Composites: Part A 30 (1999) 299–304
Fatigue lifetime of glass fabric/epoxy composites
G. Caprino, G. Giorleo
Department of Materials and Production Engineering, University of Naples ‘‘Federico II’’, Piazzale Tecchio, 80, 80125 Naples, Italy
Received 18 March 1998; accepted 26 June 1998
Abstract
Monotonic and fatigue tests were carried out in four-point bending on a glass fabric/epoxy composite, using two different stress ratios.
Ultimate failure both in monotonic tests and in fatigue was precipitated by microbuckling phenomena happening at the compression side of
the specimens. The experimental results were evaluated adopting a fatigue model statistically implemented, based on the hypothesis of a twoparameter Weibull distribution of the monotonic strength, previously assessed for random glass fibre reinforced plastics failed in tension. The
fatigue model was able to account for the effect of the stress ratio on the fatigue life, accurately predicting the classical S–N curve. By the
model, the virgin strength for each specimen failed in fatigue was evaluated, and the distributions of the measured and calculated monotonic
strength were compared. Some discrepancies between the two distributions, resulting in poor agreement in the tail portions of the curves,
were noted. It is shown that the inconsistencies found are probably attributable to the inadequancy of a two-parameter Weibull curve to
describe the actual material trend. Better results were obtained by using a three-parameter Weibull distribution. ! 1999 Published by Elsevier
Science Ltd. All rights reserved.
Keywords: B. Fatigue; A. Fabrics/textiles; Stress ratio
1. Introduction
Mechanical structures are often subjected to in-service
fatigue loadings, resulting in damage nucleation and propagation. In metal components a single crack, starting from a
structural discontinuity, usually grows with cycle evolution,
eventually reaching critical dimensions. In this case, fracture mechanics concepts have been successfully employed
to predict fatigue life [1–3].
Unlike metals, composite materials exhibit a peculiar
behaviour in fatigue, undergoing different damage types,
such as matrix cracking, fibre–matrix debonding, fibre
breakage, delamination, quite evenly distributed within the
entire material volume. Although some attempts have been
made to resort to the fatigue response by modelling the
actual damage states [4,5], semi-empirical models, disregarding the microscopic material damage, have proven to
be more efficient to the task. Some of them [6–8] have been
devoted to the prediction of stiffness changes, whereas
others [9–12] have addressed the problem of residual
strength, both of which are crucial mechanical properties
to be considered in design.
It has long been recognised that the stress ratio R, i.e. the
ratio of the minimum to the maximum applied stress in
fatigue, has a strong influence on fatigue response [11–
13]. Nevertheless, this problem has not found a satisying
solution up to now, because no predictive formulae are
available with a general applicability.
In D’Amore et al. [12], a two-parameter model aiming to
predict the fatigue lifetime of composite materials was
proposed, taking into account the stress ratio. The formula,
together with a statistical development presented in Caprino
and D’Amore [14] based on the assumption of a two-parameter Weibull distribution of the virgin strength, was
assessed for random glass fibre reinforced plastics subjected
to four-point loading [12,14,15], all of which failed at the
tension side.
In this work, the model developed in Refs [12,14] was
used to treat the data deriving from flexural fatigue tests
carried out adopting two different stress ratios on glass
fabric/epoxy composites, failed by microbuckling at the
compression side. Despite the difference in material and
failure modes with respect to the cases previously considered, an excellent correlation was found between theory and
experiments, for what concerns the classical S–N curve. On
the basis of the statistical analysis, the probability of failure
of the monotonic strength was calculated from the specimens failed in fatigue, and compared with the measured
one. Some discrepancies, concerning especially the tail
portions of the curves, were noted. This suggested the adoption of a three-parameter Weibull distribution, which seems
to yield more consistent results.
1359-835X/99/$ – see front matter ! 1999 Published by Elsevier Science Ltd. All rights reserved.
PII: S1359-835 X( 98)00 124-9
300
G. Caprino, G. Giorleo / Composites: Part A 30 (1999) 299–304
2. Analysis
In order to predict the tensile fatigue life of polymer
matrix composites reinforced with randomly oriented fibres,
in D’Amore et al. [12] the following assumptions were
made: (a) the tensile strength undergoes a power law
decrease with increasing fatigue cycles; (b) the strength
decay is linearly dependent on the stress amplitude. By
integration of the strength decay expression, and using the
boundary condition n ! 1 " !n ! !o , where !o is the
monotonic tensile strength of virgin material and !n the
residual strength after n cycles, the relationship
!o ! !n ! " !max #1 ! R$ #n# ! 1$
#1$
was obtained, where R is the stress ratio (i.e. the ratio of
minimum-to-maximum applied stress), and ", # two
material parameters to be experimentally evaluated.
With the further hypothesis that final failure takes place
when !n ! !max, the critical number of cycles to failure, N,
was calculated from Eq. (1) as
#
N ! 1"
!
!
"$ 1
#
1
!o
!1
"#1 ! R$ !max
A useful form for Eq. (2) is
"
!o
1
! "#N # ! 1$
!1
!max
1!R
#2$
#3$
which can be utilised to calculate the constants appearing in
the model. In fact, from Eq. (3) all the experimental results
should converge to a single straight line passing through the
origin, when the quantity on the left side is plotted against
(N # ! 1). From the equation of such a straight line, calculated by the best fit method, both " and # are obtained. This
procedure was successfully used in D’Amore et al. [12],
where Eq. (2) was first assessed.
The previous fatigue model was statistically developed in
Caprino and D’Amore [14]. It was assumed that the probability of failure of the virgin material follows a two-parameter Weibull distribution. Moreover, in agreement with
the strength-life equal rank assumption [16], it was postulated that the scatter in fatigue is correlated with the scatter
in monotonic strength, in the sense that, for fixed test conditions, a lower ultimate strength results in lower fatigue life.
Of course, in this case N is also a statistical variable, as
appears from Eq. (2). The expected number of cycles to
failure N* pertaining to a given probability of failure
FN(N*) was obtained as
(%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
'
&
1
$
#
1!%
!
!
!ln%1 ! FN #N $&! ! 1 #4$
N ! 1"
"#1 ! R$ !max
where $ and % are the scale parameter (or characteristic
strength) and the shape parameter of the two-parameter
Weibull distribution of the monotonic strength.
An extensive experimental program was carried out in
Refs [14, 15], where the fatigue behaviour of randomly
oriented glass fibre composites, different in resin content
and fibre form, was examined. From the results, the parameters " and # appearing in Eq. (1) were determined,
according to the method previously described. Then, Eq.
(3) was solved for !o, giving
!o ! !oN ! !max %1 " "#1 ! R$#N # ! 1$&
#5$
From Eq. (5), the virgin material strength for each specimen failed in fatigue (indicated by the symbol !oN in the
equation to distinguish it from that, !o, directly measured in
a monotonic characterisation test), was evaluated. Finally,
the distribution of !oN was compared with that of !o,
obtained from monotonic tests, and an excellent agreement
was found. This was an effective way to assess both the
fatigue model and the statistical formulation, supporting
the hypotheses made. From a practical point of view, the
results obtained demonstrated that the probability of failure
in fatigue can be predicted from less expensive and less
time-consuming monotonic tests.
It can be easily verified that, if the case of fatigue in
compression is considered, using the same hypotheses
made previously the following relevant formulae,
equivalent to Eqs. (2)–(5), are obtained:
#
!
"$1!#
1
!oc
!1
#2 ' $
N ! 1"
"#1 ! 1!R$ !maxc
!
"
)
*
!oc
1
! " N# ! 1
!1
1 ! 1!R
!maxc
#3 ' $
(%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
'
&
1
$
#
1!%
!ln%1 ! FN #N ! $&! ! 1
N ! 1"
"#1 ! 1!R$ !maxc
!
#4 ' $
!oc ! !ocN ! !maxc %1 " "#1 ! 1!R$#N # ! 1$&
#5 ' $
In Eqs. (2 ' ), (3 ' ), (4 ' ) and (5 ' ), !oc is the absolute value of
the material compression strength, and !maxc the absolute
value of the highest compression stress applied in fatigue.
Of course, it is expected that the ", #, $, % values will be
different in tension and compression, because in general the
S–N curve has a different shape under the two types of
loadings, and the same happens for the distribution of the
static strength.
3. Materials and experimental methods
From prepreg layers by Ciba Composites, made of
305 g m !2 plain weave glass fabric and M9/M10 modified
epoxy resin, four square plates 200 mm in side and 3.9 mm
in nominal thickness were fabricated by hot press moulding,
following the procedure recommended by the supplier. The
fibre content by volume in the cured material, measured by
resin burning, was Vf ! 40%.
From the plates, a hundred specimens 80 mm in length
G. Caprino, G. Giorleo / Composites: Part A 30 (1999) 299–304
301
variation of the compression stress within a single cycle at
the upper surface of the specimen, where the highest stress
level occurs. Two different stress ratios, namely R ! 10 and
R ! 1.43, were adopted. Four specimens were tested for
each experimental condition. In all cases, the complete
sample collapse was assumed as a failure criterion. After
tests, the specimens were visually examined to assess failure
modes. Some of the failure surfaces were also observed by
scanning electron microscopy, in order to identify the
damage mechanisms resulting in final collapse. However,
the discussion of this topic is beyond the scope of the
present work.
4. Results and discussion
Fig. 1. Load–displacement curves for the material under study.
and 12 mm in width were cut by a diamond saw. The specimen length coincided with the warp direction of the
reinforcing fabric.
All the tests were carried out in four-point bending,
adopting an outer span 66 mm and an inner span 22 mm,
on an INSTRON 8501 servo-hydraulic machine.
To measure virgin strength, 16 specimens were chosen
from the available population by a random procedure and
loaded up to failure in stroke control, at a crosshead speed of
v ! 100 mm min !1, to approximately match the speed
conditions adopted in fatigue.
The fatigue tests were performed on randomly selected
specimens in load control, using a sinusoidal waveform,
with frequencies in the range 0.8–2 Hz. The load was
applied in such a way that the upper half-thickness of the
beam was subjected to a variable compression stress during
fatigue, whereas a tensile stress state was generated in the
lower half-thickness. As will be discussed later, the final
failure was always found at the compression side. Therefore, the stress ratio R was defined with reference to the
Fig. 2. Probability of failure of the monotonic strength. Symbols: experimental data. Lines: best-fit two-parameter (dashed line) and three-parameter (solid line) Weibull curves.
Under monotonic test conditions, the material exhibited a
linear response up to about 70% of its ultimate load (Fig. 1).
Then, a progressive loss in rigidity was recorded, until ultimate failure took place suddenly. The deviation from
Hookean behaviour was quite limited; in fact, the secant
rigidity at break (dashed line in Fig. 1) was 3–7% lower
than the initial one. The departure from linearity was
accompanied by small kinks forming at the compression
surface of the specimen (probably due to fibre microbuckling at the crossover points of the fabric bundles), evenly
distributed along the inner span of the beam, and signalled
by local resin whitening. Final collapse was apparently
provoked by the coalescence of some kinks, followed by a
rapidly propagating delamination, about one-quarter thickness far from the compression surface, and resulting in the
macroscopic buckling of the upper sublaminate.
From a macroscopic viewpoint, the final failure of fatigued specimens was similar to that previously described for
the monotonically loaded samples. In fact, in this case too
the sudden death was due to compression. However, the
resin whitening was observed within the entire material
volume, emerging also at the tension side. This phenomenon, indicating a damage accumulation in fatigue more
diffused than under monotonic conditions, was consistent
with the loss of rigidity measured at break: the latter was
higher than the one found in monotonic tests, reaching about
15% of initial rigidity (Fig. 1).
From the results of monotonic tests, the mean value of the
virgin material strength was calculated, yielding !oc !
616.2 N mm !2. The subscript c affecting the strength indicates that the observed failure mode was compression,
although it must be recognised that the actual material
strength measured in a conventional compression test is
generally different from that obtained in flexure.
The probability of failure of the virgin material as a
function of the applied stress was evaluated, and plotted
in Fig. 2 (symbols). The dashed line in the same figure
represents the two-parameter Weibull curve best-fitting
the experimental data. It can be seen that a sufficiently
good agreement exists between the latter and theory. This
302
G. Caprino, G. Giorleo / Composites: Part A 30 (1999) 299–304
Fig. 3. Semi-log plot of the non-dimensional maximum applied stress,
!maxc/!oc, against number of cycles to failure, N. Continuous lines: theoretical predictions.
indicates that a two-parameter Weibull law can be reasonably assumed to analytically model the statistical distribution of monotonic strength, supporting the hypothesis
made in developing the statistical model. However, from
the data in Fig. 2 it is also seen that the agreement
between the theoretical prediction and the test results is
less good in the lower tail portion of the distribution
function, which is relevant to design. Although this discrepancy may be attributed to a sampling fluctuation due to
the small sample size, it will be shown later in this paper
that a different interpretation can be given for the
observed phenomenon.
From the equation of the theoretical curve plotted in Fig.
2, a characteristic strength $ ! 636.2 N mm !2 and a shape
parameter % ! 16.24 were obtained.
The usefulness of a two-parameter Weibull distribution in
treating composite material strength has long been recognised [9–12,17,18]. Looking at the shape parameter, very
different % values in bending, ranging from 10 [15] to 43
[17], have been reported in the literature, depending on the
particular material considered. The present shape parameter
is in good agreement with those found in [15], where % was
in the range 10–20 for randomly oriented glass fibre reinforced plastics.
The results of the fatigue tests are shown in Fig. 3, where
the ratio of the highest compression stress applied in flexure
to the virgin material strength, !maxc/!oc, is plotted against
the number of cycles to failure, N, on a semi-log scale. As
also observed elsewhere [11–15], it is evident that the stress
ratio R has a strong influence on the fatigue life.
Although the effectiveness of Eq. (1) in describing both
the fatigue life and the effect of R was demonstrated in other
works [12,14,15], two main differences between the results
reported in Refs [12,14,15] and in the present paper must be
emphasised: (a) in all the materials considered previously,
the reinforcement was randomly oriented, and (b) a tensile
failure was noted, rather than a compression one, as in this
Fig. 4. Semi-log plot of the term on the right of Eq. (2), K, against number
of cycles to failure, N. Continuous line: theoretical prediction.
case. This explains why it was preferred, in the experimental
program, to carry out fatigue tests using two different stress
ratios, in order to validate the fatigue model.
According to Eq. (3 ' ), all the fatigue data pertaining to
different R values should converge to a single master curve,
when the term on the left side (indicated by the symbol K in
the following) is plotted against N. From Fig. 4, where the
experimental K values are plotted against N on a semi-log
scale, it is seen that this actually happens. Therefore, the
efficiency of Eq. (3 ' ) in modelling the effect of the stress
ratio is demonstrated. The present results, together with
those discussed in Refs [12,14,15], indicate that the hypothesis of a strength decrease linearly dependent on stress
amplitude, adopted here, can account for the influence of
stress ratio on fatigue life, when a state of pure tension or
pure compression is generated within the material volume
where the dominant crack causing final collapse develops.
It is worth noting that the results in Fig. 4 do not confirm
the validity of the theoretical model in describing the classical S–N curve. To achieve this, it is necessary to show
that, as predicted by Eq. (3 ' ), a suitable value of the constant
# can be found, such that the (N # ! 1) values are well fitted
by a straight line passing through the origin, when plotted
against K. Of course, in this case the constant " will be
given by the slope of the straight line.
From the experimental results, the quantity (N # ! 1) was
evaluated by successive approximations of #, until the bestfit straight line actually passed through the origin. The
results of such analysis are shown in Fig. 5. From the figure,
it is appreciated that all the data points, although affected by
a quite large scatter, actually follow a linear trend.
The continuous best fit straight line in the figure provided
the values " ! 0.0205 and # ! 0.453. Substituting them in
Eq. (2 ' ), the solid lines in Figs 3 and 4, representing the
theoretical predictions, were drawn. The agreement between
theory and experimental points is excellent, demonstrating
the applicability of the fatigue model also for the case under
investigation.
It is important to note that, in using the procedure
G. Caprino, G. Giorleo / Composites: Part A 30 (1999) 299–304
Fig. 5. Graph for the evaluation of the parameters ", # appearing in the
fatigue model. Continuous line: best-fit straight line.
depicted in Fig. 5 for the evaluation of " and #, only the
fatigue data are utilised. Consequently, all the quantities
appearing in Eq. (5) ((5 ' )) do not contain any implicit information on the scatter in monotonic strength. Therefore,
when Eq. (5) ((5 ' )) is used for the calculation of !oN
(!ocN) starting from a set of fatigue results, the scatter in
the calculated !oN (!ocN) values only depends on the scatter
in fatigue. Because of this, in order to verify the strength–
life equal rank assumption, in Refs [14, 15] Eq. (5) was
employed to calculate the monotonic material strength !oN
of randomly oriented glass fibre reinforced plastics, starting
from the fatigue data. Comparing the !oN values with those,
!o, measured by monotonic tests, it was found that the
Weibull distributions of !oN and !o were in close agreement
with each other. This result experimentally supported the
hypothesis that the scatter in fatigue life strictly derives
from the scatter in monotonic strength. At the same time,
the effectiveness of the fatigue model, on which Eq. (5)
relies, was highlighted.
The procedure followed in [14, 15] was applied also in
Fig. 6. Probability of failure of the calculated monotonic strength, !ocN.
Dashed line: two-parameter Weibull distribution of the measured monotonic strength. Continuous line: three-parameter Weibull distribution of the
measured monotonic strength.
303
Fig. 7. Probability of failure of the calculated monotonic strength, !ocN.
Dashed line: best-fit two-parameter Weibull curve. Continuous line: best-fit
three-parameter Weibull curve.
this work, where !ocN was evaluated for each of the specimens failed in fatigue substituting in Eq. (5 ' ) the " and #
values resulting from the fatigue tests. Fig. 6 (symbols)
reports the probability of failure of the calculated monotonic
strength. The dashed line in the figure is the two-parameter
Weibull curve directly obtained from the monotonic tests
(see Fig. 2), drawn for comparison. It is seen that the correlation between the probability of failure of the measured and
calculated monotonic strength is reasonable. However, as
noted also for the measured static strength (Fig. 2), some
deviation in the theoretical curve from the experimental
trend is noted in the lower tail portion of the distribution
function, where the dashed line tends to predict a higher
probability of failure than the data points, for a given applied
stress.
In an attempt to improve the ability to predict the probability of failure in the lower part of the diagram in Fig. 6, it
was assumed that a three-parameter Weibull distribution,
rather than a two-parameter one, could represent the trend
of the data in Fig. 2. From them, the values $ !
112.2 N mm !2, % ! 2.14 and & ! 518.1 N mm !2, where &
is the location parameter, were calculated by the method of
best-fit. By these values, the solid line in Fig. 2, also plotted
in Fig. 6 for comparison, was drawn. Looking at Fig. 6, it is
seen that assuming a three-parameter Weibull distribution
results in a better agreement between theory and experiments in correspondence of the tails. However, the
correlation is poorer in the central portion of the curve.
Probably, the previous discrepancies are attributable to
the small sample size of the monotonic tests, which does
not allow for a reliable estimate of the parameters characterising the distribution function. Assuming that this is
the case, some valuable information on the actual virgin
strength distribution can be derived from the distribution
of !ocN. In Fig. 7, the same data points shown in Fig. 6 are
reported. The dashed and solid line refer to the best-fit
two-parameter and three-parameter Weibull distributions,
304
G. Caprino, G. Giorleo / Composites: Part A 30 (1999) 299–304
respectively, evaluated on the basis of the !ocN results.
From the calculations, the values $ ! 639.4 N mm !2,
% ! 18.09 were obtained for the two-parameter curve,
and $ ! 108.5 N mm !2, % ! 2.62, & ! 524.5 N mm !2
for the three-parameter one. From Fig. 7, the three-parameter distribution is in excellent agreement with the data
points, whereas the two-parameter curve is unable to
follow the actual trend, especially when the tails are
considered. This result suggests that the selection of a
suitable probability function must be carefully evaluated,
in order to resort to reliable estimates in the design of
structures, where the lower tail portion of the distribution
function is usually of topical importance.
It is worth noting that, if a three-parameter Weibull curve
is employed to model the probability of failure of the monotonic strength, Eqs. (4) and (4 ' ), which are particularly
useful in design, no longer hold, because their development
directly involves the hypothesis of a two-parameter Weibull
distribution [14].
5. Conclusions
From the results presented and discussed in this paper,
concerning monotonic and fatigue tests carried out in fourbending on glass fabric/epoxy composites, the following
conclusions can be drawn:
the fatigue model, based on the hypothesis of a strength
decay linearly dependent on the stress amplitude, is able
to account for the effect of stress ratio on the fatigue
lifetime, provided the dominant crack determining final
failure grows under a pure tension or pure compression
stress state;
if a two-parameter Weibull distribution is used to model
the probability of failure of the virgin strength for the
material considered, the actual probability of failure in
fatigue will be lower than the one calculated on the
basis of the lower tail portion of the curve;
more reliable theoretical estimates are probably achieved
by the adoption of a three-parameter Weibull curve for
the distribution function of the monotonic strength.
References
[1] Singer E. Fracture mechanics in design of pressure vessels.
Engineering Fracture Mechanics 1969;1:507–517.
[2] Broek D. Elementary engineering fracture mechanics. Leyden:
Noordhoff, 1974.
[3] Austin TSP, Webster GA. Application of a creep-fatigue crack growth
model to type 316 stainless steel, In Ainsworth RA, Skelton RP,
editors. Behaviour of defects at high temperatures, ESIS 15. London:
Mechanical Engineering Publications, 1993: 219–37.
[4] Reifsnider KL, Highsmith AL. The relationship of stiffness changes in
composite laminates to fracture-Related damage mechanisms. In:
Proceedings of the Second USA-URSS Symposium. The Hague:
Nijhoff, 1982: 279–90.
[5] Wang SS, Chim ES-M, Suemasu H. Mechanics of fatigue damage and
degradation in random short-fiber composites, part II—analysis of
property degradation. Journal of Applied Mechanics 1986;53:347–
357.
[6] Hwang W, Han KS. Fatigue of composites—fatigue modulus concept
and life prediction. Journal of Composite Materials 1986;20:125–
153.
[7] Yang JN, Yang SH, Jones DL. A stiffness-based statistical model for
predicting the fatigue life of graphite/epoxy laminates. Journal of
Composite Technology and Research 1989;11:129–134.
[8] Lee LJ, Fu KE, Yang JN. Prediction of fatigue damage and life for
composite laminates under service loading spectra. Composite
Science and Technology 1996;56:635–648.
[9] Yang JN, Liu MD. Residual strength degradation model and theory of
periodic proof tests for graphite/epoxy laminates. Journal of Composite Materials 1977;11:176–203.
[10] Hwang W, Han KS. Statistical study of strength and fatigue life of
composite materials. Composites 1987;18(1):47–53.
[11] El Kadi H, Ellyin F. Effect of stress ratio on the fatigue of unidirectional glass fibre/epoxy composite laminates. Composites
1994;25(10):917–924.
[12] DAmore A, Caprino G, Stupak P, Zhou J, Nicolais L. Effect of stress
ratio on the flexural fatigue behaviour of continuous strand mat reinforced plastics. Science and Engineering of Composite Materials
1996;5(1):1–8.
[13] Mandell JF, Meier U. Effects of stress ratio, frequency, and loading
time on the tensile fatigue of glass-reinforced epoxy. Long-Term
Behaviour of Composites, ASTM STP 1983;813:55–77.
[14] Caprino G, D’Amore A. Flexural fatigue behaviour of random continuous fibre reinforced thermoplastic composites. Composite Science
and Technology 1998; in press.
[15] Caprino G, D’Amore A, Facciolo F. Fatigue sensitivity of random
glass fibre reinforced plastics. Journal of Composite Materials 1998;
in press.
[16] Hahn HT, Kim RY. Proof testing of composite materials. J. Compos.
Mater. 1975;9:297–311.
[17] Whitney JM, Knight M. The relationship between tensile and flexure
strength in fiber-reinforced composites. Experimental Mechanics
1980;20:211–216.
[18] Wisnom MR. Relationship between strength variability and size
effect in unidirectional carbon fibre/epoxy. Composites
1991;22(1):47–52.