Application of an Elongated Kelvin Model to Space · PDF fileApplication of an Elongated...

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National Aeronautics and Space Administration

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Application of an Elongated Kelvin Model to Space Shuttle Foams

Spray-on foam insulation is applied to the exterior of the Space Shuttle’s External Tank to limit propellant boil-off and to prevent ice formation. The Space Shuttle foams are rigid closed-cell polyurethane foams. The two foams used most extensively on the Space Shuttle External Tank are BX-265 and NCFI24-124. Since the catastrophic loss of the Space Shuttle Columbia, numerous studies have been conducted to mitigate the likelihood and the severity of foam shedding during the Shuttle’s ascent to space. Due to the foaming and rising process, the foam microstructures are elongated in the rise direction. As a result, these two foams exhibit a non-isotropic mechanical behavior. In this paper, a detailed microstructural characterization of the two foams is presented. The key features of the foam cells are summarized and the average cell dimensions in the two foams are compared. Experimental studies to measure the room temperature mechanical response of the two foams in the two principal material directions (parallel to the rise and perpendicular to the rise) are also reported. The measured elastic modulus, proportional limit stress, ultimate tensile stress and the Poisson’s ratios for the two foams are compared. The generalized elongated Kelvin foam model previously developed by the authors is reviewed and the equations which result from this model are presented. The resulting equations show that the ratio of the elastic modulus in the rise direction to that in the perpendicular-to-rise direction as well as the ratio of the strengths in the two material directions is only a function of the microstructural dimensions. Using the measured microstructural dimensions and the measured stiffness ratio, the foam tensile strength ratio and Poisson’s ratios are predicted for both foams. The predicted tensile strength ratio is in close agreement with the measured strength ratios for both BX-265 and NCFI24-124. The comparison between the predicted Poisson’s ratios and the measured values is not as favorable.

https://ntrs.nasa.gov/search.jsp?R=20080047339 2018-05-17T20:00:03+00:00Z

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Application of an Elongated Kelvin Model to Space Shuttle Foams

Roy M. SullivanLouis J. Ghosn

Bradley A. Lerch

Structures and Materials DivisionNASA Glenn Research Center

Cleveland, OH

April 7, 2008

49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference

Renaissance Schaumburg Hotel & Convention Center Schaumburg, IL

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Spray-on Foam Insulation is used on the External Tank to reduce propellant boil-off and prevent ice formation on the external surfaces.

Acreage foam is NCFI24-124 (Machine sprayed)

Close-outs are hand sprayed BX-265 or PDL-1034

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Typical Flight Loads on ET Foam Applications

Aerodynamic Heating

Aerodynamic Shear

Cryogenic Temperatures(-423F/-297F)

Substrate Flexure and Membrane Forces

Tank WallFoam Insulation

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Figure 3.5 – Cell Geometry, NCFI24-124

• 97% air; γ = 0.03• polymeric cell walls• due to its microstructure, material is anisotropic (possess different material properties in different directions)

Foam Microstructure

Parallel-to-rise direction

Perpendicular-to-rise direction

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BX-265

NCFI24-124

The foam microstructure can be approximated by an elongated tetrakaidecahedron (a fourteen-sided polyhedron)

Average number of faces per cell: 12.4*

Average number of faces per cell: 13.7*

_________________* Wright L. S. and Lerch B. A., 2005. Characterization of space shuttle insulative materials, NASA/TM-2005-213596.

8 hexagonal faces4 diamond-shaped faces2 square faces36 edges

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Zhu H.X., Knott J.F. and Mills N.J., 1997. Analysis of the elastic properties of open-cell foams with tetrakaidecahedral cells. Journal of the Mechanics and Physics of Solids 45 (3), 319-343.

Warren, W.E. and Kraynik, A.M., 1997. Linear elastic behavior of a low-density Kelvin foam with open cells. Journal of Applied Mechanics 64, 787-794.

Dement’ev, A.G. and Tarakanov, O.G., 1970. Model analysis of the cellular structure of plastic foams of the polyurethane type. Mekhanika Polimerov 5, 859-865. translation Polymer Mechanics 6, 744-749.

Gong, L., Kyriakides, S. and Triantafyllidis, N., 2005. On the stability of Kelvin cell foams under compressive loads. Journal of the Mechanics and Physics of Solids 53, 771-794.

Ridha, M., Shim,V.P.W. and Yang, L.M., 2006. An elongated tetrakaidecahedral cell model for fracture in rigid polyurethane foam. Key Engineering Materials 306-308, 43-48.

Equi-axial Unit Cell Models

Elongated Unit Cell Models

Some Significant Previous Studies

Thomson, W. (Lord Kelvin), 1887. On the division of space with minimum partitionalarea. Phil. Mag. 24, 503-514.

William Thomson (Lord Kelvin) determined that the tetrakaidecahedron (with slightly curved faces) was nature’s preferred shape for soap bubbles and other foams since it is the shape that minimizes the surface area per unit volume and packs to fill space.

The tetrakaidecahedron foam model is commonly referred to as the Kelvin foam model after:

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H

D

L

b

θ

θ

L

The size and shape of an elongated tetrakaidecahedron are uniquely defined by specifying three independent dimensions.

θsin4LH =

bLD 2cos2 += θ

bLL

DHR

2cos2sin4

+==

θθ

Aspect Ratio of Unit Cell

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~20 μm

Simplifying Assumptions:1) The structural rigidity of the cell faces is assumed to contribute little to the foam mechanical behavior.2) The mass of the cell faces are only a small fraction of the total solid mass.

Face thickness in the middle of the faces~ 0.1 μm to 1.0 μm

The majority of the solid mass resides in the cell edges (where the faces come together).

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C

B

D

G

F

H

X

Y

Z (rise direction)

Y

Z (rise direction)

2/cos bL +θ

θsin2L

B

H

G

F

D

C

Xθ

θL

L

2/cos bL +θ

The mechanical behavior can be accurately modeled by considering the deformation of the cell edges only. The cell edges are assumed to act like struts possessing axial and flexural rigidity.

( ) ( )( )22

2cos2sin

2228

bLL

bLAHD

bLAVV

total

solid

s +

+=

+===

θθρργ

H

D

L

b

θ

θ

L

Relative Density

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Cell edge cross-sections are approximated as three-cusp hypocycloids (Plateau Borders)

y

x

r

( )( )

( )( ) 12/11320

24/31160

24/11320

2/3

3

3

4

2

rS

rS

rII

rA

y

x

yx

π

π

π

π

−=

−=

−==

−=Cross-section properties

bending moments of inertiaCross-sectional areaSection moduli

are a function of the cross-section radius only.

yx II ,A

yx SS ,

Thus, four microstructural dimensions are required:

3 to define the unit cell + 1 to specify the edge cross-section dimension

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[ ]22

222 cos2sin12cos

sin24

bLAL

IL

EIEz

+⎥⎦

⎤⎢⎣

⎡+

=

θθθ

θ

( )[ ] [ ]bLALIL

IALLzyzx ++

−==

θθθθθυυ

cos2cossin12sincos122

232

222

( )⎥⎦⎤

⎢⎣⎡ +++

==bL

AIbLL

EIEE yxθθθ 2323 cos212sin2sin

12

( )( ) ( )3232

2

sin2cos21212

bLAbLIIAbb

yxxy+++

−==

θθυυ

( )( )( ) ( )[ ]3232

2

sin2cos2122cos2cos212

bLAbLIbLIAL

yzxz+++

+−==

θθθθυυ

Y

Z (rise direction)

B

H

G

F

D

C

X

L

G′

H ′

F ′

B′

yyσ

yyσ

[ ]bLS

LA

LLx

ssyy

sxx

2cos22sinsincos 22

+⎥⎥⎦

⎤

⎢⎢⎣

⎡+

==

θθθθ

σσσ

Young’s moduli:

Poisson’s ratios:

Strengths:

[ ]bLS

LbA

Lbz

ssyy

sxx

2cos222sin

2sin

+⎥⎥⎦

⎤

⎢⎢⎣

⎡+

==

θθθ

σσσ

[ ]2cos24cos

2sin bL

SL

A Lx

sszz

+⎥⎥⎦

⎤

⎢⎢⎣

⎡+

=

θθθ

σσ

Failure of the b-length edges

Failure of the L-length edges

Careful consideration of the unit cell deformation leads to convenient algebraic equations for the Young’s modulus, Poisson’s ratio and strengths in the principal material directions.

based on peak stress in any edge reaching solid material strength σs

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The ratio of the rise direction modulus to the normal-to-rise direction modulus and the ratio of the strengths are indicative of the amount of elongation.

( ) ( )

⎥⎦⎤

⎢⎣⎡ +

⎥⎦⎤

⎢⎣⎡ +++

===θθ

θθ

22

2

22

322

sin12cos

/cos212/sin2

4AL

I

LbAL

ILbR

EE

EER

y

z

x

zE

Stiffness Ratio

Strength Ratio

θθθθ

σσ

σσ

σ cossin2sincos2

ALSALS

RR Lx

Lx

syy

szz

sxx

szz

+

+===

Doesn’t involve the properties of the solid material. Only a function of the unit cell dimensions and the edge cross-section dimension.

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Microstructural Characterization

rD

H

r

DHR /=

1.752726.0100142100248NCFI24-124

1.42NA*100136100193BX-265

nAvg., μm

nAvg., μm

nAvg., μm

Aspect Ratio

Edge Cross-section Radius

Cell Width DCell Height H

* Data not available

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Mechanical Testing

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Strain

Stre

ss (M

Pa)

Rise

Perpendicular-to-rise

BX-265

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.01 0.02 0.03 0.04

Strain

Stre

ss (M

Pa)

NCFI24-124

Rise

Perpendicular-to-rise

375.4

555.8

Rise

Ult. Tensile Strength (kPa)

248.1

315.4

Rise

Proportional Limit (kPa)

RatiosInitial Modulus (MPa)

20.80

13.53

Rise

7.07

7.03

Normal-to-rise

110.3

173.5

Normal-to-rise

188.2

319.4

Normal-to-rise

1.992.252.94NCFI24-124

1.741.821.92BX-265

Ultimate Tensile Strength

Proportional Limit

Stiffness

BX-265NCFI24-124

Average Properties

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Mechanical Testing

Poisson’s Ratios

0.0310.0373NCFI24-124

0.0310.0369BX-265

Relative Density*

Average Density (g/cc)

Average Density

0.100.641νzx

0.040.183νxz

0.140.382νxy

NCFI24-124

0.290.17

0.5360.675

νzx

0.00070.273νxz

0.060.355νxy

BX-265

Standard Deviation

Avg. Measured Poisson’s Ratio

* Calculated assuming density of polymer is 1.2 g/cc.

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AQQ = BQQ =

BA QQ >

and the shape parameter

The shape is defined by the aspect ratioDHR =

θcosLbQ =

BA RR >

ARR = BRR =

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( ) ( )

⎥⎦⎤

⎢⎣⎡ +

⎥⎦⎤

⎢⎣⎡ +++

⎟⎟⎠

⎞⎜⎜⎝

⎛

+=

θθ

θθ

θθ

22

2

22

322

sin12cos4

/cos212/sin2

2cos2sin4

ALI

LbAL

ILb

bLLRE

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+=

θθ

θθ

θθ

σ

cossin2

sincos2

2cos2sin4

ALS

ALS

bLLR

Lx

Lx

( )( )( )

( )( ) ⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

++++

+++

+++⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

++

=γ

γ

2222

52

3

1

2222

2232

122

322

2

~16~1624

~816

~16~1624

~16432~8~16

64~2

4

RQRQQ

QCRC

RQRQQ

RQQQRCC

RQ

QRQRRE

QQ 22~+=

( ) ( )

( ) ( ) ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

++++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

++++

=

5.0225.0

22

5.05.15.23

1

5.0225.0

22

5.05.05.13

1

~16~1624

~244

~16~1624

~216~

RQRQQ

RQCC

RQRQQ

RQCRQC

RRγ

γ

σ

in terms of L, b, θ, I/A

in terms of R, Q, γ

bLLR

2cos2sin4

+=

θθ

θcosLbQ =

( )( )22cos2sin

22

bLL

bLA

+

+=

θθγ

QQ 22~+=

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Previous research studies using the elongated Kelvin model set an artificial restriction on the unit cell shape,

This reduces the number of unit cell dimensions required to exercise the theory and apply the equations by one. The unit cell shape is now described by specifying only two dimensions.

However, this restriction on the unit cell geometry seems arbitrary and it reduces the generality of the model and limits its applicability to a narrower range of foams.

θcos2/ =Lb

b2

D

which is equivalent to assuming that bD 22=

2=Q

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0

1

2

3

4

5

6

7

8

1.0 1.2 1.4 1.6 1.8 2.0Aspect Ratio R

Stif

fnes

s R

atio

RE

Gupta et al. (1986), Polyisocyanurate

Hilyard (1982), Flexible Polyurethane

Huber & Gibson (1988), Flexible Polyurethane

Huber & Gibson (1988), Rigid Polyurethane

Mehta & Colombo (1976), PolystyreneGong et al. (2005a), Polyester-Urethane

BX-265 Rigid Polyurethane

NCFI 24-124 Rigid Polyurethane

Q = (2)0.5

Q = 1

Q = 2

1%

5%

10%

1%5%

10%

γ = 1%

5%

10%

Plateau Border Cross-section

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Micro-mechanicsModel

)( mD μ

2.90.0311.75142NCFI24-124

1.90.0311.42136BX-265

R γ ER

NCFI24-124BX-265

1.097.811

.373.422

.060.177

26.025.218.0*22.8

NA77.263.0*61.8

NA35.635.0*41.5

2.25 PL1.99 ult.

2.341.82 PL1.74 ult.

1.75

--0.7765--1.0747

MeasuredPredictedMeasuredPredicted

Q

σR

)( mL μ

)( mb μ

)( mr μ

xyυ

xzυ

zxυ

* Measurements from another block of BX-265.NA Data not available.

06.0355. ±

0007.0273. ±

14.0382. ±

04.0183. ±

10.0641. ±29.0536. ±17.0675. ±

Q

Db22 +

= ( )Q

RQbL

42216 22

++=

( )bLCRDr

8161

3

+=

γ

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Open Cell

Closed Cell

0.2300.120νxz

1.24%1.62%γ

1.7270.256νzx

0.0240.621νxy

Open cellClosed cell

mrmL

mbmD

R

μμ

μμ

2.206.90

24144

72.1

=====

Finite Element Analysis of an Elongated Kelvin Model

3.02

==

νGPaE

Unit cell dimensions

Solid Properties

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Summary and Conclusions

• Model was successful in predicting the measured strength ratios and edge cross-section radii using measured average cell height, cell width, relative density and stiffness ratios as input.

• Prediction of Poisson’s ratios was not as successful.• Shuttle foams have a microstructure such that , so a micromechanics model

derived from a general elongated Kelvin unit cell is needed.• Future work should include performing finite element analysis of BX-265 and

NCFI24-124 unit cells with faces included and predict Poisson’s ratios.

2≠Q

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Backup Slides

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Finite Element Analysis


Foam Elastic Constants

Average Foam Stresses

Description of Foam Microstructure

Edge (Strut) Stresses and Failure Initiation

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Intertank Flanges

Manual Spray Close-outs of BX-265

Protrusion Air Load (PAL) Ramp

Redesign eliminated the LOxand LH2 PAL Ramps

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Bipod Fitting and Ramp Closeout

Current design

BX-265

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Ice Frost Ramps

Cover the pressurization line and cable tray support fittings to prevent ice formation.

PDL-1034

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Structural Analysis of the Foam Applications under launch loads and environmentsis performed by Finite Element Analysis

Finite element analysis treats the foam material as a homogeneous, temperature dependant and orthotropic material• elastic constants and strengths obtained during material characterization testing• the effect of a varying microstructure on the foam structural integrity is neglected

Analyses are not used for flight qualification, but as a tool to study possible foam shedding mechanisms and to guide proposed design changes.

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