The Relationship Between Tensile Strength and Flexure Strength in Fiber.reinforced Composites,

F lexure - and tens i le -coupon data on un id i rec t iona l g raph i te -epoxy compos i tes

a re compared to a Weibu l l two-parameter s ta t i s t i ca l - s t rength mode l

by J.M. Whitney and M. Knight

ABSTRACT--Tensile data on unidirectional composites generated from a flexure test usually yield a higher strength than observed from a standard tensile coupon. According to a statistical-strength theory based on a Weibull distribution, the presence of a stress gradient in the flexureqest results in an apparent increase in tensile strength as compared to the tensile test under uniform stress. In the present paper, this concept is explored by utilizing data from unidirectional graphite-epoxy composites to compare with theoretical results generated from a two-parameter Weibull distribution. A larger variation in tensile strength is observed from tensile- coupon data than from flexure data. Such differences are not in accordance with strength theories based on a uniform flaw distribution and raise questions concerning variability of the test methods, as well as sources of material variability.

List of

B(S) = b= L=

h

S= Sb = So =-

So Sb =

S t .=

X=

P~o = Xo=

Symbols

risk of rupture at stress S width of test specimen, mm (in.) length of tensile specimen or span of beam specimen, mm (in.) thickness of test specimen, mm (in.) maximum stress, MPa (psi) characteristic bending strength, MPa (psi) scale parameter in two-parameter Weibull distribution maximum likelihood estimate of So adjusted maximum likelihood estimate for scale parameter under bending load, MPa (psi) characteristic tensile strength, MPa (psi) adjusted maximum-likelihood estimate for scale parameter under tensile load, MPa (psi) normalized strength scale parameter for normalized strength maximum likelihood estimate of normalized- scale parameter shape parameter in Weibull distribution

J.M. Whitney and M. Knight are Materials Research Engineers, Non- metallic" Materials Division, Air Force Materials Laboratory, Wright- Patterson Air Force Base, OH 45433. Paper was presented at 1979 SESA Spring Meeting hem in San Francisco, CA on May 20-25. Original rnanuscripl submitted: October 12, 1979. Final version received: January 29, 1980.

= maximum likelihood estimate of shape parameter & = unbiased estimate of pooled-shape parameter for

tensile loading ~b = unbiased estimate of pooled-shape parameter

for bending

Introduction Tensile data on unidirectional composites are often

used as one of the key factors in materials selection, and also provide basic ply properties which are used in laminate design. Such data generated from a unidirectional- flexure test usually yield higher strength than data ob- tained from a standard tensile coupon. It is primarily for this reason that flexure data are not considered appropriate for design purposes. This difference in apparent tensile strengths can be accounted for, however, if one considers the brittle nature of most polymeric-matrix composites. In particular, a statistical-strength theory based on a Weibull distribution' can be used to explain the difference between unidirectional-tensile data generated from a flexure test and a standard tensile coupon. Tbe presence of a stress gradient in the flexure-test results in an apparent increase in tensile strength compared to the tensile test under uniform stress. Establishment of a viable relation- ship between the flexure test and standard-tensile-coupon test would provide a potential basis for use of the flexure test in the generation of unidirectional-design data. Since flexure tests are easy to run and relatively inexpensive, a large statistical-data base obtained with this method rather than tensile coupons is far more economical.

A two-parameter Weibull model was used by Bullock 2 in correlating 4-point flexure and tensile-coupon data for unidirectional graphite-epoxy composites. Excellent agreement was obtained between theory and experiment. The Weibull model has been previously applied to ceramic materials '.4 and more recently to randomly oriented short-fiber composites.'

In the present work, unidirectional graphite-epoxy tensile data are obtained on both 3-point and 4-point flexure tests, as well as on straight-sided tensile coupons. The influence of specimen thickness on tensile strength is investigated in addition to the effect of stress gradient. Thus, a much broader data base for comparison to Weibull statistical theory is available in the current work than presented by Bullock. 2 Unlike the experimental

Experimental Mechanics 9 211

P

if- b.~

results discussed in Ref. 2, however, a significantly larger variation in tensile strength vs. flexure strength is obtained with the current data. This trend is observed in two entirely different graphite-epoxy material systems. Such differences are not in accordance with statistical-strength theories based on a uniform flaw distribution. Possible sources of this departure from classical brittle-failure theory are discussed in detail.

Statistical.strength Model According to the Weibull statistical-strength theory for

brittle materials,' the probability of survival, P, at a maximum stress level S for a uniaxial-stress field in a homogeneous material governed by a volumetric flaw distribution is given by

P(S / >_ S) = R(S) = exp [ -B (S) ] (1)

where S/is the value of the maximum stress at failure, R is the reliability, and B is the risk of rupture. A non- uniform stress field, o, can always be written in terms of the maximum stress in the following manner

o(x ,y ,z ) = S f (x ,y ,z ) (2)

For a two-parameter Weibull model, the risk of rupture is of the form

B(S) = A (So, c~ > 0) (3) o0

where

A = Jv [ f (x ,Y ,Z) l '~dV (4)

and So is the scale parameter, sometimes referred to as the characteristic strength, and c~ is the shape parameter which characterizes the flaw distribution in the material. Both of these parameters are considered to be material properties independent of size. Thus, the risk to break will be a function of the stress distribution in the test specimen.

Equation (3) can also be written in the form

~_2 9cm(9in )

J - P/2

P/2

L=15.2 cm (6 in)

b = 1.27 cm (0.5 in.)

~L /2 "q J L ~.~- b -,,.{

P/2

P/2

t

P/2

L /h=52, b=l .27cm(O.5in. ) h = 0 0127 cm/PLY (0 005 in / PLY )

Fig: 1--Geometry of test spec imens

TABLE 1--TEST MATRIX

Material Tension 4-PT Flex. 3-PT Flex.

T300/5208,8 Ply 25 T300/5208,16 Ply 20 AS/3501-5,16 Ply 36

- - 28 21 25 30 28

where

B(S) = (5)

SA = SoA -'/~ (6)

and the reliability function, eq (1), can be written as a two-parameter Weibull distribution

e(s) = exp [ - (~)~ (7)

Thus, tensile tests from specimens containing different stress fields can be represented by a two-parameter Wei- bull distribution with the same shape parameter, but with a scale parameter which will shift according to eq (6).

For the case of a simple tensile test under uniform stress, eq (6) takes the form

SA = S, = So(V,) -' '~ (8)

where the subscript t denotes simple tension. Thus, the scale parameter for uniform tension is a function of specimen volume.

For flexural loading, the integration in eq (4) can be performed in closed form and results in the following relationships between the scale parameters for tension and flexure

__ = ( V t ] ] l / ,~ Sb [2 (c~ + 1) 2 . - -~- O-point) (9) S,

S~ _ [4(c~+ 112 ~ ~,~ S, (a + 2) ( )] (4-point) (10)

where the subscript b denotes bending. The results for 4-point flexure correspond to loading at quarter points. In order to illustrate the effect of nonuniform stress distribution, consider the case V, = Vb. For values of c~ = 15 and 25, eqs (9) and (10) yield

Sb 1.52, a = 15 - - = (3-point) (11) St 1.33, c~ = 25

Sb 1.31, a = 15 - - = (4-point) (12) St 1.20, c~ = 25

These values of c~ are typical of currently utilized com- posites such as glass-epoxy and graphite-epoxy. Thus, the flexure test can, in theory, produce significantly higher tensile strengths than the tensile test, with the 3-point loading producing the highest strength. This is due to the fact that the maximum stress is produced at the outer surface in the center of the beam, while the 4-point loading produces the maximum stress at the outer surface throughout the center section. In particular, the smaller

212 9 June 1980

TABLE 2--WEIBULL PARAMETERS

..T300/5208, Test Si, MPa (kSl)

8 Ply, ^T300/5208, A oq Sol, MPa (kSl)

16 Ply, ^AS/3501-5, 16 Ply, (~i Sol, MPa (kSI) ^ ot i

Tension 1790 (259) 17.7 1665 (241) 18.5 1506 (218) 13.3 Tension * 1776 (257) 20.4 . . . . Tensiont . . . . 1506 (218) 13.2 4-Pt Flex - - - - 1734 (251) 29.3 1624 (235) 29.2 4-Pt Flex* - - - - 1741 (252) 28.7 - - - - 4-Pt Flext . . . . 1624 (235) 32.7 3-Pt Flex 2377 (344) 41.4 1790 (259) 36.7 1617 (234) 22.9 3-Pt Flex* 2377 (344) 42.6 1797 (260) 36.2 - - - -

*Based on a reduced sample size of 20, tBased on a reduced sample size of 28.

the volume under maximum stress, the higher the local strength.

It should be noted that eqs (9) and (10) are based on the assumption that failure in the flexure test is a direct function of normal stress on the tension side of the beam. The effect of interlaminar stresses is neglected.

Specimen-thickness effects, as well as stress-gradient effects, are also of interest. For pure tension, eq (8) becomes

S, = So(Lbh) . . . . (13)

where L, b and h are gage length, width and thickness, respectively, of the tensile coupon. For specimens of thickness h, and h~, eq (13) yields

S,, = (h ,} , , , , S,~ .~, . , h~ > h, (14)

Thus, the thin specimens will have a higher characteristic strength compared to the thick specimens. In the case of flexural loading, the beam span, L, must also be adjusted for any thickness change in order to assure a constant span-to-depth ratio in the flexure test. For 3-point loading, eq (6) becomes

Sb = So [ 2(oe + 1)~_ (?)],/ ,* (3-point) (15) bh 2 L

For specimens of thickness h, and hz, with L /h constant, eq (15) yields

&~ U~" , h, > h, (16)

Because of the requirement for a constant L /h ratio, any thickness change will have greater effect on the flexure test than on the tensile-coupon test. Again, thin specimens should yield a higher characteristic strength than thick specimens. It is also obvious that eq (16) holds for 4- point bending as well as 3-point bending.

Experimental Procedure and Data Reduction Two graphite-epoxy material systems were chosen for

this investigation, T300/5208 (Narmco) and AS/3501-5A (Hercules). Unidirectional panels were fabricated in an autoclave according to each manufacturer's recommended cure cycle. The average fiber-volume content was 70

percent for T300/5208 and 65 percent for AS/3501-5A. Both 8-ply and 16-ply panels were processed for T300/ 5208, while only 16-ply panels were fabricated for AS/ 3501-5A. Test specimens were cut from the large panels with a diamond wheel.

Specimen geometry and dimensions for both tension and flexure are shown in Fig. 1. For tension, a straight- sided coupon was utilized in accordance with ASTM Standard D-30397 The flexure tests were run in accor- dance with ASTM Standard D-7907 with the following deviations. Loads were applied at a distance of L /4 from the supports, rather than at a distance of L /3 as required by the ASTM standard. In addition, the specimens were 13 mm (0.5 in.) wide rather than 25 mm (1.0 in.). These deviations have become accepted practice for graphite- epoxy composites. A test matrix is shown in Table 1.

The Weibull parameters for each data set were estimated from the maximum likelihood estimator (MLE). 8 A two- sample test" was utilized for testing the equality of shape parameters in a two-parameter Weibull distribution with unknown scale parameters.

Weibull parameters are shown in TabJe 2 for each data

1.0 ~ - [ TENSION

0 8 L " v ~ [U] [ ' "

\ 06 2~

0 4 =~ = 184

X. 0.2 0 I 1 I

08 09 1.0 X

Fig. 2- -Pooled Weibull distribution for tension loading, T300/5208 graphite-epoxy composites

Exper imenta l Mechan ics 9 213

Fig. 3--Weibull distributions for tension loading, T300/5208 graphite-epoxy composites

I 0

0.8

06

u3

0.4

0.2

0171

S(MPa)

1400 1600 1800

TENSION

T500/5208 tol

S t = !665 MPa '~ X ~

"~ ~ / -S~ :I790MPa (259 KSI)

5~ =18 4

8 PLY

2;o S(KSI)

2000 I

:500

set. In order to use the two-sample test results tabulated in Ref. 9, it was necessary to reduce the number of replicates in some data sets so that equal sample sizes could be obtained within each material system. This was accomplished by numbering the failures in each data set to be reduced from lowest strength to highest strength and using a table of random numbers to discard the appropriate number of specimens. WeibuU parameters were then determined from the reduced sets by use of MLE. The resulting shape parameters, ~i, represented estimates for equal sample sizes within each material system and the tabulated data from Ref. 9 could then be applied. Weibull parameters associated with equal-sample- size sets are also shown in Table 2.

For cases in which the shape parameters satisfied the two-sample test criterion for a confidence level of 98 percent, a data-pooling technique was used to estimate a shape parameter for the pooled population. The approach utilized in the present paper involves normalizing each data set included in the pooling procedure by its estimated characteristic strength ~~ and the resulting normalized strength data was fit to a two-parameter Weibull dis- tribution, again utilizirig MLE. For a perfect fit to the data-pooling scheme, the scale parameter of the pooled distribution should be exactly unity. Each scale parameter of the individual distributions were adjusted, however, to produce an exact value of unity for the pooled-scale parameter. In particular

A rSpi = XoSo i (17)

where Spi, Xo, and So i are adjusted values of the scale parameter for the ith data set, the MLE estimate of the pooled-location parameter, and the MLE estimate of the location parameter for the ith data set, respectively. It should be noted that MLE is asymptotically unbiased, i.e., it is a biased estimator for small sample sizesP Un- biasing factors are tabulated in Ref. 11. Values of pooled parameters were adjusted for bias and denoted by &v-

Discussion of Results

Application of the two-sample test to all of the data within each material system failed to indicate a constant value of ai . Application of the two-sample test to flexure

I0 ~ BENDING ~

0.8 ~ [0]

06 T3_OOJ52o8

t I t "~,q~ 00% o ss 090 095 too

x Fig. 4--Pooled Weibull distribution for flexural loading, T300/5208 graphite-epoxy composites

1.05

data and tensile-coupon data separately, however, indicated that pooling procedures would be appropriate for each of these test methods. Since tensile-coupon data on AS/ 3501-5A composites were obtained for only one specimen geometry, data pooling could only be accomplished on flexure strength for this material.

Comparison between strength data and Weibull dis- tributions obtained from the data-reduction procedures are shown in Figs. 2-9. Data points are converted to probabilities of survival from the "median rank" (MR) defined as

MR - j - 0.3 (18) n+0.4

where j is the survival order number (data listed in de- creasing order of strength) and n is the total number of samples tested. Pooled-shape parameters are denoted by

214 9 June 1980

I 014OO

08

06

04

02

0 200

S(MPo) 1600 1800 2000 2200 2400

O BENDING ~_\ \O T300/5208 \o ~

~_~ g O 3PT, 16 PLY r~c~ ~ 3PT, 8PLY

gb =1734 MP(] ~] OI Sb = 2377 MISo "~

~[3 O~r = 1790 MPo

I f , , L , \q J r I ~ 225 250 275 300 325 350

StKSI)

Fig. 5--Weibull distributions for f lexural loading, T300/5208 graphite-epoxy composites

E

1.0

0.8

0.6

04

0.2

S(MPa) I100 1300 1500 1700

TENSION

AS/3501-5A ~)~

?,--128 g, : 1499 MPa ~3

(217 KSI)

I I I I- ~ 150 175 200 225 250

S(KSI)

Fig. 6--Weibull distribution for tension loading, AS/3501-5A graphite-epoxy composites

~ 0 ~ ' ~ 0 ~ m - - i r T

O608 ~ ~ S E N D I N G

g ab :251

04

02

O I J I, I O 80 O 85 090 095 IOO 105

X

Fig. 7--Pooled Weibull distribution for flexural loading, AS/3501-5A graphite-epoxy composites

JO

~t and ~b where the subscripts l and b denote tension and bending, respectively.

Note that, for both graphite-epoxy material systems, the ratio of bending-shape parameters to tension-shape parameters, ~b/~,, is approximately 2, which is a departure from classical Weibull theory. The characteristic flexure strengths are, however, consistently higher than the characteristic tensile strengths as predicted by the Weibull failure model.

The difference obtained in shape parameters between tensile coupons and flexure tests suggests that their failures are governed by two different flaw distributions. Typical failure modes for tension and flexure demonstrate the same brooming type of failure mode for both loading methods. Tension tends to produce a more catastrophic failure due to the uniform stress field, while flexure loading produces a more localized failure due to stress gradients. Similar modes could lead one to believe that the failures are governed by the same flaw distribution. This can, however, be misleading as further discussion will show.

Experimental Mechanics 9 215

S(MPe) 1300 1500 1700

I0 , I ~ (3 I 0 I

~Q.~ BENDING 0.8 ~ (4PT)

AS/3501- 5A ~ _

E Sb = 1624 M P o ~ 04 (235 KSI) ~

02

I I i 017! 200 225 .250 S(KSI)

Fig. 8--Weibull distribution for 4-point flexural loading, AS/3501-5A graph i te -epoxy compos i tes

1900

275

S (MP~) 1300 1500 17OO

tO p I I

BENDING 08 ~ (SPT)

AS/SSOL-~ b \

O6 ~t~ = 25 I

Sb =1617 MPo (~ (234KS~1 ~

0.4

02

I I , 175 200 225 250 S(KSI)

Fig. 9--Weibull distribution for 3-point flexural loading, AS/3501-5A graph i te -epoxy compos i tes

1900

275

A possible source of apparent scatter in tensile-coupon data is specimen misalignment which induces bending and/or a nonuniform stress field. The straight-sided geometry associated with composite tensile coupons makes them particularly sensitive to misalignment, with uni- directional composites being the most sensitive due to the high ratio of axial to transverse strength and stiffness. It can easily be seen from eqs (3) and (4) that a constant nonuniform stress field will change the characteristic strength but not the shape parameter. Misalignment, however, is likely to induce a nonuniform stress field which varies from specimen to specimen depending on the degree and nature of the misalignment. Such variations can reduce the estimated value of the shape parameter, ~g, by producing artificially large scatter in measured tensile strengths.

Another anomaly associated with the experimental data is the extremely high tensile-strength values obtained from 3-point flexural loading of 8-ply T300/5208 uni- directional composites. It is possible that the ratio of load-nose radius-to-specimen thickness is too large, producing a distributed load rather than a concentrated load.

One final observation concerning the data, is the obvious possibility that the Weibull distribution is not a valid model for describing failure in the fiber direction for a unidirectional fiber-reinforced composite.

It should be noted that Figs. 2-9 do not indicate a minimum strength at the origin, but are simply for con- venience plotted with an initial point corresponding to a very high reliability. For example, the origin of Fig. 2 (x = 0.75) corresponds to a reliability of 0.9950 and not to a reliability of 1. For any value o fx > 0, R(x) < 1.

Conclusions

It is obvious from the data presented that the experi- mental results do not correlate with a two-parameter Wei- bull statistical model. This lack of correlation may be a result of test methodology or may simply be an indication that the Weibull distribution is an inadequate model for describing failure in the fiber direction for a unidirectional fiber-reinforced composite. It is important from a design standpoint to determine the precise source of the dis- crepancy. This can only be accomplished by establishing failure mechanisms. Until this is done, it appears that any attempt to predict tensile-coupon data from flexural data for design purposes is premature.

Acknowledgments The authors wish to acknowledge Ran Cornwall, Ran

Esterline, and Chuck Fowler of the University of Dayton Research Institute for the fabrication and testing of composite specimens.

References 1. Weibull, W., "'A Statistical Theory of the Strength of Materials,"

lng. Vetenskaps Akad. Hand/ (Roy Swedish Inst. Eng. Research Proc.), NR 151 (1939).

2. Bullock, R.E., "'Strength Ratios of Composite Materials in Flexure and in Tension, "' J. of Camp. Marls, 8, 200 (1974).

3. Daniel, I.M. and 14"eil, N.A., "The Influence of Stress Gradient Upon Fracture of Brittle Materials," ASME Paper ~,o. 63-H'A-228, presented at Winter Annual Meeting, Amer, Sac. of Mech. Eng., Phil- adelphia, PA, Nov. 17-22, 1963.

4. Weil, N.A. and Daniel, I.M., "Analysis of Fracture Probabilities in Uniformly Stressed Brittle Materials, " J. of the Amer. Car. Sac., 47, 268 (1964).

5. Knight, M. and Hahn, H.T., "'Strength and Elastic Modulus of a Randomly-Distributed Short Fiber Composites," J. of Camp. Marls, 9, 77 (1975),

6. ASTM Standard D-3"039, "Standard Test Method for Tensile Properties of Oriented Fiber Composites, "" Book of ASTM Standards, Part 36, 721 (1978).

7, ASTM Standard D-790, "'Standard Test Method for Flexural Properties of Plastics and Electrical Insulating Materials, " Book of ASTM Standards, Part 35, 321 (1979).

8. Mann, N.R., Schafer, R.E. and Singpurwalla, N.D., Method for Statistical Analysis of Reliability and Life Data, John H 'i/ey and Sons, New York (1974).

9. Thoman, Darrel R. and Bain, Lee J., "Two Sample Tests in the Weibull Distribution," Technometrics, I 1, 805 (1969).

10. Wolf, R.V. and Lemon, G.H., Reliability Prediction for Com- posite Joints--Bonded and Bolted Air Force Technical Report AFML-TR- 74-197 (March 1976).

11. Thoman, Darrel R., Bain, Lee J. and Antle, Charles E., "'In- ferences on the Parameters of the 14eibull Distribution, " Technonletrics, 11 (3), 445-460 (Aug. 1969),

216 9 June 1980