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Terry Hause, Ph.D.,
Research Mechanical Engineer
And
Sudhakar Arepally
Deputy Associate Director
U.S. Army RDECOM-TARDEC, CASSI Analytics
Warren, MI 48397
Laminated Composite Sandwich Plates with a Weak Compressible Core Impacted by Blast Loading
Physics-Based Modeling in Design & Development for
U.S. Defense Conference
Physics-Based Modeling in Design &
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1. REPORT DATE 03 NOV 2011
2. REPORT TYPE Briefing Charts
3. DATES COVERED 03-11-2011 to 03-11-2011
4. TITLE AND SUBTITLE LAMINATED COMPOSITE SANDWHICH PLATES WITH A LEAKCOMPRESSIBLE CORE IMPACTED BY BLAST LOADING
5a. CONTRACT NUMBER
5b. GRANT NUMBER
5c. PROGRAM ELEMENT NUMBER
6. AUTHOR(S) Terry Hause; Sudhakar Arepally
5d. PROJECT NUMBER
5e. TASK NUMBER
5f. WORK UNIT NUMBER
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) U.S. Army TARDEC ,6501 E.11 Mile Rd,Warren,MI,48397-5000
8. PERFORMING ORGANIZATIONREPORT NUMBER #22408
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) U.S. Army TARDEC, 6501 E.11 Mile Rd, Warren, MI, 48397-5000
10. SPONSOR/MONITOR’S ACRONYM(S) TARDEC
11. SPONSOR/MONITOR’S REPORT NUMBER(S) #22408
12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution unlimited
13. SUPPLEMENTARY NOTES Physics-Based Modeling in Design and Devlopment for U.S. Defense Conference
14. ABSTRACT High bending stiffness and strength to weight ratio Excellent thermal and sound insulation Increaseddurability under a thermo-mechanical loading environment Tight thermal distortion tolerancesLightweight in structure
15. SUBJECT TERMS
16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT Same as
Report (SAR)
18. NUMBEROF PAGES
38
19a. NAME OFRESPONSIBLE PERSON
a. REPORT unclassified
b. ABSTRACT unclassified
c. THIS PAGE unclassified
Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18
ACKNOWLEDGEMENTS
The author would like to express thanks to the U.S. Army RDECOM
TARDEC for their support and funding under the Independent
Laboratory In-House Research Program (ILIR)
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**Disclaimer: Reference herein to any specific commercial
company, product, process, or service by trade name, trademark,
manufacturer, or otherwise, does not necessarily constitute or imply
its endorsement, recommendation, or favoring by the United States
Government or the Department of the Army (DoA). The opinions of
the authors expressed herein do not necessarily state or reflect
those of the United States Government or the DoA, and shall not
be used for advertising or product endorsement purposes.**
DISCLAIMER
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OUTLINE
1. Motivation
2. Basic Assumptions and Preliminaries
3. Theoretical Developments
4. Solution Methodology
5. Blast Loading
6. Results
7. Concluding Remarks
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MOTIVATION
• High bending stiffness and strength to weight ratio
• Excellent thermal and sound insulation
• Increased durability under a thermo-mechanical loading
environment
• Tight thermal distortion tolerances
• Lightweight in structure
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BASIC ASSUMPTIONS AND PRELIMINARIES
1. The face sheets fulfill the Love-Kirchoff assumptions and are thin
compared with the core.
2. The bonding between the face sheets and the core is assumed to be
perfect.
3. The kinematic boundary conditions at the interfaces between the core
and the facings are satisfied.
4. The core is assumed to be a weak orthotropic transversely
compressible core carrying only the transverse strains and the normal
strain.
5. The shock wave pressure is uniformly distributed on the front face of
the sandwich plate.
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6 UNCLASSIFIED
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Fig 1b. An asymmetric sandwich plate under blast loading
top face sheets
bottom face sheets
core
Detonation
Standoff Distance
z
y x
tft
bft
ct
Fig 1a. Incident pressure profile
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7 UNCLASSIFIED
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THEORETICAL DEVELOPMENTS
dα
tfca
α
tfcd
αaα
tα u
ttxu
ttxuuυ ,33,33
22
dat uuυ 333
dα
bfca
α
bfcd
αaα
bα u
ttxu
ttxuuυ ,33,33
22
dab uuυ 333
cα
c
dα
c
bf
tfa
αc
bf
tfd
αc
dα
bf
tfa
α
bf
tfa
αcα
t
xux
t
ttux
t
ttu
t
xu
ttu
ttuυ Φ1
4
22
2
44 2
23
,33,333
,3,3
d
c
ac ut
xutzyxυ 3
333
2),,,(
Top Face
Bottom Face
Core
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Displacement Field
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Note:
the Greek indices have the range 1, 2, while the Latin indices have the range 1, 2, 3 and
unless otherwise stated, Einstein’s summation convention over the repeated indices is
assumed. Also, denotes partial differentiation with respect to the coordinates , while
superscripts t and b indicate the association with the top and bottom facings respectively.
Also,
)(2
1),(
2
1 bi
ti
di
bi
ti
ai uuuuuu
represent the average and the half difference of the face sheet mid-surface displacements while, the core
displacements, cαΦ warping functions of the core.
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9 UNCLASSIFIED
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Non-Linear Strain-Displacement Relationships
21,31,111 )(
2
1υυγ
22,32,222 )(
2
1υυγ
23,33,333 )(
2
1υυγ
3,32,32,33,2232
1)(
2
1υυυυγ
3,31,31,33,1132
1)(
2
1υυυυγ
2,31,31,22,1122
1)(
2
1υυυυγ
The strain-displacement relationships given by the Lagrangian Strain-Displacement
Relationships used in conjunction with the Von-Karman assumptions is given in
indicial notation as
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10 UNCLASSIFIED
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Substitution of the displacement relationships gives:
dαβ
tfca
αβ
tfcd
αβaαβ
tαβ κ
ttxκ
ttxγγγ
2233
dαβ
bfca
αβ
bfcd
αβaαβ
bαβ κ
ttxκ
ttxγγγ
2233
Where,
)(2
1),(
2
1 bαβ
tαβ
dαβ
bαβ
tαβ
aαβ γγγγγγ
)(2
1),(
2
1 bαβ
tαβ
dαβ
bαβ
tαβ
aαβ κκκκκκ
Top Layer
Bottom Layer
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In the above expressions, are referred to as the average and half difference of
tangential or membrane strains of the top and bottom facings; while, are referred to as
the average and half difference of the bending strains of the top and bottom facings. The
expressions for the membrane and bending strains are not provided here.
),( daαβγ
),( daαβκ
For the core, the strain-displacement relationships take the form
333ci
ci
ci κzγγ
In these expressions, and are the membrane and bending strains, respectively. These
expressions are not provided here.
ciγ 3
ciκ 3
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12 UNCLASSIFIED
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Both the top and bottom face sheets are considered to be constructed from unidirectional
fiber reinforced anisotropic laminated composites, the axes of orthotropy not necessarily
being coincident with the geometrical axes. The stress-strain relationships for each lamina
of the facings becomes
12
22
11
66
2622
161211
12
22
11
2Sym γ
γ
γ
Q
QQQ
τ
τ
τ
Where, for i, j = (1, 2, 6) are the Transformed plane-stress reduced stiffness measures. ijQ
The stress-strain relationships for the orthotropic core with the geometrical and material
axes coincident are expressed as
ccccccccc γGτγGτγEτ 2323231313133333 ,,
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Constitutive Equations
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Hamilton’s Variational Principle
0)(1
0
dtTδWδUδt
t
U = strain energy,
W = represent the work done by external forces
T = represent the kinetic energy
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14 UNCLASSIFIED
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dAdxδγτdxδγτdxδγτUδbfc
c
c
c
c
tfc
tt
t
bαβ
bαβ
t
t
ci
ciA
t
tt
tαβ
tαβ
2
2 3
2
2 333
2
2 3
A
bbbccctttbbtt dAδυυCδυυCδυυCδυtxxqδυtxxqWδ 33333332133213 222),,(ˆ),,(ˆ
dAdtdxδυυρdxδυυρdxδυυρTdtδbfc
c
c
c
c
tfc
tt
t
bbbf
t
t A
t
t
ccct
tt
tttf
t
t
2
2 333
2
2 333
2
2 3331
0
1
0
Where are the tensorial components of the second Piola-Kirchoff stress tensor, while
A is attributed to the area of the sandwich plate. ijτ
Where denotes the transverse pressure loading from a spherical air-blast
and C is the structural damping coefficient per unit area of the plate.
),,( 21 txxqt
Where and are the mass densities of the core and the top and bottom face
sheets, respectively, and denotes the transverse acceleration.
cρ bf
tf ρρ ,
υ
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15 UNCLASSIFIED
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0: , aβαβ
aα Nuδ
0: 3,
c
cαd
βαβdα
t
NNuδ
0:Φ 3 cα
cα Mδ
02
ˆˆ
2
222
2
4
21:
333
333
3,3,33,3,,33
btd
bt
acbt
dbb
ftt
fac
cbb
ftt
f
cα
dα
c
ααα
dbf
tfc
c
dαβ
dαβ
aαβαβ
aαβ
aαβ
a
qqu
CC
uCCC
uρtρt
uρtρtρt
Nut
Nuttt
tNuMNuuδ
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Equations of Motion
Eqs. (1) , (2)
Eqs. (3) , (4)
Eqs. (5) , (6)
Eq. (7)
16 UNCLASSIFIED
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02
ˆˆ
22232
1
4
21:
33
3333
,3333,3,,33
bt
abt
dbt
abb
ftt
fdc
cbbf
ttf
cαα
c
bf
tfcd
c
aαβ
dαβ
dαβαβ
dαβ
aαβ
d
uCC
uCC
uρtρt
uρt
ρtρt
Nt
ttNu
tNuMNuuδ
Where, the global stress resultants and stress couples are defined as
bαβ
tαβ
bαβ
tαβ
aαβ
aαβ MMNNMN ,
2
1,
bαβ
tαβ
bαβ
tαβ
dαβ
dαβ MMNNMN ,
2
1,
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Eq. (8)
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Where the local stress resultants and stress couples are given as:
3
2
2 32
,1, dxtt
xτMNc
tfc
t
tt
tfct
αβtαβ
tαβ
3
2
2 32
,1, dxtt
xτMNbfc
c
tt
t
bfcb
αβbαβ
bαβ
,),1(,2
2 33333
c
c
t
t
ci
ci
ci dxxτMN )3,2,1( i
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18 UNCLASSIFIED
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For the case of simply supported boundary conditions, the boundary conditions become:
Along the edges ),0( nn Lx
033 dadnn
ann
dnt
ant
dnn
ann uuMMNNNN
n and t are the normal and tangential directions to the boundary. When ,1n 2tand when ,2n 1t
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Boundary Conditions
19 UNCLASSIFIED
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Special Case: Symmetric orthotropic single layer facings
In fulfillment of the geometric boundary conditions, a suitable representation for da uu 33 and,
is given by:
)sin()sin()( 213 xμxλtwu nmamn
a
)sin()sin()( 213 xμxλtwu nmdmn
d
21 , LπnμLπmλ nm Where,
The transverse explosive loading is represented as
),sin()sin()(),,( 2121 xμxλtqtxxq nmmnt
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Solution Mthodology
20 UNCLASSIFIED
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which implies through integration of both sides over the plate area that
2 1
0 0 21212121
)sin()sin(),,(4
)(L L
nmtmn dxdxxμxλtxxqLL
tq
Letting,
]/)(exp[]/)(1)[()(),,( 0021 papatt tttαtttqqtqtxxq S
And integrating gives
2
)(16)(
πmn
tqtq t
mn
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21 UNCLASSIFIED
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The first two Equations of Motion can be satisfied by a stress potential in conjunction with a
compatibility equation not provided here. Equations of Motion (3) through (6) can be shown
to be satisfied by expressing these equations of motion in terms of displacements and
assuming appropriate functional forms in terms of unknown constant coefficients and the
amplitudes as a function of time. The unknown constants are determined by substitution
and comparing coefficients.
At this point the Extended-Galerkin Method is utilized by retaining the last two Equations of
Motion within the energy functional and carring out the indicated integrations results in two
nonlinear coupled second order ordinary differential equations in terms of the modal
amplitudes. These are given as:
2)()( 3
302
1211101mna
mnad
mnamn
admn
amn
aamn
aamn
amn
qwCwwCwwCwCwCwm
2)()()()( 2
212
203
032
02012mnd
mnamn
damn
ddmn
ddmn
ddmn
ddmn
dmn
qwwCwCwCwCwCwCwm
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22 UNCLASSIFIED
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The coefficients 21200301301210 ,,,, CCCCCCC are expressions which depend on the
material and geometrical properties of the structure.
These two governing differential equations are then solved using the 4th Order runge-Kutta
Method.
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23 UNCLASSIFIED
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For a free in-air spherical air burst, the pressure profile over time is given
in figure 2 as
Fig 2. Incident Profile of a blast wave
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Blast Loading
24 UNCLASSIFIED
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The wave form shown in figure 4 is given by an expression known as
The Friedlander equation and is give as
p
a
t
tt
p
aosot e
t
ttPPtP
1)()(
ambientoverreOverpressuPeak1081141772
23
ZZZPso
Where,
kilograms of terms in TNT of
charge of weightequivalent the is Distance Standoff the is
distancescaled31 WR
WRZ
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25 UNCLASSIFIED
UNCLASSIFIED
time the is
waveblast the of duration phase positive the is
arrival of time the is
pressure ambient the is
t
t
t
P
p
a
o
For conditions of STP at sea level, the time of arrival and the positive
phase duration can be determined from
scaling root Cube3
1
111
W
W
R
R
t
t
Arrival time or positive phase duration for a reference explosion of charge weight, 1W
Arrival time or positive phase duration
It should be noted that the standoff distances are themselves scaled
According to the cube root law
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26 UNCLASSIFIED
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To validate the present approach, the dynamic response of a simply supported plate
impacted by a uniform pressure pulse was chosen from R.S. Alwar et Al.
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Results-Validation
Fig 3. The nondimensional global deflection-time response of
a simply supported sandwich plate impacted by a uniform
pressure pulse 27 UNCLASSIFIED
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Physics-Based Modeling in Design &
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Results-Present
Fig. 4 The effect of the transverse modulus of the core on the global
response of a sandwich plate with orthotropic facings.
28 UNCLASSIFIED
UNCLASSIFIED
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Fig. 5 The counterpart of Fig. 4 for the wrinkling response of a
sandwich plate.
29 UNCLASSIFIED
UNCLASSIFIED 0.1 2
0.08 ············-- Ec=0.3 GPa
0.04
,-..._ ...... ~
~~ 0 ::r: . 1::: . \ . . . . . . . • • . . . . .. .. ..
-0.04 .. .. -
-0.08
[0/Core!O]
-0.12 0 3 6 9 12 15
Time (mscc)
TECHNOLDGY DRNEN. WARRGHTER FOOJSED.
Physics-Based Modeling in Design &
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Fig. 6 The effect of the rate-of-decay parameter on the global
response of a sandwich plate with orthotropic facings.
30 UNCLASSIFIED
UNCLASSIFIED
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
-I 0 6
Time (msec)
---- ~=0.3
~=0 .6
., ... fj . . ..... ·· .. l \ • I ·. l '• I \ i \, i \ I \ i \ l \,. ,•
.i •. ; '"·" . .
[0/CoreiO]
9 12 15
TECHNOLDGY DRNEN. WARRGHTER FOOJSED.
Physics-Based Modeling in Design &
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Fig. 7 The counterpart of Fig. 6 for the wrinkling response.
31 UNCLASSIFIED
UNCLASSIFIED 15
12
9
6
3
~ 8 .... '--' 0 '1:! :: 0
~E -5: -3
-6
-9
-12
- 15 0 3 6 9
Time (msec)
--- J3=0.3
··- ·····--- ~=0.6
[0/Core/0]
12 15
TECHNOLDGY DRNEN. WARRGHTER FOOJSED.
Physics-Based Modeling in Design &
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Fig. 8 The effect of the core thickness on the global deflection-time
history of a sandwich plate with orthotropic facings.
32 UNCLASSIFIED
UNCLASSIFIED 1.5
1.25
0.75
0.5
0.25 ~ ...... '-'
0 ::! ~
~:; :r: -0.25
-0.5
-0.75
-I
-1 .25
-1.5 0 2 4
Time (msec) 6
·······- tc=30mm
---- tc=50mm
. I .
I : : : I
:
8
I . : . . . . . \ .•. l·
[OiCore/0]
10
TECHNOLDGY DRNEN. WARRGHTER FOOJSED.
Physics-Based Modeling in Design &
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Fig. 9 The counterpart of Fig. 8 for the wrinkling response.
33 UNCLASSIFIED
UNCLASSIFIED
20
16
12
8
4
Q 0 '-" 0
0 \:~ 0 ...... ';;;) ......
-4
-8
-1 2
-16
-20 0
.. ::
.. U I ... •• • f
2
.. i: . ... .• f •• •• . :::: ••••• &: .. :: ., :: · :: ::·:, f . n' .
4
·······-···· t(..=30mm
' .. . .. .. •• j,
::u~ :: u .. l: :: !i A ..... ······· • ·: :· : :1 • . . : .... .... ••
6
tc=SOmm
[0/Core/0]
10 Time (mscc)
TECHNOLDGY DRNEN. WARRGHTER FOOJSED.
Physics-Based Modeling in Design &
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Fig. 10 The effect of the stacking sequence of the facings on
the global response of a sandwich plate.
34 UNCLASSIFIED
UNCLASSIFIED
-0.8 0 2
• . . : • . • • • • . : • • . • . • • • : • • .
4
• I . • . • • . : • • . • . . . . .
J • . .
Time (msec) 6
[0/Core/0]
[0/90/0/Core/0!90/0]
10
TECHNOLDGY DRNEN. WARRGHTER FOOJSED.
Physics-Based Modeling in Design &
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Fig. 11 The counterpart of Fig. 10 for the wrinkling
response.
35 UNCLASSIFIED
UNCLASSIFIED
15
12.5
10
7.5
5
~ 0
'-" 0 ">::: ::: 0 2.5 ~a; ----::c
0
-2.5
-5
-7.5
- 10
' : J i! : ! : s . •• .. .. =:: • :a::;;::: I It I II tl Ill
0
: :: •: :: . ...... . I II I II. I II I • .... . . . . . . . . . . . . . . . : . .. ..
i
q ~ .: .t
~ :: ::iii :: 1: ::
I ' •' '• ,, ~ :: :: ·: : :: I ' :f ... ..
. •: : : i: I : I t l ... ~ . . . ..
! : :
2
: . :: : ·· :
.. : i :: : : . ~ :: : .. f :: : : ~ = .. :• .. .. :: 'thh'
4
············- fO!Corc/0] [Oi90/0/Core/0/90/0]
6 8 10 Time (mscc)
TECHNOLDGY DRNEN. WARRGHTER FOOJSED.
Physics-Based Modeling in Design &
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Fig. 12 The effect of the core shear modulus ratio on the deflection-time
history of cross-ply laminated sandwich plate.
36 UNCLASSIFIED
UNCLASSIFIED
0.3 ~' --'=0.5 (j'
23
0.2 d. ___!l=JO c"
23
0.1
0
81 . ., - . . ~~ :I: \ ...
-0.1 '···
-0.2
-0.3
f Oi90/0!Core/0/90/0l
-0.4 0 3 6 9 12 15
Time (msec)
TECHNOLDGY DRNEN. WARRGHTER FOOJSED.
Physics-Based Modeling in Design &
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Fig. 13 The counterpart of Fig. 12 for the wrinkling response.
37 UNCLASSIFIED
UNCLASSIFIED
~~c .:::,.,c ._ :: c ~ - -;;;:.._<:: -. ...... ::r:
6
4
2
. ~ ' .. io • • •
0
-2 .. ~ . . 1
-4
-6
dl ' ----~.5 c·
23
Gc
····-·······-·· ~3-10 G' 23
:
(0/90/0/Core/0/90/0)
_g L-..... ~------L-----~----------L------~------L-----~~-----~-----~-----~ 0 3 6 9 12 15
Tin1e (msec)
TECHNOLDGY DRNEN. WARRGHTER FOOJSED.
Concluding Remarks
Physics-Based Modeling in Design &
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The governing theory of asymmetric sandwich plates with a first-order compressible core
impacted by a Friedlander-type of blast has been presented and simplified for the case of
symmetric cross-ply and single-layered orthotropic facings. In all cases, it was mentioned
that all four edges are simply supported and freely movable. Results were then presented
for this simplified case and validated against results found in the literature from
R. S. Alwar et al. It was found that for the incompressible core case that there was close
agreement among the results. In regards to the compressible core case, no appropriate
results have been found in the literature for the theory presented in this paper for the
simply supported case with all edges freely movable. The effect of a number of important
geometrical and material parameters were analyzed with conclusions drawn. Some of the
important conclusions were that wrinkling response seems to be diminished as the young’s
modulus of the core is increased. The same is the case for larger rates of decay. Also, for
thicker cores, both the global and wrinkling responses are less severe. It was also revealed
that the compressibility of the core has only a marginal effect upon the global response of
the sandwich plate. Finally, the cross-ply type layup when compared with single-layered
facings seemed to have a large effect on the global response and less effect on the
wrinkling response.
One should keep in mind that both the stress and strain profiles should be determined to
determine possible failure of the structure.
38 UNCLASSIFIED
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