojectileedge . 22
2.1.5 Applicationofconvolutiontotheunitsolutions . . . . . . 24
2.1.6 Deceleration of the projectile due to membrane reaction
forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Multi-layermodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Resultsanddiscussion . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.1 Keyparameters . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.2 Singlelayercase . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.3 Multi-layercases . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 Single-andmulti-layermodelsforbiaxialmaterials 53
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Introductiontosingle-layermodel . . . . . . . . . . . . . . . . . . 54
3.2.1 Reviewofprojectileimpactinto1Dtape . . . . . . . . . . . 55
3.2.2 Difficulties in extending results from 1D tape impact to
impactintoabiaxialmembrane . . . . . . . . . . . . . . . . 57
3.2.3 In-plane tension wave and out-of-plane cone wave pro-
filesofdeformation . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.4 Verticalforcesactingtodeceleratetheprojectile . . . . . . 64
3.2.5 Angles at the cone wave front and ratio of strains at the
projectileandtheconewavefront . . . . . . . . . . . . . . 72
3.2.6 Calculatingthekeyintegraldirectlyusinganaccurateex-
pansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.2.7 Strain decay along tension wave arms by wave induced
dragofcrossingyarns . . . . . . . . . . . . . . . . . . . . . 88
3.2.8 Application of Duhamel convolution to unit solutions to
obtaingeneralsolutions . . . . . . . . . . . . . . . . . . . . 91
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3.2.9 Yarnlengthcompatibilityalongtheyarnaxisinthecone . 92
3.2.10 Fundamental formula for strain at the cone wave-front
andotherkeyvariables . . . . . . . . . . . . . . . . . . . . 94
3.2.11 Deceleration of the projectile due to membrane reaction
forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.2.12 Average and peak strains around the projectile edge and
attheconewave-frontnose . . . . . . . . . . . . . . . . . . 101
3.2.13 Out-of-planevelocityprofileintheconewave . . . . . . . 108
3.2.14 Explicit formulas for initial strains, velocities and angles
upontheimpactofprojectile . . . . . . . . . . . . . . . . . 110
3.3 Multi-layermodelforbiaxialmaterials . . . . . . . . . . . . . . . . 112
3.4 Resultsanddiscussions . . . . . . . . . . . . . . . . . . . . . . . . 114
3.4.1 Verificationandcomparisons . . . . . . . . . . . . . . . . . 114
3.4.2 Keyparameters . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.4.3 Single-layerresults: analysisandverification . . . . . . . . 118
3.4.4 Furtherexplorationofthesingle-plymodel . . . . . . . . . 149
3.4.5 Three-layersystemwithgaps . . . . . . . . . . . . . . . . . 154
3.4.6 Aten-layersystemwithgaps . . . . . . . . . . . . . . . . . 171
4 Conclusions 186
4.1 Summaryofthethesis . . . . . . . . . . . . . . . . . . . . . . . . . 186
4.2 Suggestionsonfuturework . . . . . . . . . . . . . . . . . . . . . . 191
A Dimensionlessframeworkfornumericalsolutionsinisotropicmateri-
als 194
A.1 Analysisandequations . . . . . . . . . . . . . . . . . . . . . . . . . 194
A.2 Initialconditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
A.3 Summaryofkeyequationstosolvenumerically . . . . . . . . . . 198
B Dimensionless framework for numerical solutions in bi-axial mem-
branes 200
B.1 Analysisandequations . . . . . . . . . . . . . . . . . . . . . . . . . 200
B.2 Dimensionlessinitialconditions . . . . . . . . . . . . . . . . . . . . 206
Bibliography 208
vii
LISTOFTABLES
2.1 AlistofkeyparametersofKevlarandDyneema . . . . . . . . . . 33
2.2 A comparison of critical layer gapGn;cr, critical velocity Vn;cr and
relativeefficiencyfromn = 1to10inKevlarandDyneemacases 46
3.1 A list of key parameters implemented in the Kevlar and
Dyneema cases (Dyneema SK76 is assumed in all the following
simulations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.2 A comparison of the strain decrease from X = 1 to X = 2 and
thento X = R ((cid:28))atdifferenttimes. . . . . . . . . . . . . . . . . . . 135
c
3.3 A comparison of the strain decrease from X = r to X = 2r and
p p
X = r atdifferenttime . . . . . . . . . . . . . . . . . . . . . . . . . 137
c
3.4 Comparisonofsomeimportantquantitiescalculatedfromstudy
ofthesingle-layer,three-layerandten-layercases. . . . . . . . . . 177
3.5 Comparison of some important quantities obtained from study
ofsingle-layer,three-layerandten-layercases. . . . . . . . . . . . 183
viii