Prediction of on-axes elastic properties of plain weave fabric composites

Composites Science and Technology 45 (1992) 135-152

Prediction of on-axes elastic properties of plain weave fabric composites N. K. Naik & V. K. Ganesh Aerospace Engineering Department, Indian Institute of Technology, Powai, Bombay 400 076, India (Received 14 February 1991; revised version received 4 September 1991; accepted 2 October 1991)

Two fabric composite models are presented for the on-axes elastic analysis of two-dimensional orthogonal plain weave fabric lamina. These are twodimensional models taking into account the actual strand cross-section geometry, possible gap between two adjacent strands and undulation and continuity of strands along both warp and fill directions. The shape functions considered to define the geometry of the woven fabric lamina compare well with the photomicrographs of actual woven fabric lamina cross-sections. There is a good correlation between the predicted results and the experimental values. Certain modifications are suggested to the simple models available in the literature so that these models can also be used to predict the elastic properties of woven fabric laminae under specific conditions. Some design studies have been carried out for graphite/epoxy woven fabric laminae. Effects of woven fabric geometrical parameters on the elastic properties of the laminae have been investigated.

Keywords: woven fabric lamina, prediction, two-dimensional, plain weave, elastic constants

Gap between the adjacent strands Maximum strand thickness h Thickness of matrix at hm x=0, y=0 Thickness of matrix and hx,(x, y), strands in X - Z plane at a i=1,2,3,4 point as defined by coordinates x and y (Fig. 6) hyi(y), i = 1, 2, 3, 4 Thickness of matrix and strands in Y - Z plane at x = 0 (Fig. 5) Total thickness of WF H lamina Transverse bulk modulus k Reduced stiffness of Qij, i , j = 1, 2, 6 lamina Local reduced and aver%, aged compliance constants i,j=1,2,6 Volume V Fibre volume fraction Matrix volume fraction Vm Cartesian co-ordinates x, y, z

g

NOTATION Strand width Extensional, coupling and bending compliance constants Parameters as defined in ax,, Zx, Fig. 6 Parameters as defined in ay,, z r` Fig. 5 Extensional, coupling and Ao, B j, Dij bending stiffness constants i , j = l , 2, 6 EL, ET, ~/LT, GET, Grr UD composite elastic properties along the fibre and transverse fibre directions eL(o), eT(o), vLT(o), Local reduced elastic constants for undulation angle CLT(O), C (O) 0 Elastic constants of unit Ex, vxy, cell/WF lamina

a

a~*, b~*,d~* i,j=1,2,6

Composites Science and Technology 0266-3538/92/$05.00 © 1992 Elsevier Science Publishers Ltd. 135

136

N. K. Naik, V. K. Ganesh

ZX i(x, y),

zyi(y),

i = 1, 2

i = 1, 2

O(x), O(y) O

Subscripts f L T W

Shape functions of strand undulation in X - Z plane (Fig. 6) Shape functions of strand undulation in Y - Z plane (Fig. 5) Local off-axis angle of the undulated strand Maximum off-axis angle of the undulated strand

Quantities in fill direction Quantities in fibre direction Quantities in transverse fibre direction Quantities in warp direction

Superscripts Quantities of element Quantities of fibre Quantities of matrix WF composite overall O properties pm Quantities of pure matrix s Quantities of strand sl Quantities of slice * Quantities of UD crossply laminate Overbars indicate average values/quantities transformed to global direction el f m

Abbreviations CCA CLT EAM MKM MMPM PS SAM SP UD WF 1-D 2-D 3-D

Composite cylinder assemblage (model) Classical laminate theory Element array model Modified Kabelka's model Modified mosaic parallel model Parallel-series (model) Slice array model Series-parallel (model) Unidirectional Woven fabric One-dimensional Two-dimensional Three-dimensional

1 INTRODUCTION

The increasing use of composite materials has revolutionised the aerospace industry over the past two decades. The ability to vary the properties and performance of composite materials has been in large measure responsible for the great impact that these materials have had. Traditionally, advanced composite structures have been fabricated from tape prepregs which were systematically stacked to form a laminate. This type of construction tends to give optimal in-plane stiffness and strength. Since the primary loads usually are in-plane, the use of such composites appeared logical. However, there are many situations where neither primary nor secondary loads are in-plane. In such situations tape prepreg laminates may not be the most appropriate. The future for composites is undergoing a transition. The aerospace performance criteria consisting of high specific stiffness and high specific strength are being supplemented with high toughness and efficient manufacturability. With this, textile structural composites in general and woven fabric (WF) composites in particular are finding increasing use in primary as well as secondary structural applications along with unidirectional (UD) tape composites. Making use of the unique combination of light weight, flexibility, strength and toughness, textile structures like wovens, knits, braids and nonwovens have now been recognised as attractive reinforcements for structural applications. Woven fabric is formed by interlacing two mutually perpendicular sets of yarns. The lengthwise threads are called warp and the crosswise threads fill or weft. The interlacing pattern of the warp and fill is known as the weave. Two-dimensional (2-D) fundamental weaves are plain, twill and satin. The micromechanical behaviour of woven fabric laminates depends on the fabric properties, which in turn depend on the fabric structure. The parameters involved in determining the fabric structure are weave, fabric count, fineness of yarn, fibre characteristics, yarn structure, degree of undulation, etc. The architecture of a WF lamina is complex and therefore the parameters controlling the mechanical and thermal properties of WF composites are too numerous. This makes it impractical to characterise the WF composites through tests alone, necessitating analytical

Prediction of on-axes elastic properties o f plain weave fabric composites

models which can predict the mechanical and thermal properties of the WF composites. A variety of analytical models (Raju et al. ~) has been proposed for the prediction of the thermo-elastic properties of WF laminae. The models are based on the classical laminate theory (CLT) 2-4 or finite element analysis, s-7 Halphin et al. 2 extended the laminate analogy developed to predict the elastic stiffness of a randomlyoriented, short-fibre composite to 2-D and 3-D woven fabric composites. The weave geometry considered here represents the fabric in 1-D only and also the circular geometry of the strand cross-section considered here is not realistic. Chou & Ishikawa 3 have presented three models to predict the elastic properties of WF lamina. These are the mosaic model,S the fibre undulation model s and the combination of the above two, the bridging model. 9 The mosaic model idealises the WF lamina as an assemblage of asymmetric crossply laminates. Depending on whether the pieces of the crossply laminate are in parallel or in series, i.e. isostrain or isostress condition, respectively, the bounds of the stiffness as predicted by the mosaic model can be evaluated. This model does not consider the strand continuity and stress disturbance at the interface of the assemblage. The fibre undulation model considers the strand continuity and undulation, but it is a 1-D model as it considers the undulation of the strand in the loading direction only. The combination of mosaic and fibre undulation models, called the bridging model, was proposed to analyse satin weave fabrics. The model considers the bridging effect present in the satin weave fabric due to the presence of non-interlacing regions. The bridging model considers the fibre continuity and is a 2-D model for satin weave, but reduces to fibre undulation model in the case of plain weave. This model considers the undulation in the loading direction, as in the case of the fibre undulation model, but the strand undulation in the transverse direction and its actual cross-sectional geometry are not considered. These models were later extended to evaluate the thermal properties and to analyse hybrid WF laminae. ~°-~2 In general, the analytical predictions did not correlate well with the experimental results ~3 for plain weave fabric composites. Kabelka 4 suggested a method of evaluating the elastic and thermal properties of a plain weave fabric lamina. This is a 2-D model taking into

137

consideration the undulation in both warp and fill directions, but the actual strand cross-sectional geometry was not considered. The properties of the undulated warp and fill strands were evaluated under the constant stress condition in the strand and then the classical laminate theory was used to predict the overall properties. A 3-D finite element analysis was presented by Raju et al. ~ to predict the thermal expansion coefficients of the WF lamina. Here, again the WF lamina was idealised as an assemblage of asymmetric crossply laminates. Whitcomb 6 also used the 3-D finite element analysis to analyse WF lamina. Here, the undulation and continuity of the strands were considered in order to study the effect of various weave parameters on the mechanical properties of the WF lamina. The undulation shapes at the interlacing considered in the above studies were very approximate and may not present an accurate behaviour of a plain weave fabric lamina. Zhang & Harding 7 and Dow & Ramnath 14 presented fabric models based on energy principles. Zhang & Harding 7 used the strain energy equivalence principle to predict the elastic properties of a plain weave lamina. The finite element method was used to evaluate the strain energies of the constituent phases for the analysis. In these studies (Refs 7 and 14), the undulation of the strand was considered in the loading direction only and therefore all the inherent discrepancies present in 1-D models would also be present.

1.1 The choice of a 2-D model

A single layer WF composite is designated as WF lamina. The woven fabric can be in the form of an open weave or a close weave. In the case of the open weave, there may be gaps between two adjacent strands, whereas close weave fabrics are tightly woven without any gap between two adjacent strands. There can also be certain fabrics made of twisted strands which would invariably have a certain amount of gap even if they are tightly woven. It is obvious that the presence of a gap between the adjacent strands would affect the stiffness of the WF lamina and hence should be accounted for while evaluating the thermo-mechanical properties. The experimentally determined fibre volume fraction, Vf, of the WF lamina is the overall Vf,

138

N. K. Naik, V. K. Ganesh z

z

ho

~

,:~::~

,

I ho SECTION

SECTION S~- S~

S o- S O

h 0 = hm/2

,z

., ?z

~..

.',

i

I

,

SECTION S z - S z

t1 1 •

SECTION S 3 - S 1

-t-o~12

i SECTION S 4 - S~ h4=(hm+ hf) /2

WARF

lllfl .---_54 5a Ii ---5Z

" " " "

j~_~11" S0 E

IIIJlIF

.

PLAN

"---51

,----~o x

Fig. 1. Plain weave fabric lamina structure--cross-sections at different intervals. V~', but for the analysis of the WF lamina the strand Vf, V~, forms the input. It is therefore necessary to evaluate the strand Vf from the overall Vf determined experimentally. The available methodologies do not take into account the gap between the adjacent strands, the actual cross-sectional geometry of the strand, and strand undulation transverse to the loading direction. Mathematically, the series model 3 should give the lower bound of stiffness due to the assumption of the isostress condition and thereby higher complementary energy. But owing to the gross simplification of not considering the above mentioned parameters, the 1-D series model predicts higher stiffness than the expected lower bound. Also, for the evaluation of the strand Vf from the overall Vf, the information about the gap and strand undulation in both warp and fill directions is necessary.

Figure 1 presents the cross-sections of a plain weave fabric lamina at different sections from the midpoint of the fill strand (So-S0) to the midpoint

Fig. 2. Optical micrograph---cross-sectionalview of a plain weave fabric laminate.

Prediction of on-axes elastic properties of plain weave fabric composites

Fig. 3. Plain weave fabric structure.

of the gap ($4-$4). Figure 2 is an optical micrograph showing the typical cross-sections of the plain weave fabric lamina at different sections. A typical plain weave fabric structure is shown in Fig. 3. It is seen that the thickness of the fill strand decreases gradually from the midpoint of the strand to zero in the gap region. This reduction due to the strand cross-sectional geometry would reduce the overall stiffness of the WF lamina. Therefore, the geometry of the strand cross-section should be considered while evaluating the stiffness and this requires a 2-D model. The available 1-D models predict higher stiffness as the maximum strand thickness is considered in these models.

2 FABRIC COMPOSITE MODELS The plain weave fabric composite models presented here are 2-D in the sense that they consider the undulation and continuity of the strand in both the warp and fill directions. The models also account, for the presence of the gap between adjacent strands and different material and geometrical properties of the warp and fill strands.

2.1 Refined models Two refined models are presented in this section. In the first model, the unit cell is discretised into slices along the loading direction. The individual slices are analysed separately and the unit cell elastic properties are evaluated by assembling the slices under the isostrain condition. Such a model

139

is called a slice array model, abbreviated SAM. In the second model, the unit cell is discretised into slices either along or across the loading direction. The slices are further subdivided into elements. The individual elements are analysed separately. The elements are then assembled in parallel or series to obtain the slice elastic constants. Further, the slices are assembled either in series or parallel to obtain the elastic constants of the unit cell. This scheme of discretising the unit cell into slices and further into elements is called an element array model, abbreviated EAM.

2. 1.1 Slice array model (SAM) In the analysis, the strand is taken to be transversely isotropic and its elastic properties are evaluated from the transversely isotropic fibre and matrix properties at strand Vf. It should be noted that owing to the presence of pure matrix pockets in the WF lamina, the strand Vf would be much higher than the composite overall Vf. The strand properties are evaluated using the composite cylinder assemblage (CCA) model (Refs 15 and 16) which is briefly presented in the Appendix. The details of evaluation of strand Vf from composite overall Vf is discussed later. The representative unit cell of a WF lamina is taken as shown in Fig. 4(a). By virtue of the symmetry of the interlacing region in plain weave fabric, only one quarter of the interlacing region is analysed. The analysis of the unit cell is then performed by dividing the unit cell into a number of slices as shown in Fig. 4(b). These slices are then idealised in the form of a four-layered laminate i.e. an asymmetric crossply sandwiched between two pure matrix layers as shown in Fig. 4(c). The effective properties of the individual layer considering the presence of undulation are used to evaluate the elastic constants of the idealised laminate. This, in turn, is used to evaluate the elastic constants of the unit cell/WF lamina. In order to define the undulation and geometry of the strand cross-section the following shape functions are used. The below mentioned expressions are with reference to Figs 5 and 6. In the Y - Z plane, i.e. along the warp direction (Fig. 5)

hf :~y zy,(y) = - ~- cos - ayt

(1)

N. K. Naik, V. K. Ganesh

140

kZ hm/2 .

jJ''

"

~

hZ f

_ _ _ ~ ~ ( y )

~

..............

i •; - "

(a)

WARP

hw

~

hY2 [Y)

UNIT CELL hf

c

L_

0

__

$ hm/2 - - o f f 2 - -

Fig.

5. Plain

weave

-~ gfl2

fabric lamina direction.

cross-section:

warp

and hyl(y) - hf ~ + hm

zy2(y)

hy2(y) = hw (b)

ACTUAL SLICES

hy3(y) = zy2(y) - zy~(y)

y = O--~afl2

=0

y = ae/2----~ (af + g,)12

hy4(y) = hf ~ + hm

zy~(y) (3)

In the X - Z plane, i.e. along the fill direction (Fig. 6) hw :rx hm zxl(x, y) = -~- cos - - - hy~(y) + - (4) ax, 2

2

*

hw ~rx zx2(x, y) = - -~- cos (aw + gw)

FACTORED h w

hm

hyjy) + ~

(5)

(c) IOEALISEDSLICES Fig.

4. Plain

weave

fabric lamina idealisation.

unit

zy2(y)

hf :ry = ~ c o s ia r + gf)

cell

and

its

and (2) I

where

hx3!(x,y)

FILL

i

hf

r

Jtaf ay t =

2 [ ~ - cos-,(2zY'~ ] \hf/J h, ( ~ae ~ zy, = + ~- cos \2(af + &) /

J J i

hm/2 Fig.

~ .

.

.

.

4

6. Plain

axt/2---

aw/2

weave

fabric lamina direction.

cross-section:

fill

Prediction of on-axes elastic properties of plain weave fabric composites

141

where $'t'aw axt =

_~[2z~;~ t~ \hw/ h~ [ n:a. - - ~ COS~, - - ~ 2 2(a. + g . ) // 2 cos

Zx, =

0.09rnrn--=~

O-&8mm

Fig. 8. Actual geometry of the plain weave fabric lamina cross-section: simulated.

and

hXl(X, y) = hw + h.,_ _ zxl(x, y) 2 hxz(x, y ) = zx,(x, y ) - zx2(x, y) x = 0---~awl2 = 0 x = aw/2--->(aw + gw)/2

(6)

hx3(x, y) = hy3(y) hx4(x, y) = zxz(x, y) - hx3(x, y) + (hw + h,,)/2 + hf The validity of the above expressions can be ascertained by comparing the optical micrograph of the actual WF lamina cross-section along the fill direction (Fig. 7) and the simulated plot making use of the same strand parameters (Fig.

8). It can be seen in the above expressions that the parameter z~, would reduce to zero and ax, to aw when the gap between the adjacent warp strands is zero ( X - Z plane). Similarly, in the Y - Z plane, zy, would reduce to zero and ay, to a~. The idea of introducing these parameters in the shape functions is to simulate the gap between the

strands mathematically, which was otherwise not possible if the same expression is used for zx~ and zx2 and similarly for zy~ and zyz. These parameters only steepen the outer contour of the strand cross-section without disturbing the overall undulation of the strand. The slope of the fill strand is so maintained that at a given point in all the sections across the loading direction the slope of the strand is the same, i.e. the local off-axis angle of the fill strand, Of, is not a function of y. Similarly, the local off-axis angle of the warp strand, Ow, is not a function of x. The steepening of the outer contour and maintaining the same off-axis angle in all the planes across the loading direction at a given point make the cross-section of the WF lamina unit cell unsymmetrical about its midplane. This can be seen in Fig. 1 which presents the cross-sections of two adjacent unit cells. Only the cross-sections at the midpoint of the strand (S0-S0) and the gap ($4-$4) are symmetric about their midplanes. Here, asymmetry or symmetry indicates the presence or absence of averaged coupling stiffness terms of that cross-section, respectively. In all the other sections it is seen that the thickness of the top pure matrix layer is less than that of the bottom pure matrix layer. With this, h~>h~ and h~'>h~ and ha'> h~. This is the behaviour in the region AB (Fig. 1), whereas the behaviour is assumed to be the reverse in the region BC. In other words, the thickness of the pure matrix layer would be more at the top than at the bottom in the region BC. Mathematically, it means that the coupling effect of region AB and BC are balancing each other. This exercise is done to see that the averaged coupling terms are zero for the unit A C D E and this is true as the plain weave fabric composites do not twist globally on extension. This apparent twisting of the fabric on extension was seen because of the shape function considered. The other way of eliminating the coupling terms is to pull the entire fabric in such a way that h ~ - h l in section S~-S1, h 2 - h 2 in section Sz-S2 and so on. It

Fig. 7. Actual g e o m e t r y of the plain w e a v e fabric lamina cross-section: scanning electron micrograph.

__

?

tf

~

t

142

N. K. Naik, V. K. Ganesh

The volume of the pure matrix region in the unit cell can be evaluated by calculating the thicknesses in the pure matrix region for the shape functions considered and then integrating to get the volume of the matrix in the pure matrix region. The thickness ordinates in the pure matrix region are given by hx,(x, y) and hx4(x, y) as given in expression (6). Knowing the overall Vf of the WF lamina the strand Vf can be calculated. The strand fibre volume fraction is given by vw V~-~- W ° -

written as (Ref. 17) 1 s,,(8)

-

-

EL(S)

(7)

The transversely isotropic strand elastic constants can be evaluated from V~ and the fibre and matrix properties. It should be noted here that these properties are the properties of the straight strand, i.e. the properties of the equivalent UD lamina. The elastic constants of the undulated strands along the global axes are to be determined in order to evaluate the global elastic constants of the WF lamina. In the case of warp strands (Fig. 4), it is done by transforming the compliance of the warp strand for the off-axis angle at the midpoint of that slice. In the case of fill strands, the effective mean value of the compliance is calculated by considering sections of infinitesimal thicknesses along the fill strand and transforming the compliance of these infinitesimal sections along the global direction and then integrating them in the interval ( 0 - ~ O 0 , Here, O~ is the off-axis angle at x = (aw+gw)/2 i.e. the maximum off-axis angle. The local off-axis angle in the fill strand Of(x) is expressed as 0f(x) = tan -l d [zx2(x, y)] =tan- / _ --sin } \2(aw + gw) (aw + gw)

(8)

and in the warp strand it is expressed as d

El.

2VLT]m2n2+ n4

1 &2(o) =

I

i

ET(0)

ET

ET(0) s

(o) = _ _

1

ET m2

--

GLT(0)

GET

ET n2

..[_ _ _

Gvr

where m = cos 8, n = sin 0. For the fill strand the mean value of the compliance is expressed as

afo°

S~j=~

Sij(O) dO

(11)

In an actual WF lamina O is very small, and therefore the functions sin 0 and cos 0 can be substituted by the first term of their Taylor series in the integration of eqn (11). Integrating eqn (11), the effective elastic constants of the fill strand are EL

1+5Oz

(12)

GET (~fLT = (~2( G L T - 1) 1 + 3 \GTr After evaluating the reduced elastic constants of the warp strand as explained earlier and of the fill strand by using eqn (12) in the slice, and also considering the presence of pure matrix layers, the extensional stiffness of that slice can be expressed as 1

Ow(y) = tan-'-d-y [Zy2(y)]

(lO)

S12(0 ) ~-----VTL(0) _ VTL 2 _ m + --V'r'rn2

o V pm

m4

-

4

A~(y) = ~ ~ hXk(X, Y)(Q_.ij)k

(13)

k=l

1[

=tan-/

:rhf

~y

--sin ] \2(af + g0 (ae--+gf)

(9)

The respective off-axis angle reduces the effective elastic constants in the global X and Y directions. The reduced compliance can be

where, hxk(x, y) and (Qij)k are the thicknesses and mean transformed stiffness of the kth layer in the nth slice. Here, hxk(x, y) is evaluated at constant x, for different values of y. The thickness of the warp strand is maximum

Prediction of on-axes elastic properties of plain weave fabric composites at x = 0 and zero from x = a , / 2 to x = ( a , + gw)/2. Therefore the mid thickness of the warp strand is taken for the extensional stiffness calculations i.e. the thickness hw is multiplied by a factor [0.71a,/(a, + gw)]. The balance of the thickness is assumed to be filled with pure matrix. From the extensional stiffness of the slices the elastic constants of the unit cell are evaluated by assembling the slices together under the isostrain condition in all the slices, i.e. the averaged in-plane extensional stiffness is evaluated. The averaged in-plane extensional stiffnesses of the unit cell/WF lamina can be expressed as 2

c

(a) SERIES-PARALLEL COMBINATION C'

S'

~(.,+go/2

Aij = (af + gf) Jo

A~!(y)dy

(14)

It can be seen from Fig. 4 that the unit cell is not symmetric about its midplane and therefore the coupling stiffness terms are present. But owing to the nature of interlacing of the strands in the plain weave fabric the coupling terms in two adjacent unit cells of the WF lamina would have opposite signs and therefore are zero for the WF lamina as a whole. The elastic constants of the unit cell/WF lamina can then be obtained from the expressions: TM

Ex=A~,(1

A~2 )

Gxy=A66

143

(15)

A~2

Vyx A22 In the case of balanced plain weave fabrics the Young's moduli in both fill and warp directions, i.e. Ex and Ey, are the same. For an unbalanced plain weave fabric, the Young's modulus in the warp direction should be calculated by the same procedure as in the fill direction.

2. 1.2 Element cirray model (EAM) The limitations of SAM are that this method approximates the stiffness contribution of the warp strand and accounts for the gap between the adjacent warp strands approximately. It should also be noted that when the maximum off-axis angle, O, is substantially high such that the first term of the Taylor series would not be accurate enough to define the sine and cosine functions, SAM would fail to give accurate results. In E A M these constraints are overcome by

Z

X

A"

z¢" (b) PARALLEL-SERIES COMBINATION Fig. 9. Plain weave fabric lamina unit ceil discretised into slices and elements.

subdividing the slices into elements (1, 2, 3) of infinitesimal thickness (Fig. 9). Then, within these elements, the elastic constants of the warp and fill strands are transformed for the local off-axis angle (Fig. 9) and CLT is used to evaluate the stiffness of that element. The average in-plane compliance of the slices are evaluated under the constant stress condition in every element of that slice, i.e. the mean integral value of the element compliance over the length of the slice along the fill strand are evaluated. From the compliances of the slices the stiffnesses of the slices are calculated and then the elastic constants of .the unit cell are evaluated considering a constant strain state in all the slices. This procedure where the elements in the slices are assembled in series (isostress condition) and then the slices are considered in parallel (isostrain condition) is one way of evaluating the overall stiffness (Fig. 9(a)). Such a scheme is referred to as a series-parallel (SP) combination. The other way is to make the slices across the loading direction as shown in Fig. 9(b). The slices A', B' and C' are subdivided into elements. Then the elements in the slices A ' , B' and C' are assembled with isostrain condition to obtain the slice stiffness. The slice stiffnesses are inverted to

144

N. K. Naik, V. K. Ganesh

obtain the slice compliances. The slices A', B' and C' are placed in series along the loading direction. The unit cell compliance is obtained by the integrated average of the slice compliances. The unit cell stiffnesses are obtained by inverting the unit cell compliances. Thus is the parallelseries (PS) combination. Here, the expressions used to define the undulation and the geometry of the strand cross-section are the same as the ones used in SAM, i.e. eqns (1)-(6). The strand Vf and the local off-axis angle in fill and warp directions are calculated from eqns (7), (8) and (9), respectively. The elastic constants of the warp and fill strands within the element are transformed using eqn (10). Then the stiffnesses of the elements are calculated from CLT. The elastic constants of the unit cell/WF lamina are then evaluated as described earlier, i.e. by either a series-parallel or parallel-series combination. In the SP combination (Fig. 9(a)), the average of the slice coupling compliance, (b*~ , a, ~, in the n t h slice would be nullified by a similar slice in the adjacent unit cell. But, the element coupling stiffness, (Bij) e~, and bending stiffness, (D~j)e~, would increase the value of the element extensional compliance, taxi) " *'~ , on inversion. This would amount to local softening of the element and hence a reduction in the stiffness of the slice and finally the WF lamina. But in a PS combination (Fig. 9(b)), the average of the slice coupling stiffness in the nth slice, (B~j)S~, would be nullified by a similar slice in the adjacent unit cell. In a PS combination, since the slice coupling stiffnesses are zero, the slice extensional compliances,"~aij) *'~ , are not affected by the coupling and bending stiffness terms on inversion. A PS combination would therefore predict a higher value of stiffness compared to a SP combination. In a WF lamina, locally induced moment resultants would be present as a result of the application of the in-plane stress resultants. For a plain weave fabric lamina, owing to the nature of interlacing the induced moment would be such that it constrains the local bending deformation. This would amount to setting the element curvature terms to zero. When this is done, both SP and PS combinations would give the same results. In SAM the mean integral value was calculated by using an exact integration. But in E A M the integration becomes complex and the integral should therefore be evaluated numerically.

3 MODIFIED SIMPLE MODELS The simple models available in the literature are not accurate for the prediction of the elastic constants of 2-D plain weave fabric laminae. Here, modifications are suggested to the existing simple models, 3'4 which make the results of these models comparable with the refined model predictions.

3.1 Modified mosaic parallel model (MMPM) In the 1-D parallel model, 3 the fabric is idealised as an assemblage of units of antisymmetric crossply laminates placed in parallel across the loading direction. Here, the continuity and undulation of the strands are not considered. A constant midplane strain is assumed in order to evaluate the stiffness of the WF lamina. From this assumption the equations for the in-plane stiffness for a plain weave fabric lamina reduce to Aij = A~

(16)

In the above model the crossply stiffnesses are calculated from the elastic properties of the UD lamina at the strand Vt. Therefore the stiffnesses predicted by the mosaic parallel model are much higher than the experimental results. If the overall Vf of the WF lamina experimentally determined is used to evaluate the elastic properties of the UD lamina and then the mosaic parallel model is used to predict the WF lamina stiffness properties, the results are in good agreement with the predictions of the refined models as well as the experimental results.

3.2 Modified Kabeika's model (MKM) In this model (Ref. 4), the WF lamina is idealised as a three-layered laminate consisting of two undulated laminae in the crossply configuration (Fig. 10) and one pure matrix layer. Here, the UD elastic properties of the undulated laminae in the crossply configuration are reduced for undulation and then CLT is used to evaluate the elastic properties of the WF lamina. The local off-axis angle of the fill and warp strands are expressed a s : 4 d [hw

z~x~

\2

aw/

0f(x) = tan -1 ~ t - - cos - - t Ow(y) = t a n - l - v uy

cos

(17) (18)

Prediction of on-axes elastic properties of plain weave fabric composites J

stiffness of the WF lamina is evaluated using CLT. In this method, the thicknesses of the warp and fill laminae are taken as the thicknesses of the respective strands. But, in an actual case, normally the strands are elliptically shaped with maximum thickness at the mid section. This model would therefore give a higher stiffness because the maximum strand thickness is considered for calculations. Secondly, the presence of a gap is not accounted for. The presence of a gap can be approximately taken into account by replacing the strand width by strand width plus the gap between the corresponding adjacent strands in eqns (17)(19). In order to account for the elliptical shape of the strand, the strand thickness is factored to its mid value while calculating the in-plane stiffness. The remaining thickness is taken as pure matrix layer. The ordinate of the strans thickness follows the sine function, therefore the mid thickness of the warp and fill strand would be

Y

YJ

~X

IZ

h.,,,

,-",,,,(y)

e~

,y Fig. 10. Plain weave fabric lamina--representation of interlacing. The ratios hw/aw and hf/ae can be considered to be very small for the actual strand configurations. Hence the maximum off-axis angles in the fill and warp directions are Of = Ow

hf = 0.707hf /~w= 0-707hw

(21)

If the gap is present, the average thickness can be approximated as ae /~f = 0"707h'((af + gf))

(22)

:rhw

2aw :thf ~-2ae

(19)

S0(t~) d 0

hw= 0"707hw((aw+wgw)) 4 EXPERIMENTAL WORK

The reduced elastic properties of the equivalent warp and fill laminae are evaluated by finding the mean integral value of the local compliance of the respective lamina. This is done by transforming the compliance for the local off-axis angle and then integrating the transformed compliance. The average compliance may be expressed as S0= ~

145

(20)

The above expression can be used for both warp and fill strands by inserting the respective strand geometrical and elastic parameters. Inverting the compliance, the effective elastic properties can be found. Knowing the effective elastic properties of the equivalent laminae, the

The experimental programme was designed to determine the elastic moduli of the WF lamina along the warp and fill directions. The experiments were carried out on E-glass/epoxy and carbon/epoxy laminae. The thickness of the E-glass fabric was 0.2 mm, and the warp and fill thread counts were 15 per cm, while the thickness of the carbon fabric was 0.16 mm and its warp and fill thread counts were 8-8 per cm. It may be noted that even though the number of counts are the same along the warp and fill directions, the fabrics are not balanced because of different gaps along the warp and fill directions and hence a different degree of undulation. The epoxy resin LY556 with hardener HY951 supplied by Cibatul, India, was used and the

146

N. K. Naik, V. K. Ganesh

laminae were prepared at room temperature in a specially designed matched die mould. The overall fibre volume fractions of the laminae were determined as described in the ASTM specification D 3171. Static tensile test specimens were prepared according to ASTM specification D 3039. The laminae thicknesses of the E-glass/epoxy and carbon/epoxy composites were less than the minimum required by ASTM D3039. Since no other standards are available for such testing, the same standard was used for the specimens made from WF laminae. The tests were performed on a Lloyd M50K machine. The specimens were tested at room temperature (27°C) at a crosshead speed of 1 mm/min. A total of 40 specimens was tested. The scatter range for carbon/epoxy for Ey was 56-61 GPa and for Ex it was 47-50 GPa. For E-glass/epoxy, the scatter range was 17-21 GPa. The mean values of the test results are presented in the next section. The geometrical parameters of the fabric were determined by means of an optical microscope at a magnification of 20.

5 RESULTS A N D DISCUSSION Two fabric composite models have been presented for the on-axes elastic analysis of 2-D orthogonal plain weave fabric laminae. The models consider the actual strand cross-sectional geometry and the presence of a gap between the adjacent strands. An analytical technique to evaluate VI from V~' determined experimentally is also presented. The shape functions considered are compared with a scanning electron micrograph. The shape functions agree well with the actual geometry of the WF lamina. Some approximations are incorporated in SAM in order to reduce the computational complexity without compromising on the final results for actual WF lamina configurations. These approximations would predict slightly higher stiffness compared to EAM. In E A M two combinations of assembling the element stiffness are presented. In the SP combination, the local bending deformations can be considered or they can be assumed to be constrained by locally induced moments. The assumption that the local bending deformations are constrained is realistic considering the nature of interlacing of the plain weave fabric composites.

Table 1. Elastic properties of fibre and matrix Material

Fibre Carbon t9 E-glass~ Graphite 19 Matrix epoxy~

(GPa)

(OPa)

(GPa)

(GPa)

230-0 72.0 388.0

40-0 72.0 7.2

24-0 27-7 6.8

14-3 27.7 2-4

0-26 0-30 0-23

3.5

3.5

1.3

1.3

0-35

Isotropic.

In order to examine the micromechanical approaches for the prediction of the elastic constants of a WF lamina, three material systems with different weave geometries were considered. The elastic properties of the fibres and matrix are given in Table 1. Assumed geometry within the practical range was taken for the graphite/epoxy material system in order to study the sensitivity of the fabric geometry on the elastic properties of the WF lamina. The fabric geometrical parameters of carbon/epoxy and E-glass/epoxy WF laminae are the actual dimensions measured with an optical microscope (Table 2). These two material systems were considered to compare the results of the proposed models with the experimental results. Table 3 presents the measured V~' and the corresponding calculated V~. In the case of the graphite/epoxy WF lamina, V~' was calculated from a V~ of 0.8. The maximum V~ was assumed to be 0.8 to ensure that the fibres do not become contiguous. In the above calculations the V~ values of the warp and fill strands were assumed to be the same and this assumption is valid as the diameter of the fibres in the warp and fill strands are the same and the processing conditions are the same. It is seen from Table 3, that V~' is almost half of V~. This is possible in the case of a plain weave fabric lamina because of the number and size of pure resin pockets present. It may not be possible to achieve a V~' of about 0.5 and above in the case of plain weave fabric laminae. With this the maximum V~ of 0-7-0.8 would have been attained. Table 4 presents the elastic properties of the UD lamina at VI and V~' calculated from the composite cylinder assemblage (CCA) model. 15,16 With these U D lamina properties and different models described in the preceding sections and Ref. 4, the elastic properties of the WF laminae considered were predicted. In the case of the

Prediction of on-axes elastic properties of plain weave fabric composites

147

Table 2. Plain weave fabric lamina strand and weave geometrical parameters Material

Fill strand

Carbon/epoxy E-Glass/epoxy Graphite/epoxy

W a r p strand

af (mm)

hf (mm)

gf (mm)

aw (mm)

hw (mm)

gw (mm)

0.96 0-62 2.00

0.08 0-10 0-50

0-18 0-05 0,50

1.10 0.62 2-00

0-08 0.10 0.50

0.04 0.05 0.50

H (mm)

V~

0.16 0-20 1-00

0.44 a 0.42 a 0.41 b

a Determined experimentally. b Calculated from strand Vf of 0.80.

Table 3. Overall Vf and the corresponding strand Vf Material

Overall Vf (Vf°)

Carbon/epoxy E-Glass/epoxy Graphite/epoxy

Table 4. Elastic properties of UD lamina using CCA model

Strand Vf (V~)

0-44 0.42 0-41

Material

0.78 0-70 0.80

EL

ET

GLT

Vf

(GVa)

(GPa)

Gaa(GPa)

VLT

(GPa)

7-55 3.00 5.80 2.85 4-40 2-27

6.70 3.00 6.60 3.10 2,10 1,70

0.28 0.40 0.31 0.39 0.25 0-30

0.78 0-44 0.70 0.42 0.80 0.41

Carbon/epoxy

182.50 18-50 105.40 8.60 E-Glass/epoxy 5 1 . 5 0 17-50 32.25 8.55 Graphite/epoxy 311.00 6.30 161.00 5.00

E-glass/epoxy WF lamina, the strand appeared, when seen through an optical microscope, to have a slight twist. Ideally, the UD lamina properties evaluated using the CCA model should be multiplied by the fibre-to-strand property translation efficiency factor and then these properties should be used for further

calculations. But, in this case, as the angle of twist was small the translation efficiency factor was taken as unity. The predicted results are tabulated in Table 5.

Table 5. Elastic properties of plain weave fabric lamina: Comparison of predicted and experimental results Material Carbon/epoxy

E-Glass/epoxy

Graphite/epoxy

Er

Ex

Gxy

(GPa)

(GPa)

(GPa)

58.9 57-1 35.6 57.6 64.8 60.3

52-4 51-2 35.4 57.6 58.5 49.3

5.1 4-7 4.7 3.0 5.3 --

0.07 0.10 0-10 0.07 0.04 --

89.3

89-3

7,0

0.04

SAM EAM--PS raSP MMPM MKM Experiment Kabeika's method (Ref. 4)

20-3 19.6 17-9 20.7 21.7 19-3

20.3 19.6 17.9 20-7 21.5 19-3

3,7 3.7 3-7 2.9 3.9 --

0-23 0.20 0.20 0.17 0.13 --

29.2

29.2

4.9

0-12

SAM EAM--PS --SP MMPM MKM Experiment Kabelka's method (Ref. 4)

28.8 23-7 16.2 80.1 32.2 .

28-8 23.7 16.2 80-1 32.2 .

2.8 2.8 2.8 2-3 3.0

0.08 0-12 0.12 0.02 0-04

3.7

0.03

Model SAM EAM--PS --SP MMPM MKM Experiment Kabelka's method (Ref, 4)

MMPM--Modified mosaic parallel model, MKM--Modified Kabeika's model.

44-3

.

44.3

Wx

.

V7

0.44

0.42

0.41

148

N. K. Naik, V. K. Ganesh

The unit cell was subdivided into 50 slices parallel to the loading direction in SAM. In E A M , the unit cell was subdivided into 50 slices. The slices were along the loading direction in the SP combination whereas they were across the loading direction in the PS combination. For both the SP and PS combinations, each slice was further subdivided into 50 elements. Hence, the total n u m b e r of elements was 2500. This was arrived at after a convergence study. In the case of carbon/epoxy and E-glass/epoxy, the experimental values of Young's moduli along the warp and fill directions are also presented in Table 5. The comparison of the predicted results by SAM, PS, the modified mosaic parallel model (MMPM) and the modified Kabelka's model (MKM) exhibit good agreement with the experimental results in the case of carbon/epoxy and E-glass/epoxy. The local bending deformation is considered in SP and therefore the results of SP are lower compared to the results of PS. But, in an actual plain weave fabric lamina, local bending deformations due to the coupling effect in each unit cell can be assumed to be constrained. The results of PS in which the element coupling terms do not affect the slice compliance are therefore taken as realistic. It should be noted that the results of PS with or without the coupling terms would be the same and the results of SP without the coupling terms would be the same as the results of PS. The results presented in Table 5 against SP combination are the results obtained considering the local bending deformations. Comparing the results of PS and SP of E A M , it can be seen that the coupling terms have affected the carbon/epoxy WF lamina results more than the E-glass/epoxy or graphite/epoxy WF lamina results. This is due to less undulation in carbon/epoxy WF lamina and larger EL/ET ratio of the equivalent carbon/epoxy U D lamina. This would increase the absolute value of the coupling terms and thereby lead to greater local softening. In the case of the graphite/epoxy WF lamina, although its equivalent U D EL/E.r ratio is high, the effect of coupling terms is smaller, by comparison with the carbon/epoxy WF lamina, because of the greater undulation of the strands. It may be noted that the E-glass and graphite plain weave fabrics considered are balanced whereas the carbon plain weave fabric is unbalanced (Table 2). Hence, the elastic moduli along the warp and fill directions are the same for

the E-glass/epoxy and graphite/epoxy WF laminae whereas they are different for the carbon/epoxy WF lamina. For the carbon/epoxy WF lamina, since aw > af and gw < g~, one would expect E,. to be greater than Ex. Such results are obtained from SAM and PS. For SP, the trend can be different depending upon the effect of coupling terms. The contribution of the absolute values of the coupling terms is more in the overlap region than in the gap region. In the case of the carbon/epoxy and E-glass/epoxy WF laminae the difference in results of SAM and PS is less, but the results of PS in the case of a graphite/epoxy WF lamina are considerably lower than the results predicted by SAM. This is due to the lower strand thickness to strand width (h/a) ratio in the case of carbon/epoxy and E-glass/epoxy WF laminae and higher h/a for graphite/epxoy WF lamina. Recalling SAM, in order to consider the net effect of the warp strand, warp strand thickness was factored to its mid value and the effect of the gap was taken into account approximately. The difference in the results of SAM and PS is due to this approximation in SAM. The approximation seems to be valid when the h/a ratio is low as the slope of the strand outer contour would be small and the thickness of the strand would be nearly uniform along the strand width. In PS, the slices are subdivided further into elements and the thickness of the warp strand at the midpoint of that element is considered while calculating the stiffness. Therefore, this m e t h o d would give consistent results for all h/a ratios and the results would always be less than that of SAM. The only drawback of PS is that it involves more calculations and therefore consumes more computational time. General evaluation of approximate methods is, of course, impossible. The validity of the modified simple models, i.e. M M P M and MKM, is therefore assessed only on the basis of individual cases. Two actual WF laminae configurations and one assumed geometry case were considered to assess the modified simple models. The assumed geometry case was considered to evaluate the modified simple models at a high level of undulation and for the sake of generality. The results obtained from the refined models, modified simple models and a simple model (Ref. 4) are tabulated in Table 5. The simple model presented in Ref. 4 is only for close weave and

Prediction of on-axes elasticproperties of plain weavefabric composites hence it does not take the presence of a gap into account. For the present calculations, therefore, the gap was assumed to be equal to zero. The results of MMPM compare well with the results of refined models in the case of carbon/epoxy and E-glass/epoxy WF laminae and is grossly inaccurate for a graphite/epoxy WF lamina. But the results of MKM compare well for the E-glass/epoxy WF lamina and not so well for graphite/epoxy and carbon/epoxy WF laminae. MMPM gives an accurate prediction for carbon/epoxy and E-glass/epoxy WF laminae, because these WF laminae configurations have lower h/a ratios. It may be noted that while formulating this model the presence of undulation was ignored and lower h/a amounts to undulation angle tending to zero. The authenticity of this can be verified by comparing the results of MMPM with the results of the refined models in the case of a graphite/epoxy WF lamina. Here, it is seen that MMPM gives very high modulus values as it does not consider the undulation whereas the undulation is the principal parameter which reduces the stiffness in the case of a graphite/epoxy WF lamina. The prediction of MKM compares well with the results of refined models compared to the results of MMPM for the fabric structures having higher h/a, i.e. for graphite/epoxy. For the case of a carbon/epoxy WF lamina, though the prediction is not as accurate as that of MMPM, the results are also not grossly inaccurate. This is because MKM mainly considers the stiffness reduction due to the presence of undulation and the undulation is quite small for a carbon/epoxy WF lamina. The stiffness reduction in a WF lamina is mainly due to the lower V~' and the presence of undulation in the strands as compared to the U D crossply laminates. MMPM considers the reduced V~' by considering the strand properties at V~', but does not consider the undulation of strands. This model is therefore applicable for fabric structures having very much less undulation. It is worth noting here that most of the actual woven fabrics used in structural applications and made of high modulus fibres have lower h/a ratios. There can be a combination of Vf and undulation which would give practically the same results by MMPM and MKM. This can be seen in the case of E-glass/epoxy WF lamina. Figure 11 shows the variation of V~' and V~ as a function of h/a ratio. Here, it is seen that both V~' and V~

149

are constant for all h/a ratios and a given gap. This is true because the variation of h/a would correspondingly reduce the total thickness of the lamina and the volume of pure matrix regions, thereby keeping V~' constant. This clearly indicates that MMPM gives the same result for all h/a ratios for a given material system and gap. Figure 12 shows the variation of V~' and V~ as a function of the gap width to strand width (g/a) ratio. Here, it is seen that V~ reduces with the same V~ as the gap increases. Now, with this observation it can be shown that MMPM considers the gap indirectly, i.e. with the presence of the gap, V~' falls and correspondingly alters the equivalent U D elastic constants. MKM considers the effect of the reduced Vf by considering the balance of the factored warp and fill layer thicknesses as a pure matrix layer. The predicted values of Gxy and vxy are also presented in Table 5. In general, the refined and modified simple models give lower values of WF lamina G~y, whereas these models give higher values of Vxy compared to the simple model. The lower values of G~y are due to the lower value of V~' and the presence of undulation. The higher values of Vxy occur for the same reasons. The values of Gxy were evaluated by carrying out the analysis along the warp and fill directions separately in the case of carbon/epoxy WF lamina. The same results were obtained in both cases. The degree of undulation depends on the h/(a + g) ratio. Lower h/(a + g) ratio indicates a lower degree of undulation and vice versa. Figure 11 presents the effect of the hw/aw ratio on E~ as a function of gap (gw) for a balanced, plain weave fabric lamina with aw = af, hw = hf and gw = gf. It is obvious from the plot that as the hw/aw ratio increases for a given gap, Ex reduces. This is attributed to the fact that as the hw/aw ratio increases, the effect of undulation is increased. The reduction in E~ is steeper for larger values of gw. As seen from Fig. 11, the presence of larger gw can further reduce the undulation and consequently the higher value of E~ is obtained than with lower values of gw until an optimum hw/aw value is reached. The trend would be the reverse above the optimum value of the hw/aw ratio. In other words, in this range, lower values of Ex would be obtained with higher values of gw than with lower values of gw. The variation of Ex as a function of gw/aw for different hw/aw ratios is presented in Fig. 12 for a

150

N. K. Naik, V. K. Ganesh 100

GRAPHITE /EPOXY

k

-

~,,B----- gw l a w = 0-5

1-0

Ow= 2'0 mm

90

I

80 --

_

~

0.8

~ ' S T R AND Vf gw = 0"5 mm

" ~ 70

0'6 o O.

60

-o

/)'\

50 t,L,l

\~. ~N~

_

40

0.4

--OVERALL Vf gw = 0"5mm

30

0,2

20 10 0"0

1

0-1

I

0"2

l

I

0"3

0"4

O-0 05

hw/a w ]Fig. 1]. Variation of K~ and Vr as a function of h~/a,~.

balanced, plain weave fabric lamina. The effect of gwiS twofold. As the gap is increased, obviously V~' would decrease with the same V~, in turn the elastic moduli would reduce. On the other hand, the presence of a gap would reduce the degree of undulation and hence the elastic moduli would increase. From this it is obvious

that the optimum gap would give the maximum possible elastic moduli. In addition to this, the fabrics with gaps between adjacent strands, i.e. open weave fabrics, provide better wettability and in turn better performance of the WF lamina/laminate. It is seen from Fig. 12 that as gw/awincreases, Ex increases until an optimum gw

40-

GRAPHITE /EPOXY

-1-0

a w = 2"0 mrn hw/aw= 0.15

= "r-~

3s

A

o (1.

0"8

""STRAND V~

0'6

h w / a W = 0-2

__~_ .,c ~

X

uJ

0'4

0'2 hw/aw=

0'0

0"3

1

I

I

I

0"2

0,4

0"6

0"8

gw/aw

Fig. 12. Variation of Ex and Vt as a function of gw/aw.

0.0

1"0

Prediction of on-axes elasticproperties of plain weave fabric composites is reached, and thereafter it decreases. For larger h/a ratios, the numerical value of the optimum gap is greater than for lower values of h/a ratios. For certain combinations of fabric geometrical parameters, the optimum gap can be zero, as seen for hw/aw=O.15. Even though the percentage gain in Ex due to optimum gw may not be considerable, the magnitude of gap achieved can be significant enough to facilitate better wettability and formability. The gain in Ex as presented in Fig. 12 is the absolute value. The gain in terms of specific modulus would be much higher owing to the difference in densities of fibre and matrix. It may be noted that Figs 11 and 12 are plotted by using SAM. For the balanced plain weave fabric lamina, the properties along the warp and fill directions are the same. Hence, the discussion relating to Ex along the fill direction and Ey along the warp direction are the same. For the unbalanced, plain weave fabric lamina, the same analysis can be used along the warp direction to obtain Ey.

6 CONCLUSIONS The predictions of the refined models have been evaluated by comparison with the results of an experimental programme. It is seen that the predictions of the refined models match well with the experimental results. It should be noted that certain limitations are inherent in the use of modified simple models in terms of the range of applicability. The results obtained from the modified simple models, however, certaintly indicate that these techniques, when used with some judgment, are very satisfactory engineering tools. The refined model, SAM, was used to study the effect of h/a and g/a on the WF lamina longitudinal modulus and V~'. It is seen that there is a significant effect of the h/a ratio on the longitudinal modulus, but V~' is constant for all h/a ratios. With the optimum gap between the two adjacent strands, the specific stiffness would be the highest. The overall Vf of the WF lamina reduces with the increase in the gap between the adjacent strands with the same strand Vf.

Ministry of Defence, Government of India, Grant No. Aero/RD-134/100/10/90-91/659. REFERENCES 1. Raju, I. S., Foye, R. L. & Avva, V. S., A review of analytical methods for fabric and textile composites.

2. 3.

4.

5.

6.

7.

8. 9. 10. 11. 12. 13.

14. 15. 16. 17. 18.

ACKNOWLEDGEMENT This work was supported by the Structures Panel, Aeronautics Research & Development Board,

151

19.

Proceedings of the Indo-US Workshop on Composites for Aerospace Application: Part I, Indian Institute of Science, Bangalore, July 1990. pp. 129-59. Halpin, J. C., Jerine, K. & Whitney, J. M., The laminate analogy for 2 and 3 dimensional composite materials. J. Composite Mater., 5 (1971) 36-49. Chou, T. W. & Ishikawa, T., Analysis and modeling of two-dimensional fabric composites. In Textile Structural Composites, ed. Chou, T. W. & Ko, F. K. Elsevier Science Publishers, Amsterdam, 1989, pp. 209-64. Kabelka, J., Prediction of the thermal properties of fibre-resin composites. In Developments in Reinforced Plastics--3, ed. Pritchard, G. Elsevier Applied Science Publishers, London, 1984, pp. 167-202. Raju, I. S., Craft, W. J. & Avva, V. S., Thermal expansion characteristics of woven fabric composites. Proceedings of the International Conference on Advances in Structural Testing, Analysis and Design, Vol. 1, Bangalore, July 1990. Tata McGraw-Hill, New Delhi, pp. 3-10. Whitcomb, J. D., Three-dimensional stress analysis of plain weave composites. Paper presented at the 3rd Symposium on Composite Materials: Fatigue and Fracture, Orlando, Florida, October 1989. Zhang, Y. C. & Harding, J., A numerical micromechanics analysis of the mechanical properties of a plain weave composite. Computers and Structures, 36 (1990) 839-44. Ishikawa, T. & Chou, T. W., One-dimensional micromechanical analysis of woven fabric composites. AIAA J., 21 (1983) 1714-21. Ishikawa, T. & Chou, T. W., Stiffness and strength behaviour of woven fabric composites. J. Mater. Sci., 17 (1982) 3211-20. Ishikawa, T. & Chou, T. W., Elastic behavior of woven hybrid composites. J. Composite Mater., 16 (1982) 2-19. Ishikawa, T. & Chou, T. W., In-plane thermal expansion and thermal bending co-efficients of fabric composites. J. Composite Mater., 17 (1983) 92-104. Ishikawa, T. & Chou, T. W., Thermo-mechanical analysis of hybrid fabric composites. J. Mater. Sci., 18 (1983) 2260-8. Ishikawa, T., Matsushima, M., Hayashi, Y. & Chou, T. W., Experimental confirmation of the theory of elastic moduli of fabric composites. J. Composite Mater., 19 (1985) 443-58. Dow, N. F. & Ramnath, V., Analysis of Woven Fabrics for Reinforced Composite Materials, NASA-CR178275, (1987). Hashin, Z., Theory of Fibre Reinforced Materials. NASA-CR-1974, (1972). Hashin, Z., Analysis of composite materials--A survey. J. Appl. Mech., 50 (1983) 481-505. Lekhnitskii, S. G., Theory of Elasticity of an Anisotropic Body. Holden-Day, San Francisco, 1963. Tsai, S. W. & Hahn, H. T., Introduction to Composite Materials. Technomic Publishing Co., Lancaster, PA 1980. Engineered Materials Handbook, Vol. 1. American Society for Metals, Metals Park, Ohio, 1989.

152

N. K. Naik, V. K. Ganesh

transverse Young's modulus are bracketed by close bounds. The bounds of G17- are given by

APPENDIX

The composite cylinder assemblage (CCA) model (Refs 15 and 16) gives simple closed form analytical expressions for the effective composite moduli EL, GeT, VET and k, while the moduli GTr and ET are bracketed by close bounds. Here, the UD composite is modelled as an assemblage of long composite cylinders consisting of the inner circular fibre and the outer concentric matrix shell. The fibre and matrix are considered to be transversely isotropic. The transverse bulk modulus of the UD composite is given by

v,

G-rr~-~ = G ~ +

1

(km + 2G~-r)Vm

+

G~r - G~r

2G~n~r(km+ G~r)

v - v ~ 1 + (,~v~ + W when G ~ > G-~

and

k' > k m

Whereas

km(k e + G ~ ) ( 1 - G) + U ( k m + G ~ ) V f k-

v~

G'rr(+) = G~r +

(k' + G~'-r)(1 - Vr) + (k m + G~r)Vf

1 GfTr -- G-~r

Here 1

4

4V~T

1

ke

E~-

E[

G~

1

4 E~

km

P-'\ G~ < G~

+

V~T)2Vm

fl, -

Vf

V.,/U + Vf/k m + 1 / G ~

Y+

fi~ -

k m + 2G~

f m y = G'rr/G'rr

flz = U

G ~ = G~mT.

The bounds of ET are given by 4kG~±)

ET(~) -- k + mG-r-r(~)

G~V~. + G'~T(1 + Vd

where

G~T(1 + V,) + GfLTVm

transverse

shear

modulus

4kV2T

and

the

kf + 2G~-v

Vm = 1 -- Vf

( v [ T - v'~O(1/k ~ - 1/U)VW,,, Vm/k f + Vf/k m + |/G~-r

fl,

P= y-I

k~

VLT= v~TV,+ v~TVm

The

k f < k"

yfl~

c~= l + y f l 2

The longitudinal Poisson's ratio and shear modulus are given by

+

and

Here

EL = EfLVf + E'~Vm

]

(av~-fl,))J

when

The longitudinal Young's modulus of the UD composite is given by -

2G-~r(k m + G-~-r)

(1 + t~,)v,

4VLmT 1 E~ G~

4(VII

(km + 2G~)Vm

+

m = l + ~ EL