Anti-symmetric elastic properties of composite plates of satin weave cloth

FibreScienceand Technology15(1981) 127-145

ANTI-SYMMETRIC ELASTIC PROPERTIES OF COMPOSITE PLATES OF SATIN WEAVE CLOTH

TAKASHI ISHIKAWA

Composite Structure Section, 1st Airframe Division, National Aerospace Laboratory, 1880 Jindaiji, Chofu, Tokyo 182 (Japan)

SUMMARY

Composite plates made of satin weave fabric show macroscopic anti-symmetry and coupling effects. The essential mechanism of such behaviour is explained. Closedform solutions of the upper and lower bounds of the stiffness and compliance are obtained by the simplest series model in connection with uniform stress and strain assumptions. Finite element analyses using three-dimensional isoparametric elements are also conducted. The solutions fall between both bounds within a certain limit. The deviation of the actual quasi-micromechanical conditions from the simple assumptions can be seen from the FEM results. Rough estimation of thermal warping coefficients is carried out for carbon-epoxy and glass-polyimide 8th harness satin composites by simple measurements. More accurate results are provided by experiments for measuring thermal deflection under a small temperature change. These experimental results confirm the theoretical predictions made by both methods. Several comments are made for the purpose of preventing such phenomena, based on the geometry of the fabric structures. INTRODUCTION

Woven cloth has frequently been used as a raw material for modern composite structures. In composite aircraft structures, the recent design policy suggests that the woven cloth should be used for complicated or curved parts, such as corrugated webs and stiffeners of various sectional shapes. In the lay-up process of those parts, it is much easier to handle the woven cloth than unidirectional (UD) laminae. However, for simpler shaped parts laminate plates of UD plies are usually employed. For example, outer panels and spar flange caps are composed of the UD laminate, which has a relatively large load-carrying capacity. 127 Fibre Science and Technology 0015-0568/81/0015-0127/$02.50 © Applied Science Publishers Ltd, England, 1981 Printed in Great Britain

128

TAKASHI ISH|KAWA

Of the many types of woven cloth, satin cloth, particularly 8th harness satin cloth, is most widely used. The most likely reason for this being that the satin cloth is very pliable and easy to fit on moulds. Another advantage of the satin cloth was reported in reference 1, namely that it has greater strengths after curing than plain weave cloth. These properties can probably be ascribed to the texture of the satin weave (see Fig. 1).

warp

IIIIII IIIIII Illli

illlli

filling

IIIII1

IlllU /' ,i Uilil : ililii , iliill i -' lii ' Iliili

IIIIII

II1111

IIIIII

Fundamental Region ...........

Fig. 1.

,.~--~..

. .~ ; ~ ~ ; ,-: ~ .

~. ., ..........

, :.,:.,~..~,,,

F a b r i c s t r u c t u r e of 8th h a r n e s s satin w e a v e ; n = 8.

However, it has recently been discerned that composite plates made of the satin cloth show macroscopic anti-symmetry and coupling effects caused by its weaving pattern. Due to these characteristics, the plate will be deformed by a temperature variation, and will have fewer buckling loads. Although such phenomena are crucially important, no research papers which directly treat this problem could be found by the author. h order to provide a basic understanding of this problem, closed-form solutions of the macroscopic coupling moduli, Bu, and the thermal warping coefficients were calculated based on the simplest model, referred to here as a 'series model'. It should also be pointed out that such solutions provide the simplest upper and lower bounds of the stiffness and compliance. The finite element analyses were conducted for the idealised mosaic model of the satin weave composite plate. Measurement of the thermal bending deflection was carried out, using electro-optical equipment, for the strip specimen of the satin plate with the purpose of experimental verification. The experimental results were compared with the theoretical results obtained by the two methods. A lamination procedure for preventing these characteristics will be recommended in the present paper.

ANTI-SYMMETRIC ELASTIC PROPERTIES OF COMPOSITE PLATES

129

FABRIC STRUCTURE OF SATIN WEAVE CLOTH

There exist many types of weaving patterns which can be essentially classified by the number of the 'repeat', n, as illustrated in Fig. 2. The number of n w, (n in the filling direction) is defined such that a filling thread is interlaced with every nw-th warp thread. In other words, the same weaving pattern will appear after every n w warp

n:2

n=3

n=4

%--%

lllL-qlll l l ~

<]1111I>IIIIII IllllL"llllll I r'lllllt,' ! III111 IIII11 ~

I

IIIIII ~,

(~ain Weove) (Twill Weave)

(ClawFoot Satin)

n=5

II1111 ,

n=6

IIIIII

I..

IIIIII

I11111 ,

i,, IIIIII' IIIlll ',,~--'" IIIIII L lllll I III111 IIIIII Fig. 2.

IIIIII illlll

]11111

111111 ,'" : IIIIII ~,'" IIIIII i f IIIIII ,," ii ¢// " IIIIII r_ _ llllll___ I11111

IIIIII

Various types of weaving pattern.

thread in the direction of the filling. In general weaving structures, nf, for the filling can differ from nw. However, it is assumed that n w - - n f - - n in the present paper. A m o n g the patterns in Fig. 2, the weave of n = 2 is usually called plain weave and n = 3 is one type of twill weave. The minimum number o f n for the satin weave is 4, and this satin is usually called 'claw foot satin' or 'Turkish satin'. This type is also often used for composite aircraft structures. As mentioned above however, the most popular satin for pre-preg sheets is the 8th harness satin shown in Fig. 1, where n = 8; this is sometimes known as 'long shaft satin'. The weave of n = 6 is an example of a quasi-regular satin. 2 The polygons enclosed by dotted lines in Figs l and 2 indicate one of the arbitrary candidate figures of the minimum repeating, or fundamental, regions of each weave. The essential reason for the anti-symmetry of the satin composite is that the filling covers a wider area on one side than the warp, and that the opposite occurs on another side. The side where the filling is predominant will be considered as a top

130

TAKASHI ISHIKAWA

side, for the sake of convenience. The ratio of the area of the warp in the top side is approximately equal to the inverse number of n, i.e. it is 1:8 for the 8th harness satin (see Figs 1 and 3). On the other hand, it is 1:2 for the plain weave, and this is the very reason why the macroscopic anti-symmetry vanishes for the plain weave composite plate. A two-layered cross-ply laminate of U D plies can be considered as

Fig. 3.

Photograph of actual weaving pattern of 8th harness satin (a = 1 mm).

the special case for the satin plate where the ratio becomes equal to zero, or to unity on another side. In other words, the satin plate might be considered as a cross-ply laminate in which the top and bottom sides alternate with each other in different pitches. This kind of simplification of the fabric structure gives rise to the basic concept of both the series model and the mosaic model introduced in the analyses.

CONSTITUTIVE EQUATION

The essential equation for solving the present problem is the constitutive equation of an asymmetric laminate subjected to external forces and temperature variation. It is assumed that Bernoulli-Euler's hypothesis holds for the laminate. Therefore, shear deformation in the thickness direction is automatically neglected here. It is also assumed that all material properties remain constant within a small temperature change denoted AT. Moreover, a uniform temperature distribution in the thickness direction is postulated. Then, the constitutive equation can be derived such that 3'4

ANTI-SYMMETRIC ELASTIC PROPERTIES OF COMPOSITE PLATES

{N} = [A

B ] {e°} - A T { g }

131

(1)

where N, M, t ° and x are the membrane stresses, moments, strains at a certain reference plane, and curvature, respectively. The geometrical mid-plane is chosen as the reference plane in the present analysis. The plate moduli of A, B, D, A, and B are given in the following form: N

(Aij , Bij, Dij ) =

(1, z, z2)E,~-~° dz

(2)

al "~ dz

(3)

al"~z dz

(4)

hra- 1 m=l N

/]i =

7 f? 2f?

m--I

m--1 N

Bi =

m

1

m=l

where ,

= Eij

sj

(5)

and i , j = 1,2, 6. As a concrete example, unabridged expressions of those plate moduli of a 2-layered cross-ply laminate illustrated in Fig. 4 are written here, using the engineering elastic constants of the constituent UD lamina, E L etc., such that A 11 = ( E l l + E22)h/2 = (EL + E x ) h / { 2 ( l - VLVv)} = A22

(6a)

A lz = E12h = VLEx/(I -- VLVT)

(6b)

A66 = E66h = GLT h

(6c)

Bll

=

(Ell

--

E22)h2/8 = (E L - Ex)h2/{8(! - VLVT)} = --B22

D11 = (E11 + E22)h3/24 = (EL + Ex)h3/{24( 1 -

I"LVT) } ~---0 2 2

D12 = E l z h 3 / 1 2 = VLE.rh3/{12(1 - VLVT) }

D66 = E66h3/! 2 = GL-rh3/12

(7) (8a)

(8b) (8c)

"41 = {El lsl. + E12(SL + ST) + E 2 2 s v } h / 2 = {ELSe + Ex~-r + v e E x ( s e + ST) }h/{2(l - VeVx)} = "]2 BI = {El

(9a)

lSL + EI2(~T -- ~L) -- E22sT}h2/8

= {EL~L -- E x s v + VLEv(~T -- ~L)}h2/{8(l -- VLVT) } = - - B 2

(9b)

132

TAKASHI ISHIKAWA GLT

.-"

L --

~.:..:5:'2 :'I.":.."

,"

.1

--EL.~

~

,¢.. : :

•

h/2

~2 i! ~.ET[

~r

Fig. 4.

i-~.'~ :i i i :'"

yx

Basic two-layered cross-ply laminate.

where the underlined equations represent the coupling terms. For such a laminate, B~s and/~i, namely, the coupling terms, do not vanish because E,+ # E 1 and :q+ 4: O(T. It should be noted that this laminate can be regarded as a basic component of the satin weave composite plate. Equations (6) (9) will be directly used for the closed-form solution in the next section.

SOLUTIONS

B A S E l ) O N A SERIES M O D E L

A one-dimensional model which is composed of the pieces of the basic laminate placed alternately as indicated in Fig. 5 will now be considered. This model will be referred to as a 'series model'. The essential point of the model is that the twoInterface b ET

EL

t

NI

N1

Fig. 5. Series model. dimensional extent of the plate is neglected. Moreover, the disturbance of stress and strain around the interface between the top and bottom regions will also be neglected. The model will be subjected to an in-plane force, N,, in the longitudinal direction. As the first approach, it is assumed that N l is uniform everywhere in the model. Let an average curvature, #1, be defined by an average along the 1-direction. Then, from the following equation: #1 = - - l f ~ ° ~¢tdx= 1 ( ~ ' r/a

/./a

te~dx+ ["" K+ldX ) .ja

=

( 1 - ! ) b t,,, N l

(10)

ANTI-SYMMETRIC ELASTIC PROPERTIES OF COMPOSITE PLATES

133

where a denotes the unit length, namely, the width of the warp or filling. Equation (10) implies that the coupling compliance is modified such that

where the prime refers to the satin plate. Geometrical interpretation of these equations tells us that the rest of the curvature after slight cancellation yields to the total coupling effect of the satin plate. By considering a*' = a'bit and d *tlt-- dl./*b, a,, _ a , t ., it - - - - i j dij = di*' (12) are obtained. Taking the inverse of the compliance matrix and heeding eqns (11) and (12), the following equations are obtained. , Dij..~

1-

di*

1--

B i j =- --

A i j' = ( a * )

1+

bik(akt )

a*) 1 --

-1

btj*

}1

(13)

lh*l-)tUkl~lj

( a i ,k )

1 bktDo,.b,.,,(a,~fl , t , , - 1

It follows that eqns (11) and (12) give the simplest upper bound of the compliance and that eqn (13) gives the simplest lower bound of the stiffness according to the parallel approaches in many references concerning the elastic moduli of composites, (e.g. by Paul 5 and Tsai, 6 etc.). In the alternative method the strains in the wide sense, that is, go and K are assumed to be uniform everywhere in the model. Then, an average membrane stress, NI, is given by

N'~dx )

?q =~oo N'dx---(ff na =Atllg°-~-A12g2"b

1-

Btl 1K1

(14)

It follows that

A,'j = AI~

8,'~ =/[1 - 2) 8~ \

HI

(15)

U

In a similar manner D~'j = D~j.

(t 6)

These equations represent the simplest upper bounds of the stiffness of the satin plate. Taking the inverse of the stiffness matrix, will give equations similar to eqn (13). Such equations are the lower bound of the compliance.

134

TAKASHI ISHIKAWA

The relationships between these upper and lower bounds and 1In are shown in t ~t Figs 6 and 7 for b*~ and A~I ( a ~ ) , respectively. In the calculation, the following material properties are adopted for the U D carbon-epoxy v'8 EL

11500

=

VL = 0 . 3

E~ = 900 0[L = 0. 0

455

GLT =

(kg/mm

0~T= 3.0 X 10_ 5

2)

( o c _ l)

) ~

(17)

Since the limit of l/n---, 0 is equivalent to the basic two-layered laminate, both bounds coincide with each other at the limit. For the plain weave, 1In = 0.5, the coupling ~Cllt m '

a~'("10-4

--3

b n "10

:~-1)

-2.0.

0-0

An"104 -(kg/mr~

~] FEM (h/a=0.4) (h/a=0.2)

UpperBound

-6.0-

-4.0- ~

----An'

- Cmrn/kg)

\ 10.0- \ CFRP (h=0.4mm)

~

~

x'xUxpPerB°und

.

o.o o111 o.2

t 8thH.Satin 2plyLIDI..am.

,

. ""N1

1In

o.5

P~inW.

Fig. 6. Relationships between upper and lower bounds of b*'1 and inverse number of repeat, l/n. of satin weave.

,ol

o.o

I

\

.,

U.B.

.

•

.

-2-0

to.o

010 0.11 0.2 0.3 04 0.5 tin I 8thH.Satin PlainW. 2_plyUDLam. CFRP, h=O.Zanm Fig. 7. Relationships between both bounds of a*'1 and A',I, and l/n of satin weave.

effect vanishes as mentioned previously. Both bounds of bi*' (Bi'j), therefore, are identical to zero, whereas those of a*' (A;j) do not coincide with each other even for the plain weave. The FEM results are also shown in Figs. 6 and 7 and will be explained later. Thermal expansion coefficients and thermal warping coefficients are also derived by using this model, where the thermal warping coefficients are defined by ~ / A T ( = -~:2/AT). The state that the model is deformed by only a temperature change without any external forces is now considered. If it is assumed that the one-

ANTI-SYMMETRIC ELASTIC PROPERTIES OF COMPOSITE PLATES

135

dimensional average of the curvature and strain represents the effective deformation of the plate, then

1=

I~(A'I~ +A~)-A~BI~I

(Ai, +AI2)(D'11 _Di2)_B,121

(18)

and

(D]I - D ~' z ) A~,, - B',,B] l

!

(At11 + A I 2 ) ( D 1 1 -

(19) 12

Dtlz)- Bll

Thus, the thermal warping coefficient decreases linearly to l/n. On the other hand, the thermal expansion coefficient is the same as for the basic laminate. Numerical examples of these coefficients will be shown later and compared with experimental and FEM results. FINITE ELEMENT ANALYSIS

In order to examine the above-mentioned bound theories, three-dimensional finite element analyses were conducted for the idealised mosaic model of the satin plate. A slightly modified version of SAP IV, which had been developed by Bathe et al.,9'l° was utilised. The purpose of the modification was to make the program suitable for the present problem. A three-dimensional isoparametric element was chosen from the element library. The maximum node number per element is 16, while 21 is admissable in the program. The mesh subdivision is shown in Fig. 8, where 'middle' means the regular mesh with 867 nodes and 512 elements, and 'fine' means the irregular one with 993 nodes and 584 elements. The objective materials are 8th harness satin plates made of carbon~epoxy and glass-polyimide. According to the mosaic idealisation, one element consists of a U D composite. Therefore the material properties in eqn (! 7) are adopted for the carbon-epoxy; the following values are used for the glass-polyimide: 1~ E L = 4400

E.t = 1300

VL= 0"34

CtL =6"59 × 10 - 6

GL-r = 450

(kg/mm 2)

ctx = 2-7 × 10 -5

(°C -1)

(20)

It is now planned to apply this glass-polyimide heat-resistant composite to the S T O L aircraft as components of the upper surface blown flaps and the heat shield plates of the wing box.

136

TAKASHI ISHIKAWA Fine : 993 Nodes, 584Elements Middle: 867 Nodes,512 Elements

IIIII

nnmmmmmmmmmmm lmmmmQ nnnnmm lnmnnmmmmmnmm lnmmnmmmmnn,mm m[]annmmmmmmmlmnmm

IIII IIII

1'1!1111111,

l!lLIt**tllll illm**mllll

'~ 1111111 "k IIIIIII

IIIIIIIII

IIIIlillL

ill

IIII Itll Illl IIII

!11,!,, !tttl

IIII

L

• z-Trcmslotion Fixed Points 1",:;I I

I

I

I

I

I

1

I

1

I

1

I

I::.'J.'.-.'.I;:.'J'.'.'.I.'.'.I:::I'.::I.'.;'[.'.'.'I.::tI.::I.'.'.I:.:'L:;:I

/

|

I:.'.-J. I

(3-D Isoparametric Elements) Fig. 8.

Mesh subdivision of one-ply 8th harness satin (8a x 8a).

The results are shown in Figs 9 to 13. Figures 9 and l0 indicates the pattern of the coupling deformation by N 1. The region enclosed by dotted lines represents the bottom region. Figure 10 clearly shows that ~2w/?x2 varies along the centreline of the repeating region so considerably that its sign changes. An oscillatory character of 02w/~x 2 along the edge lines of the repeating region is caused by the mesh irregularity. A membrane stress distribution is described in Fig. 11, and the extent of the deviation of the FEM solution from the constant stress series model can be seen. Figures 12 and 13 show the results for the thermal warping. The feature of an anticlastic curvature is easily observed in Fig. 12. The tendency of the variation of /)2w/~?x2 is very similar to Figure 10, although it is inverted. The result expressed by the broken lines corresponds to the relatively thick case where a/h = 2.5. Note that the curvature becomes larger for the smaller a/h by virtue of consideration of the shearing deformation in the thickness direction in the FEM analyses. The results of the coupling compliance and the thermal warping coefficients are listed in Table 1. The main point of this table is a comparison of the solutions obtained by the series model and by the FEM. The values titled'FEM' are the results

ANTI-SYMMETRIC ELASTIC PROPERTIES OF COMPOSITE PLATES

]37

Contour

T__

-058~ ~/ ,

8th Horness - 0 6 ~r~

Scole at Middle Surfoce

Fig. 9.

o/h= 5 N1 = 7.492hg/mm Fine Mesh

F E M solution of bending deformation caused by m e m b r a n e stress, N 1, for 8th harness carbon~epoxy.

obtained by a numerical integration of the deformation within one unit of the repeating region. It should be noted that the results from the fine mesh fall between the upper and lower bounds for both materials. However, this consistency may be apparent because of the shear deformation in the thickness direction. If the shear rigidity in this direction is considered in the calculation of the bounds, the converged results by FEM can not exceed such bounds. Table 1 implies that the coupling deformation of the satin plate can be predicted in practice, by using the series model solutions. In other words, a somewhat adventurous idealisation of the series model works out fairly well.

EXPERIMENTS ON THERMAL WARPING

A positive experimental verification of this theoretical prediction would be indispensible here. A direct determination of the coupling moduli, however, is rather difficult. Therefore, at first an indirect route has to be used, i.e. measurement of the thermal warping coefficients.

Fine(993Nodes) Middle(867Nodes) (xl0 ax2-2mm-~)

z 0 20/.,06002040 0 204060

~ + ~ 11~~"

tN1/ ==:~h ~/mm ~

!

CFRP q~{, 00- "¢/ -,,¢ 8thH~ness o / h = 5 , N1 =7.492kg/rnm Fig. 10.

.

"

~ , ., ~ o

Variation of bending curvature by N~ along its direction.

CFRP FineMesh,o/h=5 Fig. 11. Membrane stress (Nl/h) distribution in fundamental region under macroscopic Nl/h= 18.73 kg/mm 2 (for carbon~epoxy).

001i~-~.~-.~_~\///~, 003 002 ~,,,

~\ "~/~00

Isovalue Contour

k

Ii/

k

xlO-2m W

002

] 1,1

at Middle SurfQce

Ol?a

0.

I0"

8th Harness. FineMesh o/h= 5 Fig. 12. FEM solution of thermal bending deformation caused by temperature descent of 160 °C for 8th harness carbon-epoxy.

139

A N T I - S Y M M E T R I C E L A S T I C P R O P E R T I E S OF COMPOSITE PLATES o/h

Fine

....

•

2.5

Fine 1 5-0 Middle j

O0 ~ J,~ccz-~~

x ~

F-z'' .. . .. . . . . 0,0 31---.-

CFRP 8th H~ness Fig. 13.

ax L

(xlO-2 ram-l) (at Middle Surface

t j "v

~

,5" .`5..

~

~ .

~ .

.

.

.

o

.

,

V a r i a t i o n o f t h e r m a l b e n d i n g c u r v a t u r e a l o n g the x-direction.

TABLE 1 THEORETICAL RESULTS OF COUPLING COMPLIANCES AND THERMAL WARPING COEFFICIENTS OF CARBON-EPOXY AND GLASS-POLYIMIDE8TH HARNE~ SATIN PLATES BY THE F E M AND SERIES MODEL

Material

Carbonepoxy

Glasspolyimide

Thickness h (ram)

Series model

U.B." L.B)

Series U.B. model L.B. F E M (middle) F E M (fine) F E M (fine) F E M (fine) Series U.B. model L.B. F E M (fine)

h/a

b*'l ( k g - 1 )

Thermal warping coefficient rx/AT (mm - 1)

--

- 3 . 6 7 x 10 -3 - 2 . 3 9 × 10 - 3

- 4 . 6 4 × 10 -5

-4.28 -2.79 -2.11 -2.98 -3-75 -3.39 -9.24 -8.18 -8-38

- 5 . 0 1 x 10 -5

0.432 0.400 0.400 0.400 0.400 0.432 0-244

-0.200 0-200 0.400 0.432 __

0.244

0.610

x × x x × × x × x

10 -3 10 - 3 10 -3 10 -3 10 -3 10 -3 10 -3 10 3 10 -3

-2.70× -3.93 × -4.48 × -4.23 × -8.49 x

10 -5 10 -5 10 -5 10 - s 10 -5

- 7 - 6 4 x 10 -5

" U.B.; Upper Bound b L.B.; L o w e r B o u n d .

Source plates of test specimens were prepared by Mitsubishi Rayon Co. Ltd and Kawasaki Heavy Industries Ltd. for carbon-epoxy and glass-polyimide, respectively. The raw materials for both composites are pre-preg sheets of the 8th harness satin. Some of these consist of a few plies of the satin pre-preg. The plates of the one-ply satin of both composites were extremely deformed because of a

140

TAKASHI ISHIKAWA

temperature difference between cure and room temperatures as shown in Figs 14 and 15. In the case of asymmetric laminated plates of two-ply satin, there still remained a considerable thermal warping, although the curvature was much smaller than for the one-ply. Such a thermal bending deformation almost completely vanished for the two-ply semi-symmetric* plates. However, a slight twisting deformation then appeared in these plates. A convincing explanation for such a twisting deformation

Fig. 14. Source plates of carbon-epoxy specimens (note the thermal deformation caused by temperature descent from the curingpoint to room temperature), a, One-ply;b, two-plyanti-symmetric: c, two-ply semi-symmetric. was found and is reported in the next section. By simple measurement of the curvatures of the one-ply plates, approximate values of the thermal warping coefficient can, of course, be obtained. The results are indicated in Table 2 under the title 'simple measurements'. The bending deformation, however, is sensitively affected by the variation of elastic moduli and thermal expansion coefficients over a * The precise meaning of 'semi-symmetric"is explained in the next section.

ANTI-SYMMETRIC ELASTIC PROPERTIES OF COMPOSITE PLATES

141

C

I

1

l-~g. 15. Source plates of glass polyimide specimens (note the thermal deformation especially of the one-ply plate which deforms so extremely that it becomes a cylinder), a, One-ply; b, two-ply antisymmetric; c, two-ply semi-symmetric.

142

TAKASHI ISHIKAWA

TABLE 2 EXPERIMENTAL RESULTS OF THERMAL WARPING COEFFICIENTS OF CARBON EPOXY AND GLASSPOLYIMIDE ONE-PLY 8TH HARNESS SATIN PLATES WITH SOME THEORETICALPREDICTIONS

Material

Carbon epoxy

Classification

Experimental

Theoretical Glasspolyimide

Experimental

Theoretical

Simple measurements No, 1 Exp, N o , 2 Series M. F E M (fine) Simple measurements Exp. N o . 1 No. 2 Series M. F E M (fine)

Thickness h (ram)

h/a

Thermal warping coefficient G/A T (ram - 1)

0.432

0.432

0.431 0-431 0.432 0.432 0-244

0,431 0,431 0,432 0.610

0.233 0.233

0.583 0.583

- 3.70 × 10 - 5 ( A T = 140 °C) - 4 . 9 2 × 10 5 - 5 - 0 4 x 10 - s - 4 . 6 4 × 10 -5 - 4 - 2 3 x 10 5 - 7 . 6 9 × 10 5 ( A T = 250 °C) - 8 . 4 4 x 10 - s - 8 . 1 0 × 10 -5

0-244 0-244

0.610

- 8 . 4 9 × 10 5 - 7 . 6 4 x 10 s

wide temperature range. 12 An experiment, therefore, was conducted in order to determine xI/AT under a small temperature change. Strip specimens were cut out from the source plates. They were fixed as cantilevers and heated on both sides by electric lamps. Edge displacements were measured using an electro-optical transducer, Zimmer model 100B. This instrument is highly suitable for such measurements because of its non-contact property. A schematic photo of the experiment is shown as Fig. 16. The target heating condition was a uniform temperature increase in the specimen. For checking purposes, as a first step the temperature distribution along the longitudinal direction was measured using thermocouples. The deviation from the uniform distribution was taken into account in a data reduction process. It was found that the many leads to the thermocouples slightly impeded the thermal deflection. As a second step, therefore, only one thermocouple was used in the measurement of the edge displacements. The results are listed in Table 2 with some theoretical predictions. The simple solutions by the series model are a little larger than the FEM solutions. They are also close to the experimental results. This comparison tells us that the series model is quite suitable for predicting the thermal warping coefficient. It is not necessary to conduct an elaborate computation by the F E M in order to estimate design indices. I f more accurate information is needed, the FEM analyses or other elegant techniques should be used as the present mosaic model will not be satisfactory.

NOTE ON S T A C K I N G P R O C E D U R E FOR P R E V E N T I N G C O U P L I N G EFFECTS

At first, it should be noted that there are two types of usual weaving pattern for the 8th harness satin. They can be identified by the shapes of the fundamental region.

ANTI-SYMMETRIC ELASTIC PROPERTIES OF COMPOSITE PLATES

143

The shape of the first one is drawn in Fig. 1 and that of the second one is drawn in Fig. 8. Considering the directions of the warp and filler, it is not possible to obtain one by the other by rotating 90 ° or turning upside down. It should also be noted that rotating 90 ° and turning upside down are geometrically identical for the case of the usual (regular) satin. Therefore, a two-layered laminate, which is composed of only one type of satin with one upside-down layer, is not completely symmetric. This laminate was previously referred to as 'semi-symmetric'. The simple lamination yields to an 'asymmetric laminate'. As mentioned above, the thermal bending deflection will practically vanish for a semi-symmetric laminate. If both thermal bending and twisting are to be repressed completely, then the two types of satin must be employed to ensure that the plate is strictly 'symmetric'.

egulator

Fig. 16.

Schematic view ofan experiment for measuring thermal warping coefficients of strip specimens using an electro-optical transducer.

The reason for such a thermal twisting is stated here. By the FEM analyses, small values of the cross-elasticity terms, al6 or a26 , were obtained for the 8th harness satin. Although we cannot know for certain that these values are accurate because of the relatively coarse mesh used for this purpose, it is clear that they do not vanish in contrast with the values for cross-ply laminates. The signs o f a 16 and a26 are changed by turning the 8th harness satin upside down. Thus, it can easily be understood that a semi-symmetric laminate of the satin has torsional-extensional coupling terms and it is slightly twisted by a temperature change.

144

TAKASHI 1SHIKAWA CONCLUSIONS

The essential mechanism of the anti-symmetric elastic properties of the satin composite plates is now understood. The closed-form expressions of the upper and lower bounds of the plate moduli were obtained by using the constant stress and strain assumptions in connection with the series model. If the simplicity of the series model is taken into consideration, it is remarkably efficient for predicting the coupling effect of the satin plate. The finite element solutions fell between both bounds within a certain range of the ratio of the width of the threads to the thickness. It should be noted that this ratio has a considerable effect upon the effective coupling moduli of the plate. Quasimicromechanical curvature and stress distributions were calculated through the FEM analyses. These analyses demonstrated how the actual situation deviates from the assumptions made for the bounds. The thermal warping coefficients were roughly determined by straightforward measurements of deformed specimens under a large temperature change. An experiment to measure a thermal deflection under a small temperature variation was also carried out in order to attain more accurate results. The experimental verification suggests that the theoretical prediction provides fairly useful information about the thermal warping coefficient even if it is derived from the series model. Some comments for preventing such coupling effects were made based on a geometrical interpretation of the fabric structures. The torsional deformation of the semi-symmetric two-ply satin laminate was ascribed to the non-vanishing crosselasticity terms. It should be mentioned here that this problem is of a very complicated nature and that the present paper is only an initial and basic approach. Many extended investigations have to be conducted. For example, the propriety of the mosaic model should be examined for more detailed aspects, or the geometrical non-linearlity must be included in the analysis. These points remain to be solved in the near future. ACKNOWLEDGEMENTS

The author acknowledges the preparation of the source plates of the specimens by Mitsubishi Rayon Co. Ltd. and Kawasaki Heavy Industries Ltd. He wishes to thank Mr T. Furuta of his section for the arrangements with the former company. He is also indebted to Mr Y. Aoki for his assistance in the experiment. REFERENCES I. A. NISHIMURA,"Design and Material Properties of Woven Fabric of Carbon Fibers, Proceedings of 10th Symposium on Materials in Aerospace Technology, Tokyo, Japan, Feb. 1980, p. 1, (in Japanese). 2. J. H. STRONG,Foundations of Fabric Structure, National Trade, 1953.

ANTI-SYMMETRIC ELASTIC PROPERTIES OF COMPOSITE PLATES

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3. R. M. Jones, Mechanics of Composite Materials, Scripta, Washington DC, 1975: 4. J. M. WHITNEYand A. W. LEISSA,Analysis of heterogeneous anisotropic plates, J. Appl. Mech., 36 (1969) p. 261. 5. B. PAUL, Prediction of elastic constants of multiphase materials, Trans. AIME, 218 (1960) p. 36. 6. S. W. TSAI, 'Structural Behaviour of Composite Materials', NASA CR 71, 1964. 7. T. ISmKAWA, K. KOVAMAand S. KOBAYASHI, Elastic moduli of carbon-epoxy composites and carbon fibers, J. Comp. Mat., 11 (1977) p. 332. 8. Mitsubishi Heavy Industries Ltd, Unpublished data. 9. K.-J. BATHE and E. L. WlLSON, Numerical Methods in Finite Element Analysis, Prentice-hall, Englewood Cliffs, 1976. 10. K.-J. BATHE,E. L. WILSONand F. E. PETERSON, User's Manual of SAP IV, University of California, Berkeley, 1973 (revised in 1974). 11. RhOne-Poulenc Corp., 'Catalogue of Polyimide Resin'. 12. T. lSHIKAWA, K. KOYAMA and S. KOBAYASHI, Thermal expansion coefficients of unidirectional composites, J. Comp. Mat., 12 (1978) p. 153.