1

Analysis of TPS Sandwich Panel with Foam Core

Huadong Zhu1 and Bahavani V. Sankar2 Department of Mechanical & Aerospace Engineering, University of Florida, Gainesville, FL 32611-6250

The method of Fourier analysis is combined with the Galerkin Method for solving the two-dimensional elasticity equations for a sandwich thermal protection system (TPS) insulation panel with foam core subjected to transverse loads. The variation of the Young’s modulus through the thickness is given by a polynomial in the thickness coordinate and the Poisson’s ratio is assumed to be constant. The Fourier series method is used to reduce the partial differential equations to a pair of ordinary differential equations, which are solved using the Galerkin method. The method will be useful is analyzing functionally graded TPS structures with arbitrary variation of thermo-mechanical properties in the thickness direction. The analysis was also performed by using one-dimensional laminated plate theory. Significant differences were found in the results suggesting that the laminate theory may not be suitable for the analysis of thick TPS.

Nomenclature b = displacement coefficient C = constitutive matrix c = displacement coefficient Ef = Young’s modulus of foam material h = thickness L = length p = surface traction S = single layer stiff matrix t = surface traction φ = base function ρ = solidity θ = temperature field

I. Introduction hermal protection systems (TPS) on reusable launch vehicles have to be designed such that the maximum temperature and stresses of the reusable launch vehicles (RLV) tank structure is kept below a specified safe

limit. A significant component of the total vehicle weight in addition to structural weight of the tanks and vehicle is the weight of the thermal protection system. Sandwich structure can offer high stiffness with relatively much weight savings compared with widely used laminated structures. A recent study of TPS modeling and performance issues performed by Blosser1,2 has shown that currently existing Saffil-foam-filled TPS tile weighs 1.2 5.85 to 19.3 Kg/m2 (1.2 to 3.96 lb/ft2), while the structural weight ranges from 4.64 to 8.54 Kg/m2 (0.95 to 1.75 lb/ft2). The volume enclosed is filled with the Saffil (fibrous foam type) insulation. Since the Saffil insulation is flexible and cannot be attached directly, it needs encapsulation in a foil and the secondary TPS support structure. Metallic foams, which have low density, relatively low conductivity and some load-bearing ability, seem to be a promising candidate for insulation material besides Saffil insulation, if their relative density profiles are designed appropriately.

Functionally graded foams can improve the performance of the insulation. For steady state heat flow, Venkataraman et al.3 optimized the density profile of the foam in order to minimize the transmitted heat for a given mass or minimize the thickness for a specified transmitted heat. In a second paper Zhu et al.4 minimized the TPS 1 Graduate student, Department of Mechanical & Aerospace Engineering, P.O. Box 116250, University of Florida, AIAA Student Member 2 Professor, Department of Mechanical & Aerospace Engineering, P.O. Box 116250, University of Florida, AIAA Associate Fellow

T

45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference19 - 22 April 2004, Palm Springs, California

AIAA 2004-1935

Copyright © 2004 by ZHU. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

2

mass for specified heat transfer. Later a transient problem was solved in which the maximum structure temperature of a two-layer insulation was minimized by varying the solidity profile for a given total thickness and mass5. The two layers were assumed homogeneous with density being constant in each layer. It is shown that the cooler inner layer of optimal design has high density, while the hotter outer layer has low density. This is in contrast to the steady-sate optimum, where the density profile is the reverse.

The objective of this paper is to develop a thermal-structural analysis of a sandwich TPS panel subjected to transient heat conduction during reentry. The material properties such as Young’s modulus, yield strength, thermal conductivity and coefficient of thermal expansion are assumed to be a function of temperature. Hence approximate methods such as finite difference for heat conduction, and Galerkin method for structural analysis have to be used. The sandwich panel is divided into four elements (two face sheets and two layers of core with different densities). An elasticity solution is derived for each layer based on the procedures developed by Sankar et al.6,7. The solution for strains and stresses in the entire sandwich panel is obtained by enforcing the compatibility of tractions and displacements at the interfaces. The Fourier series method is used to reduce the partial differential equations to a pair of ordinary differential equations, which are solved using the Galerkin method. For the purpose of comparison, the panel is also analyzed by one dimensional plate theory.

II. Analysis In this section we identify the necessary information and modeling details required to perform structural analysis

of the TPS insulation. The heat transfer in the TPS is assumed to be one-dimensional. The finite width effects of the TPS insulation and the heat shorts resulting from the support structure around the perimeter of the TPS tile are ignored. The structural mass on the inside will correspond to the mass of the stiffened panel shell used for the RLV tank construction. The insulation itself is made of a titanium open cell foam material.

The heat transfer in foams proceeds by three modes: conduction through the solid materials, conduction in the gas filling this foam and radiation inside the foam. The model used to calculate the heat transfer coefficient in the foam is discussed in Venkataraman et al. [3]. To minimize radiation we require higher density foams (smaller foam pore sizes) while to minimize conduction we need low density foams (large foam pore sizes). Since there is a temperature gradient through the insulation, an optimum insulation requires different densities in different regions. Optimum density profiles of metallic foam insulations that minimizes heat transmitted to the inside under transient steady-state heat transfer conditions are presented in Venkataraman et al.3.

Figure 1 shows a schematic of the simplified sandwich TPS panel. The length of the panel is 0.4572 m (18 inch). The top facesheet is made of homogeneous titanium plate with thickness 0.5 mm. The foam core is divided into two layers with total thickness 90 mm (3.54 inch). The solidity in each layer of foam core can vary continuously. The foam is idealized as having rectangular cells of uniform size. The variation in solidity is achieved by tailoring the cell size while keeping the strut diameter (ds) fixed at 0.05 mm (0.002 inch). The structural mass on the inside will correspond to the mass of the stiffened panel shell used for the RLV tank construction which is made of aluminum with thickness 2.2 mm (about 0.0866 inch). The areal density of structure is 6.1 Kg/m2 (1.25 lb/ft2). The temperature dependent material properties of titanium and aluminum are shown in Appendix.

Figure 1: Schematic of the heat transfer in the insulation on a reusable launch vehicle structure

III. Structural Analysis of the TPS Panel The TPS panel is assumed to be simply supported and in a state of plane strain normal to the z-x plane (one

dimensional plate). The governing equations are formulated for each layer. The solutions of strains and stresses of entire sandwich are obtained by enforcing the compatibility displacements and continuity of tractions at the

z

x

Structure

Top half core

Top face sheet

Bottom half core

q Qrad

3

interface, which is analogous to assembling element stiffness matrices in FEM analysis. We will derive the elasticity solution for a typical layer first.

A. Analysis of a Single Layer

Figure 2: FG Beam subjected to symmetric loading

Consider a functionally graded (FG) one-dimensional plate of height h and length L as shown in Fig. 2. The

boundary conditions at the two edges of the plate, x=0 and x=L, are similar to that of a simply supported plate. We assume that the FGM is isotropic at every point and the Poisson’s ratio v is a constant through the thickness. The variation of Young’s modulus E in the thickness direction is given by a polynomial in z as:

+

+

+=

3

4

2

3210)(hza

hza

hzaaEzE (1)

There are normal and shear tractions applied on both top and bottom surfaces. The tractions are assumed to be symmetric about the centerline (x=L/2):

K531

22

22

11

11

,,LL

nxcosTtxsinPp

xcosTtxsinPp

==

====

πξ

ξξξξ

(2)

The temperature distribution θ(x,z) at bottom surface is also assumed symmetric about the centerline (x=L/2). The temperature variation in the thickness direction follows same pattern for each x, which can be expressed in polynomial as:

KK 321531

)(1

1

1

,,,,nL

n

zdxsinTz,xn

n

===

⋅

= ∑∑

∞

=

−∞

=

λπξ

ξθλ

λλ

(3)

We will solve the problem for one value of ξ. Let

x

z p1, t1

p2, t2

4

xsinzTdz,x nn ξθ λλ

λλ

14

1)( −

=∑= (4)

Assume the displacements are in the form of:

xsinzWz,xwxcoszUz,xuξξ

)()()()(

==

(5)

The strains are derived as:

xcosW'Uxsin'W

xsinU

xz

zz

xx

ξξγξεξξε

)( +==−=

(6)

Assuming that the principal material directions coincide with the x and z axes, the constitutive equations are:

−

=

0)()(

)(00

00

55

3313

1311

zz

z,xc

cccc

z

x

xz

zz

xx

xz

zz

xx

αα

θγεε

τσσ

(7)

where [C] can be expressed as:

−

−

=−

13

3311

13

11

13

11

1

100

01

01

G

EE

EE

Cν

ν

(8)

Equation (6) can be rewrite as:

xsinzTdzz

ccccc

nz

x

xz

zz

xx

xz

zz

xx

ξββ

γεε

τσσ

λλ

λ

14

155

3313

1311

0)()(

0000

−

=∑⋅

−

=

(9)

where the β’s can be expressed as:

=

0)()(

0000

0 55

3313

1311

zz

ccccc

z

x

z

x

αα

ββ

(10)

Substitute the Eq. (6) into Eq. (9), we have,

5

xcosZTxsinzSxsinzS

zxz

zzz

xxx

ξτξσξσ

)()()(

===

(11)

where

( )( )

( )ξβξ

βξλ

λ

λλ

W'UcTzzTd'WcUcS

zzTd'WcUcS

z

znz

xnx

+=−+−=

−+−=−

−

55

13313

11311

)(

)(

(12)

The governing equilibrium equations are (body forces are neglected):

0

0

=∂∂

+∂∂

=∂∂

+∂∂

zx

zxzzxz

xzxx

στ

τσ

(13)

Substituting Eq. (11) into Eq. (13), we obtain,

0)()(0)()(

=−=+

zTz'Sz'TzS

zz

zx

ξξ

(14)

Equation (14) are solved by Galerkin method by assuming:

4,1)()(

)()(

==

=

jzbzW

zczU

jj

jj

φ

φ (15)

where,

34

2321 )(;)(;)(;1)( zzzzzzz ==== φφφφ (16)

Substituting the approximate solution in the governing differential equations, we obtain the residuals. The residuals are minimized by equating their weighted averages to zero:

( )

( ) 410)(

410)(

0

0

,i,dzzTS

,i,dzz'TSh

iz'

z

hizx

==−

==+

∫

∫φξ

φξ (17)

Using integration by parts we can rewrite Eq. (17) as:

6

( ) 4,10)0()0()()(

4,10)0()0()()(

0 0

'

0 0

'

==−−+

==−−+

∫ ∫

∫ ∫

iShhSdzTdzS

idzTThhTdzS

h h

iziziziz

h h

izizizxi

φφξφφ

φφφξφ (18)

Substituting for Sx(z), Sz(z) and Tz(z) from Eq. (12) into Eq. (18) and using the approximate solution for U(z) and W(z) in Eq. (15) we obtain:

=

)2(

)1(

)4()3(

)2()1(

i

i

ijij

ijij

ff

cb

KKKK

(19)

where,

∫∑

∫∑

∫∫

∫∫

∫∫

∫∫

−

=

−

=

−−=

+−=

−=

−−=

−−=

−=

h

iznzizi)(

i

h

ixnzizi)(

i

h'jij

'i

h)(

ij

'j

'i

h

ji

h)(

ij

h

ji'j

h'i

)(ij

j

h'i

'j

h

i)(

ij

dz'zzTdhShSf

dzzzTdhThTf

dzcdzcK

dzcdzcK

dzcdzcK

dzcdzcK

0

14

1

2

0

14

1

1

055

013

4

033

055

23

011

2

055

2

055

013

1

)()()()0()0(

)()()()0()0(

φβφφ

φβξφφ

φφξφφξ

φφφφξ

φφξφφ

φφξφφξ

λλ

λ

λλ

λ

(20)

)ccccbbbb(cb

DT

43214321=

= (21)

The boundary conditions at both top and bottom surface are shown as following: At top surface

xsinPp

xcosTtξξ

11

11

==

(22)

Hence we have

7

1

1

)()(

ThTPhS

z

z

==

(23)

At bottom surface

xsinPp

xcosTtξξ

22

22

==

(24)

which yields

2

2

)0()0(

TTPS

z

z

−=−=

(25)

Substituting Equations (23) and (25) into Equation (20),[f] in Equation (20) can be rewritten as:

−

−

−

−+

⋅

−=

∫

∫

∫

∫

∫

∫

∫

∫

−

−

−

−

−

−

−

−

h

z

h

z

h

z

h

z

h

x

h

x

h

x

h

x

n

dz'zz

dz'zz

dz'zz

dz'zz

dzzz

dzzz

dzzz

dzzz

Td

PTPT

hhhh

hhhh

ffffffff

04

1

03

1

02

1

01

1

04

1

03

1

02

1

01

1

2

2

1

1

44

33

22

11

44

33

22

11

)2(4

)2(3

)2(2

)2(1

)1(4

)1(3

)1(2

)1(1

)(

)(

)(

)(

)(

)(

)(

)(

)0(0)(0)0(0)(0)0(0)(0)0(0)(0

0)0(0)(0)0(0)(0)0(0)(0)0(0)(

φβ

φβ

φβ

φβ

φβξ

φβξ

φβξ

φβξ

φφφφφφφφ

φφφφφφφφ

λ

λ

λ

λ

λ

λ

λ

λ

λ

(26)

or,

MPf +Φ= (27)

Equation (15) can be expanded as:

8

=

4

3

2

1

4

3

2

1

32

32

2

2

1

1

000000010001000000001

10000

ccccbbbb

hhhhhh

WUWU

(28)

or,

HDU = (29)

Now we have three Equations (19), (27) and (29). From these three equations, we obtain,

[ ] ** MPKMPHKU +=+Φ= −1 (30)

or,

RSUP −= (31)

where

[ ][ ] **

*

MKR

KS1

1

−

−

=

= (32)

B. Analysis of the Whole Panel Assume displacements are in the form:

K53151 ,,L,iL

nxcosTtxsinPp

ii

ii

===

==

πξ

ξξ

(33)

For given temperature distribution in each layer:

)i()i()i()i( RUSP −= (34)

In order to satisfy equilibrium, the contribution of the different tractions at each interface should sum to zero. Enforcing the balance of force and displacement at the interface, we got global stiff matrix SG:

9

=

−

−

−

−

⋅

5

5

1

1

)4(4

)4(3

)4(2

)4(1

)3(4

)3(3

)3(2

)3(1

)2(4

)23

)2(2

)2(1

)1(4

)1(3

)1(2

)1(1

5

5

4

4

3

3

2

2

1

1

000000

000000

00

0000

0000

00

000000

PT

PT

RRRR

RRRR

RRRR

RRRR

WUWUWUWUWU

S(

G (35)

Solving the above Eq. (35), we can obtain the displacement field for each layer. The displacement field and the constitutive equations can be used to obtain the stress filed in each layer.

Sankar7 derived the one dimensional plate theory following the Euler-Bernoulli beam assumption. He also neglected the normal stresses in thickness direction σzz. We followed a similar procedure to derive a sandwich beam theory for the composite sandwich panel.

IV. Numerical Results The analysis procedures described in the preceding section was used to calculate the stresses in a TPS panel

during the reentry of the vehicle. Based on our previous work of minimizing the maximum temperature of a structure4, we found that the cooler inner layer of the optimal two-layer design has high solidity, while the hotter outer layer has low solidity. So we vary the solidity of foam linearly from 0.11 at bottom to 0.01 at top surface. It should be noted that the elastic constants of metal foams depend on the relative density. Choi8 derived an expression for the Young’s modulus of a cellular solid as:

sf EE ρ31

= (36)

where ρ is the solidity and the subscripts f and s, respectively, refer to the homogenized foam and the strut material. In our study, the heat flux and pressure on the vehicle during reentry are obtained from Blosser et al.1,2. The heat flux varies significantly over the surface. For our study we choose a location on the windward surface referred to as station 1199 (STA 1199) as a representative point for the point design. The heat flux and pressure histories for that location are reproduced in Appendix A.

The stress variations with respect to time at various locations of the TPS are presented in Figs. 3 through 6. It can be noted that both laminate and Galerkin solutions agree well with each other, which means laminate theory can provide relatively accurate results for slender beam (L/H=4.9). We also compared the two solutions for beam with length L=225 mm (9 in.) (L/H=2.46). The results are shown in Figures 7-10. The maximum difference between theses two solution for stresses in structure is 18%, which suggests that the laminate theory may not be suitable for the analysis of thick TPS.

10

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-14

-12

-10

-8

-6

-4

-2

0

2x 107

Time (S)

σxx

(Pa)

σxx at top surface by laminate solutionσxx at bottom surface by laminate solutionσxx at top surface by Galerkin methodσxx at bottom surface by Galerkin method

Figure 3 History of normal stress (σxx) distribution in top facesheet (L=18 inch)

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5x 106

Time (S)

σxx

(Pa)

σxx at top surface by laminate solutionσxx at bottom surface by laminate solutionσxx at top surface by Galerkin methodσxx at bottom surface by Galerkin method

Figure 4 History of normal stress (σxx) distribution in top half core (L=18 inch)

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-2

-1

0

1

2

3

4

5

6

7

8x 106

Time (S)

σxx

(Pa) σxx at top surface by laminate solution

σxx at bottom surface by laminate solutionσxx at top surface by Galerkin methodσxx at bottom surface by Galerkin method

Figure 5 History of normal stress (σxx) distribution in bottom half core (L=18 inch)

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-6

-5

-4

-3

-2

-1

0

1x 107

Time (S)

σxx

(Pa)

σxx at top surface by laminate solutionσxx at bottom surface by laminate solutionσxx at top surface by Galerkin methodσxx at bottom surface by Galerkin method

Figure 6 History of normal stress (σxx) distribution in structure (L=18 inch)

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-14

-12

-10

-8

-6

-4

-2

0

2

4x 107

Time (S)

σxx

(Pa)

σxx at top surface by laminate solutionσxx at bottom surface by laminate solutionσxx at top surface by Galerkin methodσxx at bottom surface by Galerkin method

Figure 7 History of normal stress (σxx) distribution in top facesheet (L=9 inch)

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5x 106

Time (S)

σxx

(Pa)

σxx at top surface by laminate solutionσxx at bottom surface by laminate solutionσxx at top surface by Galerkin methodσxx at bottom surface by Galerkin method

Figure 8 History of normal stress (σxx) distribution in top half core (L=9 inch)

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-2

-1

0

1

2

3

4

5

6

7

8x 106

Time (S)

σxx

(Pa) σxx at top surface by laminate solution

σxx at bottom surface by laminate solutionσxx at top surface by Galerkin methodσxx at bottom surface by Galerkin method

Figure 9 History of normal stress (σxx) distribution in bottom half core (L=9 inch)

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-6

-5

-4

-3

-2

-1

0

1x 107

Time (S)

σxx

(Pa)

σxx at top surface by laminate solutionσxx at bottom surface by laminate solutionσxx at top surface by Galerkin methodσxx at bottom surface by Galerkin method

Figure 10 History of normal stress (σxx) distribution in structure (L=9 inch)

11

We also investigate the performance of functionally graded foam insulation for transient heat transfer condition by comparing it with the uniform solidity foam insulation. For the purpose of comparison, the problem of a uniform foam with same weight is also solved. Figures 11 and 12 show the temperature histories at top facesheet and structure, respectively. It is shown in Figure 12 that maximum structural temperature can be reduced by 30 K by linearly varying the insulation foam solidity. This is achieved by increasing the top facesheet temperature faster for linear design compared with uniform design, as shown in Figure 11. The histories of stress ratio, which is defined as the ratio between the stresses in each layer and temperature-dependent yield stress, are also plotted for each layer in Figures 13 through 16. These four plots indicate that the maximum stress ratio in the base structure and foam core can be reduced 25% or more by linearly varying the foam solidity with only 5% increase in stress ratio in the top face sheet.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000300

400

500

600

700

800

900

Time (s)

Top

face

shee

t tem

pera

ture

(K)

Uniform designLinear design

TLmax=850.7 K

TUmax=848.3 K

Figure 11 Temperature history at top facesheet for both linear and uniform designs

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000300

320

340

360

380

400

420

440

460

480

500

Time (s)

Bot

tom

face

shee

t tem

pera

ture

(K)

Uniform designLinear design

TLmax=456.6 K

TUmax=487.5K

Figure 12 Structural temperature history for both linear and uniform designs

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

Time (S)

Stre

ss ra

tio in

top

face

shee

t

stress ratio at top surface of linear designstress ratio at bottom surface of linear designstress ratio at top surface of uniform designstress ratio at bottom surface of uniform design

Figure 13 History of normal stress (σxx) distribution in top face sheet (L=18 inch)

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Time (S)

Stre

ss ra

tio in

top

half

core

stress ratio at top surface of linear designstress ratio at bottom surface of linear designstress ratio at top surface of uniform designstress ratio at bottom surface of uniform design

Figure 14 History of normal stress (σxx) distribution in top half core (L=18 inch)

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Time (S)

Stre

ss ra

tio b

otto

m h

alf c

ore

stress ratio at top surface of linear designstress ratio at bottom surface of linear designstress ratio at top surface of uniform designstress ratio at bottom surface of uniform design

Figure 15 History of normal stress (σxx) distribution in bottom half core (L=18 inch)

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

Time (S)

Stre

ss ra

tio in

stru

ctur

e

stress ratio at top surface of linear designstress ratio at bottom surface of linear designstress ratio at top surface of uniform designstress ratio at bottom surface of uniform design

Figure 16 History of normal stress (σxx) distribution in structure (L=18 inch)

12

V. Conclusions and Future Work The analysis procedures developed in this paper will be useful in optimizing the multilayer TPS

structure subjected to temperature and stress constraints. The results suggest that classical lamination theory may not be suitable for the analysis of thick TPS and more accurate plate theories, e.g., higher order theories or sandwich plate theories, need to be developed. One of the major issues in the TPS for RLVs is that the temperature attains the maximum after the vehicle has landed when the mechanical loads are minimum or almost do not exist. On the other hand during the beginning phase of the reentry the mechanical loads are high but the thermal loads are low. This situation offers an opportunity to optimize the TPS and minimize the mass. This problem will be considered in the future.

Appendix

300 400 500 600 700 800 90070

75

80

85

90

95

100

105

110

115

120

Temperature (K)

You

ng's

mod

ulus

of T

i (G

Pa)

Figure 17 Young’s modulus of titanium

300 400 500 600 700 800 900200

300

400

500

600

700

800

900

1000

Temperature (K)

Yie

ld s

tress

of T

i (M

Pa)

Figure 18 Yield stress of titanium

300 400 500 600 700 800 9009.2

9.4

9.6

9.8

10

10.2

10.4x 10-6

Temperature (K)

CTE

of T

i (K-1

)

Figure 19 CTE of titanium

300 400 500 600 700 800 900550

600

650

700

Temperature (K)

Spe

cific

hea

t of T

i (J/

Kg-

K)

Figure 20 Specific heat of titanium

300 350 400 450 500 55054

56

58

60

62

64

66

68

70

72

74

Temperature (K)

You

ng's

mod

ulud

of A

l (G

Pa)

Figure 21 Young’s modulus of aluminum

300 350 400 450 500 550 6000

50

100

150

200

250

300

350

Temperature (K)

Yie

ld s

tress

of A

lum

inum

(MP

a)

Figure 22 Yield stress of aluminum

13

300 350 400 450 500 550 600 650 7002.25

2.3

2.35

2.4

2.45

2.5

2.55

2.6

2.65x 10-5

Temperature (K)

CTE

of A

l (K-1

)

Figure 23 CTE of aluminum

300 350 400 450 500 550 600 650 700800

850

900

950

1000

1050

1100

1150

Temperature (K)

Spe

cific

hea

t of A

l

Figure 24 Specific heat of aluminum

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.5

1

1.5

2

2.5

3

3.5x 104

Time (s)

App

lied

heat

flux

(J/m

2 -s)

Heat flux history at spot 1199

Figure 25 Re-entry heat flux history at windward spot 1199

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

2

4

6

8

10

12x 104

Time (s)

Pre

ssur

e (P

a)

Pressure history for spot 1199

Figure 26 Re-entry pressure history at windward spot 1199

Acknowledgments This work was funded by the Metals and Thermal Structures Branch at NASA Langley Research Center

through grant NAG-1-01046 to the University of Florida. Partial support of URETI sponsored by NASA under NCC3-994, managed by NASA Glenn Research Center, is gratefully acknowledged.

References

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