Physics-Based Modeling in Design & Development for … Documentation Page Form Approved ... SUPPLEMENTARY NOTES Physics-Based Modeling in Design and Devlopment for U.S. Defense Conference

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 &

Development for U.S. Defense Conference

UNCLASSIFIED: Distribution Statement A. Approved for public release.

1 UNCLASSIFIED

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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)



2 UNCLASSIFIED

UNCLASSIFIED

**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



3 UNCLASSIFIED

UNCLASSIFIED

OUTLINE

1. Motivation

2. Basic Assumptions and Preliminaries

3. Theoretical Developments

4. Solution Methodology

5. Blast Loading

6. Results

7. Concluding Remarks

4 UNCLASSIFIED

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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



5 UNCLASSIFIED

UNCLASSIFIED

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.



6 UNCLASSIFIED

UNCLASSIFIED

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



7 UNCLASSIFIED

UNCLASSIFIED

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



Displacement Field

8 UNCLASSIFIED

UNCLASSIFIED

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.



9 UNCLASSIFIED

UNCLASSIFIED

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



10 UNCLASSIFIED

UNCLASSIFIED

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



11 UNCLASSIFIED

UNCLASSIFIED

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



12 UNCLASSIFIED

UNCLASSIFIED

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

QQ

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 ,,



Constitutive Equations

13 UNCLASSIFIED

UNCLASSIFIED

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



14 UNCLASSIFIED

UNCLASSIFIED

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



15 UNCLASSIFIED

UNCLASSIFIED

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



Equations of Motion

Eqs. (1) , (2)

Eqs. (3) , (4)

Eqs. (5) , (6)

Eq. (7)

16 UNCLASSIFIED

UNCLASSIFIED

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

qq

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,



Eq. (8)

17 UNCLASSIFIED

UNCLASSIFIED

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



18 UNCLASSIFIED

UNCLASSIFIED

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



Boundary Conditions

19 UNCLASSIFIED

UNCLASSIFIED

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



Solution Mthodology

20 UNCLASSIFIED

UNCLASSIFIED

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



21 UNCLASSIFIED

UNCLASSIFIED

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



22 UNCLASSIFIED

UNCLASSIFIED

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.



23 UNCLASSIFIED

UNCLASSIFIED

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



Blast Loading

24 UNCLASSIFIED

UNCLASSIFIED

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



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



26 UNCLASSIFIED

UNCLASSIFIED

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.



Results-Validation

Fig 3. The nondimensional global deflection-time response of

a simply supported sandwich plate impacted by a uniform

pressure pulse 27 UNCLASSIFIED

UNCLASSIFIED



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



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.



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




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




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





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)




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




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)




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)





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)


Concluding Remarks



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

UNCLASSIFIED

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