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Static and cyclic biaxial bending of CFRP panels
Composites Science and Technology 52 (1994) 267-273
0266-3538(94)00026-3
ELSEVIER
~) 1994 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0266-3538194/$07.00
STATIC A N D CYCLIC B I A X I A L B E N D I N G OF CFRP PANELS A. S. Chen* & F. L. Matthews$ Centre for Composite Materials, Imperial College of Science, Technology and Medicine, Prince Consort Road, London, UK, SW7 2BY (Received 4 February 1994; accepted 20 May 1994) the advantage of being similar to practical situations of tubular components in engineering applications. The latter provide experimental results which are relatively convenient to interpret. A detailed review of the subject of biaxial loading tests on composites can be found elsewhere. 1 Biaxial bending of plates is also a biaxial loading situation which often exists in engineering structures, for instance wing structures, automobile bodies and floor panels. However, there is very limited information concerning biaxial bending of composite panels, especially bending introduced by a lateral cyclic load. The present work is an experimental study of the behaviour of rectangular composite plates subjected to biaxial bending under monotonic and cyclic lateral loads. It is hoped that a design procedure will be developed for predicting microdamage and failure induced by biaxial cyclic bending on the basis of the above investigation. The aim of this paper is to describe and discuss the experimental results obtained on some rectangular laminated plates subjected to lateral central loading with all edges simply supported.
Abstract
Biaxial bending of panel components is a loading situation which exists in engineering structures. The behaviour of composite plates subjected to biaxial flexing has yet to be studied in detail. In the present investigation, static and cyclic biaxial bending were applied to rectangular carbon-fibre~epoxy-resin simplysupported laminated plates with three geometric aspect ratios. Series damage mechanism studies, mainly by N D T methods, and mechanical property evaluations were carried out. It was found that various damage modes could occur below about 20% of the ultimate load~deflection levels. This damage, mainly matrix cracks and delamination, developed during fatigue. The amount of damage indicated by the area of the damage zone was mainly dependent on the load level, but not on the aspect ratios. Despite their low monotonic load-bearing ability, plates with a higher aspect ratio could have a higher resistance to biaxial bending fatigue. Keywords: biaxial bending, flexural fatigue, NDT, CFRP
EXPERIMENTAL ARRANGEMENT INTRODUCTION
Design of the biaxial flexural test jig used in this work was mainly influenced by the three point-bending test rig recommended in the CRAG specifications.2 A schematic central cross-section view of the jig is illustrated in Fig. 1. The central load indenter is a circular section bar with a 12.5 mm radius hemispherical loading nose. The supporting frame is a combination of two pairs of changeable supporting columns sited on a pre-positioned base plate, which enables one to have biaxial bending spans of various aspect ratios. The radius of the supporting corner is 5 mm. The plate specimen is simply supported all round by the 10-mm-wide flat top of the supporting columns. The material used was XAS/913 carbonfibre/epoxy-resin composite (Ciba-Geigy). All composite laminates were laid up from unidirectional
With increasingly widespread utilisation of composite materials in various engineering structural systems such as aircraft, automobiles and power plants, which are typically subjected to multiaxial/biaxial loading conditions, the understanding of behaviour of composite materials under such conditions has received a great deal of attention. A considerable amount of experimental and theoretical work is associated with tubular specimens under various biaxial stresses. Other research programmes have been carried out with flat cruciform specimens with biaxial loading applied to the arms. The former has * Present address: School of Materials Science, University of Bath, Bath, UK, BA2 7AY. ~tTo whom correspondence should be addressed. 267
A. S. Chen, F. L. Matthews
268
Plate
~-~ I I ~
I
indenter
speo,men~-
~
Supporting
I /
K
Loading
8, B=I
B=2
4
2
I
"
I
0 0
J
i
i
2
4
6
Deflection, m m
Fig. 1. A schematical central cross-section of the biaxial flexing jig.
prepreg with [02, 90212s 16-ply stacking sequence and autoclaved as 430mm by 300mm panels with a nominal cure thickness of 2 mm. The basic mechanical properties, which were determined by more than six individual tests under C R A G specifications, of the composite laminate are listed in Table 1. Plate specimens were cut to three sizes, 170 x 70, 120 x 70 and 70 x 70 mm, for testing on the biaxial jigs with spans 150 x 50, 100 x 50 and 50 × 50 mm, respectively. The aspect ratio, B, is defined as the unsupported length parallel to the surface fibre direction/the unsupported length perpendicular to the surface fibre direction. Three aspect ratios, B = 3, 2 and 1, were therefore designated for the above specimens, respectively. The biaxial bending jig was used for both monotonic and cyclic tests in a Mayes servo-hydraulic test machine. The monotonic biaxial bending tests were carried out with a loading rate of about 0.1 kN/s. The fatigue experiments were performed in load control mode at about 4 H z frequency. The large specimen deformation during fatigue tests prevents any higher frequency being applied. With sine form constant amplitude cyclic loading, the load ratio R--minimum load/maximum load (Lmax/Lmi~)--was set as 0.1, which keeps the load indenter always in contact with the specimens in order to eliminate any impact reaction during the fatigue test. A real-time computer data acquisition system was used to monitor the changes of the deflection as a result of the cyclic loading during fatigue. Six or more specimens from each aspect ratio were used in monotonic loading tests. More than five specimens were used for every
Fig. 2. Typical load-deflection curves of the static biaxial bending tests. load level in fatigue proof tests, and a single specimen was tested for post-fatigue residual property determination in every condition. A combination of non-destructive and destructive damage analysis methods, namely untrasonic C-scan and X-radiography techniques, and monitoring stiffness change, were employed to assess the damage condition and its growth in the specimens during loading. C-scans were carried out in a Meccasonics ultrasonic scanning equipment with a 10Hz probe. The X-radiographs were taken in a Faxitron X-ray machine with source-specimen distance of about 40mm, at 15kV and 5-min exposure time. The specimens were submerged in radio-opaque zinc iodide solution for a few minutes before being X-rayed. The interface C-scan and stereo Xradiography techniques were used with some degree of sophistication to look into the damage states in the through-thickness direction. Monitoring the change of dynamic stiffness as an indirect observation of damage development of the plate subjected to biaxial fatigue was made possible by means of the computer data acquisition system.
RESULTS
AND
DISCUSSION
M o n o t o n i c loading
Load/deflection relations Under static loading all plates showed similar behaviour. Typical load/deflection (L/D) curves for the plates with all three aspect ratios are presented in Fig. 2. The vertical axis is a measure of central load,
Table 1. Basic mechanical properties of the composite material
Tensile strength (GPa)
Tensile failure strain
Tensile modulus (GPa)
Poisson's ratio (v~2)
Flexural strength (GPa)
Flexural modulus (GPa)
1.19 + 0.02
1-53 + 0-03
73.39± 1-86
0-04 4- 0.01
1-24 ± 0-06
71-44 ± 3-99
(%)
269
Static and cyclic biaxial bending of CFRP panels Table 2. Results of the monotonic biaxiai bending tests
Aspect ratio (B)
Ultimate failure load (kN)
Ultimate failure deflection (ram)
First crack load (kN)
First crack deflection (mm)
Initial secant stiffness (kN/mm)
Final secant stiffness (kN/mm)
1 2 3
7.64+0.73 6.73 ± 0.68 6.04 ± 0-23
6.13+0.39 7-48 ± 0.32 7.50 ± 0.35
2.16+0.18 2-23 + 0.15 2.04 + 0.02
1.73+0.21 2.80 + 0-08 2.93 ± 0.07
1.08±0.14 0.67 ~- 0.04 0.59 ± 0.01
1.24±0.08 0.90 ± 0-09 0.80 ± 0.01
and the horizontal of central deflection, and also the maximum deflection. A linear response of the deflection to the load exists in all curves until the deflection exceeds approximately one-half of the plate thickness. Then, the large deflection effect on a bent plate appears as load-carrying capacity increasing non-linearly with deformation. This increasing stiffness is affected by several sudden load-drops which are visible as kinks in the L/D curves. These load-drops were closely associated with onset and sudden growth of damage, which will be discussed in a following section. The damage changed the continuity of the composite material and reduced the stiffness of the composite plates as a whole. It is clear that two effects are present in the change of stiffness of the plates, increasing stiffness due to the membrane stress caused by large deflection and decreasing stiffness due to accumulation of damage caused by the loading. The final failure of the plates happens when the damaged plate has no more load-bearing ability, with the loading indenter cutting through the plates. The statistical results (mean + standard deviation) of the monotonic tests are tabulated in Table 2. The ultimate failure load (UFL) and failure deflection which were measured at the point where the maximum load was reached change with the aspect ratio in reverse manner, i.e. the former decreases and the latter increases with an increase of the aspect ratio. Two measurements of secant stiffness are determined from the L/D curves. Defined as the ratio of failure load/failure deflection, the final secant stiffness decreases, as expected, with an increase of the aspect ratio, coinciding with the change of initial secant stiffness, defined as load at deflection = 1 mm, half of the plate thickness. The change of both secant stiffnesses with aspect ratio indicates, as expected, that they are structurally related parameters (change with span of support) rather than just material-related properties, such as the plate flexural rigidity defined by classical plate theory) It is interesting to see that there is no significant difference between the first crack loads of the plates with different aspect ratios, though the corresponding first crack deflection tends to increase with the aspect ratio. This is a reflection of the changing tendency of the initial secant stiffness with aspect ratio. It is also noted that differences in these parameters are much smaller between B --- 2 and
3 than between B = 1 and 2. This seems to suggest that beyond some point, the aspect ratio could have no significant influence on the performance of plates subjected to biaxial flexing.
Damage assessment As mentioned before, the final failure of the plate subjected to lateral central loading will start with the indenter cutting through the plate. The macro failure mode consists of quite localised damage, mostly shear deformation at the edges of the indenter. Although this failure mode seems to relate well to the loading arrangement, in a composite laminate more widespread microdamage inside the plate is expected. The examination of damage prior to final failure has indeed shown that the damage area inside the plate is more extended than the central loading area. Individual plates were loaded up to several different predetermined load/deflection levels, below the final failure point, and then released to be examined by NDT methods. Figures 3(a)-(c) show the Xradiography examination results of the B = 2 plate, as a set of examples. Figure 3(a) shows the damage state of the plate unloaded immediately after the first load-drop in the L/D curve occurred. Although the total damage zone is relatively small, there are two main damage mechanisms involved. Some matrix cracks travelled not far away from the central region along the fibre orientation in both 0° and 90° directions, and several delamination areas starting from the above cracks can also be clearly seen in the X-radiography image. These cracks and delaminations grow with increasing load levels (Fig. 3(b) and (c)), 0
I~
C
Fig. 3. Damage detected by X-radiography on the B = 2 plates subjected to different load levels: (a) 2.05 kN, (b) 4-00 kN, and (c) 6.00 kN.
270
A. S. Chen, F. L. M a t t h e w s 4000
d
10-
3000 ¸
--a-....o--------it--
B=I B=2 B=3
/a ,"
8¸
/'
e~
//
6'
/'
200o .d l ooo
4" o n A
e
¸
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2
4
B=I B=2 B=3
0 i-,-7---7 ,-7 ',--7--~---7-.7 .... l 0 "1 10 0 101 10 2 10 3 10 4 10 5 10 6 10 7
6
Load, kN
Number of cycles
Fig. 4. The apparent projected damage areas of the plates subjected to static loading.
Fig. 5. The Weibull median fatigue lives of the plates under the criterion of fatigue fracture.
and coincide with a very distinct 'cross' appearance until the final failure stage. This 'cross' appearance is possibly a combined result of 0/90 cross-ply lay-up sequence and the corner-lifting effect of a simply supported rectangular plate, 4 although the magnitude of the corner lifting is unknown. The projected damage areas for all three aspect ratios were measured, and are plotted as functions of the load at which the damage occurred in Fig. 4. It appears that the apparent projected damage areas is a linear function of the load applied, regardless of the aspect ratio until the load reaches very high levels. The most important result from the stereo X-radiography and interfacial C-scan studies is that the longest matrix cracks and the largest delamination areas are always associated with the layers closest to the supported surface of the specimens. The damage form in the through-thickness direction in all aspect ratio specimens has a 'pyramid' shape, the 'base' being on the supported surface of the plate. This observation of the damage condition corresponds well to the through-thickness stress distribution of the laminated plate subjected to biaxial bending: In the loaded surface layer, the maximum biaxial compressive stress could only cause localised compressive fibre fracture. In the through-thickness direction, the magnitude of the tensile stress increases towards the supported layer. The biaxial tensile stress may not be high enough to induce fibre tension fracture, but is high enough to break the relatively weak interfibre links, and cause fibre/matrix interracial failure or matrix cracks parallel to fibres. Delamination could start from the matrix cracks due to interlaminar shear and normal stress introduced by the occurrence of the matrix cracks. The largest damage zone therefore occurs in the supported layer subjected to the maximum biaxial tensile stress.
or 106 cycles if unfailed, at several different load levels under two failure criteria: fatigue fracture or when the maximum deflection reached a certain value. The load-cycles data were analysed with the two parameter Weibull distribution model:
Cyclic loading L o a d - c y c l e s ( L / l o g N) curves Series fatigue proof tests were conducted on all three aspect ratio plates. Plates were cycled to final failure,
P = 1 - exp[-(N/b)m],
where N is specimen life, m is the Weibull modulus, and b is the Weibull characteristic value. In the case of the fatigue life, b is the life point with the failure probability P =0.632. For each test population (a minimum of five specimens at a given load level) the median fatigue life, which is the life with 50% fatigue failure probability, 6 was calculated. Figures 5 and 6 show L/log N plots of the Weibull median fatigue lives (the experimental data are omitted for clarity) of all three aspect ratio plates under the criteria of fracture and maximum deflection, respectively. The curves are drawn as the best-fit straight lines through the monotonic data at the first half cycle and median life at different maximum cyclic load levels. It can be seen that under both criteria, the curves for the plates with higher aspect ratios appear flatter than those for the square plates, which therefore suggests that, despite their lower static load-bearing abilities, plates with higher aspect ratio have higher resistance to biaxial bending fatigue. Under the criterion of the maximum deflection (Fig.
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o
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10 3 10 4 Number of cycles
.
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.
.
10 5 10 6
.
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Fig. 6. The Weibull median fatigue lives of the plates under the criterion of maximum deflection, Dm=x= 3 mm (solid symbols) and Dm.x= 3 mm (open symbols).
Static and cyclic biaxial bending of CFRP panels 6), especially for the maximum deflection = 3 mm, one and half thickness of the plates, the differences between B = 2 and 3 are smaller than between B = 1 and 2. This confirmed the suggestion from the static tests that the aspect ratio could lose its significant influence on the performance of the plates subjected to biaxial flexing at higher aspect ratios. It is also apparent that the rates of decay for the L/logN curves corresponding to the different maximum deflections are slower than those for fracture failure, and the rates are even slower for those with the smaller deflection criterion. This tendency has strongly indicated that a fatigue limit exits, although this must be at quite a low load level.
Dynamic performance Changes of dynamic stiffness, which is defined as the ratio of maximum load/maximum deflection, were monitored by the computer system during fatigue. Results of three specimens, one from each aspect ratio, which failed as a result of fatigue at about 80% of their UFL are shown in Fig. 7. The stiffness of the plates was normalised with respect to the reference initial value of the individual specimen at the start of fatigue, and presented as functions of the proportional fatigue life N/Nf, where N is the number of cycles, and Nf the number of cycles to failure. The variations of the dynamic stiffness are dramatic. At the beginning of the fatigue, within about 10% of the fatigue time, it sharply decreased to about 80-90% of its original value. The change then flattens over about the next 80% of the life time until the final stage, which covers about the last 10% of the whole life. This distinct fast-slow-fast dynamic stiffness change, to some extent, is a reflection of degradation of the plate material. In the case of laminated plates, as will be discussed in a following section, the rigidity of the plate is reduced by the accumulation and progression of various damage mechanisms during fatigue. It is understandable that the total amount of stiffness change is not only closely linked to the aspect ratio, but also depends on the maximum fatigue load applied for one particular aspect ratio plate. However, 100-
271
the l o a d dependence of the stiffness change in the plates subjected to biaxial bending has been complicated by the large deflection effect on bent plates. This complication is beyond the scope of this paper, and will be the subject of a further paper by the authors. Discussion here is simply concentrated on different aspect ratio plates subjected to similar proportional maximum fatigue load, 80% of their UFL. Under this loading condition, the larger stiffness reduction at the same proportional fatigue life occurs to plates with the lower aspect ratio. The difference between the two high aspect ratios is small, whereas the difference between them and the square plates is obvious. Despite the larger initial value at the very beginning of fatigue, the dynamic stiffness of the square plate decreased at a higher rate than that of the two high aspect ratios. This is in good agreement with the discussion based on the L/log N curves under the deflection criterion (see Fig. 6). Although in the tested region (<10 6 cycles), at the same fatigue load, the high aspect ratio specimens have shorter lives under a deflection criterion, the faster stiffness decrease of the square specimen means that it will certainly suffer an earlier failure than the other plates after a large number of cycles, if the stiffness change tendencies of all plates stays unchanged.
Residual properties Static residual property tests were carried out on some unfailed specimens after cyclic loading. The Lmax and number of cycles were predetermined inside the envelope drawn by fatigue fracture L/logN curves (see Fig. 5). Residual failure load, deflection and initial secant stiffness were measured from the L/D curves. However, none of these measured parameters show very clear trends, except for the initial secant stiffness. The effect of large deflection on the plates with fatigue-induced damage could be a possible reason for the scattered test results. Figure 8 illustrates the large deflection effect by plotting L/D curves from the residual property tests for the B = 3 plates after 105 cycles at different Lmax. The curve of 8 Lmax
~-
..... .........
6
1.50k_N 3.37kN 4.00iN
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~" 80"
~ 7o-
2
B=I
z 6 0
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0
.
20
-
,
-
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-
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40 60 80 Proportional life, N/Nf
-
100
Fig. 7. Changes o f the normalised dynamic stiffness as functions of the proportional fatigue life.
0
2
4 Deflecnoe, ram
6
Fig. g. L o a d / d e f l e c t i o n c u r v e s o f the r e s i d u a l p r o p e r t y tests
on the B = 3 plates after 103 cycles.
272
A. S. Chen. F. L. Matthews 1.2
t.o .~ 0.8
o.+: o.4:::
0.2-"
--
--'-"
l0 °
,-~
---2-'.
B=3
101 102 103 104 105 106 10 7 Stop cycles
Fig. 9. Changes of the residual initial stiffness of the plates after fatigue at L .... = 1.5kN (open symbols) and L ...... = 50% UFL (solid symbols).
the monotonic test is superposed for comparison. Although the residual initial secant stiffness is obviously lower after the higher Lmax fatigue, the final parts of the curves did not show any clear indication of a fatigue effect on the mechanical behaviour, but only the large deflection effect. Comparing to the monotonic test curve, no load-drop can be seen in any L I D curves of residual property tests unless the loading level exceeds the L .... of the previous cyclic loading process. These load-drops are related to the sudden occurrence or progression of damage during loading. Since the damage state is irreversible, after the first few cycles no more sudden occurrence of damage is expected until the load level exceeds the previous maximum load. The variation of the residual initial secant stiffness after fatigue can be clearly seen for all aspect ratio plates in Fig. 9. The higher rate of decrease coincides with the higher fatigue loading level for plates with the same aspect ratio, B. The rate of the initial stiffness decrease with fatigue is also affected by the aspect ratio. At the same absolute fatigue load, L .... = 1.5kN (open symbols), despite the higher residual initial stiffness of the square plate after a short fatigue process, its residual initial stiffness reduces with fatigue much faster than that of the B = 2 and 3 specimens. This fast decreasing rate of the residual intial secant stiffness of the square plates can also be seen in a comparison based on fatigue with the same proportional cyclic load, L,,~ = 50% of UFL (solid symbols), although the absolute differences between them are comparatively small. The difference between plates with B = 2 and 3 is relatively small, as seen in the static tests, L / l o g N curves and dynamic stiffness changes.
Fig. 10. Damage detected by C-scan on the B : 1 plates after fatigue at L,.... = l . 5 k N up to (a) 1(?, (b) 104. (c! 105 and (d) 10'~ cycles. delamination. The fatigue-induced damage progression involved occurrences of new matrix cracks and growth of existing matrix cracks and delamination. Figures 10(a)-(d) show damage detected by C-scan on the square plates after fatigue at L ...... - 1 . 5 k N for different numbers of cycles. Growth of the damage zone with fatigue can be seen clearly. From interfacial C-scan and stereo X-radiography results, one can confirm that the through-thickness 'pyramid' damage zone distribution associated with static damage has become a 'truncated pyramid' in specimens subjected to fatigue. Figure 11 plots the apparent projected damage areas of two groups of specimens: those cycled at L ..... = 50% of their U F L (solid symbols) and those at the same fatigue load L .... = 1-5 kN (open symbols), for each aspect ratio. The growth of the fatigueinduced damage area is very obvious under all loading conditions for plates with the same geometry. The higher damage growth rates are with larger fatigue loads. The effect of the aspect ratio on fatigue-induced damage growth at the same fatigue loading level, however, is very small except after a large number of cycles. This agrees with the argument from the static loading damage assessment, of which the damage areas is mostly affected by loading levels but not the geometry of the plates. The damage condition in the plates will obviously affect their mechanical performance. To assess the relationship between damage state and mechanical response of the composite plates, the following discussion will be based on comparison of the normalised projected damage areas and the normalised dynamic stiffness changes, which are plotted as --; ~ 4000 - - l ~ --A-- A
B=I B=2 B=3
,i1 .'"
.',Jd
Fatigue-induced damage The damage condition of the specimens for the residual property tests were monitored by means of ultrasonic C-scan and X-radiography before they were subjected to the final monotonic loading. The damage mechanisms are still mainly matrix cracking and
.
l0 0 l01
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
102 103 104 105 106 107 Stop cycles
Fig. 11. The apparent projected damage areas of the plates after fatigue at L .... = 1-5kN (open symbols) and L ..... = 50% UFL (solid symbols).
Static and cyclic biaxial bending of CFRP panels 100
dynamic stiffness ,g
80
~-"''--:
........
.... o .... ~
Z
.......
g=l B=2
p t o g ~ t damage area
..o .......... ......................
20-
0 o
2o
"
4'0
"
6'0
"
80
100
Proportional life, %
Fig. 12. Changes of the normalised dynamic stiffness and normalised projected areas of fatigue-induced damage for plates subjected to fatigue with L m a x = 1'5 kN tO 1 0 6 cycles. functions of the proportional fatigue life in Fig. 12. The data of the normalised projected damage areas are replots of the data presented in Fig. 11 for the samples subjected to L m a x = 1.5kN. The projected damage areas were normalised with respect to the area of the whole plate within the simply supported edges. The normalised dynamic stiffness was monitored on the plates subjected to the same fatigue load level, Lmax= l ' 5 k N , and not failed as a result of fatigue up to 106 cycles. The plots of the data from the plates subjected to a low maximum fatigue load will have deliberately avoided the large deflection effect, for convenience. Although the two sets of data were obtained from different individual specimens, their tendencies with cycling have shown a remarkable similarity. During the early stages of fatigue, damage areas grew and stiffness decreased at a high rate. The stiffness reduction also flattened over the rest of the fatigue life when the damage areas grew at moderate rates. This reversed tendency strongly suggests that the change of stiffness is a mirror image of the change of damage development in the plates. The larger difference in the dynamic stiffness between B = 1 and B = 2 plates than between B - 2 and B = 3 ones is also consistent with the mirror image of the differences of the damage areas between different aspect ratios. However, a more rigorous analysis is needed before a firm relationship of damage condition and stiffness change can be drawn. It is clear that stiffness reduction as a reflection of damage development in composite plates subjected to biaxial bending has the potential of being treated as a quantitative parameter to indicate global damage in the bulk of the material. It can be easily employed for an engineering application as it only requires a simple mechanical response, namely deformation of the plates under biaxial flexing, to be recorded. CONCLUSIONS As one of the testing methods which introduce biaxial stress/strain into composite materials, biaxial bending exhibits its own advantages over other methods. Most
273
of all, the biaxial bending test is very easy to set up. The simiiarlity of biaxial bending to some engineering practices where panel components are employed marks the importance and necessity of the test method. However, the complexity of the biaxial bending stress distribution forces the composite plates to behave in ways which may not be easily interpreted, especially when the large deflection effect of the bent plate has to be accounted for. On completion of the experimental work on biaxial bending fatigue of the CFRP composite plates, with three different aspect ratios and simply supported edges, the following conclusions can be drawn: 1. The aspect ratio clearly affects the performance of the plates. In the region studied, the static mechanical properties decrease with an increase of the aspect ratio. 2. Plates with a larger aspect ratio can have a higher fatigue resistance to biaxial cyclic flexing. 3. Various damage mechanisms, mainly matrix cracks and delamination, occur under quite a low load level, about 20% of the ultimate failure load, where the plate is about to change from a small deflection to a large deflection regime. 4. The damage grows with static loading and with fatigue according mostly to loading levels, but not the aspect ratio, and strongly affects mechanical performance of the plates. 5. The growth of damage during fatigue can be monitored by change of the dynamic stiffness, and the presence of fatigue damage could be identified by a measurable parameter, namely the residual initial secant stiffness. ACKNOWLEDGEMENTS The research reported in this paper was supported by Dr G. Sims of the National Physical Laboratory, as part of the 'Materials Measurement Programme'. This programme of underpinning research is financed by the Department of Trade and Industry, UK. REFERENCES 1. Chen, A. S. & Matthews, F. L. A review of multiaxial/biaxial loading tests for composite materials. Composites, 24 (1993) 295-306. 2. Curtis, P. T. (ed.), Royal Aerospace Establishment, Farnborough, UK, Technical Report 88012, 1988. 3. Timoshenko, S. & Woinowsky-Krieger, S. Theory of Plates and Shells, 2nd edn. McGraw-Hill, New York, 1965. 4. Silard, R. Theory and Analysis of Plates. Prentice-Hall, New Jersey, 1974. 5. Lekhnitskii, S. G. Anisotropic Plates. Gordon and Breach Science Publisher, New York, 1968. 6. Little, R. E. & Jebe, E. H., Manual on Statistical Planning and Analysis of Fatigue Experiments. ASTM STP588, 1975.
0266-3538(94)00026-3
ELSEVIER
~) 1994 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0266-3538194/$07.00
STATIC A N D CYCLIC B I A X I A L B E N D I N G OF CFRP PANELS A. S. Chen* & F. L. Matthews$ Centre for Composite Materials, Imperial College of Science, Technology and Medicine, Prince Consort Road, London, UK, SW7 2BY (Received 4 February 1994; accepted 20 May 1994) the advantage of being similar to practical situations of tubular components in engineering applications. The latter provide experimental results which are relatively convenient to interpret. A detailed review of the subject of biaxial loading tests on composites can be found elsewhere. 1 Biaxial bending of plates is also a biaxial loading situation which often exists in engineering structures, for instance wing structures, automobile bodies and floor panels. However, there is very limited information concerning biaxial bending of composite panels, especially bending introduced by a lateral cyclic load. The present work is an experimental study of the behaviour of rectangular composite plates subjected to biaxial bending under monotonic and cyclic lateral loads. It is hoped that a design procedure will be developed for predicting microdamage and failure induced by biaxial cyclic bending on the basis of the above investigation. The aim of this paper is to describe and discuss the experimental results obtained on some rectangular laminated plates subjected to lateral central loading with all edges simply supported.
Abstract
Biaxial bending of panel components is a loading situation which exists in engineering structures. The behaviour of composite plates subjected to biaxial flexing has yet to be studied in detail. In the present investigation, static and cyclic biaxial bending were applied to rectangular carbon-fibre~epoxy-resin simplysupported laminated plates with three geometric aspect ratios. Series damage mechanism studies, mainly by N D T methods, and mechanical property evaluations were carried out. It was found that various damage modes could occur below about 20% of the ultimate load~deflection levels. This damage, mainly matrix cracks and delamination, developed during fatigue. The amount of damage indicated by the area of the damage zone was mainly dependent on the load level, but not on the aspect ratios. Despite their low monotonic load-bearing ability, plates with a higher aspect ratio could have a higher resistance to biaxial bending fatigue. Keywords: biaxial bending, flexural fatigue, NDT, CFRP
EXPERIMENTAL ARRANGEMENT INTRODUCTION
Design of the biaxial flexural test jig used in this work was mainly influenced by the three point-bending test rig recommended in the CRAG specifications.2 A schematic central cross-section view of the jig is illustrated in Fig. 1. The central load indenter is a circular section bar with a 12.5 mm radius hemispherical loading nose. The supporting frame is a combination of two pairs of changeable supporting columns sited on a pre-positioned base plate, which enables one to have biaxial bending spans of various aspect ratios. The radius of the supporting corner is 5 mm. The plate specimen is simply supported all round by the 10-mm-wide flat top of the supporting columns. The material used was XAS/913 carbonfibre/epoxy-resin composite (Ciba-Geigy). All composite laminates were laid up from unidirectional
With increasingly widespread utilisation of composite materials in various engineering structural systems such as aircraft, automobiles and power plants, which are typically subjected to multiaxial/biaxial loading conditions, the understanding of behaviour of composite materials under such conditions has received a great deal of attention. A considerable amount of experimental and theoretical work is associated with tubular specimens under various biaxial stresses. Other research programmes have been carried out with flat cruciform specimens with biaxial loading applied to the arms. The former has * Present address: School of Materials Science, University of Bath, Bath, UK, BA2 7AY. ~tTo whom correspondence should be addressed. 267
A. S. Chen, F. L. Matthews
268
Plate
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Fig. 1. A schematical central cross-section of the biaxial flexing jig.
prepreg with [02, 90212s 16-ply stacking sequence and autoclaved as 430mm by 300mm panels with a nominal cure thickness of 2 mm. The basic mechanical properties, which were determined by more than six individual tests under C R A G specifications, of the composite laminate are listed in Table 1. Plate specimens were cut to three sizes, 170 x 70, 120 x 70 and 70 x 70 mm, for testing on the biaxial jigs with spans 150 x 50, 100 x 50 and 50 × 50 mm, respectively. The aspect ratio, B, is defined as the unsupported length parallel to the surface fibre direction/the unsupported length perpendicular to the surface fibre direction. Three aspect ratios, B = 3, 2 and 1, were therefore designated for the above specimens, respectively. The biaxial bending jig was used for both monotonic and cyclic tests in a Mayes servo-hydraulic test machine. The monotonic biaxial bending tests were carried out with a loading rate of about 0.1 kN/s. The fatigue experiments were performed in load control mode at about 4 H z frequency. The large specimen deformation during fatigue tests prevents any higher frequency being applied. With sine form constant amplitude cyclic loading, the load ratio R--minimum load/maximum load (Lmax/Lmi~)--was set as 0.1, which keeps the load indenter always in contact with the specimens in order to eliminate any impact reaction during the fatigue test. A real-time computer data acquisition system was used to monitor the changes of the deflection as a result of the cyclic loading during fatigue. Six or more specimens from each aspect ratio were used in monotonic loading tests. More than five specimens were used for every
Fig. 2. Typical load-deflection curves of the static biaxial bending tests. load level in fatigue proof tests, and a single specimen was tested for post-fatigue residual property determination in every condition. A combination of non-destructive and destructive damage analysis methods, namely untrasonic C-scan and X-radiography techniques, and monitoring stiffness change, were employed to assess the damage condition and its growth in the specimens during loading. C-scans were carried out in a Meccasonics ultrasonic scanning equipment with a 10Hz probe. The X-radiographs were taken in a Faxitron X-ray machine with source-specimen distance of about 40mm, at 15kV and 5-min exposure time. The specimens were submerged in radio-opaque zinc iodide solution for a few minutes before being X-rayed. The interface C-scan and stereo Xradiography techniques were used with some degree of sophistication to look into the damage states in the through-thickness direction. Monitoring the change of dynamic stiffness as an indirect observation of damage development of the plate subjected to biaxial fatigue was made possible by means of the computer data acquisition system.
RESULTS
AND
DISCUSSION
M o n o t o n i c loading
Load/deflection relations Under static loading all plates showed similar behaviour. Typical load/deflection (L/D) curves for the plates with all three aspect ratios are presented in Fig. 2. The vertical axis is a measure of central load,
Table 1. Basic mechanical properties of the composite material
Tensile strength (GPa)
Tensile failure strain
Tensile modulus (GPa)
Poisson's ratio (v~2)
Flexural strength (GPa)
Flexural modulus (GPa)
1.19 + 0.02
1-53 + 0-03
73.39± 1-86
0-04 4- 0.01
1-24 ± 0-06
71-44 ± 3-99
(%)
269
Static and cyclic biaxial bending of CFRP panels Table 2. Results of the monotonic biaxiai bending tests
Aspect ratio (B)
Ultimate failure load (kN)
Ultimate failure deflection (ram)
First crack load (kN)
First crack deflection (mm)
Initial secant stiffness (kN/mm)
Final secant stiffness (kN/mm)
1 2 3
7.64+0.73 6.73 ± 0.68 6.04 ± 0-23
6.13+0.39 7-48 ± 0.32 7.50 ± 0.35
2.16+0.18 2-23 + 0.15 2.04 + 0.02
1.73+0.21 2.80 + 0-08 2.93 ± 0.07
1.08±0.14 0.67 ~- 0.04 0.59 ± 0.01
1.24±0.08 0.90 ± 0-09 0.80 ± 0.01
and the horizontal of central deflection, and also the maximum deflection. A linear response of the deflection to the load exists in all curves until the deflection exceeds approximately one-half of the plate thickness. Then, the large deflection effect on a bent plate appears as load-carrying capacity increasing non-linearly with deformation. This increasing stiffness is affected by several sudden load-drops which are visible as kinks in the L/D curves. These load-drops were closely associated with onset and sudden growth of damage, which will be discussed in a following section. The damage changed the continuity of the composite material and reduced the stiffness of the composite plates as a whole. It is clear that two effects are present in the change of stiffness of the plates, increasing stiffness due to the membrane stress caused by large deflection and decreasing stiffness due to accumulation of damage caused by the loading. The final failure of the plates happens when the damaged plate has no more load-bearing ability, with the loading indenter cutting through the plates. The statistical results (mean + standard deviation) of the monotonic tests are tabulated in Table 2. The ultimate failure load (UFL) and failure deflection which were measured at the point where the maximum load was reached change with the aspect ratio in reverse manner, i.e. the former decreases and the latter increases with an increase of the aspect ratio. Two measurements of secant stiffness are determined from the L/D curves. Defined as the ratio of failure load/failure deflection, the final secant stiffness decreases, as expected, with an increase of the aspect ratio, coinciding with the change of initial secant stiffness, defined as load at deflection = 1 mm, half of the plate thickness. The change of both secant stiffnesses with aspect ratio indicates, as expected, that they are structurally related parameters (change with span of support) rather than just material-related properties, such as the plate flexural rigidity defined by classical plate theory) It is interesting to see that there is no significant difference between the first crack loads of the plates with different aspect ratios, though the corresponding first crack deflection tends to increase with the aspect ratio. This is a reflection of the changing tendency of the initial secant stiffness with aspect ratio. It is also noted that differences in these parameters are much smaller between B --- 2 and
3 than between B = 1 and 2. This seems to suggest that beyond some point, the aspect ratio could have no significant influence on the performance of plates subjected to biaxial flexing.
Damage assessment As mentioned before, the final failure of the plate subjected to lateral central loading will start with the indenter cutting through the plate. The macro failure mode consists of quite localised damage, mostly shear deformation at the edges of the indenter. Although this failure mode seems to relate well to the loading arrangement, in a composite laminate more widespread microdamage inside the plate is expected. The examination of damage prior to final failure has indeed shown that the damage area inside the plate is more extended than the central loading area. Individual plates were loaded up to several different predetermined load/deflection levels, below the final failure point, and then released to be examined by NDT methods. Figures 3(a)-(c) show the Xradiography examination results of the B = 2 plate, as a set of examples. Figure 3(a) shows the damage state of the plate unloaded immediately after the first load-drop in the L/D curve occurred. Although the total damage zone is relatively small, there are two main damage mechanisms involved. Some matrix cracks travelled not far away from the central region along the fibre orientation in both 0° and 90° directions, and several delamination areas starting from the above cracks can also be clearly seen in the X-radiography image. These cracks and delaminations grow with increasing load levels (Fig. 3(b) and (c)), 0
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Fig. 3. Damage detected by X-radiography on the B = 2 plates subjected to different load levels: (a) 2.05 kN, (b) 4-00 kN, and (c) 6.00 kN.
270
A. S. Chen, F. L. M a t t h e w s 4000
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Number of cycles
Fig. 4. The apparent projected damage areas of the plates subjected to static loading.
Fig. 5. The Weibull median fatigue lives of the plates under the criterion of fatigue fracture.
and coincide with a very distinct 'cross' appearance until the final failure stage. This 'cross' appearance is possibly a combined result of 0/90 cross-ply lay-up sequence and the corner-lifting effect of a simply supported rectangular plate, 4 although the magnitude of the corner lifting is unknown. The projected damage areas for all three aspect ratios were measured, and are plotted as functions of the load at which the damage occurred in Fig. 4. It appears that the apparent projected damage areas is a linear function of the load applied, regardless of the aspect ratio until the load reaches very high levels. The most important result from the stereo X-radiography and interfacial C-scan studies is that the longest matrix cracks and the largest delamination areas are always associated with the layers closest to the supported surface of the specimens. The damage form in the through-thickness direction in all aspect ratio specimens has a 'pyramid' shape, the 'base' being on the supported surface of the plate. This observation of the damage condition corresponds well to the through-thickness stress distribution of the laminated plate subjected to biaxial bending: In the loaded surface layer, the maximum biaxial compressive stress could only cause localised compressive fibre fracture. In the through-thickness direction, the magnitude of the tensile stress increases towards the supported layer. The biaxial tensile stress may not be high enough to induce fibre tension fracture, but is high enough to break the relatively weak interfibre links, and cause fibre/matrix interracial failure or matrix cracks parallel to fibres. Delamination could start from the matrix cracks due to interlaminar shear and normal stress introduced by the occurrence of the matrix cracks. The largest damage zone therefore occurs in the supported layer subjected to the maximum biaxial tensile stress.
or 106 cycles if unfailed, at several different load levels under two failure criteria: fatigue fracture or when the maximum deflection reached a certain value. The load-cycles data were analysed with the two parameter Weibull distribution model:
Cyclic loading L o a d - c y c l e s ( L / l o g N) curves Series fatigue proof tests were conducted on all three aspect ratio plates. Plates were cycled to final failure,
P = 1 - exp[-(N/b)m],
where N is specimen life, m is the Weibull modulus, and b is the Weibull characteristic value. In the case of the fatigue life, b is the life point with the failure probability P =0.632. For each test population (a minimum of five specimens at a given load level) the median fatigue life, which is the life with 50% fatigue failure probability, 6 was calculated. Figures 5 and 6 show L/log N plots of the Weibull median fatigue lives (the experimental data are omitted for clarity) of all three aspect ratio plates under the criteria of fracture and maximum deflection, respectively. The curves are drawn as the best-fit straight lines through the monotonic data at the first half cycle and median life at different maximum cyclic load levels. It can be seen that under both criteria, the curves for the plates with higher aspect ratios appear flatter than those for the square plates, which therefore suggests that, despite their lower static load-bearing abilities, plates with higher aspect ratio have higher resistance to biaxial bending fatigue. Under the criterion of the maximum deflection (Fig.
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Static and cyclic biaxial bending of CFRP panels 6), especially for the maximum deflection = 3 mm, one and half thickness of the plates, the differences between B = 2 and 3 are smaller than between B = 1 and 2. This confirmed the suggestion from the static tests that the aspect ratio could lose its significant influence on the performance of the plates subjected to biaxial flexing at higher aspect ratios. It is also apparent that the rates of decay for the L/logN curves corresponding to the different maximum deflections are slower than those for fracture failure, and the rates are even slower for those with the smaller deflection criterion. This tendency has strongly indicated that a fatigue limit exits, although this must be at quite a low load level.
Dynamic performance Changes of dynamic stiffness, which is defined as the ratio of maximum load/maximum deflection, were monitored by the computer system during fatigue. Results of three specimens, one from each aspect ratio, which failed as a result of fatigue at about 80% of their UFL are shown in Fig. 7. The stiffness of the plates was normalised with respect to the reference initial value of the individual specimen at the start of fatigue, and presented as functions of the proportional fatigue life N/Nf, where N is the number of cycles, and Nf the number of cycles to failure. The variations of the dynamic stiffness are dramatic. At the beginning of the fatigue, within about 10% of the fatigue time, it sharply decreased to about 80-90% of its original value. The change then flattens over about the next 80% of the life time until the final stage, which covers about the last 10% of the whole life. This distinct fast-slow-fast dynamic stiffness change, to some extent, is a reflection of degradation of the plate material. In the case of laminated plates, as will be discussed in a following section, the rigidity of the plate is reduced by the accumulation and progression of various damage mechanisms during fatigue. It is understandable that the total amount of stiffness change is not only closely linked to the aspect ratio, but also depends on the maximum fatigue load applied for one particular aspect ratio plate. However, 100-
271
the l o a d dependence of the stiffness change in the plates subjected to biaxial bending has been complicated by the large deflection effect on bent plates. This complication is beyond the scope of this paper, and will be the subject of a further paper by the authors. Discussion here is simply concentrated on different aspect ratio plates subjected to similar proportional maximum fatigue load, 80% of their UFL. Under this loading condition, the larger stiffness reduction at the same proportional fatigue life occurs to plates with the lower aspect ratio. The difference between the two high aspect ratios is small, whereas the difference between them and the square plates is obvious. Despite the larger initial value at the very beginning of fatigue, the dynamic stiffness of the square plate decreased at a higher rate than that of the two high aspect ratios. This is in good agreement with the discussion based on the L/log N curves under the deflection criterion (see Fig. 6). Although in the tested region (<10 6 cycles), at the same fatigue load, the high aspect ratio specimens have shorter lives under a deflection criterion, the faster stiffness decrease of the square specimen means that it will certainly suffer an earlier failure than the other plates after a large number of cycles, if the stiffness change tendencies of all plates stays unchanged.
Residual properties Static residual property tests were carried out on some unfailed specimens after cyclic loading. The Lmax and number of cycles were predetermined inside the envelope drawn by fatigue fracture L/logN curves (see Fig. 5). Residual failure load, deflection and initial secant stiffness were measured from the L/D curves. However, none of these measured parameters show very clear trends, except for the initial secant stiffness. The effect of large deflection on the plates with fatigue-induced damage could be a possible reason for the scattered test results. Figure 8 illustrates the large deflection effect by plotting L/D curves from the residual property tests for the B = 3 plates after 105 cycles at different Lmax. The curve of 8 Lmax
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on the B = 3 plates after 103 cycles.
272
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the monotonic test is superposed for comparison. Although the residual initial secant stiffness is obviously lower after the higher Lmax fatigue, the final parts of the curves did not show any clear indication of a fatigue effect on the mechanical behaviour, but only the large deflection effect. Comparing to the monotonic test curve, no load-drop can be seen in any L I D curves of residual property tests unless the loading level exceeds the L .... of the previous cyclic loading process. These load-drops are related to the sudden occurrence or progression of damage during loading. Since the damage state is irreversible, after the first few cycles no more sudden occurrence of damage is expected until the load level exceeds the previous maximum load. The variation of the residual initial secant stiffness after fatigue can be clearly seen for all aspect ratio plates in Fig. 9. The higher rate of decrease coincides with the higher fatigue loading level for plates with the same aspect ratio, B. The rate of the initial stiffness decrease with fatigue is also affected by the aspect ratio. At the same absolute fatigue load, L .... = 1.5kN (open symbols), despite the higher residual initial stiffness of the square plate after a short fatigue process, its residual initial stiffness reduces with fatigue much faster than that of the B = 2 and 3 specimens. This fast decreasing rate of the residual intial secant stiffness of the square plates can also be seen in a comparison based on fatigue with the same proportional cyclic load, L,,~ = 50% of UFL (solid symbols), although the absolute differences between them are comparatively small. The difference between plates with B = 2 and 3 is relatively small, as seen in the static tests, L / l o g N curves and dynamic stiffness changes.
Fig. 10. Damage detected by C-scan on the B : 1 plates after fatigue at L,.... = l . 5 k N up to (a) 1(?, (b) 104. (c! 105 and (d) 10'~ cycles. delamination. The fatigue-induced damage progression involved occurrences of new matrix cracks and growth of existing matrix cracks and delamination. Figures 10(a)-(d) show damage detected by C-scan on the square plates after fatigue at L ...... - 1 . 5 k N for different numbers of cycles. Growth of the damage zone with fatigue can be seen clearly. From interfacial C-scan and stereo X-radiography results, one can confirm that the through-thickness 'pyramid' damage zone distribution associated with static damage has become a 'truncated pyramid' in specimens subjected to fatigue. Figure 11 plots the apparent projected damage areas of two groups of specimens: those cycled at L ..... = 50% of their U F L (solid symbols) and those at the same fatigue load L .... = 1-5 kN (open symbols), for each aspect ratio. The growth of the fatigueinduced damage area is very obvious under all loading conditions for plates with the same geometry. The higher damage growth rates are with larger fatigue loads. The effect of the aspect ratio on fatigue-induced damage growth at the same fatigue loading level, however, is very small except after a large number of cycles. This agrees with the argument from the static loading damage assessment, of which the damage areas is mostly affected by loading levels but not the geometry of the plates. The damage condition in the plates will obviously affect their mechanical performance. To assess the relationship between damage state and mechanical response of the composite plates, the following discussion will be based on comparison of the normalised projected damage areas and the normalised dynamic stiffness changes, which are plotted as --; ~ 4000 - - l ~ --A-- A
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Fatigue-induced damage The damage condition of the specimens for the residual property tests were monitored by means of ultrasonic C-scan and X-radiography before they were subjected to the final monotonic loading. The damage mechanisms are still mainly matrix cracking and
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Fig. 11. The apparent projected damage areas of the plates after fatigue at L .... = 1-5kN (open symbols) and L ..... = 50% UFL (solid symbols).
Static and cyclic biaxial bending of CFRP panels 100
dynamic stiffness ,g
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Fig. 12. Changes of the normalised dynamic stiffness and normalised projected areas of fatigue-induced damage for plates subjected to fatigue with L m a x = 1'5 kN tO 1 0 6 cycles. functions of the proportional fatigue life in Fig. 12. The data of the normalised projected damage areas are replots of the data presented in Fig. 11 for the samples subjected to L m a x = 1.5kN. The projected damage areas were normalised with respect to the area of the whole plate within the simply supported edges. The normalised dynamic stiffness was monitored on the plates subjected to the same fatigue load level, Lmax= l ' 5 k N , and not failed as a result of fatigue up to 106 cycles. The plots of the data from the plates subjected to a low maximum fatigue load will have deliberately avoided the large deflection effect, for convenience. Although the two sets of data were obtained from different individual specimens, their tendencies with cycling have shown a remarkable similarity. During the early stages of fatigue, damage areas grew and stiffness decreased at a high rate. The stiffness reduction also flattened over the rest of the fatigue life when the damage areas grew at moderate rates. This reversed tendency strongly suggests that the change of stiffness is a mirror image of the change of damage development in the plates. The larger difference in the dynamic stiffness between B = 1 and B = 2 plates than between B - 2 and B = 3 ones is also consistent with the mirror image of the differences of the damage areas between different aspect ratios. However, a more rigorous analysis is needed before a firm relationship of damage condition and stiffness change can be drawn. It is clear that stiffness reduction as a reflection of damage development in composite plates subjected to biaxial bending has the potential of being treated as a quantitative parameter to indicate global damage in the bulk of the material. It can be easily employed for an engineering application as it only requires a simple mechanical response, namely deformation of the plates under biaxial flexing, to be recorded. CONCLUSIONS As one of the testing methods which introduce biaxial stress/strain into composite materials, biaxial bending exhibits its own advantages over other methods. Most
273
of all, the biaxial bending test is very easy to set up. The simiiarlity of biaxial bending to some engineering practices where panel components are employed marks the importance and necessity of the test method. However, the complexity of the biaxial bending stress distribution forces the composite plates to behave in ways which may not be easily interpreted, especially when the large deflection effect of the bent plate has to be accounted for. On completion of the experimental work on biaxial bending fatigue of the CFRP composite plates, with three different aspect ratios and simply supported edges, the following conclusions can be drawn: 1. The aspect ratio clearly affects the performance of the plates. In the region studied, the static mechanical properties decrease with an increase of the aspect ratio. 2. Plates with a larger aspect ratio can have a higher fatigue resistance to biaxial cyclic flexing. 3. Various damage mechanisms, mainly matrix cracks and delamination, occur under quite a low load level, about 20% of the ultimate failure load, where the plate is about to change from a small deflection to a large deflection regime. 4. The damage grows with static loading and with fatigue according mostly to loading levels, but not the aspect ratio, and strongly affects mechanical performance of the plates. 5. The growth of damage during fatigue can be monitored by change of the dynamic stiffness, and the presence of fatigue damage could be identified by a measurable parameter, namely the residual initial secant stiffness. ACKNOWLEDGEMENTS The research reported in this paper was supported by Dr G. Sims of the National Physical Laboratory, as part of the 'Materials Measurement Programme'. This programme of underpinning research is financed by the Department of Trade and Industry, UK. REFERENCES 1. Chen, A. S. & Matthews, F. L. A review of multiaxial/biaxial loading tests for composite materials. Composites, 24 (1993) 295-306. 2. Curtis, P. T. (ed.), Royal Aerospace Establishment, Farnborough, UK, Technical Report 88012, 1988. 3. Timoshenko, S. & Woinowsky-Krieger, S. Theory of Plates and Shells, 2nd edn. McGraw-Hill, New York, 1965. 4. Silard, R. Theory and Analysis of Plates. Prentice-Hall, New Jersey, 1974. 5. Lekhnitskii, S. G. Anisotropic Plates. Gordon and Breach Science Publisher, New York, 1968. 6. Little, R. E. & Jebe, E. H., Manual on Statistical Planning and Analysis of Fatigue Experiments. ASTM STP588, 1975.