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Constrained buckling tests show increasing compressive strain to failure with increasing strain gradient
Composites Purr A 28A ( 1997) 959-964 0 1997 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-835X/97/$17.00
PII:S1359-835X(97)00067-5
ELSEVIER
Constrained buckling tests show increasing compressive strain to failure with increasing strain gradient
M. R. Wisnom
and J. W. Atkinson
Department of Aerospace Engineering, Bristol, UK (Received 70 February 1997; accepted
University 12 June
of Bristol,
University
Walk, BS8 ITR
79971
A new buckling rig was developed where the ends of the specimen are initially free to rotate, but lock up after a certain amount of rotation. By varying the rotation angle allowed, the ratio between compression and bending can be altered, permitting tests with different strain gradients through the thickness. Four sets of tests were carried out with 16 ply unidirectional T800/924 carbon fibre-epoxy specimens of two different lengths, each at two different rotation angles. Consistent gauge length compressive failures were obtained in all cases, with the strain at failure increasing with increasing strain gradient. 0 1997 Elsevier Science Limited. (Keywords:
compressive
failure: flexural failure: strain gradient; buckling; size effects)
INTRODUCTION Compressive strength is a critical issue for carbon fibre composite structures, and is often cited as a factor limiting the exploitation of these materials. Flexural loading is important in many applications, and it has been shown that very high values of compressive strength can be achieved in bending. For example, tests on unidirectional T800/924 intermediate modulus carbon fibre-epoxy in a pin-ended buckling rig gave compressive strains at failure of up to 21600 microstrain’, of the order of double typical quoted values of compressive failure strain. Reconciling this discrepancy is important at a fundamental level in understanding the basic failure mechanism in these materials. But from a practical viewpoint it is also important in order to know under what conditions advantage can be taken of the higher strength under flexural loading to produce lighter, more efficient structures. The previous pin-ended buckling tests showed that there is a significant size effect for compressive failure of unidirectional carbon fibre-epoxy under flexural loading’. Scaled specimens of T800/924 showed an increase in compressive strain at failure of 47% for a factor of eight decrease in linear dimensions. A similar trend was also observed from compressive failures in scaled four point bending tests on XAS/913 carbon fibre-epoxy’. The results of the pin-ended tests suggested that the size effect was predominantly due to the change in strain gradient through the thickness rather than the different stressed volumes. A number of theoretical studies have shown that higher compressive stresses are required to cause microbuckling under a non-uniform stress field than under pure
compression3-‘. Higher flexural strengths than compressive strengths are predicted, with the difference increasing with decreasing thickness4m7. To investigate this effect further experimentally. a test was required which would allow the ratio between compressive and flexural loading to be varied whilst maintaining the thickness constant. This can be done with the pin-ended buckling rig by varying the length of the specimen, and hence the buckling load. However, this leads to significant changes in the stressed volume as well as the strain gradient, and so it is hard to separate the two effects. The objective of this study was to devise a test allowing the strain gradient to be varied independently of the specimen dimensions, and hence to verify whether the strain gradient is the critical factor responsible for the observed changes in compressive strain at failure. Very few experimental studies have been published on varying the strain gradient in compressive tests. One interesting approach is that of Grandidier et al.. who developed an original device to apply combined bending and compressive loadings. The specimen was fixed in two blocks which were loaded by a system of levers. By varying the geometry of the mechanism, different ratios of bending and compressive loading could be applied. Initial results with the rig reported so far look quite promising. In this study a different approach was taken based on a variation of the pin-ended buckling tests. A new rig was developed where the ends of the specimen are initially free to rotate, but lock up after a certain amount of rotation. By varying the rotation angle allowed, the ratio between compression and bending can be altered with the same specimen. With no rotation the rig would correspond
959
Constrained
buckling
tests: M. R. Whom
and J. W. Atkinson
to one with fixed end conditions. With no restriction of rotation it would effectively be a pin-ended buckling rig. Locking the rig at different angles allows variations between these two extremes. Since there is a factor of four difference in Euler buckling load between fixed and pinned end conditions, there is scope for considerable variation in the amount of direct compression with the same specimen dimensions. Tests were carried out at the maximum and minimum rotation angles which gave gauge length failures. To increase the range of compressive to bending stress ratios, two different lengths of specimen were used. Surface strains were measured with strain gauges, allowing the effect of the strain gradient on compressive strains at failure to be determined. Figure 1
Diagram of the rig
EXPERIMENTAL Constrained buckling rig Figure 1 shows the rig. It consists of two identical mild steel fixtures (1) which are clamped in the jaws of the test machine via spigots (2). Rotating arms (4) are attached to the fixed ends by means of silver steel bearing pins (3). The bearing journals were machined to close tolerance and the diameter was kept large (20 mm) in order to minimise bearing stresses and prevent deformation of the pins. Each end of the specimen (12) is clamped to a block (10) by means of a clamping plate (11) and four bolts (13). The blocks have slots through which they are attached to the rotating arms (4) by means of six bolts. This allows the position of the blocks to be varied, enabling the offset between the loading and specimen axes to be controlled. Rotation of each end of the specimen is restricted by means of a high tensile steel tie rod (5) which attaches via bearing pins to the fixed arm (6) and rotating arm (7). The maximum rotation allowed is varied by adjusting the lock nuts (9). Test specimens A 16 ply plate of unidirectional Ciba T800/924 carbon fibre-epoxy was laid up and cured according to the manufacturer’s instructions. Two different sizes of specimens were cut from the same plate using a diamond saw. The dimensions were 60 mm X 10 mm and 80 mm X 10 mm. The clamped section at each end is 20 mm long, so the gauge lengths were 20 and 40 mm. The thickness of the specimens was 2.02 mm. It is important that the ends of the specimens are square, especially when the direct compressive stresses are high, and the risk of failure initiating from the ends is greater. The ends were therefore wet ground on a surface table using emery paper, with an arrangement of V blocks to keep the specimens square to the surface. Strain gauges 1 mm long were bonded to the centre of both faces of the specimens using cyanoacrylate adhesive. Additional gauges were also fitted on one surface of the
960
specimen at 5 mm from the centre in order to measure the variation of strain along the length. For most of the specimens, only one extra gauge was fitted, but in some cases two additional gauges were fitted on the same surface, one towards each end of the specimen. The reasons for this will be explained later. Test procedure Great care was taken to align the rig and the test pieces accurately. The specimens were firmly clamped in place with a piece of thin steel shim on either side to protect the surfaces from the sharp comers at the ends of the clamping blocks. Thin pieces of PTFE sheet were placed between the shims and the specimen to provide further protection. An offset of 16 mm was used between the mid-plane of the specimen and the loading axis, large enough to provide sufficient bending to ensure smooth rotation of the rig. It was desired to test with the maximum range of angles before lock-up whilst ensuring failures in the gauge section. Failure could initiate at the ends due to stress concentrations at the clamps if the load was too high, or due to excessive shear stresses if the rotation angle was too large. Initial tests were therefore carried out, and a range of 1- 13’ for the 20 mm specimens and 5-20” for the 40 mm ones were found to produce consistent gauge length failures. It was decided to do two sets of tests on each length specimen at the extremes of these ranges of angles. Care was taken to ensure that the two ends of the rig locked up at the same point, to try to avoid any asymmetry in the tests. The four sets of specimens were loaded under displacement control at a rate to produce failure in about 90 s. Load and strain gauge readings were logged on a computer.
RESULTS 40 mm gauge length specimens Figure 2 shows typical load-strain
plots for the centre
Constrained
gauges with 5” and 20” maximum rotation angles. The initial curves are shallow as the specimens are effectively undergoing pin-ended buckling. During this portion of the test the response is the same for the two cases. With 5” of free rotation the rig locks up at a compressive strain of about 3000 microstrain. The response then suddenly becomes steeper, with the load increasing much more rapidly until catastrophic failure occurs. The specimens with 20” free rotation behaved very similarly, except that they locked up much later, when the compressive strains were already about 15000 microstrain. All specimens broke completely into two pieces, with the failure in the central region. The fracture surfaces were inclined at an angle of approximately 75” to the fibre direction, and had the smooth appearance typical of compressive failures. There was also some splitting along the length, and fibre tensile failure close to the tension surface. The results for the maximum strains at the centre of the specimens are summarised in Table 1. Mean values of the compressive strains at 5 mm from the centre of the specimens are also shown, giving an indication of the strain gradient along the length. In all cases the strains reduced going away from the centre, with the effect being
buckling
tests: M. R. Wisnom
more noticeable were higher.
for the 5” rotation
20 mm gauge length
-20000
-15000
-10000
-5000 Strain
Figure 2 specimens
Typical
load-strain
Table 1
Summary
of results
0
5000
10000
15000
20000
case where the loads
specimens
The specimens with 13” rotation behaved very similarly to the longer ones. However the ones where only 1” of rotation was allowed showed asymmetry in their response. These specimens therefore had two strain gauges attached at 5 mm from the centre, one towards each end, to allow the asymmetry to be monitored. It was found that in some cases the strain readings were actually higher at one of these gauges than at the centre plane of symmetry. Where this was the case the strains at the gauges on the other side of the centre were much lower. This anomaly had been noticed in preliminary tests, but despite taking great care in setting up and aligning the rig it was not possible to eliminate it. Some additional tests were done whereby the rig was adjusted whilst the specimen was under load until the strain gauge readings were symmetric. However, on further loading they would again diverge. The forces are quite high in this case, and as a result of the rotation, significant shear stresses arise. It is thought that shear deformations occurring to a greater extent at one end of the specimen than the other may be the cause of the asymmetry. As a result of this behaviour, the highest compressive strain may have occurred between the gauges. To get a better estimate of the maximum value, a curve was fitted through the three strain gauge readings, and the maximum obtained by differentiating the resulting function. A quadratic variation of strain along the length was assumed. the simplest possible to go exactly through three data points. For example, for one specimen showing considerable asymmetry, the fitted equation for strain as a function of distance, X. from the centre of the gauge length was: E=134x7-757x-
-25000
and J. W. Atkinson
14130
(1)
3 shows the fit graphically. In this case the maximum strain is -15 200 microstrain compared with - 14 130 at the centre and - 14 570 and -6990 at 5 mm on either side. Figure
(microstrain)
plots for the centre
gauges
of 40 mm
Strains at failure (microstrain)
Length (mm) 40 40 20 20 20
Rotation angle
At centre
At centre
5 mm away
Failure load
(dcg)
Compression
Tension
Compression
(W
No. of spec.
-19390 (7.5%) -17700 (4.8%) -17960 (3.7%) -15000 (9.8%) -15 810” (6.9%) -16 100b (5.2%)
15290 (6.0%) II 130 (8.7%) 12370 (7.2%) 1290 (73.7%)
-18640 (13.8%) -15310 (7.8%) -16290 (2.4%) -10680 (35.4%)
2.2 (17.1%) 7.6 (1.7%) 3.84 (15.6%) 18.6 (7.3%)
5
20 (C.V.) 5 (C.V.) 13 (C.V.) I (C.V.)
I (C.V.)
20
I (C.V.)
“Highest strain from three gauges along length ‘Estimated maximum strains based on fit from three measured
6 5 6 6
2530h (60.9%)
~11970b
6
values
961
Constrained
buckling
tests: M. I?. Wisnom
and J. W. Atkinson
-2eew
Figure 3 Curve fitted to the three strain readings, obtained at the centre and 5 mm from the centre of 20 mm specimen with 1” rotation
the deflections and offset loading. It was assumed that the tensile strain due to bending varied in the same manner along the length as the compressive strain due to bending, whilst the direct strain remained constant along the length. The direct compressive strain was estimated from the load at failure assuming a modulus of 160 GPa. This strain was then subtracted from the measured strains at the centre of the specimen to give the bending components of the compressive and tensile surface strain E&, and E&. The direct strain was also subtracted from the previously calculated maximum strain to give the bending component of the maximum compressive strain, E&,,. The corresponding tensile strain due to bending was determined by multiplying by the ratio of tensile to compressive strain at the centre of the specimen: t Emax = &x
Maximum compressive strain (microstrain) 25,om 2Qmm13° 20,m
01
/
0
5,ow
-
I
10,m
&I
J &I
(2)
The direct compressive strain was then added back in to give the total strain on the tensile surface at the position of maximum compressive strain. This procedure would be expected to give very good results if the stress-strain response was linear elastic. In practice non-linearity may introduce a small degree of inaccuracy, but a reasonable estimate of the maximum tensile strain should still be obtained.
DISCUSSION
15,Klo
Strain gradient (microstrain/mm) Figure 4 The maximum compressive strains at failure from all four sets of tests, plotted as a function of the strain gradient through the thickness
Table I shows the maximum strains on both surfaces at the centre of the gauge length and the value on the compression side 5 mm away. For the 20 mm specimens with 1” rotation there were gauges at 5 mm from the centre towards both ends, and the value shown is the average of the two readings. The highest strain was not necessarily at the centre gauge, and so the maximum measured strain taken from the highest reading of the three gauges on the compressive surface is shown as well. Finally for this case the maximum based on the curve fits to the three strain gauge readings is given. The difference between the latter two values is only 1.9%, showing that the assumptions involved in the correction procedure are not that critical in determining the maximum strain. The correction improves the consistency of the results, as seen from the lower coefficient of variation for the corrected strains than for the raw measurements. The estimated compressive strain at 5 mm away from the point of highest strain from a best fit to all the data is also shown. The estimated value of the maximum tensile strain is included in the table, although this involved further assumptions since there was only one strain gauge on the tension side at the centre of the specimen. The procedure adopted involved separating out the direct compressive strain due to the end load, and the bending strain due to
Effect of strain gradient Figure 4 shows the maximum compressive strains at failure from all four sets of tests plotted as a function of the strain gradient through the thickness, defined as the difference between the compressive and tensile surface strains divided by the measured specimen thickness. For the 20 mm case with 1” of rotation, the estimated maximum strain was used. The error bars correspond to one standard deviation on each side of the mean. The results show a consistent trend of increasing strain at failure with increasing strain gradient, and a least squares line through the data gives a very good fit. The increase in compressive strain between the 20 mm long specimens with 1” rotation and the 40 mm ones with 20” rotation is 20%. The corresponding increase in strain gradient through the thickness between these two cases is 85%. In previous buckling tests a significant reduction in strain at failure was found with increasing size for scaled specimens from the same batch of T800/924 carbon fibreepoxy as used in this study’. The results suggested that the effect was predominantly due to the difference in strain gradient through the thickness. Figure 5 shows the data for the three different sized specimens together with the best fit straight line. The results from the constrained buckling tests are also shown, and are close to the line for the scaled specimens. The strain gradient effect appears to be slightly greater in the constrained tests, but given the scatter in the data, and the narrower range of strain gradients, the comparison is reasonable.
Constrained
Maximum compressive strain (microstrain) 25,LxQ
20.m
15.m
10,000
5.ooo
OL 0
10,ooo
20,am
30,m
40,ooo
50,ooO
Strain gradient (microstrain/mm) consb;ained scakddnded
buckling
tests: M. R. Wisnom
and J. W. Atkinson
volume effect. For pure compression the whole thickness is strained uniformly. However, where there is a strain gradient, the high strain is limited to the surface region, and the effective thickness of strained material is therefore less. An analysis was therefore carried out to confirm that these strained volume effects are not significant. This was done by assuming that failure follows Weibull statistical strength theory, and examining if this alternative hypothesis was able to explain satisfactorily the experimentally observed trends. Using a two parameter model based on strain, the probability of survival under a strain field E is given by:
Figure 5 Comparison of constrained buckling tests with scaled pin-ended buckling results
P(s) = exp[ -
The results from the present series of constrained buckling tests are all above the line from the scaled tests. However, the constrained specimens were from a 2 mm thick plate, whereas the scaled ones were all cut from a plate 8 mm thick. In the previous study specimens from 2 mm thick plates also produced higher strains at failure, possibly as a result of better fibre alignment in the thinner plate. These results are consistent with finite element modelling studies which show a strong effect of stress gradient on failure of flexural specimens of different thicknesse?. Failure of specimens with fibre waviness was shown to occur due to microbuckling initiating at the surface where the compressive stress is highest. As the stress gradient increases, the less highly loaded fibres just below the surface are able to provide greater support against microbuckling. A similar trend of increasing failure strain with increasing strain gradient through the thickness has now been demonstrated in two independent series of tests. In the present set the thickness was kept constant, and the ratio of compressive to bending stress was varied, whereas previously the thickness was changed whilst maintaining similar ratios of compression to bending. The fact that the trends obtained are similar in both cases supports the previous conclusion that the strain gradient through the thickness is the critical parameter responsible for the observed differences in compressive failure strains. Effect oj’stressed
volume
A possible alternative explanation for the size effect found in the previous scaled buckling tests is that it is a statistical effect due to differences in stressed or strained volume. The present tests were designed to minimise any volume effects by keeping the thickness and width constant, and varying the strain gradient by changing the constraints on specimens of the same length. However, two different length specimens have been used, and there are also differences in the effective strained length for the different rotation angles due to changes in the strain gradient along the length. A further point is that the different strain gradients through the thickness also indirectly produce a strained
(E/Q)” d\/]
(3)
where E(, is the characteristic strain and m is the Weibull modulus. The equivalent constant strain C producing the same probability of failure in a volume V,, as the actual variable strain distribution over the whole specimen can then be calculated by integration: E=
,’
s
Em dv]‘/”
V,
V. is a reference volume, which was arbitrarily taken as 1000 mm’. The strain distribution along the length of the specimens was estimated by separating the compressive and bending components, and curve fitting the compressive strains due to bending at the centre and 5 mm away as discussed in Section 3.2 for the 20 mm specimens with 1” of rotation. For the other specimens where there was only one strain gauge away from the centre, the strain distribution was assumed to be symmetric. The tensile strain due to bending was taken as being proportional to the compressive strain due to bending all along the length, consistent with the assumption in eqn (2) used to estimate the maximum value of tensile strain. The distribution of strain through the thickness was assumed to be linear. The values of equivalent strain were calculated from eqn (4) by numerical integration over the domain where strains were compressive. This was done for all four sets of tests for a reasonable range of assumed Weibull moduli. Figure 6 shows the results for values of 15 and 35 compared with the raw data with no correction. For reference, a Weibull modulus of 16.8 was calculated from the previous scaled tests by assuming that all the variation in strength with size was due to the change in volume’. If the differences in strength in the constrained buckling tests were due to changes in volume, then correcting them with the appropriate Weibull modulus should give a constant value of failure strain for all the different sets of data. However, it can be seen that the correction makes hardly any difference to the trend of increasing strain with increasing strain gradient. The effect of the differences in strained volume between the four series of tests is negligible, and the variation in compressive strains at failure can therefore be attributed to the different strain gradients through the thickness.
963
Constrained
buckling
tests: M. R. Wisnom
and J. W, Atkinson
CONCLUSIONS
0
5000
10000
15000
20000
Strain gradient (microstrain/mm) Figure 6 Correction moduli, m
for strained
volume
effect for different
The constrained buckling rig was successful in producing gauge length compressive failures at different ratios of compressive to bending stress. The results showed a 20% increase in compressive strain at failure with an increase of 85% in the strain gradient through the thickness. Calculations based on Weibull theory show that this cannot be explained in terms of a stressed volume effect. The results provide further evidence of the importance of the strain gradient in determining strains at failure under compressive loading.
Weibull
ACKNOWLEDGEMENTS
Effect of shear stresses One further complicating factor in the tests is the presence of interlaminar shear stresses due to the end loads and the rotation of the specimens. For the 40 mm specimens with the 20” rotation angle the shear stress is relatively low, and the strain is reasonably constant over the central portion of the specimen, as shown by the relatively small difference in strain between the gauges 5 mm apart. On the other hand for the 20 mm case with 1” rotation, the shear stress is much higher, and the variation of strain along the length is considerable. It has been assumed that failure occurs at the position of maximum compressive strain where the shear stress is zero. Moving along the specimen the compressive strain decreases, but the shear stress increases. It is therefore conceivable that failure actually initiates away from the centre due to interaction between compression and shear. However, in most cases one end of the angled failure surface was quite close to the centre of the specimen. Also the shear stresses are a maximum at the mid-plane where the compressive stresses are low, and are zero on the surface where the bending stresses are a maximum. It is therefore considered unlikely that there would be a significant interaction.
964
This work was funded by EPSRC/MOD under contract no. GR/H75017 in collaboration with Westland Helicopters and DERA Farnborough.
REFERENCES 1.
2.
3.
4.
5.
6.
7.
8.
Wisnom, M. R., Atkinson, J. W. and Jones, M. I., Reduction in compressive strain to failure with increasing specimen size in pinended buckling tests. Composites Science and Technology, in press. Wisnom, M. R., The effect of specimen size on flexural strength of unidirectional carbon fibre-epoxy. Composite Structures, 1991,18, 41-63. Lessard, L. B. and Chang, F. K., Effect of load distribution on the fiber buckling strength of unidirectional composites. Journal of Composite Materials, 1991, 25, 65-87. Gardin, C., Grandidier, J. C. and Potier-Ferry, M., Compressive behaviour of composite materials: a theoretical and experimental studv. In Proc. ICCh49. Vol. III. Madrid, Julv 1993, Woodbead Publishing, Cambridge, pp. 301-307. _ Wisnom, M. R., The effect of fibre waviness on the relationship between compressive and flexural strengths of unidirectional composites. Journal of Composite Materials, 1994, 28, 66-76. Swanson, S. R., Constraint effects in compression failure of fiber composites. In Proc. ICCMIO, Vol. I, Whistler, Canada, August 1995, Woodhead Publishing, Cambridge, pp. 739-746. Drapier, S., Gardin, C., Grandidier, J. C. and Potier-Ferry, M., Structure effect and microbuckling. Composites Science and Technology, 1996, 56, 861-867. Grandidier, J.C., Ferron, G. and Potier-Feny, M., Microbuckling and strength in long-fiber composites: theory and experiments. Znternational Journal of Solids and Structures, 1992, 29, 1753-1161.
PII:S1359-835X(97)00067-5
ELSEVIER
Constrained buckling tests show increasing compressive strain to failure with increasing strain gradient
M. R. Wisnom
and J. W. Atkinson
Department of Aerospace Engineering, Bristol, UK (Received 70 February 1997; accepted
University 12 June
of Bristol,
University
Walk, BS8 ITR
79971
A new buckling rig was developed where the ends of the specimen are initially free to rotate, but lock up after a certain amount of rotation. By varying the rotation angle allowed, the ratio between compression and bending can be altered, permitting tests with different strain gradients through the thickness. Four sets of tests were carried out with 16 ply unidirectional T800/924 carbon fibre-epoxy specimens of two different lengths, each at two different rotation angles. Consistent gauge length compressive failures were obtained in all cases, with the strain at failure increasing with increasing strain gradient. 0 1997 Elsevier Science Limited. (Keywords:
compressive
failure: flexural failure: strain gradient; buckling; size effects)
INTRODUCTION Compressive strength is a critical issue for carbon fibre composite structures, and is often cited as a factor limiting the exploitation of these materials. Flexural loading is important in many applications, and it has been shown that very high values of compressive strength can be achieved in bending. For example, tests on unidirectional T800/924 intermediate modulus carbon fibre-epoxy in a pin-ended buckling rig gave compressive strains at failure of up to 21600 microstrain’, of the order of double typical quoted values of compressive failure strain. Reconciling this discrepancy is important at a fundamental level in understanding the basic failure mechanism in these materials. But from a practical viewpoint it is also important in order to know under what conditions advantage can be taken of the higher strength under flexural loading to produce lighter, more efficient structures. The previous pin-ended buckling tests showed that there is a significant size effect for compressive failure of unidirectional carbon fibre-epoxy under flexural loading’. Scaled specimens of T800/924 showed an increase in compressive strain at failure of 47% for a factor of eight decrease in linear dimensions. A similar trend was also observed from compressive failures in scaled four point bending tests on XAS/913 carbon fibre-epoxy’. The results of the pin-ended tests suggested that the size effect was predominantly due to the change in strain gradient through the thickness rather than the different stressed volumes. A number of theoretical studies have shown that higher compressive stresses are required to cause microbuckling under a non-uniform stress field than under pure
compression3-‘. Higher flexural strengths than compressive strengths are predicted, with the difference increasing with decreasing thickness4m7. To investigate this effect further experimentally. a test was required which would allow the ratio between compressive and flexural loading to be varied whilst maintaining the thickness constant. This can be done with the pin-ended buckling rig by varying the length of the specimen, and hence the buckling load. However, this leads to significant changes in the stressed volume as well as the strain gradient, and so it is hard to separate the two effects. The objective of this study was to devise a test allowing the strain gradient to be varied independently of the specimen dimensions, and hence to verify whether the strain gradient is the critical factor responsible for the observed changes in compressive strain at failure. Very few experimental studies have been published on varying the strain gradient in compressive tests. One interesting approach is that of Grandidier et al.. who developed an original device to apply combined bending and compressive loadings. The specimen was fixed in two blocks which were loaded by a system of levers. By varying the geometry of the mechanism, different ratios of bending and compressive loading could be applied. Initial results with the rig reported so far look quite promising. In this study a different approach was taken based on a variation of the pin-ended buckling tests. A new rig was developed where the ends of the specimen are initially free to rotate, but lock up after a certain amount of rotation. By varying the rotation angle allowed, the ratio between compression and bending can be altered with the same specimen. With no rotation the rig would correspond
959
Constrained
buckling
tests: M. R. Whom
and J. W. Atkinson
to one with fixed end conditions. With no restriction of rotation it would effectively be a pin-ended buckling rig. Locking the rig at different angles allows variations between these two extremes. Since there is a factor of four difference in Euler buckling load between fixed and pinned end conditions, there is scope for considerable variation in the amount of direct compression with the same specimen dimensions. Tests were carried out at the maximum and minimum rotation angles which gave gauge length failures. To increase the range of compressive to bending stress ratios, two different lengths of specimen were used. Surface strains were measured with strain gauges, allowing the effect of the strain gradient on compressive strains at failure to be determined. Figure 1
Diagram of the rig
EXPERIMENTAL Constrained buckling rig Figure 1 shows the rig. It consists of two identical mild steel fixtures (1) which are clamped in the jaws of the test machine via spigots (2). Rotating arms (4) are attached to the fixed ends by means of silver steel bearing pins (3). The bearing journals were machined to close tolerance and the diameter was kept large (20 mm) in order to minimise bearing stresses and prevent deformation of the pins. Each end of the specimen (12) is clamped to a block (10) by means of a clamping plate (11) and four bolts (13). The blocks have slots through which they are attached to the rotating arms (4) by means of six bolts. This allows the position of the blocks to be varied, enabling the offset between the loading and specimen axes to be controlled. Rotation of each end of the specimen is restricted by means of a high tensile steel tie rod (5) which attaches via bearing pins to the fixed arm (6) and rotating arm (7). The maximum rotation allowed is varied by adjusting the lock nuts (9). Test specimens A 16 ply plate of unidirectional Ciba T800/924 carbon fibre-epoxy was laid up and cured according to the manufacturer’s instructions. Two different sizes of specimens were cut from the same plate using a diamond saw. The dimensions were 60 mm X 10 mm and 80 mm X 10 mm. The clamped section at each end is 20 mm long, so the gauge lengths were 20 and 40 mm. The thickness of the specimens was 2.02 mm. It is important that the ends of the specimens are square, especially when the direct compressive stresses are high, and the risk of failure initiating from the ends is greater. The ends were therefore wet ground on a surface table using emery paper, with an arrangement of V blocks to keep the specimens square to the surface. Strain gauges 1 mm long were bonded to the centre of both faces of the specimens using cyanoacrylate adhesive. Additional gauges were also fitted on one surface of the
960
specimen at 5 mm from the centre in order to measure the variation of strain along the length. For most of the specimens, only one extra gauge was fitted, but in some cases two additional gauges were fitted on the same surface, one towards each end of the specimen. The reasons for this will be explained later. Test procedure Great care was taken to align the rig and the test pieces accurately. The specimens were firmly clamped in place with a piece of thin steel shim on either side to protect the surfaces from the sharp comers at the ends of the clamping blocks. Thin pieces of PTFE sheet were placed between the shims and the specimen to provide further protection. An offset of 16 mm was used between the mid-plane of the specimen and the loading axis, large enough to provide sufficient bending to ensure smooth rotation of the rig. It was desired to test with the maximum range of angles before lock-up whilst ensuring failures in the gauge section. Failure could initiate at the ends due to stress concentrations at the clamps if the load was too high, or due to excessive shear stresses if the rotation angle was too large. Initial tests were therefore carried out, and a range of 1- 13’ for the 20 mm specimens and 5-20” for the 40 mm ones were found to produce consistent gauge length failures. It was decided to do two sets of tests on each length specimen at the extremes of these ranges of angles. Care was taken to ensure that the two ends of the rig locked up at the same point, to try to avoid any asymmetry in the tests. The four sets of specimens were loaded under displacement control at a rate to produce failure in about 90 s. Load and strain gauge readings were logged on a computer.
RESULTS 40 mm gauge length specimens Figure 2 shows typical load-strain
plots for the centre
Constrained
gauges with 5” and 20” maximum rotation angles. The initial curves are shallow as the specimens are effectively undergoing pin-ended buckling. During this portion of the test the response is the same for the two cases. With 5” of free rotation the rig locks up at a compressive strain of about 3000 microstrain. The response then suddenly becomes steeper, with the load increasing much more rapidly until catastrophic failure occurs. The specimens with 20” free rotation behaved very similarly, except that they locked up much later, when the compressive strains were already about 15000 microstrain. All specimens broke completely into two pieces, with the failure in the central region. The fracture surfaces were inclined at an angle of approximately 75” to the fibre direction, and had the smooth appearance typical of compressive failures. There was also some splitting along the length, and fibre tensile failure close to the tension surface. The results for the maximum strains at the centre of the specimens are summarised in Table 1. Mean values of the compressive strains at 5 mm from the centre of the specimens are also shown, giving an indication of the strain gradient along the length. In all cases the strains reduced going away from the centre, with the effect being
buckling
tests: M. R. Wisnom
more noticeable were higher.
for the 5” rotation
20 mm gauge length
-20000
-15000
-10000
-5000 Strain
Figure 2 specimens
Typical
load-strain
Table 1
Summary
of results
0
5000
10000
15000
20000
case where the loads
specimens
The specimens with 13” rotation behaved very similarly to the longer ones. However the ones where only 1” of rotation was allowed showed asymmetry in their response. These specimens therefore had two strain gauges attached at 5 mm from the centre, one towards each end, to allow the asymmetry to be monitored. It was found that in some cases the strain readings were actually higher at one of these gauges than at the centre plane of symmetry. Where this was the case the strains at the gauges on the other side of the centre were much lower. This anomaly had been noticed in preliminary tests, but despite taking great care in setting up and aligning the rig it was not possible to eliminate it. Some additional tests were done whereby the rig was adjusted whilst the specimen was under load until the strain gauge readings were symmetric. However, on further loading they would again diverge. The forces are quite high in this case, and as a result of the rotation, significant shear stresses arise. It is thought that shear deformations occurring to a greater extent at one end of the specimen than the other may be the cause of the asymmetry. As a result of this behaviour, the highest compressive strain may have occurred between the gauges. To get a better estimate of the maximum value, a curve was fitted through the three strain gauge readings, and the maximum obtained by differentiating the resulting function. A quadratic variation of strain along the length was assumed. the simplest possible to go exactly through three data points. For example, for one specimen showing considerable asymmetry, the fitted equation for strain as a function of distance, X. from the centre of the gauge length was: E=134x7-757x-
-25000
and J. W. Atkinson
14130
(1)
3 shows the fit graphically. In this case the maximum strain is -15 200 microstrain compared with - 14 130 at the centre and - 14 570 and -6990 at 5 mm on either side. Figure
(microstrain)
plots for the centre
gauges
of 40 mm
Strains at failure (microstrain)
Length (mm) 40 40 20 20 20
Rotation angle
At centre
At centre
5 mm away
Failure load
(dcg)
Compression
Tension
Compression
(W
No. of spec.
-19390 (7.5%) -17700 (4.8%) -17960 (3.7%) -15000 (9.8%) -15 810” (6.9%) -16 100b (5.2%)
15290 (6.0%) II 130 (8.7%) 12370 (7.2%) 1290 (73.7%)
-18640 (13.8%) -15310 (7.8%) -16290 (2.4%) -10680 (35.4%)
2.2 (17.1%) 7.6 (1.7%) 3.84 (15.6%) 18.6 (7.3%)
5
20 (C.V.) 5 (C.V.) 13 (C.V.) I (C.V.)
I (C.V.)
20
I (C.V.)
“Highest strain from three gauges along length ‘Estimated maximum strains based on fit from three measured
6 5 6 6
2530h (60.9%)
~11970b
6
values
961
Constrained
buckling
tests: M. I?. Wisnom
and J. W. Atkinson
-2eew
Figure 3 Curve fitted to the three strain readings, obtained at the centre and 5 mm from the centre of 20 mm specimen with 1” rotation
the deflections and offset loading. It was assumed that the tensile strain due to bending varied in the same manner along the length as the compressive strain due to bending, whilst the direct strain remained constant along the length. The direct compressive strain was estimated from the load at failure assuming a modulus of 160 GPa. This strain was then subtracted from the measured strains at the centre of the specimen to give the bending components of the compressive and tensile surface strain E&, and E&. The direct strain was also subtracted from the previously calculated maximum strain to give the bending component of the maximum compressive strain, E&,,. The corresponding tensile strain due to bending was determined by multiplying by the ratio of tensile to compressive strain at the centre of the specimen: t Emax = &x
Maximum compressive strain (microstrain) 25,om 2Qmm13° 20,m
01
/
0
5,ow
-
I
10,m
&I
J &I
(2)
The direct compressive strain was then added back in to give the total strain on the tensile surface at the position of maximum compressive strain. This procedure would be expected to give very good results if the stress-strain response was linear elastic. In practice non-linearity may introduce a small degree of inaccuracy, but a reasonable estimate of the maximum tensile strain should still be obtained.
DISCUSSION
15,Klo
Strain gradient (microstrain/mm) Figure 4 The maximum compressive strains at failure from all four sets of tests, plotted as a function of the strain gradient through the thickness
Table I shows the maximum strains on both surfaces at the centre of the gauge length and the value on the compression side 5 mm away. For the 20 mm specimens with 1” rotation there were gauges at 5 mm from the centre towards both ends, and the value shown is the average of the two readings. The highest strain was not necessarily at the centre gauge, and so the maximum measured strain taken from the highest reading of the three gauges on the compressive surface is shown as well. Finally for this case the maximum based on the curve fits to the three strain gauge readings is given. The difference between the latter two values is only 1.9%, showing that the assumptions involved in the correction procedure are not that critical in determining the maximum strain. The correction improves the consistency of the results, as seen from the lower coefficient of variation for the corrected strains than for the raw measurements. The estimated compressive strain at 5 mm away from the point of highest strain from a best fit to all the data is also shown. The estimated value of the maximum tensile strain is included in the table, although this involved further assumptions since there was only one strain gauge on the tension side at the centre of the specimen. The procedure adopted involved separating out the direct compressive strain due to the end load, and the bending strain due to
Effect of strain gradient Figure 4 shows the maximum compressive strains at failure from all four sets of tests plotted as a function of the strain gradient through the thickness, defined as the difference between the compressive and tensile surface strains divided by the measured specimen thickness. For the 20 mm case with 1” of rotation, the estimated maximum strain was used. The error bars correspond to one standard deviation on each side of the mean. The results show a consistent trend of increasing strain at failure with increasing strain gradient, and a least squares line through the data gives a very good fit. The increase in compressive strain between the 20 mm long specimens with 1” rotation and the 40 mm ones with 20” rotation is 20%. The corresponding increase in strain gradient through the thickness between these two cases is 85%. In previous buckling tests a significant reduction in strain at failure was found with increasing size for scaled specimens from the same batch of T800/924 carbon fibreepoxy as used in this study’. The results suggested that the effect was predominantly due to the difference in strain gradient through the thickness. Figure 5 shows the data for the three different sized specimens together with the best fit straight line. The results from the constrained buckling tests are also shown, and are close to the line for the scaled specimens. The strain gradient effect appears to be slightly greater in the constrained tests, but given the scatter in the data, and the narrower range of strain gradients, the comparison is reasonable.
Constrained
Maximum compressive strain (microstrain) 25,LxQ
20.m
15.m
10,000
5.ooo
OL 0
10,ooo
20,am
30,m
40,ooo
50,ooO
Strain gradient (microstrain/mm) consb;ained scakddnded
buckling
tests: M. R. Wisnom
and J. W. Atkinson
volume effect. For pure compression the whole thickness is strained uniformly. However, where there is a strain gradient, the high strain is limited to the surface region, and the effective thickness of strained material is therefore less. An analysis was therefore carried out to confirm that these strained volume effects are not significant. This was done by assuming that failure follows Weibull statistical strength theory, and examining if this alternative hypothesis was able to explain satisfactorily the experimentally observed trends. Using a two parameter model based on strain, the probability of survival under a strain field E is given by:
Figure 5 Comparison of constrained buckling tests with scaled pin-ended buckling results
P(s) = exp[ -
The results from the present series of constrained buckling tests are all above the line from the scaled tests. However, the constrained specimens were from a 2 mm thick plate, whereas the scaled ones were all cut from a plate 8 mm thick. In the previous study specimens from 2 mm thick plates also produced higher strains at failure, possibly as a result of better fibre alignment in the thinner plate. These results are consistent with finite element modelling studies which show a strong effect of stress gradient on failure of flexural specimens of different thicknesse?. Failure of specimens with fibre waviness was shown to occur due to microbuckling initiating at the surface where the compressive stress is highest. As the stress gradient increases, the less highly loaded fibres just below the surface are able to provide greater support against microbuckling. A similar trend of increasing failure strain with increasing strain gradient through the thickness has now been demonstrated in two independent series of tests. In the present set the thickness was kept constant, and the ratio of compressive to bending stress was varied, whereas previously the thickness was changed whilst maintaining similar ratios of compression to bending. The fact that the trends obtained are similar in both cases supports the previous conclusion that the strain gradient through the thickness is the critical parameter responsible for the observed differences in compressive failure strains. Effect oj’stressed
volume
A possible alternative explanation for the size effect found in the previous scaled buckling tests is that it is a statistical effect due to differences in stressed or strained volume. The present tests were designed to minimise any volume effects by keeping the thickness and width constant, and varying the strain gradient by changing the constraints on specimens of the same length. However, two different length specimens have been used, and there are also differences in the effective strained length for the different rotation angles due to changes in the strain gradient along the length. A further point is that the different strain gradients through the thickness also indirectly produce a strained
(E/Q)” d\/]
(3)
where E(, is the characteristic strain and m is the Weibull modulus. The equivalent constant strain C producing the same probability of failure in a volume V,, as the actual variable strain distribution over the whole specimen can then be calculated by integration: E=
,’
s
Em dv]‘/”
V,
V. is a reference volume, which was arbitrarily taken as 1000 mm’. The strain distribution along the length of the specimens was estimated by separating the compressive and bending components, and curve fitting the compressive strains due to bending at the centre and 5 mm away as discussed in Section 3.2 for the 20 mm specimens with 1” of rotation. For the other specimens where there was only one strain gauge away from the centre, the strain distribution was assumed to be symmetric. The tensile strain due to bending was taken as being proportional to the compressive strain due to bending all along the length, consistent with the assumption in eqn (2) used to estimate the maximum value of tensile strain. The distribution of strain through the thickness was assumed to be linear. The values of equivalent strain were calculated from eqn (4) by numerical integration over the domain where strains were compressive. This was done for all four sets of tests for a reasonable range of assumed Weibull moduli. Figure 6 shows the results for values of 15 and 35 compared with the raw data with no correction. For reference, a Weibull modulus of 16.8 was calculated from the previous scaled tests by assuming that all the variation in strength with size was due to the change in volume’. If the differences in strength in the constrained buckling tests were due to changes in volume, then correcting them with the appropriate Weibull modulus should give a constant value of failure strain for all the different sets of data. However, it can be seen that the correction makes hardly any difference to the trend of increasing strain with increasing strain gradient. The effect of the differences in strained volume between the four series of tests is negligible, and the variation in compressive strains at failure can therefore be attributed to the different strain gradients through the thickness.
963
Constrained
buckling
tests: M. R. Wisnom
and J. W, Atkinson
CONCLUSIONS
0
5000
10000
15000
20000
Strain gradient (microstrain/mm) Figure 6 Correction moduli, m
for strained
volume
effect for different
The constrained buckling rig was successful in producing gauge length compressive failures at different ratios of compressive to bending stress. The results showed a 20% increase in compressive strain at failure with an increase of 85% in the strain gradient through the thickness. Calculations based on Weibull theory show that this cannot be explained in terms of a stressed volume effect. The results provide further evidence of the importance of the strain gradient in determining strains at failure under compressive loading.
Weibull
ACKNOWLEDGEMENTS
Effect of shear stresses One further complicating factor in the tests is the presence of interlaminar shear stresses due to the end loads and the rotation of the specimens. For the 40 mm specimens with the 20” rotation angle the shear stress is relatively low, and the strain is reasonably constant over the central portion of the specimen, as shown by the relatively small difference in strain between the gauges 5 mm apart. On the other hand for the 20 mm case with 1” rotation, the shear stress is much higher, and the variation of strain along the length is considerable. It has been assumed that failure occurs at the position of maximum compressive strain where the shear stress is zero. Moving along the specimen the compressive strain decreases, but the shear stress increases. It is therefore conceivable that failure actually initiates away from the centre due to interaction between compression and shear. However, in most cases one end of the angled failure surface was quite close to the centre of the specimen. Also the shear stresses are a maximum at the mid-plane where the compressive stresses are low, and are zero on the surface where the bending stresses are a maximum. It is therefore considered unlikely that there would be a significant interaction.
964
This work was funded by EPSRC/MOD under contract no. GR/H75017 in collaboration with Westland Helicopters and DERA Farnborough.
REFERENCES 1.
2.
3.
4.
5.
6.
7.
8.
Wisnom, M. R., Atkinson, J. W. and Jones, M. I., Reduction in compressive strain to failure with increasing specimen size in pinended buckling tests. Composites Science and Technology, in press. Wisnom, M. R., The effect of specimen size on flexural strength of unidirectional carbon fibre-epoxy. Composite Structures, 1991,18, 41-63. Lessard, L. B. and Chang, F. K., Effect of load distribution on the fiber buckling strength of unidirectional composites. Journal of Composite Materials, 1991, 25, 65-87. Gardin, C., Grandidier, J. C. and Potier-Ferry, M., Compressive behaviour of composite materials: a theoretical and experimental studv. In Proc. ICCh49. Vol. III. Madrid, Julv 1993, Woodbead Publishing, Cambridge, pp. 301-307. _ Wisnom, M. R., The effect of fibre waviness on the relationship between compressive and flexural strengths of unidirectional composites. Journal of Composite Materials, 1994, 28, 66-76. Swanson, S. R., Constraint effects in compression failure of fiber composites. In Proc. ICCMIO, Vol. I, Whistler, Canada, August 1995, Woodhead Publishing, Cambridge, pp. 739-746. Drapier, S., Gardin, C., Grandidier, J. C. and Potier-Ferry, M., Structure effect and microbuckling. Composites Science and Technology, 1996, 56, 861-867. Grandidier, J.C., Ferron, G. and Potier-Feny, M., Microbuckling and strength in long-fiber composites: theory and experiments. Znternational Journal of Solids and Structures, 1992, 29, 1753-1161.