An analytical method for predicting in-plane and interlaminar thermal expansion coefficients of laminated orthotropic rings

Composites Science and Technology 45 (1992) 111-116

An analytical method for predicting in-plane and interlaminar thermal expansion coefficients of laminated orthotropic rings Ajit K. Roy University of Dayton Research Institute, 300 College Park, Dayton, Ohio 45469-0168, USA (Received 25 March 1991; revised version received 22 July 1991; accepted 9 September 1991)

An analytical method capable of predicting the in-plane (hoop) and interlaminat (radial) components of effective coefficients of thermal expansion (CTE) of laminated orthotropic rings is presented. This method is based on the linear theory of elasticity assuming the plane-stress condition in the (r, 0) plane of the ring. This method is applicable to any aspect ratio, a / ( b - a), (a: inner radius, b: outer radius) of the ring. A comparative study of the effective CTE for thin rings (a/(b - a) >> 1) indicates that although 2D lamination theory can predict the in-plane CTE quite accurately, it over-predicts the values of the interlaminar CTE, &3, by a large amount. As an example, for a thin ring made with T300/5208 [0J908]s laminates, the 2D theory predicts a value of tk3 that is 29% higher than that predicted by the present method. This difference may be attributed to the fact that the interlaminar normal stress, o3, is ignored in the 2D lamination theory, whereas Oa is included in the present analysis. As a result, the 2D theory overestimates the values of the interlaminar CTE.

Keywords: effective CTE, orthotropic ring, in-plane CTE, interlaminar C'I'E, graphite/epoxy

INTRODUCTION

maintain the structural integrity of the joints. The objective of this work is to present a method to predict accurately the effective values of the in-plane and interlaminar coefficients of thermal expansion (CTE) of laminated orthotropic rings. Pagano 1 presented a method for predicting the effective CTE of composite laminates. The 2D lamination theory prediction of the effective CTE, incidentally, is a special case of Pagano's method. Furthermore, the effective CTE of a ring is different from that of a flat laminate when the ring aspect ratio, a/(b- a) (a: inner radius, b: outer radius) is a small number. Thus Pagano's method is not expected to give a good prediction of the effective CTE for ring geometry of small aspect ratio. The method presented here is capable of predicting the effective CTE of rings of any aspect ratio. In the following sections, the analytical approach used for the present method is first discussed. A parametric study of the effects of the ring aspect ratio and the lamina

Circular rings are, in many cases, used for joining structural components of cylindrical cross sections, for instance, truss members of circular cross sections. In some applications these structures are exposed to temperature fluctuations during their operational life. For example, truss members in space structures are exposed to a wide temperature variation between the Earth's surface and in space. As a result, the designer must pay special attention to the thermal response of the rings that join the truss members together. When the tings are made with composite materials, owing to material orthotropy of composites both the radial (interlaminar) and hoop (in-plane) thermal response of the rings at the joints become very important to Composites Science and Technology 0266-3538/92/$05.00 © 1992 Elsevier Science Publishers Ltd. 111

112

Ajit K. R o y

stacking sequence on the values of effective CTE of the ring is then presented. A comparative study between the values of effective CTE predicted by the present method and the 2D lamination theory is also presented.

ANALYTICAL APPROACH In this method, the effective CTE of a laminated orthotropic ring is predicted by replacing the laminated ring with an equivalent orthotropic ring of the same geometry and undergoing the same temperature change as that of the laminated ring. The in-plane (hoop) and interlaminar (radial) components of the effective CTE of the ring are determined by matching the inner and outer radial boundary displacements of the equivalent ring with that of the laminated ring. The expressions for the radial displacement field of the laminated ring and the homogeneous ring are obtained by using the linear theory of elasticity for circular symmetry. As a result, the analysis is applicable to rings of any aspect ratio. In the following sections the displacement field of the laminated ring is first obtained, and then the boundary displacements of the laminated and equivalent homogeneous rings are matched to determine the effective CTE of the ring.

laminae are assumed to behave as an orthotropic layer. At the curing temperature, T~, the ring is assumed to be stress free. At any other temperature, To, below the curing temperature, the thermal stresses are induced as a result of mismatch of the coefficients of thermal expansion of the layers. The co-ordinate system used is shown in Fig. l(a). The ring is assumed to be free from axial stresses. Thus the ring is considered to be in plane stress in the (r, 0) plane. If there is no temperature variation in the cross section of the ring, then the radial displacement in the i 'h layer of the laminated ring, due to a temperature change A T, is given by 2 u(i)(r)

--

(

q'- ,c),+,_q,

° -

~ii

"tO, ( k, - "'")~

-

+ (1 -

k

)(1 -

[ ck'+l - 1

+

c79

"~

1

-

k2

AT

ATr

Ci

fli-I

ki :

a, '

~i/ C(i)'

/ S~ir)

~?(i) ~,(i) vrO __ -- - - - '-'tO -7.

--~,oo

S~)o'

AT=TO-T Here, qi is the normal traction acting at the interface between the i n and (i + 1) th layer (Fig. l(b)), and CCr') and cr~) are the CTE in radial and hoop directions, respectively, of the i th layer. S}~,) (j, k = r , 0) and ai are the elements of the compliance matrix and the inner radius of the i 'h layer, respectively. If the lamina of a layer is oriented in the axial

r, radial ~,~~. r , radial

~

,

~

~

.

~"

J

b=an

~I

IRing Cross Sectionl

(a)

(1)

where

Displacement field of laminated orthotropic ring The linear theory of elasticity applicable to circular symmetry is used for the stress analysis. The use of the linear theory of elasticity allows the analysis to be applicable to any thickness of ring as long as the ring behaves linearly and elastically. The layers of the ring are considered to be cylindrically orthotropic. For any off-axis (4)) lay-up in a layer, it is assumed there are adjacent (+4)) laminae, and the adjacent (-l-qb)

1S-~ icff'

(b)

Fig. 1. (a) Geometry and co-ordinate system for the laminated ring; (b) cross section of the ring.

Thermal expansion coefficients of laminated orthotropic rings direction (~ = 0), then k~ = 1 and a~° = a~g~. The expression for the radial displacement of the layer reduces to

ut°(r)= f---~2~ (q,-,c2-q,) ( 1 - v~,~) + (q,-,-q,)c2(l + v'~) ( ~ ) l + tr"'r AT

(2) The expressions for the radial displacement, u ~o, obtained above are expressed in terms of the interfacial normal tractions, q , acting on the interfaces between the layers. The interfacial normal tractions, q , are determined by satisfying the continuity condition at each interface,

u ~i)=u <~+l) at r = a ~

i=1,2 .....

Determination of effective CTE of orthotropic ring To calculate the values of the effective coefficients of thermal expansion, 0:, and 0:0, the laminated ring is replaced by an equivalent homogeneous orthotropic ring of the same geometry and undergoing the same temperature change as that of the laminated ring. & and 0:0 of the ring are determined by matching the inner and outer displacements of the equivalent ring with that of the laminated ring. The expression for the radial displacement, t~, for the equivalent ring is obtained by following the same solution procedure as for the laminated ring.

(n-l)

(3)

/~(r) =

where n is the number of layers in the ring. The substitution of either eqn (1) or (2) into eqn (3) yields a set of simultaneous equations to determine q, as follows

Y~ + Z~+, - Z~)a~

(0:, - 0:o)(/~2 - 92o)b (1 -/~2)(1 - c 2~)

rc

_

'

x t -~+ % +

qi+lfli+lai+l + qiYiai + qi_lfliai _, = (X~+~ -

113

~ - 7,~0 ',71 _JAT

&,(1 - 9 , o ) - 0:0(/~2 - 9r0) 1 --/~2 r AT

AT

for i = 1, 2 , . . . ,

( n - 1)

(4)

where a

c=a,

where

i 2kicki ~, = s~'~i--_~

r,= s~o (v~ - k. ] +-c~--2] ' 1 - c2k'/

-

-2ki+lX [,xi+~) 1 + Ci+'i "} Sgg"t-,o + k,+, 1 - 15i+1 -~'<'+'/

If ki :# 1 X , = a~'/)7-_'kteg) ~- [ vt'0"+ k, 1 + 1----c-/2~c 2k'- 2cki'-' ]]

if
z,=

(5)

1 + c 2k'- 2c k'+' ]

try')(1 - v~) - trg)(k~- "i,(i)'l VrOI

L0

Here, ~¢/j (i, ] = r, 0) are the elements of the compliance matrix of the equivalent ring. The compliance of the equivalent ring is taken as the effective compliance of the laminated ring whose values are obtained by using the technique developed by Roy & Tsai. 3 The effective coefficients of thermal expansion of the equivalent ring, as mentioned above, are determined by matching the inner and outer displacements of the equivalent ring with that of the laminated ring. Thus, after using eqn (5) and either eqns (1) or (2), the expressions for the two components of the effective coefficients of thermal expansion are

l-k, ~

1

for k~= 1

0:, - ~ [uo(t~2 + (~2 _ ~rO) ~2)

X,=Y~=0,

and

Z , = a ~ i)

To determine the effective thermal expansion coefficients, we need only consider the stresses due to a temperature change AT. In that case, qo and q,+~ are both zero.

-- Ub(t~, + (~2 -- ~rO) ~,)]

&o =

1

-~

(6)

[ua(fl2 + (1 - 9.o) t~2)

- Ub(fl, +

(1 - 9,o) 81)]

(7)

Ajit K. Roy

114 where

000

....

,

'

'

'

'

' ''

I

,

,

,

,

T~r-

,

[T3OOlS2os]

6,1[fl2 +

= Eft1 + (1 -

_

621

0.072

......... ?.'~ ..........................

Lami n o t e (;ross S e c t i o n

; ..............................

" " ' " " + . , . . " ""X,

_ [H, +

(£2

_

_

rO) 6,1[/ 2 + (1 --

p2o)b

[ c 2r"÷'

-

c;'

621

c

-

c~ ]

f l 1 = ( 1 - / ~ 2 ) ( 1 - c 2 ~ ) L /¢+9~o 4 /~---~o J AT (/~2_9%)b

[c ~ + ' - 1

N2

0.054

etT

0036

c ' + ' - c 2~]

0.018 0

a

61=1_~'-----5AT,

1

b 62 = 1 _/~2 AT

u. and ub are respectively the values of the radial displacements for r = a and r = b in eqns (1) or (2). The effective CTE, &~ and &o, in two perpendicular directions of the ring can be determined from eqns (6) and (7). It will be shown later that for large values of the ring aspect ratios &~ and &o respectively represent &3 and &2 of a flat laminate for in-plane extension. Thus this method can also be used to determine the effective CTE for flat symmetric laminates in extension.

DISCUSSION OF RESULTS The T300/5208 graphite/epoxy composite is chosen as the representative composite to discuss the results. Incidentally, the three-dimensional unidirectional lamina properties are needed as the input data for the present analysis. The lamina of a graphite/epoxy unidirectional composite is transversely isotropic. The threedimensional elastic properties of a transversely isotropic lamina contain five independent constants; four are associated with its in-plane (two-dimensional) properties and the other one can be taken as one of its interlaminar properties, for instance the transverse-transverse Poisson's ratio, v32 (or Vrr). The twodimensional lamina material data are obtained from Ref. 4. The value of vrr for T300/5208 composites is taken as 0.52 as reported by Knight? For the purpose of discussion, two cross-ply symmetric laminates, [0/90],~ and [90/0],~, are chosen as the representative laminates in Figs 2 and 3. The subscript r used above in the laminate notation indicates the number of sublaminates used in the half-laminate. In Figs 2 and 3 the effects of the ring aspect ratio (inner

10

100

INNER RADIUS THICKNESS

Fig. 2. Effect of ring aspect ratio (inner radius/thickness) and number of sublaminates (r) on the effective in-plane (hoop) CTE, &2, of laminated orthotropic rings. The two cross-ply laminates are: [0/90],~ and [90/0],~, where r is the number of sublaminates.

radius/thickness, a / ( b - a ) ) , stacking sequence of sublaminates, and the number of sublaminates are studied. Here, &2 and ~3 are the effective coefficients of thermal expansion of the ring in the hoop and radial directions, respectively. The values of &2 and ~3 in the following figures are normalized with respect to the transverse thermal expansion coefficient, trr, of the constituent lamina. It is observed in Fig. 2 that the value of &2, in general, depends on the stacking sequence of the sublaminate, the number of sublaminates present in the laminate, and the ring aspect ratio. At low aspect ratio, the values of &2 of the ring are highly dependent on the stacking sequence of the sublaminates and the number of sublaminates, r, present in the laminate. However, for a/(b - a) >- 30, the values of &2 converge

1,6

'

'

'

'

'

, , ' !

_~.(T3001S20eJ]

[

,

.

.

.

.

.

.

:: [,mi.,t**

I.S

•

1.4

........\ ..................................J r - 1 ' ~ , /r-3

I

L,=inate

Cross

][0/90lr" 1190/0lr$] ]W I~" 'x"l

-~ ~

Jr:3

Section

r=2

.-.s---, I

1.3 et T

1,2

..:iTi:.i..i.i;.ii2

............ i.....~ ............. -

hoop

1.l 1, ~xJa] ,

1 1

,

,

.....

i

10 INNER R A D I U S THICKNESS

. . . . . . . .

100

Fig. 3. Effect of ring aspect ratio (inner radius/thickness) and number of sublaminates (r) on the effectiveinterlaminar (radial) CTE, &a, of laminated orthotropic rings. The two cross-ply laminates are: [0/90],~ and [90/0]~s, where r is the number of sublaminates.

Thermal expansion coefficients of laminated orthotropic rings asymptotically to a single value, and thus both the effects of stacking sequence and ring aspect ratio disappear. Furthermore, for a given aspect ratio, as the number of sublaminates increases, the effect of stacking sequence decreases. In addition, it is also known that if the ring is made with a homogeneous material, the value of &2 will be independent of the ring aspect ratio, and &2 of the ring will be the same as that of a flat laminate. Moreover, the data in Fig. 2 reveal that, when r - 3 and the ring aspect ratio a/(b - a) >-10, the value of (~12¢ of the ring represents that of a flat laminate. Thus with increasing number of sublaminates, &2 of the laminated converges to that of the flat laminate. Similar data for ~3 are presented in Fig. 3. It is observed in this figure that the rate of convergence of the value of ~3 towards its homogeneous value is slower than that of &2- A comparison of the nature of convergence of the data in Figs 2 and 3 indicates that the effect of the ring aspect ratio, a / ( b - a ) , is more prominent in the value of &3 than that of &2. This can easily be understood by observing the variation of the values of &3 with increasing the ring aspect ratio, especially for laminates [0/90]~o~ and [90/011o~ (when r = 1 0 ) . The effect of reversed stacking sequence of the two laminates, [0/90]~0~ and [90/0]~o~, on the value of &3 does not disappear until the ring aspect ratio, a/(b -a)>30. For &2 such an effect practically disappeared at a/(b-a)>-10. Thus in the case of &3, the laminated ring behaves like a homogeneous ring only at a / ( b - a)---30. Consequently, for these two cross-ply laminates, convergence of the values of &3 towards its asymptotic value (equivalent to homogeneous material) is about three times slower than that for &2. The asymptotic values of &2 and &3 occur when the ring aspect ratio is large, i.e. when rings are considered to be thin. Thus the asymptotic values of &2 and &3 represent those for thin rings. Moreover, the asymptotic values of &2 and &3 in Figs 2 and 3 are independent of ring aspect ratio and stacking sequence of the sublaminates, and the asymptotic values of &2 and ?r3 therefore represent those for the fiat and symmetric laminated bar in extensional deformation. Furthermore, the 2D lamination theory can predict the in-plane effective coefficients of thermal expansion of flat and symmetric laminates quite accurately. Thus to check the accuracy of the present method, the asymptotic value of the

115

in-plane component of the effective CTE, &2, of the ring predicted by the present method is compared with that predicted by the 2D lamination theory in Fig. 4. The laminate [90m/01-,,]s is chosen for the comparative study. Here, m is the ratio of the thickness of the 90° layer to thickness of the half laminate. Thus, m, in this paper, is termed the fractional layer thickness of the corresponding layer. When rn = 0 and 1, the laminate [90,,/01-,,L becomes two unidirectional laminates [0]r and [90]r, respectively. For m 4:0 or 1, the values of &2 predicted by the present method are somewhat lower than those predicted by the 2D theory. However, the agreement between these two methods seems to be reasonably good. Next, the effective interlaminar CTE, &3, of thin rings (i.e. asymptotic values of &3) is compared with that predicted by 2D theory in Fig. 5. The two laminates chosen for this study are [0,,/901_,,]~ and [90,,,/01_,,]~. These two laminates represent two cross-ply laminates of reversed stacking sequence. The values of &3 in this figure are again normalized with respect to the transverse CTE, trr, of the constituent unidirectional lamina. The values of &3 for these two laminates predicted by the present method are symmetric about m being equal to 0.5. This ensures that ~3 is not dependent on stacking sequence for thin rings. The predicted values of (t3 for these two laminates, [0,,/90~_,,]s and [90,,/01_,,]~, by the 2D lamination theory are incidentally the same. But the 2D predictions of the values of &3 are in general much higher than those predicted by the present method. As an

[90m/Ol_m ] s

Laminate: 1

'

'

'

1 '

I

! 0.8

I I

o

'

I

i

'

I

'

I

'

I

'

I

}....i...........i ...........i...........i ...........iI:

[] 2DAna ysis I ii

'~

Lo mi note

!IT?0015208] f i!

~

i!

i

Cross Section

""

O.G

~2

J 0.4

3, rldi~l

~axia

~T

" '~ ' ~

0.2

0 0.2

0.4

0.6

T

hoop 1

0.8

Layer Thickness Fraction, m [0]

2

I" [go]

T

Fig. 4. Comparison of the values of &2 predicted by the present method and the 2D lamination theory. Laminate is [90m/0,_,.]s. When m = 0, the laminate becomes [0]r, and when m = 1, the laminate is [90It.

116

Ajit K. Roy

!

...................... i

1

~3

i

i

- ...................

!

~= i

I[ Laminate ~[Cr055 Section

~ - ~

i

i !

"

i--mi~iiiiiii 3, radial

otT

05

......................i......................1

o 0

[0l t

10 ,90, m .I ................ I

0.2 0.4 0.6 0.8 L a y e r T h i c k n e s s Fraction, m

or

2.h00p

1

[901 t or

[901 t [ol r Fig. 5. Comparison of the values of &3 predicted by the present method and the 2D lamination theory. Laminates are [90,,/0~_,.]~ and [0,,/90~_,,]s. When m = 0, the laminates respectively become [90]r, and [0]r, and when m = 1, the laminates are [0]r and [90]r respectively.

example, for laminate [0/90]s or [90/0]s (i.e. for m = 0-5), the 2D theory predicts a value of &3 that is 18% higher than the present method. Furthermore, in the 2D lamination theory, stress analysis is based on considering the stresses in the plane of the laminate. Hence, the interlaminar normal stress, o3, is assumed to be zero in the 2D theory, whereas in the present m e t h o d o3 is included in the analysis. As a result, the 2D theory overestimates the values of the interlaminar CTE, &3. The above comparison between the present method and the 2D lamination theory (that is made in Figs 4 and 5) is applicable to thin rings (i.e. for rings of large aspect ratio). It is known that the 2D lamination theory is not applicable to predicting C T E for thick rings (i.e. for rings of small aspect ratio). The present m e t h o d is, however, applicable to the prediction of the C T E of orthotropic rings of any aspect ratio (either thick or thin).

CONCLUSIONS An analytical m e t h o d capable of predicting the in-plane and interlaminar c o m p o n e n t s of

effective coefficients of thermal expansion (CTE) of laminated orthotropic rings is presented. This method is applicable to any aspect ratio of the ring. Thus, the effective CTE of both thick and thin rings can be predicted by this method. A comparative study of effective C T E for thin rings indicates that although 2D lamination theory can predict in-plane C T E quite accurately, it over-predicts the interlaminar CTE, &3, by a large amount. As an example, for a thin ring made with [02/908]s laminates, the 2D theory predicts a value of &3 which is 29% higher than that predicted by the present method. Incidentally, in the 2D lamination theory the stress analysis is based on considering the stresses in the plane of the laminate. Hence, the interlaminar normal stress, a3, is assumed to be zero in the 2D theory, whereas in the present m e t h o d 03 is included in the analysis. As a result, the 2D theory overestimates the values of the interlaminar CTE, &3.

ACKNOWLEDGMENT

This work was sponsored by the WL Materials Laboratory under contract n u m b e r F33615-87-C5239. The author is indebted to Professor S. W. Tsai, Stanford University, for suggesting the problem.

REFERENCES

1. Pagano, N. J., Exact Moduli of Anisotropic Laminates, Mechanics of Composite Materials, Vol. 2, ed. G. P. Sendeckyj. Academic Press, New York, 1974. 2. Roy, A. K., Response of Thick Laminated Composite Rings Due to Thermal Stresses. Composites Structures 18 (1991) 125-38. 3. Roy, A. K. & Tsai, S. W., Three-Dimensional Effective Moduli of Orthotropic and Symmetric Laminates. Journal of Applied Mechanics (ASME), 59 (1) (1992) 39-47. 4. Tsai, S. W., Composites Design, Appendix B, Fourth Ed. Think Composites, Dayton, 1988. 5. Knight, M., Three-Dimensional Elastic Moduli of Graphite/Epoxy Composites. Journal of Composite Materials, 16 (1982) 153-59.