application of an elongated kelvin model to space · pdf fileapplication of an elongated...
Post on 15-Mar-2018
242 Views
Preview:
TRANSCRIPT
1
National Aeronautics and Space Administration
www.nasa.gov
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
2
National Aeronautics and Space Administration
www.nasa.gov
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
3
National Aeronautics and Space Administration
www.nasa.gov
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
4
National Aeronautics and Space Administration
www.nasa.gov
Typical Flight Loads on ET Foam Applications
Aerodynamic Heating
Aerodynamic Shear
Cryogenic Temperatures(-423F/-297F)
Substrate Flexure and Membrane Forces
Tank WallFoam Insulation
5
National Aeronautics and Space Administration
www.nasa.gov
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
6
National Aeronautics and Space Administration
www.nasa.gov
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
7
National Aeronautics and Space Administration
www.nasa.gov
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:
8
National Aeronautics and Space Administration
www.nasa.gov
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
9
National Aeronautics and Space Administration
www.nasa.gov
~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).
10
National Aeronautics and Space Administration
www.nasa.gov
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
11
National Aeronautics and Space Administration
www.nasa.gov
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
12
National Aeronautics and Space Administration
www.nasa.gov
[ ]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
13
National Aeronautics and Space Administration
www.nasa.gov
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.
14
National Aeronautics and Space Administration
www.nasa.gov
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
15
National Aeronautics and Space Administration
www.nasa.gov
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
16
National Aeronautics and Space Administration
www.nasa.gov
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.
17
National Aeronautics and Space Administration
www.nasa.gov
AQQ = BQQ =
BA QQ >
and the shape parameter
The shape is defined by the aspect ratioDHR =
θcosLbQ =
BA RR >
ARR = BRR =
18
National Aeronautics and Space Administration
www.nasa.gov
( ) ( )
⎥⎦⎤
⎢⎣⎡ +
⎥⎦⎤
⎢⎣⎡ +++
⎟⎟⎠
⎞⎜⎜⎝
⎛
+=
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~+=
19
National Aeronautics and Space Administration
www.nasa.gov
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
20
National Aeronautics and Space Administration
www.nasa.gov
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
21
National Aeronautics and Space Administration
www.nasa.gov
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
22
National Aeronautics and Space Administration
www.nasa.gov
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
23
National Aeronautics and Space Administration
www.nasa.gov
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
24
National Aeronautics and Space Administration
www.nasa.gov
Backup Slides
25
National Aeronautics and Space Administration
www.nasa.gov
Micro-mechanicsModel
Finite Element Analysis
Micro-mechanicsModel
Foam Elastic Constants
Average Foam Stresses
Description of Foam Microstructure
Edge (Strut) Stresses and Failure Initiation
26
National Aeronautics and Space Administration
www.nasa.gov
Intertank Flanges
Manual Spray Close-outs of BX-265
Protrusion Air Load (PAL) Ramp
Redesign eliminated the LOxand LH2 PAL Ramps
27
National Aeronautics and Space Administration
www.nasa.gov
Bipod Fitting and Ramp Closeout
Current design
BX-265
28
National Aeronautics and Space Administration
www.nasa.gov
Ice Frost Ramps
Cover the pressurization line and cable tray support fittings to prevent ice formation.
PDL-1034
29
National Aeronautics and Space Administration
www.nasa.gov
30
National Aeronautics and Space Administration
www.nasa.gov
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.
top related