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On the axial splitting and curling of circular metal tubes
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International Journal of Mechanical Sciences 44 (2002) 2369 – 2391
On the axial splitting and curling of circular metal tubes X. Huanga , G. Lua;∗ , T.X. Yub a b
School of Engineering and Science, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, People’s Republic of China Received 27 May 2002; received in revised form 18 November 2002; accepted 26 November 2002
Abstract The present paper investigates the axial splitting and curling behaviour of circular metal tubes. Mild steel and aluminum circular tubes were pressed axially onto a series of conical dies each with di3erent semi-angle. By pre-cutting eight 5 mm slits which were distributed evenly at the lower end of each tube, the tube split axially and the strips curled outward. Experiments showed that this mechanism results in a long stroke and a steady load. An approximate analysis is presented which successfully predicts the number of propagated cracks, the curling radius and the force applied. This analysis takes into account ductile tearing of the cracks, plastic bending/stretching and friction. E3ects of tube dimensions, semi-angle of the die and friction are discussed in detail. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Circular tube; Conical die; Splitting; Curling; Energy absorption
1. Introduction Energy absorbing devices are employed where collision may cause serious consequences, so as to protect human as well as other important structures [1,2]. Thin-walled structures are e?cient in absorbing energy and widely used as structural members and energy absorbing devices. Thin-walled tubes have attracted much attention owing to the wide range of deformation modes which can be generated. Circular tubes may be axially crushed progressively in an axisymmetric or a non-axisymmetric (or called diamond) mode by two @at plates. It is a very e?cient energy absorption arrangement. Alexander [3] was the Arst researcher to propose a theory for the axisymmetric collapse mode based on a balance of external and internal work done. Afterwards, Abramowicz and Jones [4,5], Grzebieta ∗
Corresponding author. Fax: +61-3-9214-8264. E-mail address: [email protected] (G. Lu).
0020-7403/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0020-7403(02)00191-1
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[6], Wierzbicki et al. [7] and Singace et al. [8,9] further developed a model to predict the crush force. Recently, Guillow et al. [10] and Huang and Lu [11] reAned this work experimentally and theoretically. Another experimental arrangement is that a tube may be compressed against a shaped die, where several di3erent types of collapse mode can be established. Within a compatible range of parameters, an inversion mode has been observed. This has been studied by Reddy [12] and Reid and Harrigan [13]. Another collapse mode is splitting and curling of tubes. From the viewpoint of energy absorption, this collapse mode has a long stroke of over 90 per cent of the total tube length. Stronge et al. [14,15] conducted experiments with square tubes split against a radius/@at die. Huang et al. [16,17] further studied the splitting and curling behaviour of square tubes axially compressed between a plate and a pyramidal die. Lu et al. [18] investigated the tearing energy involved in splitting square metal tubes. Tearing energy in thin metal sheets was studied by Lu et al. [19,20] and Yu et al. [21], among others. In a comparative study of energy absorbers for improving occupant survivability in aircraft crashes, Ezra and Fay [22] identiAed circular tube splitting and curling as an e?cient system based on speciAc energy dissipation. Reddy and Reid [23] studied the splitting behaviour of circular tubes compressed axially between a plate and a radius die. The advantage of this mechanism lies not only in having a long stroke but also a steady crush force. In the present study, splitting and curling behaviour of circular tubes is further investigated experimentally and theoretically. Circular metal tubes were axially compressed between a plate and a series of conical dies which are di3erent from those used in Ref. [23]. Tubes were observed to have a number of cracks propagating along the axial direction. The strips so formed by cracks rolled up into curls with an almost constant radius. The crush force became steady after some initial @uctuations. Three energy dissipation mechanisms were involved: (1) the “near-tip” tearing associated with tube splitting; (2) the “far-Aeld” deformation associated with the plastic bending and stretching of curls; (3) the friction as the tube interacted with the die. A correlation between the crack number and the curl radius is found through a force analyses. An approximate analysis is performed to predict the force at the steady stage. As a result, the crack number is predicted independently by the minimum energy approach involving a competition between the plastic bending and fracture energy. Thereby, the curl radius and the applied force are determined using the predicted crack number. The numerical results are compared with the present experiments in detail. The analysis is also applied to the previous experiments and equations for the number of cracks and force are obtained. 2. Experiments All the experiments were performed in a Shimadzu Universal testing machine under quasi-static conditions. The experimental set-up is sketched in Fig. 1. A cone-shaped die was Axed to the bottom bed of the testing machine. A short cylindrical mandrel was used inside the tube to prevent the tube from tilting. The axes of the die, tube and testing machine were carefully aligned. The cross-head of the testing machine then pressed the tube onto the conical die at a constant rate of 0:0333 mm=s. Three di3erent semi-angles (); 45◦ ; 60◦ and 75◦ , were selected for the conical die. All dies were made from mild steel and heat-treated to increase their surface hardness. All the specimens
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Fig. 1. Sketch of the experimental set-up, with 8 evenly spaced 5 mm initial saw-cuts around lower circumference. Table 1 Summary of tests and theoretical results for mild steel tubes Test
D (mm)
t (mm)
(◦ )
ni
nf
Fmax1 (kN)
Fmax2 (kN)
Favg: (kN)
Ravg: (mm)
CS1 CS2 CS3 CS4 CS5 CS6 CS7 CS8 CS9 CS10 CS11 CS12
49.4 49.4 49.4 61.9 61.9 61.9 74.0 74.0 74.0 47.2 47.2 47.2
1.6 1.6 1.6 1.6 1.6 1.6 2.0 2.0 2.0 3.3 3.3 3.3
30.9 30.9 30.9 38.7 38.7 38.7 37.0 37.0 37.0 14.3 14.3 14.3
45 60 75 45 60 75 45 60 75 45 60 75
8 8 8 8 8 8 8 8 8 3 4 4
8 8 8 8 8 8 8 8 8 3 4 4
14.6 24.9 73.7 16.7 28.1 50.3 29.5 51.3 83.9 — — 83.0
23.7 31.8 39.6 24.2 33.8 44.4 35.3 39.5 51.8 71.5 76.8 95.3
14.2 16.9 21.9 16.0 19.8 24.8 23.3 27.0 33.8 38.8 48.8 60.0
12.2 8.1 6.5 13.6 8.3 6.5 17.9 11.5 8.3 29.8 12.6 11.2
tested were commercially available circular tubes of 200 mm long. The ratio of the diameter over the thickness ranges from 15 to 50. The detailed dimensions of all the specimens are listed in Table 1 for the mild steel tubes and Table 2 for the aluminium tubes. In order to establish the split and curl mode while preventing other collapse modes, initial 5 mm saw-cuts for all specimens were made
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Table 2 Summary of tests and theoretical results for aluminium tubes Test
D (mm)
t (mm)
(◦ )
ni
nf
Fmax1 (kN)
Fmax2 (kN)
Favg: (kN)
Ravg: (mm)
AL1 AL2 AL3 AL4 AL5 AL6 AL7 AL8 AL9 AL10 AL11 AL12 AL13 AL14 AL15 AL16 AL17 AL18
48.25 48.25 48.25 47.0 47.0 47.0 57.75 57.75 57.75 56.8 56.8 56.8 77.9 77.9 77.9 76.95 76.95 76.95
1.3 1.3 1.3 3.0 3.0 3.0 1.85 1.85 1.85 3.2 3.2 3.2 1.95 1.95 1.95 2.85 2.85 2.85
37.1 37.1 37.1 15.7 15.7 15.7 31.2 31.2 31.2 17.8 17.8 17.8 39.9 39.9 39.9 27.0 27.0 27.0
45 60 75 45 60 75 45 60 75 45 60 75 45 60 75 45 60 75
9 9 10 5 6 7 8 8 8 6 7 7 8 8 8 8 8 8
9 9 10 5 5 5 8 8 8 6 6 5 8 8 8 8 7 7
4.4 6.7 15.7 14.6 27.9 49.5 8.9 15.8 24.6 19.7 33.9 62.3 12.1 19.5 43.2 19.0 30.3 60.5
5.3 5.7 9.7 18.4 21.9 — 11.7 12.9 15.5 — — — 12.0 11.9 15.0 — — —
3.3 4.0 6.9 14.3 18.8 23.8 8.8 10.8 12.1 18.0 24.5 32.0 8.5 11.0 13.0 15.6 20.0 27.5
13.9 9.4 6.6 28.7 16.3 10.6 16.3 10.9 8.0 28.7 16.6 12.2 21.6 15.1 10.9 23.8 17.8 12.6
Fig. 2. Average stress–strain curves of the both materials.
which were evenly spaced around the lower circumference. A total of eight saw-cuts was made in each tube. Stress–strain relationships for the two materials were obtained from standard coupon tests for each thickness. The curves for a given material were close and their average is shown in Fig. 2. These two curves are approximated by a rigid, perfectly plastic relationship with @ow stress Y = 450 MPa for mild steel, and Y = 180 MPa for aluminium. In the beginning of a typical test, the strips between initial saw-cuts buckled and @ared as guided by the die. This led to the circumferential stretching of the tube. When this extension had reached
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Fig. 3. Typical force-displacement curves for mild steel tubes with D = 74:0 mm; t = 1:8 mm against dies with semi-angle = 45◦ ; 60◦ and 75◦ , respectively.
a certain level, cracks occurred at some initial saw-cut locations and propagated along the axial direction due to continuous ductile tearing. The strips so formed by the cracks rolled up into curls as the end of these strips was free to bend. These curls have an almost constant radius (R denotes the middle radius of the curl), the value of which mainly depends on the semi-angle of the die and the dimensions of the tube. When these curls had completed one revolution, the front edges of the curls contacted the tube wall. This caused the radius of the next roll to be R + t due to the thickness of the wall, t. In the following sections, the curl radius refers to the Arst curl unless noted otherwise. When the crushing distance reached 120 mm (out of the original total length 200 mm), the test was stopped. In some tests there was scu?ng and ploughing of the tubes, especially in the aluminium ones, as they were pressed onto the dies. This resulted in debris stuck to the die after the removal of the tube, so the die was cleaned with a cloth after each test. Typical force-compression curves for mild steel tubes (D = 76:0 mm and t = 2:0 mm) with three di3erent dies are shown in Fig. 3 and the corresponding specimen photographs after tests are shown in Fig. 4 with side and end views. In these three cases, cracks were parallel to the tube axis without branching or merging. The force initially increased with the cross-head movement until it reached the Arst peak, Fmax1 , which corresponded to the onset of inversion of strips from the initial saw-cuts. A second peak force, Fmax2 , then occurred, and this corresponded to the initiation of cracks. After approximately another 10 mm displacement the force reached a steady state. This force Favg remained almost constant, and is used for later analyses. When the front edge of the Arst revolution touched the tube wall, the applied force surged notably before dropping to another steady state, which corresponded to the second revolution. Because less plastic bending energy was dissipated due to the increment of the radius of the second revolution, the average force at the second steady stage was a little lower than that of the Arst steady state. Fig. 5 shows typical force-compression curves for the aluminium tubes pressed onto three di3erent semi-angled dies. Corresponding specimen photographs after tests are shown in Fig. 6. The aluminium tubes had similar deformation modes to those of the mild steel tubes, except that the average force
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(a)
(b)
(c)
Fig. 4. Photographs of typical mild steel specimens (D=74:0 mm; t =1:8 mm) after tests: from left to right CS7 (=45◦ ), CS8 ( = 60◦ ) and CS9 ( = 75◦ ).
Fig. 5. Typical force-displacement curves for aluminium tubes with D = 77:9 mm; t = 1:9 mm against dies with semi-angle = 45◦ ; 60◦ and 75◦ , respectively.
was at the same lever before and after the front edges of the curls touched the tube wall. Probably, the decrease in the applied force due to the increasing radius of the next roll was o3set by an increase in friction between the tube and the inside mandrel. When the strips of tubes began to curl up, branching or merging of cracks was observed in some experiments. That is, in some cases, cracks which developed from initial saw-cuts ceased to propagate; but in some other cases, one original crack split into two. Two typical tubes with branching or merging of cracks after tests are shown in Fig. 7, where the initial number of saw-cuts
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(a)
(b)
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(c)
Fig. 6. Photographs of typical aluminium specimens (with D = 77:9 mm; t = 1:9 mm) after tests: from left to right Al13 ( = 45◦ ), Al14 ( = 60◦ ) and Al15 ( = 75◦ ).
(a)
(b)
Fig. 7. Photographs of typical specimens after tests: (a) with crack merging (test CS12 for the mild steel tube with D = 47:2 mm; t = 3:3 mm and die semi-angle = 75◦ ); and (b) with crack branching (test AL3 for the aluminium tube with D = 48:25 mm; t = 1:3 mm and die semi-angle = 75◦ ).
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Fig. 8. Comparison of the applied force with and without lubrication for mild steel tubes with D = 74:0 mm; t = 1:8 mm and die semi-angle = 60◦ .
was 8. The numbers of cracks after initial branching or merging (ni ) are given in Tables 1 and 2 (Note they are not the initial saw-cut numbers). For aluminium tubes, further merging of cracks was also observed when the strips produced subsequent revolutions. On the contrary, the mild steel tubes kept the original cracks to the end of tests. The Anal crack numbers (nf ) are also given in Tables 1 and 2. These branching and merging of cracks resulted in a little @uctuation in the applied force. After a durable re-orientation of these cracks, the strips formed by the cracks tended to have the same width. Reddy and Reid [23] pointed out that friction is present as a multiplying factor of the applied force in splitting and curling tubes. In order to obtain a reasonable frictional factor in the present cases, a typical tube was also tested under lubricated conditions. Fig. 8 shows, however, that the di3erence in the applied force under the conditions with and without lubrication was minor. One possible reason is that the frictional factor for the present series of tests without lubrication is also very small. In later analysis, a frictional coe?cient = 0:2 is adopted for all the cases. Detail experimental results such as the average applied force (Favg: ) and average curl radius (Ravg: ) are also listed in Table 1 for the mild steel tubes and Table 2 for the aluminium ones. 3. Theoretical analysis 3.1. Analytical model and basic assumptions Consider a tube with diameter D and thickness t pressed axially and split against a die with a semi-angle . A simple kinematic model is proposed as shown in Fig. 9. Point A is the starting point of the curl, AC, with a doubly curved face of radius R in the axial direction and r0 , neglecting the increment of the tube radius in the circumferential direction (as shown in Fig. 9). The crack tip is located at B with an angle 0 as deAned in Fig. 9. C denotes the contact point between the curl
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Fig. 9. A kinematic model for splitting and curling of a circular tube.
and the die. A number of other simplifying assumptions are made to facilitate an e3ective analysis, as given below. (a) The material is regarded as rigid, perfectly plastic with an average @ow stress. This @ow stress has the same value in both the bending and membrane deformations. There is no interaction between the resultant membrane force and the bending moments in yielding. (b) There is no variation in the tube thickness during collapse. From experimental observation there is an abrupt change in the meridional curvature near the 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, see Eq. (10) later. (c) Plastic work in curling is calculated by considering meridional bending of a curved strip with a transversal radius r0 . Changes in transversal radius are neglected. (d) The e3ect of out-of-plane shear stress is neglected. (e) Fracture of the tube is caused by circumferential stretching of the tube wall. A critical crack opening displacement (COD) parameter is adopted in assessing the crack initiation and tearing energy. The Anal deformed strips are clearly doubly curved with a non-zero Gaussian curvature while originally they are cylindrical and thus of a zero Gaussian curvature. This change in Gaussian curvature requires stretching/compression of the surface [24]. Here, assumption c above leads to a stretching in the meridional direction only. Fracture of the tube due to ductile tearing may be conveniently described in terms of the COD criterion. The COD approach was identiAed as a possible way in characterizing fracture properties of ductile sheets [25,26]. The crack tip of the tube shown in Fig. 9 is not at A but at B due to the ductility of the material; the ligament holds these assumed strips together until a critical separation or crack opening displacement is reached. Clearly the parameter depends on the tube thickness,
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the fracture strain of the material as well as the stress distribution. In the present analysis, we use the non-dimensional critical separation = =t as a parameter and take = 1:0 unless speciAed otherwise. Thus, the crack tip angle 0 can be uniquely determined from the geometry of the problem and the value of , as follows: nt −1 0 = cos ; (1) 1− 2R where n is the number of cracks (or strips). 3.2. Correlation between curl radius and number of cracks The wall of the tube curls up into a constant radius, R, due to the bending moment resulting from the applied force F. The deformed cylindrical tube ahead of the crack tip (AC) has a gradually increasing radius circumferentially. The development of the tube radius r is deAned in Fig. 9. Thus the circumferential strain can be estimated as r = − 1; (2) r0 where r0 is the initial radius of the tube. Neglecting the e3ect of shear force, the equilibrium equation in the meridional direction within segment AB is given by dNx Rt =− sin ; d r
(3)
where is an angular coordinate (0 6 6 0 ); is the hoop stress and Nx is the membrane force per unit circumferential length along the meridional direction. The development of the radius of the tube can be written, from the geometry of the model, as r = r0 + R(1 − cos ):
(4)
Substituting Eq. (4) into Eq. (3), the meridional equilibrium equation is re-written as −
YRt sin dNx = : d r0 + R(1 − cos )
(5)
The meridional force at A and B can be obtained by considering the equilibrium of BC and AC separately corresponding to one of the n strips. Note the circumferential width of the strip at B and C is equal, which is assumed to be the same as that at A. From the equilibrium of BC where = 0 , the meridional membrane force per unit circumferential length at the crack tip B is − NxB = N sin( − 0 ) + N cos( − 0 );
(6)
where N is normal force per unit circumferential length at the contact point C. From the equilibrium of AC, the meridional membrane force at A where = 0 can be expressed by − NxA = N sin + N cos :
(7)
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From Eqs. (5)–(7), the normal force per unit length is found as nt Yt ln 1 + 2r 0 : N= sin − sin( − 0 ) + [cos − cos( − 0 )]
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(8)
The cross-section of a curved strip is shown in Fig. 9 with the corresponding stress distribution. The full plastic bending moment for this strip normalized with respect to the strip arc width is =n =2n nYr0 t 2 − cos d + d cos cos − cos Mp = 2 2n 2n =2n 0 nYr0 t = 2 sin : (9) − sin 2n n Let the bending moment at the crack tip be equal to this full plastic bending moment, NR sin( − 0 ) − NR[1 − cos( − 0 )] = Mp : Combining Eqs. (8) and (10), the curl radius is given as 2r02 sin − sin( − 0 ) + [cos − cos( − 0 )] 2 sin − sin : R= sin( − 0 ) − [1 − cos( − 0 )] 2n n t
(10)
(11)
By introducing the following non-dimensional parameters =
2r0 ; t
!=
2R ; t
(12)
the above equations can be re-written as !=
sin − sin( − 0 ) + [cos − cos( − 0 )] 2 2 sin − sin ; sin( − 0 ) − [1 − cos( − 0 )] 2n n
(13)
where 0 = cos−1 (1 − n=!). This is an implicit non-linear equation. Numerical results in terms of ! and n are shown in Figs. 10(a) – (c) for = 45◦ ; 60◦ and 75◦ , respectively. For small values of n; ! decreases rapidly. ! increases slowly when n is large. 3.3. Energy absorption analysis The applied force can be calculated approximately from energy balance. The rate of external work is equated to that of the plastic energy dissipated in plastic bending/stretching, the “near-tip” tearing energy associated with tube splitting and the frictional work between the die and the tube. When the tube is moving downward (whilst cracks are propagating) with a speed v, the energy balance leads to Fv = W˙P + W˙T + W˙F ;
(14)
where W˙P ; W˙T and W˙F denote the rate of energy dissipation in plastic bending, tearing and friction, respectively.
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Fig. 10. The correlation between curl radius, !, and crack number, n when: (a) = 45◦ ; (b) = 60◦ ; and (c) = 75◦ , respectively.
According to the assumed kinematics of the problem all plastic bending/stretching is conAned to the curved strips with a transversal radius r0 bending with a radius R in meridional direction. The rate of plastic bending becomes 2r0 Mp v; W˙P = (15) R + r0 1 − cos 2n where R + r0 (1 − cos =2n) is the bending radius of the neutral axis (refer to Fig. 9). Substituting Eq. (9) into Eq. (15), the rate of plastic bending dissipation is 2 sin 2n − sin n 2 v: W˙P = 2nYr0 t (16) R + r0 1 − cos 2n
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The tearing energy is involved in plastic work within the near-tip zone, which is predominantly dissipated by a di3used mode. It is mainly contributed by plastic circumferential stretching ahead of crack tip in the present consideration. Hence, the rate of tearing energy dissipation is determined by integration of the product of the elemental circumferential stress and strain rates over the volume of the deforming region; that is, ˙ WT = ˙ dV: (17) v
Thus, using Eq. (2), the rate of tearing energy dissipation is expressed as W˙T = Ynt 2 v:
(18)
Friction is a notable energy dissipation mechanism in this system. The rate of energy dissipated by frictional force is W˙ f = 2r0 Nv;
(19)
where N is the normal force per unit length, and it may be related to the applied force F and the die semi-angle, as follows: N=
F : 2r0 (sin + cos )
Therefore, the applied force required to split and curl the tube at any slow speed is − sin n 2nYr02 t 2 sin 2n sin + cos 2 + nYt : F= sin − (1 − cos ) R + r0 1 − cos 2n
(20)
(21)
By introducing a non-dimensional force f=
F ; 2r0 tY
the above equation can be recast into 2 sin 2n − sin n sin + cos n + 2 f= : sin − (1 − cos ) ! + 1 − cos 2n
(22)
(23)
Clearly, the applied force is a function of both the non-dimensional curl radius (!) and the crack number (n). Using the relationship between the curl radius and the crack number (Eq. (13), the applied force can be expressed as a function of the crack number only. Accordingly, a minimum energy approach suggests that stable cracks will propagate when the number of cracks is such that the total force renders a minimum, i.e. df = 0: dn
(24)
A small computer program has been written to calculate the crack number, and then this value is substituted into Eq. (13) to obtain the curl radius and into Eq. (23) to obtain the applied force.
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4. Numerical results and discussion 4.1. Crack number Numerical results of the crack number are shown for all the cases in Fig. 11. The experimentally observed values are also plotted with symbols. The latter seems undistinguishable for dies of di3erent semi-angles, but the test values are within the theoretical curves for the semi-angles considered. The theoretical results show that a die with larger semi-angle produces more cracks, and a relative thick tube has fewer cracks. In the experiments, only integer crack numbers occur, although the predictions are continuous lines. In practice, competition between plastic bending and fracture only takes place between the present crack number and adjacent numbers. The number of cracks formed during axial splitting of ductile metal tubes has been previously investigated by Atkins [27] after the experimental work conducted by Reddy and Reid [23]. Using a minimum energy approach, Atkins suggested that stable cracks would propagate as a result of competition between plastic @ow and fracture. Thus, in addition to the tube dimensions, three material properties are relevant to crack numbers, namely, the strength, the toughness and the hoop fracture strain. However, application of Eq. (14) in Ref. [27] is possible only when an accurate knowledge of the strength-to-toughness ratio is available, since the number of cracks is very sensitive to this value. 4.2. Curl radius The curl radius is another main parameter of this energy absorption system. The correlation between the curl radius and crack number was shown in Figs. 10(a) – (c). For a given die semi-angle () and dimensions of the tube, once the crack number is determined uniquely, the curl radius can be assessed. Figs. 12(a) – (c) show the variations of non-dimensional curl radius versus the ratio of the tube diameter to the thickness, for di3erent semi-angles = 45◦ ; = 60◦ and = 75◦ , respectively.
Fig. 11. Comparison between predicted crack number and experiments (error bar, if any, indicates the range of the values observed).
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Fig. 12. Variation of the non-dimensional curl radius, !, for a die with semi-angle: (a) = 45◦ ; (b) = 60◦ ; and (c) = 75◦ , respectively.
The theoretical predications are shown with solid lines and the experimental values are also marked with symbols. A general agreement between the predictions and experiments is obtained. It is seen that the curl radius increases with the ratio of the tube diameter to the thickness. Theoretically, yield stress does not a3ect the non-dimensional curl radius, which agrees with the experiments for =45◦ and 75◦ , though results for =60◦ may somehow suggest otherwise. The curl radius depends signiAcantly on the die semi-angle as shown in Fig. 13 for a typical tube with = 31. 4.3. Applied force The applied force relates directly the capacity of energy absorption. When the curl radius and the crack number are determined theoretically as above, the applied force can be calculated according
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Fig. 13. Variation of the non-dimensional curl radius, !, with die semi-angle for a typical tube = 31.
to Eq. (23). This is plotted against the ratio of the tube diameter to the thickness in Figs. 14(a) – (c) for dies with a semi-angle = 45◦ ; 60◦ and 75◦ , respectively. A general agreement between the theoretical prediction and experiments is obtained for all cases. Relatively thick tubes require a larger force because of larger plastic deformation and/or tearing energy, although the number of cracks may be fewer. Another factor which has a signiAcant e3ect on the applied force is the die semi-angle. The force almost linearly increases with the die semi-angle in the range of 30◦ –90◦ , as shown in Fig. 15 for a typical tube with = 31. 4.4. Energy absorption e9ciency In order to compare the energy absorption capability of circular tubes having di3erent material properties and dimensions, the energy per unit mass, w, is deAned. It is a convenient design parameter where weight is an important consideration. If the entire tube is available for deformation, the energy per unit mass is written as follows: fY w= ; (25) ' where material density ' is 7860 kg=m3 for steel and 2710 kg=m3 for aluminium, respectively. These experimental values of w are shown in Fig. 16. Aluminium tubes dissipate a larger amount of the energy per unit mass than steel ones when ¡ 40. 4.5. Friction and energy partition The e3ect of friction on the curl radius and the applied force can be investigated theoretically for a typical tube with = 31, as shown in Figs. 17(a) – (c), respectively. It can be seen that with a change in the coe?cient of friction from 0.2 to 0.6, the crack number increases by 1 or 2, and the
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Fig. 14. Variation of the non-dimensional applied force, f, with for a die semi-angle: (a) = 45◦ ; (b) = 60◦ ; and (c) = 75◦ , respectively.
curl radius decreases slightly. However, the applied force increases signiAcantly. So the friction is favourable not only in frictional work but also in plastic bending and fracture energies. Its e3ect on the applied force comes from the combination of these factors. In this system, the total energy is dissipated by three mechanisms: plastic bending/stretching, tearing and friction. The contribution of each component may be assessed to guide the design of this kind of energy absorbers as shown in Figs. 18(a) – (c) for = 45◦ ; 60◦ and 75◦ , respectively. These three energy fractions are almost invariable with the ratio of diameter to thickness, and the fracture energy and the plastic bending energy are prevailing. The fraction of fracture energy increases with the semi-angle of the die, and the portion of the fracture energy is over half of the total energy when = 75◦ .
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Fig. 15. Variation of the non-dimensional force, f, with die semi-angle for a typical tube = 31.
Fig. 16. Comparison of the speciAc energy, w, for tubes with di3erent materials.
4.6. Comparison with Reddy and Reid’s experiments [23] The present theory may also be applied to the experiments by Reddy and Reid [23]. In their experiments, the deformation of curls was guided by a radius die. Hence, for dies of a small radius, the curl radius may be assumed to be the same as the die radius. It is also assumed that the normal resultant force of the pressure between the tube and the radius die acts at 45◦ to the horizontal. Neglecting the increase of bending radius due to the position of the neutral axis, the force applied may be predicated as, from the energy balance alone, 2 1+ n √ f= + : (26) 1 − ( 2 − 1) 8n2 !
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Fig. 17. Theoretical investigation about e3ects of friction on: (a) crack number; (b) curl radius; and (c) applied force.
From df=dn = 0,
2 1=3 : n= 4!
(27)
Substituting Eq. (27) into Eq. (26) and taking the friction coe?cient = 0:2, the force applied may be re-written as
2 1=3 : (28) f = 1:23 ! The coe?cient would be 1.58 when =0:4. Eqs. (27) and (28) are plotted against the non-dimensional die (or curl) radius as shown in Figs. 19(a) and (b) for the crack number and non-dimensional force, respectively. Because a series of materials were involved in their experiments, three values of the
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Fig. 18. Partition of three energy components for: (a) = 45◦ ; (b) = 60◦ ; and (c) = 75◦ .
critical opening displacement parameter, , are used in these plots. From the comparison in n and f, it would suggest that annealing enhance the ductility of both aluminium and mild steel tubes with a larger value of : about 2 for annealed aluminium tubes and 0:5 ∼ 1:0 for the rest. In these cases, the tearing energy dominates the total dissipated energy and is twice as much as the plastic bending energy. 4.7. A @nal remark The analytical model presented is based on the observation that, for a given tube and die dimensions, the number of cracks seems a characteristic number independent of the number of initial pre-cuts. When this number is larger, the tearing energy increases, but the bending/stretching energy of the curved strips decreases due to a reduced full plastic bending moment. Conversely, a smaller
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Fig. 19. Comparison with Reid and Reddy’s experiments with radius dies [23] (tubes with D = 49:2 mm and t = 1:6 mm): (a) crack number n (note that two points overlap at 10 for ! = 12:5); and (b) non-dimensional force f.
number of cracks lead to a larger plastic energy, though the total tearing energy is smaller. The number of cracks minimizes the total force; this forms the basis of our theoretical analysis. However, this analysis is approximate and it should be noted that such a minimization procedure may lead to unrealistic results, as Calladine has demonstrated [28]. It would be interesting to obtain more information on the detailed strain Aeld within the doubly curved deforming zone and to attempt to explain the number of cracks without invoking a force/energy minimization procedure. 5. Conclusion The energy absorption capacity has been investigated experimentally and theoretically for circular metal tubes which axially split and curled with three conical dies. This collapse mode results in a
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long stroke and a steady force after initial @uctuations. The crack number, curl radius and applied force are properly predicted by an approximate analysis. The numerical results indicate the e3ect of the tube dimensions, the semi-angle of the die and the friction between the tube and the die. Three energy mechanisms are involved: plastic bending energy, tearing energy and frictional work in this type of energy absorbers. Partition of these three components of energies provides guidelines for a designer. Formulae for number of cracks and force are given for tubes with a radius die. Acknowledgements The authors wish to thank the Australian Research Council for the Anancial support to undertake this work. The authors would also like to thank Professor S.R. Reid for a helpful discussion on the topic between him and the third author (T.X. Yu) during the preparation of this manuscript. References [1] Johnson W, Reid SR. Metallic energy dissipating systems. Applied Mechanics Review 1978;31:277–88. [2] Johnson W, Reid SR. Update to: metallic energy dissipating systems. Applied Mechanics Update 1986;39:315–9. [3] Alexander JM. An approximate analysis of collapse of thin-walled cylindrical shells under axial loading. Quarterly Journal of Mechanics and Applied Mathematics 1960;13:10–5. [4] Abramowicz W, Jones N. Dynamic axial crushing of circular tubes. International Journal of Impact Engineering 1984;2:263–81. [5] Abramowicz W, Jones N. Dynamic progressive buckling of circular and square tubes. International Journal of Impact Engineering 1986;4:243–69. [6] Grzebieta RH. An alternative method for determining the behaviour of round stocky tubes subjected to axial crush loads. Thin-Walled Structures 1990A;9:66–89. [7] Wierzbicki T, Bhat SU, Abramowicz W, Brodkin D. Alexander revisited—a two folding element model of progressive crushing of tubes. International Journal of Solids and Structures 1992;29:3269–88. [8] Singace AA, Elsobky H, Reddy TY. On the eccentricity factor in the progressive crushing of tubes. International Journal of Solids and Structures 1995;32:3589–602. [9] Singace AA, Elsobky H. Further experimental investigation on the eccentricity factor in the progressive crushing of tubes. International Journal of Solids and Structures 1996;33:3517–38. [10] Guillow SR, Lu G, Grzebieta RH. Quasi-static axial compression of thin-walled circular aluminium tubes. International Journal of Mechanical Sciences 2001;43(9):2103–23. [11] Huang X, Lu G. Axisymmetric progressive crushing of circular tubes. International Journal of Crashworthiness, in press. [12] Reddy TY. Tube inversion-an experiment in plasticity. International Journal of Mechanical Engineering Education 1989;17:277–91. [13] Reid SR, Harrigan JJ. Transient e3ects in the quasi-static and dynamic internal inversion and nosing of metal tubes. International Journal of Mechanical Sciences 1998;40(2–3):263–80. [14] Stronge WJ, Yu TX, Johnson W. Long stroke energy dissipation in splitting tubes. International Journal of Mechanical Sciences 1983;25(9 –10):637–47. [15] Stronge WJ, Yu TX, Johnson W. Energy dissipation by curling tubes. In: Morton J, editor. Structural impact and crashworthiness, vol. 2. London: Elsevier, 1984. p. 576 –87. [16] Lu G, Huang X, Yu TX. Axial splitting of square tubes. In: Zhao XL, Grzebieta RH, editors. Structural failure and plasticity. Oxford: Pergamon, 2000. p. 457–62. [17] Huang X, Lu G, Yu TX. Energy absorption in splitting square metal tubes. Thin-Walled Structures 2002;40:153–65. [18] Lu G, Ong LS, Wang B, Ng HW. An experimental study on tearing energy in splitting square metal tubes. International Journal of Mechanical Sciences 1994;36(12):1087–97.
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[19] Lu G, Fan H, Wang B. An experimental method for determining ductile tearing energy of thin metal sheets. Metals and Materials 1998;4(3):432–5. [20] Fan H, Wang B, Lu G. On the tearing energy of a ductile thin plate. International Journal of Mechanical Sciences 2002;44:407–21. [21] Yu TX, Zhang DJ, Zhang Y, Zhou Q. A study of the quasi-static tearing of thin metal sheets. International Journal of Mechanical Sciences 1988;30:193–202. [22] Ezra AA, Fay RJ. An assessment of energy absorbing devices for prospective use in aircraft impact situations. In: Herrmann G, Perone N, editors. Dynamic behaviour of structures. London: Pergamon, 1972. p. 225–46. [23] Reddy TY, Reid SR. Axial splitting of circular metal tubes. International Journal of Mechanical Sciences 1986;28(2):111–31. [24] Calladine CR. Theory of shell structures. Cambridge: Cambridge University Press, 1983. [25] Parks DM, Freund LB, Rice JR. Running ductile fracture in a pressurized line pipe. Mechanics of crack growth, ASTM STP 590. American Society for Testing and Materials, Philadelphia, 1976. p. 2543– 62. [26] Wierzbicki T, Thomas P. Closed-form solution for wedge cutting force through thin metal sheets. International Journal of Mechanical Sciences 1993;35:209–29. [27] Atkins AG. On the number of cracks in the axial splitting of ductile metal tubes. International Journal of Mechanical Sciences 1987;29:115–21. [28] Calladine CR. Some problems in propagating plasticity. Symposium on Plasticity and Impact Mechanics, I.I.T. Delhi, 11–14 December, 1993.
International Journal of Mechanical Sciences 44 (2002) 2369 – 2391
On the axial splitting and curling of circular metal tubes X. Huanga , G. Lua;∗ , T.X. Yub a b
School of Engineering and Science, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, People’s Republic of China Received 27 May 2002; received in revised form 18 November 2002; accepted 26 November 2002
Abstract The present paper investigates the axial splitting and curling behaviour of circular metal tubes. Mild steel and aluminum circular tubes were pressed axially onto a series of conical dies each with di3erent semi-angle. By pre-cutting eight 5 mm slits which were distributed evenly at the lower end of each tube, the tube split axially and the strips curled outward. Experiments showed that this mechanism results in a long stroke and a steady load. An approximate analysis is presented which successfully predicts the number of propagated cracks, the curling radius and the force applied. This analysis takes into account ductile tearing of the cracks, plastic bending/stretching and friction. E3ects of tube dimensions, semi-angle of the die and friction are discussed in detail. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Circular tube; Conical die; Splitting; Curling; Energy absorption
1. Introduction Energy absorbing devices are employed where collision may cause serious consequences, so as to protect human as well as other important structures [1,2]. Thin-walled structures are e?cient in absorbing energy and widely used as structural members and energy absorbing devices. Thin-walled tubes have attracted much attention owing to the wide range of deformation modes which can be generated. Circular tubes may be axially crushed progressively in an axisymmetric or a non-axisymmetric (or called diamond) mode by two @at plates. It is a very e?cient energy absorption arrangement. Alexander [3] was the Arst researcher to propose a theory for the axisymmetric collapse mode based on a balance of external and internal work done. Afterwards, Abramowicz and Jones [4,5], Grzebieta ∗
Corresponding author. Fax: +61-3-9214-8264. E-mail address: [email protected] (G. Lu).
0020-7403/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0020-7403(02)00191-1
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[6], Wierzbicki et al. [7] and Singace et al. [8,9] further developed a model to predict the crush force. Recently, Guillow et al. [10] and Huang and Lu [11] reAned this work experimentally and theoretically. Another experimental arrangement is that a tube may be compressed against a shaped die, where several di3erent types of collapse mode can be established. Within a compatible range of parameters, an inversion mode has been observed. This has been studied by Reddy [12] and Reid and Harrigan [13]. Another collapse mode is splitting and curling of tubes. From the viewpoint of energy absorption, this collapse mode has a long stroke of over 90 per cent of the total tube length. Stronge et al. [14,15] conducted experiments with square tubes split against a radius/@at die. Huang et al. [16,17] further studied the splitting and curling behaviour of square tubes axially compressed between a plate and a pyramidal die. Lu et al. [18] investigated the tearing energy involved in splitting square metal tubes. Tearing energy in thin metal sheets was studied by Lu et al. [19,20] and Yu et al. [21], among others. In a comparative study of energy absorbers for improving occupant survivability in aircraft crashes, Ezra and Fay [22] identiAed circular tube splitting and curling as an e?cient system based on speciAc energy dissipation. Reddy and Reid [23] studied the splitting behaviour of circular tubes compressed axially between a plate and a radius die. The advantage of this mechanism lies not only in having a long stroke but also a steady crush force. In the present study, splitting and curling behaviour of circular tubes is further investigated experimentally and theoretically. Circular metal tubes were axially compressed between a plate and a series of conical dies which are di3erent from those used in Ref. [23]. Tubes were observed to have a number of cracks propagating along the axial direction. The strips so formed by cracks rolled up into curls with an almost constant radius. The crush force became steady after some initial @uctuations. Three energy dissipation mechanisms were involved: (1) the “near-tip” tearing associated with tube splitting; (2) the “far-Aeld” deformation associated with the plastic bending and stretching of curls; (3) the friction as the tube interacted with the die. A correlation between the crack number and the curl radius is found through a force analyses. An approximate analysis is performed to predict the force at the steady stage. As a result, the crack number is predicted independently by the minimum energy approach involving a competition between the plastic bending and fracture energy. Thereby, the curl radius and the applied force are determined using the predicted crack number. The numerical results are compared with the present experiments in detail. The analysis is also applied to the previous experiments and equations for the number of cracks and force are obtained. 2. Experiments All the experiments were performed in a Shimadzu Universal testing machine under quasi-static conditions. The experimental set-up is sketched in Fig. 1. A cone-shaped die was Axed to the bottom bed of the testing machine. A short cylindrical mandrel was used inside the tube to prevent the tube from tilting. The axes of the die, tube and testing machine were carefully aligned. The cross-head of the testing machine then pressed the tube onto the conical die at a constant rate of 0:0333 mm=s. Three di3erent semi-angles (); 45◦ ; 60◦ and 75◦ , were selected for the conical die. All dies were made from mild steel and heat-treated to increase their surface hardness. All the specimens
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Fig. 1. Sketch of the experimental set-up, with 8 evenly spaced 5 mm initial saw-cuts around lower circumference. Table 1 Summary of tests and theoretical results for mild steel tubes Test
D (mm)
t (mm)
(◦ )
ni
nf
Fmax1 (kN)
Fmax2 (kN)
Favg: (kN)
Ravg: (mm)
CS1 CS2 CS3 CS4 CS5 CS6 CS7 CS8 CS9 CS10 CS11 CS12
49.4 49.4 49.4 61.9 61.9 61.9 74.0 74.0 74.0 47.2 47.2 47.2
1.6 1.6 1.6 1.6 1.6 1.6 2.0 2.0 2.0 3.3 3.3 3.3
30.9 30.9 30.9 38.7 38.7 38.7 37.0 37.0 37.0 14.3 14.3 14.3
45 60 75 45 60 75 45 60 75 45 60 75
8 8 8 8 8 8 8 8 8 3 4 4
8 8 8 8 8 8 8 8 8 3 4 4
14.6 24.9 73.7 16.7 28.1 50.3 29.5 51.3 83.9 — — 83.0
23.7 31.8 39.6 24.2 33.8 44.4 35.3 39.5 51.8 71.5 76.8 95.3
14.2 16.9 21.9 16.0 19.8 24.8 23.3 27.0 33.8 38.8 48.8 60.0
12.2 8.1 6.5 13.6 8.3 6.5 17.9 11.5 8.3 29.8 12.6 11.2
tested were commercially available circular tubes of 200 mm long. The ratio of the diameter over the thickness ranges from 15 to 50. The detailed dimensions of all the specimens are listed in Table 1 for the mild steel tubes and Table 2 for the aluminium tubes. In order to establish the split and curl mode while preventing other collapse modes, initial 5 mm saw-cuts for all specimens were made
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Table 2 Summary of tests and theoretical results for aluminium tubes Test
D (mm)
t (mm)
(◦ )
ni
nf
Fmax1 (kN)
Fmax2 (kN)
Favg: (kN)
Ravg: (mm)
AL1 AL2 AL3 AL4 AL5 AL6 AL7 AL8 AL9 AL10 AL11 AL12 AL13 AL14 AL15 AL16 AL17 AL18
48.25 48.25 48.25 47.0 47.0 47.0 57.75 57.75 57.75 56.8 56.8 56.8 77.9 77.9 77.9 76.95 76.95 76.95
1.3 1.3 1.3 3.0 3.0 3.0 1.85 1.85 1.85 3.2 3.2 3.2 1.95 1.95 1.95 2.85 2.85 2.85
37.1 37.1 37.1 15.7 15.7 15.7 31.2 31.2 31.2 17.8 17.8 17.8 39.9 39.9 39.9 27.0 27.0 27.0
45 60 75 45 60 75 45 60 75 45 60 75 45 60 75 45 60 75
9 9 10 5 6 7 8 8 8 6 7 7 8 8 8 8 8 8
9 9 10 5 5 5 8 8 8 6 6 5 8 8 8 8 7 7
4.4 6.7 15.7 14.6 27.9 49.5 8.9 15.8 24.6 19.7 33.9 62.3 12.1 19.5 43.2 19.0 30.3 60.5
5.3 5.7 9.7 18.4 21.9 — 11.7 12.9 15.5 — — — 12.0 11.9 15.0 — — —
3.3 4.0 6.9 14.3 18.8 23.8 8.8 10.8 12.1 18.0 24.5 32.0 8.5 11.0 13.0 15.6 20.0 27.5
13.9 9.4 6.6 28.7 16.3 10.6 16.3 10.9 8.0 28.7 16.6 12.2 21.6 15.1 10.9 23.8 17.8 12.6
Fig. 2. Average stress–strain curves of the both materials.
which were evenly spaced around the lower circumference. A total of eight saw-cuts was made in each tube. Stress–strain relationships for the two materials were obtained from standard coupon tests for each thickness. The curves for a given material were close and their average is shown in Fig. 2. These two curves are approximated by a rigid, perfectly plastic relationship with @ow stress Y = 450 MPa for mild steel, and Y = 180 MPa for aluminium. In the beginning of a typical test, the strips between initial saw-cuts buckled and @ared as guided by the die. This led to the circumferential stretching of the tube. When this extension had reached
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Fig. 3. Typical force-displacement curves for mild steel tubes with D = 74:0 mm; t = 1:8 mm against dies with semi-angle = 45◦ ; 60◦ and 75◦ , respectively.
a certain level, cracks occurred at some initial saw-cut locations and propagated along the axial direction due to continuous ductile tearing. The strips so formed by the cracks rolled up into curls as the end of these strips was free to bend. These curls have an almost constant radius (R denotes the middle radius of the curl), the value of which mainly depends on the semi-angle of the die and the dimensions of the tube. When these curls had completed one revolution, the front edges of the curls contacted the tube wall. This caused the radius of the next roll to be R + t due to the thickness of the wall, t. In the following sections, the curl radius refers to the Arst curl unless noted otherwise. When the crushing distance reached 120 mm (out of the original total length 200 mm), the test was stopped. In some tests there was scu?ng and ploughing of the tubes, especially in the aluminium ones, as they were pressed onto the dies. This resulted in debris stuck to the die after the removal of the tube, so the die was cleaned with a cloth after each test. Typical force-compression curves for mild steel tubes (D = 76:0 mm and t = 2:0 mm) with three di3erent dies are shown in Fig. 3 and the corresponding specimen photographs after tests are shown in Fig. 4 with side and end views. In these three cases, cracks were parallel to the tube axis without branching or merging. The force initially increased with the cross-head movement until it reached the Arst peak, Fmax1 , which corresponded to the onset of inversion of strips from the initial saw-cuts. A second peak force, Fmax2 , then occurred, and this corresponded to the initiation of cracks. After approximately another 10 mm displacement the force reached a steady state. This force Favg remained almost constant, and is used for later analyses. When the front edge of the Arst revolution touched the tube wall, the applied force surged notably before dropping to another steady state, which corresponded to the second revolution. Because less plastic bending energy was dissipated due to the increment of the radius of the second revolution, the average force at the second steady stage was a little lower than that of the Arst steady state. Fig. 5 shows typical force-compression curves for the aluminium tubes pressed onto three di3erent semi-angled dies. Corresponding specimen photographs after tests are shown in Fig. 6. The aluminium tubes had similar deformation modes to those of the mild steel tubes, except that the average force
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(a)
(b)
(c)
Fig. 4. Photographs of typical mild steel specimens (D=74:0 mm; t =1:8 mm) after tests: from left to right CS7 (=45◦ ), CS8 ( = 60◦ ) and CS9 ( = 75◦ ).
Fig. 5. Typical force-displacement curves for aluminium tubes with D = 77:9 mm; t = 1:9 mm against dies with semi-angle = 45◦ ; 60◦ and 75◦ , respectively.
was at the same lever before and after the front edges of the curls touched the tube wall. Probably, the decrease in the applied force due to the increasing radius of the next roll was o3set by an increase in friction between the tube and the inside mandrel. When the strips of tubes began to curl up, branching or merging of cracks was observed in some experiments. That is, in some cases, cracks which developed from initial saw-cuts ceased to propagate; but in some other cases, one original crack split into two. Two typical tubes with branching or merging of cracks after tests are shown in Fig. 7, where the initial number of saw-cuts
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(a)
(b)
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(c)
Fig. 6. Photographs of typical aluminium specimens (with D = 77:9 mm; t = 1:9 mm) after tests: from left to right Al13 ( = 45◦ ), Al14 ( = 60◦ ) and Al15 ( = 75◦ ).
(a)
(b)
Fig. 7. Photographs of typical specimens after tests: (a) with crack merging (test CS12 for the mild steel tube with D = 47:2 mm; t = 3:3 mm and die semi-angle = 75◦ ); and (b) with crack branching (test AL3 for the aluminium tube with D = 48:25 mm; t = 1:3 mm and die semi-angle = 75◦ ).
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Fig. 8. Comparison of the applied force with and without lubrication for mild steel tubes with D = 74:0 mm; t = 1:8 mm and die semi-angle = 60◦ .
was 8. The numbers of cracks after initial branching or merging (ni ) are given in Tables 1 and 2 (Note they are not the initial saw-cut numbers). For aluminium tubes, further merging of cracks was also observed when the strips produced subsequent revolutions. On the contrary, the mild steel tubes kept the original cracks to the end of tests. The Anal crack numbers (nf ) are also given in Tables 1 and 2. These branching and merging of cracks resulted in a little @uctuation in the applied force. After a durable re-orientation of these cracks, the strips formed by the cracks tended to have the same width. Reddy and Reid [23] pointed out that friction is present as a multiplying factor of the applied force in splitting and curling tubes. In order to obtain a reasonable frictional factor in the present cases, a typical tube was also tested under lubricated conditions. Fig. 8 shows, however, that the di3erence in the applied force under the conditions with and without lubrication was minor. One possible reason is that the frictional factor for the present series of tests without lubrication is also very small. In later analysis, a frictional coe?cient = 0:2 is adopted for all the cases. Detail experimental results such as the average applied force (Favg: ) and average curl radius (Ravg: ) are also listed in Table 1 for the mild steel tubes and Table 2 for the aluminium ones. 3. Theoretical analysis 3.1. Analytical model and basic assumptions Consider a tube with diameter D and thickness t pressed axially and split against a die with a semi-angle . A simple kinematic model is proposed as shown in Fig. 9. Point A is the starting point of the curl, AC, with a doubly curved face of radius R in the axial direction and r0 , neglecting the increment of the tube radius in the circumferential direction (as shown in Fig. 9). The crack tip is located at B with an angle 0 as deAned in Fig. 9. C denotes the contact point between the curl
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Fig. 9. A kinematic model for splitting and curling of a circular tube.
and the die. A number of other simplifying assumptions are made to facilitate an e3ective analysis, as given below. (a) The material is regarded as rigid, perfectly plastic with an average @ow stress. This @ow stress has the same value in both the bending and membrane deformations. There is no interaction between the resultant membrane force and the bending moments in yielding. (b) There is no variation in the tube thickness during collapse. From experimental observation there is an abrupt change in the meridional curvature near the 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, see Eq. (10) later. (c) Plastic work in curling is calculated by considering meridional bending of a curved strip with a transversal radius r0 . Changes in transversal radius are neglected. (d) The e3ect of out-of-plane shear stress is neglected. (e) Fracture of the tube is caused by circumferential stretching of the tube wall. A critical crack opening displacement (COD) parameter is adopted in assessing the crack initiation and tearing energy. The Anal deformed strips are clearly doubly curved with a non-zero Gaussian curvature while originally they are cylindrical and thus of a zero Gaussian curvature. This change in Gaussian curvature requires stretching/compression of the surface [24]. Here, assumption c above leads to a stretching in the meridional direction only. Fracture of the tube due to ductile tearing may be conveniently described in terms of the COD criterion. The COD approach was identiAed as a possible way in characterizing fracture properties of ductile sheets [25,26]. The crack tip of the tube shown in Fig. 9 is not at A but at B due to the ductility of the material; the ligament holds these assumed strips together until a critical separation or crack opening displacement is reached. Clearly the parameter depends on the tube thickness,
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the fracture strain of the material as well as the stress distribution. In the present analysis, we use the non-dimensional critical separation = =t as a parameter and take = 1:0 unless speciAed otherwise. Thus, the crack tip angle 0 can be uniquely determined from the geometry of the problem and the value of , as follows: nt −1 0 = cos ; (1) 1− 2R where n is the number of cracks (or strips). 3.2. Correlation between curl radius and number of cracks The wall of the tube curls up into a constant radius, R, due to the bending moment resulting from the applied force F. The deformed cylindrical tube ahead of the crack tip (AC) has a gradually increasing radius circumferentially. The development of the tube radius r is deAned in Fig. 9. Thus the circumferential strain can be estimated as r = − 1; (2) r0 where r0 is the initial radius of the tube. Neglecting the e3ect of shear force, the equilibrium equation in the meridional direction within segment AB is given by dNx Rt =− sin ; d r
(3)
where is an angular coordinate (0 6 6 0 ); is the hoop stress and Nx is the membrane force per unit circumferential length along the meridional direction. The development of the radius of the tube can be written, from the geometry of the model, as r = r0 + R(1 − cos ):
(4)
Substituting Eq. (4) into Eq. (3), the meridional equilibrium equation is re-written as −
YRt sin dNx = : d r0 + R(1 − cos )
(5)
The meridional force at A and B can be obtained by considering the equilibrium of BC and AC separately corresponding to one of the n strips. Note the circumferential width of the strip at B and C is equal, which is assumed to be the same as that at A. From the equilibrium of BC where = 0 , the meridional membrane force per unit circumferential length at the crack tip B is − NxB = N sin( − 0 ) + N cos( − 0 );
(6)
where N is normal force per unit circumferential length at the contact point C. From the equilibrium of AC, the meridional membrane force at A where = 0 can be expressed by − NxA = N sin + N cos :
(7)
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From Eqs. (5)–(7), the normal force per unit length is found as nt Yt ln 1 + 2r 0 : N= sin − sin( − 0 ) + [cos − cos( − 0 )]
2379
(8)
The cross-section of a curved strip is shown in Fig. 9 with the corresponding stress distribution. The full plastic bending moment for this strip normalized with respect to the strip arc width is =n =2n nYr0 t 2 − cos d + d cos cos − cos Mp = 2 2n 2n =2n 0 nYr0 t = 2 sin : (9) − sin 2n n Let the bending moment at the crack tip be equal to this full plastic bending moment, NR sin( − 0 ) − NR[1 − cos( − 0 )] = Mp : Combining Eqs. (8) and (10), the curl radius is given as 2r02 sin − sin( − 0 ) + [cos − cos( − 0 )] 2 sin − sin : R= sin( − 0 ) − [1 − cos( − 0 )] 2n n t
(10)
(11)
By introducing the following non-dimensional parameters =
2r0 ; t
!=
2R ; t
(12)
the above equations can be re-written as !=
sin − sin( − 0 ) + [cos − cos( − 0 )] 2 2 sin − sin ; sin( − 0 ) − [1 − cos( − 0 )] 2n n
(13)
where 0 = cos−1 (1 − n=!). This is an implicit non-linear equation. Numerical results in terms of ! and n are shown in Figs. 10(a) – (c) for = 45◦ ; 60◦ and 75◦ , respectively. For small values of n; ! decreases rapidly. ! increases slowly when n is large. 3.3. Energy absorption analysis The applied force can be calculated approximately from energy balance. The rate of external work is equated to that of the plastic energy dissipated in plastic bending/stretching, the “near-tip” tearing energy associated with tube splitting and the frictional work between the die and the tube. When the tube is moving downward (whilst cracks are propagating) with a speed v, the energy balance leads to Fv = W˙P + W˙T + W˙F ;
(14)
where W˙P ; W˙T and W˙F denote the rate of energy dissipation in plastic bending, tearing and friction, respectively.
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Fig. 10. The correlation between curl radius, !, and crack number, n when: (a) = 45◦ ; (b) = 60◦ ; and (c) = 75◦ , respectively.
According to the assumed kinematics of the problem all plastic bending/stretching is conAned to the curved strips with a transversal radius r0 bending with a radius R in meridional direction. The rate of plastic bending becomes 2r0 Mp v; W˙P = (15) R + r0 1 − cos 2n where R + r0 (1 − cos =2n) is the bending radius of the neutral axis (refer to Fig. 9). Substituting Eq. (9) into Eq. (15), the rate of plastic bending dissipation is 2 sin 2n − sin n 2 v: W˙P = 2nYr0 t (16) R + r0 1 − cos 2n
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The tearing energy is involved in plastic work within the near-tip zone, which is predominantly dissipated by a di3used mode. It is mainly contributed by plastic circumferential stretching ahead of crack tip in the present consideration. Hence, the rate of tearing energy dissipation is determined by integration of the product of the elemental circumferential stress and strain rates over the volume of the deforming region; that is, ˙ WT = ˙ dV: (17) v
Thus, using Eq. (2), the rate of tearing energy dissipation is expressed as W˙T = Ynt 2 v:
(18)
Friction is a notable energy dissipation mechanism in this system. The rate of energy dissipated by frictional force is W˙ f = 2r0 Nv;
(19)
where N is the normal force per unit length, and it may be related to the applied force F and the die semi-angle, as follows: N=
F : 2r0 (sin + cos )
Therefore, the applied force required to split and curl the tube at any slow speed is − sin n 2nYr02 t 2 sin 2n sin + cos 2 + nYt : F= sin − (1 − cos ) R + r0 1 − cos 2n
(20)
(21)
By introducing a non-dimensional force f=
F ; 2r0 tY
the above equation can be recast into 2 sin 2n − sin n sin + cos n + 2 f= : sin − (1 − cos ) ! + 1 − cos 2n
(22)
(23)
Clearly, the applied force is a function of both the non-dimensional curl radius (!) and the crack number (n). Using the relationship between the curl radius and the crack number (Eq. (13), the applied force can be expressed as a function of the crack number only. Accordingly, a minimum energy approach suggests that stable cracks will propagate when the number of cracks is such that the total force renders a minimum, i.e. df = 0: dn
(24)
A small computer program has been written to calculate the crack number, and then this value is substituted into Eq. (13) to obtain the curl radius and into Eq. (23) to obtain the applied force.
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4. Numerical results and discussion 4.1. Crack number Numerical results of the crack number are shown for all the cases in Fig. 11. The experimentally observed values are also plotted with symbols. The latter seems undistinguishable for dies of di3erent semi-angles, but the test values are within the theoretical curves for the semi-angles considered. The theoretical results show that a die with larger semi-angle produces more cracks, and a relative thick tube has fewer cracks. In the experiments, only integer crack numbers occur, although the predictions are continuous lines. In practice, competition between plastic bending and fracture only takes place between the present crack number and adjacent numbers. The number of cracks formed during axial splitting of ductile metal tubes has been previously investigated by Atkins [27] after the experimental work conducted by Reddy and Reid [23]. Using a minimum energy approach, Atkins suggested that stable cracks would propagate as a result of competition between plastic @ow and fracture. Thus, in addition to the tube dimensions, three material properties are relevant to crack numbers, namely, the strength, the toughness and the hoop fracture strain. However, application of Eq. (14) in Ref. [27] is possible only when an accurate knowledge of the strength-to-toughness ratio is available, since the number of cracks is very sensitive to this value. 4.2. Curl radius The curl radius is another main parameter of this energy absorption system. The correlation between the curl radius and crack number was shown in Figs. 10(a) – (c). For a given die semi-angle () and dimensions of the tube, once the crack number is determined uniquely, the curl radius can be assessed. Figs. 12(a) – (c) show the variations of non-dimensional curl radius versus the ratio of the tube diameter to the thickness, for di3erent semi-angles = 45◦ ; = 60◦ and = 75◦ , respectively.
Fig. 11. Comparison between predicted crack number and experiments (error bar, if any, indicates the range of the values observed).
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Fig. 12. Variation of the non-dimensional curl radius, !, for a die with semi-angle: (a) = 45◦ ; (b) = 60◦ ; and (c) = 75◦ , respectively.
The theoretical predications are shown with solid lines and the experimental values are also marked with symbols. A general agreement between the predictions and experiments is obtained. It is seen that the curl radius increases with the ratio of the tube diameter to the thickness. Theoretically, yield stress does not a3ect the non-dimensional curl radius, which agrees with the experiments for =45◦ and 75◦ , though results for =60◦ may somehow suggest otherwise. The curl radius depends signiAcantly on the die semi-angle as shown in Fig. 13 for a typical tube with = 31. 4.3. Applied force The applied force relates directly the capacity of energy absorption. When the curl radius and the crack number are determined theoretically as above, the applied force can be calculated according
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Fig. 13. Variation of the non-dimensional curl radius, !, with die semi-angle for a typical tube = 31.
to Eq. (23). This is plotted against the ratio of the tube diameter to the thickness in Figs. 14(a) – (c) for dies with a semi-angle = 45◦ ; 60◦ and 75◦ , respectively. A general agreement between the theoretical prediction and experiments is obtained for all cases. Relatively thick tubes require a larger force because of larger plastic deformation and/or tearing energy, although the number of cracks may be fewer. Another factor which has a signiAcant e3ect on the applied force is the die semi-angle. The force almost linearly increases with the die semi-angle in the range of 30◦ –90◦ , as shown in Fig. 15 for a typical tube with = 31. 4.4. Energy absorption e9ciency In order to compare the energy absorption capability of circular tubes having di3erent material properties and dimensions, the energy per unit mass, w, is deAned. It is a convenient design parameter where weight is an important consideration. If the entire tube is available for deformation, the energy per unit mass is written as follows: fY w= ; (25) ' where material density ' is 7860 kg=m3 for steel and 2710 kg=m3 for aluminium, respectively. These experimental values of w are shown in Fig. 16. Aluminium tubes dissipate a larger amount of the energy per unit mass than steel ones when ¡ 40. 4.5. Friction and energy partition The e3ect of friction on the curl radius and the applied force can be investigated theoretically for a typical tube with = 31, as shown in Figs. 17(a) – (c), respectively. It can be seen that with a change in the coe?cient of friction from 0.2 to 0.6, the crack number increases by 1 or 2, and the
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Fig. 14. Variation of the non-dimensional applied force, f, with for a die semi-angle: (a) = 45◦ ; (b) = 60◦ ; and (c) = 75◦ , respectively.
curl radius decreases slightly. However, the applied force increases signiAcantly. So the friction is favourable not only in frictional work but also in plastic bending and fracture energies. Its e3ect on the applied force comes from the combination of these factors. In this system, the total energy is dissipated by three mechanisms: plastic bending/stretching, tearing and friction. The contribution of each component may be assessed to guide the design of this kind of energy absorbers as shown in Figs. 18(a) – (c) for = 45◦ ; 60◦ and 75◦ , respectively. These three energy fractions are almost invariable with the ratio of diameter to thickness, and the fracture energy and the plastic bending energy are prevailing. The fraction of fracture energy increases with the semi-angle of the die, and the portion of the fracture energy is over half of the total energy when = 75◦ .
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Fig. 15. Variation of the non-dimensional force, f, with die semi-angle for a typical tube = 31.
Fig. 16. Comparison of the speciAc energy, w, for tubes with di3erent materials.
4.6. Comparison with Reddy and Reid’s experiments [23] The present theory may also be applied to the experiments by Reddy and Reid [23]. In their experiments, the deformation of curls was guided by a radius die. Hence, for dies of a small radius, the curl radius may be assumed to be the same as the die radius. It is also assumed that the normal resultant force of the pressure between the tube and the radius die acts at 45◦ to the horizontal. Neglecting the increase of bending radius due to the position of the neutral axis, the force applied may be predicated as, from the energy balance alone, 2 1+ n √ f= + : (26) 1 − ( 2 − 1) 8n2 !
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Fig. 17. Theoretical investigation about e3ects of friction on: (a) crack number; (b) curl radius; and (c) applied force.
From df=dn = 0,
2 1=3 : n= 4!
(27)
Substituting Eq. (27) into Eq. (26) and taking the friction coe?cient = 0:2, the force applied may be re-written as
2 1=3 : (28) f = 1:23 ! The coe?cient would be 1.58 when =0:4. Eqs. (27) and (28) are plotted against the non-dimensional die (or curl) radius as shown in Figs. 19(a) and (b) for the crack number and non-dimensional force, respectively. Because a series of materials were involved in their experiments, three values of the
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Fig. 18. Partition of three energy components for: (a) = 45◦ ; (b) = 60◦ ; and (c) = 75◦ .
critical opening displacement parameter, , are used in these plots. From the comparison in n and f, it would suggest that annealing enhance the ductility of both aluminium and mild steel tubes with a larger value of : about 2 for annealed aluminium tubes and 0:5 ∼ 1:0 for the rest. In these cases, the tearing energy dominates the total dissipated energy and is twice as much as the plastic bending energy. 4.7. A @nal remark The analytical model presented is based on the observation that, for a given tube and die dimensions, the number of cracks seems a characteristic number independent of the number of initial pre-cuts. When this number is larger, the tearing energy increases, but the bending/stretching energy of the curved strips decreases due to a reduced full plastic bending moment. Conversely, a smaller
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Fig. 19. Comparison with Reid and Reddy’s experiments with radius dies [23] (tubes with D = 49:2 mm and t = 1:6 mm): (a) crack number n (note that two points overlap at 10 for ! = 12:5); and (b) non-dimensional force f.
number of cracks lead to a larger plastic energy, though the total tearing energy is smaller. The number of cracks minimizes the total force; this forms the basis of our theoretical analysis. However, this analysis is approximate and it should be noted that such a minimization procedure may lead to unrealistic results, as Calladine has demonstrated [28]. It would be interesting to obtain more information on the detailed strain Aeld within the doubly curved deforming zone and to attempt to explain the number of cracks without invoking a force/energy minimization procedure. 5. Conclusion The energy absorption capacity has been investigated experimentally and theoretically for circular metal tubes which axially split and curled with three conical dies. This collapse mode results in a
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long stroke and a steady force after initial @uctuations. The crack number, curl radius and applied force are properly predicted by an approximate analysis. The numerical results indicate the e3ect of the tube dimensions, the semi-angle of the die and the friction between the tube and the die. Three energy mechanisms are involved: plastic bending energy, tearing energy and frictional work in this type of energy absorbers. Partition of these three components of energies provides guidelines for a designer. Formulae for number of cracks and force are given for tubes with a radius die. Acknowledgements The authors wish to thank the Australian Research Council for the Anancial support to undertake this work. The authors would also like to thank Professor S.R. Reid for a helpful discussion on the topic between him and the third author (T.X. Yu) during the preparation of this manuscript. References [1] Johnson W, Reid SR. Metallic energy dissipating systems. Applied Mechanics Review 1978;31:277–88. [2] Johnson W, Reid SR. Update to: metallic energy dissipating systems. Applied Mechanics Update 1986;39:315–9. [3] Alexander JM. An approximate analysis of collapse of thin-walled cylindrical shells under axial loading. Quarterly Journal of Mechanics and Applied Mathematics 1960;13:10–5. [4] Abramowicz W, Jones N. Dynamic axial crushing of circular tubes. International Journal of Impact Engineering 1984;2:263–81. [5] Abramowicz W, Jones N. Dynamic progressive buckling of circular and square tubes. International Journal of Impact Engineering 1986;4:243–69. [6] Grzebieta RH. An alternative method for determining the behaviour of round stocky tubes subjected to axial crush loads. Thin-Walled Structures 1990A;9:66–89. [7] Wierzbicki T, Bhat SU, Abramowicz W, Brodkin D. Alexander revisited—a two folding element model of progressive crushing of tubes. International Journal of Solids and Structures 1992;29:3269–88. [8] Singace AA, Elsobky H, Reddy TY. On the eccentricity factor in the progressive crushing of tubes. International Journal of Solids and Structures 1995;32:3589–602. [9] Singace AA, Elsobky H. Further experimental investigation on the eccentricity factor in the progressive crushing of tubes. International Journal of Solids and Structures 1996;33:3517–38. [10] Guillow SR, Lu G, Grzebieta RH. Quasi-static axial compression of thin-walled circular aluminium tubes. International Journal of Mechanical Sciences 2001;43(9):2103–23. [11] Huang X, Lu G. Axisymmetric progressive crushing of circular tubes. International Journal of Crashworthiness, in press. [12] Reddy TY. Tube inversion-an experiment in plasticity. International Journal of Mechanical Engineering Education 1989;17:277–91. [13] Reid SR, Harrigan JJ. Transient e3ects in the quasi-static and dynamic internal inversion and nosing of metal tubes. International Journal of Mechanical Sciences 1998;40(2–3):263–80. [14] Stronge WJ, Yu TX, Johnson W. Long stroke energy dissipation in splitting tubes. International Journal of Mechanical Sciences 1983;25(9 –10):637–47. [15] Stronge WJ, Yu TX, Johnson W. Energy dissipation by curling tubes. In: Morton J, editor. Structural impact and crashworthiness, vol. 2. London: Elsevier, 1984. p. 576 –87. [16] Lu G, Huang X, Yu TX. Axial splitting of square tubes. In: Zhao XL, Grzebieta RH, editors. Structural failure and plasticity. Oxford: Pergamon, 2000. p. 457–62. [17] Huang X, Lu G, Yu TX. Energy absorption in splitting square metal tubes. Thin-Walled Structures 2002;40:153–65. [18] Lu G, Ong LS, Wang B, Ng HW. An experimental study on tearing energy in splitting square metal tubes. International Journal of Mechanical Sciences 1994;36(12):1087–97.
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[19] Lu G, Fan H, Wang B. An experimental method for determining ductile tearing energy of thin metal sheets. Metals and Materials 1998;4(3):432–5. [20] Fan H, Wang B, Lu G. On the tearing energy of a ductile thin plate. International Journal of Mechanical Sciences 2002;44:407–21. [21] Yu TX, Zhang DJ, Zhang Y, Zhou Q. A study of the quasi-static tearing of thin metal sheets. International Journal of Mechanical Sciences 1988;30:193–202. [22] Ezra AA, Fay RJ. An assessment of energy absorbing devices for prospective use in aircraft impact situations. In: Herrmann G, Perone N, editors. Dynamic behaviour of structures. London: Pergamon, 1972. p. 225–46. [23] Reddy TY, Reid SR. Axial splitting of circular metal tubes. International Journal of Mechanical Sciences 1986;28(2):111–31. [24] Calladine CR. Theory of shell structures. Cambridge: Cambridge University Press, 1983. [25] Parks DM, Freund LB, Rice JR. Running ductile fracture in a pressurized line pipe. Mechanics of crack growth, ASTM STP 590. American Society for Testing and Materials, Philadelphia, 1976. p. 2543– 62. [26] Wierzbicki T, Thomas P. Closed-form solution for wedge cutting force through thin metal sheets. International Journal of Mechanical Sciences 1993;35:209–29. [27] Atkins AG. On the number of cracks in the axial splitting of ductile metal tubes. International Journal of Mechanical Sciences 1987;29:115–21. [28] Calladine CR. Some problems in propagating plasticity. Symposium on Plasticity and Impact Mechanics, I.I.T. Delhi, 11–14 December, 1993.