The shape factor of composite material filament-wound flywheels

The shape factor of composite material filament-wound flywheels GIANCARLO GENTA

One of the problems which limit the potential applications of composite material flywheels built by filament winding is that of the delamination failure which occurs if the rim is not very thin. In order to compare the various solutions which can be suggested for this problem, the use of the shape factor is herein generalized for composite material rings, using a suitable failure criterion. Some methods of reducing the delamination problem are then studied. One of the most effective appears to be the varying of material properties with radius, which can be achieved with an opportune control of the winding parameters.

NO TA TION

c

subscript for circumferential direction

J

moment of inertia

e

energy

K

shape factor

f

subscript for fibre properties

R

ultimate strength

m

subscript for matrix properties

V

fibre content by volume

m

mass

a

fibre content by weight

r

subscript for radial direction

a

thermal expansion coefficient

r

radius

t3

ratio q/re

inner radius

13t

ratio q/re for multirim disks

re

outer radius

0

y

thickness

angle between the direction of the fibres and the circumferential direction

v

volume

u

Poisson's ratio

B

ratio Rc/Rr

p

density

E

Young's modulus

a

stress

F

ratio Er/Ec

6o

angular velocity

The shape factor, defined by the well known formula:(1)

elm = K o / p

is the parameter which links the energy density e l m of a flywheel with the ratio between the maximum equivalent stress a (calculated using any failure criterion) and the density p of the material. If the material shows an elastic behaviour, K is a constant, depending only on the geometrical configuration of the rotor. If the density is a constant, the shape factor can be calculated from the equation: 1 J p~2r2e K - 2 mr2e Omax

(2)

For isotropic materials a is usually calculated using Lame's maximum stress criterion. 0010-4361/81/020129-06 COMPOSITES. APRI L 1981

In the case of orthotropic materials the most common failure criteria (maximum stress, maximum strain and distortional energy, as generalized by Azzi and Tsai 1,2,3) lead to the following expression of the equivalent stress:o = max lac, BOr]

(3)

a = max [(Oc - pcrOr), B (Or - utter ac/Ec)}

(4)

a = d o c ( o c - Or) + B 2 Or2

(5)

Equations (3), (4) and (5) can be used for rotating disks built by filament winding in which the radial and circumferential directions are principal directions both for stresses and strains. The equivalent stress so calculated has to be compared with the circumferential strength of the material.

$02.00 © 1981 I PC Business Press Ltd 129

The material is assumed to be elastic, axisymmetric and orthotropic and the stress state is considered plane 3'4's. SHAPE FACTOR OF A CONSTANT THICKNESS HOMOGENEOUS RING

As it is well known: the shape factor of a very thin ring built using any kind of material is 0.5. The shape factor of an isotropic ring decreases with decreasing values of/3 until it reaches the value 0.303 for/~ -+ 0. This trend is far stronger with filament wound rings as it is shown in Fig. 1. The experimental points were obtained with burst tests on rotors similar to the one shown in Fig. 2, whose shape factor was calculated taking into account the rim only. The characteristics of the material of the rotor in Fig. 2 are very close to the ones of Fig. 1 and the agreement with distortional energy and maximum stress failure criteria is good, as in several other tests. Rotors of the type shown in Fig. 2 failed by delamination, as could be predicted from the graph in Fig. 1. The distortional energy criterion was used as it is the most conservative and was found to be in good agreement with experimental results. Failure in the right part of the curve K(fl) (for/3 > 0.93) is due to the circumferential stress, the left part being where the radial stresses are the limiting factor (failure by delamination). As it is useful to reduce the value of r, both for increasing the energy per unit volume and in order to simplify the rim-hub connection problems, the most important goal is to raise the left part of the curve K(fl). As fatigue behaviour must also be accounted for and usually the transverse properties of fibre composites are most affected by fatigue, this consideration is even more important.

Fig. 2 R o t o r w i t h E glass/epoxy r i m (a = f l y w h e e l ; b = a f t e r burst test, showing failure b y d e l a m i n a t i o n )

EFFECT OF THE CHARACTERISTICS OF THE MA TERIA L

0.6

flywheel

B=40 = 025

Various functions K(~) for materials with different ratios F = ErIEe are shown in Fig. 3a. The effect of the ratio

0.4

between circumferential and radial strength B is shown in Fig. 3b. To obtain high values of shape factor with low values of/~ the behaviour of the material must be highly anisotropic as far as elastic moduli are concerned (to keep radial stresses as low as possible) and exhibit as little anisotropy as possible in strength (to withstand the radial stresses).

0.2

o

0.1

~

I

~ 2 5 ~ 0 0.6

i

I

,

I

a =

l

,

I

i

F=O.25

0.6 ¸

v

0.4

0.4

Mox •

o2

(~ 016

0 7I

018

____~~~,oo '

0~2

'

0!4

'

0.6

'

d.8

b

'

019

Fig, 3

Fig. I

Shape f a c t o r K as a f u n c t i o n of/3 f o r an E ring ( F = ErIE c = 0.2, B = Rc/R r = 150, Vcr = 0.25)

130

glass/epoxy

Shape factor as a f u n c t i o n o f ~ for different materials (a ", materials w i t h Pcr= 0.25 B = 4 0 F = varied; b -= materials w i t h Vcr = 0.25 F = 0 . 2 5 B = varied)

COMPOSITES. APR I L 1981

The strength of the material in radial directions is lower than that of the matrix alone, due to the stress concentration ~'~. The radial strength can therefore be increased by reducing the fibre content or by using a low modulus matrix. Increasing the shape factor by reducing the fibre content is constrained due to an accompanying decrease in circumferential specific strength. The maximum attainable energy density KRe/p will accordingly show a maximum value at a certain fibre content. To solve this optimization problem, the material elastic parameters can be expressed as functions of the fibre content by volume V by the expressionsS'94°:-

.~--.,"'~/ 0.41"-

',\ /' /\

, \

o.zl-',\

VA

,'

v"

/',

/k

~I

.,,~-

"7<. / ' - 2

X\

~

IX.,\ /

.-<\/×V

4

//

,>"\

?~,

k../\×/\~/\~,i

4)"

~ /

?

× ~/

~

, I I / ",e.-,' /

l' :t.

Ec = E f V + Em(1 - V) + 4(1 - l0 (~m - ~f): EmEf/e*

(6) Er =

2Em/( )a* +7if* 8. +* 2(1 ~v vm) (1 +lXm)

(7) (s)

vet = 2Vm(Vm - uf) (l - 2Vm2)EfV/e *

V

~

x

/ ~,~,.,.

'if./

"

o.r"o.2 a

where = Era(1 -2vf)12(1 + Vm) - El(1 -2vm)12(1 + vf) = Em/2(1 + Vm) + El(3 -4Vm)12(1 + vf)

I00

3'* = a* +Er(1 - Vm)/(I +re) 8" = [El/(1 +vf) -- Em/(1 +Vm)](V/2 - 2 + 2Vm) e*

=

Em(1

_

90

2

V = 0.53 [Rc = IO00MN/m 2 (a = 0 ' 7 ) t Rr 20 M N / m 2 V =0.24 (a = 0 . 4 )

JR c = 4 8 0 M N / m 2 [Rr

40MN/m2 .-

8o

[7) (1 -- Vr -- 2re =) +El[V(1 - Vm - 2Vm 2)

+l+vm] The circumferential and radial strengths can be approximated by polynomial expressions:N = ~ , Ai Vi Re

Ef = 7 0 0 0 0 M N / m Em= 3 5 0 0 M N / m Z ~f = 0 2 um = 0 . 3 4 Pf = 2 5 0 0 k g / r n 3 Pm = 1200 k g / m 3

~=j.:::(.~;

o

J

A~ 70

ii: ? 85

,09",

~ 5o 4O

(9) 2O

i=0

M

R, =

(lO) i=O

o',

o2

o s'

o4'

I

os

I

o6

o7

V

Equation (9) is a good approximation of the experimental results even with N = 1"6. The function Rr(V) is not well known for thick rings built by t'ilament winding. The strength in directions perpendicular to the fibres appears to be lower in this case than in flat laminates with parallel fibres. Equation (8) will be assumed to be linear (M = 1), at least until experimental evidence allows the choice of a better law. There is no problem in performing the optimization using any function Rr(V). The density is obviously expressed by:p = Vpf + ( 1 -

D %

lOpm

(11)

The shape factor and the energy density of a glass fibrereinforced epoxy ring are plotted against V, with various values of/~, in Fig. 4. The value of V which gives the maximum energy density decreases as the thickness of the ring increases. As the values of the radial strength assumed in Fig. 4 are higher than those which will be obtained experimentally, this effect will probably be even greater. The use of a low-modulus matrix gives a twofold advantage; allowing a decrease in both ratio B (without affecting the strength Re) and F. The latter effect is useful, except where

COMPOSITES. APR I L 1981

Fig. 4 a = shape f a c t o r Its ~ its fibre c o n t e n t ( V ) , f o r glass/epoxy c o m p o s i t e : b = e n e r g y d e n s i t y its fibre c o n t e n t ( V ) , f o r same

F is very low (Fig. 3). Low-modulus matrices usually have the associated problem of being less effective in preventing fibre instability but in this case no fibre is under compression, and instability is not a problem. PRESTRESSED RINGS

It is possible to prestress the ring in such a way that the radial stresses are kept low. Any stress state which can cause compression in the inner fibres and tension in the outer ones will be useful. Limits to this practice are the maximum value of the stress state which can be given to the disc, the maximum compressive stress that the inner layers can withstand. Obviously the use of low modulus matrices is impossible in this case. It is also difficult to prestress the ring simply by winding under tension, as the stretching of the fibres would be very high, and the winding tension, together with the temperature and the other winding parameters is already ftxed by other requirements, chiefly by the required fibre content.

131

It is possible to build a multilayer disk and to make an interference fitting using a difference of temperature during the assembly. Unfortunately the low thermal expansion coefficient, together with the low Young's modulus, of the more common composite materials compels the use of very high temperature differences. The shape factor of two-layer thermally prestressed discs is plotted as a function of/3 in Fig. 5. Each point on the curves is obtained by ffmding the thickness of the layers which leads to the maximum shape factor. The curves are plotted for a peripheral speed of 500 m/s, with the usual plane stress state assumption. TM The use of the shape factor is in this case not really correct: it depends on the material properties and on the angular velocity (namely on the ratio pw2re2/EeATae). To obtain a significant improvement of the shape factor, impractical temperatures must be used. It should be noted that the temperatures used are those necessary for the interference fitting without taking into account any assembly clearance, and that it is not possible to have radial compression at the interface when the disc is rotating. Some connection between the layers must be provided. As the prestressing appears to be an impractical way to improve the shape factor, no multilayer discs with more than two layers are considered. CURING TEMPERA TURE

When an orthotropic body is uniformly cooled a stress state is originated. If the radial thermal expansion coefficient is bigger than the circumferential one, radial tensile stresses are produced when the ring is cooled from curing temperature, a As these tensile stresses are added to those produced by rotation, the curing temperature should be as low as possible. The effect of disc cooling is shown in Fig. 6 for glass/epoxy rings built with the same material and rotated at the same speed as that disc represented in Fig. 5. This effect is not very important as the ratio (~r/a c between the thermal expansion coefficients is only 3.14. In the case of materials with bigger ar[ac ratio or even with a negative value of(~e (eg a Kevlar/epoxy with ar/a e = - 15 2 or a graphite/epoxy with ~/ae = - 45) and a higher Young's modulus the decrease of K would have been far bigger.

0.6 E c = 50 GN/m 2 E r = I0 GN/m 2 l¢cr = 0.3 B = 40 o c = 3 . 5 x 10-6K "1

0.4

~ r = I I x 10-6K -I p =2000 kg/m 3

J ~

o~re: 5 0 0 m / s

AT--O~ / / / /

0.2 ~

q 2 Fig. 6

-

~

J /

4

/

/

-~oo -400 -600

i 0.4

i I 06 0.9 B Effect of a uniform cooling on glass/epoxy rings

VARIABLE THICKNESS HOMOGENEOUS RINGS

In general, the stresses are lowered in a ring which decreases in thickness from the inside to the outside radius, but the ratio J/mr~ (Equation (2)) also decreases and the resultant effect on the shape factor must be carefully studied. For isotropic rotors the convenience of using a conical or a uniform-stress shape is well known. As it is practically impossible to build uniform-stress discs with orthotropic materials 3, only conical discs will be considered. The shape factor of Fibreglass/epoxy rings with various values of ratios/3 and Yi/Ye and with straight taper is shown in Fig. 7. The shape factor is computed with the usual assumption of plane stress statel2: the reliability of the results decreases when the values of the ratios Yi/Ye, [Jand ye/re increase. The effect of the ratio Yi/Ye on the shape factor is very small, as can be seen from the graph. The same results, particularly for high values of ~, could be obtained for hyperbolic disks (Fig. 4 in [3] ). RINGS WITH VARIABLE ELASTIC PARAMETERS

During the winding process it is possible to change some of the parameters, such as the winding speed, the fibre pull, the temperature or the angle between the fibres and the circumferential direction. In this way it is possible to have a fibre content and an angle 0 which are functions of the radius.

o6J

Ec = 5 0 G N / r n 2 Er = I 0 G N / m z

I ~'AT= = 0 [AT2=O

Per = 0 . 3

IATt = -lOOK 2 ~ A T 2 = lOOK

B = 40 Q¢ = 3 . 5 x 10"6K "1

0.6

~: 0.4

p

= 2000kg/m

. ,. . oo , .

J

0.9

Or = II x 10"6K -I

0.4

0.8

s

/ / / / 0.7

0.2

7 7 / /

0.2

,

" LATz = I O 0 0 K

0 Fig. 5

132

I

o.,

0.6 0.5

. ,o o ,

o'B

Shape factor for two layer thermally prestressed rings

[

0

I

0.2 -/

I L0,3 2 Yi I Ye

fo.4 I

3

4

Fig. 7 Shape factor of straight tapered rings vs Yl/Ye with varying values of/3

C O M P O S I T E S . A P R I L 1981

It is even possible to obtain a disc in which the radial stresses are very low or even equal to zero, at any point. This can be performed using particular functions V(r) and 0(r). 13 Unfortunately the interest in this solution is limited by the fact that the circumferential strength of the material is lower in cases where 0 has a value other thanzero. It is possible to show that a reduction of the shape factor will be obtained in this way if the generalized distortional energy failure criterion can be directly used for each pair of contiguous layers with directions 0 and - 0 . Only the completion of the extensive material tests which are in progress will show if it is useful to wind rings with 04=0.

0.5 Constant V Linear V( r )

0.,

The full line in Fig. 8 gives the values of K as a function of /3 for rings having a linearly varying V(r) with V = 0.53 (a = 0.7) at the outer radius, and the value lying between V = 0.17 (a = 0.3) and V = 0.53 at the inner radius which leads to the maximum value of K. The graph of Fig. 8 is calculated using the same material and the same assumptions used for Fig. 4. The advantage is evident, particularly for 0.6
Rr. It is possible to build a ring using different materials, and particularly using materials with increasing stiffness or decreasing density from the inside to the outside of the ring. In particular, from the inside of the ring glass, Kevlar 29, Kevlar 49 and carbon fibres can be used. The biggest limit of this approach is that high modulus fibres, like Kevlar and graphite, which are more expensive than glass fibres must be used for the outer layers. OTHER METHODS TO IMPROVE THE SHAPE FACTOR

Those methods already seen which can be used to improve the shape factor have the primary goal of reducing the radial stresses in a fairly thick rim. The rim must be thick in order to reduce the rim/hub connection problems and to allow a fairly high energy/volume ratio.

/i

VC, I

//~/

-° )2

-1----0.4

/

/7

/./

i 0.6

I

I

J

O3

02

Alternatively, increasing V with the radius is always useful. The increase of the Young's modulus E c from the inside to the outside of the ring unloads the inner layers at the expense of the outer ones and causes a decrease in the radial stresses. The increase of the density (as usually the fibres are heavier than the matrix) produces an increase of the ratio J/mr2e. Both effects are useful to increase K.

/

/

/

t 0.8

B Fig. 8 Glass/epoxy rings (same material as in Fig. 4) with variable fibre content

eg to wrap the outside of the rim with a composite structure ~4'ls or to use very flexible spokes./~ It is possible to use different thin rims connected by elastomeric spacers or springs ~7 in such a way that radial stresses are not transferred from one rim to the others. The problem with this method is that only the outer rims are really working with a high energy density. If we have an infinite set of rims of thickness which tends to zero we have:-

K

1

=

/

(10)

1 (1 - ~ ) e/v = pco2r2e -~ Adding weight to the inner rims does not solve the problem. In the case in which the density of the rims is proportional to 1/r ~ (ie all rims are loaded with the same circumferential stress) we have:K

=

(1

-

~ ) / 4 In (1//3t) (11)

e/v = Pere2CO2(1 - 13~/2 As can be seen from Fig. 9 the last solution is better than the previous one from the point of view of the ratio energy• volume (e/v) but is worse for the energy density (elm). 0.6 _

_

p - I /r 2

0.4-

~'-. Constant ~

~ / ~ I ~ . . \

It must be noted that, if the ring has constant thickness and density,

ely = K(1 - / 7 2 )

(9)

and therefore it is useful to decrease ~3only if the decrease of K is smaller than the increase of 1 -/32. A completely different approach is to use a very thin rim, which works in the right part of the curve of Fig. 1, in such a way that delamination does not occur. To connect the rim to the hub different solutions have been suggested,

COMPOSITES.

APRI L 1981

,If

0 I/ 0

i 02

I ...... 04

e / V O'max 0.6

1 08

~'~'%'%

Fig. 9 Shape factor and ratio elvon.~x for multi-layer rings with elastomeric interlayers as a function of ratio St- The density is considered constant and proportional to 1/r 2 . Theoretical case with an infinite set of -* zero thickness rings

133

Only the use o f materials with different elastic modulus, strength and density would give optimum results from this solution.

REFERENCES 1 2

CONCLUSIONS

3

The main problem with composite material flywheels built b y filament winding is that o f delamination. All the solutions studied succeeded in reducing the radial stresses or increasing the radial strength, but only the use of low modulus matrices and the variation o f the fibre content with the radius appeared to be b o t h effective and practical.

4

As the radial strength o f the material ,appears to be the governing factor, only an extensive experimental study on filament wound thick rings to measure the effects on the radial strength o f the fibre content, the matrix properties, the winding and curing parameters, would allow optimization o f the material.

7

These considerations, which are already important for the static strength o f the rim, would have been ever more important had fatigue behaviour been taken into account.

5 6

8 9 10 11 12

ACKNOWLEDGEMENT This work was sponsored b y the CNR (Italian National Research Council).

AUTHOR Professor Giancarlo Genta is with the Politecnico di Torino, Instituto della Motorizzazione, 10100 Torino, Corso Duca Degli Abruzzi 24, Italy.

134

13 14 15 16

Azzi V.D. and Tsal S.W. 'Anisotropic strength of composites', Experimental Mechanics 5 (1965) p 283 Vicario A.A. Jr and Toland R.H. 'Failure criteria and failure analysis of composite structural components', Composite Mater 7 (Academic Press, New York, 1975) Belingardi G. and Genta G. 'Sull analisi delle tensioni in dischi rotanti in materiale ortotropo soggetti a variazione di temperatura', 11I CongressoNazionale AIMETA, Cagliari (13-16 October, 1976) MorganthalerG.F. and Bonk S.P. 'Composite flywheel stress analysis and material study', 12th National SAMPE Symp (1967) Tang S. 'Elastic stresses in rotating anisotropic disks', Int J Mech Sci 11 (1969) pp 509-517 Chamis C.C. 'Design of composite structural components' , Composite Mater 8 (Academic Press, New York, 1975) p 238 Chamis C.C. 'Mechanics of load transfer at the interface', Composite Mater 6 (Academic Press, New York, 1975) pp 31-77 Whitney J.M. and Riley M.B. 'Elastic properties of fibre reinforced composite materials', AIAA J (Settembre 1966) Sendeekyj J.P. 'Elastic behaviour of composites', Composite Mater 2 (Academic Press, New York 1975) Crivelli Viseonti I. 'Materiali compositi, tecnologie e progettazione' (Tamburini, Milano, 1975) Belingatdi G. Genta G. and Gola M. 'Optimization of orthotropic multilayer cylinders and rotating discs', L 'AerotecnieaMissili e spazio 4 (1977) Belingardi G. and Genta G. 'Generalizzazione del metodo di Manson per il calcolo di corpi a simmetria cilindrica in parete spessa in materiale ortotropo assialsimmetrico non lineare', La meecaniea italiana (June 1977) Maroeco S. Italian patent application No 23781A/76 Knigth C.E. Jr 'Deltawarap flywheel design', Flywheel Tech Symp, San Francisco [5-7 October 1977) Poubeau P.C. Communication at 1st European flywheel energy storage syrup, Thielle, Switzerland (14-15 September 1976) Proc to be published Younger F.C. 'Tension balanced spokes for fibre composite flywheel rims', Flywheel Tech Symp, San Francisco (5-7 October 1977)

COMPOSITES. APRI L 1981