1

Collapse of square metal tubes in splitting and curling mode X Huang1, G Lu1
, and T X Yu2 1 School of Engineering and Science, Swinburne University of Technology, Hawthorn, Australia 2 Department of Mechanical Engineering, Hong Kong University of Science and Technology, Hong Kong, People’s Republic of China The manuscript was received on 13 April 2004 and was accepted after revision for publication on 13 October 2005. DOI: 10.1243/095440606X78254

Abstract: This paper presents a further investigation into the energy absorbing behaviour of axially splitting and curling square mild steel and aluminium tubes. Both quasi-static and dynamic tests were conducted. A simple quasi-static kinematic model is developed which describes all the main features of the deformation process. It is assumed in this model that cracks start from the four corners and propagate along the axial direction due to continuous fracture/tearing. The free side plates so formed by cracks roll up into four curls with a constant radius. Formulas for the average crush force, curl radius, and the energy absorption are achieved by analysing the ‘far-field’ and ‘near-tip’ deformation events and the bending moment at the crack tip. This quasi-static model is also extended and used for dynamic cases to explore strain-rate effects in splitting square metal tubes. Solutions are presented and detailed comparisons are made between theoretical predictions and experimental results. Keywords: square tubes, splitting and curling mode, energy absorption, strain-rate effect

1

INTRODUCTION

With the rapid development of transportation, road safety is becoming of increasing concern to the general public due to the danger of impact incidents. Energy absorption devices can be employed during collisions for dissipating kinetic energy so as to protect the occupant from injury [1]. Thin-walled metal tubes have been identified to be effective energy absorption structures and are frequently used as impact energy absorbers [2– 5]. Thin-walled metal tubes can generate a wide range of deformation modes which depend on its geometries and collision conditions. Under axial compression, tubes can be deformed in progressive collapse [6– 11], internal and external inversion [12, 13], and splitting [14 –20]. These tubes are commonly constructed of ductile metal material which can dissipate the energy by the process of irreversible plastic deformation. However, significant changes in


Corresponding author: School of Engineering and Science,

Swinburne University of Technology, Hawthorn, Victoria 3122, Australia.

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energy absorption will be experienced if a change in the material failure mode occurs, from plastic deformation to fracture/tearing. Although the tube may lose its integrity in part, it is still capable of considerable energy absorption. Circular metal tube splitting tests and analysis have been conducted by Reddy and Reid [18] and Huang et al. [20]. Square metal tubes can also be split and curled when they were pressed axially against a die or a flat plate [14 –17]. They identified several energy dissipation mechanisms: plastic bending deformation associated with the development of curls; fracture/tearing energy associated with tube splitting and frictional work as the tube interacted with the die/plate. From the viewpoint of energy absorption, this splitting and curling collapse mode has a long stroke of over 90 per cent of the total tube length and results in a relative steady crush force. Square metal tube splitting tests were also conducted by Lu et al. [21] to measure the energy associated with ductile tearing. The values of the fracture toughness Rc (the tearing energy per unit torn area) for both mild steel and aluminium tubes were

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X Huang, G Lu, and T X Yu

successfully derived experimentally. It was found that the fracture toughness involved in tube splitting is much higher than the toughness for in-plane extension or out-of-plane tearing of a plate, due to severe local plastic deformation near the crack tip [22– 24]. In fact, the fracture energy includes the Gibbs’ thermodynamic minimum work and the dissipated energy in plastic deformation near the crack tip without which plastic fracture does not occur. Since the first part of energy only occupies a small portion of the total ductile fracture/tearing energy, Wierzbicki and Thomas [25] deals with the ductile tearing by calculating the ‘near-tip’ fracture energy through the predominated stress and strain distribution near the crack tip. Although they neglected the Gibbs’ thermodynamic minimum work which only occupies a small portion of the total ductile tearing energy, they predicted the correct functional dependence of the dissipated energy on the plate thickness and the length of cut. In the present paper, further investigation into splitting square tubes is presented. Both quasi-static and dynamic tests were conducted. The metal tube was compressed axially between a plate and a pyramidal die with various semi-angles. Tubes were observed to have cracks propagating along the four corners. Four free-side plates then rolled up into curls with an almost constant radius. The applied force became constant after the initial peak load. Theoretical analysis identified that the energy is dissipated by bending, fracture/tearing, and/or friction. It should be noted that calculation of fracture/ tearing energy near the crack tip in this analysis is based on the crack opening displacement (COD) parameter. Further analysis of the curl radius gives an explicit formula which can be used to predict the curls before testing. Dynamic analysis demonstrated the strain-rate effects in splitting and curling of square tubes. Comparisons are made between the experimentally observed characteristics and the theoretical predictions.

2 2.1

EXPERIMENTS Specimens

All the specimens tested were commercially available square hollow section tubes. The length of all the tubes was 200 mm, which was adequate to bend the tube sidewall into one curl. The external dimensions were 50  50 mm2. Two different materials were selected: mild steel and aluminium. For all mild steel tubes, at the corners there is a smooth small mean radius of approximately 2 –3 mm as shown in Fig. 1(a). For all aluminium tubes, the Proc. IMechE Vol. 220 Part C: J. Mechanical Engineering Science

Fig. 1

Cross-sections of specimens: (a) mild steel tubes and (b) aluminium tubes

four corners were trimmed as shown in Fig. 1(b) to prevent the fracture propagating to the side of the tube. The detailed dimensions of the tubes are listed in Table 1. Also, in order to prevent buckling and to establish the split and curl mode, an initial 5 mm saw-cut was made in each corner for all specimens. Mechanical properties of all specimens used were obtained from standard coupon tests. The average values of yield stress (or 0.2 per cent proof stress in the case of aluminium) sy , ultimate tensile strength su are as follows. For the mild steel tubes, sy ¼ 437.3 MPa, su ¼ 498.8 MPa. For the aluminium tubes sy ¼ 187.2 MPa, su ¼ 221.1 MPa. Table 1 Summary of quasi-static test results Test no.

Material

a (8)

t (mm)

t1 (mm)

Favg. (kN)

Ravg. (mm)

QM1 QM2 QM3 QM4 QM5 QM6 QM7 QM8 QM9 QM10 QM11 QM12 QM13 QM14 QM15 QM16 QM17 QM18 QM19 QA1 QA2 QA3 QA4 QA5 QA6 QA7 QA8 QA9 QA10 QA11

Mild steel Mild steel Mild steel Mild steel Mild steel Mild steel Mild steel Mild steel Mild steel Mild steel Mild steel Mild steel Mild steel Mild steel Mild steel Mild steel Mild steel Mild steel Mild steel Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium

45 45 45 45 45 60 60 60 60 60 60 75 75 75 75 75 75 75 75 30 30 30 45 45 45 60 60 60 75 75

1.6 1.6 2.5 2.5 3.0 1.6 1.6 2.5 2.5 3.0 3.0 1.6 1.6 2.5 2.5 2.5 3.0 3.0 3.0 2.0 2.5 3.0 2.0 2.5 3.0 2.0 2.5 3.0 2.0 2.5

1.6 1.6 2.5 2.5 3.0 1.6 1.6 2.5 2.5 3.0 3.0 1.6 1.6 2.5 2.5 2.5 3.0 3.0 3.0 1.0 1.5 2.0 1.0 1.5 2.0 1.0 1.5 2.0 1.0 1.5

11.25 13.70 30.60 31.25 47.50 17.50 17.75 39.38 34.79 50.63 44.75 18.25 15.95 41.25 42.88 39.50 44.25 48.13 46.75 3.05 6.25 10.19 3.85 8.15 12.39 6.84 9.90 19.19 5.22 11.88

20.73 19.55 18.39 18.15 20.55 11.82 10.98 11.62 12.87 12.81 13.88 6.95 7.38 8.26 7.68 8.26 9.18 9.23 8.92 40.16 37.82 35.44 24.35 23.29 22.53 18.11 13.93 13.64 13.84 10.0

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2.2

3

Quasi-static experiments

All quasi-static experiments were performed in a Shimadzu Universal testing machine, the experimental set-up is sketched in Fig. 2. The pyramidal die was fixed to the bottom bed of the testing machine. A short cuboid mandrel was used inside the bottom section of the tube to prevent the tube from tilting. The axes of the die, tube, and testing machine were carefully aligned. The crosshead of the testing machine then forced the tube slowly moving down against the pyramidal die. Four different pyramid semi-angles, a (refer Fig. 2), were selected for the die: 30, 45, 60, and 758, respectively. All dies were made from mild steel and heat-treated to increase their surface hardness. The crosshead speed was 5 mm/min for all quasi-static tests. Applied load, F, versus crosshead movement, DL, were recorded automatically. When the test started, the initial saw-cuts at the four corners divided the tube into four plates. Further movement of the crosshead resulted in four fractures at the four corners. These cracks propagated along the corners by ductile tearing/fracture. At the same time, curling of the tube began because the end of the tube consisted of four plates, which were free to bend outward into rolls. When the edge of one of the four rolls touched the flank of the tube, the experiment was stopped so that this additional contact force would not reduce the original radius of the curl. Typical force – displacement traces together with the corresponding energy– displacement plots for three different cases are shown in Fig. 3. The metal tubes tested were made of mild steel with a thickness of 2.5 mm, but the dies had a different semi-angle. In each case, the force initially increased with the crosshead movement until it reached a peak, which

Fig. 3

corresponded to the initiation of the four cracks at the corners. After that, the load decreased rapidly as the cracks propagated along the tube by ductile tearing/fracture. The four free sides then began to roll into curls. With increasing plastic deformation, the load increased again. Eventually, the curls formed with a constant radius as the plastic bending and load reached a steady state. This load remained constant with little fluctuation for the remainder of the test. The corresponding specimens after testing are shown in Fig. 4. On inspection, the curling radius of these specimens can be seen to decrease with an increase in the semi-angle of the die, a. This point will be elaborated later. Table 1 lists the test results such as the average crush force at the plateau stage and the average curl radius of the four sides at the mid-surface after testing for mild steel and aluminium tubes. 2.3

Fig. 2

Sketch of quasi-static experimental set-up

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Typical force – displacement and energy – displacement traces for mild steel square tubes of t ¼ 2.5 mm against dies of semi-angle a ¼ 45, a ¼ 60, and a ¼ 758, respectively

Dynamic experiments

All the dynamic experiments were conducted on a drop hammer testing machine as shown in Fig. 5. A load cell was fixed on the base of the testing machine and connected to the used die by auxiliaries. The specimen was put on top of the die; and the axes of the load cell, die, specimen, and testing machine were aligned carefully. When an impact test is conducted, the drop hammer with variable mass was raised by a pulley to a certain height (according to the needed impact velocity, V0 ) and released by an electromagnetic hook. The instantaneous impact velocity was measured by the velocimeter which was located near the top of the undeformed specimen. The load signal was amplified by a charge amplifier, digitized by a recording oscilloscope, and finally stored in a computer as numerical data. Proc. IMechE Vol. 220 Part C: J. Mechanical Engineering Science

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Fig. 4

Photographs of typical specimens after tests: (a) a ¼ 458, (b) a ¼ 608, and (c) a ¼ 758

It is supposed that the collision between the striker with mass M and the tube with mass m is completely inelastic, i.e. the two bodies remain together after collision. In all cases M  m, so it is reasonably assumed that there is no loss of the velocity and

kinetic energy in the initial collision process. After the initial collision, the kinetic energy of the system (the striker and tube) is gradually absorbed by the deformation of the tube. The instantaneous acceleration of the striker is a(t) ¼

F(t) M

(1)

where F(t) is the recorded instantaneous crushing force. Integrating a(t) yields the instantaneous velocity, and integrating twice yields the crushing distance ðt v(t) ¼ a(t) dt, and v(0) ¼ v0 (2) 0 ðt s(t) ¼ v(t) dt, s(0) ¼ 0 (3) 0

Fig. 5 Sketch of dynamic testing set-up

Proc. IMechE Vol. 220 Part C: J. Mechanical Engineering Science

Typical curves of the deceleration, velocity, and displacement against time are shown in Figs 6(a) and (b). It can be seen that velocity varied almost linearly from V0 to 0. Combining the above relationships, the force– displacement curve F(s) as well as the dissipated energy–displacement curves can be obtained, as shown in Fig. 7 for a typical case. In the load – displacement curve (Fig. 7), the deformation of the tube can be divided into two phases. The first phase corresponds to the initial elastic and plastic buckling and self-adjustment of the curl radius. The force in this stage is characterized by an initial peak and then fluctuation. In the second phase, the curls form with an almost constant radius and cracks steadily propagate along the axial C07204 # IMechE 2006

Square metal tubes in splitting and curling mode

Fig. 7

3

3.1

Fig. 6

(a) Deceleration and velocity curves in a typical dynamic test; (b) displacement histories in typical dynamic tests

direction. In this second phase, the force remains almost constant. Integrating force with respect to displacement, the total impact energy dissipated by the specimen is ð DL E¼

F(s) ds

(4)

0

where DL is the total shortening of the specimen. However, the total impact energy imparted to the specimen is 1 Ei ¼ Mv02 þ MgDL 2

(5)

Table 2 lists the experimental results for the average steady force, curl radius, total absorbed energy, and total input energy. In all tests, (Ei 2 E)/ E , 5 per cent, which confirms that the total energy loss can be reasonably neglected. C07204 # IMechE 2006

5

A typical dynamic force –displacement curve (for test: DAS45C)

THEORETICAL ANALYSIS ON SPLITTING OF SQUARE TUBES Theoretical model and basic assumption

Figure 8 sketches a simple theoretical model for the splitting and curling of a square tube pressed with a velocity V against a pyramidal die of semi-angle a. The tube is split and curled outward with an assumed constant radius R. Point A is the starting point of the curl. Point B denotes the crack tip with an angle b0 as defined in Fig. 8. Point C is the contact point between the curl and the die. A number of other simplifying assumptions are made to facilitate an effective analysis, as follows. 1. The material is regarded as rigid, perfectly plastic with an average flow stress s0 which is taken to be the ultimate stress su. This flow stress has the same value in both the bending and membrane deformation. There is no interaction between the resultant membrane force and the bending moment in yielding. 2. There is no variation in the tube thickness during compression and bending. From experimental observation, there is an abrupt change in the meridional curvature near crack tips. Nevertheless, for simplicity it is assumed that all the strips curl into rolls with a constant radius, R, and the bending moment at the crack tip is equal to the fully plastic bending moment of the side wall. 3. The distribution of stresses, in the region where cracks initiate, is complicated owing to the combination of the interfacial tension, out-of-plane shearing and bending. Here, it is assumed that the fracture of the tube at four corners is caused by in-plane extension and out-of-plane tearing, and the tearing energy is dominated by the interfacial tension under the prevailing state of stress. Proc. IMechE Vol. 220 Part C: J. Mechanical Engineering Science

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Table 2 Summary of dynamic test results Test no.

Material

a (8)

Striker mass (kg)

Impact velocity (m/s)

Ravg. (mm)

Favg. (kN)

E (J)

Ei (J)

ðEi  EÞ=E (%)

DMS45A DMS45B DMS45C DMS60A DMS60B DMS60C DMS75A DMS75B DMS75C DAS45A DAS45B DAS45C DAS60A DAS60B DAS60C DAS60D DAS75A DAS75B DAS75C DAS75D

Mild steel Mild steel Mild steel Mild steel Mild steel Mild steel Mild steel Mild steel Mild steel Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium

45 45 45 60 60 60 75 75 75 45 45 45 60 60 60 60 75 75 75 75

145.6 125.7 95.9 145.6 125.7 95.9 125.7 95.9 75.9 75.92 55.34 36.29 66.34 56.34 46.32 36.29 46.32 56.34 66.34 86.35

5.94 6.95 7.87 6.04 6.73 7.61 5.81 6.67 7.65 4.64 5.36 6.24 4.40 5.33 6.16 6.95 4.34 5.35 6.18 6.93

15.36 16.43 18.34 11.66 10.95 11.00 6.31 6.78 6.61 20.61 24.20 24.80 14.68 17.54 16.86 14.45 11.03 10.80 10.88 10.00

43.27 39.85 33.86 40.04 44.47 41.40 58.35 57.17 59.17 7.56 6.70 6.44 9.20 7.45 7.84 8.54 10.5 12.09 10.52 12.24

2648 3124 3054 2747 2917 2840 2136 2148 2227 892 883 747 690 864 933 914 883 1000 1136 1154

2656 3150 3055 2752 2928 2842 2157 2166 2246 933 884 748 691 865 933 914 885 1004 1135 1155

0.3 0.8 0.0 0.2 0.4 0.1 1.0 0.8 0.9 4.6 0.1 0.1 0.1 0.1 0.0 0.0 0.2 0.4 20.1 0.1

Fracture of the tube due to ductile tearing may be conveniently described in terms of the COD criterion. The COD approach was identified as a possible way in characterizing fracture properties of ductile sheets [25, 26]. The crack tip of the tube shown in Fig. 8 is not at A but at B due to ductility of the material; the ligament holds four strips together and the cross-section of the tube is expanded until a critical separation or COD is reached. Here, the COD parameter, d, is conveniently defined as shown in Fig. 8. Clearly, the parameter d depends on the tube thickness, the fracture strain of the material, as well as the stress distribution. Thus, the crack tip angle b0 can be uniquely

determined by the geometry of the model and the value of d, as follows
 d b0 ¼ cos1 1  R

(6)

where R is the average curling radius.

3.2

Determination of crush force and curling radius in quasi-static condition

In this quasi-static energy absorption system, there are three primary sources of energy dissipation: plastic bending of tube walls, fracture propagation along the four corners, and friction. Plastic bending energy is resulted from plastic bending of four side curls. When the radius of curling is R which is assumed to be same for the four side curls, the rate of plastic bending energy for the four sides is 2

_ P ¼ 4M0 V ¼ s0 Bt V W R R

(7)

where M0 ¼ s0 Bt 2 =4 is the fully plastic bending moment, B the side length of the cross-section, and t the thickness. The tearing energy associated with this specific problem can be assessed through considering the extension of four sides of the tube. The rate of tearing energy is defined by the integration of the product of the elemental extensional stress and the strain rates over the volume of the deforming region; i.e. Fig. 8

A kinematic model for splitting and curling of a square tube, where point A is the start of the curl, B is the crack tip, and C is the contact point

Proc. IMechE Vol. 220 Part C: J. Mechanical Engineering Science

_T¼ W

ð

s1_ dV

(8)

V

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7

Fig. 9 Stresses distribution at a cross-section of the square tube: (a) normal stress, (b) shear stress, and (c) stresses at the corner

Figure 9 shows the stress distribution of a crosssection of the square tube near the crack tip, which neglects the effect of the axial force. It is assumed that a uniform outward pressure, q, is applied by the die. It is also assumed that there is a uniform extension of the four tube walls and the strain is uniform along the side wall of the tube. Thus, the resultant normal stress is a constant, s, and shear stress varies linearly from t ¼ s at the corner to t ¼ 0 at the middle of the tube wall, as shown in Fig. 9. The stress yield condition is used to relate the components of the stress tensor under the plane stress assumption

s2x  sx sy þ s2y þ 3t2xy ¼ s20

(9)

At the four corners of the tube, the following applies sx ¼ txy ¼ s, sy ¼ 0. Therefore, the plastic extension stress near the cracks tip, before the crack initiation is approximately equal to s0 =2. Referring to Fig. 8, the extension strain for each side is

energy loss due to friction is _ F ¼ 4mNV W

where N is the normal force for each side of the tube. Under the quasi-static condition, it can be expressed as follows according to the vertical equilibrium of the system N¼

2R(1  cos b) B

(10)

where b is an angular coordinate (0 4 b 4 b0). Substituting it into equation (8) gains the rate of the tearing energy for four corners as _ T ¼ 4 s0 t 1 d W

(11)

Friction is an important energy dissipation mechanism under quasi-static conditions. It is known that the value of the frictional coefficient m would depends on the materials, contact conditions and other factors such as the die surface temperature due to severe deformation of the tube. The rate of C07204 # IMechE 2006

F 4(sin a þ m cos a)

(13)

_e¼ Thus, the rate of work done by the axial force W FV is balanced as follows _ PþW _ TþW _F _ e¼W W

(14)

Substituting the rate of plastic bending energy, tearing energy and friction energy into the above equation, the force required to split and curl the tube by a die of semi-angle a is obtained as F¼

1¼

(12)

s0 (Bt 2 =R þ 4t1 d) 1  m=(sin a þ m cos a)

(15)

In the above equation, the value of curl radius is unknown and it needs to be determined theoretically. According to assumption (2), the bending moment at the crack tip would be equal to the full plastic bending moment, i.e. FR ½sin(a  b0 ) þ m cos(a  b0 )  m ¼ M0 4 sin a þ m cos a

(16)

Combining with equation (15), the average curl radius is predicated by R¼

  Bt 2 sin a þ m cos a  m 1 4dt1 sin(a  b0 ) þ m cos(a  b0 )  m (17)

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By introducing the following non-dimensional parameters f ¼

F d t B R ; g¼ ; l¼ ; b¼ ; r¼ 4Bt s0 t1 t t t1

(18)

The non-dimensional curl radius can be expressed by   bl2 sin a þ m cos a  m r¼ 1 4g sin(a  b0 ) þ m cos(a  b0 )  m

(19)

where b0 ¼ cos1 (1  g=r l). It is found that the results of curling radius so calculated are insensitive to the coefficient of friction. Thus, by neglecting the effect of the friction on the curling radius, the non-dimensional curling radius can be simply expressed by r¼

  bl2 sin a 1 4g sin(a  b0 )

(20)

This is an implicit non-linear equation because b0 is a function of the curling radius r. Substituting equation (20) into equation (15), the non-dimensional crush force is expressed by f ¼

k2 þ gl k1 bl2

(21) Fig. 10

with

m sin a þ m cos a sin(a  b0 ) k2 ¼ sin a  sin(a  b0 )

k1 ¼ 1 

(22) (23)

where k1 is a constant which indicates the effect of the friction, k2 is another well-defined constant which reflects the ratio of plastic bending energy to the tearing energy. The values of k1 and k2 with g ¼ 1.0 are depicted in Figs 10(a) and (b) for the experimental mild steel tubes and aluminium tubes, respectively, indicating that k1 depends on the frictional coefficient m and the die semi-angle a, while k2 depends on the die semi-angle and the dimensions of the tube. 3.3

Strain-rate effects

Among various phenomenological rate-dependent constitutive equations for engineering materials, the Cowper –Symonds relation has been most popularly employed in structural problems. This relation represents a rigid, perfectly plastic material with a dynamic yield or flow stress that depends on strain Proc. IMechE Vol. 220 Part C: J. Mechanical Engineering Science

The variation of constants k1 and k2 for (a) mild steel tubes and (b) aluminium tubes. In the figures, the non-dimensional COD parameter, g, is taken to be 1.0

rate. Thus, the ratio of dynamic flow stress sd0 to static flow stress ss0 is expressed as
1=p sd0 1_ ¼ 1 þ D ss0

(24)

where ss0 denotes the quasi-static flow stress of the material, sd0 denotes the dynamic flow stress, 1_ represents the strain rate. D and p are material constants, which can be obtained from experiments [27, 28]. D and p are 40 s21 and 5 for mild steel, 6500 s21 and 4 for aluminium, respectively. This indicates that mild steel is a strain rate sensitive material, whereas aluminium is insensitive. To take account of strain rate, the characteristic strain rate may be defined as the average rate of circumferential strain. From equation (10), the instantaneous circumferential strain rate can be expressed as 1_ u ¼

2 sin b v B

(25)

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It shows that circumferential strain rate develops with b. And the average circumferential strain rate is Ð b0 1_ av ¼

0

1_ u db 2(1  cos b0 ) ¼ v b0 Bb0

1  cos b0 v0 ¼ Cv0 B b0

4.1

(27)

Here, C is a constant depending on the tube geometry and the critical opening angle, b0; and thus depending on the die used. According to identify ‘the calculation’ with these quasi-static parameters, the value of C is about 5 for all cases. Hence, the strain-rate effect in splitting a square tube can be written as
1=p sd0 5v0 ¼1þ s s0 D

quasi-static frictional coefficient m is taken to be 0.3 between a mild steel tube and a mild steel die, and 0.2 between an aluminium tube and a mild steel die.

(26)

It is evident that the average circumferential strain rate varies with time during dynamic deformation of the structure due to the variation of crush velocity. Furthermore, in order to simplify the calculation, it is assumed that the crush velocity linearly decreases from v0 to 0 (see Fig. 6(a)). Thus, the characteristic strain rates can be roughly assessed as 1_ ¼

9

Curl radius

The curl radius may be determined independently by equation (20) while neglecting the effect of friction. Figures 11(a) and (b) show a comparison of the non-dimensional curl radius between theoretical predictions and the tests for mild steel tubes and aluminium tubes, respectively. Good agreements are obtained. It is seen that the curl radius decreases with the die semi-angle. Apart from the die semiangle, the curl radius depends on the width of the tube (b) and the thickness ratio (l). Dynamic test results are also shown in these figures with hollow symbols; and these are very close to the corresponding quasi-static ones.

(28)

The quasi-static model can be extended to the impact loading with the strain-rate effect being considered, i.e. replacing the quasi-static flow stress with the dynamic flow stress. However, the energy absorption due to friction is low because the coefficient of dynamic friction is much lower than that of quasi-static friction. Hence, ignoring the effect of the friction and equating the energy dissipation to the work of external force, one obtains fd ¼

k2 þ gl bl2

(29)

where the non-dimensional dynamic crush force is f d ¼ F d/4Btsd0 . Thus, the strain-rate effect in our tests can be expressed by

sd0 Fd ¼ s s0 k1 F s

(30)

where F s is the corresponding quasi-static crush force. Fig. 11

4

RESULTS AND DISCUSSION

In the following analysis, the non-dimensional COD parameter, g, is taken to be 1.0. The value of C07204 # IMechE 2006

Variation of the average curl radius against the semi-angle of the die for (a) mild steel tubes and (b) aluminium tubes, where the lines denote the theoretical predications and dots are the corresponding test values

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In the above analysis, the curl radius is determined by the global equilibrium of forces in the axial direction, rather than by minimizing the dissipated energy, which has been often used in the previous analyses of similar problems to determine some unknown parameters [9, 11, 25]. As a matter of fact, our analysis assumes the curls form cylindrical surfaces instead of doubly curved surfaces, which result from complex normal and shear stress fields near the crack tips. As a result of this simplified assumption on the deformation field, the curl radius appears to be independent from the tearing/fracture energy, and thus it cannot be determined by minimizing the energy dissipation of the system. However, keeping in mind that this kind of deformation field with any curl radius is kinematically admissible, the determination of the curl radius by the global equilibrium condition in fact leads to an optimum kinematically admissible deformation field, i.e. an optimum upper bound solution for the problem.

4.2

Crush force under quasi-static loading condition

The crush force under quasi-static loading condition is the most important index for an energy absorption device. Figures 12(a) and (b) show the non-dimensional crush force, f, against the die semi-angle, a, for mild steel tubes and aluminium tubes, respectively. In general, good agreement between the theory and experiments has been obtained. Only in a few cases, experimental values are significantly lower or higher than the theoretical ones, for example, in the case of the mild steel tube with t ¼ 3.0 mm and a ¼ 758, and the aluminium tube with t ¼ 3.0 mm and a ¼ 608. This deviation from the theory may be attributed to the variations in friction and uneven radius of the four curls. Also, the analysis shows a dependence of the crush force upon the value of the die semi-angle. Therefore, a given tube can be designed to perform over a wide range of load levels by simply changing the die semi-angle.

4.3

Fig. 12

4.4

Variation of the non-dimensional quasi-static crush force, f, with the die semi-angle, a, for (a) mild steel tubes and (b) aluminium tubes

Energy partitioning under quasi-static loading condition

In this energy absorbing device, several energy dissipation mechanisms are operative: plastic deformation

Effect of friction on force and curl radius

From the above analysis, it is clear that increasing friction can raise the crush force considerably. For example, compared with theoretical force for m ¼ 0, the crush force increases by about 25 per cent for the frictional coefficient m ¼ 0.2, and about 40 per cent for m ¼ 0.3. However, the curl radius is insensitive to friction as shown in Fig. 13; for the case of mild steel tubes of t ¼ 2.5 mm, the reduction in the nondimensional curl radius is less than 1.0 for the most extreme case. Proc. IMechE Vol. 220 Part C: J. Mechanical Engineering Science

Fig. 13

The effect of friction on the curl radius for a mild steel tube of t ¼ 2.5 mm

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associated with the bending of the curls, tearing energy associated with tube splitting, and friction of the tube passing over the pyramidal die. The total energy absorbed can be partitioned into three components due to these three mechanisms. An example of the energy partitioning versus the die semi-angle is shown in Fig. 14 for the case of b ¼ 20.0, l ¼ 1.0, and m ¼ 0.3. It is seen that the tearing energy and the friction energy are dominant for small die semi-angles, while the plastic bending energy becomes more important as the die semi-angle increases.

4.5

Strain-rate effects in dynamic loading case

Figure 15 depicts a plot of the experimental and theoretical results for dynamic tests. The experimental data are shown by the dots according to the scheme described in section 3, together with the theoretical predictions from equation (28) by a solid line for mild steel tubes and a dotted line for aluminium tubes. With regard to the different materials, the experimental data are divided into two groups (mild steel and aluminium tubes) by their sd0 =ss0 values. This is expected as the mild steel tubes exhibited stronger sensitivity to the impact velocity, so as to the strain rate than the aluminium tubes. On the whole, Fig. 15 shows a satisfactory agreement: the points lie fairly close to the corresponding lines. This conclusion is insensitive to the precise choice of the value of C.

4.6

Energy absorption efficiency

The energy absorbed per unit mass, w, is often used to indicate the energy absorption efficiency and it is also a design parameter in mass-limited system.

Fig. 14

Partitioning of three energy components for b ¼ 20.0, l ¼ 1.0, and m ¼ 0.3, where WF is the friction energy, WT is the tearing energy, and WP is the plastic bending energy

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Fig. 15

Plot of the ratio between the dynamic and static flow stresses (tests data are obtained according to equation (30)) against the impact velocity

In this case, the specific energy absorbed can be written as w¼

F 4rBt

(31)

where the material density r is 7860 kg/m3 for mild steel and 2710 kg/m3 for aluminium, respectively. Figure 16 shows a comparison between the energy absorption capacity under dynamic and quasi-static loading conditions for tubes of t ¼ 2.5 mm. In general, the specific energy absorption improves with an increase of the die semi-angle. For aluminium tubes, the specific energy absorption in quasi-static cases is higher than that in dynamic cases. It is known that aluminium is a strain-rate insensitive material and hence we may anticipate its specific energy absorption under static and dynamic loading to be similar. However, in dynamic cases less energy

Fig. 16

Comparison of the energy absorption capacity between the quasi-static and dynamic tests

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X Huang, G Lu, and T X Yu

is absorbed by friction (due to reduction of the friction coefficient), resulting in a decrease in the specific energy absorption. For mild steel tubes, the specific energy absorption in quasi-static cases is much lower than that in dynamic cases, because mild steel is strongly sensitive to the strain rate and this effect compensates for the reduction of the frictional energy dissipated in dynamic cases. Generally, the mild steel tubes have better energy absorbing capacity than the aluminium ones. However, it should be noted that all the aluminium tubes were trimmed at the four corners (Fig. 1(b)), which reduced the total energy they can absorb.

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CONCLUSIONS

The splitting and curling of a square metal tube against a pyramidal die is an efficient energy absorbing system for a crashworthy system. Our experiments showed that the square tube split along the four corners and curled outward with a constant radius under a steady applied force. Based on the experimental observations, a quasi-static theoretical model is formulated. There are three main energy dissipating mechanisms: plastic bending, fracture/ tearing, and friction. However, in dynamic cases, frictional energy is negligible. Plastic bending energy and frictional energy both increase with the die semi-angle, but fracture/tearing energy is unaffected by die semi-angle. The applied force is determined through the energy conservation. With the help of the above analysis, force analysis is employed to independently determine the curl radius. The present theoretical predictions provide good agreement with experiments in all aspects. Finally, the quasistatic model is extended to a dynamic analysis of the splitting of square tubes against pyramidal dies to incorporate the strain-rate effects. As expected, the performance of mild steel square tubes shows a strong dependence on the impact velocity, whereas that of aluminium square tubes exhibits a very weak dependence.

REFERENCES 1 Johnson, W. and Mamalis, A. G. Crashworthiness of vehicles, 1978 (Mechanical Engineering Publication Ltd, Bury St Edmunds and London). 2 Jones, N. Structural impact, 1988 (Cambridge University Press, Cambridge). 3 Lu, G. and Yu, T. X. Energy absorption of structure and material, 2003 (Woodhead Publishing, Cambridge). 4 Johnson, W. and Reid, S. R. Metallic energy dissipating systems. Appl. Mech. Rev., 1978, 31, 277 –288.

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5 Johnson, W. and Reid, S. R. Update to: metallic energy dissipating systems. Appl. Mech. Update, 1986, 39, 315– 319. 6 Alexander, J. M. An approximate analysis of collapse of thin-walled cylindrical shells under axial loading. Q. J. Mech. Appl. Math., 1960, 13, 10– 15. 7 Abramowicz, W. and Jones, N. Dynamic progressive buckling of circular and square tubes. Int. J. Impact Eng., 1986, 4, 243 – 269. 8 Grzebieta, R. H. An alternative method for determining the behaviour of round stocky tubes subjected to axial crush loads. Thin Wall. Struct., 1990A, 9, 66 – 89. 9 Wierzbicki, T., Bhat, S. U., Abramowicz, W., and Brodkin, D. Alexander revisited – a two folding element model of progressive crushing of tubes. Int. J. Solids Struct., 1992, 29, 3269 – 3288. 10 Guillow, S. R., Lu, G., and Grzebieta, R. H. Quasi-static axial compression of thin-walled circular aluminium tubes. Int. J. Mech. Sci., 2001, 43(9), 2103 – 2123. 11 Huang, X. and Lu, G. Axisymmetric progressive crushing of circular tubes. Int. J. Crashworthiness, 2003, 8(1), 87 – 95. 12 Reddy, T. Y. Tube inversion – an experiment in plasticity. Int. J. Mech. Eng. Edu., 1989, 17, 277– 291. 13 Reid, S. R. and Harrigan, J. J. Transient effects in the quasi-static and dynamic internal inversion and nosing of metal tubes. Int. J. Mech. Sci., 1998, 40(2 – 3), 263 – 280. 14 Stronge, W. J., Yu, T. X., and Johnson, W. Long stroke energy dissipation in splitting tubes. Int. J. Mech. Sci., 1983, 25(9– 10), 637 – 647. 15 Stronge, W. J., Yu, T. X., and Johnson, W. Energy dissipation by curling tubes. In Structural Impact and Crashworthiness (Ed. J. Morton), 1984, Vol. 2, pp. 516 –587 (Elsevier, London). 16 Lu, G., Huang, X., and Yu, T. X. Axial splitting of square tubes. In Structural failure and plasticity (Eds X. L. Zhao and R. H. Grzebieta), 2000, pp. 457 – 462 (Elsevier, Oxford). 17 Huang, X., Lu, G., and Yu, T. X. Energy absorption in splitting square metal tubes. Thin Wall. Struct., 2002, 40, 153 – 165. 18 Reddy, T. Y. and Reid, S. R. Axial splitting of circular metal tubes. Int. J. Mech. Sci., 1986, 28(2), 111 – 131. 19 Atkins, A. G. On the number of cracks in the axial splitting of ductile metal tubes. Int. J. Mech. Sci., 1987, 29, 115 – 121. 20 Huang, X., Lu, G., and Yu, T. X. On the axial splitting and curling of circular metal tubes. Int. J. Mech. Sci., 2002, 44(11), 2369 – 2391. 21 Lu, G., Ong, L. S., Wang, B., and Ng, H. W. An experimental study on tearing energy in splitting square metal tubes. Int. J. Mech. Sci., 1994, 36(12), 1087 – 1097. 22 Lu, G., Fan, H., and Wang, B. An experimental method for determining ductile tearing energy of thin metal sheets. Met. Mater., 1998, 4(3), 432 – 435. 23 Fan, H., Wang, B., and Lu, G. On the tearing energy of a ductile thin plate. Int. J. Mech. Sci., 2002, 44, 407 – 421. 24 Yu, T. X., Zhang, D. J., Zhang, Y., and Zhou, Q. A study of the quasi-static tearing of thin metal sheets. Int. J. Mech. Sci., 1988, 30, 193 – 202.

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25 Wierzbicki, T. and Thomas, P. Closed-form solution for wedge cutting force through thin metal sheets. Int. J. Mech. Sci., 1993, 35, 209 – 229. 26 Parks, D. M., Freund, L. B., and Rice, J. R. Running ductile fracture in a pressurized line pipe. In Mechanics of crack growth, 1976, ASTM STP 590, pp. 2543 – 2562 (American Society for Testing and Materials, Philadelphia). 27 Su, X. Y., Yu, T. X., and Reid, S. R. Inertia-sensitive impact energy-absorbing structures. Part I: effects of inertia and elasticity. Int. J. Impact Eng., 1995, 16(4), 651 – 672. 28 Su, X. Y., Yu, T. X., and Reid, S. R. Inertia-sensitive impact energy-absorbing structures. Part II: effects of strain rate. Int. J. Impact Eng., 1995, 16(4), 673 – 689.

APPENDIX Notation a B C D, p E

instantaneous acceleration of the striker width of square tubes strain-rate constant, equation (27) dynamic material constants, equation (24) total input energy

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Ei F k1, k2 m M M0 N R s t, t1 v, v0 _e W _F W _P W _T W

total absorbed energy axial force constants, equations (22) and (23) mass of specimens mass of strikers fully plastic bending moment normal force curl radius displacement wall thickness, Fig. 1 velocity rate of work done by axial force rate of work done by friction rate of plastic bending energy rate of tearing/fracture energy

a b b0 d DL m s0 su sy

die semi-angle, Fig. 2 angular coordinate, Fig. 8 crack tip angle, equation (6) and Fig. 8 COD parameter, Fig. 8 total shortening of tubes frictional coefficient flow stress ultimate stress yield stress

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