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The Energy-absorbing Properties of Composite Tube-Reinforced Aluminum Honeycomb A. Al Antali 1 , R. Umer 1 , J. Zhou 2 and W.J. Cantwell 1 , 1 Aerospace Research and Innovation Center, Khalifa University of Science and Technology (KUST), 127788 Abu Dhabi, United Arab Emirates. 2 School of Engineering, University of Liverpool, Liverpool, L69 3GH. United Kingdom. Abstract This paper investigates the energy-absorbing characteristics of a lightweight honeycomb core containing embedded carbon fiber reinforced plastic (CFRP) tubes. Initial tests are undertaken on the plain aluminum honeycomb material in order to characterize its specific energy absorption (SEA) capability and to identify the prevailing failure mechanisms. Tests are then conducted on honeycomb cores reinforced with increasing numbers of composite tubes in order to establish the influence of varying the density of the tubular array on the measured SEA. Finally, a series of drop-weight impact tests are conducted in order to characterize the dynamic response of these materials and assess their overall rate-sensitivity. Tests on the plain aluminum honeycomb cores resulted in the characteristic plastic wrinkling of the cell walls, yielding an average value of SEA of 14 kJ/kg. Embedding CFRP tubes into the honeycomb served to greatly enhance the energy-absorbing properties of the core, with quasi-static values

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The Energy-absorbing Properties of Composite Tube-Reinforced Aluminum Honeycomb

A. Al Antali1, R. Umer1, J. Zhou2 and W.J. Cantwell1,

1 Aerospace Research and Innovation Center, Khalifa University of Science and Technology (KUST),

127788 Abu Dhabi, United Arab Emirates.

2 School of Engineering, University of Liverpool, Liverpool, L69 3GH. United Kingdom.

Abstract

This paper investigates the energy-absorbing characteristics of a lightweight honeycomb core containing embedded carbon fiber reinforced plastic (CFRP) tubes. Initial tests are undertaken on the plain aluminum honeycomb material in order to characterize its specific energy absorption (SEA) capability and to identify the prevailing failure mechanisms. Tests are then conducted on honeycomb cores reinforced with increasing numbers of composite tubes in order to establish the influence of varying the density of the tubular array on the measured SEA. Finally, a series of drop-weight impact tests are conducted in order to characterize the dynamic response of these materials and assess their overall rate-sensitivity.

Tests on the plain aluminum honeycomb cores resulted in the characteristic plastic wrinkling of the cell walls, yielding an average value of SEA of 14 kJ/kg. Embedding CFRP tubes into the honeycomb served to greatly enhance the energy-absorbing properties of the core, with quasi-static values of SEA reaching as high as 105 kJ/kg. An examination of the failed samples indicated that the previously-observed wrinkling mode of failure was largely absent, with the composite tubes being reduced to fine debris. Increasing the areal density of tubes in the honeycomb to relatively high values proved to be counterproductive, due to unwanted interactions between the individual tubes.

Dynamic tests on the tube-reinforced honeycombs yielded SEA values that exceeded the quasi-static results with values reaching 112 kJ/kg for a moderately-reinforced core. Given the simplicity of the preparation process, it is believed that these lightweight structures represent an attractive cost-effective energy-absorbing material for use in dynamic applications.

Introduction

It is now well-established that when correctly designed and loaded under well-defined conditions, composite materials can offer outstanding energy-absorbing characteristics over a wide range of strain-rates [1-10]. Work by Farley [11] showed that when loaded in compression, carbon fibre reinforced epoxy tubes fail in splaying and fragmentation modes, leading to significant energy dissipation and measured values of specific energy absorption (SEA) up to 80 kJ/kg [1,2]. Hull [12] proposed a unified approach to progressive crushing of composite tubes based on brittle fibers and matrices. He also argued that the crushing process occurs by splaying of the fibers as well as by fragmentation. He suggested that the micro-mechanisms of crush and the resulting load-bearing capacity of the tube could be changed by varying the distribution of fibers within the structure. Increasing the percentage of hoop fibers in the tube reduces the ease of splaying, whilst serving to increase the likelihood of shear fracture and increased fragmentation.

A number of investigations have established the importance of the tube geometry in determining the energy-absorbing characteristics of a composite tube [13-15]. Thornton and Edwards tested a number of tube geometries based on carbon, Kevlar and glass fibre reinforcements and showed that circular tubes exhibit a superior energy-absorbing capacity relative to rectangular and square tubes [15]. Mamalis et al [14] conducted crushing tests on samples with square, circular and conical cross-sections based on a glass fiber reinforced epoxy composite and established that circular tubes exhibited the most impressive energy-absorption characteristics. Farley [13] conducted tests on a range of Kevlar and carbon fibre reinforced epoxy tubes with different diameters and thicknesses and showed that the D/t ratio (tube inner diameter/thickness) determines the specific crushing stress (SCS). Farley showed that the value of this parameter almost doubled as D/t was decreased from approximately 120 to 3.8 and linked this to a decrease in interlaminar cracking.

A number of workers have investigated the potential benefits of filling composite tubes with traditional core materials, such as metal foams and aluminum honeycombs [16, 17]. Liu et al [16] introduced an aluminum honeycomb filler into square CFRP tubes and showed that the peak load and absorbed energy increased by more than ten percent relative to the plain composite tubes. In contrast, Sun et al [17] observed that incorporating an aluminum honeycomb into empty circular CFRP tubes resulted in a slight decrease in SEA, although the authors did note that these modified composite tubes still outperformed their metallic counterparts at quasi-static rates of strain. Similar conclusions were drawn by Hussein et al following tests over a range of strain rates on square CFRP tubes filled with aluminum honeycomb [18]. For example, the authors recorded values of SEA of 64 kJ/kg at a very low strain-rates for a plain CFRP tube and 55 kJ/kg for a filled tube, tested under the same conditions.

In contrast to these studies, where a low density core material is embedded within the energy-absorber, a number of researchers have investigated the effect of positioning small tubular structures in a foam substrate [19-21] Alia et al [19] conducted quasi-static and dynamic compression tests on PVC foams containing small aluminum and steel tubes and showed that the specific energy absorption does not vary with foam density, equaling approximately 69 kJ/kg for the former and 42 kJ/kg for the latter. The steel-based system exhibited a modest increase in SEA as the strain-rate was increased from quasi-static to dynamic rates, whereas the aluminum-reinforced cores were largely insensitive to strain-rate. This work was extended to investigate the energy-absorbing response of PVC foams containing an array of small diameter composite tubes [20]. Here, carbon fibre reinforced epoxy tubes with a range of D/t values were embedded in polystyrene and PVC foams with densities ranging between 15 and 225 kg/m3. An examination of the failed tubes removed from the foam substrate, indicated that the crushing process had reduced the composite to fine debris. The resulting values of SEA exceeded 93 kJ/kg, although the dynamic values were slightly lower.

Honeycomb structures could potentially represent an excellent substrate for embedding composite tubes. Aluminium honeycombs on their own have been shown to offer attractive energy-absorbing characteristics under both quasi-static and dynamic loading conditions [22-24].The aim of the work presented in this paper is therefore to investigate the potential offered by aluminum honeycomb core materials reinforced with carbon fibre reinforced epoxy tubes. Particular attention is given to assessing the effect of increasing the density of the tubular array in the honeycomb on the SEA when subjected to both quasi-static and dynamic rates of strain. The failure mechanisms occurring in both the embedded tubes and the surrounding honeycomb core are elucidated and discussed.

Experimental Procedure

The reinforced cores investigated in this study were based on an aluminum honeycomb supplied by the Hexcel Corporation and a roll-wrapped carbon fibre reinforced plastic tube supplied by Easy Composites Ltd. The aluminum honeycomb, with a cell size of 6.35 mm, a wall thickness of 0.04 mm and a nominal density of 37 kg/m3 (Hexweb CRIII based on a 5052 aluminum alloy), was supplied in the form of 20 mm thick sheets with length and width dimensions of 2 and 1 meters, respectively. Square samples, with an edge length of 45 mm, were removed from the honeycomb sheets in preparation for testing. The majority of CFRP tubes tested here were manufactured using a roll-wrapping procedure to yield tubes with an outer diameter, Do, of 6 mm, an inner diameter, D, of 4 mm and a nominal wall thickness, t, of 1 mm. The composite is based on a 2/2 twill weave, consisting of 3k carbon fibre tows. The tubes were supplied in one metre lengths and cut to size using a bandsaw. Prior to insertion in the honeycomb, one end of the tube was chamfered using a grinding wheel and the other was ground flat using the same grinding process. A thin adhesive film was applied to the outside of the tubes to ensure that they remained in place once inserted into the aluminum core. Figure 1a shows photographs of a sample containing eight tubes. In the initial part of this study, a limited number of tests were also conducted on unidirectional pultruded tubes with similar outer and inner diameters to their roll-wrapped counterparts. These tests were undertaken in order to ascertain which type of tube (wrapped or pultruded) would represent the most effective energy-absorbing reinforcement.

The energy-absorbing capability of the cores was initially investigated though a series of compression tests on an Instron 5969 universal test machine. The samples were placed between two circular steel plattens and compressed at a crosshead displacement rate of 10 mm/minute until the densification threshold was just exceeded. The load and crosshead displacement were recorded and subsequently used to determine the energy expended in crushing the cores. The specific energy absorption (SEA) of the samples was determined from the energy under the trace up to the onset of densification, divided by the mass of the sample. The effect of varying the areal density of the tubular array (equivalent to the percentage of cells occupied by tubes) on the SEA of the cores was subsequently investigated through a series of tests on 45 mm square samples containing between one and nine tubes (corresponding to an areal density of tubes, AD, between 2.5% and 21%). Additional tests were conducted on a sample based on seventeen tubes (AD = 40%) in order to investigate the effect of enhanced reinforcement.

The possibility of controlling the force associated with the crushing process was investigated by conducting tests on samples containing nine tubes of differing length. Here, three core designs were investigated, details of which are given in Figure 2. Core A contains four 20 mm long tubes positioned at the corners, four 10 mm tubes at the mid-points along the edges and one 15 mm tubes at the center. Core A has four 10 mm tubes at the corners, four 15 mm tubes along the edges and a full length tube at the center. Finally, Core C has 15 mm tubes in the corners, four full-length tubes at the mid-edges and a 10 mm tube at its center.

Dynamic Crushing Tests

The energy-absorbing properties of the tube-reinforced honeycomb and the plain core were investigated through drop-weight impact tests at impact velocities up to 3 m/s. The dynamic crushing tests were undertaken on square samples with an edge length of 45 mm, using an instrumented drop-weight impact rig. The test samples were placed on a steel platen located on a 100 kN load cell. The specimens were loaded by a falling carriage with a similarly-sized steel platten on its lowermost (contact) surface. The mass of the falling carriage was increased up to 47 kg and the release height up to 1.6 meters in order to completely crush the samples during the test. The crushing force and the displacement of the carriage were recorded using a piezoelectric load-cell and a high-speed video camera, respectively.

Results and Discussion

Energy Absorption of the Constituent Materials

Crushing tests were initially conducted on individual 6 mm diameter composite tubes containing a small chamfer at one end. Figure 3 shows a typical load-displacement trace following a compression tests on an individual roll-wrapped tube, where the force rises up to approximately 3.3 kN, before stabilising at a crushing load between 3.2 and 3.6 kN. During the final stages of the test, the force starts to decrease rapidly as the tube begins to finally disintegrate. The resulting average SEA value, determined from the area under the load-displacement trace and the mass of the sample was found to 113.5 kJ/kg. This value is compared with previously-published data on similar carbon fibre/epoxy tubes from the same supplier in Figure 4, where SEA is plotted as a function of the tube D/t ratio. It is interesting to note that the trend of increasing energy-absorption with decreasing D/t continues, with the value for the current tube (shown as a larger symbol) exceeding those associated with the other six tubes, all of which have higher values of D/t [20]. The evidence suggests that additional gains could be achieved by further reducing the size of the tubes, although difficulties associated with the manufacture of such tubes, and their buckling under load, may limit this capability. Tests were also conducted on the unidirectional pultruded tubes with the same nominal dimensions as the woven tubes. The load-displacement traces these cylinders were similar to those associated with the wrapped tubes, leading to an average SEA of 108.2 kJ/kg.

Figure 5 highlights the failure mechanisms in the two types of CFRP tube during the crushing process. From Figure 5ai, it is evident that crushing of the wrapped tube results in the uppermost region of the tubes splaying outwards against the crosshead of the test machine, fracturing fibres and matrix , as well as creating delaminations between the layers in the tube. Figure 5bi shows the residual debris remaining on the platen at the end of the test. The image indicates that a significant amount of energy has been dissipated in fracturing the tubes, highlighting, in a qualitative fashion, the excellent energy-absorbing behavior of these cylindrical structures. The pultruded tube exhibits similar failure characteristics to its woven-wrapped counterpart, although the photograph of the sample after testing suggests that there is slightly less debris. Given the similarity in the measured values of SEA for the wrapped and pultruded tubes, subsequent testing focused solely on aluminum honeycombs containing wrapped CFRP tubes.

Compression tests were conducted to evaluate the energy-absorbing characteristics of the aluminum honeycomb core. Figure 3 includes a typical load-displacement trace following compression testing on a square plain honeycomb sample with an edge length of 45 mm. Here, the force initially increases to a peak value of approximately 3.2 kN before dropping sharply. The curve then exhibits a number of smaller peaks, associated with the occurrence of localised instabilities in the cell walls within the honeycomb. Finally, the curve stabilizes at a force between 1.2 and 1.3 kN as the crushing process continues. In the final stages of the test, the load increases rapidly as the sample begins to densify between the plattens of the test machine. The average SEA value for the plain aluminium honeycomb samples was determined to be 14.6 kJ/kg. Figure 6 shows XCT images of a partially-crushed and fully-crushed plain aluminium sample. The X-ray images clearly highlight the wrinkling mechanism frequently observed in crushed honeycombs of this type. The honeycomb has collapsed in a well-defined manner, absorbing energy through the local plastic collapse mechanism. Closer inspection highlights the repeatability of the collapse process, with each web exhibiting a total of thirteen folds.

Energy Absorption in the Tube-Reinforced Aluminium Honeycomb

Figure 3 also includes a typical load-displacement trace following a test on a honeycomb sample reinforced with a single 6 mm diameter CFRP tube. The trace initially increases in a linear fashion up to approximately 3.7 kN before dropping sharply and rising once more to a plateau value of approximately 5.2 kN. The initial drop in force is clearly associated with the load-drop observed in the plain honeycomb. It is interesting to note that the crushing force for the tube/honeycomb combination is slightly higher than the sum of the average values for the individual constituents. Similar observations have been made following tests on tube-reinforced foams, where it was concluded that the presence of the foam served to modify the failure mechanisms in the composite tubes. Here, it is also possible that the walls of the honeycomb cell served to constrain the splaying process in the tubes, this will be discussed further below. The average SEA value for this single tube system was 45.4 kJ/kg, a value that is significantly higher than the previously-reported value for the plain honeycomb core.

Figure 7a shows a photo of a fully-crushed tube-reinforced honeycomb, where it is clear that the tube has been pressed into a flat disc. It is interesting to note that the honeycomb surrounding the tube has failed in a less regular manner than observed in its unreinforced counterpart, suggesting that the plastic folding mode has not been fully triggered in these hybrid systems. Clearly, the presence of the crushing tube, failing in a splaying mode introduces a more complex loading regime into the surrounding honeycomb, leading to a change in failure mode. Finally, Figure 7b shows the remains of a composite tube after removal from the surrounding honeycomb core. Here, it is evident that the crushing process has produced a large amount of fine debris, as well a smaller number of long strands, similar to that observed in the plain tubes. The visual evidence suggests that the amount of energy absorbed by the honeycomb is less than in the corresponding plain sample, whereas the converse appears to be true for the tube.

Compression Tests on Multi-tube Arrays

Figure 8 shows load-displacement traces following compression tests on honeycomb samples reinforced with three, five, seven and nine tubes. As expected, increasing the number of tubes serves to increase the crushing force, with the average value for the three tube system being 12.5 kN and that for its nine tube counterpart being 41 kN. Towards the end of the plateau region, all of the traces exhibit an increase in force followed by one or more drops that may be associated with the final disintegration of the tube, as shown previously in Figure 3. Once again, a rapid increase in force occurs during the latter stages of the test as the strain exceeds the densification threshold of the tube/honeycomb combination.

Figure 9 shows XCT images of a partially-crushed sample containing five CFRP tubes. Figure 9a shows the initial failure processes in the sample, where the splaying of the fibres is clearly evident. It is clear that the aluminium honeycomb has begun to collapse, although not in the clearly defined wrinkling process that was observed in the plain honeycomb samples. Figure 9b shows an XCT image taken from under the sample where the chaotic deformation pattern of the honeycomb is apparent. In spite of the movement of the supporting core, the tubes have remained in a vertical plane, with the splaying process being evident around their circumferences.

The influence of varying the number of tubes in the honeycomb on the energy absorption characteristics of the core is shown in Figure 10. Here, the total absorbed energy (in Joules) and the specific energy absorption values are presented for comparison. Also included in the figure are the corresponding values for the plain, unreinforced core. The absolute energy absorbed by the samples increases in a roughly linear manner from 21 Joules for the plain aluminum to 620 Joules for the nine tube system. It is also worth noting that the average crushing force also increased in a roughly linear manner with increasing AD, facilitating the prediction of the loads encountered during failure.

From Figure 10, it is clear that the SEA values increase rapidly as the number of tubes is increased. Here, the SEA increases from 15 to 105 kJ/kg in passing from the plain honeycomb to the nine tube core. This latter value is considered to be particularly high for a core material. The highest values in this figure approach the SEA of the plain tube (113.5 kJ/kg), suggesting that further, although modest increases may be achievable. Figure 11 show photos of a number of fully-crushed reinforced honeycombs, from where it is clear that the tubes have failed in a similar fashion to the single tube system. It is interesting to note that there is little or no apparent interaction between the reinforcements in the three and five tube systems, whereas the failure fronts clearly meet in the two higher density cores. Closer examination of the centers of the crushed tubes highlights the presence of compacted composite discs, associated with delamination and subsequent inward-splaying of part of inner part of the tube wall.

The energy absorption characteristics of the honeycombs were characterised using a simple rule of mixtures approach using the crushing response of the plain aluminum and the individual tube.

SEA = (Eh + Et.nt)/(mh + mt.nt)

Where Eh and Et represent the energies absorbed by the honeycomb and tubes respectively, nt is the number of tubes and mh and mt are the masses of the honeycomb and an individual tubes respectively. Inserting the energy values (Eh = 23 J and Et = 56.8 J) and masses of the constituent materials (mh = 1.6 g and mt =0.5 g) yields similar trends to those apparent in Figure 10, although the predicted values underestimate the measured values. For example, the rule of mixtures underestimates the SEA of the nine tube system by almost 15%. Good agreement is achieved by increasing the value of Et to 66.1 J and these predictions are shown by the solid line in Figure 10. Adopting the value of Et yields a value for the SEA of a single tube of 132 kJ/kg. This value is clearly higher than that measured on a simple unsupported tube and may reflect the increased support applied to the tube by the cell walls during the crushing procerss.

Figure 12 shows a load-displacement trace following a test on a sample with the higher density of tubes shown in Figure 1b (seventeen tubes and AD = 40%). Here, the force increases rapidly to 90 kN, a value close to that which would be expected, based on predictions from the lower density systems. However, rather than stabilizing at this value, the force starts to decline steadily as the crosshead advances, dropping to a value of only 13 kN prior to the onset of densification. The value of SEA for this denser system is 71.6 kJ/kg, significantly below that observed in some of the lower density samples, suggesting that there is an optimum value of density between the nine tube and seventeen tubes. The inset in Figure 12 shows a photograph of a sample following testing. Closer examination of this specimen suggested that the distinctive splaying mechanism observed in the samples shown in Figure 11 is largely absent in the current specimen. The visual evidence suggested that some tubes had tilted during the test and it is therefore likely that the first tubes to fail and splay outwards pushed adjacent tubes sideways before the previously-observed failure mechanisms had been triggered. The current failure process is therefore less effective than those observed in the earlier samples, leading to lower values of SEA.

Compression Tests on the Graded Structures

In this section, tubes of differing length were inserted into the honeycomb in order to control the crushing force during the compression process. Three specimen configurations were investigated, as summarized in Figure 2. Figure 13 shows typical load-displacement traces following tests on each of these three designs, where a controlled increase in the force profile is apparent. For example, Core A, with four 20 mm long tubes, one 15 mm long tube and four 10 mm tubes exhibits a number of distinct plateaus. Initially, the force increases steadily to a constant value of approximately 18 kN as the four longest tubes are loaded. At a displacement of approximately 5 mm, the single 15 mm long tube is loaded and the force increases by a relatively small amount to 22 kN. A final and pronounced increase in force occurs as the four shortest tubes are engaged and begin to crush. Shortly before the densification threshold, the force drops by approximately 7 kN, as the tubes lose their crush resistance, a phenomenon observed previously in Figure 8. In contrast, Core B exhibits a small initial increase in load to 3.8 kN, where the aluminum honeycomb starts to collapse and the load drops slightly before increasing to a crush force of approximately 5.5 kN, a value associated with compressing a single tube. Continued loading leads to two large steps as the four 15 mm and four 10 mm tubes are engaged, before the force drops again prior to densification. The resulting SEA values for the Cores A, B and C are approximately 75, 68 and 85 kJ/kg, respectively, varying roughly according to the cumulative length of the tubes within each of the cores. These data points, and their associated average crushing forces, are shown in Figure 14. Also included in the figure are the values resulting from the tests on samples with varying number of full-length (20mm) tubes). From the figure, it is clear that the values follow the trends observed in the original full length tubes. Grading the tubes in this way simply serves to move the reinforced core along this curve. This evidence suggests that the crushing force can be controlled in order to yield a desired deceleration during dynamic crushing conditions.

Impact Tests on the Constituent Materials and the Reinforced Cores

Figure 15 shows typical load-displacement traces following impact tests on an individual tube, the plain honeycomb core and the combination of the two. Included in the figure are the corresponding traces following quasi-static tests on these three materials. As before, the plain honeycomb exhibits an initial oscillatory response before stabilizing at a crushing force that is clearly higher than that measured at 10 mm/minute. The resulting value of SEA is approximately 20 kJ/kg, over 30% higher than the quasi-static value. The force response oscillates around a value close to the quasi-static trace, although the average SEA for the single tubes was 108 kJ/kg, slightly lower than the value of 113 kJ/kg measured previously. Schmuesser and Wickliffe [25] reported reductions in SEA of up to 30%, relative to quasi-static properties, following crushing tests on composite tubes, whereas Thornton [2] did not observe any significant rate-sensitivity following tests on carbon fiber epoxy tubes over a range of strain rates. The combined tube and honeycomb sample again exhibit a pronounced dynamic response close to that measured at quasi-static rates, although the average SEA was approximately ten percent higher at impact rates.

Figure 16 shows load-displacement traces following impact tests on samples based on 1, 3, 5, 7 and 9 tubes inserted in an aluminum honeycomb. Included in the (shown as dashed lines) are the corresponding traces associated with quasi-static testing. All of the impact tests exhibit an oscillatory response, with the level of oscillation increasing with the number of tubes embedded in the core. Such dynamic effects are likely to be associated with the vibration of the impact carriage and ringing effects in the piezo-electric load cell. From the figure, it is clear that the dynamic load-displacement traces closely follow their quasi-static counterparts, except during the initial loading phase, where several dynamically-loaded samples are stiffer than their quasi-static counterparts. It is interesting to note that the drop in load prior to densification is largely absent in the impact-loaded samples. Additionally, the densification thresholds, where the force begins to rise rapidly, are similar under both extremes of loading.

Figure 17 presents the SEA values following the drop-weight crushing tests on the reinforced cores, where it is evident that the dynamic values of energy-absorption are higher than the corresponding quasi-static values. This is in contrast to the tests on the plain tube, where increasing the strain-rate has been shown to result in a reduction in the measured SEA. It is possible that the support applied by the walls of the honeycomb, serves to constrain the tubes during the crushing process, thereby modifying the failure process. Interestingly, an examination of the figure indicates that the SEA of the plain honeycomb is also slightly higher than that measured at quasi-static rates. The figure clearly begins to plateau at higher values, tending towards a limiting value that reflects the energy-absorbing capability of the individual tubes. It is worth noting that the value for the nine tube system is approximately 112.3 kJ/kg, a value that is higher than that measured on the individual tube (108 kJ/kg). This value is considered to be high for an energy-absorbing core material and highlights the potential that these reinforced honeycombs offer for applications in dynamically-loaded structures.

Figure 18 shows photographs of a plain honeycomb sample and two reinforced systems following dynamic crushing. The plain sample has largely failed in the plastic folding mechanism observed following quasi-static testing, although there is evidence of a lower energy failure mode in one corner. The tubes in the reinforced systems have been crushed to a solid pulp, with there being little evidence of gross fragmentation of the composites. This is in contrast to the observations following the dynamic tests on the individual tubes, Figure 5bii, where individual strands and fronds are apparent on the platen. The pronounced change in failure mode may explain the increase in SEA values following dynamic tests on the modified cores.

Conclusions

A lightweight, energy-absorbing material has been developed by embedding small composite tubes in an aluminium honeycomb core. Compression tests at quasi-static rates of strain have shown that the specific energy absorption (SEA) is as high as 105 kJ/kg when 21% of the honeycomb cells are filled with tubes. An examination of the failed tubes indicated that the composite had been reduced to fine debris, suggesting that the composite tubes have indeed absorbed considerable energy. Significantly increasing the density of tubes beyond this value is counter-productive, due to the close proximity of the tubes to each other. Here, tubes that trigger first serve to destabilize their neighbours, causing them to tilt and fail in a low energy mode. It has been shown that it is possible to control the crushing force by varying the lengths of the tubes in the core. Although this has the effect of lowering the SEA of the core, it can be an attractive option if it is necessary to control deceleration forces in crash conditions.

Tests at impact rates of strain have shown that the SEA increases with strain-rate, with values as high as 112 kJ/kg being recorded. Here, the CFRP tubes were reduced to flat disc-like structures that extended well beyond the initial circumference of the reinforcing cylinders. The evidence from these dynamic tests, combined with the fact that these materials are simple to prepare, suggests that tube-reinforced honeycombs may offer potential for use in extreme conditions, such as those associated with blast loading.

Acknowledgements

This research was performed as part of the Aerospace Research and Innovation Center (ARIC) program which is jointly-funded by Mubadala Aerospace and Khalifa University.

References

[1]. G.C. Jacob, J.F. Fellers, S. Simunovic and J.M. Starbuck, Energy absorption in polymer composites for automotive crashworthiness, J. Composite Materials, 36, 2002, pp813-849.

[2]. P.H. Thornton, Energy absorption in composite structures, Journal of Composite Materials 13, 1979, pp247–262.

[3]. P.H. Thornton, J.J. Harwood and P. Beardmore, Fiber reinforced plastic composites for energy absorption purposes, Composites Science and Technology, 24, 1985, pp275-298.

[4]. C.H. Chiu, C.K. Lu and C.M. Wu, Crushing characteristics of 3-D braided composite square tubes, J. Composite Materials, 31, 1997, pp2309-2327.

[5]. H. Hamada, J.C. Coppola, D. Hull, Z. Maekawa and H. Sata, Comparison of energy absorption of carbon/epoxy and carbon/PEEK composite tubes, Composites, 23, 1992, pp245-252.

[6]. G.L. Farley, The effects of crushing speed on the energy-absorption capability of composite tubes, Journal of Composite Materials, 25, 1991, pp1314-1329.

[7]. D. Siromani, G. Henderson, D. Mikita, K. Mirarchi, R. Park, J. Smolko, J. Awerbuch and T.-M. Tan, An experimental study on the effect of failure trigger mechanisms on the energy absorption capability of CFRP tubes under axial compression, Composites: Part A, 64, 2014, pp25-35.

[8]. A. Brighton, M. Forrest, M. Starbuck, D. Erdman and B. Fox, Strain rate effects on the energy absorption of rapidly manufactured composite tubes, Journal of Composite Materials, 43, 2009, pp2183-2200.

[9]. F. Tarlochan and S. Ramesh, Composite sandwich structures with nested inserts for energy absorption application, Composite Structures, 94, 2012, pp904-916.

[10]. J. Ma and Y. Yan, Quasi-static and dynamic experiment investigations on the crashworthiness response of composite tubes, Polymer Composites, 34, 2013, pp1099-1109.

[11]. G.L. Farley and R.M. Jones. Crushing characteristics of continuous fiber-reinforced composite tubes, Journal of Composite Materials, 26, 1992, pp37-50.

[12]. D. Hull, A unified approach to progressive crushing of fiber reinforced composite tubes, Composites Science and Technology, 40, 1991, pp 377–421.

[13]. G.L. Farley, Effect of specimen geometry on the energy absorption capability of composite materials, J. Composite Materials, 20, 1986, pp390-400.

[14]. A.G. Mamalis, Y.B. Yuan and G.L Viegelahn, Collapse of thin walled composite sections subjected to high speed axial loading, Int. J. of Vehicle Design, 13, 1992, pp564-579.

[15]. P.H. Thornton and P.J. Edwards, Energy absorption in composite tubes, J. Composite Materials, 16, 1982, pp521-545.

[16]. Q. Liu, Z. Mo, Y. Wu, J. Ma, G.C.P. Tsui, Crush response of CFRP square tube filled with aluminum honeycomb, Composites Part B, 98, 2016, pp406-414.

[17]. G. Sun, S. Li, Q. Liu, G. Li and Q.Li, Experimental study on crashworthiness of empty/aluminum foam/honeycomb-filled CFRP tubes, Composite Structures 152, 2016, pp969-993.

[18]. R.D. Hussein, D. Ruan, G. Lu and I. Sbarski, Axial crushing behavior of honeycomb-filled square carbon fibre reinforced plastic (CFRP) tubes, Composite Structures, 140, 2016, pp166-179.

[19]. R.A. Alia, Z. Guan, N. Jones and W.J. Cantwell, The Energy-absorption Characteristics of Metal Tube Reinforced Polymer Foams, Journal of Sandwich Structures and Materials, 17, 2015, pp74-94.

[20]. R.A. Alia, W.J. Cantwell, G.S. Langdon, S.C.K. Yuen and G.N. Nurick, The Energy-absorbing Characteristics of Composite Tube-reinforced Foam Structures, Composites Part B: Engineering, 61, 2014, pp127-135.

[21]. D Karagiozova, DW Shu, G Lu, X Xiang, On the energy absorption of tube reinforced foam materials under quasi-static and dynamic compression, International Journal of Mechanical Sciences, 105, 2016, pp102-116.

[22]. A. P. Meran, T. Toprak and A. Mugan, Numerical and experimental study of crashworthiness parameters of honeycomb structures, Thin-Walled Structures, 78, 2014, pp87-94.

[23]. H. Zhao and C. Gary, Crushing behavior of aluminum honeycombs under impact loading, International Journal of Impact Engineering, 21, 1998, pp827-836.

[24]. M. Gotoh, M. Yamashita and A. Kawakita, Crush behavior of honeycomb structure impacted by drop-hammer and its numerical analysis, Materials Science Research International, 1996, 2, pp261-266.

[25]. D.W. Schmuesser and L.E. Wickliffe, Impact energy absorption of continuous fiber composite tubes, J. of Eng. Materials and Technology, 109, 1987, pp72-77.

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Figure 1. Examples of tube-reinforced honeycomb cores with (a) 8 CFRP tubes and (b) 17 CFRP tubes.

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Core A Core B Core C

Figure 2. Schematic showing the heights (in mm) of the nine tubes in the three graded core designs. The overall core thickness is 20 mm.

Figure 3. Typical quasi-static load-displacement traces for the plain CFRP, the plain aluminum and single tube-reinforced aluminum. H = plain honeycomb, T = single tube and H + T = single tube embedded in honeycomb.

Figure 4. The variation of the specific energy absorption of the composite tubes at quasi-static and dynamic rates with diameter to thickness ratio. The current tubes are shown as larger symbols (D/t=4) and the remaining data is taken from [20].

(i) (ii)

(a)

(i) (ii)

(b)

Figure 5. (a) Failure processes during crushing of the 6 mm diameter (D/t = 4) CFRP tubes and (b) the failed samples after testing. The left-hand photos (i) correspond to woven-wrapped tubes and the right-hand images (ii) to unidirectional pultruded tubes.

(a) (b)

Figure 6. X-CT images of the central sections of (a) a partially-crushed and (b) a fully crushed plain aluminum honeycomb sample.

(a) (b)

Figure 7. (a) Photograph of a crushed sample containing a single tube and (b) photograph of the remnants of the CFRP removed from the sample.

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Figure 8. Typical force-displacement traces following quasi-static tests on samples containing different numbers of tubes. The integers indicate the number of tubes in the sample.

(a) (b)

Figure 9. XCT images showing (a) the uppermost surface of the test sample following partial crushing and (b) a view from the lower surface of the same sample.

Figure 10. The variation of the specific energy absorption and the energy with number of tubes following quasi-static tests on the tube-reinforced honeycomb cores. The grey shaded area represents an upper-limit for the SEA based on the value for a single unsupported tube.

(a) (b)

(c) (d)

Figure 11. Photographs of crushed samples following quasi-static testing (a) 3 tubes (b) 5 tubes (c) 7 tubes and (d) 9 tubes.

Figure 12. Force-displacement trace following a compression test on a sample containing 17 tubes (areal density of tubes = 40%). The inset shows a specimen after testing.

Figure 13. Typical quasi-static compression force-displacement traces following tests on the graded cores shown in Figure 2.

Figure 14. The variation of SEA with average crushing force following quasi-static tests. The filled circles highlight the graded cores, the open circles the full length tube data and the grey filled symbol the aluminum honeycomb.

Figure 15. Typical force-displacement traces following dynamic tests on the composite tube (T), the aluminum honeycomb (H) and tube-reinforced honeycomb (H + T). The dotted lines are the equivalent quasi-static tests.

Figure 16. Typical force-displacement traces following dynamic crushing tests on honeycombs containing different numbers of tubes (solid lines). The quasi-static traces are included for comparison (dashed lines). The numbers of tubes are indicated.

Fig. 17. Comparison of the dynamic values of SEA with the quasi-static values for different numbers of embedded composite tubes.

(a) (b) (c)

Figure 18. Photographs of samples following impact testing (a) a plain honeycomb specimen (b) a specimen based on 3 tubes and (c) one based on 6 tubes.

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