en

n

w thso

ersucal m fruisplael. Srialsical

del

Thin-walled structura l shell elements suconical shells, and domes are commonly uelements in crashworthi ness applications .behavior has received considerable attentin the last four decades. Experimental , antional studies on these structural elements

c loadi of col

], multAmongta com

comparison with available experimental results was made. In 2000, for the rst time, inversion process of the frusta was

introduced and tested as a novel crushing mode of frusta by Aljawi and Alghamdi [9]. The deformation modes of capped-end frustum were investigated, experimental ly and numerically . Then, Algham- di et al. [10] performed experime ntal and numerica l studies to

during axial crushing between two parallel plates. The experimen-

simulatio n was performed under quasi-static loading using ANSYS software and compared with the experimental measure ments that showed a good correlation. Hosseini et al. [15] modeled the axisymmetri c axial crushing of thin metallic frusta by straight fold model with partly inside and partly outside folding. Change in the thickness of frusta during the formation of fold was incorporate d to explain the reasons of the inside portion of fold being smaller than the outside. The results were compared with experiments that showed a good agreement.

Corresponding author. Tel.: +98 741 2229889; fax: +98 741 2221711.

Materials and Design 49 (2013) 6575

Contents lists available at

an

elsE-mail address: Aniknejad@yu.ac.ir (A. Niknejad).are employed over a broad range of applicati ons, especially in the applications of aerospace and armaments as the conical noses of missiles and aircrafts [1]. Mamalis et al. [8] studied crumplin g of thin-walled frusta under axial compression in the concertina mode, theoretical ly. They develope d a theoretical model to predict the axial load during the folding process on an empty frusta. Also, they performed some experiments on the thin-walled frusta and investigated effects of slenderness , t/D, and semi-apical angle of the frusta. Thinning of the cross-section due to stretchin g was ne- glected. Good qualitative agreement in trends was exhibited when

treatment on the collapse behavior and energy absorption was investiga ted. Gupta and Abbas [13] introduced a theoretical model of plastic deformation during the folding process on the frusta sub- jected to quasi-static axial loading. Then, Gupta et al. [14] carriedout some experiments on conical aluminum frusta with different thicknesses and semi-apical angles. The specimens were axially compress ed and the loaddeformation curves and deformed shapes of specimens were recorded. The experiments show that frustum is deformed in axisymmetri c concertina mode and non- symmetr ic diamond modes. A three dimensional numerica lunder both quasi-static and dynamidirections [1]. The geometrical shapeers can be circular [2], square [3,4honeycomb [6,7] and frusta [1].absorbing elements, thin-walled frus0261-3069/$ - see front matter 2013 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.matdes.2013.01.014ch as cylindrical shells, sed as energy absorber Study of their collapse ion of the researchers alytica l and computa- have been carried out ngs in axial and lateral lapsib le energy absorb- i corner [5], multi-cell the different energy monly known as frusta

tal results showed that there are ve different deformation modes: outward inversion, limited inward inversion followed by outward inversion, full inward inversion followed by outward inversion, limited extensible crumpling followed by outward inversion, and full extensible crumpling. Also, they investigated effects of impact loading on inversion behavior of aluminum frusta [11]. El-Sobky et al. [12] experimentally studied energy absorption performance of right circular frusta subjected to dynamic axial load and com- pared with results of quasi-static tests. Frusta of different geomet- rical ratios and end constraints were axially crushed using a drop hammer at initial velocities in the range of 25 m/s. Effect of heat 1. Introduction classify deformation modes of unconstrai ned capped-end frusta Technical Report

Axial compression of the empty capped-

Abbas Niknejad , Ahmad Tavassolimanesh Mechanical Engineering Department, Yasouj University, P.O. Box 75914-353, Yasouj, Ira

a r t i c l e i n f o

Article history: Received 17 November 2012 Accepted 7 January 2013 Available online 20 January 2013

a b s t r a c t

This paper introduce s a necapped-end frusta using ainstantaneous axial load vis predicted. In the analytiplastic deformation of theenergy versus the axial dusing the theoretical modtubes with different matement between the theoretof the new theoretical mo

Materia ls

journal homepage: www.ll rights reserved. d frusta during the inversion progress

eoretical model of plastic deformation during the inversion process on the lid cylindrical punch. Based on the theoretical deformation model, the s the displacement during the inversion process of the capped-end frusta

odel, the inversion process is divided into two different stages including sta bottom and inversion of the frusta wall. Also, instantaneous absorbed cement and maximum inversion load during the process are estimated

ome axial inversion experiments were carried out on capped-end conical and geometrical dimensions to verify the present theory. A good agree- predictions and the experimen tal results afrms precision and accuracy and the analytical formulas.

2013 Elsevier Ltd. All rights reserved.

SciVerse ScienceDi rect

d Design

evier .com/locate /matdes

Ahmad and Thambiratnam [16] considered the foam-lled thin- walled frusta as a desirable energy absorber under axial loading due to their higher energy absorption, comparing with empty tubes. They investigated axial crushing and energy absorption capacity of foam-lled conical tubes under quasi-static axial load- ing, using non-linear nite element models. Results indicated that the crush and energy absorption performanc es of conical tubes are signicantly enhanced by foam lling. Then, they studied mechan- ical behavior of the foam-lled frusta under the impact loading and the research showed that the foam-lled frustum is useful in the impact applications [17]. Ahmad et al. [18] investigated crush behavior and energy absorption response of foam-lled conical tubes subjected to oblique impact loading by experime ntal and

the new theoretical model, the inversion process is divided into

er 1 d cos h 1 4

By assuming the frusta as a rigid-perfectl y plastic material, the absorbed energy due to the radial expansion is calculated as:

EExp r0 er V p4 r0dt1d d0 1

cos h 1

; 0 6 h 6 hmax

5In the above equation, V is volume of the bottom end plate. There- fore, summati on of the absorbed energies by two plastic bending modes and the radial expansio n mode in the circular plate of the frusta bottom end results in the following relation to estimate the total absorbed energy during the rst stage of the inversion process:

E1 p4 r0t21d d0h

p4r0dt1d d0 1cos h 1

; 0 6 h 6 hmax

6In the above relation, angle of h varies versus the axial displace ment during the rst stage of inversion process as the following:

66 A. Niknejad, A. Tavassolimanesh / Matetwo different stages:

Stage 1: plastic deformat ion of circular plate of bottom end. Stage 2: plastic inversion deformation of the frusta wall.

2.1. Stage 1: Bottom plate deformation

When the axial load is applied on the rigid upper punch, edges of circular plate of the bottom part of frusta bend downward and during the bottom bulging, radial expansion occurs in the plate, according to Fig. 2. The end point of the rst stage is indicated by hmax. During the rst stage, two bending processes happen numerical methods. Niknejad and Moeinifard [19] introduce d anew theoretical model of deformation during the inversion process of circular tubes and predicted the axial load of inversion process of the circular tubes, analytically.

Reviewing of the previous published works indicates absence of a theoretical analysis to predict mechanical behavior of the frusta during the inversion process. This article introduce s a new theoret- ical model of deformation and derives some formulas to predict the instantaneo us inversion force, maximum inversion force and instantaneo us absorbed energy by the metal capped-end frusta subjected to the quasi-static axial load using a solid cylindrical punch.

2. Theory

A frusta is a truncated circular cone and in this article, quasi- static axial compression load is performed on a capped-end frusta. The inversion process occurs during the loading. Fig. 1 shows geo- metrical dimensions of a capped-end frusta before deformation. In the gure, t1 and t2 are thickness of the bottom plate and wall thickness of the frusta, respectively . Also, L is initial height of frusta and a indicates semi-apical angle of the frustum. According to the mechanical behavior and plastic deformation of the metal frusta, in Fig. 1. Geometry of a capped-end frusta. around perimeters of two circles with diameters of d0 and d. The absorbed energy due to the rst plastic bending around the circle perimeter with diameter of d0 is calculated as:

EB M0pd0h; 0 6 h 6 hmax 1where d0 is diameter of the upper cylindrical punch and M0 is fully plastic bending moment per unit of bending length and is equal to M0 = r0t2/4 where t indicate s thickne ss and r0 is ow stress of frus- ta material and is estimated as following [20]:

r0 ryru1 n

r2

where ry, ru, and n are yield stress, ultimate stress, and work hard- ening exponen t of frusta mater ial, respectively. Also, the absorbed energy by the plastic bending around the circle perim eter with diamete r of d is equal to:

EC M0pdh; 0 6 h 6 hmax 3where d indicate s the initial average diameter of the bottom end plate. During the rst stage, the circular plate of the bottom end is expanded in the radial direction and therefore, absorbs the en- ergy. Strain component of the circular plate during the radial expan- sion is calculated as following relatio n:

d0

1

Fig. 2. A capped-end frusta during the rst stage of the theoretical deformation model.

rials and Design 49 (2013) 6575h tan1 2Dd d0

7

Therefore, the instantaneous axial load during the rst stage of the inversion process on a capped-end frusta is derived as:

P1 pr0t21h

4Dd d0 pr0dt14D d d0

1cos h

1

8

At commence of the process, D is zero and the above relation is equal to:

Pini pr0t21d d0

2d d0 9

The above relation estimates initial load at start point of the plastic deformation . Therefore, theoretical diagram of the axial load versus the displacement crosses vertical axis of the coordina te system at Pini.

2.2. Stage 2: Inversion of the frusta wall

Fig. 3 shows schematic of the inversion process on a frusta in According to Fig. 3, bending of the second stage happens around the circle perimeter with diameter of D. Based on the geometrical

Fig. 4. Schematic of regular polygon and radius of bending.

A. Niknejad, A. Tavassolimanesh / Materials and Design 49 (2013) 6575 67the second stage of the present theoretical deformat ion model. As shown in Fig. 3, during the second stage, a bending occurs in the point G with the curvature radius of R. Also, due to creation of the curvature radius of R, the circumferen tial contraction ap- pears in the frusta wall. According to geometri cal dimension and as shown in Fig. 3, the following relation is obtained:

l cosa l0 cosa D 10D indicates total displace ment during the inversion process on the metal frusta and l and l0 are shown in Fig. 3. Also, by neglecting the radial expansio n in the frusta bottom end during the rst stage comparing with the length of l and l0, the following relation is ob- tained between l and l0 during the second stage:

l Rp 2a l0 d d02

11

In the above equation, R is radius of the arc CG as shown in Fig. 3.Accordin g to Fig. 4, by consid ering the arc CG as a part of a regular polygon, R is obtained as follows, approx imately:

R t2 sin a2

12

By calculatin g the l0 from Eq. (11) and substituting in Eq. (10),the following relation is obtained:

l D2 cos a

R p2 a

d d0

413Fig. 3. A capped-end frusta during the second stage of the theoretical deformation model. relation, D is obtained as following:

D d D tana Rp 2a sina d d02

sina 14

Dissipated energy due to the plastic bending during the second stage of the new theoretical model of deformation is calculated as following by considering EG = r0eV:

EG pr0t22

4RZ l

02y sina ddy

pr0t22

4R D

2 cos a R p

2 a

d d0

4

D2 cos a

R p2 a

d d0

4

sina d

15

where y is shown in Fig. 5 and value of l was substitu ted from Eq. (13). Fig. 5 shows a part of frusta during the inversion process. In the gure, point I shows initial position of point H before the inver- sion. Therefor e, initial length of IJ = R(p 2a). Also, based on the geome trical relations , the following relation is obtained to calculate reduction of diameter of frusta cross-sectio n during the inversion process:

d00 d0 4R cosa 2Rp 2a sina 16Fig. 5. Circumferential contraction in the frusta wall.

ateIt is obvious that always R(p 2a)sina 6 2Rcosa. Therefore, in Fig. 5, always point H is in the left side of point I, and the circum- ferential contraction happens in the frusta wall during the inver- sion. The circumferen tial strain due to the contraction is calculated as:

eh d00 d0

D 4R cosa 2Rp 2a sina

D17

Therefore, consideri ng the assumption of rigid-perfectl y plastic material, the absorbed energy due to the contraction is derived as following:

Econt Z l

0r0ehpDt2 dy

pr0Rt22

2 p 2a tana 2D 2Rp 2a cosa d d0 cosa 18

Summation of the absorbed energy during the second stage is obtained as:

E2 pr0Rt22 2 p 2a: tana

2D 2Rp 2a cosa d d0 cosa pr0t22

4R

D2 cos a

R p2 a

d d0

4

D2 cos a

R p2 a

d d0

4

sina d

19

Instantaneous axial load is obtained by dividing summation of the absorbed energies during the different stages to displacement. Therefore, the following relation is derived to predict the axial load during the second stage of the inversion process on a capped-end frusta:

P2 pr0Rt22D 2 p 2a tana 2D 2Rp 2a cosa

d d0 cosa pr0t22

4RD

D2 cos a

R p2 a

d d0

4

D2 cos a

R p2 a

d d0

4

sina d

20

Totally, the instantaneo us axial force versus the axial displace- ment during the inversion process on a capped-end frusta is theo- retically predicted by Eqs. (8) and (20). Also, theoretical diagram of the instantaneo us absorbed energy versus the displacemen t is ob- tained by Eqs. (6) and (19). At commence of the inversion process on a capped-end frusta, the axial load intensive ly increases from zero to a maximum value and then, the axial load decreases, smoothly. Maximum axial load occurs at the beginning of the inversion creation in the frusta wall. On the other hand, during the inversion process, when the axial load that is applied on the upper punch reaches to the required load for inversion formation in the frusta wall, the maximum point in the loaddisplacement diagram appears. Therefore, by equating Eqs. (8) and (20) that pre- dict the axial force during the rst and second stages of the inver- sion process, a certain value is obtained for the axial displacemen t, D12 and by substituting the value of D12 in one of the Eqs. (8) and (20), the maximum inversion load is estimate d, theoreticall y.

68 A. Niknejad, A. Tavassolimanesh / MEquating the axial load of the rst and second stage of the theoret- ical deformation model results in the following relation to calcu- late the D12:Rt21hd d0 Rdt1d d0 1

cos h 1

t22 A2 sina

Ad 2R2t2 2 p 2a tana 2D f2Rp 2a d

d0g cosa 21where A is:

A D2 cos a

R p2 a

d d0

422

In Eq. (21), value of angle h is substituted from Eq. (7).

3. Experimen ts

The loaddeformation curves of energy absorber devices are the key in measuring their crashwor thiness performance. To verify predictio ns of the theoretical analysis, some inversion tests were carried out on the circular frusta with different dimensio ns and dif- ferent punches. All the tests were performed by a DMG machine, model 7166.

Specimen s were made of different materials: stainless steel and zinc alloy. To obtain the material propertie s, a dumbbell shape specimen of each frusta material was prepared and used in a static tension test accordin g to standard ASTM: E8M. Material propertie sof the specimens are given in Table 1. Table 2 gives the geometri cal dimensio ns of the specimens. All the capped-end conical tubes were axially compress ed between a constant rigid plate and a cylin- drical solid punch with constant velocity of 5 mm/min. Diagrams of instantaneo us axial load and absorbed energy versus the axial dis- placemen t of each specimen was sketched by the machine recorder.

4. Results and discussion

Four geometri cal groups of capped-end frustums were prepared and used in the axial compress ion tests with different diameters, wall thicknesses and die radiuses to verify the new theoretical deformat ion model of the inversion process and the suggested ana- lytical formulas. Three similar specimens of each dimensio n were prepared and tested to conrm repeatability of the experiments .The new deformat ion model and the derived theoretical formula determine the main operative parameters of the internal inversion process on the frusta. Theoretical Eqs. (8) and (20) estimate the loaddisplacement curve of the rst and second stage, respectively .Therefore, Eqs. (8) and (20) predict the instantaneo us axial load versus the axial displacemen t during the inversion process on the frusta. According to the theoretical analysis, the instantaneo us axial load during the inversion process of the capped-end frusta depends on material type, bottom end thickness, wall thickness, semi-apical angle, and diameter of frusta and also, punch diameter.

Figs. 6 and 7 illustrate experime ntal diagrams of axial load ver- sus axial displacemen t of steel specimens FSQ-01 and FSQ-03, respectivel y. Also, the gures compare experimental loaddis-placemen t curves with the corresponding theoretical predictions. In some parts, comparison of the theoretical predictions with the correspond ing experime ntal results shows a difference, but, the gures show that the theoretical analysis can predict the instanta- neous inversion force with a fair agreement, comparing with the correspond ing experiments . A reasonable correlation between the theoretical predictions and the experime ntal results shows that the general form of Eqs. (8) and (20) is correct and in agreement with the real created deformat ion in the frusta during the inversion process.

rials and Design 49 (2013) 6575Also, Figs. 8 and 9 compare the experimental and theoretical curves of the axial loaddisplacement of the zinc alloy specimens FZQ-01 and FZQ-02, respectively. The gures show that the

suggested theoretical relations can predict the inversion loaddis-placement diagram of the zinc alloy specimens as well as the stain- less steel specimens. In the theoretical model of deformation , some different mechanisms of energy absorber are considered such as bending of the bottom end, bulging or expansion of the bottom end plate and bending and contraction in the frusta wall. A reason- able correlation between the theoretical results and experiments afrms that the mentioned deformation modes of the introduced

model are the main energy absorber mechanisms during the inver- sion process on a frusta.

Figs. 10 and 11 compare experime ntal and theoretical diagrams of the absorbed energy versus the axial displacemen t during the inversion process on the steel specimens FSQ-01 and FSQ-03, respectively . The absorbed energy is measured by calculating the area under the loaddisplacement diagram. Theoretical Eqs. (6)and (19) estimate the diagram of absorbed energy by the frusta dur- ing the inversion process versus the axial displacemen t. Also, Figs.12 and 13 illustrate comparison of theoretical and corresponding experime ntal curves of the absorbed energyaxial displacemen tof the zinc alloy specimens FZQ-03 and FZQ-04, respectively .Comparis on of the analytical and experimental results shows a very good correlation. In design of an energy absorber system, the main paramete r is energy absorption capability by the structure and the introduce d theoretical analysis of this article can predict the instan- taneous absorbed energy by the inverted frustums versus the axial

Table 2Geometrical dimens ions of the specimens.

Specimens code d (mm) t1 (mm) t2 (mm) a (rad.) d0 (mm) L (mm) m (g) d/d0

FSQ-01 50 0.56 0.49 0.1365 36 110 63.53 1.39 FSQ-02 34 0.40 0.40 0.1570 21 70 29.37 1.62 FSQ-03 34 0.45 0.40 0.1570 25 70 29.54 1.36 FZQ-01 52 1.3 1.25 0.1653 30 92 52.28 1.73 FZQ-02 52 1.3 1.25 0.1653 35 92 53.12 1.49 FZQ-03 61.5 1.0 0.85 0.1412 39 102 44.63 1.58 FZQ-04 61.5 1.0 0.85 0.1412 43 102 44.25 1.43

ial f

Table 1Material properties of the specimens.

Material type Yield stress (MPa)

Ultimate stress (MPa)

Strain hardening exponent

Flow stress (MPa)

Stainless steel 782.63 860 0.593 650 Zinc alloy 58.6 80 0.55 55

A. Niknejad, A. Tavassolimanesh / Materials and Design 49 (2013) 6575 69Fig. 6. Experimental and theoretical diagrams of the axFig. 7. Experimental and theoretical diagrams of the axial force versus the displacement of the specimen FSQ-01. orce versus the displacement of the specimen FSQ-03.

ate70 A. Niknejad, A. Tavassolimanesh / Mdisplacemen t with a negligible error. The good agreement afrmsprecision and accuracy of the introduce d theory.

In the theoretical analysis, cross point of axial loads of two stages results in the maximum inversion load and this quantity is important in view of structures design. On the other hand, when the axial load that is applied on a capped-end frusta reaches to the maximum inversion force, the inversion process appears in the specimen. Table 3 gives the theoretical and experimental values of the maximum inversion force of the different specimens. The ta- ble shows that error percentages of the maximum inversion load predictions by the introduced theory for the all specimens except

Fig. 8. Experimental and theoretical diagrams of the axial f

Fig. 9. Experimental and theoretical diagrams of the axial f

Fig. 10. Experimental and theoretical diagrams of the absorbedrials and Design 49 (2013) 6575FSQ-02 are less than 11% and it is another proof of the theoretical analysis verity.

Also, Eq. (9) estimate s initial load of start point of the plastic deformat ion in a frusta during the inversion process. A reasonable correlation in Figs. 6 and 7 and a very good agreement in Figs. 8and 9 between the rst point of the theoretical diagram and the start point of the plastic zone in the experimental curves afrmthe theoretical analysis.

Fig. 14 shows a section of specimen FZQ-02 after the inversion test. The gure shows that the angle between the inverted part of the frusta wall and the vertical direction is equal to the semi-apical

orce versus the displacement of the specimen FZQ-01.

orce versus the displacement of the specimen FZQ-02.

energy versus the displacement of the specimen FSQ-01.

ateA. Niknejad, A. Tavassolimanesh / Mangle of the frusta. In the theoretical analysis, the same assumpti on was considered as shown in Fig. 3.

When the axial displacement is equal to the initial height of the frusta, the punch reaches to the lowest cross-section of the speci- men. Fig. 15 shows the specic absorbed energy (absorbed energy per unit of frusta mass) by the different specimens. Total absorbed energy by the steel specimen FSQ-03 with the mass of 29.54 g is equal to 277.92 J and therefore, the specic absorbed energy by the specimen from beginning up to D = L is equal to 9408 J/kg. Also, mass and total absorbed energy of the zinc alloy specimen FZQ-03 are 44.63 g and 144.3 J, respectivel y and so, the specic absorbed

Fig. 11. Experimental and theoretical diagrams of the absorbed

Fig. 12. Experimental and theoretical diagrams of the absorbed

Fig. 13. Experimental and theoretical diagrams of the absorbedrials and Design 49 (2013) 6575 71energy by the specimen is equal to 3233 J. Also, the specic ab- sorbed energies by the specimens FSQ-01 and FZQ-04 from begin- ning up to D = L are 11,066 and 3772 J/kg, respectively .Experime ntal results show that in the studied cases, the specicabsorbed energy by the steel frusta is higher than the correspond- ing value by the zinc alloy specimens during the inversion process.

Singace et al. [21] studied energy absorption performance of the folding process on circular frusta subjected to dynamic axial load and compared with the results of quasi-static tests. Frusta of differ- ent geometrical ratios and end constraints were axially folded using a drop hammer at initial velocities in the range of 25 m/s.

energy versus the displacement of the specimen FSQ-03.

energy versus the displacement of the specimen FZQ-03.

energy versus the displacement of the specimen FZQ-04.

ateTable 3Theoretical and experimental values of the maximum inversion force of the capped- end frustums.

Specimens code Maximum inversion force (N) Error percentage (%)

Experimental Theoretical

FSQ-01 10019.67 10935.16 9.13 FSQ-02 3349.75 4582.18 36.79 FSQ-03 6001.45 6165.53 10.23 FZQ-01 4149.43 3967.15 4.39 FZQ-02 4241.35 3952.58 6.81 FZQ-03 2309.62 2438.13 5.56 FZQ-04 2620.89 2760.17 5.31

72 A. Niknejad, A. Tavassolimanesh / MDue to inertia effects, the absolute values of the absorbed energy by similar folded frusta were higher under dynamic loads than un- der quasi-static loads. Their experimental measurements showed that [21] the specic absorbed energy by the folding process on acapped-end aluminum frusta with the semi-apical angle 15 , top diameter of 49.3 mm, initial height of 188.1 mm and wall thickness of 0.4 mm is equal to 2400 J/kg and also, the specic absorbed en- ergy by a capped-end aluminum frusta with the semi-apical angle, top diameter, initial height and wall thickness of 30 , 51.4 mm, 85.6 mm, and 0.65 mm, respectively is equal to 3660 J/kg. Compar- ison of the absorbed energy per unit of frusta mass during the inversion process with the corresponding value during the folding process shows that the specic absorbed energy by an inverted frusta is considerable, comparing with the folding process.

Also, axial loaddisplacement diagrams of the inversion process on the frusta in Figs. 69 show that the axial load increases from zero to a maximum value and then, smoothly decrease s. After-

Fig. 14. A section of the specimen FZQ-02 after the inversion test.

Fig. 15. Specic absorbed energy by the different specimens. wards, during a wide zone of axial displacement, the axial load re- mains constant, approximately. Therefore, the area under the loaddisplacemen t diagram that is equal to the energy absorption by the structure has a considerable value and by increasing the axial dis- placemen t and continuing the process, the specimen dissipates anoticeab le energy. It is because, during the inversion process on frusta, all particles of the frusta material are deformed in the plas- tic zone and therefore, all particles of the specimen participates in the energy absorbing mechanism and thus, a high capacity of the energy absorption by the structure is used. Therefore, inversion process of the frusta is a suitable deformation mode in view point of energy absorber s design.

Fig. 16 compares the axial loaddisplacement diagrams of the specimens FZQ-01 and FZQ-02. The mentioned specimens are the same geometrical characteristics and the same material but, spec- imens FZQ-01 and FZQ-02 were tested by different punches with diameter of 30 and 35 mm, respectively . The gure shows that when the punch diameter increases, the instantaneo us axial load of the frusta during the inversion process increases, too. For better conclusio n, Fig. 17 illustrates the loaddisplacement diagram of the specimens FZQ-03 and FZQ-04. Material and geometrical char- acteristics of the both specimens are the same but, they were in- verted by the different punches with diameter of 39 and 43 mm, respectivel y. Fig. 17 shows the same result of Fig. 16 . Comparison of the area under the both curves in Figs. 16 and 17 and also, con- sidering the clustered column diagram in Fig. 15 shows that by increasing the punch diameter during the inversion process on the frusta, absorbed energy and also, specic absorbed energy by the structure increase. However , during the inversion process punch diameter cannot be selected larger than diameter of the frusta bottom and for the punches with the larger diameter defor- mation mode of the frusta transforms from the inversion to the folding. Fig. 18 illustrates a theoretical schematic diagram of loaddisplacement of two same capped-end frustums during the inversion process by different punches with diameters of 30 and 35 mm. The gure shows that the theoretical relations of the pres- ent article results in the same result of the experiments .

Experime ntal results show that between the tested specimens that are given in Table 2, just specimen FSQ-02 was torn during the inversion process. Ratio of frusta bottom diameter to punch diameter is given in Table 2 and it is considered bottom diame- ter/punch diameter ratio of the specimens FSQ-02 and FZQ-01 are higher than the others and are equal to 1.62 and 1.733, respec- tively and their wall thicknesses are 0.4 and 1.25 mm, respectively .Fig. 19 illustrates the specimens FSQ-02 and FZQ-01 after the inversion test. The gure shows that specimen FSQ-02 with the high bottom diameter/punc h diameter ratio and low thickness has the irregular and non-axisym metric deformation mode and it was torn during the inversion process, but the specimen FZQ-01 with the high bottom diameter/punc h diameter ration and large thickness was not torn. Therefore, the experime ntal investigatio ns show that the capped-end frustums with the high ratios of bottom diameter/w all thickness and bottom diameter/p unch diameter are torn during the inversion deformat ion, probably.

One of the most important purposes of using the thin-walled sections is to increase the energy absorption capability of the structure s [22]. Nowadays, many researchers all over the world study mechanical behavior of the thin-walled structures with the different geometri es and materials during the different loading condition s [2326]. For example, Yin et al. [23] established meta- models of minimum peak crushing force and specic absorbed en- ergy (SAE) by honeycomb -lled single and bitubular polygona ltubes with enneagonal conguration. Based on these meta-mo dels,

rials and Design 49 (2013) 6575the honeycomb -lled single and bitubular enneagonal tubes were optimized to achieve maximum SEA and minimum peak crushing load. In the optimizati on process, multiobject ive particle optimiza-

ateA. Niknejad, A. Tavassolimanesh / Mtion (MOPSO) algorithm was employed. The circumcircle diameter and wall thickness of the tubes were chosen as the design param- eters. They found that a honeycomb-lled single or bitubular enne- agonal tube has the maximum SEA, when its peak crushing load is constrained under a certain value [23]. Then, Hu et al. [24] exam-ined the effects of the cell shape (or the cellwall angle) on the in-plane crushing behaviors of hexagonal honeycombs by employ- ing both experime ntal and numerica l methods. The crushing strength of honeycombs in both the x and y directions was dis- cussed in terms of the impact velocity, the cellwall angles and the relative density of honeycombs; and it was comprehens ively

Fig. 16. Comparison of the axial loaddisplacement

Fig. 17. Comparison of the axial loaddisplacement

Fig. 18. Schematic theoretical comparison of the loaddisplacement diagrials and Design 49 (2013) 6575 73evaluated by dening a crushing strength ratio and the average crushing strength.

Niknejad et al. [22] investiga ted the specic absorbed energy by circular grooved tubes made of brass alloy during the folding pro- cess under the axial loading by the theoretical and experime ntal methods . In the experimental part of their work, the circular bra- zen tubes with the external diameter of 50 mm, wall thickness of 1.8 mm and material ow stress of 220.26 MPa were compress ed axially in the quasi-static condition. The grooves were created in the form of annular patterns inside and outside on tubes surfaces alternatel y. In the all specimens, width of the grooves was selected

diagrams of the specimens FZQ-01 and FZQ-02.

diagrams of the specimens FZQ-03 and FZQ-04.

rams of the same inverted frustums with different punch diameters.

the same and equal to 3 mm but, the grooves in the different spec- imens were created with the different grooves distances of 6, 8, 9and 11 mm. Their experime ntal measure ments showed that the specic absorbed energy by the grooved tubes with the polyure- thane foam-ller is between 8600 and 10,240 J/kg. Fig. 15 showsthat the specic absorbed energy by the specimen FSQ-01 during the inversion process is equal to 11,066 J/kg. Therefore, experimen- tal results of the present article afrm that the inversion process on the empty frusta absorbs the kinetic energy as the same order of the folding process on the circular grooved tubes with the polyure- thane foam-ller. This comparis on shows advantage of using the

frusta during the inversion process as a good energy absorber with the high capacity.

Niknejad and Moeinifard [19] studied the inversion process of the circular metal tubes subjected to the quasi-static axial loading by the theoretical and experimental methods. Fig. 20 shows exper- imental diagram of the axial load versus the axial displacemen t of two different specimens that were compressed during the inver- sion process by using a die with the llet radius of 4 mm, based on the performed experiments by Niknejad and Moeinifard [19].Both of the specimens had the same wall thickness of 1 mm and initial length of 100 mm and different inner diameters of 38 and 28 mm, respectively. Calculation of the area under the loaddis-placemen t curves of two specimens shows that the absorbed en- ergy by the specimens are 1840 and 1808 J, respectively . Fig. 21 illustrate s absorbed energy versus the axial displacemen t during the inversion process by the mentioned specimens. The tubes masses are equal to 104.56 and 77.75 g, respectivel y and so, the specic absorbed energies by the inverted tubes are 17,597 and 23,254 J/kg, respectivel y. The specic absorbed energy by the in- verted tubes is higher than the correspondi ng value of the inver- sion process of the frusta but, the inversion process of the circular tubes is a more difcult process, comparing with the other energy absorption processes and also, comparing with the inver- sion process on the frusta [19]. Experimental results shows that when llet radius of the die is very small or very large, the inver- sion process on the circular tubes does not occur and usually, the deformat ion mode changes but, it is not considered in the inver- sion process on the frusta due to the conical angle of the frustums. On the other hand, the conical angle of the frustums causes the

Fig. 19. Two different specimens after the inversion tests, (a) specimen FZQ-01 and (b) specimen FSQ-02.

74 A. Niknejad, A. Tavassolimanesh / Materials and Design 49 (2013) 6575Fig. 20. Axial load versus the axial displacement during the inversion process of the ci

Fig. 21. Absorbed energy versus the axial displacement during the inversion process of thrcular brazen tubes, based on the Niknejad and Moeinifards experiments in [19].e circular brazen tubes, based on the Niknejad and Moeinifards experiments in [19].

inversion process on the conical tubes occurs more accurately and with the more repeatability , comparing with the circular tubes with the homogenous cross-sec tion along by the length.

Zhang et al. [25] performed the theoretical and numerical analyses to optimize crashworthines s of the aluminum foam-lledbitubal square columns. They simulated folding process of the thin-walled bitubal square columns subjected to the impact load- ing in two different conditions: empty and lled by aluminum foam-ller. Sectional dimensions of the outer column and inner column were 82 82 2.0 mm and 42 42 2.0 mm, respec- tively. A rigid block with the mass of 400 kg impacted onto the foam-lled bitubal column at an initial velocity of 15 m/s. The inner and outer columns were made of aluminum alloy AA6063- T6. Their numerica l simulatio ns showed that the specic absorbed

[3] Niknejad A, Liaghat GH, Moslemi Naeini H, Behravesh AH. Experimental and theoretical investigations of the rst fold creation in thin-walled columns. Acta Mech Solida Sin 2010;23(4):35360.

[4] Abedi MM, Niknejad A, Liaghat GH, Zamani Nejad M. Theoretical and experimental study on empty and foam-lled columns with square and rectangular cross section under axial compression. Int J Mech Sci 2012;65:13446.

[5] Niknejad A, Liaghat GH, Moslemi Naeini H, Behravesh AH. A theoretical formula for predicting the instantaneous folding force of the rst fold in asingle cell hexagonal honeycomb under axial loading. Proc IMechE Part C: JMech Eng Sci 2010;224:230815.

[6] Zhang X, Cheng G. A comparative study of energy absorption characteristics of foam-lled and multi-cell square columns. Int J Impact Eng 2007;34:173952.

[7] Chen A, Davalos JF. A solution including skin effect for stiffness and stress eldof sandwich honeycomb core. Int J Solid Struct 2005;42:271139.

[8] Mamalis AG, Manolakos DE, Saigal S, Viegelahn G, Johnson W. Extensible plastic collapse of thin-wall frusta as energy absorbers. Int J Mech Sci 1986;28(4):21929.

A. Niknejad, A. Tavassolimanesh / Materials and Design 49 (2013) 6575 75energy by the empty and aluminum foam-lled bitubal columns are equal to 10,670 and 13,170 J/kg, respectively. Therefore, reviewing of the recent published studies on the thin-walled struc- tures in the empty and lled conditions during the plastic defor- mation subjected to the different loadings reveals that the inversion process of the frusta is a suitable mechanism to absorb a considerable amount of the kinetic energy. This article introduces the inverted frusta as a good energy absorber with the long stroke and high load. The inverted frusta are suggested as a suitable mechanism to x on the automob ile bumper and to increase safety of the moving vehicles.

5. Conclusion

This article presents a new theoretical model of inversion defor- mation of the capped-end frusta and based on the introduce d mod- el, the instantaneous axial force and the absorbed energy by the frusta during the inversion process are predicted versus the axial displacemen t, theoreticall y. The theoretical analysis shows that the inversion load is depende nt on thickness of the bottom end and frusta wall, semi-apical angle, bottom end diameter and mate- rial properties of frusta and also, diameter of upper punch. A rea- sonable correlation between the theoretical predictions and experimental results afrms precision and accuracy of the sug- gested deformat ion model and the theoretical formulas. Also, investigatio n of the loaddisplacement diagrams of the different specimens with different materials and geometri cal dimensions shows that the inversion process on a frusta has a considerabl e en- ergy absorption capacity.

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Axial compression of the empty capped-end frusta during the inversion progress1 Introduction2 Theory2.1 Stage 1: Bottom plate deformation2.2 Stage 2: Inversion of the frusta wall

3 Experiments4 Results and discussion5 ConclusionReferences