Composite Structures 9 (1988) 173-188

Poisson's Ratios in Glass Fibre Reinforced Plastics

P. D. Craig* and J. Summerscalest

Royal Naval Engineering College, Manadon, Plymouth PL5 3AQ, UK

ABSTRACT

The characterisation of the mechanical properties of an orthotropic composite material generally requires nine interdependent elastic constants: three Young's moduli, three shear moduli and three Poisson's ratios. In most papers it is the practice to quote only two orthogonal axial moduli, a shear modulus and a Poisson's ratio in the plane of the laminate. However, the value of Poisson's ratio is a function of the orientation of the loading axis relative to the principal axis of the reinforcement fibres, both in and through the plane of the laminate. In an earlier paper, the correlation of experimental and theoretically predicted Poisson's ratios was reported around the angles in the plane of the laminate. Both unidirectional and woven rovingfibreglass panels were tested. Accurate prediction of Poisson's ratio was shown to be critically dependent on the value of shear modulus used. This paper reports an extension of the previous work to consider the through-plane properties and will examine the results in the context of the Lempriere constraints.

1 INTRODUCTION

Anisotropic materials possess properties which are dependent on the direct ion in the material. In order to present a generalised Hooke 's law it will be assumed that the material is homogeneous. The stress and strain can

* Currently serving aboard HMS Dryad. t Present address: Advanced Composites Manufacturing Centre, Plymouth Polytechnic,

Drake Circus, Plymouth, Devon PL4 8AA, UK.

173 1988 Controller, Her Majesty's Stationery Office, London

174 P. l). Craig, J. Surnmerscales

be specified by second-rank tensors, with all components of the strain linearly related to all components of the applied stresses. The generalised H o o k e ' s law may be written as:

~o = S j r k ~ (1)

where subscripts i, j, k, l may each be any of the integers l(x-direction), 2(y-direct ion) and 3(z-direction). Normally the 1-direction is aligned with the fibres of a unidirectional composite and the 3-direction is the thickness of the laminate. Sijkt form a fourth-rank tensor and are termed the material compliances, and e~i and o-kt are the second-rank mathematical tensors for strain and stress, respectively. Similarly the stress--strain tensor is given by:

~rij = Cokl~kt (2)

where Cijkt are referred to as the material stiffnesses, or the terms of the modulus tensor.

Cijk~ contains 81 t e r m s (34). From the definitions of the components of the stress and strain tensors the number of independent terms reduces to 36 because:

Cabcd = Cabdc = Cbacd, e tc . (3)

The relationship between stress and strain tensors can be defined using matrices, with the four tensor suffices reduced to two matrix suffices according to:

Tensor: 11 22 33 12 13 21 23 31 32 Matrix: 1 2 3 6 5 6 4 5 4

Factors of 2 are also introduced to cater for the difference between tensorial and engineering strains:

C,jk~ = Cmn when both m and n lie between 1 and 3

2C~jkt = Cmn when either m or n lies between 4 and 6

4Cokt = Cmn when both m and n lie between 4 and 6

As an example the matrix component C45 will be equivalent to the tensor components 4C2331, 463213, etc. The tensor equations may thus be contracted to matrix and vector operations, in engineering strains rather than in mathematical strains:

ei = S~/~rj (4) ( / , j = L ,2 . . . . 6 )

~r,-- C~e~ (5)

Poisson's ratios in glass fibre reinforced plastics 175

The C 0 matrix still contains 36 terms, but due to symmetry only 21 of these are independent , and Maxwell's reciprocal theorem further reduces this number to nine independent terms for a material with three mutually perpendicular planes of symmetry. Such a material is referred to as or thogonal anisotropic, or more simply 'orthotropic'. Normally the Cq matrix is referred to the planes of material symmetry so that the components of the matrix are aligned with the principal directions of the material. In orthotropic materials the f o r m of C 0 is identical to the form of S 0, thus:

=

CII C12

C22

Symmetric

Cl3 0

C23 0

633 0

C44

m 0 0

0 0

0 0

0 0

Cs5 0

C~_

(6)

So =

I/E,, -Vl2/Ell -v,3/Eu 0 0 0

-Uel/E22 1/E22 -v23/E22 0 0 0

-v31/E33 -v32/E33 1/E33 0 0 (I

0 0 0 1/G23 0 0

0 0 0 0 1/GI3 0

0 0 (I 0 0 I/G,2

(7)

where E is the Young's modulus, G is the shear modulus, and v o is the Poisson's ratio which characterises the strain response in the j-direction when strain is applied along the/-direction. It is implicit in the symmetry of the matrices that:

This is a simple statement of Maxwell's reciprocal theorem (Ref. 1) which states that the two strains must be equal if the two stresses are of equal magnitude and sense.

In isotropic materials the shear modulus is defined in terms of the elastic modulus, E, and Poisson's ratio, v, as:

E G - - - (9)

2(1 + v)

176 P. D. Craig, J, Summerscales

and in o rder that E and G are always positive, Poisson's ratio must always be greater than - 1. Similarly the bulk modulus is defined as:

E t( - (m)

3(1 -2v )

and is positive only if E is positive and v is less than L Thus in an isotropic material , the Poisson's ratio is restricted to the range

l (11) -l<v<- 2

in order that shear or hydrostatic loading do not produce negative strain energy. The sum of the work done by all stress components must be positive in o rder to avoid the creation of energy. This latter condition provides a thermodynamic constraint on the values of the elastic constants.

The constraint was generalised by Lempriere (Ref. 2) to include ortho- tropic materials. Both the stiffness and compliance matrices must be positive-definite. Thus in terms of engineering constants:

El, E2, E3, G23, Gl3, G12 > 0 (12)

Lempriere ' s analysis yields the following results:

(1 --U23P32), (l -- P13U31), (1--V12P21 ) > 0 (13)

and

1 - u ~2 u2~ - u~3 u3~ - 1)23 p32 - - 2b'21 V32 pl3 ~> ( } (14)

and f rom the condition of symmetry of compliances as applied to Poisson's ratio (eqn (8)) it follows that the positive conditions for Poisson's ratio can be expressed:

(15)

and

1 v21 u23 u13 <2 (16)

This condit ion shows that all three Poisson's ratios cannot simultaneously have large positive values, as their product must be less than one-half.

Poisson' s ratios in glass fibre reinforced plastics 177

However , if one is negative no restriction is placed on the other two. Dickerson and Di Martino 3 have published data for cross-plied boron-epoxy composites in which the Poisson's ratios range from 0.024 to 0.878 in the or thotropic case and from -0.414 to 1.97 for a ___25 ° laminate. The reported results satisfy the condition:

(17)

for all combinations of loading direction, and hence the Poisson's ratios are reasonable within the one applicable constraint. Insufficient data exists for full validation of the results.

Garbe r 4 obtained a set of elastic property values for transversely isotropic pyrolytic graphite:

E1 = E: = 34-5 GPa (5.0 Mpsi)

E3 = 10.3 GPa (1.5 Mpsi)

V = V12 = V21 = -0"21

p r ~ /J13 = /J23 ~ 1.00

Bert 5 analysed the values obtained by Garber in respect of the Lempriere criteria and obtained:

- l < v < 1 (18)

-0.774 < v < 0.4 (19)

- 1.824 < v' < 1.824 (20)

0.35 < v' < 1.421 (21)

The most restrictive limits are inequalities (19) and (21), which are satisfied by the actual values of v and u'. The unusual Poisson's ratios of pyrolytic graphite are thus thermodynamically valid.

Jones and H e n n e m a n n 6 analysed the inequality (15) for the inplane values of a boron/ep0xy material and obtained a limiting value of v~ <__ 3.162. There was insufficient data to apply the full set of Lempriere criteria. The baseline values in their study (of the effect of Poisson's ratio on buck- ling loads where rigorous prebuckling displacements are required) were E l / E 2 = 10 and v~2 = 0.25. During the analysis the value of v was restricted to the range 0-1-1-9. Poisson's ratio was found to increase with increasing shear modulus.

178 f', D. Craig, J. Summerscales

An important practical implication of the Lempriere criteria is that a zero modulus will give a zero on the main diagonal of the stiffness matrix, and hence inversion cannot be carried out during finite element analysis. ~

This paper continues a study reported by Gulley and Summerscales ~ who examined the variation of Poisson's ratio with fibre orientation. Experi- mental and theoretically predicted values of Poisson's ratio were reported a round the angles in the plane of the laminate. Accurate prediction of Poisson's ratio was shown to be critically dependent on the value of shear modulus used in classical laminate theory: '

P12 = ~ PLT -- ~ 1 + 2PLT + ELEr Ec) sin22~ ] G (22)

where subscripts L and T represent the longitudinal (0 °) and transverse (90 °) directions and traditionally G = G,v. The nomenclature used is illustrated by Fig. 1. The opt imum correlation of theory and experiment was obtained when Huber 's equation: ~0,~

\<'El t~2 G12 = 2(1 + \.";v~2 x u2~) (23)

was used at the 45 ° orientation of fibres to loading axis. The shear moduli can easily be obtained from tensile tests with the use of appropriately positioned strain gauges or extensometers. For the perfectly symmetric case eqn (23) reduces to the isotropi¢ equation as Et = E2 and ut2 = v2~. Figures 2 to 5 illustrate the variation of Young's modulus and Poisson's ratio for the unidirectional panel (C1) and woven roving laminate panel (A2) tested in

T

z" -- "

~a

Fig. I. Nomenclature used in this paper.

R

=~

e -

i , , i i i m

2O

-16

, 2 ,0 , / . ,0 , 6,0 , 8 .0

Ang!.e between toad and fibres [degrees]

Fig. 2. Variation of Young's modulus as loading axis rotates from reintorcement direction for unidirectional panel, CI.

16

8

Poisson's ratios in glass fibre reinforced plastics 179

20 40 6,0 8,0 Angle " between toad and reference fibre direction

Fig. 3. Variation of Young's modulus as loading axis rotates from principal fibre direction for the woven roving laminate panel, A2.

180 P. D. Craig, J. Summerscales

x Theoretica[ prediction using G~,,~o~ + Theoreticat prediction us!nq 5~; ,

÷ + ×

).2 - , %

+ ~ + ÷ + +

F o Experimental measurements

2.0 , 4/) . 6.0 8D Orientation

Fig. 4. Variation of Poisson's ratio with orientation for unidirectional panel, C 1.

~3

~, t3 . .

o Experimental measurements x Theoreticat prediction using 6,2~so, ÷ Theoreficat prediction using 6~T

0.6 x

÷ + + ~ .

+ + + + ~r +

i i , ~ , 49r~entafi0690n ~ 8~)

Fig. 5. Variation of Poisson's ratio with orientation for the woven roving laminate panel, A2.

Ref . 8. The theoret ical predictions of Poisson's ratios using both GLT and G~2~45o) are inc luded in the figures.

Through- the- th ickness da ta for the Poisson's ratio of T300/5208 graphite/ epoxy laminates have been presented by Herakovich, 12 who found no o ther re fe rences on the subject. A min imum value of - 0 . 2 1 was found for the Poisson 's rat io o f a [___25°]s laminate at 25 ° (vxz) or 65 ° (v~z) to the reference axis. This cor responds to the reflection of the Poisson's ratio against angle plot abou t the 45 ° or ientat ion. Herakovich does not present in-plane da ta for the laminates examined , and hence verification of the Lempr ie re criteria is no t possible.

Poisson's ratios in glass fibre reinforced plastics 181

In this paper the three orthogonal Poisson's ratios are obtained for the same two panels tested by Gulley and Lempriere criteria are applied to the results.

2 EXPERIMENTAL DETAILS

The results reported here were obtained from two fibreglass panels which were manufactured by hand-lay-up at W & J Tod of Portland, Dorset. The 940 mm square panels were:

A2:12 layers of TBA ECK25 woven rovings (0o/90 °) in Crystic 625 TV resin, and C1:13 layers of Fothergill and Harvey Yl19 unidirectional rovings. Test specimens were cut from the panels at 10 ° intervals with additional samples at 45 ° (11 orientations per quadrant). The nominal dimensions for samples from panel A2 were 215/115 x 24 x 11 mm (sample

TABLE 1 Transducer Channels

Channel Description

30l 302 303 304 305 306

Load cell, _+ 10 kN (2518-102/UK014) Actuator , ___50 mm movement (1340-1004) Extensometer (R) 12.5 mm gauge length, _+5 mm movement (262[)-601/(193) Extensometer (Y) 12.5 mm gauge length, _+ 2-5 mm movement (2620-602/116) Extensometer (B) 10.0 mm gauge length, -4- I mm movement (2620-603/113) Extensometer (G) 10.0 mm gauge length, -+ 1 mm movement (2620-603/118)

length/gauge length × breadth × width) and for samples from panel C1 were 300/200 × 26 × 11 mm. The surface of the specimen which was furthest from the mould during fabrication was not machined flat to avoid damage to the continuity of the reinforcements.

The specimens for u~2 and u~3 were loaded in tension, within the elastic limit, in the RNEC Instron 8032 servo-hydraulic universal testing machine. Jaw pressure of the hydraulic grips was 21 kPa (300 psi) and specimen insertion in each grip was 50 mm. The actuator was positioned at the mid-point of its travel to ensure optimum accuracy. Strains were monitored using Instron extensometers as detailed above. All transducer channels were logged by a Compulog Two data acquisition system through the Instron I/O card (type A1203-1122), using Fortran II software running under FODOS. The transducer channels are listed in Table 1.

182 I'. D. Craig, Y. Summerscales

The extensometer mounted on the rough back face of the specimen was carefully positioned in shallow parallel machined grooves. The knife edges of all extensometers were adhered to the surfaces using cyanoacrylate adhesives, in order to avoid sliding of the points of contact. R and Y are used axially with B and G used transverse to the load axis.

To obtain values for v3, and/232 four layers of each material were cemented together to form a 40 mm cube which could be loaded in compression. The extensometers were mounted such that they did not cross an adhesive layer. Extreme care was necessary to ensure exact parallelism between the loading platens and the cube surfaces. The load was ramped from 0.1 to 4.0 kN to obtain the stress-strain and extensometer values.

3 DISCUSSION

The experimental axial Young's moduli and Poisson's ratios were derived from the linear portion of the stress-strain curve. The results are presented in Table 2 for the unidirectional composites and in Table 3 for the woven roving composites. In Table 3 the two orthogonal principal directions for the elastic moduli do not comply with the general assumption of transverse isotropy. A similar result for chopped strand mat and woven roving glass fibre reinforced laminate has recently been reported by Naughton et al. 13

The alignment of the fibres with the nominal principal axis was checked visually. The C1 panel was correctly aligned, but the A2 panel had 0 ° reinforcement rotated three degrees anticlockwise of the reference axis, when viewed from the front face. All results in Table 3 have been corrected to actual principal fibre orientation. The 90 ° fibres were correctly aligned in the A2 panel. The A2 panel was very slightly bowed, necessitating great care in the calculation of Young's modulus. The axial strain used in the derivation of Young's modulus was the mean of the readings from the two axial extensometers to eliminate bending effects. Only the initial linear portion of the load-strain plot, before the limit of proportionality (LOP) was used in calculating the elastic properties (Figs 2 and 3). The slight rise in Young's modulus transverse to the principal fibre direction of the C1 panel is attributed to the light weft used to hold the form of the unidirectional fabric.

In-plane Poisson's ratios were calculated from the experimental readings from the two extensometers on the front (smooth) face of the tensile specimens, using a least-squares straight line fit. Theoretical values of Poisson's ratio were calculated using eqn (22), eqn (23) and the experi- mental values of Young's modulus and Poisson's ratio on the principal axes (0 °, 90°). A mean value of the experimental EL and ET was used for the axial and transverse moduli of panel A2. Similarly a mean value of Poisson's ratio

Poisson's ratios in glass fibre reinforced plastics 183

~d

184 I'. l). ('raig, d. Suntmerscale.s

E

E

~ b ~ 2 2 2

Poisson's ratios in glass fibre reinforced plastics 185 " U,, from ~LLey x V~2 from Craig

I I V~, from Craiq , ,

0.6 "i'~"

0.4 L • •

0.;

2,0 , 4.,0 , 6,0 8,0 Orientation*

Fig. 6. Var ia t ion of Poisson's ratio With orientat ion in two planes for the unidirect ional panel , CI.

" V~2 from GuItey -~ x ~2 from Craig

.~_~ ~ from Cra

0 . 6 ~

0 . ~ ~

0.2

Orientation*

Fig. 7. Variation of Poisson's ratio with orientation in two planes for the woven roving laminate panel, A2.

for the principal axes was calculated. Huber's formula (eqn (23)) was used to calculate the shear modulus for all the specimens, as in Ref. 8.

Transverse Poisson's ratio were calculated from a through-plane extenso- meter mounted on the side of the test-piece and the mean value of axial strain between the extensometers mounted on the front and back of the specimen. Figures 6 and 7 show the variation of both in-plane and through- plane Poisson's ratio with the change of orientation between the loading axis and the principal fibre direction in-the-plane of the laminate. A bold line (somewhat arbitrarily placed) is used to indicate the trend in through-plane

186 P. D. Craig, J. Surnrnerscales

TABLE 4 Poisson's Ratio in All Three Planes--Verification of the Lempriere Criteria

Panel CI (\. Ei/Ej) Panel A2 ( \ E,IEj)

vl2 0.308 v21 0.123 v13 0.354 v31 0.124 v23 (I-417 v32 0.414

/~12 /213

/223 /221

//'31 /)32

(1.606) 0.140 ((I.942) (0.623) I). 109 (1-061) (1-687) t).408 (1.285) ((I.593) (}.247 (0-778) (1.051) 0-380 (1.364) (0.952) 0.297 (0-733)

reed' med / f / / ~ / O ~ higfi

U J// / / / / reed med

1 - v:2 v2: 0.962 0.985 1 -/213 v31 0.956 0.899 1 -/223/232 0.827 (}.887 Inequali ty (14) 0.710 0-745 v21 v23/213 0.018 0.017 /212/232/231 0"016 0'010 El (GPa) 2(}-32 15.54 E2 (GPa) 7.88 17.51 E3 (GPa) 7-14 9.41 Glz (GPa) 3.45 2.98

Indication of relative magnitudes of v

Plane of symmetry shown shaded

limit is >0

:limit is > 0

limit <.',

Poisson's ratio. The in-plane and through-plane Poisson's ratio can be seen to vary out-of-phase in both cases. The peak-in-plane Poisson's ratio occurs at around 30 ° for the unidirectional material and at around 45 ° for the woven roving. The minima in through-plane Poisson's ratio occur at similar angles.

The Poisson's ratios on all three orthogonal material axes are presented in Table 4 for both laminates. The plane of symmetry (shown shaded) has relatively low values for v in the woven roving laminate and relatively high values of v in the unidirectional roving laminate. These planes are the plane of the fabric and the plane normal to the fibres, respectively. In general, Poisson's ratio is high for contractions in the resin direction and is low when fibres lie transverse to the loading direction.

Table 4 also includes the various inequalities which define Lempriere's criteria for the thermodynamic constraint on Poisson's ratio. In all cases the experimental values satisfy the criteria.

Poisson's ratios in glass fibre reinforced plastic's 187

The materials tested were far from ideal and defects included minor warping, fibre misalignment, inhomogeneous microstructure and similar problems associated with the method of fabrication. The results must thus be regarded as indicative of the trends which should be expected and not as accurate design data.

4 SUMMA R Y

Two glass-fibre laminates (unidirectional and bidirectional) were studied experimental ly to investigate Poisson's ratio in all planes. The variation of u at various in-plane loading angles is reported for both in-plane and through- plane orientations. A full set of axial moduli and Poisson's ratio are presented for the three orthogonal planes in the material. The values obtained are shown to satisfy Lempriere 's criteria which impose a thermodynamic constraint.

R E F E R E N C E S

l. Hoff, N. J., The analysis of structures, John Wiley, New York, 1956, p. 373. 2. Lempriere, B. M., Poisson's ratio in orthotropic materials, AIAA Journal,

6(11) (1968) 2226-7. 3. Dickerson, E. O. and Di Martino, B., Off-axis strength and testing of

filamentary materials for aircraft application, lOth National Symposium, SAMPE, San Diego, 9-11 November 1966, H23-H50.

4. Garber, A. M., Pyrolytic materials for thermal protection systems, Aerospace Engineering, 22(1) (1963) 126-37.

5. Bert, C. W., Unusual Poisson's ratio of pyrolytic graphite, AIAA Journal, 7(9) (1969) 1814--15.

6. Jones, R. M. and Hennemann, J. C. F., Effect of prebuckling deformations on buckling of laminated composite circular cylindrical shells, AIAA Journal, 18(1) (1980) 110--15. (A IAA Paper no. 78-516R.)

7. Jones, R. M. and Morgan, H. S., Analysis of linear stress strain behaviour of fibre-reinforced composite materials, AIAA Journal, 15(12) (1977) 1669-76.

8. Gulley, T. J. and Summerscales, J., Poisson's ratios in glass fibre reinforced plastics I, Proc. 15th Reinforced Plastics Congress, BPF-RPG, Nottingham, 17-19 September 1986, pp. 185--9.

9. Stavsky, Y. and Hoff, N. J., Mechanics of composite structures, In: Composite engineering laminates (Dietz, A. G. H., ed.), Chapter 1, The M1T Press, Cambridge, MA and London, 1969, pp. %12.

10. Huber, M. T., The theory of crosswise reinforced ferroconcrete slabs and its application to various important constructional problems involving rectangular slabs, Der Bauingenieur, 4(12) (1923) 354--60 and 4(13) (1923) 392-5.

11. Panc, V., Theories of elastic plates, Noordhoff International Publishing, Netherlands, 1975.

188 P. D. Craig, J. Summerscales

12. Herakovich, C. T., Composite laminates with negative through-the-thickness Poisson's ratios, VPI-E-84-10, April 1984. CCMS-84-01. NASA-CR-173681. NTIS N84-27830.

13. Naughton, B. P., Panhuizen, F. and Vermuelen, A. C., The elastic properties of chopped strand mat and woven roving in GR laminae, J. Reinforced Plastics and Composites, 4(2) (1985) 195-204.