Energy-Absorbing Capacity of Cellular Textile Composites: Modeling and Optimization

Transcription

1 Energy-Absorbing Caacity of Cellular Textile Comosites: Modeling and Otimization T.X. Yu, P. Xue and X.M. Tao Deartment of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Institute of Textile and Clothing, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong SUMMARY: In our revious studies a new tye of cellular textile comosites was develoed and it demonstrated high secific energy-absorbing caacity. In the cellular structure, each cell contained a hemisherical shell connected to a truncated conical wall (Configuration ). A theoretical model is roosed to quantify the contributions to energy dissiation from different deformation stages. Theoretical and exerimental results show that large lastic deformation of the truncated conical shell contributes most to the total energy absortion. Therefore, a new cell configuration is roosed, in which each cell is a truncated conical shell only (Configuration ). In comarison with Configuration, the grid-domed textile comosite with Configuration dislays a higher energy-absorbing caacity, lower eak force and almost constant force magnitude during its large deformation rocess. The effects of the cell height, the diameter ratio of cell-to to cell-bottom and the resin added-on ercentage on the energy-absorbing caacity are also investigated. KEYWORDS: energy, absortion, textile, cellular.. INTRODUCTION Energy absortion characteristics is one of the most imortant toics in selecting materials for many engineering alications, such as crushing elements in cars, bicycle s helmets, hard hats used in construction sites and rotective ackaging of fragile goods, etc. As summarized by Johnson & Reid [] and Yu [], the materials or devices used as imact energy absorbers should utilize the inelastic deformation as its major energy-absorbing mechanism, the force required for its deformation should remain almost constant, and the deformation stroke should be long and in a stable mode. In addition to these requirements, attributes of lightweight can also lay an imortant role in the alications listed above. Cellular solids, which contain a large amount of sace and normally have a relative low density, ossess excellent energy absortion caacity. Previous studies on the behaviour of cellular materials have been mainly limited to olymer foams [,], honeycombs (under lateral load [5,6] or axial load [7]), metal ring systems and tube arrays [8,9,] and woods [,]. Only in last few years, the energy-absorbing caacity of textile comosites has attracted attention of researchers. The energy-absortion behaviour of textile comosites have been investigated through several comonents, such as the braided comosite I-beams for crushing elements in cars [], triaxially braided comosite square tubes as crushing elements [,5] and -D woven sandwich structure [6].

2 In our revious studies we investigated a range of grid-domed cellular textile comosites in term of their energy-absortion behaviour under quasi-static comression and imact condition [7-9]. The resent aer will resent a theoretical model to study the quasi-static and dynamic behaviour of grid-domed textile comosites, based on which this kind of cellular textile comosite can be otimized in view of its secific energy-absorbing caacity.. MATERIALS AND SAMPLES WITH CONFIGURATION The reinforcements for the grid-domed comosites were knitted fabrics made of multifilament textured nylon yarns. In order to characterize the quasi-static mechanical roerties of the double jersey-nylon (DJ-N) tye, flat comosite samles with similar ercentage of olymer add-on, which had been rigidized by the same rocess as grid-domed cellular comosites, were tested under tension and bending conditions [9]. For the simle tension samles with size mm mm mm, the loading-unloading curves are shown in Fig., which demonstrate significant visco-elasticity. The flat fabrics were formed into a D cellular structure by a two-ste method, forming and consolidation by a olyester resin. The cells were arranged in a grid attern, whilst each cell contained a hemisherical ca connected to a truncated conical wall (Configuration ). The schematic geometry of this configuration is shown in Fig.. Force (KN) War Weft Dislacement (mm) Fig. : Loading-unloading; Fig. : Schematic geometry of a cell of samles with Configuration.. MECHNICS MODELS OF GRID-DOMED TEXTILE COMPOSITES As observed in both quasi-static and imact tests, the major deformation mechanism for griddomed comosite samles DJ-N is the collase of the shells, including both hemisherical ca and truncated conical wall in the cell (a single cell is shown in Fig. ). In the following, three Fig. : Deformation rocess of a grid-domed comosite samle DJ-N.

3 successive theoretical models are roosed for the observed three stages of the deformation rocess, resectively, so as to redict the force-dislacement relationshi and the energyabsorbing caacity.. Stage : Local inversion of a hemisherical dome As well known, when a hemisherical dome is subjected to a concentrated force at its aex or comressed by a rigid flat late on its to, elastic or elastolastic sna-through will occur, deending on the dimension of the dome and the material s roerties. Aart from numerous elastic analyses, based on an analysis given by Udike [], a simle but rofound rigidlastic model was resented by Calladine [, ]. Figure shows, schematically, the local inversion model emloyed by Udike and Calladine, under a comression of a rigid flat late subjected to an inward-directed force P. At any instant in the rocess, the inverted region is searated from the outer ortion of the hemisherical dome by a narrow toroidal knuckle, that moves outwards as deformation roceeds. The knuckle region is connected to the two sherical ortions by two travelling circumferential lastic hinge circles, of radius e ± l/, with l being the meridian extent of the knuckle. By formulating and solving the equations which have been obtained by roer consideration of the kinematics, equilibrium and constitutive relations of the roblem, it is finally obtained that [-] l = / R h =. R h () P = π Yh w = 9. Yh w = 6. Mo ( w/ h) () where Y is the uniaxial yield stress of the material, M o = Yh / is the fully lastic bending moment er unit width of the shell, and w is the vertical dislacement of the rigid late alied on the to of the hemisherical dome. Subscrit of the alied force ertains to deformation Stage. If each grid-domed samle contains n domes, the total force alied on the samle will be (P ) total = n P =.6 nm o (w/h).5 () Accordingly, the energy dissiation from w = to a certain dislacement w can be calculated as w W = ( P ) dw =9. nm w h total o Equations () and () rovide a theoretical force-dislacement relationshi and an estimate of the energy-absorbing caacity, resectively, of the grid-domed cellular textile comosites in their early stage (i.e. Stage ) under comression or imact loading. () Fig. : Schematic diagram of local inversion model. Fig. 5: Deformed configuration in stage.

4 . Stage : Global lastic collase of a truncated sherical ca The above local inversion mechanism of the hemisherical domes is alicable only when the suort of the sherical dome at its base circle remains fixed, so this mechanism will cease to be valid when the alied force is so large that a lastic hinge circle forms at the base of the artial sherical ca followed by a multi-hinge deformation mechanism occurred in the truncated conical shell underneath the sherical ca, see Stage described below. Since the sherical dome has been deformed in Stage, the analysis of Stage is based on the deformed configuration (Fig. 5). By assuming that a lastic hinge circle forms at the base of the truncated sherical ca and assessing the lastic energy dissiation by both bending and membrane deformations, a force-dislacement relationshi can be established as P π cosβ 6πR = + tanφsinφcos( φ + β) M o cosφcos( φ + β) h (5) and (refer to Fig. 5) w= R [ sin β cos( φ + β) sin φ ] (6) where angle β is secified by the original sherical ca, and angle φ, which can be calculated from w/r by using Eqn (6), will serve as a rocess arameter. When the force required by the global lastic collase mechanism of the truncated sherical ca calculated by Eqn. 5, P, is greater than the force required by the local inversion of the sherical ca calculated by Eqn. (), P, the local inversion will continue to develo. However, when P < P, the global collase mechanism will take over and the local inversion ceases to develo.. Stage : Large lastic deformation of a truncated conical shell After a lastic hinge forms at the connecting circle between the sherical ca and the truncated conical shell, bending moments as large as the fully lastic bending moment begin to aly on the to of the truncated conical shell. When the artial sherical shell rotates about the lastic hinge circle at its base, this bending moment and the comression force roduced by the rigid flat late alied on the to of the dome will initiate a multi-hinge deformation mechanism in the truncated conical shell (shown in Fig. 6). At this deformation stage (i.e. Stage ), although the bending along the hinge lines and the bending of the conical shell in the circumferential direction will make a certain contribution to the energy dissiation, the energy-absorbing caacity of the cell is dominated by the membrane deformation of the conical shell, esecially when the total vertical dislacement becomes very large. In the membrane-dominated deformation mechanism, merely the energy dissiated by the comression of the generators of the truncated conical shell is accounted. By noting that DJ-N samles dislay a bi-linear stress-strain relation in their tensile behaviour (refer to Fig. ), this analysis concludes that, in Stage, a linear relation, as given by equation (7), can be established between the comression force and the vertical dislacement; whilst the sloe of the force-dislacement curve can be redicted accordingly by the following simle exression: ( P ) total = π nrh cos β ( Y + w E cos β / H ) (7) d( P n Rh ) total d w H E = π cos β (8) ( ) which reresents a linear relation between the force required and the increment of the vertical dislacement from the to, w.

5 .. Estimation of the energy-absorbing caacity of grid-domed comosite samles Based on our exerimental data, Stage contributes most to the total energy-absortion of DJ-N samles. It may be suggested, therefore, to emloy Eqn (8) to estimate of the total energy-absorbing caacity W w of these comosites till any dislacement w, before densification haens. Thus, integrating Eqn (8) twice leads to w πrh Ww = ( P ) total dw n E cos β w (9) H Accordingly, when the energy absortion er unit mass of the samle is concerned, the following simle exression can serve as an estimate: Ww πrh m H E = w cos β () µ where W w denotes the energy absortion of the samle until dislacement w, m denotes the mass of the samle and µ denotes the mass of a unit cell. In a more general case, E can be regarded as the average tangential modulus of the material during its lastic deformation. Eqn. () is valid for the grid-domed textile comosites of Configuration studied under quasi-static comression. To estimate the energy-absorbing caacity of these cellular comosites under imact or comression at a high rate, Eqn () can be combined with strainrate effect of this material, leading to Ww πrh m H E w B c = cos β + log! () µ A c! o where A and B are the exerimentally determined constants and!c o is a low comression-rate served as the basis of comarison. The rediction of the above deformation models was found to be in excellent agreement with our exerimental results [9]. Fig. 6: Multi-hinge deformation mechanism in Stage. Fig. 7: A samle with Configuration.. OPTIMIZING ENERGY-ABSORBING CAPACITY OF GRID-DOMED TEXTILE COMPOSITES As illustrated above, the deformation rocess of a cell with Configuration consists of three stages, but Stages and are much shorter and make much less contribution to the total energy absorbed. In other wards, the energy absortion is mainly attributed to the lastic deformation of the truncated conical shells. Accordingly, a new cell configuration has been designed (Configuration ), in which each cell consists of a truncated conical shell only, as shown in Fig. 7. The materials used here were the

6 same as those for Configuration. Imact tests were conducted under imact velocities of.-.8 m/s. Under the identical test condition for samles with mm and the same weight, the load-time curves and energy-dislacement curves for the two configurations are given in Fig. 8 and Fig. 9, resectively. The energy-absorbing caacities for both configurations are summarized in Table, where the data in the columns with unit J denote the total energy dissiated, and those with unit J/g denote the energy dissiation er unit weight of the samle. The data in Table indicate that, comared with those of Configuration, the grid-domed textile comosites of Configuration have a higher energy-absorbing caacity (i.e..7-. time u to mm vertical dislacement, and.-.8 time u to 8mm vertical dislacement), low eak force and almost constant magnitude of force during their large deformation rocess. Load (KN) Time (msec) Configuration Configuration Fig. 8: Load-time curves for samles of two configurations. Energy (joule) Configuration Configuration 6 8 Fig. 9: Energy-dislacement curves for samles of two configurations. Table. Comarison of energy-absorbing caacity for both configurations E : imact energy V : imact velocity Energy absortion U to mm Dislacement Energy absortion U to 8mm Dislacement J J/g J J/g E = J Conf V =.7 m/s Conf H=.8mm E =5±% J Conf V =.8 m/s Conf. H=.8mm INFLUENCE OF CELL GEOMETRY AND RESIN ADD-ON UPON ENERGY-ABSORBING CAPACITY In order to further otimize the characteristics and energy-absorbing caacity of the griddomed textile comosites, cell height and diameter ratio of cell to to cell bottom were selected to exlore the influence of geometry on the mechanical roerties and energyabsorbing caacity. The effect of resin added-on ercentage on the energy-absorbing caacity was also investigated. Figures and deict the curves of load vs. vertical dislacement and the energy dissiation er unit weight vs. vertical dislacement under the imact velocity of.5 m/s and imact energy of 6 joule for the samles with the same level of resin added-on and same diameter ratio of cell-to to cell-bottom, but different cell heights. Cell heights in the range

7 from 7.8 mm to 5.7 mm were examined, while the corresonding semi-aical angles of the truncated conical shell varied from 7 to. The eak force of curve was resulted from the contact between the tu and the suorting base. Hence, the cell height should not be too Load (KN) --H=7.8mm --H=.5mm 8 --H=.8mm 6 --H=5.7mm Deflection (mm) Energy er unit weight (joule/gram) --H=7.8mm --H=.5mm --H=.8mm --H=5.7mm Fig. : Comarison for load vs. vertical dislacement. Fig. : Comarison for energy er unit weight vs. vertical dislacement. small for obtaining higher energy-absorbing caacity. From other three samles, it is evident that the magnitude of load reduced and the total stroke increased with the increase of the cell height. The samles with higher cells generally ossess greater otential of energy absortion in a longer stroke, although their energy-absorbing caacity is restricted when the imact velocity is small. Load (KN) 9 6 -Td=mm -Td=7mm -Td=mm -Td=mm 5 5 Energy er unit weight (joule/gram) -Td=mm -Td=7mm -Td=mm -Td=mm 5 5 Fig. : Comarison for load vs. vertical dislacement. Fig. : Comarison for energy er unit weight vs. vertical dislacement. Figures and resent the curves of load vs. vertical dislacement and energy er unit weight vs. vertical dislacement under the imact velocity of.5 m/s and imact energy of 5.9 joules for the samles with the same ercentage of resin added-on and the same cell height, but different diameter ratio of cell-to to cell-bottom. The to diameter of a cell varied from mm to mm, whilst the bottom diameter remained constant. From the test results, it is evident that the smaller to diameter is not beneficial to energy-absorbing caacity of the comosite. The otimal to diameter within the examined arameter range is mm. Figures and 5 deict the curves of load vs. vertical dislacement and energy er unit weight vs. vertical dislacement under the imact velocity of.8 m/s and imact energy of 8.5 joules for the samles with the same cell dimensions, while the ercentage of resin added-on varied from 55% to %. From the figures, it is found that the ercentage of resin

8 added-on has little difference on the secific energy-absorbing caacity within the investigated range. Load (KN) add on=5% add on=% add on=55% 5 5 Energy er weight (Joule/gram) add on=5% add on=% add on=55% 5 5 Fig. : Comarison for load vs. vertical dislacement. Fig. 5: Comarison for load vs. vertical dislacement. CONCLUSION The energy absorbing caacity of the grid-domed textile comosite can be redicted well by the theoretical models roosed in the aer, esecially in Stage. Comared with samles of Configuration, samles of Configuration ossess lower eak force, almost constant force magnitude during its large deformation rocess and considerably higher secific energy-absorbing caacity. The higher the cell, the lower the eak load and the greater otential for energy absortion in a longer stroke. The energy-absorbing caacity also varies with the diameter ratio of cell-to to cell-bottom. Under the condition of constant cell height of.8 mm, the otimal to diameter is found to be mm for the examined arameter range. The resin added-on exhibits little influence in the energy-absorbing caacity of grid-dome textile comosites for the examined range. ACKNOWLEDGMENT The authors would like to thank the Hong Kong Research Grant Council (RGC) for the suort to this research under contract HKUST 67/98E. REFERENCES [] Johnson W. and Reid S.R., Metallic Energy Dissiating Systems, Alied Mechanics Review, ASME, 977, [] Yu T.X, Energy-Absorbing Devices Utilizing Plastic Deformation of Metals, Advances in Mechanics, Vol. 6, No., 986, [] Gibson, L.J. and Ashby, M.F., Cellular Solids: Structure and Proerties. Pergamon Press, New York, 988. [] Mills, N.J., Chater 9 in Low Density Cellular Plastics, Physical Basis of Behaviour (eds. N.C. Hilyard and A. Cunningham). Chaman and Hall, London, 99. [5] Klintworth, J.W. and Stronge, W.J., Elasto-Pastic Yield Limits and Deformation Laws for Transversely Crushed Honeycombs, Int. J. Mech. Sci., Vol., 988,. 7-9.

9 [6] Stronge, W.J., Dynamic Crushing of Elastolastic Cellular Solids, In Mechanical Behaviour of Materials IV (ICM6), Vol. (M. Jono and T. Inoue Eds.). Pergamon Press, Oxford, 99, [7] Wierzbicki, T., Crashing Analysis of Metal Honeycombs, Int. J. Imact Engng., Vol., 98, [8] Reid, S.R., Bell, W.W. and Barr, R., Structural Plastic Model for One-Dimensional Ring Systems, Int. J. Imact Engng., Vol.,98, [9] Shim, V.P.W. and Stronge, W.J., Lateral Crushing in Tightly Packed Arrays of Thin Walled Metal Tubes, Int. J. Mech. Sci., Vol. 8, 986, [] Stronge, W.J. and Shim, V.P.W., Microdynamics of Crushing in Cellular Solids, Trans. ASME, J. Engng. Mater. Technol., Vol., 988, [] Reid, S.R., Peng, C. and Ready, T.Y., Dynamic Uniaxial Crushing and Penetration of Wood, in Mechanical Proerties of Materials at High Rates of Strain (ed. J. Harding), Inst. Phys. Conf. Series No., Bristol, 989, [] Reid, S.R. and Peng, C., Dynamic Uniaxial Crushing of Wood, Int. J. Imact Engng., Vol. 9, 997, [] Hamada H., Can Braided Comosites Be Used for Crushing Elements in Cars?, 997, Proc. ICCM-, Vol. I, [] Chiu C.H. et al., Energy Absortion of Three-Dimensional Braided Comosite Tubes, 995, Proc. ICCM-, Vol. IV, [5] Chiu C.H. et al., Crush Behavior of Triaxially Braided Comosite Tubes with Different Axial Yarn Content, 997, Proc. th Asian Textile Conf [6] Dreshsler K. et al., Energy Absortion Behaviour of D Woven Sandwich Structures, 995, Proc. ICCM-, Vol. V, [7] Tao X.M., Yu T.X., Ngan K.M. and Ko F.K., Energy Absortion of Cellular Textile Comosite under Quasi-Static Comression, Hawaii, USA, July, 997, Proc. ICCE/, , [8] Yu T.X., Tao X.M. and Wu K.Q., Energy Absortion of Cellular Textile Comosite under Imact, Hawaii, USA, July, 997, Proc. ICCE/,.99-. [9] Yu T.X., Tao X.M. and Xue P., Energy-Absorbing Caacity of Grid-domed Textile Comosites, to be ublished in Comosite Science and Technology. [] Udike, D.P., On the Large Deformation of a Rigid-lastic Sherical Shell Comressed by a Rigid Plate, ASME Journal of Engineering for Industry, No. 9, 97, [] Calladine, C.R., Analysis of Large Plastic Deformations in Shell Structures, in Inelastic Behaviour of Plates and Shells (Proc. of IUTAM Symosium, Rio de Janerro, 985), L. Bevilacqua, R. Feijoo and R. Valid Eds.,. 69-, Sringer, 986. [] Calladine, C.R., Some Problems in Proagating Plasticity, in Plasticity and Imact Mechanics, N.K. Guta Ed,. 7-97, Wiley Eastern Ltd., New Delhi, 99.