A Micromechanics Finite Element Model for Studying the Mechanical Behavior of Spray-On Foam Insulation (SOFI)

Louis J. Ghosn

Ohio Aerospace Institute Brook Park, Ohio

Roy M. Sullivan and Bradley A. Lerch

NASA Glenn Research Center Cleveland, Ohio

Proposed Abstract for the

43rd Annual Technical Meeting Society of Engineering Science

August 13-16th, 2006 A micromechanics model has been constructed to study the mechanical behavior of spray-on foam insulation (SOFI) for the external tank. The model was constructed using finite elements representing the fundamental repeating unit of the SOFI microstructure. The details of the micromechanics model were based on cell observations and measured average cell dimensions discerned from photomicrographs. The unit cell model is an elongated Kelvin model (fourteen-sided polyhedron with 8 hexagonal and six quadrilateral faces), which will pack to a 100% density. The cell faces and cell edges are modeled using three-dimensional 20-node brick elements. Only one-eighth of the cell is modeled due to symmetry. By exercising the model and correlating the results with the macro-mechanical foam behavior obtained through material characterization testing, the intrinsic stiffness and Poisson’s Ratio of the polymeric cell walls and edges are determined as a function of temperature. The model is then exercised to study the unique and complex temperature-dependent mechanical behavior as well as the fracture initiation and propagation at the microscopic unit cell level.

1

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A M

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BX

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BX

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Effe

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Cal

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NC

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NC

FI-2

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y =

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0.50

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0.00

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in, m

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NC

FI-2

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y =

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0.00

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0.50

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0.00

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PDL-

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0.40

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Stra

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244.

994.

9918

.13

0.48

80.

143

0.14

3

NC

FITe

mp

E11

E22

E33

nu12

nu13

nu23

( C )

(MPa

)(M

Pa)

(MPa

)-2

519.

969.

9645

.75

0.52

80.

114

0.11

4-1

969.

969.

9645

.75

0.52

80.

114

0.11

424

4.53

4.53

20.8

00.

528

0.11

40.

114

PDL

Tem

pE1

1E2

2E3

3nu

12nu

13nu

23( C

)(M

Pa)

(MPa

)(M

Pa)

-253

31.8

531

.85

49.9

60.

362

0.24

90.

249

-196

25.0

325

.03

39.2

60.

362

0.24

90.

249

2412

.74

12.7

419

.99

0.36

20.

249

0.24

9

Foam

Eff

ectiv

e El

astic

Pro

perti

es

23

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iona

l Aer

onau

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and

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e Ad

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istra

tion

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NC

FI 2

4-12

4: R

ound

7"

Dog

bone

Spe

cim

en

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Stra

in (m

m/m

m)

Stress (MPa)

Vacu

umSA

TP

In-P

lane

Ave

rage

UTS

at S

ATP

= 0.

185

MPa

Av

erag

e U

TS u

nder

vac

uum

= 0

.137

MPa

Ris

eAv

erag

e U

TS a

t SAT

P =

0.42

MPa

Aver

age

UTS

und

er v

acuu

m =

0.4

5 M

PaR

ise

IN-P

lane

Exp.

dat

a co

urte

sy o

fK

ristin

Mor

gan

UTS

of N

CFI

at R

T FEA

Uni

t Cel

l

Sym

bols

dat

a LM

SSC

Ext

erna

l Tan

k R

etur

n to

Flig

ht, F

ract

ure

Toug

hnes

s: P

hase

2 (K

ristin

Mor

gan)

24

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iona

l Aer

onau

tics

and

Spac

e Ad

min

istra

tion

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gov

Ris

e at

1-a

tm@

0.4

2 M

Paap

plie

d st

ress

163

MPa

(23.

6Ksi

)

Uni

t Cel

lLo

adin

gIn

Ris

e D

irect

ion

MPa

Max

imum

Prin

cipa

l Stre

ss fo

r loa

ding

in th

e ris

e di

rect

ion

NC

FI (1

atm

)

X

YZ

25

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iona

l Aer

onau

tics

and

Spac

e Ad

min

istra

tion

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gov

In-P

lane

1-a

tm@

0.1

86 M

Paex

tern

al lo

ad

182

MPa

(26.

4 K

si)

Uni

t Cel

lLo

adin

gIn

-Pla

ne d

irect

ion

MPa

Max

imum

Prin

cipa

l Stre

ss fo

r loa

ding

in th

e in

-pla

ne d

irect

ion

NC

FI (1

atm

)

X

YZ

26

Nat

iona

l Aer

onau

tics

and

Spac

e Ad

min

istra

tion

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asa.

gov

Ris

e m

id-c

ell i

n va

cuum

@ 0

.45

MPa

exte

rnal

load

205

MPa

(29.

7Ksi

)U

nit C

ell

Load

ing

In R

ise

Dire

ctio

n

MPa

X

YZ

Max

imum

Prin

cipa

l Stre

ss fo

r loa

ding

in th

e ris

e di

rect

ion

NC

FI (V

acuu

m T

est)

27

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iona

l Aer

onau

tics

and

Spac

e Ad

min

istra

tion

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gov

In-P

lane

mid

-cel

l in

vacu

um@

0.1

27 M

Ps e

xter

nal l

oad

141

MPa

(20.

5 K

si)

Uni

t Cel

lLo

adin

gIn

-Pla

ne d

irect

ion

vacu

um

MPa

X

YZ

Max

imum

Prin

cipa

l Stre

ss fo

r loa

ding

in th

e ris

e di

rect

ion

NC

FI (V

acuu

m T

est)

28

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iona

l Aer

onau

tics

and

Spac

e Ad

min

istra

tion

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gov

Appl

ied

Stre

ssM

ax.

Prin

cipa

lM

PaM

PaR

ise

SATM

0.42

169

In-P

lane

SATM

0.18

618

2R

ise

Vacu

um0.

4520

5In

-Pla

neVa

cuum

0.12

714

1

NC

FI, U

TS

X

YZ

Max

. Prin

cipa

l, 44

3MPa

Load

ing

in R

ise

Dire

ctio

nW

ithou

t cel

l wal

ls (O

pen

Cel

l)

29

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iona

l Aer

onau

tics

and

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e Ad

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istra

tion

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x

y

z C

BA

θsi

n2

2L

2L

42

cos

2L

L+

θ4

2co

s2

LL

+θ

θ

24L

Ris

e

In-P

lane

Ope

n C

ell S

truct

ure

For F

ailu

re A

naly

sis

Sim

plify

mod

el (O

nly

Edge

s)

A

CB

30

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iona

l Aer

onau

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and

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e Ad

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θ

L

C

B

y

zN

VM

M

zzσ

Equi

libriu

m a

nd S

truct

ural

Ana

lysi

s of F

ram

es y

ield

s

[]

2co

s2

sin

24

2L

LN

zz+

=θ

θσ

[]

4co

s2

cos

24

22

LL

LM

zz+

=θ

θσ

[]

42

22

cos

2si

nco

sL

LL

Nyy

+=

θθ

θσ

[]

22

cos

2si

n4

22

2 2L

LL

Myy

+=

θθ

σ

θ

L

C

B

zN

V

M

M

yyσ

y

31

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iona

l Aer

onau

tics

and

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e Ad

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istra

tion

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Stre

ngth

of M

ater

ials

app

roac

h fo

r cal

cula

ting

the

max

imum

stre

ss a

t the

ou

ter f

iber

of a

n ax

ial a

nd fl

exur

al o

f the

edg

es

IMr

AN+

=m

axσ

()

θθ

θsi

nco

s2

12si

n2

12

24

22

3 42

3 2L

LL

AIL

L

EI

Ey

⎥ ⎦⎤⎢ ⎣⎡

++

+=

() 4

22

2 22

max

2co

s2

2sin

sin

cos

LL

Ir

LA

Lyy

+⎥⎥ ⎦⎤

⎢⎢ ⎣⎡+

=

θθ

θθ

σσ

Cal

cula

ting

the

aver

age

cell

stra

in a

nd st

ress

can

als

o le

ad to

det

erm

inin

g th

eef

fect

ive

cell

mod

ulus

in tw

o di

rect

ions

:

()2 4

22

max

cos

24c

os2sin

LL

Ir

LA

zz+

⎥ ⎦⎤⎢ ⎣⎡

+=

θθ

θσ

σ

()2 4

22 22

22 2

cos

2si

n12

cos

sin

24

LL

AL

IL

EI

Ez

+⎥⎥ ⎦⎤

⎢⎢ ⎣⎡+

=

θθ

θ

θ

Lea

ds to

the

rela

tion

betw

een

the

max

imum

stre

ss in

the

stru

ts

and

the

aver

age

appl

ied

stre

ss in

the

z an

d y

dire

ctio

ns:

Whe

reA

= e

dge

x-ar

eaI

= e

dge

mom

ent o

f Ine

rtia

r =

dis

tanc

e to

out

er fi

ber

32

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iona

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and

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Rat

io o

f the

Stif

fnes

s in

the

z-an

d y-

dire

ctio

ns is (

)

[]2

42

2 2

22

2

42

23 4

23 2

2

cos

2si

n12

cos

cos

212

sin

2si

n2

LL

ALIL

LL

AIL

L

EEyz

+⎥ ⎦⎤

⎢ ⎣⎡+

⎥ ⎦⎤⎢ ⎣⎡

++

+=

θθ

θ

θθ

θ

Ass

umin

g L 2

and

L 4ar

e 69

μm

and

28 μm

, res

pect

ivel

y, w

e ca

n pl

ot th

e st

iffne

ss ra

tiove

rsus

incl

inat

ion

angl

e.

Stru

t cro

ss-s

ectio

n ha

s sm

all i

nflu

ence

on

stiff

ness

ratio

ver

sus i

nclin

atio

n an

gle

Res

ults

indi

cate

that

the

incl

inat

ion

angl

e fo

r BX

-265

is ~

55O

6.3518

==

MPaMPa

EE

yz

Mea

sure

d st

iffne

ss ra

tio fo

r BX

-265

at

room

tem

pera

ture

stiff

ness

ratio

vs

incl

inat

ion

angl

e

012345678

010

2030

4050

6070

8090

100

incl

inat

ion

angl

e (d

egre

es)

stiffness ratio

circ

ular

x-s

ectio

n ra

dius

of 1

0

trian

gula

r x-s

ectio

n le

ngth

17

circ

ular

x-s

ectio

n ra

dius

of 5

circ

ular

x-s

ectio

n ra

dius

of 2

0

33

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and

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e Ad

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Rat

io o

f stre

ngth

s in

z-an

d y-

dire

ctio

ns is

[]

[][

]4

22

22

2co

s2

cos

sin

2si

nco

s2

sin

4L

LrA

LI

rAL

IL

yz

++

+=

θθ

θθ

θθ

σσ

From

exp

erim

enta

l ten

sile

test

spec

imen

the

mea

sure

d ra

tio o

f the

rise

to in

-pla

ne st

reng

ths i

s app

roxi

mat

ely

1.6.

Res

ults

indi

cate

that

the

incl

inat

ion

angl

e is

47o

stre

ngth

ratio

ver

sus

incl

inat

ion

angl

e

00.

511.

522.

533.

544.

55

010

2030

4050

6070

8090

angl

e (d

egre

es)

strength ratio

circ

ular

x-s

ectio

n ra

dius

of 1

0

circ

ular

x-s

ectio

n ra

dius

of 5

circ

ular

x-s

ectio

n ra

dius

of 2

0

trian

gula

r x-s

ectio

n le

ngth

17

34

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Pol

ymer

(Stru

t) S

treng

th,

MP

a(k

si)

Usi

ng θ

= 47

o

Pol

ymer

(Stru

t) M

odul

us,

MPa

(ksi

)

Usi

ng θ

= 55

o

Tria

ngul

ar

x-se

ctio

n

Circ

ular

x-

sect

ion

393

(57)

7,30

0 (1

,059

)h=

17 μ

m

10 (1

.5)

111

(16)

R=2

0 μm

76 (1

1)1,

450

(210

)R

=10 μm

603

(87.

5)21

,500

(3,1

20)

R=5

μm

h

Subs

titut

ing

L 2, L

4, an

d th

e in

clin

atio

n an

gle

into

the

prev

ious

equ

atio

ns fo

r foa

mm

odul

us a

nd fo

am st

reng

ths a

nd so

lvin

g fo

r the

pol

ymer

(stru

t) m

odul

us a

nd p

olym

er

(stru

t) st

reng

ths,

we

get

35

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e Ad

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istra

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gov

Futu

re W

ork

•SO

FI m

icro

stru

ctur

e w

ere

succ

essf

ully

mod

eled

as

elon

gate

d Te

traka

idec

hahe

dron

clos

ed c

ell

•E

last

ic m

odul

us a

nd P

oiss

on’s

ratio

s w

ere

estim

ated

from

the

unit

cell

mod

el•

The

estim

atio

n of

the

poly

mer

sol

id s

treng

th a

nd s

tiffn

ess

was

at

tem

pted

by

FEA

and

Fram

e m

odel

s.•

Fram

e m

odel

yie

lds

sim

ilar r

esul

ts a

s th

e fin

ite e

lem

ent m

odel

if a

ci

rcul

ar c

ross

-sec

tion

with

a ra

dius

of 1

0 m

m is

use

d.

Sum

mar

y

•R

efin

e th

e FE

A m

odel

(Non

linea

rity)

to im

prov

e on

the

failu

re s

tress

es

timat

es•

Rev

iew

the

mod

elin

g of

a u

nit c

ell i

n va

cuum