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The effect of delamination geometry on the compressive failure of composite laminates
Composites Science and Technology 61 (2001) 2075–2086 www.elsevier.com/locate/compscitech
The effect of delamination geometry on the compressive failure of composite laminates G.J. Short, F.J. Guild, M.J. Pavier* Department of Mechanical Engineering, University of Bristol, Queen’s Building, University Walk, Bristol BS8 1TR, UK Received 19 April 2001; received in revised form 24 July 2001; accepted 26 July 2001
Abstract One of the causes of a reduction in the compressive strength of a composite material containing delaminations is the buckling of the delaminated plies. To understand the effect of delamination geometry on the compressive behaviour of laminated composite materials, compression tests were carried out on glass-fibre-reinforced plastic (GRP) test specimens containing artificial delaminations of various geometry, created by inserting PTFE film into the laminate during lay-up. Finite-element modelling was also carried out to gain further understanding of the mechanisms of compressive failure. Good agreement between finite-element predictions and experimental measurements were found for the whole range of delamination geometries that were tested. Finiteelement and simple closed-form models were also developed for delaminated panels with isotropic properties. This enabled a study of the effect of delamination geometry on compressive failure without the complicating effects of orthotropic material properties. The results of this study can be used to derive a graph of non-dimensional failure load versus non-dimensional failure stress, where the results for any one delamination geometry superimpose on those for all others. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: A. Glass fibres; C. Delamination; C., Finite-element analysis; C. Buckling; Compressive strength
1. Introduction Delaminations in layered composite materials are areas of de-bonding between adjacent plies and may result from manufacturing imperfections or from low velocity impact damage whilst in service. They can restrict the performance advantage of composite materials by causing a considerable reduction in compressive strength [1]. Previous work has shown that the mechanism of compressive strength reduction is out-of-plane buckling of the groups of plies above and below the delamination [2–4]. These groups of plies are referred to here as sublaminates; when buckling of any sub-laminate occurs the remaining plies become subject to bending due to unsymmetric loading as well as in-plane compression. The stresses in the remaining plies are, therefore, greater than would exist in an undelaminated structure, resulting in a reduced failure load. The magnitude of this strength reduction can be linked to the initial buckling * Corresponding author. Tel.: +44-117-928-8211; fax: +44-117929-4423. E-mail address: [email protected] (M.J. Pavier).
of the panel, which can be categorised into two characteristic modes of behaviour dependent on delamination geometry [3,5,6]. The first mode, referred to as the local mode, occurs when the upper sub-laminate is thin and the area of delamination large. In this case the stiffuess of the upper sub-laminate is lower relative to that of the lower sub-laminate and the upper sub-laminate buckles out of plane forming a blister on the panel surface. The second mode, referred to as the global mode, is observed when the delamination has a small area and is deeper within the panel. This mode is characterised by both sub-laminates buckling in the same direction and with the same magnitude of out of plane displacement. Fig. 1 shows these buckling modes diagrammatically. Many experimental studies of the effect of delamination on the compressive behaviour of panels have been carried out using post impact compression strength tests [7,8]. This type of test in principally concerned with measuring the effect of impact energy on the residual compressive strength of a coupon. Its use in understanding the effect of delamination geometry on the compressive strength of a panel is less clear as multiple delaminations of complex shape and often unknown position are formed through the thickness of the mate-
0266-3538/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(01)00134-8
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rial. Experimental work has also been conducted using compression test specimens with embedded artificial delaminations [9,10]. Here, a single or multiple delaminations of known shape and position are implanted into the specimen, allowing the effect of delamination geometry on the compressive behaviour and residual strength of the panel to be studied in a controlled manner. The use of finite-element modelling can provide further understanding of the mechanisms of failure. Panel failure is often a sudden event with little or no propagation of damage prior to failure. Studying the mechanisms involved from experimental data alone is therefore difficult. With artificial delaminations, definition of the delamination geometry is simple and finiteelement simulations are relatively straightforward to undertake. In the simplest case of single or multiple delaminations across the width of a specimen, twodimensional analysis allows comparison with experimental data [11–13]. For embedded delaminations, three-dimensional modelling is necessary to allow the shape and size of the delamination to be represented accurately [3,4,14–16]. In general, finite-element analyses of the effect of delaminations on compressive strength show good agreement with experimental results and allow the mechanisms of failure to be investigated
and understood more easily than would be possible from experimental data alone. This paper presents the results of an investigation into the effect of size and through thickness position of single embedded delaminations on the sub-laminate response and compressive failure load of GRP specimens. An anti-buckling guide was used during testing and the laminate was instrumented with LVDTs to measure the out of plane responses of the upper and lower sublaminates to compressive loads. Three dimensional finite-element analysis was also carried out in order to compare predicted failure loads with those measured experimentally. In addition to the experimental and finite-element work on layered GRP laminates, analytical and finite-element models of delaminated isotropic panels were developed to study the effect of delamination geometry, without the complications of material orthotropy. These models enable the buckling mode to be predicted depending on the delamination size and through thickness position. The models also provide an estimation of the compressive failure load of a panel, provided the failure stress of the materials is known as a proportion of the critical buckling stress of the panel. The isotropic models provide acceptable indications of the experimental behaviour of the GRP laminates.
Fig. 1. Local and global modes of delamination induced buckling.
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2. Experimental work 2.1. Specimen preparation The experimental tests were carried out on specimens containing artificially created delaminations. The laminates were manufactured from ICI Fiberite 934 unidirectional glass fibre pre-preg. The lay-up for all specimens tested was [0,+45, 45, 0]s producing specimens with a typical thickness of 2.4 mm. All specimens were cut from panels produced by hand lay-up and cured using the vacuum bag method with the manufacturers recommended cure cycle. Delaminations were formed using a single thickness of 10 mm PTFE film laid between plies [2]. Test specimens of 200 mm long by 50 mm wide were then dry cut from these plates using a diamond encrusted edge slitting saw. After trimming, aluminium end tabs of dimensions 50 mm 50 mm by 2 mm thick were etched for 30 min in chromic sulphuric acid at 60 C and then bonded to the coupons with 3M 9323 two-part structural adhesive. Once cured, the ends of the specimens were machined parallel to minimise uneven loading during testing. The geometry of the specimens is shown in Fig. 2. Specimens were manufactured with different sizes and through thickness positions of artificial delamination. In this work, three sizes of delamination were used: 10 mm 10 mm, 15 mm 15 mm and 25 mm 25 mm. Fig. 2 shows for example a specimen with a 25 mm 25 mm delamination. In all cases the delamination area was square, as this offered a good compromise between ease of damage production and realistic representation of inservice delamination damage [2]. In addition, three different through-thickness positions were used: position ‘A’ between the first and second, 0 and +45 plies, position ‘C’ between the third and fourth, 45 and 0 plies and position ‘D’ between the two zero plies at the centre line of the laminate (Fig. 3). 2.2. Experimental method During testing, an anti-buckling guide [2] was used to prevent gross out of plane deformation of the test specimens. The stiffness of the guide in bending was much larger than that of the specimens. The guide was designed so that both faces of the panel could be viewed during testing. Brackets were mounted to the guide to allow the deflections of both surfaces of the specimen to be measured using LVDTs. The guide incorporated a 40 mm 40 mm window to allow out of plane displacement of the specimen in the central region near the delamination, as shown in Fig. 2. The window size was kept the same for all specimens. The tests were conducted on a Zwick 1478 test machine with a cross-head speed of 2 mm/min so as to achieve failure within 60–90 seconds. Panel failure was
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assumed to have occurred when the load dropped by 50%, at which point the test stopped automatically. Failure load and out of plane deflections of the panel faces were recorded continuously during each test. For each delamination geometry, three identical specimens were tested. For undamaged specimens, that is specimens without artificial delaminations present, six specimens were tested. However, of these six, only three failed in the centre of the specimen and were taken to be good test results. 2.3. Experimental results Failure load results are shown in Table 1 for each geometry of delamination. The results for specimens of the same geometry are consistent; typically results are within 5% of the average failure load. Failure loads decrease as the delamination size increases, and decrease as the delamination through thickness position increases. The greatest loss of strength was for specimens containing a 25 mm square delamination in the ‘D’ position, that is in the centre of the laminate. For these specimens the average failure load was 31% less than that for undamaged specimens. The average failure loads versus delamination through thickness positions are plotted later when compared with finite-element predictions. For all specimens, no growth of the delamination prior to failure was noticed. For specimens with delaminations in the ‘A’ position, some splitting was observed on the topmost, 0 ply near the edge of the delamination, parallel to the 0 fibre direction. No damage growth of any kind was seen in specimens with the delamination in positions ‘C’ or ‘D’. For specimens containing delamination, buckling of the sub-laminates occurred before failure. Depending on the size and through thickness position of the delamination, different modes of delamination buckling were observed. These buckling modes are termed local and global following Seikine et al. [3], as described previously. Fig. 4 shows the LVDT results for two specimens: one an example of a local buckling mode, the other an example of a global buckling mode. In Fig. 4 the designations ‘Top’ and ‘Bottom’ refer to the sub-laminates above and below the delamination, as marked in Fig. 3. Local buckling occurred for the specimen containing a 25 mm square delamination in the ‘A’ position, designated the 25A specimen. For this specimen the sublaminate above the delamination buckled out of plane at a load of about 19 kN. As load was increased above this level the out of plane deflection increased for both the sub-laminates but in opposite directions. A global mode occurred for the specimen containing a 15 mm square delamination in the ‘C’ position, designated the 15C specimen. Both sub-laminates deflect out of plane
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in the same direction but there is no sudden increase in the magnitude of the deflection as for the 25A specimen. All specimens containing delaminations in the ‘A’ position gave a local buckling mode while all those containing delaminations in the ‘C’ and ‘D’ positions gave a global mode. When a local mode occurred, buckling took place at a load of between 10 and 20 kN. This sudden buckling event is believed to be due to residual adhesion of the sub-laminates and that without such residual adhesion, out of plane displacement of the sub-laminates would occur at a significantly lower load.
3. Finite-element analysis 3.1. Finite-element models Previous work [16] has indicated that the failure of a delaminated specimen may be predicted by the load at which the maximum compressive fibre direction stress reaches a critical value. Such a prediction ignores damage growth mechanisms and any dependence of the critical fibre direction stress on laminate geometry and loading. Nevertheless, comparisons with experimental
Fig. 2. Geometry of a composite specimen with a 25 mm square delamination, showing position of delamination and size of window in anti-buckling guide.
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Fig. 3. Lay-up of composite specimen and through-thickness positions of delamination.
Table 1 Failure load results Delamination geometry
Failure loads (kN)
Size
Depth
Specimen 1
Specimen 2
Specimen 3
Average
Undamaged 25 mm square 20 mm square 25 mm square 15 mm square 15 mm square 10 mm square
– ‘A’ ‘C’ ‘D’ ‘A’ ‘C’ ‘A’
42.9 36.6 30.5 29.4 38.8 36.5 39.1
45.8 37.8 33.7 30.0 41.1 40.2 43.2
45.6 38.6 31.2 33.7 40.9 36.9 39.8
44.8 37.7 31.8 31.0 40.3 37.9 40.7
results have shown the critical stress model gives acceptable results. Prediction of maximum fibre direction stresses in a post-buckled orthortropic laminate inevitably demand finite-element analysis and in this work the ABAQUS V5.8 finite-element system was used with PATRAN V8.5 for pre-processing. To reduce the size of the finiteelement model, only part of each specimen was modelled: the 50 mm by 50 mm central part of the specimen containing the delamination as shown in Fig. 2. The finite-element model was constructed from two layers of 20 noded orthotropic brick elements, with a planar edge length of 2.5 mm, to represent the thickness of plies above and below the plane of the delamination. Each layer was given a small initial out of plane displacement of 0.01 mm at the delamination centre to initiate buckling during the analysis. The two layers of elements were joined at the interface, except for the delamination area where contact conditions were applied to prevent physically inadmissible inter-penetration of the elements during analysis. A typical finite-element mesh is shown in Fig. 5.
Fig. 4. Out of plane displacements measured at the specimen centre on the top and bottom surfaces for an undamaged specimen and specimens containing a 25 mm square delamination in the ‘A’ position and a 15 mm square delamination in the ‘C’ position.
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Fig. 5. Geometry of the finite-element mesh. Table 2 Material properties of B glass/epoxy Property E11 E22 E33 V23 V31 V12 G23 G31 G12
46.0 GPa 13.0 GPa 13.0 GPa 0.42 0.3 0.3 4.6 GPa 5.0 GPa 5.0 GPa
Layered material properties were applied to the elements according to the unidirectional linear elastic material properties shown in Table 2, rotated accordingly for the off-axis plies. Models were constrained to represent the boundary conditions provided by the antibuckling guide with the unloaded edges of the model constrained in the out of plane direction over a 5 mm wide strip along each edge and the loaded edges simply supported. Compressive load was applied to the model as a pressure over the two faces normal to the loading direction, as shown in Fig. 5. A static non-linear analysis was then conducted, including the effects of large deflections, to predict post buckling behaviour of the specimens. 3.2. Finite-element results Fig. 6 shows the out of plane displacement results at the centre of the upper and lower sub-laminates compared with the LVDT results from experiment. Fig. 6(a) shows typical results for a geometry giving a local mode
of buckling: a specimen containing a 25 mm square delamination in the ‘A’ position. Agreement is generally good once sub-laminate separation occurs. The finiteelement model does not predict the behaviour of the panel before this point since it takes no account of the residual adhesion between the sub-laminates existing in the experiments. Fig. 6(b) shows results for a 15 mm square delamination in the ‘C’ position, a geometry resulting in a global mode of buckling. Agreement is excellent. Predictions of the specimen failure loads are made by assuming the maximum fibre direction stress reaches a limiting value at failure. In this work, a value for the limiting fibre direction compressive stress of the composite was derived by measurement. Five uni-directional test specimens were manufactured and tested according to the CRAG specification [17]. The specimens were 110 mm long by 10 mm wide and 2 mm thick with 50 mm 10 mm aluminium end tabs bonded to the specimen so as to leave a gauge length of 10 mm. Testing was conducted using an Instron 8501 servo-hydraulic machine with hydraulic grips. The compressive fibre direction strengths for the five specimens varied from 665 to 686 MPa and gave an average of 674 MPa. For all models, the stress distribution in the laminate was examined to determine the maximum value of fibre direction stress in the most heavily loaded ply. The position of the maximum value of stress was typically near the centre of the specimen, either on the upper or lower most zero degree ply, depending on the direction of the out of plane deflection. Fig. 7(a) shows predicted contours of fibre direction stress for the most heavily loaded zero degree ply for the case of a 25 mm square delamination in the ‘A’ position where a local mode of buckling occurs. The maximum stress occurs in two
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bands between the centre and edge of the delamination. Fig. 7(b) shows a similar contour plot for the case of a 25 mm square delamination in the ‘D’ position. A global mode of buckling occurs and the maximum stress shifts to the centre of the delamination. In Fig. 7(a) and (b) the load applied to the model is equivalent to the experimental failure load. The maximum stress in the model is similar to that measured in the compressive strength tests, 674 MPa. Fig. 8 shows the predicted maximum compressive fibre direction stress for the most heavily loaded ply versus the applied load for 25 mm square delaminations in the ‘A’, ‘C’ and ‘D’ positions. The failure load pre-
Fig. 6. Comparison of experimental measurements and finite-element predictions of central out of plane displacements for: (a) a specimen containing a 25 mm square delamination in the ‘A’ position and (b) a specimen containing a 15 mm square delamination in the ‘C’ position.
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dicted by the model is obtained by the point at which the curves of compressive stress cross the line representing the measured compressive strength for the material. Fig. 8 suggests a decreasing specimen strength with increasing delamination depth. Fig. 9 compares the finite-element predictions of failure load with the experimental measurements. Results for 15 and 25 mm square delaminations are plotted versus through thickness delamination position. Agreement is generally good, although the finite-element models consistently under-predict the experimental results. It is notable that the failure load of the undamaged specimen is about 5% less than the predicted failure load. This implies the measured compressive strength obtained in the unidirectional tests cannot be achieved in a multi-directional laminate. Clearly, using a reduced compressive strength derived from the undamaged tests would give
Fig. 7. Finite-element prediction of fibre direction compressive stress in the most heavily stressed 0 ply at a load corresponding to the experimental failure load for: (a) a specimen containing a 25 mm square delamination in the ‘A’ position and (b) a specimen containing a 25 mm square delamination in the ‘D’ position.
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better agreement with experimental failure loads for specimens containing delaminations.
4. Isotropic modelling 4.1. Methodology The experimental and finite-element work described above has related specifically to a multidirectional laminate laid up with 8 plies of GRP pre-preg. It was
considered of interest to investigate in a more general manner the effects of delamination geometry on compressive behaviour. Therefore a programme of analytical and finite-element modelling was carried out using isotropic material properties. This modelling work considered a square delaminated plate, simply-supported around its edges and subjected to compressive loading in one direction. The geometry of the plate is shown in Fig. 10. The delamination geometry is defined by the ratio of delamination size to plate size a/b and the ratio of delamination depth to plate thickness t/T. 4.2. Prediction of buckling mode The first study carried out using the isotropic model was involved with the prediction of buckling mode. The delaminated plate geometry is considered to be formed of two laminates, one to represent the sub-laminate above the delamination, the other to represent the remainder of the plate. The model uses the analytical expression [18] which allows the critical buckling stress scr for a rectangular plate to be evaluated for different boundary conditions. 2 2 E t cr ¼ K ð1Þ 12ð1 v2 Þ b2
Fig. 8. Finite-element predictions of maximum fibre direction compressive stress versus applied compressive load for specimens containing 25 mm square delaminations in the ‘A’, ‘C’ and ‘D’ positions.
Fig. 9. Comparison of experimental and finite-element predicted failure loads versus through thickness delamination position for 25 mm square and 15 mm square delaminations.
Fig. 10. Delamination geometry for isotopic models.
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where t is the plate thickness and b the plate width. The non-dimensional factor K depends on the ratio of the length to the width of the plate and the boundary conditions. If the critical stress sTcr for the sub-laminate above the delamination is much less than the critical stress sBcr for the remainder of the plate, local buckling occurs. Conversely, if sBcr is much less than sTcr, global buckling T B takes place. Setting cr ¼ cr provides an indication of where the boundary between local and global buckling modes exists. This boundary is therefore calculated by KT
t2 ðt TÞ2 ¼ KB 2 a b2
ð2Þ
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element mesh used 20 node three-dimensional elements as before (Fig. 5). A value for Young’s modulus of E=70 GPa and Poisson’s ratio v=0.3 was used, while the plate dimensions were b=60 mm and T=3 mm. The buckling mode was assessed using a non-linear analysis with a small initial out of plane displacement as before to initiate buckling. The finite-element results are superimposed on Fig. 11 and show an acceptable level of agreement with the analytical predictions of buckling mode. In addition, Fig. 11 shows the experimental results for the GRP specimens. Despite the difference in material properties, it can be seen the isotropic analytical model is able to predict the experimentally observed buckling modes.
The boundary conditions for the remainder of the plate are simply-supported around the edge. For the sub-laminate above the delamination the boundary conditions lie somewhere between simply-supported and built-in. In Fig. 11, the expression given by Eq. (2) is plotted for these two boundary conditions as delamination size a/b versus through thickness position t/T. Fig. 11 is considered a buckling mode map, allowing the buckling mode to be predicted for any combination of delamination size and through thickness position. The map shows that local buckling is predicted for large delamination size and small through thickness position while global buckling is predicted for small delaminations with a large through thickness position. Fig. 11 also shows the results of isotropic finite-element models to predict buckling mode for different delamination geometries. Three different through thickness positions were used: t/T=0.25, 0.33 and =0.5, and three different sizes: a/b=0.25, 0.5 and 0.75. The finite-
4.3. Prediction of strength
Fig. 11. Delamination induced buckling mode map for varying delamination size and through thickness position.
Fig. 12. Maximum compressive stress versus applied compressive load for the case where t/T=0.5 and a/b=0.75.
The isotropic finite-element models used to produce the results in Fig. 11 were also used to predict maximum stress versus applied load in the same way as for the layered orthotropic finite-element models. Fig. 12 shows the maximum compressive stress in the loading direction versus the applied load for an isotropic finite-element model with a delamination geometry given by t/T=0.5 and a/b=0.75. Two curves are plotted, one for the maximum stress in the sub-laminate above the delamination, the other in the remainder of the plate, referred to as the bottom sub-laminate in the figure. If failure is predicted as before, by when the maximum stress reaches a material limit, Fig. 12 provides a method to find the failure load for any level of compressive failure stress. For a failure stress of 1000 MPa for example, a failure load of about 80 kN is predicted with failure initiating in the bottom sub-laminate.
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The maximum stress versus failure load results may now be used to provide a prediction of the failure load depending on the compressive failure stress for the material. Fig. 13 shows such results for a delamination through thickness position given by t/T=0.5 and three different delamination sizes: a/b=0.25, 0.5 and 0.75. When the compressive failure stress is low enough, buckling does not occur and the predicted failure load depends linearly on the failure stress with no effect of delamination geometry. Once the failure stress is high enough, buckling effects become important and delamination geometry influences the failure load. For a failure stress above about 500 MPa, Fig. 13 shows an increase in the failure load with reducing delamination size.
out of plane displacement in the lower sub-laminate begins to increase. Although it is straightforward to infer two buckling loads from Fig. 14(a), other delamination geometries resulting in a local buckling mode give out of plane displacement results that are more difficult to interpret. Fig. 14(b) shows results for a geometry giving global buckling, when t/T=0.5 and a/b=0.75.For this geometry a single buckling load of 64.1 kN is observed. Using the value of the critical buckling load the failure load versus failure stress graph of Fig. 13 is re-plotted in normalised form as in Fig. 15, where curves have been added for each geometry of delamination. The
4.4. Normalisation of strength predictions It has been found worthwhile to normalise the results of Fig. 13. The failure stress is normalised by dividing by the stress at which buckling occurs. The failure load is normalised by dividing by the load at which buckling occurs. This normalisation requires a value for the buckling load Pcr which was obtained from the out of plane displacement versus load results from the isotropic finite-element models. Fig. 14(a) shows such results for a geometry where local buckling occurs: t/T=0.33 and a/b=0.75.The local buckling load is 36.0 kN, the load when the out of plane displacement of the sub-laminate begins to increase. However, since the stress in the lower sub-laminate is the stress influencing failure, the buckling load used to normalise the failure load is taken to be 75.1 kN. This is the load at which the
Fig. 13. Compressive failure stress versus failure load for the cases where t/T=0.5 and a/b=0.25, 0.5 and 0.75.
Fig. 14. Central out of plane displacements on the top and bottom surface versus compressive load for the cases where: (a) t/T=0.33 and a/b=0.75 and (b) t/T=0.5 and a/b=0.75.
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failure stress is normalised using the critical buckling stress, calculated from the buckling load by cr ¼
Pcr bT
ð3Þ
Although the curves for different delamination geometries do not lie perfectly over each other, they do form a narrow band which could be used as a failure criterion for a delaminated plate. When the normalised failure stress is below 1, the failure load versus failure stress curve is linear. For normalised failure stresses above 1, buckling occurs before failure and the rate of increase of failure load with failure stress reduces.
Fig. 15. Normalised compressive failure stress versus normalised failure load.
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Fig. 15 also shows an attempt at superimposing the experimental GRP results. It is very much harder to measure an accurate buckling load from the LVDT results such as those in Fig. 6(a) and (b). Nevertheless, the normalised experimental results do support the principle that failure loads may be predicted using a master failure criterion based on normalised parameters. 4.5. Occurrence of delamination growth An interesting observation is that delamination growth typically occurs before failure for delaminated CFRP specimens of similar geometry to the GRP specimens used here [9]. It is possible to show this is expected, by comparing the load to cause delamination growth with the load to cause failure for a near surface, across-the width delamination in an isotropic plate. Bolotin [5] presents an analytical model to predict delamination growth for a near surface delamination. For the geometry shown in Fig. 16, the load Pg applied to the plate to cause delamination growth is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Gc E 1 v2 ð4Þ Pg ¼ Pd þ bT t where Pd is the load applied to the plate to cause the sub-laminate above the delamination to buckle and Gc is the inter-laminar fracture toughness. If the load Pd is negligible as it usually is, the analysis suggests that as the material properties change the applied to cause ffi pffiffiffiffiffiffiffiffiload delamination growth increases with Gc E for delaminations of the same size and depth. To predict failure it is assumed the failure stress is greater than or equal to the buckling stress and that the
Fig. 16. Geometry for isotropic across-the-width delamination model.
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increase in failure load with failure stress is small. Therefore, failure is predicted for the across-the-width geometry by the buckling load of the sub-laminate below the delamination. Taking the sub-laminate to be built in at the edge of the delamination, the failure load is P0 ¼
2 EbðT tÞ3 3a2
Acknowledgements Mr. G. J. Short was supported by EPSRC Studentship award No. 97700381. The measurements of compressive strength were carried out with the help of Dr. J. Summerscales of the University of Plymouth.
ð5Þ References
In this case, as the material properties change the applied load to cause failure increases with E for delaminations of the same size and depth. Hence, delamination pffiffiffiffiffiffiffiffiffiffiffi growth before failure becomes more likely as E=Gc increases, for delaminations of the same geometry. Comparing composite materials with different fibres but the same matrix, and therefore similar inter-laminar fracture toughnesses, delamination growth before failure is more likely for CFRP than GRP.
5. Conclusions Tests on delaminated GRP specimens have shown the failure load decreases with delamination size and decreases with the through thickness position of the delamination. In all specimens tested, no growth of the delamination occurred and failure was preceded by buckling of the specimen. Depending on the delamination geometry, two different modes of buckling were observed: a local mode where the sub-laminate above the delamination buckled out of plane, and a global mode where both the sub-laminates above and below the delamination buckled out of plane. Finite-element models of the delaminated GRP were used to predict the buckling mode and the failure load based on a maximum fibre direction stress criterion. These predictions showed good agreement with the experimental results. An analytical and finite-element study of a delaminated isotropic plate was undertaken to examine the effect of delamination geometry without the complicating effects of material orthotropy. The study allowed a buckling mode map which predicts the buckling mode, local or global, for delaminated plates of different delamination size and through thickness position. The study also allowed a master failure curve to be plotted where failure is assessed on a maximum stress criterion. The failure curve predicts the failure load normalised with respect to the buckling load for a material failure stress normalised by the buckling stress.
[1] Garg AC. Delamination—a damage mode in composite structures. Engineering Fracture Mechanics 1988;29:557–84. [2] Pavier MJ, Clarke MP. Experimental techniques for the investigation of the effects of impact damage on carbon fibre composites. Composites Science and Technology 1995;55:157–69. [3] Seikine H, Flu N, Kouchakzadeh MA. Buckling analysis of elliptically delaminated composite laminates with consideration of partial closure of delamination. Journal of Composite Materials 2000;34:551–74. [4] Peck SO, Springer GS. The behaviour of delaminations in composite plates—analytical and experimental results. Journal of Composite Materials 1991;25:907–29. [5] Bolotin W. Delaminations in composite structures: its origin, buckling, growth and stability. Composites 1996;27B:129–45. [6] Kardomateas GA, Schmueser DW. Buckling and postbuckling of delaminated composites under compressive loads including transverse shear effects. ATAA Journal 1988;26:337–43. [7] Prichard JC, Hogg PJ. The role of impact in post-impact compression testing. Composites 1990;21:503–11. [8] de Freitas M, Reis L. Failure mechanisms on composite specimens subjected to compression after impact. Composite Structures 1998;42:365–73. [9] Pavier MJ, Chester WT. Compression failure of carbon fibre reinforced coupons containing central delaminations. Composites 1990;21:23–31. [10] Greenhalgh ES. Delamination growth in carbon fibre composite structures. Composite Structures 1993;23:165–75. [11] Williams JF, Stouffer DC, Ilic S, Jones R. An analysis of delamination behaviour. Composite Structures 1986;5:203–16. [12] Gaudenzi P, Perugini P, Spadaccia F. Postbuckling analysis of a delaminated composite plate under compression. Composite Structures 1998;40:231–8. [13] Kyoung W-M, Kim C-G, Hong C-S. Modelling of composite laminates with multiple delaminations under compressive loading. Journal of Composite Materials 1998;32:951–68. [14] Gaudenzi P. On delamination buckling of composite laminates under compressive loadin. Composite Structures 1997;39:21–30. [15] Shivakumar KN, Whitcomb JD. Buckling of a sublaminate in a quasi-isotropic composite laminate. Journal of Composite Materials 1985;19:2–18. [16] Pavier MJ, Clarke MP. Finite-element prediction of post impact compressive strength in carbon fibre composites. Composite Structures 1996;36:141–53. [17] Curtis PT. CRAG test methods for the measurement of engineering properties of fibre reinforced plastics. Royal Aircraft Establishment Technical Report 88012, 1988. [18] Timoshenko SP, Gere JM. Theory of elastic stability. McGrawHill, 1963.
The effect of delamination geometry on the compressive failure of composite laminates G.J. Short, F.J. Guild, M.J. Pavier* Department of Mechanical Engineering, University of Bristol, Queen’s Building, University Walk, Bristol BS8 1TR, UK Received 19 April 2001; received in revised form 24 July 2001; accepted 26 July 2001
Abstract One of the causes of a reduction in the compressive strength of a composite material containing delaminations is the buckling of the delaminated plies. To understand the effect of delamination geometry on the compressive behaviour of laminated composite materials, compression tests were carried out on glass-fibre-reinforced plastic (GRP) test specimens containing artificial delaminations of various geometry, created by inserting PTFE film into the laminate during lay-up. Finite-element modelling was also carried out to gain further understanding of the mechanisms of compressive failure. Good agreement between finite-element predictions and experimental measurements were found for the whole range of delamination geometries that were tested. Finiteelement and simple closed-form models were also developed for delaminated panels with isotropic properties. This enabled a study of the effect of delamination geometry on compressive failure without the complicating effects of orthotropic material properties. The results of this study can be used to derive a graph of non-dimensional failure load versus non-dimensional failure stress, where the results for any one delamination geometry superimpose on those for all others. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: A. Glass fibres; C. Delamination; C., Finite-element analysis; C. Buckling; Compressive strength
1. Introduction Delaminations in layered composite materials are areas of de-bonding between adjacent plies and may result from manufacturing imperfections or from low velocity impact damage whilst in service. They can restrict the performance advantage of composite materials by causing a considerable reduction in compressive strength [1]. Previous work has shown that the mechanism of compressive strength reduction is out-of-plane buckling of the groups of plies above and below the delamination [2–4]. These groups of plies are referred to here as sublaminates; when buckling of any sub-laminate occurs the remaining plies become subject to bending due to unsymmetric loading as well as in-plane compression. The stresses in the remaining plies are, therefore, greater than would exist in an undelaminated structure, resulting in a reduced failure load. The magnitude of this strength reduction can be linked to the initial buckling * Corresponding author. Tel.: +44-117-928-8211; fax: +44-117929-4423. E-mail address: [email protected] (M.J. Pavier).
of the panel, which can be categorised into two characteristic modes of behaviour dependent on delamination geometry [3,5,6]. The first mode, referred to as the local mode, occurs when the upper sub-laminate is thin and the area of delamination large. In this case the stiffuess of the upper sub-laminate is lower relative to that of the lower sub-laminate and the upper sub-laminate buckles out of plane forming a blister on the panel surface. The second mode, referred to as the global mode, is observed when the delamination has a small area and is deeper within the panel. This mode is characterised by both sub-laminates buckling in the same direction and with the same magnitude of out of plane displacement. Fig. 1 shows these buckling modes diagrammatically. Many experimental studies of the effect of delamination on the compressive behaviour of panels have been carried out using post impact compression strength tests [7,8]. This type of test in principally concerned with measuring the effect of impact energy on the residual compressive strength of a coupon. Its use in understanding the effect of delamination geometry on the compressive strength of a panel is less clear as multiple delaminations of complex shape and often unknown position are formed through the thickness of the mate-
0266-3538/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(01)00134-8
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rial. Experimental work has also been conducted using compression test specimens with embedded artificial delaminations [9,10]. Here, a single or multiple delaminations of known shape and position are implanted into the specimen, allowing the effect of delamination geometry on the compressive behaviour and residual strength of the panel to be studied in a controlled manner. The use of finite-element modelling can provide further understanding of the mechanisms of failure. Panel failure is often a sudden event with little or no propagation of damage prior to failure. Studying the mechanisms involved from experimental data alone is therefore difficult. With artificial delaminations, definition of the delamination geometry is simple and finiteelement simulations are relatively straightforward to undertake. In the simplest case of single or multiple delaminations across the width of a specimen, twodimensional analysis allows comparison with experimental data [11–13]. For embedded delaminations, three-dimensional modelling is necessary to allow the shape and size of the delamination to be represented accurately [3,4,14–16]. In general, finite-element analyses of the effect of delaminations on compressive strength show good agreement with experimental results and allow the mechanisms of failure to be investigated
and understood more easily than would be possible from experimental data alone. This paper presents the results of an investigation into the effect of size and through thickness position of single embedded delaminations on the sub-laminate response and compressive failure load of GRP specimens. An anti-buckling guide was used during testing and the laminate was instrumented with LVDTs to measure the out of plane responses of the upper and lower sublaminates to compressive loads. Three dimensional finite-element analysis was also carried out in order to compare predicted failure loads with those measured experimentally. In addition to the experimental and finite-element work on layered GRP laminates, analytical and finite-element models of delaminated isotropic panels were developed to study the effect of delamination geometry, without the complications of material orthotropy. These models enable the buckling mode to be predicted depending on the delamination size and through thickness position. The models also provide an estimation of the compressive failure load of a panel, provided the failure stress of the materials is known as a proportion of the critical buckling stress of the panel. The isotropic models provide acceptable indications of the experimental behaviour of the GRP laminates.
Fig. 1. Local and global modes of delamination induced buckling.
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2. Experimental work 2.1. Specimen preparation The experimental tests were carried out on specimens containing artificially created delaminations. The laminates were manufactured from ICI Fiberite 934 unidirectional glass fibre pre-preg. The lay-up for all specimens tested was [0,+45, 45, 0]s producing specimens with a typical thickness of 2.4 mm. All specimens were cut from panels produced by hand lay-up and cured using the vacuum bag method with the manufacturers recommended cure cycle. Delaminations were formed using a single thickness of 10 mm PTFE film laid between plies [2]. Test specimens of 200 mm long by 50 mm wide were then dry cut from these plates using a diamond encrusted edge slitting saw. After trimming, aluminium end tabs of dimensions 50 mm 50 mm by 2 mm thick were etched for 30 min in chromic sulphuric acid at 60 C and then bonded to the coupons with 3M 9323 two-part structural adhesive. Once cured, the ends of the specimens were machined parallel to minimise uneven loading during testing. The geometry of the specimens is shown in Fig. 2. Specimens were manufactured with different sizes and through thickness positions of artificial delamination. In this work, three sizes of delamination were used: 10 mm 10 mm, 15 mm 15 mm and 25 mm 25 mm. Fig. 2 shows for example a specimen with a 25 mm 25 mm delamination. In all cases the delamination area was square, as this offered a good compromise between ease of damage production and realistic representation of inservice delamination damage [2]. In addition, three different through-thickness positions were used: position ‘A’ between the first and second, 0 and +45 plies, position ‘C’ between the third and fourth, 45 and 0 plies and position ‘D’ between the two zero plies at the centre line of the laminate (Fig. 3). 2.2. Experimental method During testing, an anti-buckling guide [2] was used to prevent gross out of plane deformation of the test specimens. The stiffness of the guide in bending was much larger than that of the specimens. The guide was designed so that both faces of the panel could be viewed during testing. Brackets were mounted to the guide to allow the deflections of both surfaces of the specimen to be measured using LVDTs. The guide incorporated a 40 mm 40 mm window to allow out of plane displacement of the specimen in the central region near the delamination, as shown in Fig. 2. The window size was kept the same for all specimens. The tests were conducted on a Zwick 1478 test machine with a cross-head speed of 2 mm/min so as to achieve failure within 60–90 seconds. Panel failure was
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assumed to have occurred when the load dropped by 50%, at which point the test stopped automatically. Failure load and out of plane deflections of the panel faces were recorded continuously during each test. For each delamination geometry, three identical specimens were tested. For undamaged specimens, that is specimens without artificial delaminations present, six specimens were tested. However, of these six, only three failed in the centre of the specimen and were taken to be good test results. 2.3. Experimental results Failure load results are shown in Table 1 for each geometry of delamination. The results for specimens of the same geometry are consistent; typically results are within 5% of the average failure load. Failure loads decrease as the delamination size increases, and decrease as the delamination through thickness position increases. The greatest loss of strength was for specimens containing a 25 mm square delamination in the ‘D’ position, that is in the centre of the laminate. For these specimens the average failure load was 31% less than that for undamaged specimens. The average failure loads versus delamination through thickness positions are plotted later when compared with finite-element predictions. For all specimens, no growth of the delamination prior to failure was noticed. For specimens with delaminations in the ‘A’ position, some splitting was observed on the topmost, 0 ply near the edge of the delamination, parallel to the 0 fibre direction. No damage growth of any kind was seen in specimens with the delamination in positions ‘C’ or ‘D’. For specimens containing delamination, buckling of the sub-laminates occurred before failure. Depending on the size and through thickness position of the delamination, different modes of delamination buckling were observed. These buckling modes are termed local and global following Seikine et al. [3], as described previously. Fig. 4 shows the LVDT results for two specimens: one an example of a local buckling mode, the other an example of a global buckling mode. In Fig. 4 the designations ‘Top’ and ‘Bottom’ refer to the sub-laminates above and below the delamination, as marked in Fig. 3. Local buckling occurred for the specimen containing a 25 mm square delamination in the ‘A’ position, designated the 25A specimen. For this specimen the sublaminate above the delamination buckled out of plane at a load of about 19 kN. As load was increased above this level the out of plane deflection increased for both the sub-laminates but in opposite directions. A global mode occurred for the specimen containing a 15 mm square delamination in the ‘C’ position, designated the 15C specimen. Both sub-laminates deflect out of plane
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in the same direction but there is no sudden increase in the magnitude of the deflection as for the 25A specimen. All specimens containing delaminations in the ‘A’ position gave a local buckling mode while all those containing delaminations in the ‘C’ and ‘D’ positions gave a global mode. When a local mode occurred, buckling took place at a load of between 10 and 20 kN. This sudden buckling event is believed to be due to residual adhesion of the sub-laminates and that without such residual adhesion, out of plane displacement of the sub-laminates would occur at a significantly lower load.
3. Finite-element analysis 3.1. Finite-element models Previous work [16] has indicated that the failure of a delaminated specimen may be predicted by the load at which the maximum compressive fibre direction stress reaches a critical value. Such a prediction ignores damage growth mechanisms and any dependence of the critical fibre direction stress on laminate geometry and loading. Nevertheless, comparisons with experimental
Fig. 2. Geometry of a composite specimen with a 25 mm square delamination, showing position of delamination and size of window in anti-buckling guide.
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Fig. 3. Lay-up of composite specimen and through-thickness positions of delamination.
Table 1 Failure load results Delamination geometry
Failure loads (kN)
Size
Depth
Specimen 1
Specimen 2
Specimen 3
Average
Undamaged 25 mm square 20 mm square 25 mm square 15 mm square 15 mm square 10 mm square
– ‘A’ ‘C’ ‘D’ ‘A’ ‘C’ ‘A’
42.9 36.6 30.5 29.4 38.8 36.5 39.1
45.8 37.8 33.7 30.0 41.1 40.2 43.2
45.6 38.6 31.2 33.7 40.9 36.9 39.8
44.8 37.7 31.8 31.0 40.3 37.9 40.7
results have shown the critical stress model gives acceptable results. Prediction of maximum fibre direction stresses in a post-buckled orthortropic laminate inevitably demand finite-element analysis and in this work the ABAQUS V5.8 finite-element system was used with PATRAN V8.5 for pre-processing. To reduce the size of the finiteelement model, only part of each specimen was modelled: the 50 mm by 50 mm central part of the specimen containing the delamination as shown in Fig. 2. The finite-element model was constructed from two layers of 20 noded orthotropic brick elements, with a planar edge length of 2.5 mm, to represent the thickness of plies above and below the plane of the delamination. Each layer was given a small initial out of plane displacement of 0.01 mm at the delamination centre to initiate buckling during the analysis. The two layers of elements were joined at the interface, except for the delamination area where contact conditions were applied to prevent physically inadmissible inter-penetration of the elements during analysis. A typical finite-element mesh is shown in Fig. 5.
Fig. 4. Out of plane displacements measured at the specimen centre on the top and bottom surfaces for an undamaged specimen and specimens containing a 25 mm square delamination in the ‘A’ position and a 15 mm square delamination in the ‘C’ position.
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Fig. 5. Geometry of the finite-element mesh. Table 2 Material properties of B glass/epoxy Property E11 E22 E33 V23 V31 V12 G23 G31 G12
46.0 GPa 13.0 GPa 13.0 GPa 0.42 0.3 0.3 4.6 GPa 5.0 GPa 5.0 GPa
Layered material properties were applied to the elements according to the unidirectional linear elastic material properties shown in Table 2, rotated accordingly for the off-axis plies. Models were constrained to represent the boundary conditions provided by the antibuckling guide with the unloaded edges of the model constrained in the out of plane direction over a 5 mm wide strip along each edge and the loaded edges simply supported. Compressive load was applied to the model as a pressure over the two faces normal to the loading direction, as shown in Fig. 5. A static non-linear analysis was then conducted, including the effects of large deflections, to predict post buckling behaviour of the specimens. 3.2. Finite-element results Fig. 6 shows the out of plane displacement results at the centre of the upper and lower sub-laminates compared with the LVDT results from experiment. Fig. 6(a) shows typical results for a geometry giving a local mode
of buckling: a specimen containing a 25 mm square delamination in the ‘A’ position. Agreement is generally good once sub-laminate separation occurs. The finiteelement model does not predict the behaviour of the panel before this point since it takes no account of the residual adhesion between the sub-laminates existing in the experiments. Fig. 6(b) shows results for a 15 mm square delamination in the ‘C’ position, a geometry resulting in a global mode of buckling. Agreement is excellent. Predictions of the specimen failure loads are made by assuming the maximum fibre direction stress reaches a limiting value at failure. In this work, a value for the limiting fibre direction compressive stress of the composite was derived by measurement. Five uni-directional test specimens were manufactured and tested according to the CRAG specification [17]. The specimens were 110 mm long by 10 mm wide and 2 mm thick with 50 mm 10 mm aluminium end tabs bonded to the specimen so as to leave a gauge length of 10 mm. Testing was conducted using an Instron 8501 servo-hydraulic machine with hydraulic grips. The compressive fibre direction strengths for the five specimens varied from 665 to 686 MPa and gave an average of 674 MPa. For all models, the stress distribution in the laminate was examined to determine the maximum value of fibre direction stress in the most heavily loaded ply. The position of the maximum value of stress was typically near the centre of the specimen, either on the upper or lower most zero degree ply, depending on the direction of the out of plane deflection. Fig. 7(a) shows predicted contours of fibre direction stress for the most heavily loaded zero degree ply for the case of a 25 mm square delamination in the ‘A’ position where a local mode of buckling occurs. The maximum stress occurs in two
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bands between the centre and edge of the delamination. Fig. 7(b) shows a similar contour plot for the case of a 25 mm square delamination in the ‘D’ position. A global mode of buckling occurs and the maximum stress shifts to the centre of the delamination. In Fig. 7(a) and (b) the load applied to the model is equivalent to the experimental failure load. The maximum stress in the model is similar to that measured in the compressive strength tests, 674 MPa. Fig. 8 shows the predicted maximum compressive fibre direction stress for the most heavily loaded ply versus the applied load for 25 mm square delaminations in the ‘A’, ‘C’ and ‘D’ positions. The failure load pre-
Fig. 6. Comparison of experimental measurements and finite-element predictions of central out of plane displacements for: (a) a specimen containing a 25 mm square delamination in the ‘A’ position and (b) a specimen containing a 15 mm square delamination in the ‘C’ position.
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dicted by the model is obtained by the point at which the curves of compressive stress cross the line representing the measured compressive strength for the material. Fig. 8 suggests a decreasing specimen strength with increasing delamination depth. Fig. 9 compares the finite-element predictions of failure load with the experimental measurements. Results for 15 and 25 mm square delaminations are plotted versus through thickness delamination position. Agreement is generally good, although the finite-element models consistently under-predict the experimental results. It is notable that the failure load of the undamaged specimen is about 5% less than the predicted failure load. This implies the measured compressive strength obtained in the unidirectional tests cannot be achieved in a multi-directional laminate. Clearly, using a reduced compressive strength derived from the undamaged tests would give
Fig. 7. Finite-element prediction of fibre direction compressive stress in the most heavily stressed 0 ply at a load corresponding to the experimental failure load for: (a) a specimen containing a 25 mm square delamination in the ‘A’ position and (b) a specimen containing a 25 mm square delamination in the ‘D’ position.
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better agreement with experimental failure loads for specimens containing delaminations.
4. Isotropic modelling 4.1. Methodology The experimental and finite-element work described above has related specifically to a multidirectional laminate laid up with 8 plies of GRP pre-preg. It was
considered of interest to investigate in a more general manner the effects of delamination geometry on compressive behaviour. Therefore a programme of analytical and finite-element modelling was carried out using isotropic material properties. This modelling work considered a square delaminated plate, simply-supported around its edges and subjected to compressive loading in one direction. The geometry of the plate is shown in Fig. 10. The delamination geometry is defined by the ratio of delamination size to plate size a/b and the ratio of delamination depth to plate thickness t/T. 4.2. Prediction of buckling mode The first study carried out using the isotropic model was involved with the prediction of buckling mode. The delaminated plate geometry is considered to be formed of two laminates, one to represent the sub-laminate above the delamination, the other to represent the remainder of the plate. The model uses the analytical expression [18] which allows the critical buckling stress scr for a rectangular plate to be evaluated for different boundary conditions. 2 2 E t cr ¼ K ð1Þ 12ð1 v2 Þ b2
Fig. 8. Finite-element predictions of maximum fibre direction compressive stress versus applied compressive load for specimens containing 25 mm square delaminations in the ‘A’, ‘C’ and ‘D’ positions.
Fig. 9. Comparison of experimental and finite-element predicted failure loads versus through thickness delamination position for 25 mm square and 15 mm square delaminations.
Fig. 10. Delamination geometry for isotopic models.
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where t is the plate thickness and b the plate width. The non-dimensional factor K depends on the ratio of the length to the width of the plate and the boundary conditions. If the critical stress sTcr for the sub-laminate above the delamination is much less than the critical stress sBcr for the remainder of the plate, local buckling occurs. Conversely, if sBcr is much less than sTcr, global buckling T B takes place. Setting cr ¼ cr provides an indication of where the boundary between local and global buckling modes exists. This boundary is therefore calculated by KT
t2 ðt TÞ2 ¼ KB 2 a b2
ð2Þ
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element mesh used 20 node three-dimensional elements as before (Fig. 5). A value for Young’s modulus of E=70 GPa and Poisson’s ratio v=0.3 was used, while the plate dimensions were b=60 mm and T=3 mm. The buckling mode was assessed using a non-linear analysis with a small initial out of plane displacement as before to initiate buckling. The finite-element results are superimposed on Fig. 11 and show an acceptable level of agreement with the analytical predictions of buckling mode. In addition, Fig. 11 shows the experimental results for the GRP specimens. Despite the difference in material properties, it can be seen the isotropic analytical model is able to predict the experimentally observed buckling modes.
The boundary conditions for the remainder of the plate are simply-supported around the edge. For the sub-laminate above the delamination the boundary conditions lie somewhere between simply-supported and built-in. In Fig. 11, the expression given by Eq. (2) is plotted for these two boundary conditions as delamination size a/b versus through thickness position t/T. Fig. 11 is considered a buckling mode map, allowing the buckling mode to be predicted for any combination of delamination size and through thickness position. The map shows that local buckling is predicted for large delamination size and small through thickness position while global buckling is predicted for small delaminations with a large through thickness position. Fig. 11 also shows the results of isotropic finite-element models to predict buckling mode for different delamination geometries. Three different through thickness positions were used: t/T=0.25, 0.33 and =0.5, and three different sizes: a/b=0.25, 0.5 and 0.75. The finite-
4.3. Prediction of strength
Fig. 11. Delamination induced buckling mode map for varying delamination size and through thickness position.
Fig. 12. Maximum compressive stress versus applied compressive load for the case where t/T=0.5 and a/b=0.75.
The isotropic finite-element models used to produce the results in Fig. 11 were also used to predict maximum stress versus applied load in the same way as for the layered orthotropic finite-element models. Fig. 12 shows the maximum compressive stress in the loading direction versus the applied load for an isotropic finite-element model with a delamination geometry given by t/T=0.5 and a/b=0.75. Two curves are plotted, one for the maximum stress in the sub-laminate above the delamination, the other in the remainder of the plate, referred to as the bottom sub-laminate in the figure. If failure is predicted as before, by when the maximum stress reaches a material limit, Fig. 12 provides a method to find the failure load for any level of compressive failure stress. For a failure stress of 1000 MPa for example, a failure load of about 80 kN is predicted with failure initiating in the bottom sub-laminate.
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The maximum stress versus failure load results may now be used to provide a prediction of the failure load depending on the compressive failure stress for the material. Fig. 13 shows such results for a delamination through thickness position given by t/T=0.5 and three different delamination sizes: a/b=0.25, 0.5 and 0.75. When the compressive failure stress is low enough, buckling does not occur and the predicted failure load depends linearly on the failure stress with no effect of delamination geometry. Once the failure stress is high enough, buckling effects become important and delamination geometry influences the failure load. For a failure stress above about 500 MPa, Fig. 13 shows an increase in the failure load with reducing delamination size.
out of plane displacement in the lower sub-laminate begins to increase. Although it is straightforward to infer two buckling loads from Fig. 14(a), other delamination geometries resulting in a local buckling mode give out of plane displacement results that are more difficult to interpret. Fig. 14(b) shows results for a geometry giving global buckling, when t/T=0.5 and a/b=0.75.For this geometry a single buckling load of 64.1 kN is observed. Using the value of the critical buckling load the failure load versus failure stress graph of Fig. 13 is re-plotted in normalised form as in Fig. 15, where curves have been added for each geometry of delamination. The
4.4. Normalisation of strength predictions It has been found worthwhile to normalise the results of Fig. 13. The failure stress is normalised by dividing by the stress at which buckling occurs. The failure load is normalised by dividing by the load at which buckling occurs. This normalisation requires a value for the buckling load Pcr which was obtained from the out of plane displacement versus load results from the isotropic finite-element models. Fig. 14(a) shows such results for a geometry where local buckling occurs: t/T=0.33 and a/b=0.75.The local buckling load is 36.0 kN, the load when the out of plane displacement of the sub-laminate begins to increase. However, since the stress in the lower sub-laminate is the stress influencing failure, the buckling load used to normalise the failure load is taken to be 75.1 kN. This is the load at which the
Fig. 13. Compressive failure stress versus failure load for the cases where t/T=0.5 and a/b=0.25, 0.5 and 0.75.
Fig. 14. Central out of plane displacements on the top and bottom surface versus compressive load for the cases where: (a) t/T=0.33 and a/b=0.75 and (b) t/T=0.5 and a/b=0.75.
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failure stress is normalised using the critical buckling stress, calculated from the buckling load by cr ¼
Pcr bT
ð3Þ
Although the curves for different delamination geometries do not lie perfectly over each other, they do form a narrow band which could be used as a failure criterion for a delaminated plate. When the normalised failure stress is below 1, the failure load versus failure stress curve is linear. For normalised failure stresses above 1, buckling occurs before failure and the rate of increase of failure load with failure stress reduces.
Fig. 15. Normalised compressive failure stress versus normalised failure load.
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Fig. 15 also shows an attempt at superimposing the experimental GRP results. It is very much harder to measure an accurate buckling load from the LVDT results such as those in Fig. 6(a) and (b). Nevertheless, the normalised experimental results do support the principle that failure loads may be predicted using a master failure criterion based on normalised parameters. 4.5. Occurrence of delamination growth An interesting observation is that delamination growth typically occurs before failure for delaminated CFRP specimens of similar geometry to the GRP specimens used here [9]. It is possible to show this is expected, by comparing the load to cause delamination growth with the load to cause failure for a near surface, across-the width delamination in an isotropic plate. Bolotin [5] presents an analytical model to predict delamination growth for a near surface delamination. For the geometry shown in Fig. 16, the load Pg applied to the plate to cause delamination growth is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Gc E 1 v2 ð4Þ Pg ¼ Pd þ bT t where Pd is the load applied to the plate to cause the sub-laminate above the delamination to buckle and Gc is the inter-laminar fracture toughness. If the load Pd is negligible as it usually is, the analysis suggests that as the material properties change the applied to cause ffi pffiffiffiffiffiffiffiffiload delamination growth increases with Gc E for delaminations of the same size and depth. To predict failure it is assumed the failure stress is greater than or equal to the buckling stress and that the
Fig. 16. Geometry for isotropic across-the-width delamination model.
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increase in failure load with failure stress is small. Therefore, failure is predicted for the across-the-width geometry by the buckling load of the sub-laminate below the delamination. Taking the sub-laminate to be built in at the edge of the delamination, the failure load is P0 ¼
2 EbðT tÞ3 3a2
Acknowledgements Mr. G. J. Short was supported by EPSRC Studentship award No. 97700381. The measurements of compressive strength were carried out with the help of Dr. J. Summerscales of the University of Plymouth.
ð5Þ References
In this case, as the material properties change the applied load to cause failure increases with E for delaminations of the same size and depth. Hence, delamination pffiffiffiffiffiffiffiffiffiffiffi growth before failure becomes more likely as E=Gc increases, for delaminations of the same geometry. Comparing composite materials with different fibres but the same matrix, and therefore similar inter-laminar fracture toughnesses, delamination growth before failure is more likely for CFRP than GRP.
5. Conclusions Tests on delaminated GRP specimens have shown the failure load decreases with delamination size and decreases with the through thickness position of the delamination. In all specimens tested, no growth of the delamination occurred and failure was preceded by buckling of the specimen. Depending on the delamination geometry, two different modes of buckling were observed: a local mode where the sub-laminate above the delamination buckled out of plane, and a global mode where both the sub-laminates above and below the delamination buckled out of plane. Finite-element models of the delaminated GRP were used to predict the buckling mode and the failure load based on a maximum fibre direction stress criterion. These predictions showed good agreement with the experimental results. An analytical and finite-element study of a delaminated isotropic plate was undertaken to examine the effect of delamination geometry without the complicating effects of material orthotropy. The study allowed a buckling mode map which predicts the buckling mode, local or global, for delaminated plates of different delamination size and through thickness position. The study also allowed a master failure curve to be plotted where failure is assessed on a maximum stress criterion. The failure curve predicts the failure load normalised with respect to the buckling load for a material failure stress normalised by the buckling stress.
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