Fibre Science and Technology 21 (1984) 319-326

Comparison between Flexural and Tensile Modulus of Fibre Composites

G6ran Tolf

Materials Laboratory, Atlas Copco MCT AB, S-104 84, Stockholm (Sweden)

and

Per Clarin

Department of Mechanics, Royal Institute of Technology, S-100 44, Stockholm (Sweden)

SUMMARY

The difference between flexural and tensile Young's modulus, as experienced experimentally, is analysed theoretically. It can be explained by large shear deformation and a heterogeneous cross section, making the formulae from elementary bending of beams incorrect. Some recom- mendations for performing three-point bending tests are given at the end of the paper.

1. INTRODUCTION

The elastic modulus of fibre composites is measured either by pure tension or by three-point bending. Three-point bending is usually preferred owing to the simplicity in manufacturing test specimens and in performing the test. However, it is a well-known fact that these two methods do not give the same result for the elastic modulus. The reason for this has been analysed earlier 1 and is attributed to shear deformation in the three-point bending test. In ref. 1 there are also recommendations on how to perform a three-point bending test. Thus, one would think, the national standards have been changed, so that the difference in results between the two test methods can be accounted for in an accurate way.

319 Fibre Science and Technology 0015-0568/84/$03.00 Elsevier Applied Science Publishers Ltd, England, 1984. Printed in Great Britain

320 Gdran Toll, Per Clarin

However, this is not the case at all. In Sweden the international standard ISO R 178 is used when performing three-point bending tests on composites, despite the fact that this standard was originally developed for non-reinforced plastics. It gives 16 as the lowest allowable length-to- width ratio of the test specimen, compared with the suggested value of 60 given in ref. 1.

The Swedish company ASEA Plast AB, manufacturing advanced reinforced plastic products, asked us to investigate this problem and to try to explain the difference between the two test methods.

The difference can be explained by two reasons. The first is that the specimens used in the three-point bending test are too short, giving rise to large shear deformations, and thus making the formulae from elementary beam bending incorrect. The other reason can be found when doing the measurement on a laminated beam. In this case the heterogeneous cross section gives a stress distribution that is not continuously linear but only piecewise linear, thus making the elementary formulae incorrect.

In the next two sections we give a short analysis of these two phenomena and compare them with experimental results in Section 4. Finally, we give some recommendations in Section 5.

2. SHEAR DEFORMATION

The easiest way to take account of shear deformation is to use the Timoshenko theory for bending of beams. 2 In this theory the cross section is assumed to remain plane but, unlike the elementary theory, it is free to rotate relative to the centre-line of the beam. This gives a constant shear stress over the cross section. It is a simple matter to calculate the deflection under the load according to Fig. 1, using this theory, and we get

P I 3 [ E /h~27 w-4~l ~ 4~----~GA-We l+~)J (1)

where E is the Young's modulus along the beam, G is the shear modulus in the plane of deformation, w e (=p13/48E I ) is the deflection from elementary theory and k is a constant taking account of the fact that the shear stress is not constant over the cross section.

For a rectangular beam we have k = 5/6. For an explanation of the remaining symbols, we refer to Fig. 1.

Flexural and tensile modulus of fibre composites 321

I P

W

L

Fig. 1. The bending test and cross section configurations.

We can see that for an isotropic beam, having E/G around 2.5, the error made by using the elementary theory is small when l/h = 16, as is suggested in ISOR 178. However, for an anisotropic beam having E/G around 50, the error in w is, for l/h = 16, over 20 ~o. The elastic modulus E is determined from

P13 I E /h'~27 I E /h'~27 E-4-~-wl 1+~-~7) J=E. 1+~) J where E e is the Young's modulus determined by ISO R 178. Thus we see that we get the same error in E as in w. The best way to correct this error would be to estimate or measure G, or simply to use specimens with l/h large enough to make the error in the second term in E small. To get the same error in E as for an isotropic beam having l/h = 16, we would have to take l/h = 60 when E/G = 50, the same value as is suggested in ref. 1. It is, of course, a disadvantage to manufacture such long specimens, but this is still the only way to keep the error small.

3. HETEROGENEOUS CROSS SECTION

Another problem is encountered when analysing laminated composite beams. Due to the different elastic moduli and principal directions between the layers, the stress distribution over the cross section is not continuous, but only continuous and linear within each layer. Without taking account of shear deformation we can, for example, suppose that we use test specimens that are long enough according to the previous

322 Gi4ran Tolf, Per Clarin

paragraph, so that Young's modulus in tension, E t, and bending, Eb, can be determined from: 3

N

E, = ~ Eit i (2)

i=1

N 4Z( ) E b = h~ Eit i 3 -2 + (3)

i=1

where E i is the Young's modulus of layer i, ti is the thickness of layer i, z i is the distance from the centre-line to layer i and N is the number of layers.

We can see that the two results generally coincide only when we have one layer. In the bending test we overestimate the importance of the outer layers.

This problem can never be accounted for, and we must accept that there is a difference between the two types of Young's modulus. If we have a composite structure subjected to bending we must use E b in the design process and conversely we must use E t for in-plane loading of the same composite.

4. COMPARISON WITH EXPERIMENTS

To compare these results with experiments, a balanced symmetrical laminate was manufactured by taking four layers of a E-glass mat, surface weight 300gm -2, three layers of E-glass roving, surface weight 900 g m-2, and using polyester as matrix.

The test specimens were cut out so that the roving was along the specimen. The tension test specimens were fabricated according to ISOR527, while the bending test specimens had width 10mm, height 4 mm and lengths between 70 mm and 260 mm. Nine or ten samples of each specimen were fabricated.

In Fig. 2 we can see how the Young's modulus calculated from ISO R 178, E e, varies with l/h. The broken line is the value calculated from Fig. 3, as explained below. We can rewrite eqn (1) as

4bhw 1(~) 2 l PI - E + kG (4)

Flexural and tensile modulus of fibre composites 323

E GPa

20

19

18

17

16

15

14

13

12

11

10 > < O

Fig. 2.

tt t

20 30 4o 50 60 1/h

Young's modulus determined from ISO R 178 as a function of l/h.

and in Fig. 3 we see 4bhw/Pl as a function of (l/h) 2. From this figure we can calculate both E and G, and we get

E = 16.9 _ 2.5 GPa

G = 0-52 _ 0.08 GPa

To investigate the cross section we used a camera-suppl ied microscope to take photographs of the cross section, see Fig. 4. F rom these photographs we could determine t i and zi. Together with the estimates from earlier experiments Ema,= 10GPa and Eroving = 30GPa, we can calculate E t and E b f rom eqns (2) and (3), with the result E t = 20 GPa and E b = 17 GPa. This result can be compared with the experimental ly determined E t = 19.0 + 1 GPa and, as we saw, E b = 16-9 -4- 2.5 GPa.

Thus, we see that the correct ion given by taking account of the shear deformat ion is quite accurate, and that we actually need, at least in this case, l/h = 40 to get a small error in E b.

324 G6ran Tolf, Per Clarin

4bhw PI

lO-3mm2/i

22.5

200 /

175

150

Z25

I00

75

5O

25

i 0.5

Fig. 3.

i o 2 I 1.5 2 ,5 5 ~

The experimentally-determined eqn (4).

([lh) 2

~xlO 3

Flexural and tensile modulus of fibre composites 325

Fig. 4. Microscope picture of the cross section.

We can also see, from the investigation of the cross section, that the predicted difference between E t and E b does exist, and can be explained by usual laminate theory.

5. RECOMMENDATIONS

In order to get small errors in Eb, it is essential to get an estimate of E/G, so that we can use eqn(1) to determine the necessary l/h ratio for the bending test specimen. If we cannot estimate E/G, we must perform several tests with different l/h ratios.

The difference between E t and E b due to heterogeneous cross section always exists. Depending on the loading of the composite, i.e. whether bending or in-plane loading is used, we must measure the design parameter that is of interest to us.

ACKNOWLEDGEMENTS

The authors thank The Swedish Board of Technical Development and ASEA Plast AB, who sponsored this work, ASEA Plast AB for manufacturing the test specimens and giving us access to their test equipment, and Professor Kurt Berglund at ASEA Plast AB for rewarding discussions during this work.

326 G6ran Tolf, Per Clarin

REFERENCES

1. C. Zweben, W. S. Smith and M. W. Wardle, Composite materials: testing and design, ASTM STP 674, 1979, pp. 228-62.

2. Y.C. Fung, Foundations of solid mechanics, Prentice-Hall, Englewood Cliffs, New Jersey, 1965.

3. R. M. Jones, Mechanics of composite materials, Scripta Book Co., Washington, D.C., 1975.