Non-linear finite element analysis of bitubal circular tubesfor progressive and bending collapses

F. Djamaluddin, S. Abdullah n, A.K. Ariffin, Z.M. NopiahDepartment of Mechanical and Materials Engineering, Faculty of Engineering and Built Environment, Universiti Kebangsaan Malaysia, 43600 UKM, Bangi,Selangor, Malaysia

a r t i c l e i n f o

Article history:Received 24 December 2014Received in revised form26 May 2015Accepted 1 June 2015Available online 9 June 2015

Keywords:Aluminium foamCrashworthinessCylindrical tubeFinite element analysisOblique impact

a b s t r a c t

This paper presents non-linear finite element analysis to carry out the dynamic response of foam-filleddouble circular tubes for axial progressive and global bending collapse modes. Some aspects wereconsidered to investigate the crashworthiness capability of cylindrical tube for instance geometry,material and loading parameters. In addition, three types of structures namely empty double tubes (ED),foam-filled double tubes (FD), and foam-filled single tubes (FS) were observed under oblique impact.Crashworthiness parameters were determined in the numerical solution after validated with experimentand theoretical results in relevant references. Moreover, it is evident that the crash ability of the foam-filled double tube is better than the other structures. Finally, the main outcome of this study is the newdesign information of different tube configurations as energy absorber where two collapse modes areexpected.

& 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Because of the great energy absorption capacity and extraordinarylightweight, foam-filled thin-walled structures have recently increasedinterest in the automotive, high-speed railway and aerospace indus-tries. Huge research efforts have been carried out by various experi-ments [1–15], analytical analyses [1,11,13,16], and numerical methods[3,6,7,9,12,15,17–20].

Foam filled thin walled structures as cellular materials assist toimprove crushing stability and collapse modes thus increasing theoverall crashworthiness. The foam density and wall thickness arevariables to enhance the energy absorption capacity. Also, the inter-action between tube and filler might change the original collapsemode of the tube into more collapse mode efficiently. Furthermore, toincrease the energy absorption capabilities of thin-walled tubes, someresearchers [21–23] used a double-cell profile arrangements withsimilar cross-section and the double tubes are empty or filled withaluminium foam. In addition, experiment and numerical simulation ofthe new topological structure i.e. double circular tubes under axial andthree point bending conditions carried out by Guo et al. [24–26].

A combination of the angle of impacting direction (or oblique/off-axis loads) is more occurred frequently in the real vehicle collisionevent and rarely encounters either fully axial or fully bending impact.Effects of oblique loading have been taken into account by Han andPark [27]. They found that angle of loading played an important role

from axial progressive collapse mode to the global bending mode forsquare column subjected to oblique loads. In addition, Reyes et al. [28–30] showed that the increasing of loading angle as the decreasing ofthe energy absorption, in the case of the empty and foam-filled squarecolumns under the quasi-static oblique loading conditions. Mean-while, the energy absorption characteristics of tapered thin-walledrectangular tubes were explored by Nagel and Thambiratnam [31–33].They concluded that the tapered tube better stability than straighttube when the oblique impact happen. Ahmad et al. [34] focused oninvestigate the benefit of foam-filled conical tubes as vehicle energyabsorber under oblique impact. Li et al. [35] studied the deformationand energy absorption of aluminium foam-filled double cylindricaltubes subjected to oblique loading and found that foam-filled doublecircular tube better crashworthiness capability than empty and foam-filled single circular tubes under quasi-static oblique loading.

The dynamic responses and the energy absorption characteristicsof different configurations circular tubes i.e. empty double tube (ED),foam-filled double tube (FD) and foam-filled single tube (FS) wereexplored in the present study. Some parameters for instance geometryparameters (tube length, tube outer diameter, tube wall thickness),material parameter (foam density) and loading parameters (loadangle) are used as parametric studies to completely understand andquantify the deformation mode and the energy absorption behaviourof tubes. The numerical simulations were validated using the experi-ment and the theoretical solutions taken from appropriate literaturesin studying the effect of parameters for axial progressive collapsemode and global bending collapse mode. Finally, this paper isexpected the new design information of foam filled double cylindrical

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journal homepage: www.elsevier.com/locate/ijmecsci

International Journal of Mechanical Sciences

http://dx.doi.org/10.1016/j.ijmecsci.2015.06.0010020-7403/& 2015 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: þ60 3 89118411; fax: þ60 3 89259659.E-mail address: shahrum@eng.ukm.my (S. Abdullah).

International Journal of Mechanical Sciences 99 (2015) 228–236

thin-walled tubes as energy absorber in application under dynamicoblique impact loading.

2. Development of finite element model

Fig. 1 shows the schematics of the circular tubes under differentimpact loadings. In this investigation, the load angle used is 101 and301 which represents the axial progressive collapse mode and globalbending collapse modes, respectively. This load angle similar with theRef. [33,34]. To model energy absorbing structures, the aluminiumfoam filled double tubes were L in length. The top of the tubes wereimpacted in longitudinal direction of tube by an additional mass blockof 400 kg.

The tubes cross-section is shown in Fig. 2, the outer and innerthicknesses of the double tube walls are to and ti respectively, and thediameter of the outer and inner tubes are do and di respectively.

The finite element (FE) models of empty double, foam-filleddouble, and foam-filled single tubes are shown in Fig. 3. TheABAQUS–Explicit was used to develop the aluminium foam-filledtubular tube models and to predict the response of thin-walledtubes that were impacted by a free falling impinging mass.

The walls of the tubular tubes were modelled using four node shellcontinuum elements with five integration points along the element'sthickness direction. Moreover, the foam-filled tube was modelledusing eight node continuum elements with a reduced integrationtechnique combined with the hourglass control. To avoid bothartificial zero energy deformation modes and volumetric locking,enhancement-based hourglass control and reduced integration were

applied. Based on a mesh convergence study of shells and foamelements, a 2 mm element size was chosen. A mesh convergence wasaddressed to ensure a sufficient mesh density and to accuratelycapture the deformation process. The contact interaction between allcomponents was the general contact algorithm used to avoid inter-penetration of tube walls, which is less intense in terms of computa-tional time. Meanwhile, the contacts between the foam and the tubewalls were modelled as a finite sliding penalty based contact algo-rithm, with contact pairs and a hard contact. The friction coefficientvalue for all contact surfaces was set at 0.3 and 0.2 for the static anddynamic cases [34] respectively.

3. Material properties

3.1. Material of thin-walled circular tubes

The thin-walled tubes were made from aluminium alloy A6060T4 with mechanical properties of density ρ¼2700 kg/m3, Young'smodulus E¼68.2 GPa, Poisson's ratio ν¼0.3, initial yielding stressσy¼80 MPa, and ultimate stress σu¼215.5 MPa. The pairs of theplastic strain and true stress were specified in Table 1 to accuratelydefine the hardening characteristic in finite element models.

An elastic–plastic material model, with Von Mises's isotropicplasticity algorithm, was used to assess the constitutive behaviourof the tubes. The direction of aluminium foam, which causedmanufacture process effect, was ignored. Piecewise lines wereperformed to define plastic hardening in the material's constitu-tive model. Moreover, the true stress and the plastic strain wereexperimentally found to determine the piecewise lines.

3.2. Material of foam filled

The aluminium closed-cell foam filler was used, with an averagemechanical property value obtained from material tests. The materi-al's behaviour was obtained from experimental testing of the foamfilled material, while the uniaxial quasi-static compression test resultswith different foam apparent densities are given in Ref. [3].

The constitutive behaviour was based on an isotropic uniformmaterial of the foam model developed by Dehspande and Fleck [36]using non-linear ABAQUS/Explicit software packages. The effect of themanufacturing process for anisotropic behaviour of aluminium foamwas not considered in this work. Table 1 shows the details of thematerial's parameters used in the FE simulation experiments.

The crushable foam and the crushable foam hardening optionswere used to calculate the plastic behaviour of the aluminiumfoam. The equation of yield criterion in this model is described as

∅¼ σ̂�Yr0 ð1Þwhere

σ̂2 ¼ 1

1þ α3

� �2h i σ2e þ α2σ2

m

� � ð2Þ

Mass block

θ = 10° and 30°

V=Vo

L

Rigid wall

Fig. 1. The schematic of bitubal circular tubes under oblique impact.

do do do

di di

ti ti

to to to

foam empty

Fig. 2. Cross-section of the circular tubes (a) empty double tube (ED), (b) foam-filled double tube (FD), and (c) foam-filled single tube (FS).

F. Djamaluddin et al. / International Journal of Mechanical Sciences 99 (2015) 228–236 229

where σe is the effective Von Mises stress and σm denotes themean stress. The yield strength [37] is defined as Y. In addition, αis the parameter used to define the shape of the yield surface(α¼2.12) and the plastic coefficient function νp. It is assumed thatthe plastic Poisson's ratio for aluminium foam is zero [38]; thus,the formula is

α2 ¼ 2ð1�2νpÞ9ð1þνpÞ

ð3Þ

The strain hardening effect equation for the initial model isdefined as

Y ¼ σpþ γε̂εD

þα2 ln1

1� ε̂εD

� �β264

375 ð4Þ

where σp is the plateau stress, the material constants are α2, γ, εDand β, and the effective strain is defined as ε̂. The strain ofdensification is derived as

εD ¼ �9þσ2

3α2 lnρf

ρf o

!ð5Þ

The density of foam and the foam base material are defined asρf and ρfo. [37,38]. Table 2 shows the material parameters of foamfiller used in the dynamic simulation are tabulated.

3.3. Crashworthiness performance indicators

To evaluate the energy-absorbance of structures, it is necessary todefine the crashworthiness indicators. The parameters, such as energyabsorption (EA), specific energy absorption (SEA), and peak crush force(PCF), can efficiently evaluate the crashworthiness of structures.Energy absorption can be calculated as

EAðδÞ ¼Z δ

0F xð Þdδ ð6Þ

where F(x) is the instantaneous crushing force with a function of thedisplacement δ.

SEA indicates the absorbed energy (EA(δ)) per unit mass (M) ofa structure as

SEAðδÞ ¼ EAðδÞMtotal

ð7Þ

where Mtotal is the structure's total mass. In this case, a highervalue indicates the higher energy absorption efficiency of amaterial.

The average crush force (Favg) is the response parameter for theenergy absorption capability:

Favg ¼EAðδÞδ

ð8Þ

where, energy is absorbed (EA(δ)) during collapse and displace-ment (δ).

Crush force efficiency is defined as the ratio of the averagecrush force (Fagv) to the peak crush force (Fmax)

CFE¼ FavgFmax

x 100% ð9Þ

Peak crush force (Fmax) is important indicator in the design ofenergy absorption structures to absorb the impact energy incollision.

4. Model validation

FE model has been validated by experimental and theoreticalmodel in the literatures to ascertain whether it was sufficiently accu-rate. The validation results of foam-filled double circular tubes underquasi-static impact loading are presented by Djamaluddin et al. [39].They carried out the comparison between the numerical simulationand the experiment result and demonstrated the deformation pat-terns [40] and force–displacement curve [41] of numeric simulationand experiment result of foam-filled double cylindrical tube subjectedto quasi-static loading [35].

The experiment test was already performed [35] which both ofends specimen were clamped, at top of tube was applied a rigidwall under quasi-static axial (01) and oblique (51, 101, 151)with respect to the longitudinal direction of tube (velocity ¼0.09 mm/s). Different cross-section of structures namely emptysingle tube, foam-filled single tube and foam-filled double tubehave been evaluated. With different fixing block, different thick-ness and diameter of inner tube of foam-filled tubular tube, thenew deformation modes were found for instance spiral foldingmode, irregular extensional folding mode and irregular axi-symmetric or diamond deformation mode. Some parametersof crashworthiness were investigated such as specific energy

Fig. 3. The finite element (FE) models of cylindrical tubes (a) empty double tube (ED), (b) foam-filled double tube (FD), and (c) foam-filled single tube (FD).

Table 1Strain hardening data for A6060 T4 [3].

Plastic strain (%) 0.0 2.4 4.9 7.4 9.9 12.4 14.9 17.4Plastic stress (Mpa) 80 115 139 150 158 167 171 173

Table 2Material parameters of aluminium foam [37].

ρf (g/cm³) σp (N/mm²) α2 (N/mm²) β γ (N/mm²) εD

0.22 2.14 169 2.942 2.45 2.50740.534 12.56 1544 3.68 1 1.62060.71 22.18 4295 4.718 6.438 1.3357

F. Djamaluddin et al. / International Journal of Mechanical Sciences 99 (2015) 228–236230

absorption, energy absorption efficiency and energy absorptioneffectiveness factor of static loading. It can be concluded that thefoam-filled circular tube has energy absorption effectivenesssignificant higher than empty tubes and the double tubes withaluminum foam core has energy absorption capacity better thanempty and foam-filled single tubes. Besides to validate thesimulation model result, this study also write the form of energyabsorption factor effectiveness of the foam-filled circular tubeunder dynamic load in the next section, refer to the form of energyabsorption factor effectiveness subjected to quasi-static load [35].Formula of energy absorption effectiveness factor can be recom-mended for further study on the experimental test of foam filleddouble tube under axial or oblique dynamic impact loadings.

For this study, thin-walled tube was only circular tube underdynamic impact loading. Table 3 displays the energy absorptionand the average crushing force of the empty circular tubes thatcompare between the FE result and experiment result [42]. It isachieved a good agreement and FE model has capability ofsimulating the numerical response of each tubes under dynamicoblique impact.

The theoretical solution of the cylindrical tube filled by foamwas used to validate the comprehensive of the numeric simula-tion. The equation of the empty cylindrical tube derived todetermine the static mean crushing force Fsavg [2] is

Fsavg ¼ 17:0σyðd�tÞ1=3t5=3 ð10Þ

where σy, d and t are the yield stress of tube, diameter andthickness of tube wall of foam-filled circular tube respectively.

Under static load case, the metal yield stresses in dynamicimpact because of plastic rate. In addition, considering the effect ofplastic strain rate ( _ε) on the yield stress of aluminum tube materialis used the Cowper–Symonds constitutive equation as [43]

σd ¼ σsþσs_εC

� 1=k

ð11Þ

where σd and σs are the yield stress of dynamic and quasi-staticstress, and constants C and k values are material parameters foraluminium material.

The theoretical solution of mean crush force was derived byAbramowicz and Jones [44] as

Fdavg ¼ FsavgþFsavg_εC

� 1=k

ð12Þ

The theoretical model of dynamic mean crushing force foamfilled is determined by considering the interaction between emptytube and foam material as [2]

Fdavgðf Þ ¼ FdavgþσpsAf þCavg

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσpsσyAt

qð13Þ

where σf, Af, At are the plateau stress of foam, the area of foamcore, and the area of tube, respectively. The coefficient of interac-tion between the filled foam core and the wall of empty tube isCavg.

Theoretical of dynamic mean crushing force is mainly interest forvalidation of foam-filled circular tube [2,45]. Table 4 presents the

differences between FE models and theoretical solutions. It can beconcluded that the errors of both results are almost similar and theFEA results are considered satisfactory under different collapse modes.

5. Results and discussion

5.1. Deformation mode and dynamic crushing force–displacement ondynamic response

Figs. 4 and 5 show the deformation modes of the numericaldynamic responses for empty double, foam-filled double andfoam-filled single tubes, under oblique loading for load angles of101 and 301, respectively [33,34]. Table 5 presents the range ofeach parametric study of material, geometry and loading para-meters used to control the tube's response.

For axial progressive collapse mode, all numerical models oftubular structures were impacted by a mass block under a 101 loadangle; whereas, for the global bending collapse mode, all numer-ical models of tubular structures were impacted by a mass blockunder a 301 load angle. Geometric and material parameters, aswell as load angle, were found to influence the deformation mode.Normally, axial progressive collapse mode and global bendingcollapse mode for foam-filled bitubal circular tubes are describedby a numerical simulation with similar collapse formations causedby the interaction between foam and inner/outer walls of thetubes. From Figs. 4 and 5 it can be seen that the predicted lobeformation and the number of lobes for the filled single tubes areless than in other structures. However, empty double circulartubes have buckled flanges, rather than the middle of the tubes forboth collapse modes.

Diagrams of numerical dynamic crushing force–displacementin each thin-walled circular tube are shown in Figs. 6–8, and arecompared by varying the collapse mode and the tube's outerdiameter, with L¼250 mm, to¼1.5 mm, ti¼1.5 mm, di¼48 mm fordouble circular tube, with block mass G¼400 kg and an impactvelocity V¼15 m/s. All thin-walled tubes were crushed whenimpacted in full contact by the mass block. Moreover, due to axialprogressive collapse mode and global bending collapse mode fortwo kinds of tube outer diameters (do¼96 mm and 102 mm), each

Table 3FE simulation and experimental solution of empty circular tubes.

Impactor Geometry parameters Experiment [45] FE Error

Velocity v (m/s) Mass M (kg) Length L (mm) Diameter d (mm) Thickness t (mm) EA (J) Fdavg (kN) EA (J) Fdavg (kN) EA (l%l) Fdavg (l%l)

6.6 104.5 180 40 2 2326 45.6 2278 43.76 2.107 4.26.6 104.5 180 40 2.5 2260 42.3 2176 41.68 3.86 1.49

10.7 91 180 50 3 5081 86 4947 83.45 2.709 3.06

Table 4FE simulation and theoretical solutions of foam filled circular tubes.

Velocity Geometryparameter

Material parameter Dynamic mean

crushing force Fdavg(kN)

v (m/s) Diameter Thickness Foamdensity

Plateaustress

Theory FEA Error

d (mm) t (mm) ρf (kg/cm³) σf (MPa) [27] (l%l)

12.9 80 1.48 0.15 0.6 28.21 27.54 2.4316.9 80 1.46 0.25 2.9 41.37 42.66 3.0317.2 80 1.5 0.26 3.4 47.25 45.96 2.8119.7 80 1.4 0.33 6.6 62.62 61.48 1.85

F. Djamaluddin et al. / International Journal of Mechanical Sciences 99 (2015) 228–236 231

figure illustrates the diagram of dynamic response for differentcross-sections of cylindrical tubes. From the figures, it is evidentthat the dynamic initial peak force of the axial progressive collapsemode was much higher than the global bending collapse mode inthe same geometrical tubes. It can also be seen that for both

Fig. 4. Deformation modes of (a) empty double tube (ED), (b) foam-filled double tube (FD), and (c) foam-filled single tube (FS) for progressive collapse mode.

Fig. 5. Deformation modes of (a) empty double tube (ED), (b) foam-filled double tube (FD), and (c) foam-filled single tube (FS) for bending collapse mode.

Table 5Range of each loading, geometry and material parameters used in oblique loadingparametric study.

Tubegeometry

θ

(deg)L (mm) do

(mm)di(mm)

to (mm) ti(mm)

ρf (g/cm³)

ED 10,30

200, 250,300

96,102

48 1, 1.5, 2,2.5

1.5 –

FD 10,30

200, 250,300

96,102

48 1, 1.5, 2,2.5

1.5 0.22, 0.534,0.71

FS 10,30

200, 250,300

96,102

– 1, 1.5, 2,2.5

– 0.22, 0.534,0.71

0

5

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100 120

dyna

mic

cru

shin

g fo

rce

(kN

)

Displacement (mm)

Progresive (do = 102 mm)Progresive (do = 96 mm)Bending (do = 102 mm)Bending (do = 96 mm)

Fig. 6. Effect of outer diameter on dynamic crushing force of empty double tubeunder progressive and bending collapses.

0

10

20

30

40

50

60

70

0 20 40 60 80 100 120

dyna

mic

cru

shin

g fo

rce

(kN

)

Displacement (mm)

Progressive (do = 102 mm)Progressive (do = 96 mm)Bending (do = 102 mm)Bending (do = 96 mm)

Fig. 7. Effect of outer diameter on dynamic crushing force of foam-filled single tubeunder progressive and bending collapses.

0

10

20

30

40

50

60

70

80

0 20 40 60 80 100 120

dyna

mic

cru

shin

g fo

rce

(kN

)

Displacement (mm)

Progressive(do = 102 mm)Progressive(do = 96 mm)Bending(do = 102 mm)Bending(do = 96 mm)

Fig. 8. Effect of outer diameter on dynamic crushing force of foam-filled doubletube under progressive and bending collapses.

F. Djamaluddin et al. / International Journal of Mechanical Sciences 99 (2015) 228–236232

progressive and bending modes, dynamic crushing force–displa-cement behaviour was similar for different outer wall diameterswith similar circular tube cross-sections. From these results, wecan summarize that foam filled double tube is better than other

structures under oblique impact loading; and therefore recom-mended for use as an energy absorber in vehicle structures.Furthermore, it can be seen from the figures that different collapsemodes did not affect the folding mechanism of structures with twoload angles.

In Figs. 9 and 10, energy absorption is plotted as a function ofdisplacement and tube outer diameter for each collapse mode. Itcan be concluded that empty double circular tube has dynamiccrushing force significant lower than the other structures fordifferent collapse modes. However, foam-filled bitubal tubes arebetter energy absorbers in both collapse mode conditions. Theyshow that a smaller outer empty double tube diameter hassignificantly lower dynamic energy absorption than the other

0

10

20

30

40

50

60

70

80

90

0 40 80 120

Dyn

amic

ene

rgy

abso

rptio

n (k

J)

Displacement (mm)

ED(do = 102 mm)

FS (do = 102 mm)

FD (do = 102 mm)

ED (do = 96 mm)

FS (do = 96 mm)

FD (do = 96 mm)

Fig. 9. Effect of outer wall diameter on energy absorption of structures underprogressive collapse.

0

10

20

30

40

50

60

70

0 40 80 120

Spec

ific

Ener

gy a

bsor

ptio

n (k

J)

Displacement (mm)

ED (do = 102 mm)

FS (do = 102 mm)

FD (do = 102 mm)

ED (do = 96 mm)

FS (do = 96 mm)

FD (do = 96 mm)

Fig. 10. Effect of outer diameter on energy absorption of structures under bendingcollapse.

200

250

300

Leng

th (m

m)

Dynamic energy absorption (kJ)0 10 20 30 40 50 60 70

FD

FS

ED

Fig. 11. Effect of length on energy absorption of empty double tube, foam-filleddouble tube, and foam-filled single tube for progressive collapse.

200

250

300

Leng

th (m

m)

0 10Dynamic energy absorption (kJ)

20 30 40 50

FD

FS

ED

Fig. 12. Effect of length on specific energy absorption of empty double tube, foam-filled double tube, and foam-filled single tube for progressive collapse.

0

200

250

300

Leng

th (m

m)

10 20 30Specific Energy Absorption (kJ/kg)

40 50

FD

FS

ED

Fig. 13. Effect of length on energy absorption of empty double tube, foam-filleddouble tube, and foam-filled single tube for bending collapse.

0

200

250

300

Leng

th (m

m)

10Specific Energy Absorption (kJ/kg)

20 30

FD

FS

ED

40

Fig. 14. Effect of length on specific energy absorption of empty double tube, foam-filled double tube, and foam-filled single tube for bending collapse.

F. Djamaluddin et al. / International Journal of Mechanical Sciences 99 (2015) 228–236 233

structures. For both loading conditions, the figures also show thatfoam-filled double tube has higher energy absorption for an outerdiameter of 102 mm. In addition, for axial progressive collapse,whole buckling occurs after a fold is formed in the axial direction;then, for the bending collapse mode, both energy efficiency anddynamic energy absorption may decrease.

5.2. Effect of tube length on dynamic response

Figs. 11–14 show the effect of tube length on dynamic energyabsorption and specific energy absorption under an obliqueimpact. With two types of collapse mode i.e., progressive andbending, tube length has more significant influence for emptydouble, foam filled double and foam-filled single circular tubes.Clearly, the dynamic energy absorption and specific energyabsorption of foam-filled bitubal tubes with longer tube lengthsis more than that of the other types of cylindrical tubes. It isevident that this foam filled double tube configuration can be anefficient energy absorber and has more room to enhance crash-worthiness. In accordance with previous studies [23,25], thenumber of separate components of the empty structure and thefoam-filler is lower than the foam filled double structure.

The compressibility of the foam-filled double tube is signifi-cantly greater than that of the foam-filled single ones; with theinner tube taking up some of the room of the foam core.Furthermore, the specific energy absorption of the foam-filleddouble circular tube improved and was higher than that of theother structures. The numerical simulation results demonstratethat the total masses of the foam-filled double and single circulartubes were relatively similar, and the energy absorption efficiencyof the foam-filled double tube was higher than that of the singleones. Again, the foam-filled double tube was more efficient thanthe empty double and foam-filled single circular tubes under anoblique impact.

Dynamic energy absorbing effectiveness factor can be derivedaccording to literatures [46,35] as

ψ 0 ¼ 3GV2o

8δf εr σ0eAeþσ0iAiþσf Af� � ð14Þ

where G is the potential energy of the block mass, Vo is the initialimpact velocity, εr is the uniaxial tensile engineering rupturestrain, and δf is the actual crushing displacement. Meanwhile,σ0e; σ0i; and σf are the static flow stresses of the materials used forthe external and internal tubes, and the plateau crushing stress ofthe foam, respectively. Meanwhile, Aoe; Aoi; and Af are the cross-sectional areas of the external and internal tubes and the cross-sectional area of the foam infilling, respectively. It can be assumed

that the dynamic energy absorbing effectiveness factor of emptydouble tube is lower than the foam-filled single and double tubes.

5.3. Effect of outer wall thickness on dynamic response

In this section, the influence of outer wall thickness and loadangle is evaluated to determine dynamic energy absorption andcrushing force efficiency. Dynamic energy absorption was calcu-lated for each tube up to a displacement of 100 mm to comparethe relative effect of outer wall thickness and load angle for emptydouble, foam-filled double and foam-filled single circular tubes.Axial compression and global bending results were chosen as anadequate comparison. The baseline dimensions of circular tubeswere adopted for four values of outer wall thicknesses i.e., 1, 1.5, 2,and 2.5 mm to determine the effect of wall thickness under anoblique loading. Figs. 15 and 16 show the effect of collapse modeand outer wall thickness on the dynamic energy absorption andcrushing force efficiency of tubes under an oblique impact loading.It can be seen in the figures that the dynamic energy absorptionand crushing force of all tubes increased continuously as the outerwall thickness increased for each collapse mode. Furthermore, fordynamic energy absorption and crushing force efficiency, the axialprogressive collapse mode was higher than the global bending

0

5

10

15

20

25

30

35

40

45

50

2.521.51

Dyn

amic

Ene

rgy

Abs

orpt

ion

(kJ)

Outer Wall Thickness (mm)

ED (Progressive)FS (Progressive)FD (Progressive)ED (Bending)FS (Bending)FD (Bending)

Fig. 15. Effect of outer wall thickness on dynamic energy absorption of emptydouble tube, foam-filled double tube, and foam-filled single tube.

0

0.2

0.4

0.6

0.8

1

1.2

1 1.5 2 2.5

Cru

shin

g Fo

rce

Effic

ienc

y

Outer Wall Thickness (mm)

ED (Progressive)FS (Progressive)FD (Progressive)ED (Bending)FS (Bending)FD (Bending)

Fig. 16. Effect of outer wall thickness on crush force efficiency of empty doubletube, foam-filled double tube, and foam-filled single tube.

20

30

40

50

60

0 0.2 0.4 0.6 0.8

Spec

ific

Ener

gy A

bsop

tion

(kJ/

kg)

Foam density (g/cm3)

ED (Progressive)FS (Progressive)FD (Progressive)ED (Bending)FS (Bending)FD (Bending)

Fig. 17. Effect of foam density on specific energy absorption of empty double tube,foam-filled double tube and foam-filled single tube under progressive and bendingcollapse modes.

F. Djamaluddin et al. / International Journal of Mechanical Sciences 99 (2015) 228–236234

collapse mode, for each circular tube. It is evident that load anglehad the most influence on the dynamic energy absorption andcrushing force efficiency of each structure. Meanwhile, foam-filleddouble circular tube had a better crashworthiness capability thanthe other structures for dynamic energy absorption and crushingforce efficiency indicators.

5.4. Effect of foam density on dynamic response

Finally, we study the effect of foam density and collapse modeon specific energy absorption and dynamic mean crushing forcefor foam-filled single and double circular tubes with ρf¼0.22 g/cm3, 0.534 g/cm3 and 0.71 g/cm3. Figs. 17 and 18 show an increasein specific energy absorption and dynamic mean crushing force forfoam-filled single and double circular tubes for foam density.Meanwhile, the foam-filled double circular tube result was higherthan single circular tube or dynamic tube. In conclusion, a foam-filled double structure with a high density foam is able to be fullyconsidered as an energy absorber. Lastly, load angle has nosignificant effect for cylindrical tubes with different aluminiumfoam-filled densities.

6. Conclusions

A numerical simulation was performed to investigate the effect ofgeometry (outer wall diameter, length, outer wall thickness), material(foam density), and loading (axial progressive collapse mode andglobal bending mode) parameters for circular tubes. Three differentconfigurations of circular tubes, namely empty–empty tube, emptyfoam filled tube, and foam filled double cylindrical tubes underoblique loadings was observed. From this study, the main conclusionand design information are summarised: First, dynamic energyabsorption and specific energy absorption are more affected by outerwall diameter and load angle, and the dynamic response of structureincrease for as outer wall diameter of tubes increase in progressiveand bending collapses. Second, the tube length has more significantinfluence for circular tube on dynamic energy absorption and specificenergy absorption. The foam-filled bitubal tube with longer tube ismore than the other type of tubes. Third, the dynamic energyabsorption and crushing force efficiency of all tubes continuouslyincreases for as the outer wall thickness increase for each collapsemode. Fourth, the dynamic energy absorbing effectiveness factor ofempty double tube is lower than the foam-filled single and dou-ble tubes.

Lastly, the density of foam slightly influences on specific energyabsorption and dynamic mean crushing force for structures. Withsimilar foam density, the foam-filled double circular tube higher thansingle circular tube on dynamic tube due to interaction between foam,inner and outer wall of structure. Finally, the foam-filled doublecircular tube as effective energy absorbing devices since it can bewithstand as effectively as axial progressive and bending modes forcrashworthy vehicle structure.

Nomenclature

Af foam core areaAt tube areaC material parameters of aluminiumCavg coefficient of the interaction between filled foam core

and the walldo, di outer and inner diameter of double tubesE Young's modulusFavg ; F

savg ; F

davg average crush force, static average mean force,

dynamic average mean forceF(x) instantaneous crushing forceG potential energy of block massk material parameters of aluminiumL tube lengthMtotal tube total masst, to, ti tube thickness, tube outer and inner thicknessesU strain hardening effectV velocity of mass blockVo initial impact velocityY yield strengthσd, σs yield stress of dynamic and quasi-static stressesσu,σy ultimate and initial yielding stress of tubeσm, σe mean stress and effective Von Mises stressσp plateau stressσ0e; σ0i static flow stresses of materials for the external and

internal tubesα shape of the yield surfaceν Poisson's ratioνp plastic coefficient functionθ load angle∅ yield criterionρ densityδ displacementδf actual crushing displacementρf, ρfo foam density and foam base material_ε plastic strain rateε̂ effective strainεr uniaxial tensile engineering rupture strainψ 0 dynamic energy absorbing effectiveness factorα2, γ, εD, β material constants

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20

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

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amic

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