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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