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Transcript
FATIGUE LIFE OF GRAPHITE/EPOXY LAMINATES SUBJECTED TO TENSIONCOMPRESSION LOADINGS
G. Caprino and A. D'Amore
Department of Materials and Production Engineering, University of Naples "Federico II",
Piazzale Tecchio, 80, 80125 Naples, Italy
ABSTRACT
A recent two-parameter fatigue model, aimed at the prediction of the fatigue behaviour of
composite materials, was applied to fatigue results available in the literature, concerning
graphite/epoxy laminates subjected to tension-compression loadings. It was found that the
model, successfully applied in the case of tension-tension and compression-compression
fatigue, also works when the sign of stress is reversed during fatigue. However, the constants
appearing in the proposed formula must be evaluated anew when the loading mode changes
from pure tension to tension-compression. The probability of failure in fatigue is predicted with
reasonable accuracy by the model, which is far conservative in calculating the residual strength
after a given number of cycles.
INTRODUCTION
Ensuring the durability of composite structures subjected to significant fatigue loadings is one
of the most intricated tasks in design. Unlike metals, where a dominant crack develops during
cycle evolution, composites suffer a diffuse damage, made of intralaminar cracks,
delaminations, and fibre breakages [1 - 3]. The interaction of the different failure modes renders
difficult the prediction of crack propagation rates, as well as the control of crack growth through
suitable non destructive inspection methods, impairing a reliable application of the "damage
tolerant" philosophy. Furthermore, cyclic stresses result in a progressive decay of the elastic
modulus in a laminate, so that the possibility of deformations exceeding design requirements or
in-service buckling must be also considered.
In the last decades, many researchers have attempted to analytically describe the fatigue
response of composite materials. The models proposed have concerned both the modulus [4, 5]
and the strength decrease [6, 7] with elapsing number of cycles. Besides, some authors have
shown that not only the mean fatigue life, but also the probability of failure in fatigue could be
predicted by a suitable application of statistical concepts [8 - 10]. Nevertheless, it must be
recognised that the models available at the time suffer some lack of generality, in that the
constants appearing in them are sensitive to the loading mode (tension-tension, tensioncompression, compression-compression), or even to the stress ratio or the stress range adopted.
2
In this work, fatigue results available in the literature [11], concerning graphite/epoxy laminates
subjected to tension-compression fatigue loadings, are evaluated in the light of a two-parameter
fatigue model proposed recently [12] and statistically implemented in [13]. Previous data [11 15] demonstrated that the parameters appearing in the model are independent of the stress ratio
when the stress field driving fatigue failure is pure tension or pure compression, and that a
reasonably accurate prediction of the fatigue life distribution can be obtained for a variety of
composite materials. The present analysis confirms the applicability of the model to the case of
tension-compression fatigue, for what concerns both the classical S-N curve and the
distribution function. However, it is shown that the constants in the model vary when the
loading mode is tension-compression. Further, predicting the residual strength of fatigued
laminates through the formula considered results in far conservative estimates compared to the
measured values.
ANALYSIS
In [12], a two-parameter model was proposed for the prediction of fatigue lifetime of Random
Glass Fibre Reinforced Plastics (RGFRP) loaded in tension. Similarly to other approaches
adopted in fatigue [4, 16], the basic hypothesis was a continuous decrease of the material
strength with increasing number of cycles, n, according to a power law:
dm tn
= <at u n <b t
dn
(1)
In eq. (1), mtn is the residual tensile material strength after n cycles, and at, bt two positive
definite constants to be experimentally determined.
To explicitly account for the well known dependence of the strength decay on the stress ratio R
(i.e. the ratio mmin/mmax of the minimum to the maximum applied stress), it was further
assumed that the constant at is linearly dependent on stress range, 6m:
at = aot u 6m
(2)
6m = m max < m min
(3)
where, as usual:
By integration of the strength decay rate, and using the boundary condition n = 1A mtn = mto,
where mto is the monotonic tensile strength of virgin material, the following relationship was
obtained for mtn:
m tn = m to < _ t u m max u (1 < R) u (n` t < 1)
with
(4)
3
aot
1 < bt
(5)
`t = 1 < bt
(6)
_t =
Finally, the critical number of cycles to failure, Nt, was calculated putting in eq. (4) n = Nt for
mtn = mmax, that is supposing that fatigue failure is precipitated when the residual material
strength in tension equals the maximum applied stress. Solving for Nt, it was found:
£
m < m max ¥
N t = ²1 + to
´
_ t u 6m ¦
¤
1/` t
(7)
It can be easily verified that, although eq. (7) may appear different from the formula presented
in [12], the two equations coincide if the stress range 6m is used in the latter, instead of the
stress ratio R.
When tension-compression or pure compression fatigue is concerned, the previous analysis
needs some modification. The case of pure compression was treated in [14], where it was
shown that eq. (7) retains its validity, if mto represents the monotonic compression strength of
the material, mmax is the highest applied stress, and R is substituted by 1/R.
It was found in previous works [12, 13, 15] that, as hypothesised in the analysis, the constants
_ t and `t are a function of the material only, being independent of the stress range adopted,
when RGFRPs failing in tension are considered. Later, similar results were presented in [14]
and [17], where it was also shown that the model could be successfully applied to the prediction
of the fatigue behaviour of glass fabric reinforced epoxy failing in compression, as well as
graphite fibre reinforced laminates subjected to pure tension. This indicates that eq. (7) can be
applied to different types of composites and loading conditions, provided the sign of stress does
not change in the material volume where the final fatigue fracture is precipitated.
If the stress sign changes during fatigue (tension-compression fatigue), a further development
of the analysis is required. This depends on the fact that the fatigue failure could happen when
the minimum applied stress, mmin, matches the residual compression strength of the material,
mcn. Assuming that the latter varies according to a law formally identical to eq. (1):
dm cn
= ac u n <b c
dn
(1')
where the index "c" refers to compression, the following relationship, similar to eq. (7), is
easily found:
£
m
< m co ¥
N c = ²1 + min
´
_ c u 6m ¦
¤
1/` c
(7')
4
It can be noted that the sign "+" affecting eq. (1') only reflects the sign convention for stresses,
which are designated as negative when compressive.
It is expected that _ t & _c and `t & ` c, because in general the compression strength decreases
following a trend different from the tensile strength. Of course, the actual critical number of
cycles to failure will be Nt or Nc, whichever the less.
Utilising the stress-life equal rank assumption [18], and assuming a two-parameter Weibull
distribution of the monotonic tensile strength, in [13] the probability P( N *t ) to find an Nt value
lower than N *t was calculated as:
bt
¨
`t
—
•
*
« m
+_ t u (N t < 1) u 6m µ
P(N *t ) = 1 < exp ©< ³ max
at
µ
« ³–
˜
ª
¬
«
­
«
®
(8)
where at and bt are the characteristic strength and the shape parameter of the Weibull
distribution of the monotonic tensile strength, respectively.
In the case of tension-compression fatigue, following the same procedure described in [13] for
Nt, a similat expression can be found for Nc:
bc
¨
¬
`c
—
•
*
«
«
m
<
_
u
(N
<
1)
u
6m
c
c
µ ­
P(N *c ) = 1 < exp ©< ³ min
ac
µ «
« ³–
˜
ª
®
(8')
Of course, the characteristic strength ac and the shape parameter bc appearing in eq. (8') refer to
the monotonic compression strength of the material, and in general are different from at and b t,
respectively.
SCOPES
As stated previously, some experimental evidence exists that the constants _ t,c, `t,c are
independent of the stress range, provided the state of stress leading to final failure is pure
tension (compression). In this case, in principle the constants can be calculated from two fatigue
points, reliably determined by experiments, and the fatigue behaviour of the material can be
predicted for a variety of loading cases through eq. (7) (eq. (7')). If, in addition, the
characteristic strength and the shape parameter in tension (compression) are determined by
monotonic tests, also the scatter in fatigue life can be calculated by eq. (8) (eq. (8')).
Unfortunately, the applicability of eqs. (7), (8), (7'), (8') was never ascertained in the case of
tension-compression, which is a loading mode particularly critical for the fatigue life of
composite laminates [6, 11]. It is expected that, when the absolute value of mmin is low
compared to the material compression strength, and mmax is significant with respect to the
material tensile strength, the fatigue life will be governed by eqs. (7) and (8). Nevertheless,
5
even in this case _ t and `t could be dependent on the applied compression stress. Physically
speaking, this behaviour would indicate (eq. (1)) that the latter influences the tensile strength
decay law during cycle evolution. Moreover, the question arises whether eq. (4) is actually able
to follow the trend of the strength decrease with increasing number of cycles.
The previous considerations outline the significance of the present work, whose scopes
were restricted to cases of tension-compression fatigue where final failure is determined by
mmax (|mmin| << |mco|), and are as follows:
a) to verify the efficiency of the model in predicting the typical S-N curve, the probability
of failure in fatigue, and the residual tensile strength of the material;
a) to ascertain whether the constants _t and `t are influenced by the presence of a compression
stress.
EXPERIMENTAL VERIFICATION
In order to ascertain the effect of a compressive stress on the fatigue behaviour of
graphite/epoxy laminates, in [11] monotonic, tension-tension and tension-compression fatigue
tests were carried out on T300/934 quasi-isotropic laminates having [0/45/90/-452/90/45/0]2
layup. The results of monotonic tests were utilised in [17], where the values at = 70.76 ksi and
bt = 22.17 were calculated for the characteristic strength and the shape parameter of the twoparameter Weibull distribution of the monotonic tensile strength, respectively, using the best fit
method. In [17], the tension-tension test data were also analysed in the light of eqs. (7) and (8).
It was shown that eq. (7) is able to efficiently describe the fatigue life of the laminate, whereas
eq. (8) is sufficiently accurate in predicting the scatter in fatigue lifetime. The parameters _ t and
`t were evaluated by a trial-and-error technique, noting that eq. (7) can be rewritten as:
m to < m max
`
= _t u N t t < 1
6m
(
)
(9)
`
so that, plotting (N t t < 1) against the quantity on the left side of eq. (9), indicated by K
hereafter, should result in a straight line passing through the origin. By this method, the values
_t = 0.0838 and `t = 0.153 were obtained for the case of pure tension fatigue.
The tension-compression fatigue tests in [11] were carried out using two different values of
mmin, namely mmin = -10 ksi (Table I) and mmin = -16 ksi (Table II), and varying mmax in the
range 34 - 62 ksi. In all cases, |mmax| was sufficiently higher than |mmin| to guarantee a tensile
failure.
To verify whether eq. (7) could be applied to the prediction of the tension-compression fatigue
life for the laminate under concern, in this work the data in Table I were used to calculate K ,
and the same procedure described in [17] was followed to evaluate _ t and `t: the quantity
`
(N t t < 1) was obtained assuming an attempt value for ` t, and the straight line best fitting the
experimental points was found; then, `t was varied until the best-fit straight line passed through
6
the origin. This happened for `t = 0.217. The results are graphically shown in Fig. 1: it is seen
that, as anticipated from eq. (7), the trend of the data points is substantially linear, witnessing
the suitability of the theoretical model. The value _ t = 0.0423 was calculated from the slope of
the best fit straight line.
Treating the data in Table II according to the procedure outlined previously, _t = 0.0366 and ` t
= 0.238 were calculated for the case mmin = -16 ksi.
Considering the values of _t and `t found in the different cases (Table III), it can be concluded
that, contrary to what expected from the results presented in [12 - 15], these constants do
undergo variations when a tension-compression loading is applied in fatigue. Moreover, a clear
trend is observed from the data in Table III: when |mmin| increases, `t also increases, whereas
_ t decreases progressively. Since, from eq. (7), an increase in both the constants results in a
poorer response in fatigue, the actual trend of the fatigue life in the case concerned cannot be
easily guessed from the mere values of _t and `t.
The results of the fatigue tests performed in [11] are shown in Fig. 2, under form of classical SN curves. The lines in the same figure are the theoretical predictions from eq. (7), where the
constants in Table III were used. It is seen that the correlation between the experimental points
and theory is satisfying, confirming the usefulness of the fatigue model. From the curves
drawn, at high cycles (low mmax) the increase in `t with increasing |mmin| prevails on the
decrease in _t, resulting in an earlier fatigue failure. The same does not happen at low cycles
(high mmax), where the contrasting effects of _ t and `t balance each other. Consequently, the
material behaviour at low cycles is practically independent of the minimum applied stress.
It must be noted that the constants of the model are expected to be sensitive not only to |mmin|,
but also to the test temperature and frequency. However, the effect of these parameters should
be different for _t and `t. To show this, a more detailed discussion illustrating the influence of
the constants on the shape of the overall S-N curve is in order.
The curves in Fig. 3a) represent the fatigue life predicted from eq. (7), assuming a fixed stress
ratio (R = 0.1), `t = 0.2, and different values of _t. The same R value was adopted to draw the
lines in Fig. 3b), where _t was held constant (_t = 0.05) and `t was varied. It is clearly seen
that, when `t is fixed (Fig. 3a)), all the curves, irrespective of the _ t value used, tend to
assume the same slope at high cycles, whereas they diverge continuously when `t is varied
(Fig. 3b)). It can be verified that this tendency is observed when different values of R are
assumed.
In recent years some data have been published [19], concerning the effect of temperature and
frequency on the fatigue life of CFRP laminates. The authors carried out fatigue tests in flexure
on a carbon cloth/epoxy composite, using a constant stress ratio (R = 0.05), and static tests up
to failure at various temperatures and loading rates. Reporting the fatigue results on a stresstime semilogarithmic plot, two stages, schematically depicted in Fig. 4, were individuated : a) at
low cycles the frequency effect was consistent with the strain rate effect on static strength, and
7
could be predicted by the time-temperature superposition principle; this indicated that
viscoelastic phenomena play a role in determining fatigue life under these conditions; b) at high
cycles and sufficiently low temperature, the fatigue life was independent of static strength, and
the time to failure decreased with increasing frequency; in this zone the data trend both in the
stress-time diagram and in the classical S-N diagram was well fitted by a straight line, whose
slope was independent of the frequency and the temperature. Interestingly, these results were
qualitatively in agreement with those reported by Mandell [20], concerning (0/90) glass/epoxy
laminates tested at various frequencies.
Recalling Fig. 3, the previous experimental data suggest that, at least in some cases, the test
temperature and frequency affect only _ t, whereas `t is practically independent of these
parameters. Further, when a change in the mode of failure from static to cyclic fatigue is
observed (Fig. 4), eq. (7) is expected to fail in describing the fatigue life at high stress levels.
It is important to note that, when the method graphically depicted in Fig. 1 is used to calculate
_t and `t, the only parameters determining the position of the best fit straight line are the mean
`
values of (N t t < 1) for each K. Therefore, the previous constants do not contain any implicit
information concerning the scatter in fatigue or in monotonic strength. On the other hand,
according to theory the virgin strength of each specimen failed in fatigue could be calculated
from eq. (7), which can be reduced to the form:
(
`
)
m to = m toN = m max + _ t u 6m u N t t < 1
(10)
In eq. (10), the symbol mtoN is used to distinguish the converted value of the monotonic
strength (i.e. the one calculated through eq. (10) utilising fatigue data) from the measured value,
m to.
If, as assumed in the development of the statistical model, the strength-life equal rank
assumption is valid, it must be concluded that the scatter in monotonic strength coincides with
the scatter in mtoN.
To verify the previous hypothesis, the results in Tables I and II were used in eq. (10), together
with the constants _ t and `t shown in Table III. All the mtoN values obtained were collected,
and the probability of failure P was evaluated by the relationship:
P=
mi
m +1
(11)
where mi is the failure order and m the total number of specimens.
In Fig. 5, the statistical distribution of the converted monotonic strength (solid symbols) is
compared to the two-parameter Weibull curve (continuous line) best fitting the measured
monotonic strength [17]. The agreement is outstanding, supporting both the strength-life equal
rank assumption and the fatigue model discussed here, on which eq. (10) relies.
8
To study the distribution function of fatigue life under assigned loading conditions, in [11] also
the experimental results in Table IV were generated, using a fixed value for mmin (mmin = - 16
ksi). The data in Table IV are plotted as solid symbols in Figs. 6 to 8, where the probability of
failure was calculated according to eq. (11).
As previously noted, _t and `t for mmin = - 16 ksi were available (see Table III), and a t, bt for
the laminate under evaluation were calculated from monotonic tests in [17]. Therefore, in this
work the probability of failure in fatigue for each of the cases in Figs. 6 - 8 could be calculated
from eq. (8), resulting in the continuous lines drawn in the same figures.
It is evident that the correlation between theory and experiments is reasonably good in Figs. 6
and 5, whereas it is quite poor in Fig. 8. This probably depends on the fact that only twelve
data points were used to evaluate the constants _t and `t for mmin = - 16 ksi, and among them
no experimental results were employed referring to mmax = 42 ksi (Table II).
Interestingly, in [10] the same results considered here were adopted to assess the Yang and Liu
three-parameter Model (YLM) [9], aimed at the prediction of the residual strength and fatigue
life of graphite/epoxy laminates. In comparing the data in Figs. 7 and 8 with the theoretical
predictions of YLM, a discrepancy was found in the lower tail portion of the distribution
function. This poor agreement was attributed to the presence of a few outliers (three data in Fig.
6 and one datum in Fig. 7) belonging to a defective population. It was verified in this work that,
censoring the outliers, the correlation becomes excellent for the data in Fig. 7, whereas it
remains reasonable for what concerns Fig. 6.
Besides the results in Table IV, in [11] tension-compression fatigue data were obtained using
mmax = 26 ksi and mmin = - 16 ksi. When a prediction of the distribution function of fatigue life
was attempted in [9] using YLM, the correlation with experimental results was poor. It was
noted in [9] that this was anticipated, because the constants of the YLM were established from
test data not including mmax = 26 ksi, so that the prediction was based on an extrapolation. For
the same reason, the results concerning mmax = 26 ksi were not used here to assess the present
model.
In principle, the model discussed in this work should be able to calculate not only the fatigue
life, but also the residual material strength after a given number of cycles, through eq. (4).
However, when this evaluation was attempted in [17] for tension-tension fatigue, theoretical
values consistently lower than the experimental ones were found.
In [11], also residual strength data after tension-compression fatigue were obtained. Twentytwo specimens were fatigued up to n = 14400 cycles with mmax = 42 ksi and mmin = - 16 ksi.
Two specimens failed before the maximum number of cycles could be reached, and the residual
tensile strength mres of the surviving sample was measured. The same procedure was applied
using n = 2150 cycles, mmax = 50 ksi and mmin = - 16 ksi. In the latter case, only one specimen
underwent a premature failure. The experimental results are plotted in Fig. 9 by solid symbols
affected by vertical bars, denoting standard deviation. The lines in the same figure are the
theoretical predictions based on eq. (4); their extreme points represent the intersection of the
9
residual strength curves with the fatigue curve, so that they can be interpreted as the residual
strength of the material when mres equals mmax.
Comparing the data points in Fig. 9 with the theoretical predictions, it is seen that eq. (4) yields
a quite conservative estimate of the residual strength, confirming the conclusions drawn in [17]
for the residual strength after tension-tension fatigue. Therefore, it can be concluded that eq. (7)
must be considered a merely empirical formula for the description of the S-N curve of a
composite laminate.
CONCLUSIONS
A two-parameter fatigue model, recently proposed, has been used to analyse tensioncompression fatigue data generated in [11], concerning T300/934 quasi-isotropic laminates.
From the results presented and discussed, the conclusions are as follows.
•• The fatigue model is able to accurately describe the classical S-N curve, provided the two
constants appearing in it are suitably calculated from tension-compression fatigue data. In
fact, the two parameters assume different values in tension-tension and in tensioncompression fatigue, and seem to be sensitive also to the compression stress level adopted
during fatigue.
•• When the model is used to predict the probability of failure in fatigue, a reasonable agreement
is found between the theoretical predictions and the experimental data. The efficiency of the
statistical analysis is confirmed by the fact that the distribution function of the measured static
tensile strength can be obtained from the distribution function of the virgin strength
calculated by the model, using the fatigue life data.
•• The model is conservative in evaluating the residual tensile strength of the material after an
assigned number of cycles. Therefore, the strength degradation law on which the model
relies is not verified in practice. This indicates that the formula for the calculation of the
critical number of cycles in fatigue is of the empirical type.
REFERENCES
[1] Wang, S. S., and Chim, E. S. M., "Fatigue Damage and Degradation in Random ShortFiber SMC Composite", J. Compos. Mater. , Vol. 17, No. 2, 1983, pp. 114-134.
[2] Charewicz, A., and Daniel, I. M., “Damage Mechanisms and Accumulation in
Graphite/Epoxy Laminates”, in Composite Materials: Fatigue and Fracture, H. T. Hahn
Ed., ASTM STP 907, 1986, pp. 274-297.
[3] Reifsnider, K. L., “Damage and Damage Mechanics”, in Fatigue of Composite Materials,
K. L. Reifsnider Ed., Elsevier Publ., Amsterdam, 1991, pp. 11-77.
10
[4] Wang, S. S., Suemasu, H., and Chim, E. S. M., "Analysis of Fatigue Damage Evolution
and Associated Anisotropic Elastic Property Degradation in Random Short-Fiber Composite",
J. Compos. Mater. , Vol. 21, Dec. 1987, pp. 1084-1105.
[5] Whitworth, H. A., "Modeling Stiffness Reduction of Graphite/Epoxy Composite
Laminates", J. Compos. Mater. , Vol. 21, April 1987, pp. 363-372.
[6] Schütz, D., and Gerharz, J. J., "Fatigue Strength of a Fiber-Reinforced Material",
Composites, October 1977, pp. 245-250.
[7] Ellyin, F., and El Kady, H., "A Fatigue Failure Criterion for Fiber Reinforced Composite
Lamina", Compos. Struct. , Vol. 15, 1990, pp. 61-74.
[8] Halpin, J. C., Jerina, K. L., and Johnson, T. A., “Characterization of Composites for the
Purpose of Reliability Evaluation”, in Analysis of Test Methods for High Modulus Fibers
and Composites, ASTM STP 521, 1973, pp. 5-64.
[9] Yang, J. N., and Liu, M. D., "Residual Strength Degradation Model and Theory of
Periodic Proof Tests for Graphite/Epoxy Laminates", J. Compos. Mater. , Vol. 11, 1977, pp.
176-203.
[10] Yang, J. N., "Fatigue and Residual Strength Degradation for Graphite/Epoxy Composites
Under Tension-Compression Cyclic Loading", J. Compos. Mater. , Vol. 12, Jan. 1978, pp.
19-39.
[11] Ryder, J. T., and Walker, E. K., "Ascertainment of the Effect of Compressive Loading on
the Fatigue Life Time of Graphite/Epoxy Laminates for Structural Applications", AFML-TR76-241, WPAFB, December 1976.
[12] D’Amore, A., Caprino, G., Stupak, P., Zhou, J. and Nicolais, L., "Effect of Stress Ratio
on the Flexural Fatigue Behaviour of Continuous Strand Mat Reinforced Plastics", Sci. and
Eng. of Compos. Mater. , Vol. 5, No. 1, 1996, pp. 1-8.
[13] Caprino, G., and D'Amore, A., "Flexural Fatigue Behaviour of Random Continuous
Fibre Reinforced Thermoplastic Composites", ", Compos. Sci. Technol., Vol. 58, No. 6,
1998, pp. 957-965.
[14] Caprino, G., and Giorleo, G., "Fatigue Lifetime of Glass Fabric/Epoxy Composites", in
press on Composites, Part A.
[15] Caprino, G., D'Amore, A., and Facciolo, F., "Fatigue Sensitivity of Random Glass
Fibre Reinforced Plastics", J. Compos. Mater., Vol. 32, No. 12, 1998, pp. 1203-1220.
11
[16] Buggy, M., and Dillon, G., “Flexural Fatigue of Carbon Fibre-reinforced PEEK
Laminates”, Composites, Vol. 22, No. 3, May 1991, pp. 191-198.
[17] Caprino, G., "Predicting Fatigue Life of Composite Laminates Subjected to TensionTension Fatigue", in press.
[18] Hahn, H. T., and Kim, R. Y., “Proof Testing of Composite Materials”, J. Compos.
Mater., Vol. 9, 1975, pp. 297-311.
[19] Y. Miyano, M. K. McMurray, J. Enyama and M. Nakada, “Loading rate and temperature
dependence on flexural fatigue behaviour of satin woven CFRP laminate”, J. Compos. Mater. ,
Vol. 28, 1994, pp. 1250-1260.
[20] J. F. Mandell, “Fatigue behavior of short fiber composite materials”, in Fatigue of
Composite Materials, K. L. Reifsnider Ed., Elsevier Publ., Amsterdam, 1990.
12
TABLES
Table I - Tension - compression fatigue results [11]. mmin = -10 ksi.
mmax
(ksi)
62
62
62
58
58
58
54
54
N
(cycles)
810
1127
10
4840
4980
1675
10500
11055
mmax
(ksi)
54
50
50
50
46
46
46
42
N
(cycles)
6997
10651
16030
10000
25520
36500
78710
130720
mmax
(ksi)
42
42
38
38
38
34
34
34
N
(cycles)
78290
188887
111600
414560
485190
1047000
1322440
2104510
Table II - Tension - compression fatigue results [11]. mmin = -16 ksi.
mmax
(ksi)
58
58
58
50
N
(cycles)
1402
3251
1010
10906
mmax
(ksi)
50
50
38
38
N
(cycles)
11445
3981
213539
55380
mmax
(ksi)
38
34
34
34
N
(cycles)
43485
123672
749444
638880
Table III - Values of the constants _t and `t appearing in eq. (7), calculated for different values
of mmin.
mmin
(ksi)
0
-10
-16
_t
`t
0.0838
0.0423
0.0366
0.153
0.217
0.238
13
Table IV - Tension - compression fatigue results for the assessment of the distribution function
of fatigue life [11]. mmin = -16 ksi.
mmax
(ksi)
58
58
58
58
58
58
58
58
58
58
58
58
58
58
58
58
58
58
58
58
N
(cycles)
5
57
71
316
330
1010
1167
1385
1402
1473
1510
1740
1930
2186
2356
2685
3251
3406
3430
3453
mmax
(ksi)
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
N
(cycles)
142
700
1650
3671
3981
5586
7002
7833
8490
8580
9274
10906
11445
11960
13655
14094
14720
16008
19320
21774
mmax
(ksi)
42
42
42
42
42
42
42
42
42
42
42
42
42
42
42
42
42
42
42
42
N
(cycles)
7980
18550
20340
20400
21610
23100
23400
23468
25121
27111
27850
30785
35764
38600
39580
40000
40100
41520
56000
72935
14
FIGURES CAPTIONS
Fig. 1 - Graph for the calculation of the constants _ t and `t in eq. (7), in the case mmin = -10
ksi.
Fig. 2 - Fatigue curves for the material under study.
Fig. 3 – Effect of varying: a) _t , and, b) `t on the fatigue trend.
Fig. 4 – Fatigue behaviour of the CFRP tested in [19].
Fig. 5 - Statistical distribution of the converted monotonic strength, mtoN.
Fig. 6 - Distribution function of fatigue life for mmax = 58 ksi and mmin = - 16 ksi.
Fig. 7 - Distribution function of fatigue life for mmax = 50 ksi and mmin = - 16 ksi.
Fig. 8 - Distribution function of fatigue life for mmax = 42 ksi and mmin = - 16 ksi.
Fig. 9 - Residual strength after tension-compression fatigue (solid symbols) and theoretical
predictions (lines).
15
24
N` - 1
18
12
6
0
0
0.25
0.5
Q
Fig. 1
0.75
1
16
80
mmax (ksi)
60
m min = 0
40
m min = -10 ksi
m min = -16 ksi
Static
20
0
2
4
LogN
Fig. 2
6
8
17
1.2
_ t = 0.02
_ t = 0.05
mmax/mto
1
_ t = 0.2
0.8
0.6
0.4
0
2.5
5
LogN
Fig. 3a)
7.5
10
18
1.2
`t = 0.15
`t = 0.2
1
mmax/mto
`t = 0.3
0.8
0.6
0.4
0
2.5
5
LogN
Fig. 3b)
7.5
10
19
Static
fatigue
Stress
Frequency
Log t
Fig. 4
20
Probability of failure
1
0.75
0.5
0.25
0
50
60
70
mtkT (ksi)
Fig. 5
80
90
21
Probability of failure
1
0.75
0.5
0.25
0
0
2
4
LogT
Fig. 6
6
22
Probability of failure
1
0.75
0.5
0.25
0
2
3
4
LogN
Fig. 7
5
23
Probability of failure
1
0.75
0.5
0.25
0
3
4
5
LogN
Fig. 8
6
24
80
mres (ksi)
60
40
20
mmax = 42 ksi
mmax = 50 ksi
0
0
2
4
LogN
Fig. 9
6