Fatigue lifetime of glass fabric/epoxy composites

Transcription

1 Composites: Part A 30 (1999) 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, 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. 999 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 X/99/$ see front matter 999 Published by Elsevier Science Ltd. All rights reserved. PII: S X(98)

2 300 G. Caprino, G. Giorleo / Composites: Part A 30 (1999) 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 ˆ 1! s n ˆ s o, where s o is the monotonic tensile strength of virgin material and s n the residual strength after n cycles, the relationship s o s n ˆ as max 1 R n b was obtained, where R is the stress ratio (i.e. the ratio of minimum-to-maximum applied stress), and a, b two material parameters to be experimentally evaluated. With the further hypothesis that final failure takes place when s n ˆ s max, the critical number of cycles to failure, N, was calculated from Eq. (1) as 1 1 s N ˆ 1 o b 2 a 1 R s max A useful form for Eq. (2) is s o 1 s max 1 R ˆ a Nb 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 b ). From the equation of such a straight line, calculated by the best fit method, both a and b 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 F N (N*) was obtained as s N b 1 g ˆ 1 ln 1 F a 1 R s N N Š=d 4 max where g and d 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 1 oriented glass fibre composites, different in resin content and fibre form, was examined. From the results, the parameters a and b appearing in Eq. (1) were determined, according to the method previously described. Then, Eq. (3) was solved for s o, giving s o ˆ s on ˆ s max 1 a 1 R N b Š From Eq. (5), the virgin material strength for each specimen failed in fatigue (indicated by the symbol s on in the equation to distinguish it from that, s o, directly measured in a monotonic characterisation test), was evaluated. Finally, the distribution of s on was compared with that of s 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 s 1=b N ˆ 1 oc 2 0 a 1 =R s maxc s oc 1 s maxc 1 =R ˆ a Nb s N b 1 g ˆ 1 ln 1 F a 1 =R s N N Š=d maxc s oc ˆ s ocn ˆ s maxc 1 a 1 =R N b Š 5 0 In Eqs. (2 0 ), (3 0 ), (4 0 ) and (5 0 ), s oc is the absolute value of the material compression strength, and s maxc the absolute value of the highest compression stress applied in fatigue. Of course, it is expected that the a, b, g, d 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 V f ˆ 40%. From the plates, a hundred specimens 80 mm in length

3 G. Caprino, G. Giorleo / Composites: Part A 30 (1999) 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 ˆ 10 and R ˆ 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. 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 ˆ 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 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. 4. Results and discussion 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 s oc ˆ 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

4 302 G. Caprino, G. Giorleo / Composites: Part A 30 (1999) Fig. 3. Semi-log plot of the non-dimensional maximum applied stress, s maxc /s 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 g ˆ N mm 2 and a shape parameter d ˆ 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 d 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 d was in the range 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, s maxc /s 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 0 ), 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 0 ) 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 0 ), a suitable value of the constant b can be found, such that the (N b ) values are well fitted by a straight line passing through the origin, when plotted against K. Of course, in this case the constant a will be given by the slope of the straight line. From the experimental results, the quantity (N b ) was evaluated by successive approximations of b, 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 a ˆ and b ˆ Substituting them in Eq. (2 0 ), 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

5 G. Caprino, G. Giorleo / Composites: Part A 30 (1999) Fig. 5. Graph for the evaluation of the parameters a, b appearing in the fatigue model. Continuous line: best-fit straight line. depicted in Fig. 5 for the evaluation of a and b, only the fatigue data are utilised. Consequently, all the quantities appearing in Eq. (5) ((5 0 )) do not contain any implicit information on the scatter in monotonic strength. Therefore, when Eq. (5) ((5 0 )) is used for the calculation of s on (s ocn ) starting from a set of fatigue results, the scatter in the calculated s on (s 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 s on of randomly oriented glass fibre reinforced plastics, starting from the fatigue data. Comparing the s on values with those, s o, measured by monotonic tests, it was found that the Weibull distributions of s on and s 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, s ocn. Dashed line: two-parameter Weibull distribution of the measured monotonic strength. Continuous line: three-parameter Weibull distribution of the measured monotonic strength. Fig. 7. Probability of failure of the calculated monotonic strength, s ocn. Dashed line: best-fit two-parameter Weibull curve. Continuous line: best-fit three-parameter Weibull curve. this work, where s ocn was evaluated for each of the specimens failed in fatigue substituting in Eq. (5 0 ) the a and b 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 g ˆ N mm 2, d ˆ 2.14 and 1 ˆ N mm 2, where 1 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 s 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,

6 304 G. Caprino, G. Giorleo / Composites: Part A 30 (1999) respectively, evaluated on the basis of the s ocn results. From the calculations, the values g ˆ N mm 2, d ˆ were obtained for the two-parameter curve, and g ˆ N mm 2, d ˆ 2.62, 1 ˆ 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 0 ), 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 [2] Broek D. Elementary engineering fracture mechanics. Leyden: Noordhoff, [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: [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: [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: [6] Hwang W, Han KS. Fatigue of composites fatigue modulus concept and life prediction. Journal of Composite Materials 1986;20: [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: [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: [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: [10] Hwang W, Han KS. Statistical study of strength and fatigue life of composite materials. Composites 1987;18(1): [11] El Kadi H, Ellyin F. Effect of stress ratio on the fatigue of unidirectional glass fibre/epoxy composite laminates. Composites 1994;25(10): [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: [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: [17] Whitney JM, Knight M. The relationship between tensile and flexure strength in fiber-reinforced composites. Experimental Mechanics 1980;20: [18] Wisnom MR. Relationship between strength variability and size effect in unidirectional carbon fibre/epoxy. Composites 1991;22(1): [1] Singer E. Fracture mechanics in design of pressure vessels. Engineering Fracture Mechanics 1969;1: