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The fatigue behaviour of orthotropic laminates under tension-compression loading
Int J Fatigue 13 No 3 (1991) pp 209-215
The fatigue behaviour of orthotropic laminates under tension-compression loading A. R o t e m
The fatigue behaviour of orthotropic laminates, [0°,~45°,0°]2s and [90°,__.45°,90°]2s, has been evaluated under alternating tension-compression loading. Even though the first laminate is much stronger than the second both started to fail by delamination. Visual damage started to show only at the very end of the fatigue life but measurement of the stiffness showed that degradation starts at about 80% of the fatigue life. The first laminate failed in compression after delamination between the 0 ° and the 45° laminae, while the second failed in tension after delamination between the +45 ° and - 4 5 ° laminae. It is shown that the interlaminar fatigue strength of both laminate structures can be correlated to the applied fatigue load. With the aid of the fatigue failure envelope concept, the behaviour of the orthotropic laminate can be characterized. The experimental results for this laminate (in the two perpendicular directions) are shown here and a method to predict the fatigue behaviour under loading in any direction is formulated.
Key words: fatigue of composites; delamination; tension-compression fatigue; interlaminar fatigue Orthotropic laminates have different characteristics in different directions. The tension and compression strengths of one orientation may also be different (in absolute values). The fatigue behaviour for loading in any direction would also depend on the applied stress ratio, that is the minimum-tomaximum stress ratio (denoted by R). Other factors such as temperature, speed of load cycling and environment 1-3 may also affect the behaviour. Most of the published fatigue data are for particular laminate constructions tested at specific orientations and stress ratios. 4-6 This is a very reasonable attitude as the variety of laminate structures, load orientations and stress ratios is enormous. Most of the fatigue tests were of the tension-tension loading type (0 < R < 1), a smaller number were of the compression-compression loading type (R > 1) and only a very few were of the tension-compression loading type (R < 0). The reason for this is probably the difficulties in performing a sound tension-compression fatigue test on thin laminates. A generalized approach had been proposed 7 to deal with the fatigue behaviour of laminates on the basis of a single lamina. It has been shown that the in-plane fatigue behaviour of a lamina (or a unidirectional laminate), subjected to any constant stress ratio R, could be described by five fatigue functions. Two functions are for the degradation of stress amplitude with number of load cycles in the fibre direction, one for tension failure and one for compression failure. Two other functions are for the transverse direction (perpendicular to the fibre direction) again for tension and compression, and the fifth is for the in-plane shear. The laminate behaviour could be described based on the behaviour of each lamina which it contains and the resolving stress amplitudes. It has been shown recently s that if the fatigue functions of a laminate,
for tension and compression failures, are known and the quasi-static strengths in tension and compression are also known, the fatigue behaviour for any stress ratio could be determined. However, for a laminate with laminae in different orientations, in some cases of loading and in the presence of free edges or open inner surfaces, high interlaminar stresses could develop upon loading. 9 These interlaminar stresses oscillate under fatigue loading and can develop interlaminar cracks. When such a crack propagates under fatigue loading it forms a delamination, splitting the laminate at some location. Then part of the laminate may fail by buckling during the compressive half cycle, l°'n regardless of the failure mode that is developed in the laminae. The final failure may be the result of either of these processes, the in-plane failure mode or part laminate buckling. There are several lay-ups of laminae that form an orthotropic laminate. Among these are the [0°,-+45°,0°],, laminate and the [90°,+45°,90°]ns laminate, which are really the same laminate with a reference direction rotated by 90°. This laminate will be used as a representative of orthotropic laminates.
Fatigue failure analysis The fatigue loading on a laminate is defined by the oscillating force per unit length Ni, and the oscillating moment per unit length Mi and the load ratio R. The index i indicates the coordinate system. Using the theory of laminated plates, the strain at each lamina can be calculated by solving the equations
("M/:(;
0142-1123/91/030209-07 © 1991 Butterworth-Heinemann Ltd Int J Fatigue May 1991
209
where A, B and D are the linear or non-linear stiffness matrices of the laminate. The stresses in each lamina can then be calculated by using the stress-strain relations of the lamina and by rotating them to the principal axis (of the lamina). The stress ratio R remains the same as the load. Three fatigue failure envelopes, s for the fibre direction transverse to it and for shear, have to be constructed. Each envelope is based on the static strength in tension S,,, in compression S,~ and two fatigue functions (except that the shear envelope has only one static strength and one fatigue function). The tensile fatigue function is derived from an S - N curve of the tensile failure mode, say with R -- 0.1, and the compressive fatigue function is derived from an S - N curve with compressive failure mode, say with R -- 10. The shear fatigue function can be derived from an S - N curve of pure shear loading with any value of R, but preferably with R---1. Once the fatigue envelopes were determined, the S - N curve for any specific R-ratio can be determined. The applicable failure modes for all the principal stresses are determined from the failure envelope and the appropriate S - N curve is calculated. Then the specific stresses and the S - N parameters are inputs in the fatigue failure criterion t2 for predicting the fatigue life. This analysis is suitable for the case where failure is dominated by in-plane stresses. When the interlaminar stresses are the dominant features of the fatigue life the problem becomes more complex. The estimate of the magnitudes of the stresses needs a numerical solution by finite-element or finite-difference methods. 13,14 However, as a crack is formed and propagates the boundary conditions are changed and a new problem has to be solved. An alternative way would be to solve the problem in a half empirical method, that is in a similar method to the solution of the in-plane fatigue problem. The main difference would be that the interlaminar stresses would be evaluated only explicitly as a linear proportion to the external load for a specific laminate structure. For this purpose we shall use here a [0°,-+45°,0°]2s laminate, but the method can be applied to any laminate structure. This laminate develops a high interlaminar stress when it is loaded in the directions of the major axes. Since the laminate fails by interlaminar stresses as the specimen is loaded in the direction of a major axis, the calculated strength for this loading is proportional to the interlaminar strength. Therefore, the measurement of the quasi-static strengths, in tension and compression, for both laminate directions (0 ° and 90°), gives the interlaminar strengths explicitly from the calculated strengths of the loadings along the directions of the major axes. Testing such laminates for fatigue behaviour at some stress ratios of tension-tension loading and compression-compression loading and calculating the S - N parameters would give S - N curves that are proportional to the interlaminar fatigue behaviour. These parameters enable the fatigue envelopes for interlaminar stresses to be plotted and the fatigue behaviour of any stress ratio to be calculated. Let S~, denote the quasi-static strength in tension and Ss~ denote the quasi-static strength in compression, and let the S - N curve be represented by a semilogarithmic formula: S = So(1 + K l o g N )
SffSo, = s,
SsffSo, = Ss,
So~Sot = sc
SJSoo = S,c
(3)
The S - N curve will be formulated as follows:
s, = 1 + K, l o g N
(4)
for the tension failure mode and sc = 1 + Kc logN
(5)
for the compression failure mode. Figure 1 shows two fatigue failure envelopes for numbers of cycles N1 and N2. In order to normalize to the same scale, the compressive mode values were multiplied by the strength ratio SoJSot. The points st(NO and s,(N2) are on the same S - N curve with some value of the stress ratio Rt, where the material failed in a tension mode. The points so(N1) and sc(N2) are on different S - N curves, with some value of the stress ratio Ro where the material failed in a compression mode. Based on the relations between the fatigue failure envelopes, the behaviour at some other stress ratio, say Rt2 for the tensile domain or Re2 for the compressive domain, could be found. For the tensile domain, the S - N parameters would be
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Sst
~"
I+K~
-s~ + i ~ - ~ (6)
for the imaginary strength and
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///
/: :Y
m = 0 ¢3
A///
Smo
//~0(N2)j///
(2)
where S is the fatigue strength for N loading cycles and So is the 'fatigue imaginary strength' defined by extrapolating the S - N curve to the ordinate of one cycle, ts The quasi-static strengths and the fatigue loading were measured in the same direction and both caused failure by delamination. For every
210
fatigue failure load level, S, there are interlaminar stresses at a proportional level. Therefore there is an S - N curve for the interlaminar stresses that is proportional to the S - N curve of the loading. Hence, normalizing the fatigue strength with respect to So (of the loading) will also give normalization of the fatigue interlaminar strength with respect to the imaginary interlaminar strength. The interlaminar stresses include a normal component and shear components that will reciprocate as the external load. The shear stress components are not sensitive to the load direction but the normal stress is sensitive. Therefore two modes of loading are defined, tensile and compressive, with respect to the external loads, which are proportional to the interlaminar normal stress. Normalizing with respect to the imaginary strength, we obtain
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NORMALIZED MEAN STRESS (Smean/Sot)
Fig. 1 Fatigue failure envelopes normalized with respect to Sot
Int J Fatigue M a y 1991
1-R /-1
R, s~
(z)
4- 1 + K~/
for the slope. For the compressive domain, denoted by R~z So¢2 = ~
(
6R~2(R~- 1) -R~-R~-~2 + R ~ ( ~ -
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- Ro -
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1)
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(8)
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for the imaginary strength and So¢ ( Rc2(Rc- 1) .) K c 2 = s ~ 2 -R¢-R-~c2+R2R22-1) - 1
(9)
for the slope. In the case of tension-compression loading, when the applicable failure mode is not known, both S - N curves should be calculated and the lower (stress or fatigue life) S - N curve should be used. Of cause, the failure mode can be estimated if the fatigue failure envelope is plotted, and then only the appropriate S - N curve can be calculated. Similar relationships can easily be deduced for the case of log-log or power law fits of the S - N curve, that is the S - N curve is represented by (10)
S = SoN K
The procedure of calculation follows the same path. Experimental
procedure
Sixteen-ply plates were fabricated from prepregs of graphite fibres in an epoxy matrix (T300/934), with 62.4% of fibre volume. The cured plates (0.9 x 0.9 m 2) were non-destructively evaluated using an ultrasonic C-scan to ensure flawfree material. Rectangular specimens were cut from the laminated plates in two perpendicular directions to form [0°, -+45°,0°]2s and [90°,-+45°,90°]2s stacking sequences. Glasscloth tabs were bonded to the specimens with 934 epoxy glue. The final dimensions of the specimens were 130 x 12.7 mm 2, with an open gauge length between the tabs of 15 mm. Figure 2 shows a typical specimen configuration. All tests were performed using a servocontrolled, electrohydraulic mechanical test system using a special fixture
designed to minimize bending moments. A special double extensiometer, with double 8 mm gauge length on both opposing sides, was attached to the specimen, to ensure measurement of any bending should it appear. The axial strain was calculated as the average of the readings of the two sides of the extensiometer and the bending was calculated as the difference between these readings. The fatigue tests were conducted at a frequency of 10 Hz and a stress ratio of R = -1. A computerized data acquisition system was employed to record the output of the load cell, the stroke measuring device (LVDT) and the two extensiometers. The computerized system scanned the four channels in 40 p.s (10 I~s per channel) and was tuned to make 100 readings in a single period of the sine wave. The readings were then processed to give the amplitude, the mean and the phase angle of the sine wave of each of the channels. This information was then stored on a diskette for future analysis. Tests were performed at various stress levels to obtain information on the fatigue life from several thousand cycles to several million cycles. Some tests were photographed with a video camera to monitor the failure process. Quasi-static tension and compression tests were also conducted to determine the tensile and compressive strengths. Results and discussion Fatigue tests of reversed amplitude (R = - 1 ) were performed on the two types of laminates at different load levels. The results, as S - N curves, are shown in Figs 3 and 4. The fatigue parameters of the S - N curve (Equations (2) and (10)) have been calculated from these data using the least-squares method of fitting. The resuks are given in Table 1. Also given there are the results of quasi-static strength tests in tension and compression. Semilogarithmic and log-log lines calculated with the fitting parameters were added to Figs 3 and 4, and also the absolute values of the quasi-static strengths. It is clearly seen in both figures that the fatigue data fit the semilogarithmic lines better, because these lines extrapolate to the static strength. However, the compressive static strength fits the fatigue data of the [0°,-+45°,0°]as laminate while the tensile static strength fits the fatigue data of the [900,-+45°, 90012s laminate. In both cases this is the lower absolute value. This does not imply that compressive stresses are responsible for the laminate failure. Examining the failure process as recorded by the video camera, we find that both laminates start to fail by delamination separation. Figures 5 and 6 are photographs of specimens taken from the video recorder.
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Int J Fatigue M a y 1991
, 300 U3
1
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lo4 LIFE
[CYCIJES]
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Static strength (MPa) Tensile, S= Compressive, S,=
TENSILESTRENGTH
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Fatigue S-N parameters (R = - 1 ) semilogarithmic fit Imaginary strength, Sot
i100
Slope, Kt Imaginary strength, Soc Slope, K~
50 101
102
103
104
105
106
107
108
FATIGUELIFE [CYC~..S] (Log scale)
Fig. 4 S - N d a t a and curve fitting for a [90°,-+45°,90°]== laminate. R = - 1 , f = 10 cycles/s
Slope, K~ Imaginary strength, So¢ Slope, K~
90° 182 -289
---
177 -0.106 51
-692 -0.095 699
log-log fit Imaginary strength, Sot
100
curves
0°
Load direction
I - - LOG-U00 S~MILOGFIT FIT[
~200
for the S-N
---
270 -0.104 48
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Fatigue S-N parameters Semilogarithmic fit, R = 10
Imaginary strength, Soc Slope, /~
m B
-290 -0.085
Semilogarithmic fit, R = 0.1
Imaginary strength, Sot Slope, Kt
Fig. 5 Fatigue failure of the laminate when loaded in the 0 ° direction by a 425 MPa amplitude. Photographs are edge views taken after: (a) 1000 cycles, (b) 37100 cycles, (c) 38300 cycles, (d) 40650 cycles, (e) 42100 cycles and (f) 42130 cycles
Figure 5 shows six frames from the failure process of a [0 °, +-450,00]2" specimen. The first frame (a) shows the specimen before damage is noticeable. The second frame (b) shows the onset of the first crack and the third frame (c) shows the crack after it has propagated to the entire gauge length. The fourth frame (d) shows the beginning of another crack, on the other side, and the fifth frame (e) shows the delamination of the two cracks on both sides of the specimen between the 0° and 45 ° laminae. The second frame was taken after 39 000
212
901 -0.047 62
m
load cycles were already imposed on the specimen. The cracks propagate very rapidly along the specimen axis until the whole gauge length is delaminated. After more cycles were imposed, the specimen finally collapses by buckling of the thin remaining laminae on the compression half cycle. Clearly, the delamination process causes this specimen to fail. Figure 6 also shows four frames of the failure process, but of the other type of laminate, that is [90°,-+45°,90°]2 s. The second frame (b) also shows delamination, but this time it is between the +45 ° and - 4 5 ° laminae. It is interesting to see that the 90 ° laminae did not crack or delaminate at this stage. This is contrary t o the behaviour under tension-tension loading, 6 where the 90 ° laminae developed many cracks, and while loading continued it separated into small fractions and peeled off the specimen. Returning to this test, after the cracks developed along the whole length of the specimen and more load cycles were imposed, a crack was formed in the middle of the specimen at the location of the 90° layer (third frame (c)). Immediately afterwards, the two thinner segments that were formed by this crack collapsed by buckling on the compression half cycle, as seen in the fourth frame (d). Thus both laminates failed by delamination, but the delamination occurred between different layers. The delamination process starts many cycles before it is seen by a simple visual instrument like a video camera. However, the effect on the strain amplitude, or the apparent modulus, is much more detectable. Figure 7 shows the strain amplitude on both sides of the specimen, as a function of the number of load cycles, for a [0°,+45°,0°]2s laminate. It is seen that the amplitude readings start to oscillate violently at about 80% of the fatigue life, probably because of the delamination under the extensiometer pointers. Therefore, the displacement of the mounting points, one relative to the other, was also measured, and the results are also plotted on Fig. 7 as an apparent strain (it is used only as a guide when the extensiometer failed to function because of layer breakage under the pointers, and is denoted by 'stroke'). Some typical behaviours of a few specimens are shown in Fig. 8, plotted for the axial modulus on normalized coordinates. It is seen that the degradation is not unique; however, all of this happened in the same region.
Int J Fatigue M a y 1991
Fig. 6 Fatigue failure of the laminate when loaded in the 90 ° direction by a 106 MPa amplitude. Photographs are edge views taken after: (a) 1000 cycles, (b) 9630 cycles, (c) 9710 cycles and (d) 9780 cycles
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with R = - I ,
f=10Hz
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stress appears between the laminae on compression loading and vice versa. Therefore the development of the crack on the fatigue loading takes place mainly on the compression half cycle. On the other hand, the [90 °, -+45°,90°]2, laminate is stronger in compression than in tension (see Table 1). This means that the interlaminar stresses developed under tension loading are more severe than under the compression loading and therefore the delamination has mainly occurred on the tension half cycle. Figure 9 shows the strain behaviour of this laminate. Here again the severe failure process starts at about 80% of the fatigue life, but the amplitude increment starts even earlier. The results for the mean of the amplitudes that were measured on both sides of the specimen are also shown on this graph, and are seen to remain unchanged throughout the loading (except for the last 20% of the fatigue life, where the measurement has no meaning). Figure 10 shows the relative movement of the mounting points, which agrees with the extensiometer readings for the 80% fatigue life, and shows a rapid amplitude increase afterwards until final fracture. This region corresponds to a massive delamination that causes stiffness degradation of the specimen. This degradation in stiffness is shown for some specimens in Fig. 11. Again we see that most of the degradation occurs in the last 20% of the
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1
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0.2
,,
0.0 0.0
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oi, FATIGUE L ~ E
oi, [~N]
Fig. 8 Change of normalized modulus of some [0°,-+45°,0 °] laminate specimens loaded in the 0° direction
This laminate has a greater quasi-static tensile strength than compressive strength (in absolute values). As the failure process starts by delamination, it can be concluded that the combination of interlaminar stresses is more severe when the laminate is loaded in compression (in the laminate plane) than in tension. This is probably so because tensile interlaminar
Int J Fatigue M a y 1991
0
5000
10000 FATIGUELIFE [CYCLES]
15000
Fig. 9 Strain amplitude response for loading with constant stress amplitude in the 0 ° direction for [900,-+45o,90o]2, laminate
with R = - 1 ,
f = 10 Hz
213
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....
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~
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~
o.o
x
~ I
x l
'~
olo
oi~
o:,
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,1o
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Fig. 11 Normalized modulus change of some [90°,--45°,90 °] specimens loaded in the 90° direction
fatigue life, but some specimens start to degrade earlier. It is seen that this does not depend on the stress level as a specimen with a moderate stress level (and moderate fatigue life) shows the greatest degradation. The results of both series of tests show that the fatigue failure starts by delamination. However, the site of the delamination is different for the two series of tests. However, in both series of tests the loading is on the same laminate, only the direction of the applied load is different. Therefore, the orthotropic laminate behaves in an orthotropic manner, also in the interlaminar zone. Referring to Fig. 12, when a cyclic load No is acting on this laminate, it can be resolved into two components, Nx and N w where Nx = No cos 0 Ny = No sin 0
(11)
and the laminate axis (0) is along the x-axis. The Nx component creates the highest interlaminar stresses between the 0° lamina and the 45 ° lamina. The Ny component creates the highest interlaminar stresses between the +45 ° and the - 4 5 ° laminae. A crack will start at the more severe location between the two. When the stress ratio is changed, the new S-N curves could be found separately for the N~ and the IVy components. Using the different fatigue failure envelopes, the [0 °, +45°,0°]2s for the Nx and [90 °,-+45 °, 90°]2, for the Ny, the failure modes can be estimated, and the S-N parameters can be calculated from Equations (6) to (9). If the failure mode is in the process of changing to the tensile mode, the location of failure initiation may also be changing and failure will occur in the laminae and not in the interlaminar
214
zone, or the location may remain unchanged. In this case, failure starts with delamination as before, but the direction of the stresses is altered. Indeed, this particular laminate failed by delamination on both failure modes, tension and compression. Hence, the failure envelopes (Fig. 1) describe the interlaminar failures in both failure modes. For the 0° direction, the tensile failure mode corresponds to compressive normal interlaminar stress and vice versa, and for the 90 ° direction, the compressive failure mode corresponds to the compressive normal interlaminar stress and vice versa. Additional fatigue tests were carried out at stress ratios of 0.1 and 10 (tension-tension and compression-compression) to enable the construction of fatigue envelopes. Using the fatigue data collected, which are listed in Table 1, and the quasistatic strengths, fatigue envelopes could be drawn. By calculating the fatigue strength for a particular number of cycles for both failure modes, tension and compression, using the S-N curve equations, a fatigue failure envelope is formed. Figure 13 shows a family of fatigue failure envelopes for this laminate when it is loaded in the 0° direction, and Fig. 14 shows it for loading in the 90 ° direction. These two figures characterize the fatigue behaviour of this laminate for loading in any stress ratio R and any direction 0 in the plane of the '
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Int J Fatigue M a y 1991
References SCALE 1.0=177 [MPa]
1.
1
2.
0 Smcan
~ .
-2
.
.
.
I
3.
N=lo7
,
,
,
,
-1
J
0
,
,
,
,
I
1
.
.
.
4.
Jamison, R.D., Schulte, K., Reifsnider, K.L. and Stinchcomb, W.W. 'Characterization and analysis of damage mechanisms in tension-tension fatigue of graphite/ epoxy laminates' Effects of Defects in Composite Materials, ASTM STP 836 (American Society for Testing and Materials, 1984) pp 21-55
5.
Adams, D.F. 'Analysis of the compression fatigue properties of a graphite/epoxy composite' Advances in Composite Materials, ICCM3, Ed A.R. Bunsell (Pergamon, Oxford, 1980) pp 81-94 Lifshitz, J.M. 'Compressive fatigue and static properties of unidirectional graphite/epoxy composite' J Composite Technol Res 10 3 (1988) pp 100-106
.
2
Fig. 14 F a t i g u e f a i l u r e e n v e l o p e s of a [90o,-+45o,900]2. l a m i n a t e
for loading in the 0° direction
laminate. The fact that failure is the result of the interlaminar stresses caused by the free edge is indicated only implicitly, and there is no need to calculate the exact interlaminar stress field in order to describe the fatigue behaviour.
Conclusions Fatigue failure of laminates with free edges, which fail by delamination, can be analysed for its behaviour using the fatigue failure envelope concept, s It has been shown that an orthotropic laminate with free edges had failed by delamination as a first stage, under any load direction (in the laminate plane) and any stress ratio. Even though the exact solution and calculation of the interlaminar stresses for all these possible loadings is very complicated, it has been shown that it can be overcome by implicitly defining them with stresses in the laminate plane. In this way, for any cyclic stress load in the laminate plane there corresponds an interlaminar cyclic stress field. In the same way, the interlaminar strength corresponds to a unique strength for loading in some direction in the laminate plane and the same is true for the fatigue strength. For an orthotropic laminate loading along the two major axes describes loading in any direction and the interlaminar stresses associated with this. The interlaminar stress field would be changed as the direction of loading will change and the critical point for delamination may shift from one site (between some laminae) to another site (between some other laminae). The total behaviour of the orthotropic laminate may be described by two fatigue failure envelopes, each one for loading along one of the major axis directions. For example, the laminate [ 0 ° , - - 4 5 ° , 0 ° ] 2 s would have one family of fatigue failure envelopes for the 0 ° direction and one family of fatigue failure envelopes for the perpendicular direction (the [90 °, +-45°,90°]2s laminate)i Resolving the load along these two directions would give two loads with the same stress ratio R as the load. The fatigue lives for each of the load components may be found from S - N curves evaluated from the two families of fatigue failure envelopes. The fatigue life of the laminate would be, statistically of course, the shorter of the two as it is the dominant one, that is interlaminar failure will start on the site that corresponds to this load component.
Int J Fatigue May 1991
Rotem, A. and Nelson, H.G. 'Fatigue behavior of graphite/epoxy laminates at elevated temperatures' Fatigue of Fibrous Composite Materials, ASTM STP 723 (American Society for Testing and Materials, 1981) pp 152-173 Chamis, C.C., Lark, R.F. and Sinclair, J.H. 'Integrated theory for predicting the hydrothermomechanical response of advanced composite structural component' Advanced Composite Materials--Environmental Effects, ASTM STP 658 Ed J.R. Vinson (American Society for Testing and Materials, 1978) pp 160-192 Reifsnider, K.L., Stinchcomb, W.W. and O'Brien, T.K. 'Frequency effects on a stiffness-based fatigue failure criterion in flawed composite specimens' Fatigue of Filamentary Composite Materials, ASTM STP 636 (American Society for Testing and Materials, 1987) pp 171-184
6.
7.
Rotem, A. 'Fatigue failure mechanism of composite laminates' IUTAM Symp on Mechanics of Composite Materials (Pergamon, Oxford, 1983) pp 421-435
8.
Rotem, A. 'Tension and compression failure modes of laminated composites loaded by fatigue with different mean stress' J Composite Technol Res 12 4 (1990) pp 201-208 Stalnaker, D.O. and Stinchcomb, W.W. 'Load historyedge damage studies in two quasi-isotropic graphite epoxy laminates' Proc Fifth Conf on Composite Materials: Testing and Design, ASTM STP 674 (American Society for Testing and Materials, 1979) Chai, H.C. and Babcock, C.D. 'Two dimensional modeling of compressive failure in delaminated laminates' J Composite Mater 19 (January 1985) pp 67-97
9.
10.
11.
12.
13.
14.
15.
Tracy, J.J. and Pardoen, G.C. 'The effect of delamination on the fundamental and higher buckling loads of laminated beams' J Composite Technol Res 11 3 (1989) pp 87-93 Rotem, A. and Hashin, Z. 'Fatigue failure criterion for fiber reinforced materials' J Composite Mater 8 2 (1974) pp 448-464 Crossman, F.W. and Wang, A.S.D. 'Some new results on edge effect in symmetric composite laminate' J Composite Mater 11 (1977) p 92 Altus, E., Rotem, A. and Shmueli, M. 'Free edge effect in angle ply laminates--A new three dimensional finite difference solution' J Composite Mater 14 (January 1980) pp 21-30 Rotem, A. 'Fatigue and residual strength of composite laminates' J Eng Fract Mech 25 5/6 (1986) pp 819-827
Author The author is with the Faculty of Mechanical Engineering, Technion--Israel Institute of Technology, Haifa 32000, Israel. He was previously an N R C senior associate at NASA Ames Research Centre, Test Engineering and Analysis Branch, Moffett Field, CA 94035, California, USA, where part of this work was carried out.
215
The fatigue behaviour of orthotropic laminates under tension-compression loading A. R o t e m
The fatigue behaviour of orthotropic laminates, [0°,~45°,0°]2s and [90°,__.45°,90°]2s, has been evaluated under alternating tension-compression loading. Even though the first laminate is much stronger than the second both started to fail by delamination. Visual damage started to show only at the very end of the fatigue life but measurement of the stiffness showed that degradation starts at about 80% of the fatigue life. The first laminate failed in compression after delamination between the 0 ° and the 45° laminae, while the second failed in tension after delamination between the +45 ° and - 4 5 ° laminae. It is shown that the interlaminar fatigue strength of both laminate structures can be correlated to the applied fatigue load. With the aid of the fatigue failure envelope concept, the behaviour of the orthotropic laminate can be characterized. The experimental results for this laminate (in the two perpendicular directions) are shown here and a method to predict the fatigue behaviour under loading in any direction is formulated.
Key words: fatigue of composites; delamination; tension-compression fatigue; interlaminar fatigue Orthotropic laminates have different characteristics in different directions. The tension and compression strengths of one orientation may also be different (in absolute values). The fatigue behaviour for loading in any direction would also depend on the applied stress ratio, that is the minimum-tomaximum stress ratio (denoted by R). Other factors such as temperature, speed of load cycling and environment 1-3 may also affect the behaviour. Most of the published fatigue data are for particular laminate constructions tested at specific orientations and stress ratios. 4-6 This is a very reasonable attitude as the variety of laminate structures, load orientations and stress ratios is enormous. Most of the fatigue tests were of the tension-tension loading type (0 < R < 1), a smaller number were of the compression-compression loading type (R > 1) and only a very few were of the tension-compression loading type (R < 0). The reason for this is probably the difficulties in performing a sound tension-compression fatigue test on thin laminates. A generalized approach had been proposed 7 to deal with the fatigue behaviour of laminates on the basis of a single lamina. It has been shown that the in-plane fatigue behaviour of a lamina (or a unidirectional laminate), subjected to any constant stress ratio R, could be described by five fatigue functions. Two functions are for the degradation of stress amplitude with number of load cycles in the fibre direction, one for tension failure and one for compression failure. Two other functions are for the transverse direction (perpendicular to the fibre direction) again for tension and compression, and the fifth is for the in-plane shear. The laminate behaviour could be described based on the behaviour of each lamina which it contains and the resolving stress amplitudes. It has been shown recently s that if the fatigue functions of a laminate,
for tension and compression failures, are known and the quasi-static strengths in tension and compression are also known, the fatigue behaviour for any stress ratio could be determined. However, for a laminate with laminae in different orientations, in some cases of loading and in the presence of free edges or open inner surfaces, high interlaminar stresses could develop upon loading. 9 These interlaminar stresses oscillate under fatigue loading and can develop interlaminar cracks. When such a crack propagates under fatigue loading it forms a delamination, splitting the laminate at some location. Then part of the laminate may fail by buckling during the compressive half cycle, l°'n regardless of the failure mode that is developed in the laminae. The final failure may be the result of either of these processes, the in-plane failure mode or part laminate buckling. There are several lay-ups of laminae that form an orthotropic laminate. Among these are the [0°,-+45°,0°],, laminate and the [90°,+45°,90°]ns laminate, which are really the same laminate with a reference direction rotated by 90°. This laminate will be used as a representative of orthotropic laminates.
Fatigue failure analysis The fatigue loading on a laminate is defined by the oscillating force per unit length Ni, and the oscillating moment per unit length Mi and the load ratio R. The index i indicates the coordinate system. Using the theory of laminated plates, the strain at each lamina can be calculated by solving the equations
("M/:(;
0142-1123/91/030209-07 © 1991 Butterworth-Heinemann Ltd Int J Fatigue May 1991
209
where A, B and D are the linear or non-linear stiffness matrices of the laminate. The stresses in each lamina can then be calculated by using the stress-strain relations of the lamina and by rotating them to the principal axis (of the lamina). The stress ratio R remains the same as the load. Three fatigue failure envelopes, s for the fibre direction transverse to it and for shear, have to be constructed. Each envelope is based on the static strength in tension S,,, in compression S,~ and two fatigue functions (except that the shear envelope has only one static strength and one fatigue function). The tensile fatigue function is derived from an S - N curve of the tensile failure mode, say with R -- 0.1, and the compressive fatigue function is derived from an S - N curve with compressive failure mode, say with R -- 10. The shear fatigue function can be derived from an S - N curve of pure shear loading with any value of R, but preferably with R---1. Once the fatigue envelopes were determined, the S - N curve for any specific R-ratio can be determined. The applicable failure modes for all the principal stresses are determined from the failure envelope and the appropriate S - N curve is calculated. Then the specific stresses and the S - N parameters are inputs in the fatigue failure criterion t2 for predicting the fatigue life. This analysis is suitable for the case where failure is dominated by in-plane stresses. When the interlaminar stresses are the dominant features of the fatigue life the problem becomes more complex. The estimate of the magnitudes of the stresses needs a numerical solution by finite-element or finite-difference methods. 13,14 However, as a crack is formed and propagates the boundary conditions are changed and a new problem has to be solved. An alternative way would be to solve the problem in a half empirical method, that is in a similar method to the solution of the in-plane fatigue problem. The main difference would be that the interlaminar stresses would be evaluated only explicitly as a linear proportion to the external load for a specific laminate structure. For this purpose we shall use here a [0°,-+45°,0°]2s laminate, but the method can be applied to any laminate structure. This laminate develops a high interlaminar stress when it is loaded in the directions of the major axes. Since the laminate fails by interlaminar stresses as the specimen is loaded in the direction of a major axis, the calculated strength for this loading is proportional to the interlaminar strength. Therefore, the measurement of the quasi-static strengths, in tension and compression, for both laminate directions (0 ° and 90°), gives the interlaminar strengths explicitly from the calculated strengths of the loadings along the directions of the major axes. Testing such laminates for fatigue behaviour at some stress ratios of tension-tension loading and compression-compression loading and calculating the S - N parameters would give S - N curves that are proportional to the interlaminar fatigue behaviour. These parameters enable the fatigue envelopes for interlaminar stresses to be plotted and the fatigue behaviour of any stress ratio to be calculated. Let S~, denote the quasi-static strength in tension and Ss~ denote the quasi-static strength in compression, and let the S - N curve be represented by a semilogarithmic formula: S = So(1 + K l o g N )
SffSo, = s,
SsffSo, = Ss,
So~Sot = sc
SJSoo = S,c
(3)
The S - N curve will be formulated as follows:
s, = 1 + K, l o g N
(4)
for the tension failure mode and sc = 1 + Kc logN
(5)
for the compression failure mode. Figure 1 shows two fatigue failure envelopes for numbers of cycles N1 and N2. In order to normalize to the same scale, the compressive mode values were multiplied by the strength ratio SoJSot. The points st(NO and s,(N2) are on the same S - N curve with some value of the stress ratio Rt, where the material failed in a tension mode. The points so(N1) and sc(N2) are on different S - N curves, with some value of the stress ratio Ro where the material failed in a compression mode. Based on the relations between the fatigue failure envelopes, the behaviour at some other stress ratio, say Rt2 for the tensile domain or Re2 for the compressive domain, could be found. For the tensile domain, the S - N parameters would be
\
Sst
~"
I+K~
-s~ + i ~ - ~ (6)
for the imaginary strength and
,/
///
/: :Y
m = 0 ¢3
A///
Smo
//~0(N2)j///
(2)
where S is the fatigue strength for N loading cycles and So is the 'fatigue imaginary strength' defined by extrapolating the S - N curve to the ordinate of one cycle, ts The quasi-static strengths and the fatigue loading were measured in the same direction and both caused failure by delamination. For every
210
fatigue failure load level, S, there are interlaminar stresses at a proportional level. Therefore there is an S - N curve for the interlaminar stresses that is proportional to the S - N curve of the loading. Hence, normalizing the fatigue strength with respect to So (of the loading) will also give normalization of the fatigue interlaminar strength with respect to the imaginary interlaminar strength. The interlaminar stresses include a normal component and shear components that will reciprocate as the external load. The shear stress components are not sensitive to the load direction but the normal stress is sensitive. Therefore two modes of loading are defined, tensile and compressive, with respect to the external loads, which are proportional to the interlaminar normal stress. Normalizing with respect to the imaginary strength, we obtain
/
/
/
/
/
i
-1
,
,
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,
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.
.
.
.
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,
1
NORMALIZED MEAN STRESS (Smean/Sot)
Fig. 1 Fatigue failure envelopes normalized with respect to Sot
Int J Fatigue M a y 1991
1-R /-1
R, s~
(z)
4- 1 + K~/
for the slope. For the compressive domain, denoted by R~z So¢2 = ~
(
6R~2(R~- 1) -R~-R~-~2 + R ~ ( ~ -
R~2(Rc - 1)
- Ro -
7£
1)
]
? -R2
(8)
- I)I
" i T6 £ -/
for the imaginary strength and So¢ ( Rc2(Rc- 1) .) K c 2 = s ~ 2 -R¢-R-~c2+R2R22-1) - 1
(9)
for the slope. In the case of tension-compression loading, when the applicable failure mode is not known, both S - N curves should be calculated and the lower (stress or fatigue life) S - N curve should be used. Of cause, the failure mode can be estimated if the fatigue failure envelope is plotted, and then only the appropriate S - N curve can be calculated. Similar relationships can easily be deduced for the case of log-log or power law fits of the S - N curve, that is the S - N curve is represented by (10)
S = SoN K
The procedure of calculation follows the same path. Experimental
procedure
Sixteen-ply plates were fabricated from prepregs of graphite fibres in an epoxy matrix (T300/934), with 62.4% of fibre volume. The cured plates (0.9 x 0.9 m 2) were non-destructively evaluated using an ultrasonic C-scan to ensure flawfree material. Rectangular specimens were cut from the laminated plates in two perpendicular directions to form [0°, -+45°,0°]2s and [90°,-+45°,90°]2s stacking sequences. Glasscloth tabs were bonded to the specimens with 934 epoxy glue. The final dimensions of the specimens were 130 x 12.7 mm 2, with an open gauge length between the tabs of 15 mm. Figure 2 shows a typical specimen configuration. All tests were performed using a servocontrolled, electrohydraulic mechanical test system using a special fixture
designed to minimize bending moments. A special double extensiometer, with double 8 mm gauge length on both opposing sides, was attached to the specimen, to ensure measurement of any bending should it appear. The axial strain was calculated as the average of the readings of the two sides of the extensiometer and the bending was calculated as the difference between these readings. The fatigue tests were conducted at a frequency of 10 Hz and a stress ratio of R = -1. A computerized data acquisition system was employed to record the output of the load cell, the stroke measuring device (LVDT) and the two extensiometers. The computerized system scanned the four channels in 40 p.s (10 I~s per channel) and was tuned to make 100 readings in a single period of the sine wave. The readings were then processed to give the amplitude, the mean and the phase angle of the sine wave of each of the channels. This information was then stored on a diskette for future analysis. Tests were performed at various stress levels to obtain information on the fatigue life from several thousand cycles to several million cycles. Some tests were photographed with a video camera to monitor the failure process. Quasi-static tension and compression tests were also conducted to determine the tensile and compressive strengths. Results and discussion Fatigue tests of reversed amplitude (R = - 1 ) were performed on the two types of laminates at different load levels. The results, as S - N curves, are shown in Figs 3 and 4. The fatigue parameters of the S - N curve (Equations (2) and (10)) have been calculated from these data using the least-squares method of fitting. The resuks are given in Table 1. Also given there are the results of quasi-static strength tests in tension and compression. Semilogarithmic and log-log lines calculated with the fitting parameters were added to Figs 3 and 4, and also the absolute values of the quasi-static strengths. It is clearly seen in both figures that the fatigue data fit the semilogarithmic lines better, because these lines extrapolate to the static strength. However, the compressive static strength fits the fatigue data of the [0°,-+45°,0°]as laminate while the tensile static strength fits the fatigue data of the [900,-+45°, 90012s laminate. In both cases this is the lower absolute value. This does not imply that compressive stresses are responsible for the laminate failure. Examining the failure process as recorded by the video camera, we find that both laminates start to fail by delamination separation. Figures 5 and 6 are photographs of specimens taken from the video recorder.
.... ,
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Fig. 2 Typical specimen dimensions
Int J Fatigue M a y 1991
, 300 U3
1
lo°
loI
lo2
lo3 FATIGUE
lo4 LIFE
[CYCIJES]
l~s (log
IP
ld
1os
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Fig. 3 S - N data and curve fitting for a [0°,--45%0°]=. laminate. R = - I , f = 1 0 cycles/s 211
300 • 4 - 250 o
T a b l e 1. P a r a m e t e r s
COMPRBSSIVES'rR]Dgn~
~
Static strength (MPa) Tensile, S= Compressive, S,=
TENSILESTRENGTH
150
894 -654
Fatigue S-N parameters (R = - 1 ) semilogarithmic fit Imaginary strength, Sot
i100
Slope, Kt Imaginary strength, Soc Slope, K~
50 101
102
103
104
105
106
107
108
FATIGUELIFE [CYC~..S] (Log scale)
Fig. 4 S - N d a t a and curve fitting for a [90°,-+45°,90°]== laminate. R = - 1 , f = 10 cycles/s
Slope, K~ Imaginary strength, So¢ Slope, K~
90° 182 -289
---
177 -0.106 51
-692 -0.095 699
log-log fit Imaginary strength, Sot
100
curves
0°
Load direction
I - - LOG-U00 S~MILOGFIT FIT[
~200
for the S-N
---
270 -0.104 48
-1015 -0.090 869
Fatigue S-N parameters Semilogarithmic fit, R = 10
Imaginary strength, Soc Slope, /~
m B
-290 -0.085
Semilogarithmic fit, R = 0.1
Imaginary strength, Sot Slope, Kt
Fig. 5 Fatigue failure of the laminate when loaded in the 0 ° direction by a 425 MPa amplitude. Photographs are edge views taken after: (a) 1000 cycles, (b) 37100 cycles, (c) 38300 cycles, (d) 40650 cycles, (e) 42100 cycles and (f) 42130 cycles
Figure 5 shows six frames from the failure process of a [0 °, +-450,00]2" specimen. The first frame (a) shows the specimen before damage is noticeable. The second frame (b) shows the onset of the first crack and the third frame (c) shows the crack after it has propagated to the entire gauge length. The fourth frame (d) shows the beginning of another crack, on the other side, and the fifth frame (e) shows the delamination of the two cracks on both sides of the specimen between the 0° and 45 ° laminae. The second frame was taken after 39 000
212
901 -0.047 62
m
load cycles were already imposed on the specimen. The cracks propagate very rapidly along the specimen axis until the whole gauge length is delaminated. After more cycles were imposed, the specimen finally collapses by buckling of the thin remaining laminae on the compression half cycle. Clearly, the delamination process causes this specimen to fail. Figure 6 also shows four frames of the failure process, but of the other type of laminate, that is [90°,-+45°,90°]2 s. The second frame (b) also shows delamination, but this time it is between the +45 ° and - 4 5 ° laminae. It is interesting to see that the 90 ° laminae did not crack or delaminate at this stage. This is contrary t o the behaviour under tension-tension loading, 6 where the 90 ° laminae developed many cracks, and while loading continued it separated into small fractions and peeled off the specimen. Returning to this test, after the cracks developed along the whole length of the specimen and more load cycles were imposed, a crack was formed in the middle of the specimen at the location of the 90° layer (third frame (c)). Immediately afterwards, the two thinner segments that were formed by this crack collapsed by buckling on the compression half cycle, as seen in the fourth frame (d). Thus both laminates failed by delamination, but the delamination occurred between different layers. The delamination process starts many cycles before it is seen by a simple visual instrument like a video camera. However, the effect on the strain amplitude, or the apparent modulus, is much more detectable. Figure 7 shows the strain amplitude on both sides of the specimen, as a function of the number of load cycles, for a [0°,+45°,0°]2s laminate. It is seen that the amplitude readings start to oscillate violently at about 80% of the fatigue life, probably because of the delamination under the extensiometer pointers. Therefore, the displacement of the mounting points, one relative to the other, was also measured, and the results are also plotted on Fig. 7 as an apparent strain (it is used only as a guide when the extensiometer failed to function because of layer breakage under the pointers, and is denoted by 'stroke'). Some typical behaviours of a few specimens are shown in Fig. 8, plotted for the axial modulus on normalized coordinates. It is seen that the degradation is not unique; however, all of this happened in the same region.
Int J Fatigue M a y 1991
Fig. 6 Fatigue failure of the laminate when loaded in the 90 ° direction by a 106 MPa amplitude. Photographs are edge views taken after: (a) 1000 cycles, (b) 9630 cycles, (c) 9710 cycles and (d) 9780 cycles
,
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,
.
/]
.
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~
4OOO
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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t
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i
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,
40000
FATIGUEl..g~
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,
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i 100000
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with R = - I ,
f=10Hz
1.0
. i
0.I
I°l
! ~ ~
stress appears between the laminae on compression loading and vice versa. Therefore the development of the crack on the fatigue loading takes place mainly on the compression half cycle. On the other hand, the [90 °, -+45°,90°]2, laminate is stronger in compression than in tension (see Table 1). This means that the interlaminar stresses developed under tension loading are more severe than under the compression loading and therefore the delamination has mainly occurred on the tension half cycle. Figure 9 shows the strain behaviour of this laminate. Here again the severe failure process starts at about 80% of the fatigue life, but the amplitude increment starts even earlier. The results for the mean of the amplitudes that were measured on both sides of the specimen are also shown on this graph, and are seen to remain unchanged throughout the loading (except for the last 20% of the fatigue life, where the measurement has no meaning). Figure 10 shows the relative movement of the mounting points, which agrees with the extensiometer readings for the 80% fatigue life, and shows a rapid amplitude increase afterwards until final fracture. This region corresponds to a massive delamination that causes stiffness degradation of the specimen. This degradation in stiffness is shown for some specimens in Fig. 11. Again we see that most of the degradation occurs in the last 20% of the
0.6
,
- --........
~
.
.
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.
,
,
N=38250 N*42130
|
1
N = 1 4 8 6 4 0
/ii i
0.2
,,
0.0 0.0
o12
oi, FATIGUE L ~ E
oi, [~N]
Fig. 8 Change of normalized modulus of some [0°,-+45°,0 °] laminate specimens loaded in the 0° direction
This laminate has a greater quasi-static tensile strength than compressive strength (in absolute values). As the failure process starts by delamination, it can be concluded that the combination of interlaminar stresses is more severe when the laminate is loaded in compression (in the laminate plane) than in tension. This is probably so because tensile interlaminar
Int J Fatigue M a y 1991
0
5000
10000 FATIGUELIFE [CYCLES]
15000
Fig. 9 Strain amplitude response for loading with constant stress amplitude in the 0 ° direction for [900,-+45o,90o]2, laminate
with R = - 1 ,
f = 10 Hz
213
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.
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.
.
i
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•
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~3000
No
i+oo
e
1000
D.-
X
0 0
50O0
1GO]O
15OO0
F A ~ O t m L m m ICYOUSSl
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0.4
....
lq,,,*lS~O
~
....... N~Se.so
~
o.o
x
~ I
x l
'~
olo
oi~
o:,
o:+
o:~
,1o
FATIGUELIFE {n/l~]
Fig. 11 Normalized modulus change of some [90°,--45°,90 °] specimens loaded in the 90° direction
fatigue life, but some specimens start to degrade earlier. It is seen that this does not depend on the stress level as a specimen with a moderate stress level (and moderate fatigue life) shows the greatest degradation. The results of both series of tests show that the fatigue failure starts by delamination. However, the site of the delamination is different for the two series of tests. However, in both series of tests the loading is on the same laminate, only the direction of the applied load is different. Therefore, the orthotropic laminate behaves in an orthotropic manner, also in the interlaminar zone. Referring to Fig. 12, when a cyclic load No is acting on this laminate, it can be resolved into two components, Nx and N w where Nx = No cos 0 Ny = No sin 0
(11)
and the laminate axis (0) is along the x-axis. The Nx component creates the highest interlaminar stresses between the 0° lamina and the 45 ° lamina. The Ny component creates the highest interlaminar stresses between the +45 ° and the - 4 5 ° laminae. A crack will start at the more severe location between the two. When the stress ratio is changed, the new S-N curves could be found separately for the N~ and the IVy components. Using the different fatigue failure envelopes, the [0 °, +45°,0°]2s for the Nx and [90 °,-+45 °, 90°]2, for the Ny, the failure modes can be estimated, and the S-N parameters can be calculated from Equations (6) to (9). If the failure mode is in the process of changing to the tensile mode, the location of failure initiation may also be changing and failure will occur in the laminae and not in the interlaminar
214
zone, or the location may remain unchanged. In this case, failure starts with delamination as before, but the direction of the stresses is altered. Indeed, this particular laminate failed by delamination on both failure modes, tension and compression. Hence, the failure envelopes (Fig. 1) describe the interlaminar failures in both failure modes. For the 0° direction, the tensile failure mode corresponds to compressive normal interlaminar stress and vice versa, and for the 90 ° direction, the compressive failure mode corresponds to the compressive normal interlaminar stress and vice versa. Additional fatigue tests were carried out at stress ratios of 0.1 and 10 (tension-tension and compression-compression) to enable the construction of fatigue envelopes. Using the fatigue data collected, which are listed in Table 1, and the quasistatic strengths, fatigue envelopes could be drawn. By calculating the fatigue strength for a particular number of cycles for both failure modes, tension and compression, using the S-N curve equations, a fatigue failure envelope is formed. Figure 13 shows a family of fatigue failure envelopes for this laminate when it is loaded in the 0° direction, and Fig. 14 shows it for loading in the 90 ° direction. These two figures characterize the fatigue behaviour of this laminate for loading in any stress ratio R and any direction 0 in the plane of the '
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.
.
.
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.
.
.
.
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.
.
.
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,
Smax-min SCALE
1.0= 90, [MPa]
1.0
0.5
0.0
Smean
-0.5
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.
-,io ....
-oi,
.
.
.
.
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.
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.
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.
.
.
.
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Fig. 13 Fatigue failure envelopes of a [0°,-+45°,0°]2. laminate for loading in the 0 ° direction
Int J Fatigue M a y 1991
References SCALE 1.0=177 [MPa]
1.
1
2.
0 Smcan
~ .
-2
.
.
.
I
3.
N=lo7
,
,
,
,
-1
J
0
,
,
,
,
I
1
.
.
.
4.
Jamison, R.D., Schulte, K., Reifsnider, K.L. and Stinchcomb, W.W. 'Characterization and analysis of damage mechanisms in tension-tension fatigue of graphite/ epoxy laminates' Effects of Defects in Composite Materials, ASTM STP 836 (American Society for Testing and Materials, 1984) pp 21-55
5.
Adams, D.F. 'Analysis of the compression fatigue properties of a graphite/epoxy composite' Advances in Composite Materials, ICCM3, Ed A.R. Bunsell (Pergamon, Oxford, 1980) pp 81-94 Lifshitz, J.M. 'Compressive fatigue and static properties of unidirectional graphite/epoxy composite' J Composite Technol Res 10 3 (1988) pp 100-106
.
2
Fig. 14 F a t i g u e f a i l u r e e n v e l o p e s of a [90o,-+45o,900]2. l a m i n a t e
for loading in the 0° direction
laminate. The fact that failure is the result of the interlaminar stresses caused by the free edge is indicated only implicitly, and there is no need to calculate the exact interlaminar stress field in order to describe the fatigue behaviour.
Conclusions Fatigue failure of laminates with free edges, which fail by delamination, can be analysed for its behaviour using the fatigue failure envelope concept, s It has been shown that an orthotropic laminate with free edges had failed by delamination as a first stage, under any load direction (in the laminate plane) and any stress ratio. Even though the exact solution and calculation of the interlaminar stresses for all these possible loadings is very complicated, it has been shown that it can be overcome by implicitly defining them with stresses in the laminate plane. In this way, for any cyclic stress load in the laminate plane there corresponds an interlaminar cyclic stress field. In the same way, the interlaminar strength corresponds to a unique strength for loading in some direction in the laminate plane and the same is true for the fatigue strength. For an orthotropic laminate loading along the two major axes describes loading in any direction and the interlaminar stresses associated with this. The interlaminar stress field would be changed as the direction of loading will change and the critical point for delamination may shift from one site (between some laminae) to another site (between some other laminae). The total behaviour of the orthotropic laminate may be described by two fatigue failure envelopes, each one for loading along one of the major axis directions. For example, the laminate [ 0 ° , - - 4 5 ° , 0 ° ] 2 s would have one family of fatigue failure envelopes for the 0 ° direction and one family of fatigue failure envelopes for the perpendicular direction (the [90 °, +-45°,90°]2s laminate)i Resolving the load along these two directions would give two loads with the same stress ratio R as the load. The fatigue lives for each of the load components may be found from S - N curves evaluated from the two families of fatigue failure envelopes. The fatigue life of the laminate would be, statistically of course, the shorter of the two as it is the dominant one, that is interlaminar failure will start on the site that corresponds to this load component.
Int J Fatigue May 1991
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Author The author is with the Faculty of Mechanical Engineering, Technion--Israel Institute of Technology, Haifa 32000, Israel. He was previously an N R C senior associate at NASA Ames Research Centre, Test Engineering and Analysis Branch, Moffett Field, CA 94035, California, USA, where part of this work was carried out.
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