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Fatigue of composite laminates under off-axis loading
International Journal of Fatigue 21 (1999) 253–262
Fatigue of composite laminates under off-axis loading T.P. Philippidis *, A.P. Vassilopoulos University of Patras, Department of Mechanical and Aeronautics Engineering, Section of Applied Mechanics, P.O. Box 1401, 265 00 Patras, Greece Received 15 June 1998; received in revised form 13 October 1998; accepted 13 October 1998
Abstract Results from an experimental program consisting of static and fatigue tests on flat coupons, cut at different off-axis directions from a multidirectional, (MD), Glass/Polyester, (GRP), laminate are presented in this paper. The material is similar to those used by GRP wind turbine rotor blade manufacturers, i.e. hand lay-up and room temperature curing. The stacking sequence of the MD laminate under consideration is [0/( ⫾ 45)2/0]T. Specimens were cut at five different off-axis directions from that laminate and over one hundred and forty tests were conducted under static and cyclic loading. Based on the test results the effect of off-axis loading on static and fatigue behaviour of the MD laminate is studied. A simple empirical model is used to predict the observed stiffness degradation and to determine stiffness based S–N curves by means of a limited number of test data. For the materials investigated in this program it is shown that E-modulus variation depends on the off-axis loading as much as on the applied cyclic stress level. Stiffness based S-N curves corresponding to 5–20% stiffness reduction are more conservative than standard S-N allowables of 95% reliability. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Composites; S-N curves; Experimental data; Reliability; Damage
Nomenclature E1 EN Eo N Nf R ␣ K, c, , b dN/do R(N) ␣f, N¯
␣ˆ f, Nˆ Nij N¯oi Xij ni
Young modulus measured at first cycle Young modulus measured at N-th cycle static Young modulus Number of cycles Number of cycles to failure Stress ratio, R ⫽ min/max Stress amplitude material constants Normalized energy loss Reliability of N Shape and scale parameters of Weibull distribution Estimated values of ␣f, N¯ Number of cycles of specimen j at i-th stress level. Characteristic number of cycles Normalized number of cycles, Xij ⫽ Nij /Nˆoi number of specimens at i-th stress level
* Corresponding author: Tel.: ⫹ 30-61-997235; fax: ⫹ 30-61997190; e-mail: [email protected]
M
Total number of specimens
1. Introduction The prediction of fatigue damage and fatigue life of laminated composite materials has been the subject of many investigations. Different approaches have been adopted, based on different damage metrics for measuring fatigue damage accumulation. The aim of such studies was to establish a process requiring a minimum of experimental data while reliably predicting the condition of the material. Existing fatigue theories can be classified in four categories [1]: 1. Macroscopic failure theories based on static strength criteria modified to account for cyclic loading. 2. Strength degradation fatigue theories, where the damage metric is the residual strength of the composite material after a cyclic program. According to these, failure occurs when the residual strength decreases to the maximum applied cyclic stress. 3. Stiffness-change fatigue theories are those assuming
0142-1123/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 9 8 ) 0 0 0 7 3 - 5
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stiffness degradation as a fatigue damage metric. The superiority of these methods, compared to strength degradation based ones, lies on their non-destructive inspection option of fatigue damage growth. Residual strength demonstrates minimal decreases with the number of cycles until a stage close to the end of lifetime when it begins to change rapidly. On the other hand, stiffness exhibits greater changes during fatigue life [1–3]. Stiffness based criteria can be established in order to use these methods as design tools [4]. 4. Finally, actual damage mechanisms fatigue theories are based on the modelling of intrinsic defects in the matrix of the composite material that can be treated as matrix cracks. The study of the development of these cracks provides information on fatigue damage. The present paper is concerned with stiffness-change based fatigue theories. Experimental data are presented from a program consisting of static and fatigue tests on flat coupons, cut at different off-axis directions from a MD GRP laminate. In addition, stiffness variation with number of cycles was continuously monitored and measured for each specimen and for all the off-axis angles. The dependence of stiffness changes on the applied stress level and off-axis angle was systematically investigated. It was observed, in general, that the higher the cyclic stress, the lower the stiffness variation with increasing number of cycles. A previously introduced [4], empirical model, is combined with the present experimental data, showing that stiffness reduction can be used to determine stiffness based S-N curves which do not correspond to failure but to a preset value of stiffness degradation. A linear relation between stiffness reduction and number of cycles is adopted. Theoretical predictions are corroborated satisfactorily by the corresponding experimental values. Finally, the area enclosed in stress-strain hysteresis loops, indicative of energy dissipation, is continuously monitored for each off-axis angle and is found to increase slightly with cumulative number of cycles. However, these changes have no obvious relation with off-axis angle or stress level.
2. Stiffness degradation model An empirical model for the prediction of stiffness based S-N curves from a minimum of experimental data has been previously introduced [4]. The model is used in the present study for predicting stiffness degradation and the corresponding S-N curves from only a few experimental data. The degree of damage in a polymer matrix composite coupon can be evaluated by measuring stiffness degra-
dation, EN/E1 where E1 denotes the Young modulus of the material measured at the first cycle, different in general from the static value Eo, and EN the corresponding one at the N-th cycle. The central, main, part of the curve expressing the variation of EN/E1 versus normalized number of cycles, N/Nf, Nf being the number of cycles to failure, is assumed linear [4]:
冉冊
EN ␣ ⫽1⫺K E1 Eo
c
N Nf
(1)
Material constants, K and c, in Eq. (1) are determined by means of the respective experimental data for stiffness degradation which is assumed to depend on the number of stress cycles, N, and the level of applied cyclic stress amplitude, ␣. Relation (1) also establishes a stiffness based design criterion since for a preset value of EN/E1 ⫽ p one can solve for ␣ to obtain an alternative form of S-N curve corresponding not to material failure but to a specific stiffness degradation percentage, (1-p)%. The experimental results derived in the course of this study corroborate satisfactorily the above deterministic material model behaviour. The determination of constants K and c, by fitting Eq. (1) to the experimental data, is generally a time consuming procedure that needs a large number of tests. Therefore, the following procedure is adopted. For a specific value of EN ⫽ EL at N ⫽ NL, Eq. (1) reads:
冉冊
EL ␣ ⫽1⫺K E1 Eo
c
NL Nf
(2)
Solving the above for (␣/Eo)c and substituting in Eq. (1) yields:
冉
冊
EN EL N ⫽1⫺ 1⫺ E1 E1 NL
(3)
In terms of the above relation it is now possible to calculate the stiffness degradation ratio, EN/E1, by using a limited amount of experimental data for each stress level, ␣. In fact, by using EL, NL values corresponding to a specific, relatively short, life interval, e.g. 60% of Nf, the ratio EN/E1 from Eq. (3) is calculated and together with Eq. (1), after determination of constants K and c, it is possible to extrapolate the expected S-N behaviour for a specific stiffness degradation level. The validity of the aforementioned procedure for the composite material investigated herein was proved by comparison with the experimental data.
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3. Experimental A comprehensive experimental program has been realized consisting of static and fatigue tests of straight edge coupons cut from a multidirectional laminate. The scope of the tests was to determine static and fatigue strength under reversed loading in the laminate principal directions. Coupon stiffness during fatigue, defined by the average slope of the stress-strain loop [5], was monitored continuously. Area included in each hysteresis loop was calculated at every cycle to account for energy dissipation effects. In addition to the tests on the MD laminated coupons, similar tests have been performed on the basic building lamina of the laminate, the unidirectional one. The material used was E-glass/polyester, E-glass from AHLSTROM GLASSFIBRE, while the polyester resin was CHEMPOL 80 THIX by INTERCHEM. This resin is a thixotropic unsaturated polyester and was mixed with 0.4%, Cobalt naphthenate solution (6% Co), accelerator and 1.5% Methyl Ethyl Ketone Peroxide, MEKP, (50% solution), catalyst. Rectangular plates were fabricated by hand lay-up technique and cured at room temperature. The stacking sequence of the MD laminate consisted of four layers, 2 ⫻ UD, Unidirectional lamina of 100% aligned warp fibers, with a weight of 700 g/m2 and 2 ⫻ stitched ⫾ 45 of 450 g/m2, 225 g/m2 in each off-axis angle. With reference direction, 0°, that of the UD layer fibers, the lay-up can be encoded as [0/( ⫾ 45)2/0]T. Specimens were cut, by a diamond wheel, at 0°, on-axis, and 30°, 45°, 60° and 90° off-axis directions. All data from specimens at 0°, 45° and 90° was used to obtain basic anisotropic mechanical properties of the material, while off-axis specimen results at 30° and 60° were used for the verification of theoretical predictions from macroscopic fatigue failure criteria [6]. The resulting stacking sequences of specimens cut off-axis at 30° and 60° were designated as [30/(75/-15)2/30]T and [60/(-75/15)2/60]T respectively. All test data was used for the study of stiffness variation during fatigue life as a function of stress level and off-axis angle. In addition to the MD laminates, UD plates were manufactured as well consisting of two layers at a thickness of 1.6 mm. The specimens were prepared according to ASTM 3039-76 standard, and aluminum tabs were glued at their ends. Coupon edges were trimmed with sandpaper. The MD specimens were 250 mm long and had a width of 25 mm. Their nominal thickness was 2.6 mm. The length of the tabs, with a thickness of 2 mm, was 45 mm leaving a gauge length of 160 mm for each one of the specimens. Static and fatigue tests were performed. The number of specimens tested, 142 in total, was partitioned as follows: 50 specimens for static tests to provide baseline data, both tensile and compressive strengths were
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obtained, while 92 specimens were tested under uniaxial reversed cyclic stress for the determination of S-N curves at various off-axis directions. Static tests were performed in tension and compression on an MTS machine of 250 kN capacity under displacement control at a speed of 1 mm/min. The specimens used for compression tests had a gauge length of 30 mm to avoid buckling. Fatigue tests of sinusoidal waveform were also carried out on the same MTS machine. Frequency was kept at 10 Hz for all coupons, while the stress ratio (R ⫽ min/max) was equal to ⫺ 1. The tests were continued until last ply failure or 106 cycles, whichever occurred first. Especially for the on-axis MD specimens loading was continued up to 5 ⫻ 106 cycles. The antibuckling jig [5], of Fig. 1, was used in this experimental program.
4. Results and discussion The S-N curves derived experimentally for on- and off-axis MD coupons under reversed cyclic loading are collectively presented in Fig. 2. The various S-N curves follow a trend similar to that exhibited by static test results shown in Fig. 3. Theoretical predictions from Tsai-Hahn quadratic failure criterion [6], solid line, are also shown in this figure. The fatigue strength of the 0° MD specimens was the highest, while was decreasing at higher off-axis angles. The controversial behaviour of the 45° specimens, exhibiting higher fatigue strength than the 30° specimens, was to be expected and was related to the stacking sequence of the laminate under consideration. When specimens are cut at 45°, the fibers of the stitched ( ⫾ 45) layers become on-axis and the prevailing failure mode is
Fig. 1.
Antibuckling device.
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Fig. 2.
Experimental S-N curves.
Fig. 4.
Fig. 3. Off-axis static strength, st, of [0/( ⫾ 45)2/0]T GRP laminate.
shifted from matrix dominated, observed in 30° and 60° off-axis coupons, to partially fiber controlled. Therefore, these fibers receive most of the fatigue load instead of the matrix. In general, mixed failure mode was observed for all the off-axis specimens.
Stiffness degradation of UD specimens.
4–6. In Fig. 4 the stiffness degradation for UD specimens is presented for each stress level used for the determination of the corresponding S-N curve. The observed stiffness degradation up to failure is less than 7%. Despite the large scatter of data, which is common to fatigue test results, it is observed that stress level is reversly proportional to stiffness degradation. In Figs. 5 and 6 stiffness changes over number of cycles for the on- and off-axis specimens of the MD laminate are presented. In Fig. 5 the variation of stiffness of the 0° MD specimens is shown, while the stiffness degradation of the 30° MD specimens is presented in Fig. 6. In general, the higher the stress level the lower the decrease of stiffness as it was also observed for a
4.1. Stiffness degradation Attempts have been made during this study to correlate the experimentally obtained data with analytical predictions, based on the aforementioned simple model, associating the stiffness reduction with cyclic stress levels. Normalizing factors have been used in order to take into account data from all specimens regardless the number of cycles to failure and the initial stiffness of each one of them. An initial value, E1, for the Young modulus of elasticity corresponding to the first cycle, different in general from the static value, Eo, has been used. The number of cycles was normalised using the value of cycles to failure, Nf. The experimentally obtained results for the stiffness changes versus the number of cycles are shown in Figs.
Fig. 5.
Stiffness degradation of on-axis MD specimens.
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Fig. 8. Edge view of a 90° off-axis MD specimen showing matrix failure.
Fig. 6.
Stiffness degradation of 30° off-axis MD specimens.
Carbon/PEEK material system [5] with a pseudoisotropic lay-up. Moreover, with increasing off-axis angle stiffness degrades more drastically. It is clearly shown in the aforementioned figures that the decrease of stiffness in the 0° MD specimens is less than that of the offaxis specimens, regardless of the stress level or number of cycles. This phenomenon originates from the different failure modes exhibited by the on- and off-axis specimens. For the coupons of the former type, fibers are always in the loading direction and matrix damage accumulation is limited. On the other hand, a greater reduction in stiffness was observed in the off-axis specimens, which was attributed to an accumulation of defects in the matrix. In Figs. 7 and 8 fracture surfaces of typical specimens are shown after final failure. A 0° MD coupon with onaxis fibers broken is shown in Fig. 7. In Fig. 8 the edge
Fig. 7.
Failure of the fibers of an on-axis MD specimen.
view of a 90° MD specimen is shown with the fibers perpendicular to the loading direction. The reason for the creation of so many matrix cracks in these layers during testing is attributed to the off-axis placement of the fibers. The experimental values for stiffness degradation shown in Figs. 5 and 6 were then compared to the respective theoretical predictions derived by means of Eq. (3). In Figs. 9 and 10 the theoretically predicted stiffness decrease lines along with the experimentally obtained curves for specimens cut at 0° and 30° from the MD laminate are shown. In Fig. 9, stiffness variation for the 0° MD specimens is presented, with the theoretical predictions shown as straight lines and the test data fitted by least square curves.
Fig. 9. Comparison of experimental values and theoretical predictions for stiffness degradation. On-axis MD specimens.
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Fig. 11.
Stiffness based S-N curves for on-axis MD specimens.
Fig. 10. Comparison of experimental values and theoretical predictions for stiffness degradation. 30° off-axis MD specimens.
Considering the limited number of tests required for the analytical calculations, the correlation between the test values and those calculated from Eq. (3) was good. The theoretical straight lines, in most of the cases shown, reflect efficiently the stiffness variations during lifetime except for the stress level at 100 MPa where a choice of EL at NL/Nf ⫽ 0.6, instead of the actual 0.4, would yield better results. Similar comments also apply for the comparison of experimental data and theoretical predictions for the 30° off-axis specimens presented in Fig. 10. The general trend observed in stiffness variation diagrams, e.g. Figs. 5 and 6, is that the higher the applied cyclic stress, the less the stiffness degradation. However, in each one of these figures a stress level exists for which the above rule does not apply. This also holds for the theoretical predictions, by means of Eq. (3), presented in Figs. 9 and 10. Despite the deterministic and empirical nature of Eq. (3), its use and effectiveness relies on the simulation of experimental behaviour since EL and NL are taken from the observed material response at each stress level. It has to be mentioned, however, that these experimentally obtained values correspond to a limited life interval, e.g. NL/Nf ⫽ 0.4 up to 0.6. Using these analytically predicted stiffness changes, S-N curves corresponding to given material degradation and not ultimate failure, can be drawn for each one of the materials used. Material constants K and c are now calculated by fitting Eq. (1) to the above data, requiring only a limited number of tests. In Figs. 11 and 12 stiffness based S-N curves are shown for 0° and 30° MD specimens respectively. In Fig. 11 theoretical S-N curves are presented corresponding to 5% and 15% stiffness degradation along with experimental failure points. S-N curve for 5% stiffness
Fig. 12.
Stiffness based S-N curves for 30° off-axis MD specimens.
decrease is on the safe side of all the experimental failure data. On the other hand, the S-N curve corresponding to 85% remaining stiffness seems to fit well the experimental failure data. This is in agreement with the information shown in Fig. 5 where the main part of the stiffness data during life time lies in a region between 0.85 and 0.95 of the initial value E1. Similar conclusions are derived by observing Fig. 12 where the stiffness based S-N curves for 30° MD specimens are presented. However, in this case, the stiffness based S-N curve beyond which no failure points are observed, corresponds to 10% stiffness degradation. Furthermore, this data seems to fit better the curve of 60% remaining stiffness, reflecting the behaviour of 30° MD specimens shown in Fig. 6.
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4.2. Statistical analysis of fatigue data
X(Xi1,Xi2,…,Xini), i ⫽ 1,2,…,m
Fatigue data from each set of on- and off-axis coupons were subjected to statistical analysis to determine 95% reliability S-N curves. For that purpose the analysis presented by Whitney [7,8] was adopted, based mainly on two assumptions: (a) a classical power law representation of the S-N curve and (b) a two parameter Weibull distribution of time to failure. The form of the S-N equation is:
where:
Nb␣ ⫽ T⇒␣ ⫽ T (1/b)N( ⫺ 1/b) ⫽ N( ⫺ 1/b)
R(X) ⫽ exp ⫺
(4)
where N is the number of cycles, ␣ is the stress amplitude level and b, are material constants. For every stress level the number of cycles to failure is described by a two parameter Weibull distribution.
冋 冉冊册
N R(N) ⫽ exp ⫺ ¯ N
(5)
Calculation of constants and b of Eq. (4) is performed as follows: A two parameter Weibull distribution is fitted to the data of each stress range:
冋 冉冊册
Ni R(Ni) ⫽ exp ⫺ ¯ Ni
␣fi
(6)
where i ⫽ 1,2,...,m with m the total number of stress levels. The parameters ␣fi and N¯i of each Weibull distribution are determined by the following Maximum Likelihood Estimators:
冘 冘 ni
N␣ijˆ fiLnNij
j⫽1
⫺
ni
N␣ijˆ fi
1 ni
冘 ni
j⫽1
LnNij ⫺
1 ⫽0 ␣ˆ fi
(7)
j⫽1
冉 冘 冊
1 Nˆi ⫽ ni
ni
N␣ijˆ fi
1 ˆ fi ␣
(9)
Nij Xij ⫽ ˆ Ni
(10)
It is assumed that this set of data also follows a two parameter Weibull distribution:
冋 冉 冊册 X Xo
␣f
(11)
The parameters of the distribution of Eq. (11) are given by:
冘冘 冘冘 ni
m
X␣ijˆ fLnXij
␣f
259
i⫽1 j⫽1 m
ni
X
1 ⫺ M
ˆf ␣ ij
冘冘 m
ni
LnXij ⫺
i⫽1 j⫽1
1 ⫽0 ␣ˆ f
(12)
i⫽1 j⫽1
冉 冘冘 冊
1 Xˆo ⫽ M
m
ni
X
ˆf ␣ ij
1 ˆf ␣
(13)
i⫽1 j⫽1
Where M is the total number of specimens, i.e., the sum of the coupons from each stress level. The value of Xˆo has to be unity for a perfect fit [7]. If Xˆo takes any value other than unity, the characteristic number of cycles can be adjusted to produce Xˆo ⫽ 1. In particular: N¯oi ⫽ XˆoNˆi
(14)
The slope of the S-N curve, 1/b, and the y-intercept, can be determined by fitting log␣i versus logN¯oi to a straight line. With , b and ␣f now determined the S-N curve for any desired level of reliability can be calculated by:
␣ ⫽ 兵[ ⫺ LnR(N)](1/␣fb)其N( ⫺ 1/b)
(15)
(8)
j⫽1
with ni the number of specimens at each stress level. Eq. (7) has only one positive root [7] which is the estimated value for the shape parameter ␣fi, and thus the estimated characteristic number of cycles is given directly by Eq. (8). The values of the number of cycles at each stress level are subsequently normalized by the corresponding estimated characteristic number, Nˆi. Thus, the following normalized data set is formed:
A computer implementation of this method and application on a wide variety of fatigue data is given elsewhere [9]. In the same reference, two other methods for statistical analysis of fatigue data are also described and applied on the same data sets, but the method given herein is adopted due to better overall performance. 4.3. Stiffness based and reliability S-N curves S-N curves at specific reliability level and stiffness based S-N curves calculated as previously described are
260
Fig. 13.
T.P. Philippidis, A.P. Vassilopoulos / International Journal of Fatigue 21 (1999) 253–262
Comparison of S-N curves for on-axis MD specimens. Fig. 15.
then compared. The S-N curves which correspond to 95% reliability level for all on- and off-axis data are quite similar to the corresponding curves for 5–20% stiffness reduction, when plotted on the S-N plane. In fact, as it is shown in Figs. 13–15, S-N curves corresponding to stiffness reduction of 5–20%, depending on off-axis configuration, are more conservative, especially for high cycle numbers, than the statistically obtained SN curves. The S-N curves for the on-axis MD specimens are presented in Fig. 13. The stiffness based S-N curve corresponding to 5% stiffness reduction is the one which lies closer to the statistically determined S-N curve. In Fig. 14 the curve for 10% decrease of stiffness is plotted together with the curve for 95% reliability level for the coupons cut at 30° off-axis. Finally, the curves for 10% and 20% stiffness decrease are compared with the S-N curve for 95% reliability for 60° off-axis MD specimens in Fig. 15. Equations of all above mentioned S-N curves are given in Table 1. As it is observed from Figs. 13–15, stiffness reduction level of stiffness based S-N curves associated with 95%
Fig. 14. Comparison of S-N curves for 30° off-axis MD specimens.
Comparison of S-N curves for 60° off-axis MD specimens.
statistical ones, correlates satisfactorily with off-axis angle; 5% for the on-axis coupons, 10% for 30° off-axis and 20% for 60° off-axis. Therefore, for the material system investigated, a 10% stiffness reduction every 30° off-axis is the appropriate figure for stiffness based S-N curve determination. Concerning the experimental results presented herein, the total time needed for the determination of a statistical S-N curve amounts to 8E ⫹ 06 cycles (16 coupons), whereas only half of it is necessary for stiffness based S-N curves. Therefore, the latter type of S-N curve can be successfully used in initial design phases, replacing 95% allowables. 4.4. Energy dissipation measurements In the course of this experimental program efforts were also directed to the definition of meaningful ‘fatigue functions’ to be used eventually as overall damage metrics. To this end, hysteresis loop area, indicative of energy dissipation, was continuously monitored for every specimen and results are presented showing the relation of normalized energy loss (dN/do), determined as the area enclosed in a hysteresis stress–strain loop, with normalized number of cycles (N/Nf). In Fig. 16 energy-loss variation versus number of cycles is shown for the on-axis MD specimens. Generally, ‘internal friction’ seems to either increase or decrease in early stages, while it remains almost constant up to failure. For any specimen type it is valid that higher stress levels induce higher energy dissipation. Results from 30° off-axis MD coupons are presented in Fig. 17. Energy dissipation is shown to increase with number of cycles and stress level. However, the above conclusions are not true for all stress levels or specimens. This kind of measurement is probably not best suited to reflect the material dissipative response.
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Table 1 Calculated S-N curve equations
(deg.)
Experimental (Linear regression)
Statistical 95%
Stiffness based*
0 30 60
␣ ⫽ 160.1N−0.05016 ␣ ⫽ 101.4N−0.05722 ␣ ⫽ 102.8N−0.07530
␣ ⫽ 150.9N−0.05121 ␣ ⫽ 107.2N−0.07285 ␣ ⫽ 122.0N−0.09809
␣ ␣ ␣ ␣
⫽ ⫽ ⫽ ⫽
180.8N−0.06636 (0.95) 130.9N−0.09217 (0.90) 93.6N−0.07981 (0.90) 98.9N−0.07981 (0.80)
* number in parenthesis correspond to residual stiffness.
5. Conclusions
Fig. 16. Dissipated energy variation for the on-axis MD specimens.
Fig. 17. Dissipated energy variation for the 30° off-axis MD specimens.
During fatigue life the stiffness of a structural element is reduced. Efforts have been made to correlate this stiffness degradation with the damage accumulated in the material. Stiffness changes of coupons cut at several offaxis angles from a multidirectional laminate [0/(⫾45)2/0]T and subjected to reversed loading were investigated in this study. It is observed in general that stiffness degradation increases with off-axis angle and decreases for higher applied cyclic stress. The aforementioned behaviour results from the failure modes observed in each specimen and is related as much to the cyclic stress as to the off-axis angle. When the cyclic stress is high the failure mode is fiber dominated and the fracture is due to tensile stresses. Under low cyclic stress the damage mode is shifted from fiber to matrix controlled while failure is driven mainly by compressive loads. This happens when on-axis specimens are tested with the fibers along loading direction being the elements that carry the imposed loads. When off-axis specimens are tested the situation is different. In this case the failure mode, irrespective of the stress level, is always matrix dominated. For this reason also, the observed stiffness degradation for off-axis coupons is higher than that of on-axis ones. An empirical method is validated herein for the determination of S-N curves which do not correspond to fatigue strength data, but to a predetermined value of stiffness reduction, e.g. 5%–10% of the initial value, by using only a portion of the fatigue data, e.g. data up to 60% of fatigue life. Furthermore, S-N curves at 95% reliability level were determined for each set of fatigue data. It is shown in this paper that these two kinds of S-N curves are comparable although stiffness based curves are more conservative, especially for higher number of cycles. This was an interesting feature of the study presented herein along with the fact that for the determination of stiffness based S-N curves only a limited amount of fatigue data is needed. The use of these curves instead of 95% reliability allowables could be anticipated for initial design phases and prototyping. The basic drawback of residual stiffness based fatigue criteria i.e., large number of tests, can be avoided by using the procedure presented herein. The necessary
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experiments can be conducted within limited time of testing, e.g. 5 days, and stiffness based S-N curves can be readily produced. By continuously monitoring stress-strain hysteresis loop area during reversed loading of each coupon and correlating with number of cycles at fracture it was concluded that energy dissipation measurements do not constitute an efficient parameter to be used as a cumulative damage metric.
Acknowledgement Financial support of the work contained in this paper through project EPET II, #573 by the General Secretariat of Research and Technology is gratefully acknowledged.
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[2] Highsmith AL, Reifsneider KL. Stiffness reduction mechanisms in composite laminates. In: Reifsneider KL, editor. Damage in Composite Materials, ASTM STP 775. American Society for Testing and Materials, 1982:103–17. [3] Talreja R. Fatigue of Composite Materials. Lancaster Pennsylvania: Technomic, 1987. [4] Andersen SI, Brondsted P, Lilholt H. Fatigue of polymeric composites for wingblades and the establishment of stiffness-controlled fatigue diagrams In: Proceedings of 1996 European Union Wind Energy Conference, Go¨teborg Sweden, 20–24 May 1996:950–53. [5] Pantelakis SG, Philippidis TP, Kermanidis TB. Damage accumulation in thermoplastic laminates subjected to reversed cyclic loading. In: Paipetis SA, Yioutsos AG, editors. High Technology Composites in Modern Applications, University of Patras, 1995:156– 64. [6] Philippidis TP, Vassilopoulos AP. Fatigue strength prediction under multiaxial stress. J Comp Mat 1998 (Accepted for publication). [7] Whitney JM, Daniel IM, Pipes RB. Experimental Mechanics of Fiber Reinforced Composite Materials. Prentice-Hall, 1984. [8] Whitney JM. Fatigue characterisation of composite materials. In: Lauraitis KN, editor. Fatigue of Fibrous Composite Materials, ASTM STP 723. American Society for Testing and Materials, 1981:133–51. [9] Philippidis TP, Vassilopoulos AP. Statistical Evaluation of Fatigue Data (1997), University of Patras, 2nd semester 1997 progress report, EPET-II #573 (in Greek).
Fatigue of composite laminates under off-axis loading T.P. Philippidis *, A.P. Vassilopoulos University of Patras, Department of Mechanical and Aeronautics Engineering, Section of Applied Mechanics, P.O. Box 1401, 265 00 Patras, Greece Received 15 June 1998; received in revised form 13 October 1998; accepted 13 October 1998
Abstract Results from an experimental program consisting of static and fatigue tests on flat coupons, cut at different off-axis directions from a multidirectional, (MD), Glass/Polyester, (GRP), laminate are presented in this paper. The material is similar to those used by GRP wind turbine rotor blade manufacturers, i.e. hand lay-up and room temperature curing. The stacking sequence of the MD laminate under consideration is [0/( ⫾ 45)2/0]T. Specimens were cut at five different off-axis directions from that laminate and over one hundred and forty tests were conducted under static and cyclic loading. Based on the test results the effect of off-axis loading on static and fatigue behaviour of the MD laminate is studied. A simple empirical model is used to predict the observed stiffness degradation and to determine stiffness based S–N curves by means of a limited number of test data. For the materials investigated in this program it is shown that E-modulus variation depends on the off-axis loading as much as on the applied cyclic stress level. Stiffness based S-N curves corresponding to 5–20% stiffness reduction are more conservative than standard S-N allowables of 95% reliability. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Composites; S-N curves; Experimental data; Reliability; Damage
Nomenclature E1 EN Eo N Nf R ␣ K, c, , b dN/do R(N) ␣f, N¯
␣ˆ f, Nˆ Nij N¯oi Xij ni
Young modulus measured at first cycle Young modulus measured at N-th cycle static Young modulus Number of cycles Number of cycles to failure Stress ratio, R ⫽ min/max Stress amplitude material constants Normalized energy loss Reliability of N Shape and scale parameters of Weibull distribution Estimated values of ␣f, N¯ Number of cycles of specimen j at i-th stress level. Characteristic number of cycles Normalized number of cycles, Xij ⫽ Nij /Nˆoi number of specimens at i-th stress level
* Corresponding author: Tel.: ⫹ 30-61-997235; fax: ⫹ 30-61997190; e-mail: [email protected]
M
Total number of specimens
1. Introduction The prediction of fatigue damage and fatigue life of laminated composite materials has been the subject of many investigations. Different approaches have been adopted, based on different damage metrics for measuring fatigue damage accumulation. The aim of such studies was to establish a process requiring a minimum of experimental data while reliably predicting the condition of the material. Existing fatigue theories can be classified in four categories [1]: 1. Macroscopic failure theories based on static strength criteria modified to account for cyclic loading. 2. Strength degradation fatigue theories, where the damage metric is the residual strength of the composite material after a cyclic program. According to these, failure occurs when the residual strength decreases to the maximum applied cyclic stress. 3. Stiffness-change fatigue theories are those assuming
0142-1123/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 9 8 ) 0 0 0 7 3 - 5
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stiffness degradation as a fatigue damage metric. The superiority of these methods, compared to strength degradation based ones, lies on their non-destructive inspection option of fatigue damage growth. Residual strength demonstrates minimal decreases with the number of cycles until a stage close to the end of lifetime when it begins to change rapidly. On the other hand, stiffness exhibits greater changes during fatigue life [1–3]. Stiffness based criteria can be established in order to use these methods as design tools [4]. 4. Finally, actual damage mechanisms fatigue theories are based on the modelling of intrinsic defects in the matrix of the composite material that can be treated as matrix cracks. The study of the development of these cracks provides information on fatigue damage. The present paper is concerned with stiffness-change based fatigue theories. Experimental data are presented from a program consisting of static and fatigue tests on flat coupons, cut at different off-axis directions from a MD GRP laminate. In addition, stiffness variation with number of cycles was continuously monitored and measured for each specimen and for all the off-axis angles. The dependence of stiffness changes on the applied stress level and off-axis angle was systematically investigated. It was observed, in general, that the higher the cyclic stress, the lower the stiffness variation with increasing number of cycles. A previously introduced [4], empirical model, is combined with the present experimental data, showing that stiffness reduction can be used to determine stiffness based S-N curves which do not correspond to failure but to a preset value of stiffness degradation. A linear relation between stiffness reduction and number of cycles is adopted. Theoretical predictions are corroborated satisfactorily by the corresponding experimental values. Finally, the area enclosed in stress-strain hysteresis loops, indicative of energy dissipation, is continuously monitored for each off-axis angle and is found to increase slightly with cumulative number of cycles. However, these changes have no obvious relation with off-axis angle or stress level.
2. Stiffness degradation model An empirical model for the prediction of stiffness based S-N curves from a minimum of experimental data has been previously introduced [4]. The model is used in the present study for predicting stiffness degradation and the corresponding S-N curves from only a few experimental data. The degree of damage in a polymer matrix composite coupon can be evaluated by measuring stiffness degra-
dation, EN/E1 where E1 denotes the Young modulus of the material measured at the first cycle, different in general from the static value Eo, and EN the corresponding one at the N-th cycle. The central, main, part of the curve expressing the variation of EN/E1 versus normalized number of cycles, N/Nf, Nf being the number of cycles to failure, is assumed linear [4]:
冉冊
EN ␣ ⫽1⫺K E1 Eo
c
N Nf
(1)
Material constants, K and c, in Eq. (1) are determined by means of the respective experimental data for stiffness degradation which is assumed to depend on the number of stress cycles, N, and the level of applied cyclic stress amplitude, ␣. Relation (1) also establishes a stiffness based design criterion since for a preset value of EN/E1 ⫽ p one can solve for ␣ to obtain an alternative form of S-N curve corresponding not to material failure but to a specific stiffness degradation percentage, (1-p)%. The experimental results derived in the course of this study corroborate satisfactorily the above deterministic material model behaviour. The determination of constants K and c, by fitting Eq. (1) to the experimental data, is generally a time consuming procedure that needs a large number of tests. Therefore, the following procedure is adopted. For a specific value of EN ⫽ EL at N ⫽ NL, Eq. (1) reads:
冉冊
EL ␣ ⫽1⫺K E1 Eo
c
NL Nf
(2)
Solving the above for (␣/Eo)c and substituting in Eq. (1) yields:
冉
冊
EN EL N ⫽1⫺ 1⫺ E1 E1 NL
(3)
In terms of the above relation it is now possible to calculate the stiffness degradation ratio, EN/E1, by using a limited amount of experimental data for each stress level, ␣. In fact, by using EL, NL values corresponding to a specific, relatively short, life interval, e.g. 60% of Nf, the ratio EN/E1 from Eq. (3) is calculated and together with Eq. (1), after determination of constants K and c, it is possible to extrapolate the expected S-N behaviour for a specific stiffness degradation level. The validity of the aforementioned procedure for the composite material investigated herein was proved by comparison with the experimental data.
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3. Experimental A comprehensive experimental program has been realized consisting of static and fatigue tests of straight edge coupons cut from a multidirectional laminate. The scope of the tests was to determine static and fatigue strength under reversed loading in the laminate principal directions. Coupon stiffness during fatigue, defined by the average slope of the stress-strain loop [5], was monitored continuously. Area included in each hysteresis loop was calculated at every cycle to account for energy dissipation effects. In addition to the tests on the MD laminated coupons, similar tests have been performed on the basic building lamina of the laminate, the unidirectional one. The material used was E-glass/polyester, E-glass from AHLSTROM GLASSFIBRE, while the polyester resin was CHEMPOL 80 THIX by INTERCHEM. This resin is a thixotropic unsaturated polyester and was mixed with 0.4%, Cobalt naphthenate solution (6% Co), accelerator and 1.5% Methyl Ethyl Ketone Peroxide, MEKP, (50% solution), catalyst. Rectangular plates were fabricated by hand lay-up technique and cured at room temperature. The stacking sequence of the MD laminate consisted of four layers, 2 ⫻ UD, Unidirectional lamina of 100% aligned warp fibers, with a weight of 700 g/m2 and 2 ⫻ stitched ⫾ 45 of 450 g/m2, 225 g/m2 in each off-axis angle. With reference direction, 0°, that of the UD layer fibers, the lay-up can be encoded as [0/( ⫾ 45)2/0]T. Specimens were cut, by a diamond wheel, at 0°, on-axis, and 30°, 45°, 60° and 90° off-axis directions. All data from specimens at 0°, 45° and 90° was used to obtain basic anisotropic mechanical properties of the material, while off-axis specimen results at 30° and 60° were used for the verification of theoretical predictions from macroscopic fatigue failure criteria [6]. The resulting stacking sequences of specimens cut off-axis at 30° and 60° were designated as [30/(75/-15)2/30]T and [60/(-75/15)2/60]T respectively. All test data was used for the study of stiffness variation during fatigue life as a function of stress level and off-axis angle. In addition to the MD laminates, UD plates were manufactured as well consisting of two layers at a thickness of 1.6 mm. The specimens were prepared according to ASTM 3039-76 standard, and aluminum tabs were glued at their ends. Coupon edges were trimmed with sandpaper. The MD specimens were 250 mm long and had a width of 25 mm. Their nominal thickness was 2.6 mm. The length of the tabs, with a thickness of 2 mm, was 45 mm leaving a gauge length of 160 mm for each one of the specimens. Static and fatigue tests were performed. The number of specimens tested, 142 in total, was partitioned as follows: 50 specimens for static tests to provide baseline data, both tensile and compressive strengths were
255
obtained, while 92 specimens were tested under uniaxial reversed cyclic stress for the determination of S-N curves at various off-axis directions. Static tests were performed in tension and compression on an MTS machine of 250 kN capacity under displacement control at a speed of 1 mm/min. The specimens used for compression tests had a gauge length of 30 mm to avoid buckling. Fatigue tests of sinusoidal waveform were also carried out on the same MTS machine. Frequency was kept at 10 Hz for all coupons, while the stress ratio (R ⫽ min/max) was equal to ⫺ 1. The tests were continued until last ply failure or 106 cycles, whichever occurred first. Especially for the on-axis MD specimens loading was continued up to 5 ⫻ 106 cycles. The antibuckling jig [5], of Fig. 1, was used in this experimental program.
4. Results and discussion The S-N curves derived experimentally for on- and off-axis MD coupons under reversed cyclic loading are collectively presented in Fig. 2. The various S-N curves follow a trend similar to that exhibited by static test results shown in Fig. 3. Theoretical predictions from Tsai-Hahn quadratic failure criterion [6], solid line, are also shown in this figure. The fatigue strength of the 0° MD specimens was the highest, while was decreasing at higher off-axis angles. The controversial behaviour of the 45° specimens, exhibiting higher fatigue strength than the 30° specimens, was to be expected and was related to the stacking sequence of the laminate under consideration. When specimens are cut at 45°, the fibers of the stitched ( ⫾ 45) layers become on-axis and the prevailing failure mode is
Fig. 1.
Antibuckling device.
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Fig. 2.
Experimental S-N curves.
Fig. 4.
Fig. 3. Off-axis static strength, st, of [0/( ⫾ 45)2/0]T GRP laminate.
shifted from matrix dominated, observed in 30° and 60° off-axis coupons, to partially fiber controlled. Therefore, these fibers receive most of the fatigue load instead of the matrix. In general, mixed failure mode was observed for all the off-axis specimens.
Stiffness degradation of UD specimens.
4–6. In Fig. 4 the stiffness degradation for UD specimens is presented for each stress level used for the determination of the corresponding S-N curve. The observed stiffness degradation up to failure is less than 7%. Despite the large scatter of data, which is common to fatigue test results, it is observed that stress level is reversly proportional to stiffness degradation. In Figs. 5 and 6 stiffness changes over number of cycles for the on- and off-axis specimens of the MD laminate are presented. In Fig. 5 the variation of stiffness of the 0° MD specimens is shown, while the stiffness degradation of the 30° MD specimens is presented in Fig. 6. In general, the higher the stress level the lower the decrease of stiffness as it was also observed for a
4.1. Stiffness degradation Attempts have been made during this study to correlate the experimentally obtained data with analytical predictions, based on the aforementioned simple model, associating the stiffness reduction with cyclic stress levels. Normalizing factors have been used in order to take into account data from all specimens regardless the number of cycles to failure and the initial stiffness of each one of them. An initial value, E1, for the Young modulus of elasticity corresponding to the first cycle, different in general from the static value, Eo, has been used. The number of cycles was normalised using the value of cycles to failure, Nf. The experimentally obtained results for the stiffness changes versus the number of cycles are shown in Figs.
Fig. 5.
Stiffness degradation of on-axis MD specimens.
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257
Fig. 8. Edge view of a 90° off-axis MD specimen showing matrix failure.
Fig. 6.
Stiffness degradation of 30° off-axis MD specimens.
Carbon/PEEK material system [5] with a pseudoisotropic lay-up. Moreover, with increasing off-axis angle stiffness degrades more drastically. It is clearly shown in the aforementioned figures that the decrease of stiffness in the 0° MD specimens is less than that of the offaxis specimens, regardless of the stress level or number of cycles. This phenomenon originates from the different failure modes exhibited by the on- and off-axis specimens. For the coupons of the former type, fibers are always in the loading direction and matrix damage accumulation is limited. On the other hand, a greater reduction in stiffness was observed in the off-axis specimens, which was attributed to an accumulation of defects in the matrix. In Figs. 7 and 8 fracture surfaces of typical specimens are shown after final failure. A 0° MD coupon with onaxis fibers broken is shown in Fig. 7. In Fig. 8 the edge
Fig. 7.
Failure of the fibers of an on-axis MD specimen.
view of a 90° MD specimen is shown with the fibers perpendicular to the loading direction. The reason for the creation of so many matrix cracks in these layers during testing is attributed to the off-axis placement of the fibers. The experimental values for stiffness degradation shown in Figs. 5 and 6 were then compared to the respective theoretical predictions derived by means of Eq. (3). In Figs. 9 and 10 the theoretically predicted stiffness decrease lines along with the experimentally obtained curves for specimens cut at 0° and 30° from the MD laminate are shown. In Fig. 9, stiffness variation for the 0° MD specimens is presented, with the theoretical predictions shown as straight lines and the test data fitted by least square curves.
Fig. 9. Comparison of experimental values and theoretical predictions for stiffness degradation. On-axis MD specimens.
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Fig. 11.
Stiffness based S-N curves for on-axis MD specimens.
Fig. 10. Comparison of experimental values and theoretical predictions for stiffness degradation. 30° off-axis MD specimens.
Considering the limited number of tests required for the analytical calculations, the correlation between the test values and those calculated from Eq. (3) was good. The theoretical straight lines, in most of the cases shown, reflect efficiently the stiffness variations during lifetime except for the stress level at 100 MPa where a choice of EL at NL/Nf ⫽ 0.6, instead of the actual 0.4, would yield better results. Similar comments also apply for the comparison of experimental data and theoretical predictions for the 30° off-axis specimens presented in Fig. 10. The general trend observed in stiffness variation diagrams, e.g. Figs. 5 and 6, is that the higher the applied cyclic stress, the less the stiffness degradation. However, in each one of these figures a stress level exists for which the above rule does not apply. This also holds for the theoretical predictions, by means of Eq. (3), presented in Figs. 9 and 10. Despite the deterministic and empirical nature of Eq. (3), its use and effectiveness relies on the simulation of experimental behaviour since EL and NL are taken from the observed material response at each stress level. It has to be mentioned, however, that these experimentally obtained values correspond to a limited life interval, e.g. NL/Nf ⫽ 0.4 up to 0.6. Using these analytically predicted stiffness changes, S-N curves corresponding to given material degradation and not ultimate failure, can be drawn for each one of the materials used. Material constants K and c are now calculated by fitting Eq. (1) to the above data, requiring only a limited number of tests. In Figs. 11 and 12 stiffness based S-N curves are shown for 0° and 30° MD specimens respectively. In Fig. 11 theoretical S-N curves are presented corresponding to 5% and 15% stiffness degradation along with experimental failure points. S-N curve for 5% stiffness
Fig. 12.
Stiffness based S-N curves for 30° off-axis MD specimens.
decrease is on the safe side of all the experimental failure data. On the other hand, the S-N curve corresponding to 85% remaining stiffness seems to fit well the experimental failure data. This is in agreement with the information shown in Fig. 5 where the main part of the stiffness data during life time lies in a region between 0.85 and 0.95 of the initial value E1. Similar conclusions are derived by observing Fig. 12 where the stiffness based S-N curves for 30° MD specimens are presented. However, in this case, the stiffness based S-N curve beyond which no failure points are observed, corresponds to 10% stiffness degradation. Furthermore, this data seems to fit better the curve of 60% remaining stiffness, reflecting the behaviour of 30° MD specimens shown in Fig. 6.
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4.2. Statistical analysis of fatigue data
X(Xi1,Xi2,…,Xini), i ⫽ 1,2,…,m
Fatigue data from each set of on- and off-axis coupons were subjected to statistical analysis to determine 95% reliability S-N curves. For that purpose the analysis presented by Whitney [7,8] was adopted, based mainly on two assumptions: (a) a classical power law representation of the S-N curve and (b) a two parameter Weibull distribution of time to failure. The form of the S-N equation is:
where:
Nb␣ ⫽ T⇒␣ ⫽ T (1/b)N( ⫺ 1/b) ⫽ N( ⫺ 1/b)
R(X) ⫽ exp ⫺
(4)
where N is the number of cycles, ␣ is the stress amplitude level and b, are material constants. For every stress level the number of cycles to failure is described by a two parameter Weibull distribution.
冋 冉冊册
N R(N) ⫽ exp ⫺ ¯ N
(5)
Calculation of constants and b of Eq. (4) is performed as follows: A two parameter Weibull distribution is fitted to the data of each stress range:
冋 冉冊册
Ni R(Ni) ⫽ exp ⫺ ¯ Ni
␣fi
(6)
where i ⫽ 1,2,...,m with m the total number of stress levels. The parameters ␣fi and N¯i of each Weibull distribution are determined by the following Maximum Likelihood Estimators:
冘 冘 ni
N␣ijˆ fiLnNij
j⫽1
⫺
ni
N␣ijˆ fi
1 ni
冘 ni
j⫽1
LnNij ⫺
1 ⫽0 ␣ˆ fi
(7)
j⫽1
冉 冘 冊
1 Nˆi ⫽ ni
ni
N␣ijˆ fi
1 ˆ fi ␣
(9)
Nij Xij ⫽ ˆ Ni
(10)
It is assumed that this set of data also follows a two parameter Weibull distribution:
冋 冉 冊册 X Xo
␣f
(11)
The parameters of the distribution of Eq. (11) are given by:
冘冘 冘冘 ni
m
X␣ijˆ fLnXij
␣f
259
i⫽1 j⫽1 m
ni
X
1 ⫺ M
ˆf ␣ ij
冘冘 m
ni
LnXij ⫺
i⫽1 j⫽1
1 ⫽0 ␣ˆ f
(12)
i⫽1 j⫽1
冉 冘冘 冊
1 Xˆo ⫽ M
m
ni
X
ˆf ␣ ij
1 ˆf ␣
(13)
i⫽1 j⫽1
Where M is the total number of specimens, i.e., the sum of the coupons from each stress level. The value of Xˆo has to be unity for a perfect fit [7]. If Xˆo takes any value other than unity, the characteristic number of cycles can be adjusted to produce Xˆo ⫽ 1. In particular: N¯oi ⫽ XˆoNˆi
(14)
The slope of the S-N curve, 1/b, and the y-intercept, can be determined by fitting log␣i versus logN¯oi to a straight line. With , b and ␣f now determined the S-N curve for any desired level of reliability can be calculated by:
␣ ⫽ 兵[ ⫺ LnR(N)](1/␣fb)其N( ⫺ 1/b)
(15)
(8)
j⫽1
with ni the number of specimens at each stress level. Eq. (7) has only one positive root [7] which is the estimated value for the shape parameter ␣fi, and thus the estimated characteristic number of cycles is given directly by Eq. (8). The values of the number of cycles at each stress level are subsequently normalized by the corresponding estimated characteristic number, Nˆi. Thus, the following normalized data set is formed:
A computer implementation of this method and application on a wide variety of fatigue data is given elsewhere [9]. In the same reference, two other methods for statistical analysis of fatigue data are also described and applied on the same data sets, but the method given herein is adopted due to better overall performance. 4.3. Stiffness based and reliability S-N curves S-N curves at specific reliability level and stiffness based S-N curves calculated as previously described are
260
Fig. 13.
T.P. Philippidis, A.P. Vassilopoulos / International Journal of Fatigue 21 (1999) 253–262
Comparison of S-N curves for on-axis MD specimens. Fig. 15.
then compared. The S-N curves which correspond to 95% reliability level for all on- and off-axis data are quite similar to the corresponding curves for 5–20% stiffness reduction, when plotted on the S-N plane. In fact, as it is shown in Figs. 13–15, S-N curves corresponding to stiffness reduction of 5–20%, depending on off-axis configuration, are more conservative, especially for high cycle numbers, than the statistically obtained SN curves. The S-N curves for the on-axis MD specimens are presented in Fig. 13. The stiffness based S-N curve corresponding to 5% stiffness reduction is the one which lies closer to the statistically determined S-N curve. In Fig. 14 the curve for 10% decrease of stiffness is plotted together with the curve for 95% reliability level for the coupons cut at 30° off-axis. Finally, the curves for 10% and 20% stiffness decrease are compared with the S-N curve for 95% reliability for 60° off-axis MD specimens in Fig. 15. Equations of all above mentioned S-N curves are given in Table 1. As it is observed from Figs. 13–15, stiffness reduction level of stiffness based S-N curves associated with 95%
Fig. 14. Comparison of S-N curves for 30° off-axis MD specimens.
Comparison of S-N curves for 60° off-axis MD specimens.
statistical ones, correlates satisfactorily with off-axis angle; 5% for the on-axis coupons, 10% for 30° off-axis and 20% for 60° off-axis. Therefore, for the material system investigated, a 10% stiffness reduction every 30° off-axis is the appropriate figure for stiffness based S-N curve determination. Concerning the experimental results presented herein, the total time needed for the determination of a statistical S-N curve amounts to 8E ⫹ 06 cycles (16 coupons), whereas only half of it is necessary for stiffness based S-N curves. Therefore, the latter type of S-N curve can be successfully used in initial design phases, replacing 95% allowables. 4.4. Energy dissipation measurements In the course of this experimental program efforts were also directed to the definition of meaningful ‘fatigue functions’ to be used eventually as overall damage metrics. To this end, hysteresis loop area, indicative of energy dissipation, was continuously monitored for every specimen and results are presented showing the relation of normalized energy loss (dN/do), determined as the area enclosed in a hysteresis stress–strain loop, with normalized number of cycles (N/Nf). In Fig. 16 energy-loss variation versus number of cycles is shown for the on-axis MD specimens. Generally, ‘internal friction’ seems to either increase or decrease in early stages, while it remains almost constant up to failure. For any specimen type it is valid that higher stress levels induce higher energy dissipation. Results from 30° off-axis MD coupons are presented in Fig. 17. Energy dissipation is shown to increase with number of cycles and stress level. However, the above conclusions are not true for all stress levels or specimens. This kind of measurement is probably not best suited to reflect the material dissipative response.
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261
Table 1 Calculated S-N curve equations
(deg.)
Experimental (Linear regression)
Statistical 95%
Stiffness based*
0 30 60
␣ ⫽ 160.1N−0.05016 ␣ ⫽ 101.4N−0.05722 ␣ ⫽ 102.8N−0.07530
␣ ⫽ 150.9N−0.05121 ␣ ⫽ 107.2N−0.07285 ␣ ⫽ 122.0N−0.09809
␣ ␣ ␣ ␣
⫽ ⫽ ⫽ ⫽
180.8N−0.06636 (0.95) 130.9N−0.09217 (0.90) 93.6N−0.07981 (0.90) 98.9N−0.07981 (0.80)
* number in parenthesis correspond to residual stiffness.
5. Conclusions
Fig. 16. Dissipated energy variation for the on-axis MD specimens.
Fig. 17. Dissipated energy variation for the 30° off-axis MD specimens.
During fatigue life the stiffness of a structural element is reduced. Efforts have been made to correlate this stiffness degradation with the damage accumulated in the material. Stiffness changes of coupons cut at several offaxis angles from a multidirectional laminate [0/(⫾45)2/0]T and subjected to reversed loading were investigated in this study. It is observed in general that stiffness degradation increases with off-axis angle and decreases for higher applied cyclic stress. The aforementioned behaviour results from the failure modes observed in each specimen and is related as much to the cyclic stress as to the off-axis angle. When the cyclic stress is high the failure mode is fiber dominated and the fracture is due to tensile stresses. Under low cyclic stress the damage mode is shifted from fiber to matrix controlled while failure is driven mainly by compressive loads. This happens when on-axis specimens are tested with the fibers along loading direction being the elements that carry the imposed loads. When off-axis specimens are tested the situation is different. In this case the failure mode, irrespective of the stress level, is always matrix dominated. For this reason also, the observed stiffness degradation for off-axis coupons is higher than that of on-axis ones. An empirical method is validated herein for the determination of S-N curves which do not correspond to fatigue strength data, but to a predetermined value of stiffness reduction, e.g. 5%–10% of the initial value, by using only a portion of the fatigue data, e.g. data up to 60% of fatigue life. Furthermore, S-N curves at 95% reliability level were determined for each set of fatigue data. It is shown in this paper that these two kinds of S-N curves are comparable although stiffness based curves are more conservative, especially for higher number of cycles. This was an interesting feature of the study presented herein along with the fact that for the determination of stiffness based S-N curves only a limited amount of fatigue data is needed. The use of these curves instead of 95% reliability allowables could be anticipated for initial design phases and prototyping. The basic drawback of residual stiffness based fatigue criteria i.e., large number of tests, can be avoided by using the procedure presented herein. The necessary
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experiments can be conducted within limited time of testing, e.g. 5 days, and stiffness based S-N curves can be readily produced. By continuously monitoring stress-strain hysteresis loop area during reversed loading of each coupon and correlating with number of cycles at fracture it was concluded that energy dissipation measurements do not constitute an efficient parameter to be used as a cumulative damage metric.
Acknowledgement Financial support of the work contained in this paper through project EPET II, #573 by the General Secretariat of Research and Technology is gratefully acknowledged.
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