Abstract
Finite element analysis and optimization design carry out for the quasi static responses of foamfilled double circular tube is presented in this paper. In the investigation of the crashworthiness capability, some aspects were considered for variations in geometry parameters of tubes and the loading condition to investigate the crashworthiness capability. Empty, foamfilled, and full foamfilled doublé tubes of thin walled structures were observed subjected to oblique impact (0˚  40˚). The numerical solution was used to determine the crashworthiness parameters. In addition, NSGA II and Radial Basis Function were used to optimize the crashworthiness capability of tubes. In conclution, the crash performaces of foamfilled double tube is better than the other structures in this work. The outcome that expected is the new design information of various kinds of cylindrical tubes for energy absorber application.
Keywords:
Aluminium foam; Crashworthiness; Cylindrical tube; Finite Element Analysis; Oblique impact; Optimization
1 INTRODUCTION
Foamfilled thinwalled structures have recently increased interest in the automotive industry because of the great energy absorption capacity and extraordinary lightweight. Huge research efforts have been carried out by various experiments (Gupta and Velmurugan, 1999Gupta NK, R. Velmurugan. Axial compression of empty and foam filled composite conical shells, J Compos Mater, 33 (1999) 567591.; Hassen et al., 2000Hanssen A.C, M. Langseth, O.S. Hopperstad, Static and dynamic crushing of circular aluminium extrusions with aluminium foam filler, Int J Impact Eng, 24 (2000) 475507.; Santosa et al., 2000Santosa S.P, T. Wierzbicki, A.G. Hanssen, M. Langseth, Experimental and numerical studies of foamfilled sections, Int J Impact Eng, 24 (2000) 509534.; Borvik et al., 2003Borvik T, O.S. Hopperstad, A. Reyes, M. Langseth, G. Solomos, T. Dyngeland, Empty and foamfilled circular aluminium tubes subjected to axial and oblique quasistatic loading, Int J Crashworthines, 8 (2003) 481494.; Meguid et al., 2004Meguid S.A, M.S. Attia, A. Monfort, On the crush behaviour of ultralight foamfilled structures, Mater Design, 25 (2004) 183189.; Reyes et al., 2004Reyes A, Hopperstad OS, Langseth M. Aluminum foamfilled extrusions subjected to oblique loading: experimental and numerical study. Int J Solids Struct 2004;41(56):164575.; Babbage and Mallick 2005Babbage J.M, P.K. Mallick, Static axial crush performance of unfilled and foamfilled aluminumcomposite hybrid tubes, Compos Struct, 70 (2005) 177184.; Mamalis et al., 2008Mamalis A. G, D.E. Manolakos, M.B. Ioannidis, K.N. Spentzas, S. Koutroubakis, Static axial collapse of foamfilled steel thinwalled rectangular tubes: experimental and numerical simulation, Int J Crashworthines, 13 (2008) 117126.; Rezadoust et al., 2008Rezadoust A. M, M. Esfandeh, S.A. Sabet, Crush behavior of conical composite shells: Effect of cone angle and diameter/wall thickness ratio, PolymPlast Technol, 47 (2008) 147151.; Taher et al., 2009Taher, R. Zahari, S. Ataollahi, F. Mustapha, S. Basri, A doublecell foamfilled composite block for efficient energy absorption under axial compression, Compos Struct, 89 (2009) 399407. 23; Ghamarian et al. 2011Ghamarian, H.R. Zarei, M.T. Abadi, Experimental and numerical crashworthiness investigation of empty and foamfilled endcapped conical tubes, Thin Wall Struct, 49 (2011) 13121319.; Niknejad et al., 2011Niknejad, G.H. Liaghat, H.M. Naeini, A.H. Behravesh, Theoretical and experimental studies of the instantaneous folding force of the polyurethane foamfilled square honeycombs, Mater Design, 32 (2011) 6975.; Kavi et al 2006Kavi, A.K. Toksoy, M. Guden, Predicting energy absorption in a foamfilled thinwalled aluminum tube based on experimentally determined strengthening coefficient, Mater Design, 27 (2006) 263269.; Seitzberger et al., 1997Seitzberger, F.G. Rammerstorfer, H.P. Degischer, R. Gradinger, Crushing of axially compressed steel tubes filled with aluminium foam, Acta Mech, 125 (1997) 93105.), analytical analyses (Gupta and Velmurugan, 1999Gupta NK, R. Velmurugan. Axial compression of empty and foam filled composite conical shells, J Compos Mater, 33 (1999) 567591.; Taher et al., 2009Taher, R. Zahari, S. Ataollahi, F. Mustapha, S. Basri, A doublecell foamfilled composite block for efficient energy absorption under axial compression, Compos Struct, 89 (2009) 399407. 23; Wang et al., 2007Wang, Z.J. Fan, L.J. Gui, Theoretical analysis for axial crushing behaviour of aluminium foamfilled hat sections, Int J Mech Sci, 49 (2007) 515521.), and numerical methods (Santosa et al., 2000Santosa S.P, T. Wierzbicki, A.G. Hanssen, M. Langseth, Experimental and numerical studies of foamfilled sections, Int J Impact Eng, 24 (2000) 509534.; Reyes et al., 2004Reyes A, O.S. Hopperstad, M. Langseth, Aluminum foamfilled extrusions subjected to oblique loading: experimental and numerical study, Int J Solids Struct, 41 (2004) 16451675.; Babbage and Mallick 2005Babbage J.M, P.K. Mallick, Static axial crush performance of unfilled and foamfilled aluminumcomposite hybrid tubes, Compos Struct, 70 (2005) 177184.; Rezadoust et al., 2008Rezadoust A. M, M. Esfandeh, S.A. Sabet, Crush behavior of conical composite shells: Effect of cone angle and diameter/wall thickness ratio, PolymPlast Technol, 47 (2008) 147151.; Niknejad et al., 20113Niknejad, G.H. Liaghat, H.M. Naeini, A.H. Behravesh, Theoretical and experimental studies of the instantaneous folding force of the polyurethane foamfilled square honeycombs, Mater Design, 32 (2011) 6975.; Seitzberger et al., 1997Seitzberger, F.G. Rammerstorfer, H.P. Degischer, R. Gradinger, Crushing of axially compressed steel tubes filled with aluminium foam, Acta Mech, 125 (1997) 93105.; Ahmad et al., 2008Ahmad Z, Thambiratnam DP. Dynamic computer simulation and energy absorption of foamfilled conical tubes under axial impact loading. Comput Struct 2009;87(34):18697., ^{2009}Ahmad, D.P. Thambiratnam, Crushing response of foamfilled conical tubes under quasistatic axial loading, Mater Design, 30 (2009) 23932403.; ZiaeiRad et al 2008ZiaeiRad, M. Salimi, L. Mirfendereski, Finite Element Modelling of FoamFilled Tapered ThinWalled Rectangular Tubes under Oblique Impact Loading, Steel Res Int, (2008) 317324.; Santosa et al 2001Santosa, J. Banhart, T. Wierzbicki, Experimental and numerical analyses of bending of foamfilled sections, Acta Mech, 148 (2001) 199213.).
The tube collapse more efficiently because the interaction between tube and filler might change the original collapse mode. Furthermore, some researchers (Seitzberger et al 1997Seitzberger, F.G. Rammerstorfer, H.P. Degischer, R. Gradinger, Crushing of axially compressed steel tubes filled with aluminium foam, Acta Mech, 125 (1997) 93105., ^{2000}Seitzberger M, F.G. Rammerstorfer, R. Gradinger, H.P. Degischer, M. Blaimschein, C. Walch, Experimental studies on the quasistatic axial crushing of steel columns filled with aluminium foam, Int J Solids Struct, 37 (2000) 41254147.; Yuen et al., 2008Yuen S, Chung Kim, Nurick GN, Starke RA. The energy absorption characteristics of doublecell tubular profiles. Lat Am J Solids Struct 2008; 5(4) : 289317.) used a doublecell profile arrangements with similar crosssection and the double tubes are empty or filled with aluminium foam to increase the energy absorption capabilities of thinwalled tubes. In addition, Guo et al (2010aGuo LW, Yu JL, Li ZB. Experimental studies on the quasistatic bending behavior of double square columns filled with aluminum foams. Acta Mech 2010; 213:34958., ^{2010b}Guo LW, Yu JL. Experimental studies on the quasistatic axial crushing behavior of double square columns filled with aluminum foams. J Exp Mech 2010;25(3):2718., ^{2011}Guo LW, Yu JL. Dynamic bending response of double cylindrical tubes filled with aluminum foam. Int J Impact Eng 2011;38(23):8594.) carried out the experiment and numerical simulation of the new topological structure i.e. double circular tubes under axial and three point bending conditions.
In the real vehicle collision event, a combination of the angle of impacting direction (or oblique/offaxis loads) is more occurred frequently and rarely encounters either fully axial or fully bending impact. Effects of oblique loading have been taken into account by Han and Park (1999Han DC, Park SH. Collapse behavior of square thinwalled columns subjected to oblique loads. Thinwalled Struct 1999; 35:16784.). They found that angle of loading played an important role from axial progressive collapse mode to the global bending mode for square column subjected to oblique loads. In addition, Reyes et al. (2002Reyes A, Langseth M, Hopperstad OS. Crashworthiness of aluminum extrusions subjected to oblique loading: experiments and numerical analyses. Int J Mech Sci 2002;44(9):196584., ^{2003}Reyes A, Langseth M, Hopperstad. OS. Square aluminum tubes subjected to oblique loading. Int J Impact Eng 2003;28(10):1077106., ^{2004}Reyes A, Hopperstad OS, Langseth M. Aluminum foamfilled extrusions subjected to oblique loading: experimental and numerical study. Int J Solids Struct 2004;41(56):164575.) showed that the increasing of loading angle as the decreasing of the energy absorption, in the case of the empty and foamfilled square columns under the quasistatic oblique loading conditions. Meanwhile, the energy absorption characteristics of tapered thinwalled rectangular tubes were explored by Nagel and Thambiratnam (2004Nagel G, Thambiratnam D. A numerical study on the impact response and energy absorption of tapered thinwalled tubes. Int J Mech Sci 2004; 46:20116., ^{2005}Nagel G, Thambiratnam D. Computer simulation and energy absorption of tapered thinwalled rectangular tubes. Thinwall Struct 2005; 43:122542., ^{2006}Nagel G, Thambiratnam D. Dynamic simulation and energy absorption of tapered thinwalled tubes under oblique impact loading. Int J Impact Eng 2006; 32:1595620.). They concluded that the tapered tube better stability than straight tube when the oblique impact happen. Ahmad et al. (2010Ahmad Z, Thambiratnam DP, Tan ACC. Dynamic energy absorption characteristics of foamfilled conical tubes under oblique impact loading. Int J Impact Eng 2010; 37:47588.) focused on investigate the benefit of foamfilled conical tubes as vehicle energy absorber under oblique impact. Li et al (2012Li ZB, Yu JL, Guo LW. Deformation and energy absorption of aluminium foam filled tubes subjected to oblique loading. Int J Mech Sci 2012:4856.) studied the deformation and energy absorption of aluminum foam filled double cylindrical tubes subjected to oblique loading and found that foamfilled double circular tube better crashworthiness capability than empty and foamfilled single circular tubes under quasistatic oblique loading.
There is limited design information about the double circular tube filled by foam, particularly under different angular impact. The aim of this paper is to determine the crushing behaviour and optimum values with respect to wall diameter, the wall thickness, and the foam density of the double circular tubes. Finite element models were validated by experiment and theory resulted in studying the effect of parameters for different tubes, namely empty double tube, foamfilled double tube, and full foamfilled double tube. Finally, crashworthiness optimization of double circular tubes under different impact angular loading (0, 10, 20, 30 and 40 degree), which serve as guidelines for efficient design of foamfilled bitubal tubes.
2 DEVELOPMENT OF FINITE ELEMENT MODEL (FEM)
Fig. 1 shows the schematics of the tubes under different impact loadings. To model energy absorbing structures, the aluminium foam filled double tubes were 250 mm in length, and the diameter of the outer and inner tubes are 100 mm and 50 mm, respectively. All of the structures fixed at the bottom and at the other end impacted by a moving rigid wall with constant velocity of 0.9 m/s in the Figure 1. The tubes crosssection is shown in Figure 2 and the outer and inner thicknesses of the double tube walls were the same at 1.8 mm.
Crosssection of the circular tubes (a) empty double tube, (b) foamfilled double tube, and (c) full foamfilled tube.
The finite element (FE) models of empty double tube, foamfilled double tube, and foamfilled single tube are shown in Fig. 3. The ABAQUSExplicit was used to develop the aluminium foamfilled tubular tube models and to predict the response of thinwalled tubes that were impacted by a free falling impinging mass.
The Finite Element (FE) models of cylindrical tubes (a) empty double tube, (b) foamfilled double tube, and (c) full foamfilled tube.
Four node shell elements were used for tube walls and eight node continuum elements were used for foam. The element size is 2 mm based on a mesh convergence study of shells and foam elements. For all contact for instance the interaction between the foam and the tube walls, the cofficient value is 0.3
3 MATERIAL PROPERTIES
3.1 ThinWalled Circular Double Tubes Material
The thinwalled tubes were made from aluminium alloy A6060 T4 with mechanical properties of density ρ=2700kg/m^{3}, the Young's modulus E = 68.2 GPa, the Poisson's ratio v = 0.3 initial yielding stress σ_{y} = 80 MPa, and ultimate stress σ_{u} = 215.5 MPa. The pairs of the plastic strain and true stress were specified in Table 1 to accurately define the hardening characteristic in finite element models.
3.2 Aluminium Foam Filled Material
With an average mechanical property value obtained from material tests, the aluminium closedcell foam filler was used. With different foam apparent densities, the material's behaviour was obtained from experimental testing of the foam filled material, while the uniaxial quasistatic compression test results are given in references (Santosa et al., 2000Santosa S.P, T. Wierzbicki, A.G. Hanssen, M. Langseth, Experimental and numerical studies of foamfilled sections, Int J Impact Eng, 24 (2000) 509534.). Nonlinear ABAQUS/Explicit software packages was used to model the constitutive behaviour based on an isotropic uniform material of the foam model developed by Dehspande and Fleck (2000Deshpande, N.A. Fleck, Isotropic constitutive models for metallic foams, J Mech Phys Solids, 48 (2000) 12531283.)
4 CRASHWORTHINESS INDICATOR
To evaluate the energyabsorbance of structures, it is necessary to define the crashworthiness indicators. The parameters, such as Energy Absorption (EA), SEA, and PCF, can efficiently evaluate the crashworthiness of structures. Energy absorption can be calculated as:
where, F(x) is the instantaneous crashing force with a function of the displacement δ.
SEA indicates the absorbed energy (EA(δ)) per unit mass (M) of a structure as:
where, M_{total} is the structure's total mass. In this case, a higher value indicates the higher energy absorption efficiency of a material.
The average crush force (F_{avg}) is the response parameter for the energy absorption capability:
where, energy is absorbed (EA(δ)) during collapse and displacement (δ).
Crush force efficiency is defined as the ratio of the average crush force (F_{agv} ) to the peak crush force (F_{max} ),
Peak crush force (F_{max}) is important indicator in the design of energy absorption structures to absorb the impact energy in collision.
5 CRASHWORTHINESS OPTIMIZATION
The surrogate model (Radial Basis Functions (RBF)) and DOptima were choosed for the relationships between the individual objective functions and the design variable vector. Nondominated sorting GA (NSGA) II was choosed because a more effective and efficient algorithm for ranking the solution, assigning ranking fitness, and benchmarking number problems. Figure 4 shows the implementation of multi objective optimization for the empty and foamfilled double cylindrical tubes. All the optimization designs were developed by MATLAB.
6 MODEL VALIDATION
To ascertain whether it was sufficiently accurate, FE model has been validated by experimental and theoretical model in the literatures. Djamaluddin et al. (2015aDjamaluddin, S. Abdullah, A.K. Ariffin, Z.M. Nopiah. Optimization of foamfilled double circular tubes under axial and oblique impact loading conditions. Thinwalled structures. 87 (2015) 111.) presented the validation results of foamfilled double circular tubes under quasistatic impact loading. The deformation patterns (Djamaluddin et al., 2014aDjamaluddin, S. Abdullah, A.K. Ariffin, Z.M. Nopiah. Multi Objective Optimization of Aluminum Foam Filled Double Tubes Subjected to Oblique Impact Loading for Automobile Bumper Beam, Applied Mechanics and Materials Vol. 663 (2014) pp 9397) and force  displacement curve (Djamaluddin et al., 2014bDjamaluddin, S. Abdullah, A.K. Ariffin, Z.M. Nopiah. Multi objective optimization of foamfilled tubular circular tubes for quasistatic and dynamic responses. Latin american journal solid and structure, 2015, ^{2015b}Djamaluddin, S. Abdullah, A.K. Ariffin, Z.M. Nopiah. Optimization of foamfilled double circular tubes under axial and oblique impact loading conditions. Thinwalled structures. 87 (2015) 111.) of numeric simulation and experiment result were carried out the comparison between the numerical simulation and the experiment result and demonstrated of foamfilled double cylindrical tube subjected to quasistatic loading (Li et al., 2012Li ZB, Yu JL, Guo LW. Deformation and energy absorption of aluminium foam filled tubes subjected to oblique loading. Int J Mech Sci 2012:4856.)
7 RESULT AND DISCUSSION
7.1 Deformation Modes
Deformation modes of the tubes can be clearly seen that the optimal form in Figure 5. All type of tubes under axial loading in 0˚ and angular loding 10˚ deform progressive buckling. Fold in all the tube that has formed the global symmetry collapse mode (Abramowicz and Jones, 1984, Qi et al. 2012). At 20˚ loading, double empty tube (ED) and the double tube that full foam (DD) failed with progressive global collapse but the double tube filled with foam (FD) failed in collapse bending mode. Some folds can be seen on the top of FD in plastic shapes flange. When the angle of loading at 30˚ and 40˚, all of tubes failure or deformation are in the bending global zone. Furthermore, the lobes are formed by compressionlipped in bending mode. Some structures have the same under oblique impact such as square tube (Reyes et al. 2002Reyes A, Langseth M, Hopperstad OS. Crashworthiness of aluminum extrusions subjected to oblique loading: experiments and numerical analyses. Int J Mech Sci 2002;44(9):196584.), rectangular tube (Nagel and Thambiratnam 2006Nagel G, Thambiratnam D. Dynamic simulation and energy absorption of tapered thinwalled tubes under oblique impact loading. Int J Impact Eng 2006; 32:1595620.) and conical tube (Ahmad et al., 2010Ahmad Z, Thambiratnam DP, Tan ACC. Dynamic energy absorption characteristics of foamfilled conical tubes under oblique impact loading. Int J Impact Eng 2010; 37:47588.).
Deformation modes (a) empty double tube (ED), (b) foam filled double tube (FD) and (c) full foam filled double tube (DD).
7.2 Effect of Thickness
Figure 6 show the effect of tube wall thickness and angle of loading are assessed to determine the mean crushing force from progressive collapse to bending collapse of tube. Mean crushing force is calculated with displacement of 120 mm for each tubes. It is expected that the mean crushing force (F_{avg} ) decrease with increase angle of load. The wall thickness of tube increase give effect to F_{agv} value in different loading angle. But, the FD tube under 30˚ And 40˚ decrease wall thickness of tube can be increase F_{agv} . Normally, the mean crushing force the in progressive collapse more sensitive compared to bending collapse. Furthermore, the increasing of load angle is more important when the large of wall thickness which it can be reduce the SEA and F_{agv} value (Qi et al. 2012).
Effect of wall thickness on mean crushing force of (a) ED, (b) FD. Effect of wall thickness on mean crushing force of (c) DD with different load angle.
7.3 Effect of Diameter
Figure 7 shows the effect of different diameter (d) of the tube wall in response to a quasistatic parameter such as specific energy absorption (SEA) each tube configuration. The results indicated that EA is more sensitive in progressive collapse than bending collapse on the diameter of the tube wall responses. However, the SEA does not have a significant impact on increasing of tube diameter and angle of loading for the two types of progressive collapse and global bending. Furthermore, it is interesting to be noted that the reduction in energy absorption and specific energy absorption are caused with increasing the load angle. This is also in contrast to the case with the increase in wall thickness as shown previously.
7.4 Effect of Foam Density
Figure 8 and 9 show the effect of varying density foam on the quasistatic response such as the crushing energy efficiency (CFE) under axial load (0˚) and oblique (10˚  40˚). In general, increasing the density of the foam effect crusher energy efficiency significantly by loading different angle. It notes that the CFE is less with lowdensity foam (0.22 g / cm3) of the tube is filled with high density foam (0.534 and 0.71 g / cm3). Ahmad and Thambiratnam (2009Ahmad, D.P. Thambiratnam, Crushing response of foamfilled conical tubes under quasistatic axial loading, Mater Design, 30 (2009) 23932403.) studied the density of the foam is an important thing and it has an effect that can be used as a parameter to control the crashworthines characteristics and behavior of thinwalled tubes.
7.5 Effect of Geometrical Imperfection
To read the buckling modes of the inner and outer tubes placed between two rigid wall. The imperfection is used in ABAQUS/Explicit. Figure 10 show that the simulations have initial geometrical imperfection and without and it is evident that the collapsing shape of the tubes by sufficient accuracy.
Collapsing mode of empty doublé tubes (ED) under axial quasistatic load (a) with initial geometrical imperfection and (b) without.
7.6 Optimization Design of Double Tubes (FD) Under Oblique Impact
In this section, the double foam filled tube are optimized under five impact angles 0°, 10°, 20°, 30° and 40°. The optimization problem can be written as the following:
Figure 11 shows the Pareto fronts of PCF against SEA for double circular foam filled tubes under five different impact angles. In addition, two objective SEA and PCF conflict with each other for all design cases. It can be seen that any increase in SEA always leads to an undesirable increase in PCF, and vice versa. From 0° to 30°, the Pareto front gradually moves to the topleft region as the impact angle increases indicating that both PCF and SEA decrease with an increase of impact angle.
Pareto fronts for double circular foam filled (FD) tubes under four different impact angles.
8 CONCLUSIONS
This study was purpose to investigate and and generate design information the effect of foam filling, on the quasi static responses and energy absorption characteristics of thinwalled circular tubes using finite element simulations. Energy absorption response was quantified with respect to variations in the parameter of foam density, wall thickness and diameter of tube.
This paper has observased the crush response and energy absorption of three different cross section of circular tubes namely emptyempty tube, empty foam filled tube and foam filledfoam filled double cylindrical tubes under axial compression and oblique impact loadings. The computer simulations, validated by experiments and theory, have been used to obtain an insight into the oblique crush response and quantify the energy absortion, specific energy absorption and peak crushing force for variations in the load angle, wall thickness, foam density and length of tubes. The main conclusion is that full foam filled double circular tubes were found to be effective energy absorbing devices since they can withstand an oblique impact load as effectively as an axial compression and bending mode to reduct the energy absorption as crashworthiness structures.
Finally, the optimization design of foam filled doublé tuve under multiple impact angles have been investigated. By maximizing the specific energy absorption and minimizing the peak crushing force simultaneously under different impact angles, so multi optimization problem is considered. It is found that the optimal design highly depends on the given impact angle for FD and it conclude that FD has much better crashworthiness capacity and it can be recommended as an efficient energy absorber for vehicles
References
 Ahmad, D.P. Thambiratnam, A.C.C. Tan, A numerical study on the axial loading and energy absorption of foamfilled conical tubes, Proceedings of the 7th International Conference on Shock & Impact Loads on Structures, (2007) 99106.
 Ahmad Z, Thambiratnam DP. Dynamic computer simulation and energy absorption of foamfilled conical tubes under axial impact loading. Comput Struct 2009;87(34):18697.
 Ahmad, D.P. Thambiratnam, Crushing response of foamfilled conical tubes under quasistatic axial loading, Mater Design, 30 (2009) 23932403.
 Ahmad Z, Thambiratnam DP, Tan ACC. Dynamic energy absorption characteristics of foamfilled conical tubes under oblique impact loading. Int J Impact Eng 2010; 37:47588.
 Babbage J.M, P.K. Mallick, Static axial crush performance of unfilled and foamfilled aluminumcomposite hybrid tubes, Compos Struct, 70 (2005) 177184.
 Borvik T, O.S. Hopperstad, A. Reyes, M. Langseth, G. Solomos, T. Dyngeland, Empty and foamfilled circular aluminium tubes subjected to axial and oblique quasistatic loading, Int J Crashworthines, 8 (2003) 481494.
 Deshpande, N.A. Fleck, Isotropic constitutive models for metallic foams, J Mech Phys Solids, 48 (2000) 12531283.
 Djamaluddin, S. Abdullah, A.K. Ariffin, Z.M. Nopiah. Optimization of foamfilled double circular tubes under axial and oblique impact loading conditions. Thinwalled structures. 87 (2015) 111.
 Djamaluddin, S. Abdullah, A.K. Ariffin, Z.M. Nopiah. Multi objective optimization of foamfilled tubular circular tubes for quasistatic and dynamic responses. Latin american journal solid and structure, 2015
 Djamaluddin, S. Abdullah, A.K. Ariffin, Z.M. Nopiah. Multi Objective Optimization of Aluminum Foam Filled Double Tubes Subjected to Oblique Impact Loading for Automobile Bumper Beam, Applied Mechanics and Materials Vol. 663 (2014) pp 9397
 Gupta NK, R. Velmurugan. Axial compression of empty and foam filled composite conical shells, J Compos Mater, 33 (1999) 567591.
 Ghamarian, H.R. Zarei, M.T. Abadi, Experimental and numerical crashworthiness investigation of empty and foamfilled endcapped conical tubes, Thin Wall Struct, 49 (2011) 13121319.
 Guo LW, Yu JL, Li ZB. Experimental studies on the quasistatic bending behavior of double square columns filled with aluminum foams. Acta Mech 2010; 213:34958.
 Guo LW, Yu JL. Experimental studies on the quasistatic axial crushing behavior of double square columns filled with aluminum foams. J Exp Mech 2010;25(3):2718.
 Guo LW, Yu JL. Dynamic bending response of double cylindrical tubes filled with aluminum foam. Int J Impact Eng 2011;38(23):8594.
 Han DC, Park SH. Collapse behavior of square thinwalled columns subjected to oblique loads. Thinwalled Struct 1999; 35:16784.
 Hanssen A.C, M. Langseth, O.S. Hopperstad, Static and dynamic crushing of circular aluminium extrusions with aluminium foam filler, Int J Impact Eng, 24 (2000) 475507.
 Kavi, A.K. Toksoy, M. Guden, Predicting energy absorption in a foamfilled thinwalled aluminum tube based on experimentally determined strengthening coefficient, Mater Design, 27 (2006) 263269.
 Li ZB, Yu JL, Guo LW. Deformation and energy absorption of aluminium foam filled tubes subjected to oblique loading. Int J Mech Sci 2012:4856.
 Mamalis A. G, D.E. Manolakos, M.B. Ioannidis, K.N. Spentzas, S. Koutroubakis, Static axial collapse of foamfilled steel thinwalled rectangular tubes: experimental and numerical simulation, Int J Crashworthines, 13 (2008) 117126.
 Meguid S.A, M.S. Attia, A. Monfort, On the crush behaviour of ultralight foamfilled structures, Mater Design, 25 (2004) 183189.
 Nagel G, Thambiratnam D. A numerical study on the impact response and energy absorption of tapered thinwalled tubes. Int J Mech Sci 2004; 46:20116.
 Nagel G, Thambiratnam D. Computer simulation and energy absorption of tapered thinwalled rectangular tubes. Thinwall Struct 2005; 43:122542.
 Nagel G, Thambiratnam D. Dynamic simulation and energy absorption of tapered thinwalled tubes under oblique impact loading. Int J Impact Eng 2006; 32:1595620.
 Niknejad, G.H. Liaghat, H.M. Naeini, A.H. Behravesh, Theoretical and experimental studies of the instantaneous folding force of the polyurethane foamfilled square honeycombs, Mater Design, 32 (2011) 6975.
 Reyes A, O.S. Hopperstad, M. Langseth, Aluminum foamfilled extrusions subjected to oblique loading: experimental and numerical study, Int J Solids Struct, 41 (2004) 16451675.
 Reyes A, Langseth M, Hopperstad OS. Crashworthiness of aluminum extrusions subjected to oblique loading: experiments and numerical analyses. Int J Mech Sci 2002;44(9):196584.
 Reyes A, Langseth M, Hopperstad. OS. Square aluminum tubes subjected to oblique loading. Int J Impact Eng 2003;28(10):1077106.
 Reyes A, Hopperstad OS, Langseth M. Aluminum foamfilled extrusions subjected to oblique loading: experimental and numerical study. Int J Solids Struct 2004;41(56):164575.
 Rezadoust A. M, M. Esfandeh, S.A. Sabet, Crush behavior of conical composite shells: Effect of cone angle and diameter/wall thickness ratio, PolymPlast Technol, 47 (2008) 147151.
 Santosa S.P, T. Wierzbicki, A.G. Hanssen, M. Langseth, Experimental and numerical studies of foamfilled sections, Int J Impact Eng, 24 (2000) 509534.
 Santosa, J. Banhart, T. Wierzbicki, Experimental and numerical analyses of bending of foamfilled sections, Acta Mech, 148 (2001) 199213.
 Seitzberger, F.G. Rammerstorfer, H.P. Degischer, R. Gradinger, Crushing of axially compressed steel tubes filled with aluminium foam, Acta Mech, 125 (1997) 93105.
 Seitzberger M, F.G. Rammerstorfer, R. Gradinger, H.P. Degischer, M. Blaimschein, C. Walch, Experimental studies on the quasistatic axial crushing of steel columns filled with aluminium foam, Int J Solids Struct, 37 (2000) 41254147.
 Shahbeyk S, Spetrinic N, Vafai A. Numerical modelling of dynamically loaded metal foamfilled square columns. Int J Impact Eng 2007;34:57386.
 Taher, R. Zahari, S. Ataollahi, F. Mustapha, S. Basri, A doublecell foamfilled composite block for efficient energy absorption under axial compression, Compos Struct, 89 (2009) 399407. 23
 Wang, Z.J. Fan, L.J. Gui, Theoretical analysis for axial crushing behaviour of aluminium foamfilled hat sections, Int J Mech Sci, 49 (2007) 515521.
 Yuen S, Chung Kim, Nurick GN, Starke RA. The energy absorption characteristics of doublecell tubular profiles. Lat Am J Solids Struct 2008; 5(4) : 289317.
 ZiaeiRad, M. Salimi, L. Mirfendereski, Finite Element Modelling of FoamFilled Tapered ThinWalled Rectangular Tubes under Oblique Impact Loading, Steel Res Int, (2008) 317324.
Publication Dates

Publication in this collection
Nov 2016
History

Received
12 Feb 2016 
Reviewed
07 June 2016 
Accepted
08 June 2016