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Axial crushing of embedded multi-cell tubes
Accepted Manuscript
Axial crushing of embedded multi-cell tubes Xiong Zhang , Kehua Leng , Hui Zhang PII: DOI: Reference:
S0020-7403(17)31428-5 10.1016/j.ijmecsci.2017.07.019 MS 3805
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
28 May 2017 2 July 2017 9 July 2017
Please cite this article as: Xiong Zhang , Kehua Leng , Hui Zhang , Axial crushing of embedded multicell tubes, International Journal of Mechanical Sciences (2017), doi: 10.1016/j.ijmecsci.2017.07.019
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Highlights A new type of embedded multi-cell (EMC) tubes is proposed and investigated.
Quasi-static axial crushing tests are performed for empty and EMC tubes.
Numerical simulations are carried out and compare well with experiment.
Theoretical expression is obtained to predict the crush resistance of EMC tubes.
Interaction effect is found to account for 40% of the total crush force.
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Axial crushing of embedded multi-cell tubes Xiong Zhang a, b,, Kehua Leng a, Hui Zhang c a
Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, PR China
b
Hubei Key Laboratory of Engineering Structural Analysis and Safety Assessment, Luoyu Road 1037, Wuhan,
c
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430074, China
School of Mechanical Engineering and Automation, Wuhan Textile University, Wuhan 430073, Hubei, PR China
Abstract
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The axial crushing resistance of a new type of embedded multi-cell (EMC) tubes is investigated in this paper. Quasi-static tests are carried out first to investigate the deformation and force response of both empty square tubes and EMC tubes. The components of the crush force of EMC tubes are analyzed quantitatively. Nonlinear explicit finite element method is then employed to simulate the crushing process and the numerical results compare well
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with experiment. By using the FE model validated by experiment, parametric study is then performed to investigate the energy absorption performance of embedded tubes with various configurations. Finally, theoretical
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analysis is conducted to analyze the components of crush resistance of the whole structure and an expression is derived to predict the mean crushing force of EMC tubes. Theoretical predictions are in good agreement with experimental and numerical results.
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Keywords: Multi-cell; Axial crush; Energy absorption; Experimental test; Embedded tubes.
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1. Introduction
Commercial hexagonal aluminum honeycombs [1-2] are widely applied in aerospace
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engineering as core of sandwich structures due to their high specific strength, rigidity and energy absorption capacity. As two-dimensional cellular structures, honeycombs are anisotropic with very much better out-of-plane strength than their in-plane property. Compared to the isotropic aluminum foam, this anisotropic feature guarantees better specific strength when loaded in the
Corresponding author. Tel.: +86 27 87543538; fax: +86 27 87543501. E-mail address: [email protected] (X. Zhang). 2
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out-of-plane direction. For instance, Santosa and Wierzbicki [3] reported that honeycomb outperformed foam material in improving specific energy absorption (SEA, energy absorption per unit mass) of box columns under axial crush when served as inside fillers. Actually, the behavior of metal tubes filled with foam materials has received enormous investigations [4-9]. The energy
expressions were derived to predict the response of them.
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absorption mechanisms of them under axial or transverse have been disclosed and theoretical
Recently, the crashworthiness performance of multi-cell sections under various load
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conditions attracted extensive interests [10-22]. Many experimental and numerical investigations validated that multi-cell sections could provide considerably higher energy absorption efficiency than single cells. Compared to filling structure with cellular material, adopting multi-cell section
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seems a much better option to improve the crashworthiness. A comparative study conducted by Zhang and Cheng [23] showed that the SEA value of aluminum multi-cell columns was about
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50-100% higher than that of simple column filled with aluminum foam, when the mass was kept
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the same.
Actually, commercial aluminum honeycombs are also a type of multi-cell structure with thin
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wall thickness and periodic structural configuration. Aluminum foils are always used to fabricate
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honeycombs by gluing and tension process while multi-cell columns are generally produced by extrusion forming. Unfortunately, the present fabrication technique is in lack of flexibility or leads to low cost-effectiveness. Only few sectional shapes of honeycombs can be produced by the gluing and tension process. Meanwhile, for the extrusion of multi-cells, new mold is required for every cross-sectional shape and any change in dimensions. This apparently increases the production cost of multi-cell sections and limits the wide applications of them. To overcome this
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drawback, a novel type of embedded multi-cell (EMC) section is proposed here to satisfy the requirement of high cost-effectiveness and easy availability. As shown in Fig. 1, EMC sections are constituted by a group of single tubes confined within an outside thin-walled envelope. The outside envelope can be a larger tube or a plate being folded
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and welded to contain the inside single tubes. Of particular interest in the present work is to investigate the crush resistance of EMC tubes and the sections in Fig. 1(A) are investigated experimentally, numerically and theoretically. The crush resistance of EMC tubes should be larger
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than the summation of that of the outside and inside empty tubes due to similar mechanisms as foam-filled columns. The interaction between the tubes will dissipate more energy and a quantitative investigation is necessitated to determine the interaction effect. Experimental study
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and numerical analysis are performed in this work to deal with this problem and a theoretical expression is also presented to predict the crush resistance of EMC tubes.
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2. Experimental study
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2.1 Scheme and specimen preparation The energy absorption characteristics of a group of EMC tubes are experimentally
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investigated in this section. As shown in Fig. 2, the EMC tubes are constituted by empty extruded
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aluminum square tubes with the outside width C of 10, 15 and 32 mm. The wall thickness t of the tube with C=32 mm is 0.95 mm while the thickness of the other two tubes is 1.0 mm. All the tubes are cut to the same length of 75 mm. Four square tubes with C=15 mm and nine those with C=10 mm are inserted into the square tubes with C=32 mm. Three specimens are prepared for each type of tubes and more specimens are tested when necessary. The structural material of the tubes is AA6063 O. Although the square tubes are provided by
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the same supplier, the material hardening data for the tubes with different dimensions are obtained by tensile tests based on the standard test methods in ASTM E8M-04. The tensile tests are performed in a 10 kN capacity Zwick Z010 universal material testing machine and the specimens for the tests are cut along the longitudinal direction of the square tubes. The representative
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engineering stress-strain curves for AA6063 O tubes are plotted in Fig. 3. As shown in the figure, the difference between the three curves is small and one of the curves is selected to be used in the subsequent numerical analysis. The material hardening data for the selected curve are listed in
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Table 1. The mechanical properties of the material AA6063 O are also presented here: Young's modulus E = 68.9 GPa, Poisson's ratio ν = 0.33, initial yield stress σy = 31.8 MPa and ultimate stress σu = 91.2 MPa.
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2.2 Experimental results
For the sake of convenience, the specimens are named by the following rules: "C", "t" and
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"L" denote the outside width, thickness and length of the tube, respectively; "M" means
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"multi-cell"; "N" is the test number of repetitions. The default length of the specimens is 75 mm and "L" can be omitted in this case. For example, C32t0.95 means empty tube with C=32 mm,
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t=0.95mm and L=75 mm. According to the specimens prepared for the tests, the default outside
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tube of EMC tubes is C32t0.95 and the status of the inside tubes is given within parentheses. For instance, M(9C10t1) represents an EMC with 9 cells and the dimensions for the inside tube is C=10 mm, t=1 mm and L=75 mm. If necessary, the dimensions for the outside tube can be added in the designation, e.g. MC34t2L60(9C10t1) means a multi-cell tube with the outside tube C34t2L60 and 9 inside tube C10t1. The deformed shapes of the empty and EMC specimens are shown in Fig. 4. For empty tubes,
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C32t0.95 and C15t1 deform in quasi-inextensional [24] or symmetric collapse mode [25] of square tube, while global buckling is present for C10t1. Only three lobes are formed for C32t0.95 and in this case, the mean crushing force may differ significantly from the tubes in normal state. The development of two end lobes will lead to big deviation as reported by Zhang et al. [26]. To
are also given in the figure. Seven lobes are formed in this case.
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remove this disturbance, tubes with the length of 150 mm are tested and the deformation modes
The outside tubes of the EMC tubes M(4C15t1) deform in inextensional mode while those of
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the multi-cell tubes M(9C10t1) develop a quite novel mode which was not reported before. One side wall of the tube bulges outward while the other side walls still fold progressively. A more detailed illustration for deformation of the multi-cell tubes is offered in Fig. 5. The front view has
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been given in Fig. 4 and the other four views are provided in Fig. 5. For the tube with four cells, the deformation is quite regular and every corner elements develop inextensional mode. The tube
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with nine cells deforms in a more complicated way. A sectional view shows that the inside nine
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cell tubes deform in global buckling mode although they are subjected to the constraint of the outside tube. The deformation of the outside tube is also to some extent affected by the inside
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tubes, but it is still sustainable and resistant to the crush loads. From the viewpoint of the whole
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structure, it can still be deemed as a progressive mode since the crush resistance of the structure does not decrease dramatically. The crushing force-displacement curves of the specimens are presented in Fig. 6. The force
responses of empty tubes C32t0.95L150 and C15t1 show all standard features of progressive folding. As expected, the short specimens of C32t0.95 exhibit some irregular force response and may result in an overestimation of the mean crushing force. The force curves of the small empty
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tubes C10t1 show representative features of the structures failing in global bending: the force increases to a peak and then drops down. As for EMC tubes, the force responses of the tubes with four cells and nine cells are slightly different. There is a fast drop after the initial peak for the tubes with four cells while only slight
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and smooth drop is observed for those with nine cells. This is apparently associated with the deformation mode. Progressive buckling always leads to a fast drop after the initial peak just as the curves in Fig. 6(C)-(E), while the global bending failure results in a smooth drop similar as the
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curves in Fig. 6(F). As for M(9C10t1), the global buckling of inside tubes is constrained by the outside tube and the force drop is hence stopped and finally reversed due to the densification of inside tubes. Furthermore, good repeatability is observed in the tests. The three repetitions show
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almost the same collapse mode and similar force responses.
Since the inside tubes of M(9C10t1) buckle globally during the axial crushing, some
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specimens with shorter length are tested. The length of empty tube C10t1 is reduced gradually and
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the global bending switches to progressive folding at L=25 mm. As shown in Fig. 7(A), extensional mode is developed for these short tubes instead of quasi-inextensional mode. The
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force curves of C10t1L25 are also given in the figure. Interestingly, the features of these curves
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are similar to those for EMC tubes with nine cells. There is no high initial peak and fast drop of the force. Multi-cell tubes M (9C10t1) with L=60 mm are also tested and the results are given in Fig. 7(B). Similar deformation and force responses are obtained but a slightly higher force level is observed.
2.3 Analysis and discussion The crushing resistance and energy absorption of the embedded tubes are analyzed in this
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section. Energy absorption E is obtained by integrating the crushing force with respect to crushing distance δ and mean crushing force Pm is defined as the ratio of energy absorption E to the distance δ. The effective crushing distance δe is determined just before the densification stage of a structure. The related energy absorption indices of the specimens tested are listed in Table 2. As
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shown in the table, for C32t0.95, the tubes with larger length do provide more stable mean crushing force Pm although the average Pm is almost the same. The mean force of C32t0.95L150 is hence employed in the later analysis. For C10t1, the tubes with L=25 mm deform in extensional
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mode and give much higher Pm than those deforming in global bending with L=75 mm.
As listed in Table 2, the mean force of EMC tubes is larger than the simple summation of the mean crushing forces of all constituent tubes. Apparently, this is caused by the interaction effect
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similar as that for foam-filled tubes. However, the interaction effect is more complicated here since it includes not only the interaction between inside and outside tubes, but also the interaction
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between different inside tubes. To analyze the interaction effect of EMC tubes, the mean crushing force can be expressed as
(1)
Pmc Psum Pint
(2)
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Pmc Po u t s id N e Pi n s i d ePi n t
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or
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Pmc, Poutside and Pinside are mean crushing force of the whole multi-cell tube, the outside tube and the inside filling tube, respectively. N is the number of inside tubes, Psum is the simple summation of the mean forces of all constituent tubes and Pint is the crushing force caused by interaction effect between the constituent tubes. The interaction effect of EMC tubes is analyzed in Table 3. As shown in the table, for M(4C15t1), the interaction effect accounts for 29.0% of the whole crush resistance. Due to the low
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crush resistance of C10t1 caused by global failure, the interaction effect of M(9C10t1) is summed up to 56.1% of the structural resistance. If C10t1L25 which develops extensional mode is adopted, the ratio of Pint to Pmc is reduced significantly to 7.4%. The mean force of ML60(9C10t1) is slightly higher and accordingly the interaction effect is increased. Apparently, the interaction
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effect is associated with the relative strength of the inside versus outside tubes. The change of geometrical parameters will definitely affect the interaction effect significantly and this will be further investigated by numerical simulation in next section.
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3. Numerical investigation 3.1 Finite element modeling
Nonlinear explicit finite element code LS-DYNA is employed to simulate the axial crushing
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of empty and EMC specimens. LS-PREPOST is used to pre- or post-process the data files. A representative finite element model of EMC tube is illustrated in Fig. 8. The specimens are loaded
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by two rigid walls with a stationary one in the bottom and one moving downward from the top
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with a prescribed velocity. Belytschko-Tsay 4-node shell elements with five integration points through the thickness are employed to mesh the structures and the characteristic element size of
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the meshes is set to 1.0 mm.
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Automatic single surface contact is employed to consider the self-contact of specimens and
automatic node to surface contact is applied to simulate the contact between specimens and rigid walls. When EMC tubes are analyzed, automatic surface to surface contact is used to simulate the interaction between outside envelope tube and inside embedded tubes. The friction coefficients in all these contacts may be different, but they are assumed to be 0.3 here for the sake of simplicity. Calculations show that only negligible difference (smaller than 2%) is observed for axial crush
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resistance of EMC tubes when the friction coefficient is varied from 0.3 to 0.8. The structural material is modeled by piecewise linear plasticity model with the material hardening data given in Table 1. To finish the quasi-static simulation in a reasonable time, the load velocity is accelerated to 1m/s and the principles proposed by Santosa et al [7] are followed. Previous quasi-static
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simulations [26-27] based on these principles showed very good agreement with experiment. To induce thin-walled structures to deform in the similar deformation mode as experiment, triggers are always necessitated. In the present analysis, indentation triggers are introduced in the
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cell walls of the tubes. The location of the triggers can be set according to deformation of specimens in experiment. For instance, as shown in Fig. 8, indentation triggers are introduced in four locations for M(4C15t1) as boxed in the figure. The position of the triggers is set to 6 mm
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distant from the top or bottom of the tube. Both inside and outside tubes are indented and the depth of the trigger is 0.3 mm. The introducing of trigger may have some influence on the initial
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peak force, but there is only negligible effect on mean crushing force and energy absorption of the
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structures.
3.2 Simulation of the tests
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The simulated deformation patterns for the specimens are plotted in Fig. 9. For EMC tube
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M(4C15t1), extensional mode is developed when trigger is not introduced. The same deformation mode as experiment is obtained when the indentation trigger in Fig. 8 is present. The outside square tube deforms in quasi-inextensional mode with seven lobes formed. However, for tube M(9C10t1), the numerical simulations with various trigger forms fail to induce the global buckling of the inside embedded tubes and can not obtain the same deformation mode as experiment. The EMC tube without trigger deforms in extensional mode and the one with trigger develops a mixed
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mode of extensional and quasi-inextensional folds. A check of the deformation for the inside tubes shows that progressive folding instead of global buckling is developed. In the meantime, the simulated deformation modes of empty tube C32t0.95L150 and C15t1 are in good agreement with experiment. The tube C10t1 deforms in global bending failure and no effort is exerted here to
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simulate its behavior. The crushing force versus displacement curves in experiment and simulation are compared in Fig. 10. It is found that the simulated results of multi-cell tube M(4C15t1) and empty tubes
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compare very well with measured data in the experimental test, although the force level is slightly underestimated. In addition, the effective crushing distance is overestimated in all simulations. The densification process is postponed in the numerical analysis. The crushing force of M(9C10t1)
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in simulation is considerably lower than that in experiment and most importantly, the features of the force curves in the two cases are quite different. Apparently, the difference in deformation
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mode should be responsible for this. In addition, it is noted that the difference in force level is very
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small for different progressive buckling mode of EMC tubes. As shown in Fig. 10(A) and (B), the difference between the curves with and without trigger is not big although the tubes deform in
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different progressive mode.
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The mean crushing force Pm of the specimens in simulation is compared with experiment in
Table 4. The effective crushing distance δe to calculate the Pm in simulation is defined as δNum and it is also given in the table. The numerical simulations generally underestimate the crushing resistance, but the results are satisfactory. The maximum error is smaller than 6% for empty tubes and it is about 8% for EMC tubes. It should be mentioned that although the error of Pm for M(9C10t1) in simulation is not big, the simulation results of it may not be meaningful since the
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features of deformation pattern and force response curve in experiment and simulation are quite different. In addition, the maximum difference in Pm between different progressive modes of multi-cell tubes is about 3%. This is favorable to perform parametric study on the crush resistance of such EMC tubes. Any type of trigger can be chosen since no big difference in Pm will be caused
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when progressive buckling mode is developed. It is concluded that the numerical model generally give a good simulation of the specimens except for the case when the inside tubes fail in global buckling.
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3.3 Parametric study on EMC tubes
The energy absorption performance and interaction effect of EMC tubes under axial crushing should be highly correlated with the geometric parameters. The influences of the number of inside
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tubes and the dimensions of the inside and outside tubes on crush resistance of EMC tubes are investigated in this section. The length of all tubes is set to 75 mm and the aspect ratio of inside
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tube is constrained to be no less than 15 to guarantee progressive collapse and avoid similar case
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as M(9C10t1).
The specimens with various configurations are listed in Table 5. Nine four-cell tubes and
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three nine-cell tubes are analyzed. For four-cell tubes, the outside width of the inside tube is set to
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15 mm while three different thicknesses: to=0.5, 0.75 and 1mm are analyzed. At the same time, the inside width of the outside tube is 30 mm with also three wall thicknesses: ti=1, 1.5 and 2 mm. For nine-cell tubes, the inside tube is invariant to be C15t1 while the above three thicknesses are employed for the outside tube. The same indentation triggers as employed for M(4C15t1) and shown in Fig. 8 are introduced here. Almost all outside tubes of four-cell embedded tubes deform in quasi-inextensional mode while those of nine-cell tubes develop extensional mode. Deformed
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shapes of two representative specimens: MC33t1.5(4C15t1) and MC48t1.5(9C15t1) are given in Fig. 11. The crushing force versus displacement curves of the EMC tubes are plotted in Fig. 12. It is noted that the force level increases significantly when the wall thickness of inside or outside tube is increased.
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The energy absorption indices including structural mass, energy absorption E, SEA and mean crushing force Pmc of the EMC tubes are given in Table 5. The values of E and Pmc are calculated at the crushing distance of 45 mm. Both SEA and Pmc are increasing quickly with the increase of
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wall thickness, no matter inside or outside tube is concerned. The correlation between the indices (SEA and Pmc) and wall thicknesses (to and ti) is illustrated in Fig. 13 for four-cell tubes. Thicker wall indicates higher SEA and Pmc. It is interesting to notice that the four-cell tubes with similar
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structural mass exhibit close values of E、SEA and Pmc, although the wall thicknesses are different. In addition, as shown in Table 5, the Pmc of nine-cell tubes is generally larger than that of four-cell
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tubes, but the SEA values of all EMC tubes with the inside tube C15t1 are comparable to each
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other. Considering that the mass of nine-cell tubes is larger, if the wall thickness of four-cell is increased to guarantee the same mass as nine-cell tubes, the SEA of four-cell tubes should be
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much larger. That is, four-cell tube is preferable than nine-cell tube in energy absorption
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efficiency.
4. Theoretical analysis Theoretical analysis on the crush resistance of EMC tubes is performed in this section.
Although no related theory was established to predict the energy absorption of EMC tubes, empty square tubes have received extensive studies.
4.1 Theoretical aspect of empty square tube
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For empty tubes deforming in quasi- inextensional mode, the mean crushing force Pm can be predicted by the theory proposed by Wierzbicki and Abramowicz [28]. With the consideration of effective crushing distance [25], the mean force can be expressed as 1 5
Pm 13.06 0C 3 t 3
(3)
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where σ0 is flow stress of the material, C and t are width and thickness of the square tube, respectively. The determination of flow stress σ0 is actually not easy and it should be a value depending on the strain and hence the geometry of the tube. Different values were adopted by
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researchers such as: yield stress σy, ultimate stress σu, the average value of σy and σu [29-30], 0.92 times σu [24] and etc.
When the average value of σy and σu is adopted here, σ0 is 61.5 MPa and the predicted mean
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forces of C32t0.95 and C15t1 are 2.32 and 1.94 kN respectively. The Pm of C32t0.95 is satisfactory while that of C15t1 is considerably underestimated. The numerical simulation of
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empty tubes in section 3.2 presents almost the same Pm value as experiment and this prompts us to
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consider that the flow stress may be associated with the geometrical parameters. Since the flow stress is ranging between σy and σu, a linear relation is proposed here to correlate the flow stress
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with the ratio γ=B/t (where B is the middle width of the tube).
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y 1 u c c
(4)
γc is the critical ratio corresponding to the case σ0=σy and it is set to 70 here. For smaller γ, the deformation is more intensive and higher flow stress should be adopted. According to Eq.(4), the flow stress should be 63.5 and 79.3 MPa for C32t0.95 and C15t1 and the predicted values of Pm are 2.42 and 2.50 kN, respectively. Apparently, the theoretical results are in good agreement with experiment data in this case. 14
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To further validate the effectiveness and accuracy of equations (3) and (4), more numerical analyses are performed for empty tubes with different dimensions. The length of the tubes is set to 150 mm and indentation trigger is introduced to guarantee quasi-inextensional mode. The numerical results are given in Table 6. Two empty tubes in Table 6: C30t1 and C30t2 are tested to
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validate the theoretical and numerical results. The tubes are made of the same material AA6063 O and two specimens are tested in each case. The force-displacement curves and deformed shapes of the empty tubes are shown in Fig. 14. The mean forces of C30t1N1 and C30t1N2 are 2.48 and
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2.59 kN, while the values of C30t2N1 and C30t2N2 are 10.34 and 10.69 kN, respectively. The average forces of C30t1 and C30t2 are hence 2.53 and 10.51 kN. Similar as the results in Table 4, numerical simulations slightly underestimate the mean force by 5%-8%, but generally give a good
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prediction. It should be mentioned that the mean force of C30t2 is larger than 4 times that of C30t1. Since the exponent of wall thickness should not be larger than 2 in the expression of mean
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force, the update of flow stress with the change of geometrical parameters is demonstrated to be
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reasonable.
In Table 6, it is found that if the flow stress σ0 is equal to the average value of σy and σu, the
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predicted mean forces by Eq. (3) are considerably lower than numerical results for tubes with low
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B/t ratio. When Eq. (4) is employed, the theoretical predictions compare well with numerical results. The error is generally smaller than 10% except for C50t1. The flow stress may be underestimated for tubes with high B/t ratio. Consequently, for B/t larger than γc/2=35, the average value (σy+σu)/2 is suggested for the flow stress and as listed in Table 6, the predicted values agree well with simulation results.
4.2 Analysis for EMC tubes
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As indicated in Eq. (1), the crush resistance of EMC tubes can be divided into three parts: Poutside, Pinside and Pint. Based on the theoretical analysis of empty square tube in the above section, Poutside and Pinside can be predicted theoretically and the interaction effect can hence be estimated. For the EMC tubes numerically investigated in Section 3.3, the components of crush resistance are
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analyzed in Table 7. Poutside and Pinside are obtained by Eqs. (3) and (4). Psum is then obtained by summing up the mean forces of all constituent tubes. By subtracting Psum from Pmc, the interaction effect item Pint is obtained. A coefficient κ is defined here as
(5)
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Pint / Pmc
As shown in Table 7, the κ values are generally ranging from 25% to 35%. For the sake of simplicity, a constant value κ=28.6% is assumed. In this case, Pint is equal to 40% of Psum, which means
Pm c 1 . 4P s u m
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or
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Pmc 1.4Poutside N Pinside
(6) (7)
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The mean forces of EMC tubes predicted by Eq. (6) or (7) are also listed in Table 7. It is surprising to find that such simple expressions offer very good theoretical predictions. The error
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between numerical result and theoretical prediction is smaller than 10% for all cases except for the
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tube MC33t1.5(4C15t0.5). The theoretical predictions for the specimens tested in Section 2 are also listed in Table 4 and the predictions compare very well with experimental results. It should be mentioned that a unified expression Eq. (3) is employed here for all possible deformation mode of square tubes in EMC tubes. In some cases, Eq. (3) and Eq. (6) may not be applicable and more investigations are required. For example, when the inside tube is changed to triangular or the whole EMC tube fails globally if it is slender.
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5. Conclusion The crush resistance of a type of EMC tubes is studied in this work. Such structures are flexible in cross-section, highly cost-effective and easily available. Experimental, numerical and theoretical analyses are carried out to study the behavior of such structures under axial crushing.
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Similar as foam-filled tubes, the total crush force of the structure can be divided into three components: crush force of outside tube, resistance of inside tubes and interaction effect. Results show that the interaction effect accounts for about 40% of the whole crush force.
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The theoretical aspect for mean crushing force of empty and EMC tubes is also analyzed in the present work. A new method to determine the flow stress of empty square tube is proposed and a formula is offered to predict the mean force of EMC tube. The theoretical predictions are in
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good agreement with experimental and numerical results. The outside and inside tubes of the EMC tubes investigated here are simply placed together. The energy absorption capacity can
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apparently be further improved by introducing adhesive to the structure. The performance is
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associated with the extent of bonding between the constituent tubes. Further investigation on
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related problems is required.
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Acknowledgements The present work was supported by National Natural Science Foundation of China (Nos.
11372115, 11672117, 11502177) and the Fundamental Research Funds for the Central Universities, HUST (No. 2015MS069).
References: 17
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foam-filled sections, Acta Mech, 148 (2001) 199-213.
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[9] H.W. Song, Z.J. Fan, G. Yu, Q.C. Wang, A. Tobota, Partition energy absorption of axially crushed aluminum foam-filled hat sections, Int J Solids Struct, 42 (2005) 2575-2600. [10] W. Chen, T. Wierzbicki, Relative merits of single-cell, multi-cell and foam-filled thin-walled structures in energy absorption, Thin Wall Struct, 39 (2001) 287-306. [11] A. Najafi, M. Rais-Rohani, Mechanics of axial plastic collapse in multi-cell, multi-corner crush tubes, Thin Wall Struct, 49 (2011) 1-12.
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[12] X. Zhang, H. Zhang, Energy absorption of multi-cell stub columns under axial compression, Thin Wall Struct, 68 (2013) 156-163. [13] A.A. Nia, M. Parsapour, An investigation on the energy absorption characteristics of multi-cell square tubes, Thin Wall Struct, 68 (2013) 26-34.
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[14] X. Zhang, H. Zhang, Energy absorption limit of plates in thin-walled structures under compression, Int J Impact Eng, 57 (2013) 81-98.
[15] Z. Tang, S. Liu, Z. Zhang, Analysis of energy absorption characteristics of cylindrical multi-cell
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columns, Thin Wall Struct, 62 (2013) 75-84.
[16] X. Zhang, H. Zhang, Axial crushing of circular multi-cell columns, Int J Impact Eng, 65 (2014) 110-125.
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[17] A. Alavi Nia, M. Parsapour, Comparative analysis of energy absorption capacity of simple and
74 (2014) 155-165.
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multi-cell thin-walled tubes with triangular, square, hexagonal and octagonal sections, Thin Wall Struct,
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[18] T. Tran, S. Hou, X. Han, W. Tan, N. Nguyen, Theoretical prediction and crashworthiness optimization of multi-cell triangular tubes, Thin Wall Struct, 82 (2014) 183-195.
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[19] A. Jusuf, T. Dirgantara, L. Gunawan, I.S. Putra, Crashworthiness analysis of multi-cell prismatic
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structures, Int J Impact Eng, 78 (2015) 34-50. [20] S. Tabacu, Axial crushing of circular structures with rectangular multi-cell insert, Thin Wall Struct, 95 (2015) 297-309. [21] H. Yin, Y. Xiao, G. Wen, Q. Qing, Y. Deng, Multiobjective optimization for foam-filled multi-cell thin-walled structures under lateral impact, Thin Wall Struct, 94 (2015) 1-12. [22] Z. Wang, Z. Li, X. Zhang, Bending resistance of thin-walled multi-cell square tubes, Thin Wall
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Struct, 107 (2016) 287-299. [23] X. Zhang, G. Cheng, A comparative study of energy absorption characteristics of foam-filled and multi-cell square columns, Int J Impact Eng, 34 (2007) 1739-1752. [24] W. Abramowicz, T. Wierzbicki, Axial crushing of multicorner sheet metal columns, J Appl Mech-t
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Asme, 56 (1989) 113-120. [25] W. Abramowicz, N. Jones, Dynamic axial crushing of square tubes, Int J Impact Eng, 2 (1984) 179-208.
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[26] X. Zhang, Z. Wen, H. Zhang, Axial crushing and optimal design of square tubes with graded thickness, Thin Wall Struct, 84 (2014) 263-274.
[27] X. Zhang, H. Zhang, Z. Wang, Bending collapse of square tubes with variable thickness, Int J
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Mech Sci, 106 (2016) 107-116.
[28] T. Wierzbicki, W. Abramowicz, On the crushing mechanics of thin-walled structures, J Appl
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Mech-t Asme, 50 (1983) 727-734.
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[29] A.G. Hanssen, M. Langseth, O.S. Hopperstad, Static and dynamic crushing of square aluminium extrusions with aluminium foam filler, Int J Impact Eng, 24 (2000) 347-383.
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[30] M. Langseth, O.S. Hopperstad, A.G. Hanssen, Crash behaviour of thin-walled aluminium
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members, Thin Wall Struct, 32 (1998) 127-150.
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Figure captions Fig. 1. Illustrations for embedded multi-cell sections. Fig. 2. Specimens of empty tubes and embedded multi-cell sections. Fig. 3. Representative engineering stress-strain curves for AA6063 O tubes. Fig. 4. Deformed shapes of the empty and embedded multi-cell tubes.
Fig. 6. Crushing force-displacement curves for the specimens.
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Fig. 5. Illustrations for the deformation of representative embedded multi-cell tubes.
Fig. 7. Force responses and deformed shapes of tubes with shorter length.
Fig. 8. Representative finite element model for embedded multi-cell tubes. Fig. 9. Deformed shapes of specimens in numerical simulation.
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Fig. 10. Simulated crushing force-displacement curves for the specimens. Fig. 11. Deformed shapes for two EMC tubes.
Fig. 12. Crushing force-displacement curves of embedded multi-cell tubes with various wall thicknesses.
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Fig. 13. Mean force and SEA of embedded four-cell tubes with various wall thicknesses.
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Fig. 14. Force responses and deformed shapes for empty tube C30t1 and C30t2.
Table 1 Material hardening data for AA6063 O.
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Table 2 Experimental results for empty and EMC tubes. Table 3 Interaction effect of EMC tubes.
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Table 4 Mean crushing forces Pm of the specimens. (Unit: kN) Table 5 Numerical results of EMC tubes. Table 6 Theoretical predictions for Pm of empty square tubes. Table 7 Theoretical analysis and predictions for Pmc of EMC tubes.
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(B)
(A)
Enveloped by a square tube
Enveloped by triangular tube
a
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Welding line
Enveloped by a square tube
Enveloped by folding and welding of plate
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Fig. 1. Illustrations for embedded multi-cell sections.
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Fig. 2. Specimens of empty tubes and embedded multi-cell sections.
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Fig. 3. Representative engineering stress-strain curves for AA6063 O tubes.
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M(4C15t1) N2
C32t0.95
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N1
C32t0.95L150
M(9C10t1) N2
N3
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C15t1
M
N1
C10t1
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Fig. 4. Deformed shapes of the empty and embedded multi-cell tubes.
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M(9C10t1)N1
A'
Top
Back
Left
A-A' sectional view
Right
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Fig. 5. Illustrations for the deformation of representative embedded multi-cell tubes.
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B
C
D
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Fig. 6. Crushing force-displacement curves for the specimens.
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Fig. 7. Force responses and deformed shapes of tubes with shorter length.
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Rigid wall
Indentation trigger
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Supported by rigid wall
x
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Fig. 8. Representative finite element model for embedded multi-cell tubes.
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W/o trigger
W/ trigger
W/o trigger
W/ trigger
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M(9C10t1)
C32t0.95L150
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Fig. 9. Deformed shapes of specimens in numerical simulation.
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Fig. 10. Simulated crushing force-displacement curves for the specimens.
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MC48t1.5(9C15t1)
MC33t1.5(4C15t1)
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Fig. 11. Deformed shapes for two EMC tubes.
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Fig. 12. Crushing force-displacement curves of embedded multi-cell tubes with various wall thicknesses.
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Fig. 13. Mean force and SEA of embedded four-cell tubes with various wall thicknesses.
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Fig. 14. Force responses and deformed shapes for empty tube C30t1 and C30t2.
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Table 1 Material hardening data for AA6063 O. 0.0 31.8 11.3 99.1
1.5 59.6 13.4 102.7
3.2 74.6 15.6 105.8
5.1 83.8 17.7 108.7
7.1 90.1 19.9 111.4
9.2 95.0 22.1 114.0
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Plastic strain (%) Plastic stress (MPa) Plastic strain (%) Plastic stress (MPa)
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C15t1
C10t1
C10t1L25
M(4C15t1)
M(9C10t1)
ML60(9C10t1)
E (J)
Pm (kN)
N1
105.0
255.9
2.44
N2
105.0
251.2
2.39
N3
105.0
252.6
2.41
N1
52.5
135.3
2.58
N2
52.5
119.2
2.27
N3
52.5
136.8
2.61
N1
50.0
128.1
2.56
N2
50.0
131.8
2.64
N3
50.0
135.4
N1
20.0
28.1
N2
20.0
29.3
N3
20.0
26.5
N1
17.0
54.8
N2
17.0
54.8
N3
17.0
56.3
3.31
N1
42.0
767.3
18.27
N2
42.0
767.0
18.27
N3
42.0
764.0
18.19
N1
37.5
1285.9
34.30
N2
37.5
1266.1
33.77
N3
37.5
1293.3
34.48
N1
30.0
1146.1
38.21
N2
30.0
1125.6
37.51
N3
30.0
1096.7
36.56
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*Default length of the specimens is 75 mm.
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Average (kN) 2.41
2.48
2.63
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C32t0.95
δe (mm)
2.71 1.41 1.46 1.32 3.22 3.22
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C32t0.95L150
Specimen
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Table 2 Experimental results for empty and EMC tubes.
1.40
Global buckling
3.25
Extensional mode
18.24
34.18
37.43
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Table 3 Interaction effects of EMC tubes. Case
Pmc (kN)
Psum (kN)
Pint (kN)
Pint/ Pmc
M(4C15t1) M(9C10t1) ML60(9C10t1) M(9C10t1)* ML60(9C10t1)*
18.24 34.18 37.43 34.18 37.43
12.95 14.99 14.99 31.67 31.67
5.29 19.19 22.43 2.51 5.76
29.0% 56.1% 59.9% 7.4% 15.4%
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* Mean crushing force of C10t1L25 is employed.
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Table 4 Mean crushing forces Pm of the specimens. (Unit: kN)
Specimen
δNum (mm)
Test
105.0 50.0 45.0 44.0
2.41 2.63 18.24 34.18
Theoretical prediction
w/o trigger
w/ trigger
Error (%)
Pm
Error (%)
/ / 17.30 31.15
2.27 2.50 16.75 31.87
-5.8 -5.3 -8.2 -6.8
2.42 2.50 17.69 33.01
0.2 -5.2 -3.0 -3.4
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C32t0.95L150 C15t1 M(4C15t1) M(9C10t1)
Numerical simulation
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Table 5 Numerical results of EMC tubes. Mass (g)
E (J)
SEA (J/g)
Pmc (kN)
MC32t1(4C15t0.5) MC32t1(4C15t0.75) MC32t1(4C15t1)
48.6 59.8 70.5
330.06 517.86 770.32
6.79 8.66 10.93
7.33 11.51 17.12
MC33t1.5(4C15t0.5) MC33t1.5(4C15t0.75) MC33t1.5(4C15t1)
61.8 73.0 83.7
484.65 798.00 1055.30
7.84 10.94 12.61
10.77 17.73 23.45
MC34t2(4C15t0.5) MC34t2(4C15t0.75) MC34t2(4C15t1)
75.4 86.5 97.3
782.59 1079.61 1407.55
10.38 12.48 14.47
17.39 23.99 31.28
M47t1(9C15t1) M48t1.5(9C15t1) M49t2(9C15t1)
139.4 158.6 178.3
1553.64 1803.22 2335.58
11.15 11.37 13.10
34.53 40.07 51.90
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Table 6 Theoretical predictions for Pm of empty square tubes. t (mm)
σ0 (MPa) by Eq. (4)
Eq. (3) Pm (kN)
Num. Pm (kN)
Error (%)
Pm (kN) σ0=(σy+σu)/2
Error (%)
30 30 30 40 40 40 50 50 50
1 1.5 2 1 1.5 2 1 1.5 2
66.6 75.1 79.3 58.1 69.4 75.1 49.6 63.8 70.8
2.70 5.99 10.22 2.60 6.09 10.65 2.39 6.03 10.82
2.51 5.73 9.68 2.72 6.13 11.61 2.99 6.10 10.16
7.7 4.5 5.6 -4.7 -0.5 -8.3 -20.2 -1.2 6.5
2.50 4.91 7.92 2.75 5.40 8.72 2.96 5.82 9.39
-0.5 -14.4 -18.1 0.9 -11.9 -24.9 -1.1 -4.7 -7.5
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C (mm)
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Table 7 Theoretical analysis and prediction for Pmc of EMC tubes.
MC32t1(4C15t0.75) MC32t1(4C15t1) MC33t1.5(4C15t0.5) MC33t1.5(4C15t0.75) MC33t1.5(4C15t1) MC34t2(4C15t0.5) MC34t2(4C15t0.75) MC34t2(4C15t1) M47t1(9C15t1) M48t1.5(9C15t1) M49t2(9C15t1)
Poutside
Pinside
Psum
Pint
Pint/ Pmc
Eq. (6)
Error
Pmc (kN)
(kN)
(kN)
(kN)
(kN)
(%)
Pmc (kN)
(%)
7.33 11.51 17.12 10.77 17.73 23.45 17.39 23.99 31.28 34.53 40.07 51.90
2.69 2.69 2.69 6.04 6.04 6.04 10.43 10.43 10.43 2.46 6.05 10.81
0.68 1.50 2.55 0.68 1.50 2.55 0.68 1.50 2.55 2.55 2.55 2.55
5.39 8.68 12.91 8.74 12.03 16.26 13.13 16.41 20.65 25.45 29.05 33.80
1.94 2.83 4.21 2.03 5.70 7.19 4.26 7.58 10.63 9.07 11.02 18.10
26.5 24.6 24.6 18.8 32.2 30.7 24.5 31.6 34.0 26.3 27.5 34.9
7.55 12.15 18.07 12.24 16.84 22.77 18.38 22.98 28.90 35.63 40.67 47.33
2.9 5.6 5.6 13.7 -5.0 -2.9 5.7 -4.2 -7.6 3.2 1.5 -8.8
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MC32t1(4C15t0.5)
Num.
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Graphical abstract
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Axial crushing of embedded multi-cell tubes Xiong Zhang , Kehua Leng , Hui Zhang PII: DOI: Reference:
S0020-7403(17)31428-5 10.1016/j.ijmecsci.2017.07.019 MS 3805
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
28 May 2017 2 July 2017 9 July 2017
Please cite this article as: Xiong Zhang , Kehua Leng , Hui Zhang , Axial crushing of embedded multicell tubes, International Journal of Mechanical Sciences (2017), doi: 10.1016/j.ijmecsci.2017.07.019
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Highlights A new type of embedded multi-cell (EMC) tubes is proposed and investigated.
Quasi-static axial crushing tests are performed for empty and EMC tubes.
Numerical simulations are carried out and compare well with experiment.
Theoretical expression is obtained to predict the crush resistance of EMC tubes.
Interaction effect is found to account for 40% of the total crush force.
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Axial crushing of embedded multi-cell tubes Xiong Zhang a, b,, Kehua Leng a, Hui Zhang c a
Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, PR China
b
Hubei Key Laboratory of Engineering Structural Analysis and Safety Assessment, Luoyu Road 1037, Wuhan,
c
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430074, China
School of Mechanical Engineering and Automation, Wuhan Textile University, Wuhan 430073, Hubei, PR China
Abstract
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The axial crushing resistance of a new type of embedded multi-cell (EMC) tubes is investigated in this paper. Quasi-static tests are carried out first to investigate the deformation and force response of both empty square tubes and EMC tubes. The components of the crush force of EMC tubes are analyzed quantitatively. Nonlinear explicit finite element method is then employed to simulate the crushing process and the numerical results compare well
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with experiment. By using the FE model validated by experiment, parametric study is then performed to investigate the energy absorption performance of embedded tubes with various configurations. Finally, theoretical
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analysis is conducted to analyze the components of crush resistance of the whole structure and an expression is derived to predict the mean crushing force of EMC tubes. Theoretical predictions are in good agreement with experimental and numerical results.
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Keywords: Multi-cell; Axial crush; Energy absorption; Experimental test; Embedded tubes.
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1. Introduction
Commercial hexagonal aluminum honeycombs [1-2] are widely applied in aerospace
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engineering as core of sandwich structures due to their high specific strength, rigidity and energy absorption capacity. As two-dimensional cellular structures, honeycombs are anisotropic with very much better out-of-plane strength than their in-plane property. Compared to the isotropic aluminum foam, this anisotropic feature guarantees better specific strength when loaded in the
Corresponding author. Tel.: +86 27 87543538; fax: +86 27 87543501. E-mail address: [email protected] (X. Zhang). 2
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out-of-plane direction. For instance, Santosa and Wierzbicki [3] reported that honeycomb outperformed foam material in improving specific energy absorption (SEA, energy absorption per unit mass) of box columns under axial crush when served as inside fillers. Actually, the behavior of metal tubes filled with foam materials has received enormous investigations [4-9]. The energy
expressions were derived to predict the response of them.
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absorption mechanisms of them under axial or transverse have been disclosed and theoretical
Recently, the crashworthiness performance of multi-cell sections under various load
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conditions attracted extensive interests [10-22]. Many experimental and numerical investigations validated that multi-cell sections could provide considerably higher energy absorption efficiency than single cells. Compared to filling structure with cellular material, adopting multi-cell section
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seems a much better option to improve the crashworthiness. A comparative study conducted by Zhang and Cheng [23] showed that the SEA value of aluminum multi-cell columns was about
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50-100% higher than that of simple column filled with aluminum foam, when the mass was kept
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the same.
Actually, commercial aluminum honeycombs are also a type of multi-cell structure with thin
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wall thickness and periodic structural configuration. Aluminum foils are always used to fabricate
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honeycombs by gluing and tension process while multi-cell columns are generally produced by extrusion forming. Unfortunately, the present fabrication technique is in lack of flexibility or leads to low cost-effectiveness. Only few sectional shapes of honeycombs can be produced by the gluing and tension process. Meanwhile, for the extrusion of multi-cells, new mold is required for every cross-sectional shape and any change in dimensions. This apparently increases the production cost of multi-cell sections and limits the wide applications of them. To overcome this
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drawback, a novel type of embedded multi-cell (EMC) section is proposed here to satisfy the requirement of high cost-effectiveness and easy availability. As shown in Fig. 1, EMC sections are constituted by a group of single tubes confined within an outside thin-walled envelope. The outside envelope can be a larger tube or a plate being folded
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and welded to contain the inside single tubes. Of particular interest in the present work is to investigate the crush resistance of EMC tubes and the sections in Fig. 1(A) are investigated experimentally, numerically and theoretically. The crush resistance of EMC tubes should be larger
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than the summation of that of the outside and inside empty tubes due to similar mechanisms as foam-filled columns. The interaction between the tubes will dissipate more energy and a quantitative investigation is necessitated to determine the interaction effect. Experimental study
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and numerical analysis are performed in this work to deal with this problem and a theoretical expression is also presented to predict the crush resistance of EMC tubes.
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2. Experimental study
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2.1 Scheme and specimen preparation The energy absorption characteristics of a group of EMC tubes are experimentally
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investigated in this section. As shown in Fig. 2, the EMC tubes are constituted by empty extruded
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aluminum square tubes with the outside width C of 10, 15 and 32 mm. The wall thickness t of the tube with C=32 mm is 0.95 mm while the thickness of the other two tubes is 1.0 mm. All the tubes are cut to the same length of 75 mm. Four square tubes with C=15 mm and nine those with C=10 mm are inserted into the square tubes with C=32 mm. Three specimens are prepared for each type of tubes and more specimens are tested when necessary. The structural material of the tubes is AA6063 O. Although the square tubes are provided by
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the same supplier, the material hardening data for the tubes with different dimensions are obtained by tensile tests based on the standard test methods in ASTM E8M-04. The tensile tests are performed in a 10 kN capacity Zwick Z010 universal material testing machine and the specimens for the tests are cut along the longitudinal direction of the square tubes. The representative
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engineering stress-strain curves for AA6063 O tubes are plotted in Fig. 3. As shown in the figure, the difference between the three curves is small and one of the curves is selected to be used in the subsequent numerical analysis. The material hardening data for the selected curve are listed in
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Table 1. The mechanical properties of the material AA6063 O are also presented here: Young's modulus E = 68.9 GPa, Poisson's ratio ν = 0.33, initial yield stress σy = 31.8 MPa and ultimate stress σu = 91.2 MPa.
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2.2 Experimental results
For the sake of convenience, the specimens are named by the following rules: "C", "t" and
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"L" denote the outside width, thickness and length of the tube, respectively; "M" means
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"multi-cell"; "N" is the test number of repetitions. The default length of the specimens is 75 mm and "L" can be omitted in this case. For example, C32t0.95 means empty tube with C=32 mm,
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t=0.95mm and L=75 mm. According to the specimens prepared for the tests, the default outside
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tube of EMC tubes is C32t0.95 and the status of the inside tubes is given within parentheses. For instance, M(9C10t1) represents an EMC with 9 cells and the dimensions for the inside tube is C=10 mm, t=1 mm and L=75 mm. If necessary, the dimensions for the outside tube can be added in the designation, e.g. MC34t2L60(9C10t1) means a multi-cell tube with the outside tube C34t2L60 and 9 inside tube C10t1. The deformed shapes of the empty and EMC specimens are shown in Fig. 4. For empty tubes,
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C32t0.95 and C15t1 deform in quasi-inextensional [24] or symmetric collapse mode [25] of square tube, while global buckling is present for C10t1. Only three lobes are formed for C32t0.95 and in this case, the mean crushing force may differ significantly from the tubes in normal state. The development of two end lobes will lead to big deviation as reported by Zhang et al. [26]. To
are also given in the figure. Seven lobes are formed in this case.
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remove this disturbance, tubes with the length of 150 mm are tested and the deformation modes
The outside tubes of the EMC tubes M(4C15t1) deform in inextensional mode while those of
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the multi-cell tubes M(9C10t1) develop a quite novel mode which was not reported before. One side wall of the tube bulges outward while the other side walls still fold progressively. A more detailed illustration for deformation of the multi-cell tubes is offered in Fig. 5. The front view has
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been given in Fig. 4 and the other four views are provided in Fig. 5. For the tube with four cells, the deformation is quite regular and every corner elements develop inextensional mode. The tube
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with nine cells deforms in a more complicated way. A sectional view shows that the inside nine
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cell tubes deform in global buckling mode although they are subjected to the constraint of the outside tube. The deformation of the outside tube is also to some extent affected by the inside
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tubes, but it is still sustainable and resistant to the crush loads. From the viewpoint of the whole
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structure, it can still be deemed as a progressive mode since the crush resistance of the structure does not decrease dramatically. The crushing force-displacement curves of the specimens are presented in Fig. 6. The force
responses of empty tubes C32t0.95L150 and C15t1 show all standard features of progressive folding. As expected, the short specimens of C32t0.95 exhibit some irregular force response and may result in an overestimation of the mean crushing force. The force curves of the small empty
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tubes C10t1 show representative features of the structures failing in global bending: the force increases to a peak and then drops down. As for EMC tubes, the force responses of the tubes with four cells and nine cells are slightly different. There is a fast drop after the initial peak for the tubes with four cells while only slight
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and smooth drop is observed for those with nine cells. This is apparently associated with the deformation mode. Progressive buckling always leads to a fast drop after the initial peak just as the curves in Fig. 6(C)-(E), while the global bending failure results in a smooth drop similar as the
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curves in Fig. 6(F). As for M(9C10t1), the global buckling of inside tubes is constrained by the outside tube and the force drop is hence stopped and finally reversed due to the densification of inside tubes. Furthermore, good repeatability is observed in the tests. The three repetitions show
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Since the inside tubes of M(9C10t1) buckle globally during the axial crushing, some
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specimens with shorter length are tested. The length of empty tube C10t1 is reduced gradually and
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the global bending switches to progressive folding at L=25 mm. As shown in Fig. 7(A), extensional mode is developed for these short tubes instead of quasi-inextensional mode. The
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force curves of C10t1L25 are also given in the figure. Interestingly, the features of these curves
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are similar to those for EMC tubes with nine cells. There is no high initial peak and fast drop of the force. Multi-cell tubes M (9C10t1) with L=60 mm are also tested and the results are given in Fig. 7(B). Similar deformation and force responses are obtained but a slightly higher force level is observed.
2.3 Analysis and discussion The crushing resistance and energy absorption of the embedded tubes are analyzed in this
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section. Energy absorption E is obtained by integrating the crushing force with respect to crushing distance δ and mean crushing force Pm is defined as the ratio of energy absorption E to the distance δ. The effective crushing distance δe is determined just before the densification stage of a structure. The related energy absorption indices of the specimens tested are listed in Table 2. As
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shown in the table, for C32t0.95, the tubes with larger length do provide more stable mean crushing force Pm although the average Pm is almost the same. The mean force of C32t0.95L150 is hence employed in the later analysis. For C10t1, the tubes with L=25 mm deform in extensional
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mode and give much higher Pm than those deforming in global bending with L=75 mm.
As listed in Table 2, the mean force of EMC tubes is larger than the simple summation of the mean crushing forces of all constituent tubes. Apparently, this is caused by the interaction effect
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similar as that for foam-filled tubes. However, the interaction effect is more complicated here since it includes not only the interaction between inside and outside tubes, but also the interaction
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between different inside tubes. To analyze the interaction effect of EMC tubes, the mean crushing force can be expressed as
(1)
Pmc Psum Pint
(2)
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Pmc Po u t s id N e Pi n s i d ePi n t
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or
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Pmc, Poutside and Pinside are mean crushing force of the whole multi-cell tube, the outside tube and the inside filling tube, respectively. N is the number of inside tubes, Psum is the simple summation of the mean forces of all constituent tubes and Pint is the crushing force caused by interaction effect between the constituent tubes. The interaction effect of EMC tubes is analyzed in Table 3. As shown in the table, for M(4C15t1), the interaction effect accounts for 29.0% of the whole crush resistance. Due to the low
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crush resistance of C10t1 caused by global failure, the interaction effect of M(9C10t1) is summed up to 56.1% of the structural resistance. If C10t1L25 which develops extensional mode is adopted, the ratio of Pint to Pmc is reduced significantly to 7.4%. The mean force of ML60(9C10t1) is slightly higher and accordingly the interaction effect is increased. Apparently, the interaction
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effect is associated with the relative strength of the inside versus outside tubes. The change of geometrical parameters will definitely affect the interaction effect significantly and this will be further investigated by numerical simulation in next section.
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3. Numerical investigation 3.1 Finite element modeling
Nonlinear explicit finite element code LS-DYNA is employed to simulate the axial crushing
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of empty and EMC specimens. LS-PREPOST is used to pre- or post-process the data files. A representative finite element model of EMC tube is illustrated in Fig. 8. The specimens are loaded
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by two rigid walls with a stationary one in the bottom and one moving downward from the top
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with a prescribed velocity. Belytschko-Tsay 4-node shell elements with five integration points through the thickness are employed to mesh the structures and the characteristic element size of
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the meshes is set to 1.0 mm.
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Automatic single surface contact is employed to consider the self-contact of specimens and
automatic node to surface contact is applied to simulate the contact between specimens and rigid walls. When EMC tubes are analyzed, automatic surface to surface contact is used to simulate the interaction between outside envelope tube and inside embedded tubes. The friction coefficients in all these contacts may be different, but they are assumed to be 0.3 here for the sake of simplicity. Calculations show that only negligible difference (smaller than 2%) is observed for axial crush
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resistance of EMC tubes when the friction coefficient is varied from 0.3 to 0.8. The structural material is modeled by piecewise linear plasticity model with the material hardening data given in Table 1. To finish the quasi-static simulation in a reasonable time, the load velocity is accelerated to 1m/s and the principles proposed by Santosa et al [7] are followed. Previous quasi-static
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simulations [26-27] based on these principles showed very good agreement with experiment. To induce thin-walled structures to deform in the similar deformation mode as experiment, triggers are always necessitated. In the present analysis, indentation triggers are introduced in the
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cell walls of the tubes. The location of the triggers can be set according to deformation of specimens in experiment. For instance, as shown in Fig. 8, indentation triggers are introduced in four locations for M(4C15t1) as boxed in the figure. The position of the triggers is set to 6 mm
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distant from the top or bottom of the tube. Both inside and outside tubes are indented and the depth of the trigger is 0.3 mm. The introducing of trigger may have some influence on the initial
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peak force, but there is only negligible effect on mean crushing force and energy absorption of the
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structures.
3.2 Simulation of the tests
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The simulated deformation patterns for the specimens are plotted in Fig. 9. For EMC tube
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M(4C15t1), extensional mode is developed when trigger is not introduced. The same deformation mode as experiment is obtained when the indentation trigger in Fig. 8 is present. The outside square tube deforms in quasi-inextensional mode with seven lobes formed. However, for tube M(9C10t1), the numerical simulations with various trigger forms fail to induce the global buckling of the inside embedded tubes and can not obtain the same deformation mode as experiment. The EMC tube without trigger deforms in extensional mode and the one with trigger develops a mixed
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mode of extensional and quasi-inextensional folds. A check of the deformation for the inside tubes shows that progressive folding instead of global buckling is developed. In the meantime, the simulated deformation modes of empty tube C32t0.95L150 and C15t1 are in good agreement with experiment. The tube C10t1 deforms in global bending failure and no effort is exerted here to
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simulate its behavior. The crushing force versus displacement curves in experiment and simulation are compared in Fig. 10. It is found that the simulated results of multi-cell tube M(4C15t1) and empty tubes
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compare very well with measured data in the experimental test, although the force level is slightly underestimated. In addition, the effective crushing distance is overestimated in all simulations. The densification process is postponed in the numerical analysis. The crushing force of M(9C10t1)
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in simulation is considerably lower than that in experiment and most importantly, the features of the force curves in the two cases are quite different. Apparently, the difference in deformation
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mode should be responsible for this. In addition, it is noted that the difference in force level is very
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small for different progressive buckling mode of EMC tubes. As shown in Fig. 10(A) and (B), the difference between the curves with and without trigger is not big although the tubes deform in
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different progressive mode.
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The mean crushing force Pm of the specimens in simulation is compared with experiment in
Table 4. The effective crushing distance δe to calculate the Pm in simulation is defined as δNum and it is also given in the table. The numerical simulations generally underestimate the crushing resistance, but the results are satisfactory. The maximum error is smaller than 6% for empty tubes and it is about 8% for EMC tubes. It should be mentioned that although the error of Pm for M(9C10t1) in simulation is not big, the simulation results of it may not be meaningful since the
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features of deformation pattern and force response curve in experiment and simulation are quite different. In addition, the maximum difference in Pm between different progressive modes of multi-cell tubes is about 3%. This is favorable to perform parametric study on the crush resistance of such EMC tubes. Any type of trigger can be chosen since no big difference in Pm will be caused
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when progressive buckling mode is developed. It is concluded that the numerical model generally give a good simulation of the specimens except for the case when the inside tubes fail in global buckling.
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3.3 Parametric study on EMC tubes
The energy absorption performance and interaction effect of EMC tubes under axial crushing should be highly correlated with the geometric parameters. The influences of the number of inside
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tubes and the dimensions of the inside and outside tubes on crush resistance of EMC tubes are investigated in this section. The length of all tubes is set to 75 mm and the aspect ratio of inside
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tube is constrained to be no less than 15 to guarantee progressive collapse and avoid similar case
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as M(9C10t1).
The specimens with various configurations are listed in Table 5. Nine four-cell tubes and
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three nine-cell tubes are analyzed. For four-cell tubes, the outside width of the inside tube is set to
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15 mm while three different thicknesses: to=0.5, 0.75 and 1mm are analyzed. At the same time, the inside width of the outside tube is 30 mm with also three wall thicknesses: ti=1, 1.5 and 2 mm. For nine-cell tubes, the inside tube is invariant to be C15t1 while the above three thicknesses are employed for the outside tube. The same indentation triggers as employed for M(4C15t1) and shown in Fig. 8 are introduced here. Almost all outside tubes of four-cell embedded tubes deform in quasi-inextensional mode while those of nine-cell tubes develop extensional mode. Deformed
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shapes of two representative specimens: MC33t1.5(4C15t1) and MC48t1.5(9C15t1) are given in Fig. 11. The crushing force versus displacement curves of the EMC tubes are plotted in Fig. 12. It is noted that the force level increases significantly when the wall thickness of inside or outside tube is increased.
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The energy absorption indices including structural mass, energy absorption E, SEA and mean crushing force Pmc of the EMC tubes are given in Table 5. The values of E and Pmc are calculated at the crushing distance of 45 mm. Both SEA and Pmc are increasing quickly with the increase of
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wall thickness, no matter inside or outside tube is concerned. The correlation between the indices (SEA and Pmc) and wall thicknesses (to and ti) is illustrated in Fig. 13 for four-cell tubes. Thicker wall indicates higher SEA and Pmc. It is interesting to notice that the four-cell tubes with similar
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structural mass exhibit close values of E、SEA and Pmc, although the wall thicknesses are different. In addition, as shown in Table 5, the Pmc of nine-cell tubes is generally larger than that of four-cell
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tubes, but the SEA values of all EMC tubes with the inside tube C15t1 are comparable to each
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other. Considering that the mass of nine-cell tubes is larger, if the wall thickness of four-cell is increased to guarantee the same mass as nine-cell tubes, the SEA of four-cell tubes should be
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much larger. That is, four-cell tube is preferable than nine-cell tube in energy absorption
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efficiency.
4. Theoretical analysis Theoretical analysis on the crush resistance of EMC tubes is performed in this section.
Although no related theory was established to predict the energy absorption of EMC tubes, empty square tubes have received extensive studies.
4.1 Theoretical aspect of empty square tube
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For empty tubes deforming in quasi- inextensional mode, the mean crushing force Pm can be predicted by the theory proposed by Wierzbicki and Abramowicz [28]. With the consideration of effective crushing distance [25], the mean force can be expressed as 1 5
Pm 13.06 0C 3 t 3
(3)
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where σ0 is flow stress of the material, C and t are width and thickness of the square tube, respectively. The determination of flow stress σ0 is actually not easy and it should be a value depending on the strain and hence the geometry of the tube. Different values were adopted by
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researchers such as: yield stress σy, ultimate stress σu, the average value of σy and σu [29-30], 0.92 times σu [24] and etc.
When the average value of σy and σu is adopted here, σ0 is 61.5 MPa and the predicted mean
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forces of C32t0.95 and C15t1 are 2.32 and 1.94 kN respectively. The Pm of C32t0.95 is satisfactory while that of C15t1 is considerably underestimated. The numerical simulation of
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empty tubes in section 3.2 presents almost the same Pm value as experiment and this prompts us to
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consider that the flow stress may be associated with the geometrical parameters. Since the flow stress is ranging between σy and σu, a linear relation is proposed here to correlate the flow stress
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with the ratio γ=B/t (where B is the middle width of the tube).
0
y 1 u c c
(4)
γc is the critical ratio corresponding to the case σ0=σy and it is set to 70 here. For smaller γ, the deformation is more intensive and higher flow stress should be adopted. According to Eq.(4), the flow stress should be 63.5 and 79.3 MPa for C32t0.95 and C15t1 and the predicted values of Pm are 2.42 and 2.50 kN, respectively. Apparently, the theoretical results are in good agreement with experiment data in this case. 14
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To further validate the effectiveness and accuracy of equations (3) and (4), more numerical analyses are performed for empty tubes with different dimensions. The length of the tubes is set to 150 mm and indentation trigger is introduced to guarantee quasi-inextensional mode. The numerical results are given in Table 6. Two empty tubes in Table 6: C30t1 and C30t2 are tested to
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validate the theoretical and numerical results. The tubes are made of the same material AA6063 O and two specimens are tested in each case. The force-displacement curves and deformed shapes of the empty tubes are shown in Fig. 14. The mean forces of C30t1N1 and C30t1N2 are 2.48 and
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2.59 kN, while the values of C30t2N1 and C30t2N2 are 10.34 and 10.69 kN, respectively. The average forces of C30t1 and C30t2 are hence 2.53 and 10.51 kN. Similar as the results in Table 4, numerical simulations slightly underestimate the mean force by 5%-8%, but generally give a good
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prediction. It should be mentioned that the mean force of C30t2 is larger than 4 times that of C30t1. Since the exponent of wall thickness should not be larger than 2 in the expression of mean
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force, the update of flow stress with the change of geometrical parameters is demonstrated to be
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reasonable.
In Table 6, it is found that if the flow stress σ0 is equal to the average value of σy and σu, the
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predicted mean forces by Eq. (3) are considerably lower than numerical results for tubes with low
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B/t ratio. When Eq. (4) is employed, the theoretical predictions compare well with numerical results. The error is generally smaller than 10% except for C50t1. The flow stress may be underestimated for tubes with high B/t ratio. Consequently, for B/t larger than γc/2=35, the average value (σy+σu)/2 is suggested for the flow stress and as listed in Table 6, the predicted values agree well with simulation results.
4.2 Analysis for EMC tubes
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As indicated in Eq. (1), the crush resistance of EMC tubes can be divided into three parts: Poutside, Pinside and Pint. Based on the theoretical analysis of empty square tube in the above section, Poutside and Pinside can be predicted theoretically and the interaction effect can hence be estimated. For the EMC tubes numerically investigated in Section 3.3, the components of crush resistance are
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analyzed in Table 7. Poutside and Pinside are obtained by Eqs. (3) and (4). Psum is then obtained by summing up the mean forces of all constituent tubes. By subtracting Psum from Pmc, the interaction effect item Pint is obtained. A coefficient κ is defined here as
(5)
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Pint / Pmc
As shown in Table 7, the κ values are generally ranging from 25% to 35%. For the sake of simplicity, a constant value κ=28.6% is assumed. In this case, Pint is equal to 40% of Psum, which means
Pm c 1 . 4P s u m
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or
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Pmc 1.4Poutside N Pinside
(6) (7)
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The mean forces of EMC tubes predicted by Eq. (6) or (7) are also listed in Table 7. It is surprising to find that such simple expressions offer very good theoretical predictions. The error
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between numerical result and theoretical prediction is smaller than 10% for all cases except for the
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tube MC33t1.5(4C15t0.5). The theoretical predictions for the specimens tested in Section 2 are also listed in Table 4 and the predictions compare very well with experimental results. It should be mentioned that a unified expression Eq. (3) is employed here for all possible deformation mode of square tubes in EMC tubes. In some cases, Eq. (3) and Eq. (6) may not be applicable and more investigations are required. For example, when the inside tube is changed to triangular or the whole EMC tube fails globally if it is slender.
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5. Conclusion The crush resistance of a type of EMC tubes is studied in this work. Such structures are flexible in cross-section, highly cost-effective and easily available. Experimental, numerical and theoretical analyses are carried out to study the behavior of such structures under axial crushing.
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Similar as foam-filled tubes, the total crush force of the structure can be divided into three components: crush force of outside tube, resistance of inside tubes and interaction effect. Results show that the interaction effect accounts for about 40% of the whole crush force.
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The theoretical aspect for mean crushing force of empty and EMC tubes is also analyzed in the present work. A new method to determine the flow stress of empty square tube is proposed and a formula is offered to predict the mean force of EMC tube. The theoretical predictions are in
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good agreement with experimental and numerical results. The outside and inside tubes of the EMC tubes investigated here are simply placed together. The energy absorption capacity can
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apparently be further improved by introducing adhesive to the structure. The performance is
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associated with the extent of bonding between the constituent tubes. Further investigation on
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related problems is required.
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Acknowledgements The present work was supported by National Natural Science Foundation of China (Nos.
11372115, 11672117, 11502177) and the Fundamental Research Funds for the Central Universities, HUST (No. 2015MS069).
References: 17
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[1] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties Cambridge University Press, Cambridge, 1997. [2] G. Lu, T. Yu, Energy absorption of structures and materials, Boca Raton : Cambridge : CRC Press ; Woodhead, 2003.
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[3] S. Santosa, T. Wierzbicki, Crash behavior of box columns filled with aluminum honeycomb or foam, Comput Struct, 68 (1998) 343-367.
[4] W. Abramowicz, T. Wierzbicki, Axial crushing of foam-filled columns, Int J Mech Sci, 30 (1988)
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263-271.
[5] A.G. Hanssen, M. Langseth, O.S. Hopperstad, Static crushing of square aluminium extrusions with aluminium foam filler, Int J Mech Sci, 41 (1999) 967-993.
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[6] S.R. Reid, T.Y. Reddy, M.D. Gray, Static and dynamic axial crushing of foam-filled sheet metal tubes, Int J Mech Sci, 28 (1986) 295-322.
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[7] S.P. Santosa, T. Wierzbicki, A.G. Hanssen, M. Langseth, Experimental and numerical studies of
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foam-filled sections, Int J Impact Eng, 24 (2000) 509-534. [8] S. Santosa, J. Banhart, T. Wierzbicki, Experimental and numerical analyses of bending of
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foam-filled sections, Acta Mech, 148 (2001) 199-213.
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[9] H.W. Song, Z.J. Fan, G. Yu, Q.C. Wang, A. Tobota, Partition energy absorption of axially crushed aluminum foam-filled hat sections, Int J Solids Struct, 42 (2005) 2575-2600. [10] W. Chen, T. Wierzbicki, Relative merits of single-cell, multi-cell and foam-filled thin-walled structures in energy absorption, Thin Wall Struct, 39 (2001) 287-306. [11] A. Najafi, M. Rais-Rohani, Mechanics of axial plastic collapse in multi-cell, multi-corner crush tubes, Thin Wall Struct, 49 (2011) 1-12.
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[12] X. Zhang, H. Zhang, Energy absorption of multi-cell stub columns under axial compression, Thin Wall Struct, 68 (2013) 156-163. [13] A.A. Nia, M. Parsapour, An investigation on the energy absorption characteristics of multi-cell square tubes, Thin Wall Struct, 68 (2013) 26-34.
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[14] X. Zhang, H. Zhang, Energy absorption limit of plates in thin-walled structures under compression, Int J Impact Eng, 57 (2013) 81-98.
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[17] A. Alavi Nia, M. Parsapour, Comparative analysis of energy absorption capacity of simple and
74 (2014) 155-165.
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multi-cell thin-walled tubes with triangular, square, hexagonal and octagonal sections, Thin Wall Struct,
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[18] T. Tran, S. Hou, X. Han, W. Tan, N. Nguyen, Theoretical prediction and crashworthiness optimization of multi-cell triangular tubes, Thin Wall Struct, 82 (2014) 183-195.
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[19] A. Jusuf, T. Dirgantara, L. Gunawan, I.S. Putra, Crashworthiness analysis of multi-cell prismatic
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structures, Int J Impact Eng, 78 (2015) 34-50. [20] S. Tabacu, Axial crushing of circular structures with rectangular multi-cell insert, Thin Wall Struct, 95 (2015) 297-309. [21] H. Yin, Y. Xiao, G. Wen, Q. Qing, Y. Deng, Multiobjective optimization for foam-filled multi-cell thin-walled structures under lateral impact, Thin Wall Struct, 94 (2015) 1-12. [22] Z. Wang, Z. Li, X. Zhang, Bending resistance of thin-walled multi-cell square tubes, Thin Wall
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Struct, 107 (2016) 287-299. [23] X. Zhang, G. Cheng, A comparative study of energy absorption characteristics of foam-filled and multi-cell square columns, Int J Impact Eng, 34 (2007) 1739-1752. [24] W. Abramowicz, T. Wierzbicki, Axial crushing of multicorner sheet metal columns, J Appl Mech-t
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Asme, 56 (1989) 113-120. [25] W. Abramowicz, N. Jones, Dynamic axial crushing of square tubes, Int J Impact Eng, 2 (1984) 179-208.
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[26] X. Zhang, Z. Wen, H. Zhang, Axial crushing and optimal design of square tubes with graded thickness, Thin Wall Struct, 84 (2014) 263-274.
[27] X. Zhang, H. Zhang, Z. Wang, Bending collapse of square tubes with variable thickness, Int J
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Mech Sci, 106 (2016) 107-116.
[28] T. Wierzbicki, W. Abramowicz, On the crushing mechanics of thin-walled structures, J Appl
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Mech-t Asme, 50 (1983) 727-734.
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[29] A.G. Hanssen, M. Langseth, O.S. Hopperstad, Static and dynamic crushing of square aluminium extrusions with aluminium foam filler, Int J Impact Eng, 24 (2000) 347-383.
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[30] M. Langseth, O.S. Hopperstad, A.G. Hanssen, Crash behaviour of thin-walled aluminium
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members, Thin Wall Struct, 32 (1998) 127-150.
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Figure captions Fig. 1. Illustrations for embedded multi-cell sections. Fig. 2. Specimens of empty tubes and embedded multi-cell sections. Fig. 3. Representative engineering stress-strain curves for AA6063 O tubes. Fig. 4. Deformed shapes of the empty and embedded multi-cell tubes.
Fig. 6. Crushing force-displacement curves for the specimens.
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Fig. 5. Illustrations for the deformation of representative embedded multi-cell tubes.
Fig. 7. Force responses and deformed shapes of tubes with shorter length.
Fig. 8. Representative finite element model for embedded multi-cell tubes. Fig. 9. Deformed shapes of specimens in numerical simulation.
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Fig. 10. Simulated crushing force-displacement curves for the specimens. Fig. 11. Deformed shapes for two EMC tubes.
Fig. 12. Crushing force-displacement curves of embedded multi-cell tubes with various wall thicknesses.
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Fig. 13. Mean force and SEA of embedded four-cell tubes with various wall thicknesses.
Tables
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Fig. 14. Force responses and deformed shapes for empty tube C30t1 and C30t2.
Table 1 Material hardening data for AA6063 O.
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Table 2 Experimental results for empty and EMC tubes. Table 3 Interaction effect of EMC tubes.
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Table 4 Mean crushing forces Pm of the specimens. (Unit: kN) Table 5 Numerical results of EMC tubes. Table 6 Theoretical predictions for Pm of empty square tubes. Table 7 Theoretical analysis and predictions for Pmc of EMC tubes.
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(B)
(A)
Enveloped by a square tube
Enveloped by triangular tube
a
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Welding line
Enveloped by a square tube
Enveloped by folding and welding of plate
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PT
ED
M
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Fig. 1. Illustrations for embedded multi-cell sections.
22
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ED
M
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Fig. 2. Specimens of empty tubes and embedded multi-cell sections.
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Fig. 3. Representative engineering stress-strain curves for AA6063 O tubes.
23
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M(4C15t1) N2
C32t0.95
N3
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N1
C32t0.95L150
M(9C10t1) N2
N3
ED
C15t1
M
N1
C10t1
AC
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Fig. 4. Deformed shapes of the empty and embedded multi-cell tubes.
24
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M(9C10t1)N1
A'
Top
Back
Left
A-A' sectional view
Right
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A
AC
CE
PT
ED
M
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Fig. 5. Illustrations for the deformation of representative embedded multi-cell tubes.
25
B
C
D
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A
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F
Fig. 6. Crushing force-displacement curves for the specimens.
AC
CE
PT
ED
M
E
26
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A
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B
AC
CE
PT
ED
M
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Fig. 7. Force responses and deformed shapes of tubes with shorter length.
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Rigid wall
Indentation trigger
o
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z
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V
y
Supported by rigid wall
x
AC
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ED
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Fig. 8. Representative finite element model for embedded multi-cell tubes.
28
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W/o trigger
W/ trigger
W/o trigger
W/ trigger
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M(9C10t1)
C32t0.95L150
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C15t1
AC
CE
PT
ED
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Fig. 9. Deformed shapes of specimens in numerical simulation.
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B
C
D
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A
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AC
CE
PT
ED
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Fig. 10. Simulated crushing force-displacement curves for the specimens.
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MC48t1.5(9C15t1)
MC33t1.5(4C15t1)
AC
CE
PT
ED
M
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Fig. 11. Deformed shapes for two EMC tubes.
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AC
CE
PT
ED
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Fig. 12. Crushing force-displacement curves of embedded multi-cell tubes with various wall thicknesses.
32
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AC
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PT
ED
M
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Fig. 13. Mean force and SEA of embedded four-cell tubes with various wall thicknesses.
33
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AC
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Fig. 14. Force responses and deformed shapes for empty tube C30t1 and C30t2.
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Table 1 Material hardening data for AA6063 O. 0.0 31.8 11.3 99.1
1.5 59.6 13.4 102.7
3.2 74.6 15.6 105.8
5.1 83.8 17.7 108.7
7.1 90.1 19.9 111.4
9.2 95.0 22.1 114.0
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M
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Plastic strain (%) Plastic stress (MPa) Plastic strain (%) Plastic stress (MPa)
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C15t1
C10t1
C10t1L25
M(4C15t1)
M(9C10t1)
ML60(9C10t1)
E (J)
Pm (kN)
N1
105.0
255.9
2.44
N2
105.0
251.2
2.39
N3
105.0
252.6
2.41
N1
52.5
135.3
2.58
N2
52.5
119.2
2.27
N3
52.5
136.8
2.61
N1
50.0
128.1
2.56
N2
50.0
131.8
2.64
N3
50.0
135.4
N1
20.0
28.1
N2
20.0
29.3
N3
20.0
26.5
N1
17.0
54.8
N2
17.0
54.8
N3
17.0
56.3
3.31
N1
42.0
767.3
18.27
N2
42.0
767.0
18.27
N3
42.0
764.0
18.19
N1
37.5
1285.9
34.30
N2
37.5
1266.1
33.77
N3
37.5
1293.3
34.48
N1
30.0
1146.1
38.21
N2
30.0
1125.6
37.51
N3
30.0
1096.7
36.56
AC
CE
PT
*Default length of the specimens is 75 mm.
36
Average (kN) 2.41
2.48
2.63
CR IP T
C32t0.95
δe (mm)
2.71 1.41 1.46 1.32 3.22 3.22
AN US
C32t0.95L150
Specimen
ED
Name
M
Table 2 Experimental results for empty and EMC tubes.
1.40
Global buckling
3.25
Extensional mode
18.24
34.18
37.43
ACCEPTED MANUSCRIPT
Table 3 Interaction effects of EMC tubes. Case
Pmc (kN)
Psum (kN)
Pint (kN)
Pint/ Pmc
M(4C15t1) M(9C10t1) ML60(9C10t1) M(9C10t1)* ML60(9C10t1)*
18.24 34.18 37.43 34.18 37.43
12.95 14.99 14.99 31.67 31.67
5.29 19.19 22.43 2.51 5.76
29.0% 56.1% 59.9% 7.4% 15.4%
AC
CE
PT
ED
M
AN US
CR IP T
* Mean crushing force of C10t1L25 is employed.
37
ACCEPTED MANUSCRIPT
Table 4 Mean crushing forces Pm of the specimens. (Unit: kN)
Specimen
δNum (mm)
Test
105.0 50.0 45.0 44.0
2.41 2.63 18.24 34.18
Theoretical prediction
w/o trigger
w/ trigger
Error (%)
Pm
Error (%)
/ / 17.30 31.15
2.27 2.50 16.75 31.87
-5.8 -5.3 -8.2 -6.8
2.42 2.50 17.69 33.01
0.2 -5.2 -3.0 -3.4
AC
CE
PT
ED
M
AN US
CR IP T
C32t0.95L150 C15t1 M(4C15t1) M(9C10t1)
Numerical simulation
38
ACCEPTED MANUSCRIPT
Table 5 Numerical results of EMC tubes. Mass (g)
E (J)
SEA (J/g)
Pmc (kN)
MC32t1(4C15t0.5) MC32t1(4C15t0.75) MC32t1(4C15t1)
48.6 59.8 70.5
330.06 517.86 770.32
6.79 8.66 10.93
7.33 11.51 17.12
MC33t1.5(4C15t0.5) MC33t1.5(4C15t0.75) MC33t1.5(4C15t1)
61.8 73.0 83.7
484.65 798.00 1055.30
7.84 10.94 12.61
10.77 17.73 23.45
MC34t2(4C15t0.5) MC34t2(4C15t0.75) MC34t2(4C15t1)
75.4 86.5 97.3
782.59 1079.61 1407.55
10.38 12.48 14.47
17.39 23.99 31.28
M47t1(9C15t1) M48t1.5(9C15t1) M49t2(9C15t1)
139.4 158.6 178.3
1553.64 1803.22 2335.58
11.15 11.37 13.10
34.53 40.07 51.90
AC
CE
PT
ED
M
AN US
CR IP T
Specimens
39
ACCEPTED MANUSCRIPT
Table 6 Theoretical predictions for Pm of empty square tubes. t (mm)
σ0 (MPa) by Eq. (4)
Eq. (3) Pm (kN)
Num. Pm (kN)
Error (%)
Pm (kN) σ0=(σy+σu)/2
Error (%)
30 30 30 40 40 40 50 50 50
1 1.5 2 1 1.5 2 1 1.5 2
66.6 75.1 79.3 58.1 69.4 75.1 49.6 63.8 70.8
2.70 5.99 10.22 2.60 6.09 10.65 2.39 6.03 10.82
2.51 5.73 9.68 2.72 6.13 11.61 2.99 6.10 10.16
7.7 4.5 5.6 -4.7 -0.5 -8.3 -20.2 -1.2 6.5
2.50 4.91 7.92 2.75 5.40 8.72 2.96 5.82 9.39
-0.5 -14.4 -18.1 0.9 -11.9 -24.9 -1.1 -4.7 -7.5
AC
CE
PT
ED
M
AN US
CR IP T
C (mm)
40
ACCEPTED MANUSCRIPT
Table 7 Theoretical analysis and prediction for Pmc of EMC tubes.
MC32t1(4C15t0.75) MC32t1(4C15t1) MC33t1.5(4C15t0.5) MC33t1.5(4C15t0.75) MC33t1.5(4C15t1) MC34t2(4C15t0.5) MC34t2(4C15t0.75) MC34t2(4C15t1) M47t1(9C15t1) M48t1.5(9C15t1) M49t2(9C15t1)
Poutside
Pinside
Psum
Pint
Pint/ Pmc
Eq. (6)
Error
Pmc (kN)
(kN)
(kN)
(kN)
(kN)
(%)
Pmc (kN)
(%)
7.33 11.51 17.12 10.77 17.73 23.45 17.39 23.99 31.28 34.53 40.07 51.90
2.69 2.69 2.69 6.04 6.04 6.04 10.43 10.43 10.43 2.46 6.05 10.81
0.68 1.50 2.55 0.68 1.50 2.55 0.68 1.50 2.55 2.55 2.55 2.55
5.39 8.68 12.91 8.74 12.03 16.26 13.13 16.41 20.65 25.45 29.05 33.80
1.94 2.83 4.21 2.03 5.70 7.19 4.26 7.58 10.63 9.07 11.02 18.10
26.5 24.6 24.6 18.8 32.2 30.7 24.5 31.6 34.0 26.3 27.5 34.9
7.55 12.15 18.07 12.24 16.84 22.77 18.38 22.98 28.90 35.63 40.67 47.33
2.9 5.6 5.6 13.7 -5.0 -2.9 5.7 -4.2 -7.6 3.2 1.5 -8.8
CR IP T
MC32t1(4C15t0.5)
Num.
AN US
Specimen
AC
CE
PT
ED
M
Graphical abstract
41