Expansion of circular tubes by rigid tubes as impact energy absorbers: experimental and theoretical investigation

Expansion of circular tubes by rigid tubes as impact energy absorbers: experimental and theoretical investigation M Shakeri, S Salehghaffari and R. Mirzaeifar Department of Mechanical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran doi:10.1080/13588260701483540 Abstract: In this study, expansion of deformable tubes by a rigid tube is introduced as a new mechanism of dissipating energy. In this mechanism of dissipating energy, there is a specific clearance between the surfaces of the rigid and deformable tubes, and the rigid tube is press-fitted onto the top end of the deformable one up to 30 mm. when this arrangement of dissipating energy is subjected to axial compression, the rigid tube is driven into the deformable one; consequently, the impact energy is absorbed by the plastic expansion energy of the deformable tube and the frictional energy at the contact interface between rigid and deformable tubes. Experimental, numerical and analytical study of this process under axially quasi-static loading is presented in this paper. Through the experimental and theoretical studies, major crashworthiness parameters in design are identified. The influence of friction coefficient between the surfaces of deformable and rigid tubes on the value of mean crush load is studied, and typical expansion modes of deformation that may occur during axial compression are characterized. Also, expansion load-displacement history and mechanics of the expansion process as an impact energy absorber are studied. It is shown that this energy absorption method has high crush force efficiency and favourable crashworthiness characteristics both in uniform and non-uniform loading conditions. Key words: Circular tubes, expansion, friction, axial compression, energy absorber. INTRODUCTION Energy-absorbing devices are employed to protect human and important structures where collision may cause serious damages [1, 2]. Thin-walled structures are efficient in absorbing energy and widely used as energy-absorbing devices. Metallic cylindrical tubes have attracted much attention due to their high stiffness and strength combined with a low weight and wide range of deformation which may be generated when they are subjected to axial loading [3]. The common method for dissipating the kinetic energy by using circular tubes is crushing of these tubes between two flat plates in which the damaging kinetic energy is dissipated by converting to the energy that is used for plastic deformation of tubes. Depending on geometrical or Corresponding Author: M Shakeri Department of Mechanical Engineering Amirkabir University of Technology (Tehran Polytechnic) Tehran, Iran Tel: +98-21-66405844; Fax: +98-21-66419736 E-mail: shakeri@cic.aut.ac.ir physical properties of the tube, or other parameters like load condition, axial progressive collapsing of thin-walled tubes between flat plates may happen in different modes such as axisymmetric, diamond and mixed modes. Despite the efficiency of this energy-absorbing mechanism, it has low crush force efficiency, and its crashworthiness parameters are strongly sensitive to external parameters like load condition. Investigation of this process has been performed by several authors, some of which are summarized by Jones in [4]. Another energy-absorbing device may be introduced by compressing a thin-walled tube against a shaped die, where several different types of collapse modes can be established. Within a specific range of geometrical dimensions for tube and die, the inversion mode occurs. This method has low stroke efficiency because the maximum length of the tube that contributes to dissipation of the energy is half of the tube length. Like the progressive collapsing, the crashworthiness parameters in inversion mode are sensitive to external parameters. A significant feature of this mode of deformation is the constant inversion force that can be obtained for a uniform tube. Ried [5, 6], Reddy [7, 8], Harrigan [9], Gupta [10] and several other researchers have studied internal and external inversion of a tube under axial load. Copyright C Taylor & Francis Group 493 pp. 493 501

M Shakeri, S Salehghaffari and R Mirzaeifar Table 1 Static true stress-strain relationship for the material of deformable tube εe P 0 0.0151 0.0332 0.042 0.061 0.139 0.190 σ e, N / mm 2 235 262 293 311 322 337 354 Figure 1 (a) Deformable tube; (b) rigid tube; (c) assembled arrangement. Another mechanism of absorbing energy by circular tubes is splitting and curling of cylindrical tubes against canonical dies. In this case, a great percent of the tube length contributes to dissipation of the external energy and the collapsing force is relatively steady, but the mean reaction force is low, and again, the performance of the shock absorber is affected by external parameters such as load direction. This phenomenon of absorbing energy is studied by several authors such as Stronge [11], Huang [12, 13] and Reddy [14]. In this paper, expansion of deformable tubes by a rigid tube under axial compression is introduced as a very efficient process for absorbing impact energy, which has not been taken into consideration so far. In this proposed method of energy absorption, two cylindrical tubes with different material properties, where one of these tubes is considered solid compared with the other one, are used. There is also a specific clearance between the surfaces of the rigid and deformable tubes, and the rigid tube is pressfitted onto the top end of the deformable one up to 30 mm by hydraulic press. Figure 1 shows the shape of both rigid and deformable tubes as well as their final assembling. When this arrangement is subjected to axial compression, the rigid tube is driven into the deformable one. Consequently, the impact energy is absorbed by the plastic expansion of the deformable tube (because of the clearance between rigid and deformable tubes) and the frictional energy between them. In this study, the axially quasi-static compression of the discussed shock absorber is simulated numerically by 494 using an implicit FE code (ABAQUS 6.4). For experimental validation of FE model, some compression tests are established. From the observations of numerical and experimental studies, expansion force-displacement history and mechanics of expansion process as impact energy absorbers are established. Also, the influence of friction coefficient at the contact interface between the deformable and rigid tubes on the value of mean crush load is well understood, and typical expansion modes of deformation that may occur during axial loading are fully characterized. An analytical model to predict the mean crush load of the energy absorber under axial compression is proposed; moreover, results of the analytical model are compared with those of FE models and experimental tests. It is shown that design of this new energy absorption arrangement is not as sensitive as inversion and axisymmetric crushing of tubes, which have been realized as collapsible energy absorbers so far. Also, this device has a considerably high crush force efficiency and safety factor; with consideration of major parameters in design, obtaining a wide range of constant crush loads, which may be required to protect the crashworthy system subjected to axial impact loading, is possible. Therefore, various designs with different values of mean crush load can be investigated by the designers. EXPERIMENTS: DESCRIPTION AND RESULTS Energy absorber specimens are composed of two different cylindrical tubes. In this study, the first one is called rigid tube and the other one is called deformable tube. In all specimens, the rigid tube is made from mild steel and heattreated to increase its surface hardness, and its upper edge is rounded by turning machine to the radius of 7.25 mm. The deformable tube is also made from steel alloy with internal diameter of 85 mm and thickness of 3.5 mm, and its top area is expanded by special tools. In order to obtain the material data of the deformable tube, a quasi-static material test is performed on a strip cut from it, using a standard tensile test machine and the resulting stress-strain relationship is shown in Table 1. The elastic modulus of this material is E = 210 Gpa and its density ρ = 7800 Kg/m 3.Itisassumedthatthemechanical properties of this steel alloy are not sensitive to strain rate at room temperature. In all specimens, there is a clearance with value of 5 mm between the external surface of the rigid tube and internal surface of the deformable one, and the rounded edge of the rigid tube is press-fitted onto the expanded area of the deformable one up to 30 mm by an axial compression. Copyright C Taylor & Francis Group

Expansion of circular tubes by rigid tubes as impact energy absorbers: experimental and theoretical investigation Table 2 Summary of test results for all experiment specimens Test no. Surface treatment µ L (mm) α(degree) P m (KN) P max (KN) B11 Blasting 0.25 190 37.27 209.79 226.48 B12 Blasting 0.25 190 37.43 209.39 227.65 B13 Blasting 0.25 190 37.11 210.10 224.32 N11 No Surface treatment 0.1 190 41.73 199.56 216.83 N12 No Surface treatment 0.1 190 41.26 199.78 215.10 N13 No Surface treatment 0.1 190 41.12 199.81 212.68 C11 Coating 0.05 190 43.19 192.86 204.67 C12 Coating 0.05 190 43.25 192.50 205.34 C13 Coating 0.05 190 43.76 191.15 208.72 N21 No Surface treatment 0.1 128 165.26 239.12 N22 No Surface treatment 0.1 142 174.67 228.40 N23 No Surface treatment 0.1 136 166.26 233.54 Figure 2 (a) Sketch of the experimental set-up number one; (b) Sketch of the experimental set-up number two. Also, the lengths of the rigid and deformable tubes in all specimens are 200 mm. All the experiment specimens were compressed in a universal testing machine with a computer controller under quasi-static condition. The latter was used to display the relationship between the holder stroke and the holder load. In order to compare the energy absorption characteristics of the shock absorber for uniform and non-uniform loading conditions, two different experimental set-ups, which are called experimental set-up number one and number two, are established. In the experimental set-up number one, the applied load is uniformly distributed along the edge of the rigid tube, while in the other experimental set-up; the applied load is not uniformly distributed. The schematic of these different experimental set-ups is sketched in Figure 2. As shown in this figure, in both of these two experimental set-ups, the stationary base of the testing machine is in contact with the bottom edge of the deformable tube, while its moving cross-head is pressing the rigid tube into the deformable one at a constant rate of 1.2 mm/s. Also, it can be seen that in experimental set-up number one, the cross-head of the testing machine is in contact with the collar part, which is assembled on the bottom area of the rigid tube to prevent it from tilting during the compression test, while in the another experimental set-up, a rigid plate is placed between the one side of the collar part and the cross-head of the testing machine and is attached to this moving cross-head. As a result, in this experimental set-up, the load is applied only to one side of the rigid tube and is not uniformly distributed along its edge, as it is in contact with the cross-head of the testing machine. A total of 12 specimens were tested: nine specimens were performed in experimental set-up number one and three specimens were performed in experimental set-up number two. To study the influence of the value of friction coefficient between the surfaces of the rigid and deformable tubes on energy-absorbing characteristics of the shock absorber, their surfaces are blasted in three specimens that are performed in experimental set-up number one and also are coated in three other specimens. The corresponding value of friction coefficient of these different surface treatments are listed in Table 2. All specimens were numbered according to the type of its surface treatment, number of its experimental set-up and number of the same test. For example, B21 indicates a shock absorber with blasting surface treatment and experimental set-up number two as the first test. C13 means a shock absorber with coating surface treatment and experimental set-up number one as the third (repeated) test. N22 also indicates a specimen with no surface treatment and experimental set-up number two as the second (repeated) test. The shape of some experiment specimens after compression tests is shown in Figure 3. Also, the test results for all specimens are listed in Table 2. These results show the favourable characteristics of the shock absorber from the viewpoint of energy absorption both in uniform and non-uniform loading conditions. Also, in this study they are used for verifying the theoretical studies of the expansion process of the deformable tube. It should be noted that in each repeated group of experimental tests, the average values of their results are considered to compare with results of analytical and FE models. Copyright C Taylor & Francis Group 495

M Shakeri, S Salehghaffari and R Mirzaeifar Figure 3 Photograph of typical specimens after tests: (a) N11; (b) B13; (c) N21. POSSIBLE EXPANSION MODES OF DEFORMATION Experimental results show that expansion of the deformable tube may occur in axisymmetic or nonaxisymmetic modes for uniform and non-uniform loading conditions. Figure 3(a) and (b) show the axisymmetic and non-axisymmetic expansion modes of specimens with no surface treatment, respectively. From this figure, it is observed that in axisymmetic expansion mode of deformation, both sides of the deformable tube are expanded almost equally to the size of clearance between deformable and rigid tubes, while in non-axisymmetic mode, only one side of the deformable tube is expanded, and the value of expansion in this mode of deformation is almost twice as much as the value of clearance between the deformable and rigid tubes. Furthermore, comparison between the crashworthiness parameters of axisymmetic and non-axisymmetic expansion modes of specimens performed in both experimental set-ups one and two, for example, specimen numbers N11 and N21 (see Table 2), shows that the value of the stroke and crush force efficiency and also mean crush load in non-axisymmetic state is lower than that in the axisymmetic state (see Table 2). Nevertheless, the characteristics of energy absorption of the shock absorber in non-axisymmetic mode are favourable. In fact, when the axial applied load is non-uniform, none of the introduced mechanisms of absorbing energy by tabular parts, such as axisymmetic crushing, inversion and splitting and curling of the cylindrical tubes, can perform as efficiently as the mechanism of the shock absorber in the present study. ANALYTICAL MODEL Consider a deformable tube with internal diameterd 1 and thickness t, expanded by the rigid tube to the new expansion diameter D 2 ; a simple kinematical model of this process is proposed as shown in Figure 4, and also a number of simplifying assumptions are made to facilitate the analysis, as given below. (1) The material is regarded as rigid, perfectly plastic with an average flow stress. (2) There is no variation in tube thickness during expansion process. (3) Horizontal and vertical axes are considered as the principal axes of stress and strain. (4) Coulomb friction is considered. By taking into consideration the selected element of Figure 4 and writing the equilibrium equation for it in the Figure 4 A kinematical model for expansion of circular tube by the rigid one. 496 Copyright C Taylor & Francis Group

Expansion of circular tubes by rigid tubes as impact energy absorbers: experimental and theoretical investigation vertical direction, we will have: (σ a + d σ a )(D + dd) πt σ a Dπt ( + P π D dx ) ( sin α + µp π D dx ) cos α cos α cos α = 0 (1) Disregarding the term of higher order in Eq. (1) and substituting dd = 2tanαdx, which is derived from the geometry of model, and also with a bit of simplifying, we will have: d σ a dd + σ a D + 1 ( ) µ + tan α P = 0 (2) t 2tanα The equilibrium equation in the horizontal direction for the element shown in Figure 4 is also given by: σ r = P (1 µ tan α) (3) Using Tresca yield criterion and simplifying assumption number 3, we will have: σ a σ r = σ o (4) Substituting Eq. (3) and Eq. (4) into Eq. (2), the equilibrium equation in the vertical direction is re-written as: d σ a dd + σ a D 1 ( ) µ + tan α σ a t 2tanα (1 µ tan α) = 1 ( ) µ + tan α σ o (5) t 2tanα (1 µ tan α) Introducing parameter A as follows: ( ) µ + tan α A = (6) 2tanα (1 µ tan α) And substituting this parameter into Eq. (5), this equation is simplified as follows: ( d σ a 1 dd + D A ) σ a = A t t σ o (7) It should be noted that the value of longitudinal stress in expanded diameter of deformable tube (D = D 2 ) is zero. Using this boundary condition, the solution of Eq. (7) is given by: ( σ a = 1 + t ) [( D2 σ o AD D + t ) ] e A t (D D 2) AD Substituting D = D 1 into Eq (8), the value of longitudinal stress in the unexpanded part of the deformable tube during expansion process is derived from the following expression: ( σ d = 1 + t ) [( D2 + σ o AD 1 D 1 t AD 1 ) (8) ] e A(D 1 D 2) t (9) Finally, the value of mean load required to expand the deformable tube by the rigid one is derived from the following expression: P m = [πt (D 1 + t) σ o ] {( 1 + t ) AD 1 [( D2 D 1 + t AD 1 ) ]} e A(D 1 D 2 ) t (10) Equation (10) indicates that the applied force is a function of the unknown parameter of expansion angle of deformation α. According to the minimum energy approach, which suggests that the stable expansion process propagates when the value of expansion angle of deformation is such that the total force is rendered to a minimum, this unknown parameter is derived from this expression: dp m d α = 0 (11) A computer program has been written to calculate the unknown value of expansion angle of deformation α from Eq. (11); then this calculated value is substituted into Eq. (9) to obtain the mean applied force. FINITE ELEMENT SIMULATION A finite element model has been proposed to simulate the expansion process of the deformable tube by the rigid one under axial compression for all specimens performed in experimental set-up number one. The numerical simulation of this event was carried out by using the FE code ABAQUS/Standard, version 6.4. Expansion of circular tube by the rigid one is a case of axisymmetric deformation; therefore, 2D-axisymmetric solid elements CAX8 (eight nodes biquadratic) are used. The schematic diagram of this proposed finite element model is shown in Figure 5. The deformable tube is modeled with 180 elements in the longitudinal direction and four elements through the tube thickness. The rigid tube is assembled on the top of the deformable tube, and a fixed rigid plate is placed under the bottom of the deformable tube. The rigid tube is pressed down at a constant velocity of 1.2mm/; thus, under this loading condition, no inertial effects on forming mechanism are likely to occur and no dynamics effects in deformation mechanics are needed to be taken into account. The material behavior of the deformable tube is modeled with an elastic-plastic material model using data from Table 1, which is based on the results of tension test. Isotropic strain hardening is also assumed for the material of the deformable tube. Contact algorithms, including friction, are activated to simulate contact between the surfaces of the rigid and deformable tubes as well as the contact between the deformable tube and the rigid plate. The value of friction coefficient between the rigid and deformable tubes is considered 0.25, 0.05 and 0.1 for specimens with blasting, coating and no surface treatment, respectively. Also, Copyright C Taylor & Francis Group 497

M Shakeri, S Salehghaffari and R Mirzaeifar to 1%. This numerical simulation was performed on a Pentium PC 2.8 GHz, and the typical simulation case took 2300 s. VERIFICATION OF FE MODELLING Figure 5 Proposed finite element model used for computational studies. this value is assumed 0.3 for the contacting surfaces of the deformable tube and the rigid plate. For analysis purposes, the total compression process was divided into a large number of small steps called increments. The incremental strain in each step was restricted The end part of the deformable tube is bent when the expansion process is completed. Figure 6 shows the true shape and FE simulation of this end part of the deformable tube for specimen number C12 after expansion process. As a measure of comparison between true and computed shape of the deformable tube after expansion process, the value of the bending angle of the end part of the deformable tube for each experimented specimen is chosen. These values, resulting from experimental tests and FE modelling, are listed in Table 3. These results show that the predicted FE and true shape of the deformable tube after expansion process are almost the same. Also, a comparison of the computed and observed forcedisplacement history for specimen number B11 during expansion process is shown in Figure 7. From this figure, it is observed that the predicted force-displacement curve is found to be somewhat smaller than the actual values in the initial stage of the expansion process. During the remainder of the expansion process, the predicted values tend to be slightly higher than the actual values. The value of mean crush load and maximum load derived from experimental and predicted force-displacement curves of the specimens with different surface treatment Figure 6 Shape of the end part and the leading edge of the deformable tube of specimen number C12 after expansion process, deriving from FE simulation and experimental test. 498 Copyright C Taylor & Francis Group

Expansion of circular tubes by rigid tubes as impact energy absorbers: experimental and theoretical investigation Table 3 Comparisons of test results with those of analytical and FE models α (Degree) P m (KN) P max (KN) η (%) µ Test FE Analytic Test FE Analytic Test FE Test FE 0.25 37.27 37.17 38.18 209.76 210.23 214.41 226.15 224.92 92.75 93.47 0.1 41.37 40.40 42.27 199.72 201.62 196.64 214.87 212.80 92.95 94.75 0.05 43.40 42.23 44.12 192.17 192.54 189.40 205.24 202.56 93.63 95.05 Figure 7 Comparison of experimental and computed force-displacement history for specimen number B11. each value of friction coefficient. Due to the previous discussion, these minimum values of load and corresponding expansion angle of deformation should be considered as a true answer to the problem. These values are given in Table 3 to compare with the results of FE models and the experimental tests. This table shows that there is a good agreement between analytical, experimental and FE modelling. It is worth mentioning that the value of expansion angle of deformation during expansion process and the bending angle of the end part of the deformable tube when expansion process is completed correspond in this study. From Figure 8, it also can be seen that, in a certain range of expansion angle of deformation, for each value of friction coefficient, the values of mean crush load are constant and closely equal to its minimum value. Furthermore, it is observed that this certain range of expansion angle of deformation is shortened by increasing the value of friction coefficient. These observations confirm that the proposed analytical model describes the expansion process of the deformable tube very well. ESSENTIAL FEATURE OF DEFORMATION Figure 8 Plots of mean crush load against expansion angle of deformation for different values of friction coefficient, based on analytic Eq. (10). are presented in Table 3. It is found that the computed and experimental values match very well. VERIFICATION OF ANALYTIC MODEL By using Eq. (10), which is proposed to calculate the required mean crush load for expansion of the deformable tube by the rigid one analytically, the plots of mean crush load against expansion angle of deformation for different values of the friction coefficient between the surfaces of the deformable and rigid tubes are shown in Figure 8. As is realized from this figure, the plot of mean load against expansion angle of deformation has a minimum value for To understand the progress of expansion process of the deformable tube under axial compression, FE simulations of the specimen with blasting surface treatment are considered for this study. Figure 9 depicts the deformed profile of the deformable tube for this specimen at twenty stages of expansion process. In this figure, the rigid tube is moved down up to 9.5 mm in each stage compared with the previous stage. Also, the corresponding force-displacement history of this specimen is shown in Figure 7. From the observation of this figure, it is understood that expansion process can be divided into two broad phases, namely the unsteady phase and the steady phase. During the unsteady or transient phase, the force-displacement curve constantly changes, while during the steady phase it becomes more or less constant. The unsteady part continues up to point B, while the steady phase is started from this point (see Figure 7). In the initial stages of the expansion process of the deformable tube, as the rigid tube moves down, the leading edge of the deformable tube bends (see part (2) of Figure 9). During this stage, the axial load increases steeply and reaches a peak point (point A in Figure 7). Then in the next stage of the expansion process, unbending takes place at the leading edge of the deformable tube while the rigid Copyright C Taylor & Francis Group 499

M Shakeri, S Salehghaffari and R Mirzaeifar Figure 9 Different stages of expansion process of specimen with blasting surface treatment, based on FE simulation. tube moves down further by axial compression (see part (3) of Figure 9); hence, the axial load starts declining and falls down from point A to point B in Figure 7. It is worth noticing that the unsteady part of the load ends at this stage of the process (see Figure 7). As the compression of the rigid tube continues, the curved profile formed on the upper area of the deformable tube, which is caused by bending and unbending of the leading edge of the deformable tube at the previous stages of deformation, tends towards a straight profile (see part (4) and (5) of Figure 9). At these stages, the value of load is approximately stable from point B to C in Figure 7. Finally, with further compression, this curved profile is converted to a straight profile (see part (6) of Figure 9). This phenomenon increases the compression between the rigid and deformable tubes; therefore, at this stage of the deformation, load is increasing from point C to D in Figure 7. During the remainder of the expansion process, the value of axial load is constant and is closely equal to its value at point D (see Figure 7). In these stages of the expansion process, the leading edge of deformable tube is bent inward, and its shape remains unchanging (see part (7) to part (21) of Figure 9). Details of the shape of this unchanging leading edge resulting from FE modelling and experimental test are shown in Figure 6. Figure 10 (a) Schematic representation of the expansion of the deformable tube at first stage of deformation; (b) expansion of deformable tube at next stage. along circumferential direction, and (3) friction between the surfaces of the deformable and rigid tubes during axial compression. Bending takes place at point B, where the leading edge of rigid tube first contacts the surface of the deformable tube, while unbending takes place at point A, where the rigid tube moves down from its previous position (see Figure 10). Stretching develops while the rigid tube is gradually moved down from its previous position to the next position (refer to AB and B C in Figure 10). ENERGY ABSORPTION MECHANISM This study also indicates that during expansion process of the deformable tube in the steady phase, plastic expansion of the deformable tube is mainly caused by three different mechanisms: (1) bending/unbending, (2) stretching 500 EFFECT OF FRICTION COEFFICIENT The variations of plastic, frictional and total absorbed energy of all specimens during axial compression versus the value of friction coefficient between deformable and rigid tubes, resulting from FE simulation, are shown in Copyright C Taylor & Francis Group

Expansion of circular tubes by rigid tubes as impact energy absorbers: experimental and theoretical investigation Figure 11 Variation of total, plastic and frictional absorbed energy with friction coefficient. In this study, expansion of the deformable tube by the rigid tube is introduced as a very efficient energy-absorbing arrangement for a crashworthy system. This proposed mechanism of absorbing energy results in high crush force efficiency and a constant load. Also, because of the uniformity of the modes of deformation, its crashworthiness parameters are not sensitive to external parameters like load condition. In this study, numerical simulations of expansion process of the deformable tube under axially quasi-static loading were performed; also, an analytical model to predict the mean crush load of the shock absorber was proposed. Comparisons were made between the experimentally observed characteristics and theoretical predictions. There was a good agreement between analytical, numerical and experimental studies. In this mechanism of absorbing energy, mean crush force can be mainly affected by the material and thickness of deformable tube, the value of friction coefficient and the value of clearance between the surfaces of the deformable and rigid tubes. However, it is only the effect of friction coefficient on energy-absorbing behaviors of the shock absorber that is studied in this paper. It is shown that the value of mean crush load is considerably affected by the friction coefficient at the contact interface between the deformable and rigid tubes; in fact, the design of the proposed shock absorber of this study based on the different required mean crush loads is very versatile and controllable. REFERENCES Figure 12 Partition of two energy components for all specimens. Figure 11. This figure indicates that the value of mean crush load and total absorbed energy of the specimens increases considerably with the value of friction coefficient. In fact, the increasing of both plastic and frictional absorbed energies with friction coefficient is the main cause of this event. However, it is worth noticing that the value of the plastic energy is little affected by the friction coefficient; theoretically, the decreasing of the expansion angle of deformation of the deformable tube during expansion process with friction coefficient (see Table 3) causes a small increasing of the plastic expansion energy. Also the contribution of each component of energy absorption, plastic expansion of the deformable tube and the frictional energy, against the friction coefficient is shown in Figure 12. This figure indicates that the percentage of frictional absorbed energy increases linearly with the value of the friction coefficient. CONCLUSION 1. W Johnson and S R Ried, Metallic energy dissipating systems, Appl Mech Rev, 1978 31 277 288. 2. W Johnson and S R Ried, Update to: metallic energy dissipating systems, Appl Mech Update, 1986 39 315 319. 3. A A A Alghamdi, Collapsible impact energy absorbers: an overview, Thin-Walled Struct, 2001 39 189 213. 4. N Jones, Structural Impact, Cambridge, Cambridge University Press, 1989. 5. S R Reid, Plastic deformation mechanisms in axially compressed metal tubes as impact energy absorbers, Int J Mech Sci, 1993 35 1035 1052. 6. S R Reid and J J Harrigan, Transient effects in the quasi static and dynamic inversion and nosing of metal tubes, Int J Mech Sci, 1998 40 263 280. 7. T Y Reddy, Tube inversion an experiment in plasticity, Intl Jl Mech Eng Educ, 1989 17 277 291. 8. A Colokoglu and T Y Reddy, Strain rate and inertial effects in free external inversion of tubes, Int J Crashworthiness, 1996 1 93 106. 9. J J Harrigan, Internal inversion and nosing of laterally constrained metal tubes. Ph.D. Thesis, UMIST, 1995. 10. P K Gupta, An investigation in to large deformation behavior of metallic tubes, Ph.D. Thesis, Department of Applied Mechanics, IIT, Delhi, 2000. 11. W J Stronge, T X Yu and W Johnson, Long stroke energy dissipation in splitting tubes, Int J Mech Sci, 1983 25 (9/10) 637 647. 12. X Huang, G Lu and T X Yu, On the axial splitting and curling of circular metal tubes, Int J Mech Sci, 2002 44 2369 2391. 13. X Huang, G Lu and T X Yu, Energy absorption in splitting square metal tubes, Thin-Walled Struct, 2002 40 153 165. 14. T Y Reddy and S R Reid, Axial splitting of circular metal tubes, Int J Mech Sci, 1986 28 (2) 111 131. Copyright C Taylor & Francis Group 501

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