Z. Hashin1

Professor, Deparlment of Solid Mechanics,

Materials, and Structures, School of Engineering,

Tel Aviv University, Tel Aviv, Israel

Mem. ASME

Analysis of Properties of Fiber Composites With Anisotropic Constituents Expressions and bounds for the five effective elastic moduli of a unidirectional fiber com-posite, consisting of transversely isotropic fibers and matrix, are derived on the basis of analogies between isotropic and transversely isotropic elasticity equations. Application of results for determination of the five elastic moduli of graphite fibers is discussed. Effec-tive thermal expansion coefficients are derived on the basis of a general theorem. Effec-tive conductivities, dielectric constants, and magnetic permeabilities are derived by use of certain mathematical analogies.

Introduction

Most of the analytical work on the subject of computation of the effective properties of fiber-reinforced materials in terms of constit-uent properties and internal geometry has been concerned with the case of isotropic phases. Assessment of the effects of fiber an-isotropy has become of considerable importance because of the widespread use of carbon and graphite fibers which are highly an-isotropic.

Polymeric matrix such as epoxy or metal matrix such as aluminum can be considered isotropic. However, in carbon-carbon composites which consist of carbon fibers in carbon matrix, the matrix can also be significantly anisotropic.

The present work is concerned with the relatively simple case of a uniaxially reinforced material in which fibers and matrix are transversely isotropic, the axis of transverse isotropy being in fiber direction. Properties to be considered are elasticity, thermal expan-sion, thermal and electrical conduction, dielectrics, and magnetics.

Certain aspects of the problem have already been considered in the literature. Hill [1] has developed general relations among some of the effective elastic moduli of a two-phase fiber composite when the

Fig. 1 Unidirectional fiber composite

1 Also, Adjunct Professor, Department of Metallurgy and Materials Science,

College of Engineering, University of Pennsylvania, Philadelphia, Pa. 19104. Contributed by the Applied Mechanics Division for presentation at the

Winter Annual Meeting, New York, N. Y., December 2-7, 1979, of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS.

Discussion on this paper should be addressed to the Editorial Department, ASME, United Engineering Center, 345 East 47th Street, New York, N. Y. 10017, and will be accepted until December 1,1979. Readers who need more time to prepare a Discussion should request an extension of the deadline from the Editorial Department. Manuscript received by ASME Applied Mechanics Division, July, 1978; final revision, January, 1979. Paper No. 79-WA/APM-6.

phases are transversely isotropic and has obtained general bounds for the axial Young's modulus, the axial Poisson's ratio, and the transverse bulk modulus for that case. Whitney [2] and Chen and Cheng [3] approached the effective elastic properties problem on the basis of certain assumptions. This will be discussed later on. Rosen and Hashin [4] have generalized Levin's [5] method of computation of effective thermal expansion coefficients to the case of anisotropic phases, thus including the present material symmetry as a special case.

Journal of Applied Mechanics SEPTEMBER 1979, VOL. 46 / 543 Copyright 1979 by ASME

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In the present work the problem is treated by establishment of mathematical analogies between uniaxially reinforced materials with isotropic and transversely isotropic phases, respectively. Most of this work has been previously reported in [6]. Certain effective elastic properties expressions derived in [6] have also been independently given by Behrens [7].

Elas t i c P r o p e r t i e s A long fiber-reinforced cylinder, Pig. 1, consists of transversely

isotropic phases whose elastic stress-strain relations are written in the form

e n = ne ii + K22 + 33) 022 = It 11 + (k + Gr)t22 + (k - GT)(SS

033 = leu + (k - GT)e22 + (k + GT)(33

a + GT1(u^ + uftbrif, = (k2 - GT2)u^na + G T 2 ( u 5 + uffljnp (14)

On the external boundary

V/3 (16)"

where 6,, is given by (8). Because of (8), (7) assumes the form

011 = 2* (22 + 33)

0a/3 = (k* - GT)eyySafi + 2G'Tea 013 = 023 = 0

(16)

In order to compute the effective elastic moduli k* and G*T it is necessary to find the displacements ua as defined by the boundary-value problem (12)-(15), to compute the phase stresses from (11) and then to compute the average stress tensor aap. Then k* and GT are determined by (16).

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In the case of isotropic phases the previous formulation remains mathematically identical. The transverse modulus k becomes the plane strain bulk modulus and the transverse shear modulus becomes the isotropic shear modulus, k is now related to other isotropic moduli by

X + G = ^ _ = - ^ (17) 1 - 2c 3G - E

It follows that

1 All analytical and numerical results for k* and G\- for isotropic phases transform into corresponding results for the case of trans-versely isotropic phases by replacement of isotropic k and G phase moduli in the former by transversely isotropic k and GT phase moduli

2 All bounds for k* and G,T for isotropic phases transform into corresponding bounds for the case of transversely isotropic phases by the same replacement scheme.

If results for isotropic phases are expressed in terms of moduli other than k and G, then those moduli must first be expressed in terms of k and G and the replacement is then carried out. Thus the replace-ment scheme for phase moduli can be summarized as follows:

Isotropic Phases

k = \ + G G X

F.

Transversely Isotropic Phases

k Gf

k-GT (3k - GT)GT (18)

-(l-GT/k)

It should be carefully noted that this analogy is valid only for the moduli k* and G'T and for the internal fields u, eap, and

k* felfe + GTl)Vl + kj(ki + GTl)l)2 (ki + GTi)ui + (ki + GTI )U 2

k,+- V2 1 .+. ^

(31)

k2 ki ki + GTI c p _i_ p 4(y,42 - vAi)2viV2 E'A = EA\Ui + EA2v2 + -; (32)

Ul/2 + U2/l + 1/Gri

, (f/12 ~ P/tlXl/fel ~ l/k2)viV2 .._. ^A= "Al" l+ K/12U2+ " : " 7TZ (33) v1/k2 + v2/ki+ 1/Gn

, _ GAIVX + GA2(1 + V2> , "2 G A I ( 1 + "2) + GA2vi 1 Vl

where if

Gj-(-) - Gri +

GT-(-) < GJ- < GT(+)

GT2 > G T I ; k2 > fei

u2

G/12 - GAi 2GAI

(34) (35)

^ fei + 2GT i + - ' - :" i

G% T(+) '

Gr2 G-pi 2Gn(/zi + G T I ) (1 + ft)^2

1 + -

p - i>2 1 H 5 ' a u i + 1

ft ~ 7182, _ 7 + ft 1 + 7/82' ^ " 7 - 1 *1 o &2

(35a)

(356)

fei + 2GTi fe2 + 2GT 2 GT2

G 71 and if

GT2 < Gri ; &2 < k\

(1 + j8i)u2 1 + -p~v2 1 +

i - ft.

T(+) = G n + "2

, fei + 2GT 1 + r r : - : - .-"i

(35c)

(35d)

GTI Gn 2 G J I ( A I + G n ) (but see the Appendix for possible modification of bound (35c)). In all of these formulas, 1 indicates matrix and 2 indicates fibers.

The bounds (35a, c) were originally [8] defined in terms of solutions of 6 X 6 determinants. The present results are their algebraic equiv-alents. Further explanation of the nature of the bounds is given in the Appendix.

The moduli I* and n* can be obtained by use of the relations (2a, 6). Bounds on Ej- and V'T can be obtained in terms of (2c, d) and the bounds (35), as follows:

4k*G'TW E% T ( )

V T{) :

k* + m*GT()

k* - m*G*Tir) k* + OT*GJ.(T)

4k*p'A2 1 + -E\

(36)

(37)

(38)

The present results have been given in [6]. Behrens [7] has inde-pendently given the results (31), (34), (35a, c) and also the correct ; results for n* and /* of the composite cylinder assemblage model with transversely isotropic phases.

In the case of isotropic phases the composite cylinder assemblage results are in good agreement with experimental data, [6]. The upper bounds (35b)-(36) can be used to represent these moduli for the case of stiff fibers. It is to be expected that in the case of anisotropic fibers there will be equally good agreement.

Effective elastic moduli of uniaxial fiber composites have also been computed by numerical methods. In this approach the composite is represented as a square or hexagonal periodic array of identical cir- * cular fibers and the analysis is carried out by finite-difference or fi- ^

. nite-element methods, e.g., [10, 11]. Computer programs for such analyses can be directly used for transversely isotropic fibers and matrix. For computation ofk* and GT isotropic phase moduli values are replaced according to the scheme (18). Once k* is known, E*A and | v'A are determined by (19). Other moduli of interest are then defined J by (2)-(3). For shear modulus G*A isotropic phase shear moduli in the | computer program are replaced by phase axial shear moduli.

General bounds for the effective elastic moduli of uniaxial fiber composites with arbitrary transverse phase geometry have been given ; in [1,9]. These bounds are easily transferred to the case of transversely isotropic phases. Lower bounds for k*, E% v*A, and G*A are the ex-pressions (31)-(34). The corresponding upper bounds are obtained by interchanging the phase indices 1 and 2 in (31)-(34) provided that phase 2 moduli appearing in the bounds (but not necessarily Poisson's ratios) are larger than phase 1 moduli.

The general lower bound on G 7- is given by (35a) and the upper bound is obtained by interchange of 1 with 2. Again, phase 2 moduli must be larger than phase 1 moduli.

The bounds thus obtained are valid for any cylindrical phase ge-ometry, thus in particular for any fiber arrangement and fiber cross sections. The bounds for k*, E*A, v\, and GA are best possible in terms of phase volume fractions. The corresponding status of the G r bounds is not known.

Bounds for ET are given in terms of the general bounds, [6], by

E n)

4&() Gr(> fe() + m\) GT

Table 1 Bounds for elastic properties of graphite/epoxy; arbitrary transverse phase geometry Epoxy Matrix CI)

E = 3.45 x 10 3 MPA

G = 1.28 x 10 3 MPa

k - 4.26 x 10 3 MPa

- .35

G r a p h i t e

E = 345 x 1 0 3 MPa A

Em = 9 . 6 6 x 1 0 3 MPa T GT = 3 . 7 2 x 1 0 3 MPa

k = 6 . 9 0 x 1 0 J MPa

F l b e r s (2 ) GR - 2 . 0 7 x 1 0 3 MP

v . - . 2 0 A

v = . 3 0

V 2

0

. 2 0

. 4 0

. 6 0

. 8 0

1 .00

E A

3 . 4 5

7 1 . 7

1 4 0 . 0

2 0 8 . 3

2 7 6 . 6

345

V A ( - )

. 3 5

. 315

. 2 8 3

. 2 5 3

. 2 2 6

. 2 0

V A ( + )

. 3 5

. 3 1 8

. 2 8 7

. 2 5 7

. 2 2 8

. 2 0

k *

4 . 2 6

4 . 6 3

5 . 0 8

5 . 5 9

6 . 1 9

6 . 9 0

k * * < + >

4 . 2 6

4 . 6 8

5 . 1 4

5 . 6 6

6 . 2 3

6 . 9 0

G ? ( - > 1 .28

1 .53

1 .85

2 . 2 8

2 . 8 6

3 . 7 2

l T ( + )

1 .28

1 .57

1 .94

2 . 3 9

2 . 9 7

3 . 7 2

F * T ( - )

3 . 4 5

4 . 5 7

5 . 4 0

6 . 4 6

7 . 8 1

9 . 6 6

F * T ( + )

3 . 4 5

4 . 6 8

5 . 6 1

6 . 7 0

8 . 0 3

9 . 6 6

T ( - )

. 3 5

. 4 8 4

. 4 4 3

. 3 9 8

. 3 5 0

. 3 0

. 3 5

. 4 9 7

. 4 6 6

. 4 2 3

. 3 6 9

. 3 0

G*

1 .28

1 .40

1 .54

1 .70

1 .88

2 . 0 7

G A

1. 28

1 . 4 1

1 .55

1 . 7 1

1 .88

2 . 0 7

Moduli 10 MPa = 1 GPa = .145 x 10 psi Poisson's ratios - nondimensional

method is more reliable than the second. Smith [13] measured the elastic moduli of a series of graphite and carbon/epoxy composites ultrasonically and used elastic moduli expressions of Behrens [14] and Halpin and Tsai [15] for elastic moduli of the fiber composite. The expressions of [14] are based on long wavelength approximations for square or hexagonal arrays of identical circular fibers while [15] is semiempirical, obtained by curve fitting for isotropic fibers case. It is not quite clear how the isotropic fiber results have been transcribed to the case of anisotropic fibers. Dean and Turner [16] have exploited the composite cylinder assemblage results for isotropic fibers with a replacement scheme which does not appear correct. Kriz and Stin-chcomb [17] also used composite cylinder assemblage results but their expressions for transverse shear modulus and consequently also for transverse Young's modulus are incorrect. They assumed that v of isotropic fiber is replaced by VT of transversely isotropic fiber, but the correct replacement is as in (18).

In reference to the problem of determination of fiber anisotropic moduli it is of great importance to observe that carbon or graphite fiber moduli k, GT, GA are of the order of corresponding G, k - X + G of the epoxy. It follows that the general bounds for arbitrary transverse phase geometry just discussed are very close. Table 1 shows such bounds for typical graphite or carbon fiber moduli and epoxy matrix. It is seen that the bounds are extremely close and thus de-termine the effective moduli with great accuracy (E'A bounds are practically equal to the first two terms in (32)). Since the bounds are valid for any cylindrical phase geometry it follows that they determine the effective moduli of unidirectional carbon or graphite/epoxy composites for any random fiber arrangement and any fibers cross section shapes. Therefore, such composites provide a very accurate means of determination of fiber elastic moduli by simple use of the expressions (31)-(34), (35a), (36)-(38) with omission of the sub-scripts.

The results given here are also of significant importance for metal matrix fiber composites. Consider a composite consisting of fibers of properties given in Table 1 and aluminum matrix. If it were erro-neously assumed that the fiber is isotropic with (the easily measured) axial Young's modulus 345 GPa2 and Poisson's ratio 0.20 it would follow that the fiber has isotropic moduli Gi = 144 GPa, &2 = 239 GPa while corresponding values for aluminum are G\ = 26.7 GPa, k\ = 78.7 GPa. It would thus follow that the composite effective moduli k*,Gr, GA,E*T are larger than those of the aluminum. In reality, however, fiber elastic moduli are as in Table 1, thus considerably lower than

1 GPa = 0.145 X 106 psi.

GPa

60

50

10

30

20

0

rE' ALUMINUM

E,,=7I GPa

,

E,2=9.7 GPa GRAPHITE

i i

FIBER VOLUME FRACTION

Fig. 2 Bounds for effective transverse Young's modulus of unidirectional graphite/aluminum (CCA model)

aluminum moduli. Therefore the moduli k*, G"T, G\, and E*T are considerably lower than those of aluminum. It is thus seen that stif-fening is only provided in fiber direction while in transverse directions the fibers are of the nature of cylindrical cavities. Fig. 2 shows E*T bounds based on the composite cylinder assemblage model for such a composite, demonstrating the reduction of stiffness by the fibers.

Thermoelastic Expansion Coefficients A general method to compute effective thermal expansion coeffi-

cients of two phase composites in terms of effective elastic properties and phase properties has been given by Levin [5] and has been ex-tended and described in detail [4, 6].

In the general case of an anisotropic two phase composite with anisotropic constituents the results are

aij = ctij+ (a - aiV) Pkirs (S*nij - Srsij) (41a) Pkirs (Srfij - SrVij) = hlij (416)

where

= effective thermal expansion coefficients rsu - effective elastic compliances

0,(1,2) C ( l , 2 ) _ rsij

phase thermal expansion coefficients phase elastic compliances

hjki fourth rank symmetric unit tensor

Journal of Applied Mechanics SEPTEMBER 1979, VOL. 46 / 547

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and an overbar denotes average over the composite. In the present case of macroscopic and microscopic transverse

isotropy the effective property tensors entering into (41) have the following components

n = otA\

S' - - -Sim~E-A

SI122 = SI133 =

>1313

22 = 33 = T (42)

S222:

E'A

1 4Gi

=

S3333 =

" 2 2 3 3 :

S9323 =

Ex

4G*g (43)

all others vanish. The components of phase expansion coefficients and compliances

are analogous. Here A and T denote fiber and transverse to fiber di-rections, respectively.

It follows from (41) that the two expansion coefficients are given by

aA = aA + (afl ~ ct$) Pkirs (Srm - Srsn)

a'T = a-r + (a{,2/ - (44) where Pkirs must be determined from (41&) by inversion of the transversely isotropic matrix Sr^}j Sr]}j.

Note that the nonvanishing components of S*rsij are given by (43) which also define in analogous fashion the nonvanishing components of the phase compliance tensor difference Sj^lj Sr]]j. The nonvan-ishing components of the phase thermal expansion tensors are as in (42). It may be shown that only the effective properties E'A, k*, and v'A enter into (44). It follows that (44) gives exact closed form ex-pressions for the CCA model.

When the phases are isotropic (44) reduces to the known results

aT = a +

a2 - ct\ 1/K2 ~ 1/Ki

2 - OL\

1/K2 - 1/tf! 3

2k*

(1 - 2v'A) 1 EA K

3(1 - 2v'A) vA E'A

1 K

(45)

where K is three-dimensional bulk modulus. Expressions (44) can be used to obtain experimentally the thermal

expansion coefficients of transversely isotropic fibers on the basis of measured expansion coefficients and elastic properties of the unidi-rectional material and known matrix and fiber elastic properties. For the case of carbon or graphite/epoxy effective elastic properties are accurately given by the general bounds. Thus when these are used in (44) the effective thermal expansion coefficients are accurately de-termined for random fiber arrangement and arbitrary fiber cross sections.

Conduction, Magnetics and Dielectrics The problems of computation of effective thermal and electrical

conductivity, effective magnetic permeability, effective dielectric constant and effective diffusion coefficient are mathematically identical. For details see e.g., [6,18]. This implies that any result for such an effective property is valid for all the others by appropriate identification of the phase properties in the expression for the effec-tive constant. The present discussion will be in terms of thermal conductivity.

Consider again a transversely isotropic fiber-reinforced or fibrous cylinder in which the phases are transversely isotropic with material axes of symmetry in cylinder axis direction, Fig. 1. The phase con-ductivities are HA, MT while the effective cylinder conductivities are HA, (I*T- The effective conductivities are defined by subjecting the external cylinder surface to the linear temperature variation

*(S) = - H-xi

Then the average temperature gradient in the composite is Hi and the average flux is

ri'ijH] (47) where /x'j are the effective conductivities. In the present case the only nonvanishing components of n]j are fi\ = ji'A and fit = ill = n*T.

In order to compute JX*A (46) is specialised to

3>(S) = -Hix1 (48) In this case the heat flow is entirely axial and trivial considerations of steady-state heat conduction show that in this case (48) is also the temperature field throughout the composite cylinder. It follows at once that

fi*A = nA1] vi + fiT v2 (49) rigorously and for any cylindrical phase geometry.

In order to compute the transverse conductivity (46) is specialized to

*(S) = - H2x2 (50)

Conduction

In this respect graphite or carbon/epoxy composites are particularly attractive since for such composites the general bounds for fiber composite properties are extremely close, thus determining properties accurately for random fiber arrangements and arbitrary fiber cross sections.

A c k n o w l e d g m e n t Support of the Penn-Israel program, of the Naval Air Systems

Command and of the Office of Naval Research under Contract N00014-78-C-0544 is gratefully acknowledged.

R e f e r e n c e s 1 . Hill, R., "Theory of Mechanical Properties of Fibre-Strengthened Ma-

terialsI. Elastic Behaviour," Journal of the Mechanics and Physics of Solids, Vol. 12,1964, pp. 199-212.

2 Whitney, J. M., "Elastic Moduli of Unidirectional Composites With Anisotropic Filaments," Journal of Composite Materials, Vol. 1, 1967, pp. 188-193."

3 Chen, C. H., and Cheng, S., "Mechanical Properties of Anisotropic Fiber-Reinforced Composites," ASME JOURNAL OF APPLIED MECHANICS, Vol. 37,1970, pp. 186-189.

4 Rosen, B. W., and Hashin, Z., "Effective Thermal Expansion Coefficients and Specific Heats of Composite Materials," International Journal of Engi-neering Science, Vol. 8,1970, pp. 157-173.

5 Levin, V. M., "On the Coefficients of Thermal Expansion of Heteroge-neous Materials," (in Russian) Mekhanika Tuerdogo Tela, No. 1, 1967, pp. 88-94; English translationMechanics of Solids, Vol. 2, No. 1,1967, pp. 58-61.

6 Hashin, Z., "Theory of Fiber-Reinforced Materials," Final Report, Contract NAS1-8818, Nov. 1970; NASA CR 1974,1972.

7 Behrens, E., "Elastic Constants of Fiber-Reinforced Composites With Transversely Isotropic Constituents," ASME JOURNAL OF APPLIED ME-CHANICS, Vol. 38,1971, pp. 1062-1065.

8 Hashin, Z., and Rosen, B. W., "The Elastic Moduli of Fiber-Reinforced Materials," ASME JOURNAL OF APPLIED MECHANICS, Vol. 31, 1964, pp. 223-232.

9 Hashin, Z., "On Elastic Behaviour of Fibre-Reinforced Materials of Arbitrary Transverse Phase Geometry," Journal of the Mechanics and Physics of Solids, Vol. 13,1965, pp. 119-134.

10 Pickett, G., "Elastic Moduli of Fiber-Reinforced Plastic Composites," Fundamental Aspects of Fiber-Reinforced Plastic Composites, Schwartz, R. T., and Schwartz, H. S., eds., Chapter 2, Interscience, 1968.

11 Adams, D. F., Doner, D. R., and Thomas R. L., "Mechanical Behaviour of Fiber-Reinforced Composite Materials," AFML-TR-67-96,1967.

12 Rosen, B. W., Private Communication. 13 Smith, R. E., "Ultrasonic Elastic Constants of Carbon Fibers and Their

Composites," Journal of Applied Physics, Vol. 43,1972, pp. 2555-2561. 14 Behrens, E., "Elastic Constants of Composite Materials," Journal of

the Acoustical Society of America, Vol. 45,1969, pp. 102-108; ibid pp. 1567-1570.

15 Ashton, J. E., Halpin, J. C, and Petit, P. H., Primer on Composite Ma-terials, Chapter 5, Technomic Publ. Co., 1969.

16 Dean, G. D., and Turner, P., "The Elastic Properties of Carbon Fibers and their Composites," Composites, Vol. 4,1973, pp. 174-180.

17 Kriz, R. D., and Stinchcomb, N. W., "Elastic Moduli of Transversely Isotropic Graphite Fibers and Their Composites," Experimental Mechanics, Vol. 19,1979, pp. 41-49.

18 Hashin, Z., "Assessment of the Self-Consistent Scheme Approximation Conductivity of Particulate Composites," Journal of Composite Materials, Vol. 2,1968, pp. 284-299.

APPENDIX

Analysis of Composite Cylinder Assemblage Model The Composite Cylinder Assemblage model (CCA) was originally

analyzed in [8] by variational methods. A much simpler direct method has been given in [6] and its main points will here be summarized.

The simplest case is k*, the effective transverse modulus. Consider for this purpose a transversely isotropic homogeneous cylinder of arbitrary cross section. If such a cylinder is subjected to the boundary conditions

ui(S) = 0 u(S) = ex (53)

which are a special case of (4) it follows trivially that the internal strain and stress fields are homogeneous and are given by

foil = 0 0 0

0 0

n.a _ G + ^2- (58a) GrV) < Gr(->; G n + ) < Gn+) (6m ^ * h A- 9C

1 |_ * ^ iyj Therefore the optimal bounds are (35a, 6) When matrix is stiffer thrin

GT2 - Gn 2GTi(ki + GTi) fibers (59) and (58) are reversed. In this event G:(IM = GT2 + l < 5 8 & ) G*rc(+) > G*r(+)

1 k2 + 2Gn 1- - V2 and therefore the optimal upper bound is (35d). No such conclusions

T 1~

r 2 2 can be drawn for the lower bounds. Numerical experience shows thai

mostly G'T(-) > Gr}-) a n d that when this inequality is reversed, tin-w e n

bounds are only slightly different. It therefore appears that from a GT2> G T I ; hi~> k\ (59) practical point of view the lower bound (35c) can always be used.

Comparison with experimental results shows that G jf(+) is in gocil When the inequalities (59) reverse the bounds (58) also reverse. agreement with experimental results for fibers stiffer than matrix

Let 1,2 be identified with matrix and fibers, respectively. If fibers while G"(_) is in good agreement in the case of matrix stiffer than i'i-are stiffer than matrix, equation (59), then it can be shown that bers (graphite or carbon/aluminum).

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