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Enhancing energy absorption of circular tubes under oblique loads through introducing grooves of non-uniform depths
International Journal of Mechanical Sciences 166 (2020) 105239
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Enhancing energy absorption of circular tubes under oblique loads through introducing grooves of non-uniform depths Kun Tian a, Yuan Zhang a, Fan Yang a,∗, Qi Zhao a, Hualin Fan b,∗ a
School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, China Research Center of Lightweight Structures and Intelligent Manufacturing, State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
b
a r t i c l e
i n f o
Keywords: Circular tubes Oblique loading Circumferential grooves Non-uniform depths
a b s t r a c t Compared with the excellent energy absorption performance under axial loading, thin-walled tubes are vulnerable to instable global bending under oblique loading condition. In this paper, a novel design was developed to improve the energy absorption performance of the thin-walled circular tubes under oblique loading by introducing multiple circumferential grooves with non-uniform depths. Quasi-static experiments and finite element simulations were carried out for tube specimens with different groove configurations. Theoretical models were developed to explain the different energy absorption performance of different tubes. The effects of the loading angle and the friction condition on the energy absorption were also investigated. The results highlight the advantages of the gradiently grooved tubes with groove depth decreasing from loading end to fixed end over the uniformly grooved tubes and the original tubes under the oblique loading condition. The work in this paper can provide a guide for the design of advanced energy absorbing devices for arbitrary loading condition.
1. Introduction With the fast development of electric and hydrogen vehicles that are more vulnerable to impact damage, and the light-weight trend of transport facility, increasing attention has been paid to the performance of energy absorbing devices that play an essential role in protecting the occupants in a collision event. The metallic thin-walled tubes have been proved to be efficient energy absorbers, and thus were widely used in automobile, highspeed railway and aerospace industries during the past decades. These tubes can undergo extensive plastic deformation to effectively absorb the kinetic energy. Regarding to various working conditions and emphases, different design criteria are proposed, leading to different configuration of thin-walled tube for optimal performance [1]. In general, a preferred energy absorber should be able to meet these demands: (i) high specific energy absorption (SEA), (ii) tolerable peak crushing force usually taking place at initial stage, and (iii) repeatable collapse mode [2]. A significant number of researches have been conducted to experimentally, numerically and theoretically investigate the energy absorbing behavior of thin-walled tubes, with the configuration being circular tube [3-5], triangular tube [6-8] and square tube [8], etc. Most of the existing efforts were dedicated to the axial loading condition. However, an energy absorber is rarely subjected to pure axial loading in actual crushing events, but rather to oblique loading. It is therefore important to in∗
vestigate the energy absorption and the crushing behavior of thin-walled tubes under the oblique loading condition. Han and Park [9] made a first effort to investigate the static and dynamic collapse behavior of square thin-walled tubes subjected to oblique loading. They found that when the loading angle is large enough, the tubes tend to collapse in global bending mode, which is less stable with a prominent reduction in energy absorption compared with the progressive folding collapse mode. In addition, they discussed the critical angle at which the progressive folding converts to the global bending. Børvik et al. [10] studied the deformation behavior of the circular thin-walled tubes under oblique loading. They also observed that both the energy absorption and the peak force decrease as the loading angle increases. Reyes et al. [11,12] analyzed the energy absorption ability of the aluminum square thin-walled tubes under oblique loading and found that the collapse mode relies on not only the loading angle but also the tube thickness. For conventional thin-walled tubes, the design parameters are very few. Only three geometric parameters can be tuned, namely, the thickness, the diameter and the length. Recently, a number of innovative configurational designs were proposed to improve the energy absorption behavior. The introduction of taper angle is one of the effective designs, and appeals to increasing number of researchers [13-16]. Nagel and Thambiratnam [13] investigated the energy absorption of tapered thin-walled tubes through dynamic simulations, and found the tapered tubes less apt to global bending under oblique loading, compared with
Corresponding authors. E-mail addresses: [email protected] (F. Yang), [email protected] (H. Fan).
https://doi.org/10.1016/j.ijmecsci.2019.105239 Received 5 June 2019; Received in revised form 4 October 2019; Accepted 12 October 2019 Available online 13 October 2019 0020-7403/© 2019 Elsevier Ltd. All rights reserved.
K. Tian, Y. Zhang and F. Yang et al.
the conventional tubes. However, the higher space requirement of the tapered tubes raised an issue for its application as the space for the crushing boxes is limited inside the vehicle. In some work [17-20], the multi-cell tubes of varying cross sections were investigated, and the related studies revealed that they were highly weight-effective. Foamfilling was also proposed [21-23]. Unfortunately, the foam filled tubes did not show any significant improvement in energy absorption under the oblique impact loading condition [10]. Functionally graded thickness tubes with varying thickness were proposed in recent years [2426]. Li et al. [27] compared energy absorption of functionally graded thickness (FGT), tapered uniform thickness (TUT) and straight uniform tubes (SUT) under oblique impact loading and confirmed that FGT is superior to TUT and SUT. Zhang et al. [28] investigated the energy absorption characters of conical tubes with gradient thickness subjected to the oblique loading. In 2002, thin-walled tubes with circumferential grooves were brought forward as a new energy absorber design [29]. An appreciable merit of the grooved thin-walled tubes is that the folding scenario and the energy absorption can be controlled by changing the distance between the grooves or the geometry of each groove. Therefore, the design space of the energy absorbers is tremendously enlarged. Abedi et al. [30] compared the SEA and the initial peak force of the foamfilled simple and grooved specimens under axial compression. Isaac and Oluwole [31] numerically investigated the crashworthiness of hexagonal grooved tubes subjected to dynamic impact. Yao et al. [32] built theoretic and numerical models of thin-walled circular tubes with nonuniform grooves considering both height and thickness variation under axial loading. They found that introducing the thickness gradient grooves can bring a significant improvement in the energy absorption. In 2014, Wei et al. [33] proposed a novel design of energy absorber consisting of an array of thin-walled circular tubes with grooves of gradient depths, which is able to adjust itself for arbitrary loading conditions. It was believed that the non-uniform groove design could coordinate the deformation of the whole structure and lead to the fully utilization of the material. Existing work in the literature provides insights into the energy absorption performance of grooved tubes under the axial loading. However, researches of the grooved tubes under oblique impact loading are still scarce. Motivated by this gap, we carried out a comprehensive investigation in this paper for the crushing behavior of the grooved thin-walled tubes under oblique loading condition. This paper focuses on the thin-walled circular tubes with grooves of varying depths subjected to oblique loading. Finite element (FE) numerical simulations and compression experiments were carried out for the simple tubes, uniformly grooved and non-uniformly grooved tubes to investigate their energy absorption performances and highlight the advantage of the gradiently grooved tubes under oblique loading condition. This paper is organized as follows. Following this introduction, a brief introduction is given for the FE model in Section 2. The details of the experiments are presented and the test results are shown to validate the FE model in Section 3. The influences of various design parameters are investigated through a parametric FE study in Section 4. Some conclusions are drawn in Section 5. 2. Finite element simulations Systematic FE simulations were carried out to investigate the crushing behaviour of the circular tubes with different groove configurations under the oblique loading condition using the general-purpose FE code ABAQUS. The details of the FE model are given below. 2.1. Geometry The geometry of the gradiently grooved thin-walled tube proposed in this study is schematically shown in Fig. 1. The grooves are equally spaced with non-uniform depth, and located alternatively on the inside and outside surfaces of the tube shell. The geometrical parameters of the
International Journal of Mechanical Sciences 166 (2020) 105239
Fig. 1. Schematic configuration of the loading scenario of the specimen of thinwalled tube with non-uniform grooves.
Fig. 2. Stress-strain curve of the specimen material obtained from the uniaxial test. Table 1 Mechanical property parameters of the tube material. Density 𝜌
Modulus E
Poisson’s ratio 𝜈
Yield stress 𝜎 y
2.7 g/mm3
70 GPa
0.334
35.5 MPa
tube include the length L, the outer radius R, and the thickness t. The geometrical parameters of the grooves include the width w, the depth d, and the inter-groove space 𝜆. In this work, attention was focused on the effect of the groove depth d. Other geometric parameters were set to L = 180 mm, R = 30 mm, t = 2 mm, w = 3 mm, 𝜆 = 20 mm, unless otherwise specified. The distance between the first groove and the loading end is 10 mm.
2.2. Material model The tube was made of Aluminium alloy A6061-T6. To obtain the mechanical properties, a uniaxial tensile test was conducted for the dogbone specimen according to Standard GB228. The stress-strain curve obtained from the test is shown in Fig. 2, with the typical material parameters summarized in Table 1. The obtained stress-strain data were incorporated into the FE model to accurately capture the strain hardening properties of the tube material, using the von-Mises plasticity law with isotropic hardening.
K. Tian, Y. Zhang and F. Yang et al.
Fig. 3. Discretized configuration of the FE model.
The 8-noded linear brick element (Type C3D8R) with reduced integration and enhanced hourglass control was selected to discretize the tubes in this work. Fig. 3 shows the meshed configuration, with average element size of 0.5 mm. This mesh size was chosen to account for both computational accuracy and computational cost based on several trial simulations. 2.3. Loading, boundary and contact conditions The tube was loaded by a rigid analytical platen that was moving at a speed of 1 m/s along an oblique direction forming an angle of 𝛼 with the tube axis. All the degrees of freedom (DOFs) of the nodes at the bottom end of the tube were constraint to model the fixed boundary condition. The rigid body movement of the tube is also avoided in this
International Journal of Mechanical Sciences 166 (2020) 105239
way. In addition, a rigid support platen was added just below the tube to prevent the tube shell from penetrating into the support base due to large deformation during the simulation. Several contact interactions may take place between the tube and the loading platen, the tube and the support platen, and the tube surface and itself. The last contact interaction was enforced to prevent selfpenetration of the developed folds. In the simulations, all the contact interactions were properly modeled using the all-inclusive general contact algorithm in ABAQUS. For axial loading condition, previous studies showed that the energy dissipation due to friction is negligible [34,35]. However, for the oblique loading condition, the friction effect cannot be ignored as shown later in this paper. The static-kinetic friction formulation was applied. In order to determine the friction coefficients, we built a number of FE models with different friction coefficients, and it turned out that the results of the numerical model with static coefficient of 0.25, kinetic coefficient of 0.10, and an exponential decay coefficient of 1 showed the closest match with the experimental results. Therefore, unless specified otherwise, the above friction coefficients were adopted for all contact interactions. 3. Experiment To validate the results of the numerical simulations, quasi-static experimental tests were carried out using the universal testing machine (Model CSS-44200, 20 tones) from Changchun Testing Machine Institute. For this purpose, we designed the grooved tube specimens and the test platform that is suitable for the oblique loading condition. The test setup is shown in Fig. 4. The platform included a support base with an oblique platen and a pressing block that could fix the specimen, and a large loading platen that could apply the compression load. The equipment was easy to assemble and could be used multiple times. The platform and the tube specimens were made of carbon steel Q235 and aluminum alloy AL6061-T6, respectively. All the components were manufactured by Shanghai Shili Machineries Co., Ltd. The tube specimens were lathed by a CNC lathe machine with a polycrystalline diamond Fig. 4. Experimental setup, (a) loading platform with the specimen, and (b) half of 3D drawing showing the details of the fixed end.
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International Journal of Mechanical Sciences 166 (2020) 105239
energy absorption performance is believed to be not much influenced. This is because that the plastic deformation is always the dominant part of energy dissipation/transformation compared with the inertia and the elastic parts, and also aluminum can be taken as a strain-rate insensitive material. 4. Theoretical modelling
Fig. 5. Three specimens tested, (a) original, (b) uniformly grooved, (c) gradiently grooved with groove depth decreasing from top to bottom. Table 2 Investigated tubes with different sequence of groove depths. Tube No.
Sequence of groove depths (mm)
1 2 3 4 5 6
1.0, 1.0, 1.0, 1.0, 1.0, 1.0 1.75, 1.50, 1.25, 1.0, 0.75, 0.50, 0.25 1.50, 1.50, 1.50, 0.75, 0.75, 0.75 1.50, 1.50, 1.50, 0.75, 0.75, 0.75, 0.25, 0.25 1.50, 1.50,1.50,1.00, 1.00, 1.00, 0.50, 0.50 0.50,0.50,0.75,0.75,1.0,1.0
To further demonstrate the mechanisms behind the energy absorption enhancement of the gradiently grooved tube under oblique loading, we developed theoretical models to calculate the crushing force and the absorbed energy of different tubes during the crushing process. The models were established from the energy viewpoint based on the kinematically admissible deformation mechanisms observed from the experiments. For the original tube, global bending is the dominant mechanism. Therefore, we followed the deductions by Wang and Qiu [36] for the energy absorption during the global buckling of a circular tube. The deformation mechanism is shown in Fig. 6. The deformation is localized to the region between the cross-sections BU and DS, where two flattened triangles ABC and ADC were formed and the ovalisation took place in the cross-sections. As per Wang and Qiu’s model, the dissipated energy E is composed of the part contributed by the bending section Ebend and the part contributed by the compression of the tube wall Ecomp . The energy by the bending section can be further decomposed into the energy dissipated at bending of the central horizontal hinge line Ehoriz , bending of the oblique hinge lines Eobliq , and lateral crushing of the wall near the bending section Ecrush . Therefore, 𝐸Bend = 𝐸horiz + 𝐸obliq + 𝐸crush + 𝐸comp The instantaneous crushing force is 𝐹Bend = 𝐹horiz + 𝐹obliq + 𝐹crush + 𝐹comp where, 𝐹horiz = 𝐹obliq =
(PCD) tool. No welding was used for the tubes. Three specimens, i.e., original, uniformly grooved and gradiently grooved tubes were manufactured for comparison purpose, as shown in Fig. 5. The groove depth of the uniformly grooved tube was 1 mm, and the groove depths of the gradiently grooved tube followed the sequence of No.2 in Table 2. Before testing, the tube specimens were annealed at a temperature of 500 Celsius degrees for 1 h followed by a gradual cooling to room temperature at a rate of approximately 100°/h. During the test, the specimen was fixed on the support base with a 15° oblique angle, and compressed by the loading platen that moved at a speed of 4 mm/min. Although the loading speed is much smaller than that in the FE simulation, the
(1)
𝐹crush = 𝐹comp =
d𝐸horiz d𝑢 d𝐸obliq d𝑢 d𝐸crush d𝑢 d𝐸comp d𝑢
(2)
(3)
(4)
(5)
In the above equations, 𝜃 is the rotation angle of the global bending, u is the displacement of the loading platen, M is the bending moment caused by different energy dissipations. The detailed expressions can be found in Appendix A. For the gradiently grooved tube, the deformation takes on a nonaxisymmetric progressive folding mode. Therefore, we calculated the Fig. 6. Deformation mechanism of the global bending mode.
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International Journal of Mechanical Sciences 166 (2020) 105239
Fig. 7. Deformation mechanism of the non-axisymmetric progressive folding mode.
Fig. 8. Deformed tube specimens in the compression experiments, (a) original, (b) uniformly grooved, and (c) gradiently grooved with groove depth decreasing from top to bottom.
Fig. 9. Deformed tube specimens in the FE simulations, (a) original, (b) uniformly grooved, and (c) gradiently grooved with groove depth decreasing from top to bottom.
energy absorption based on the modified version of the model of Singace [37] for the multi-lobe progressive collapse mode. The deformation mechanism is shown in Fig. 7. Each fold layer is composed of multiple triangular lobes, with the triangle apex rotating outward and the triangle base rotating inward. The energy dissipation E is composed of the part by bending the horizontal hinges Ehoriz , the part by bending the oblique hinges Eobliq , the part by decurving of the cylindrical wall Edecur , and the part by compression of the wall Ecomp . The horizontal hinges are assumed to coincide with the grooves, which was indeed the case in the experiments. Neglecting the change of the cross-section peripheral length (which is reasonable according to Ref. [21]), the energy dissipated at bending the horizontal hinges are
d𝐸horiz = 2𝜋𝑅𝑀p′ d(𝛽 + 𝛾)
(6)
where, 𝑀p′ =
𝜎p 𝑡 ′ 2 √ 2 3
is the plane strain plastic bending moment per unit
length of the hinge line, 𝜎p = (𝜎y + 𝜎u )∕2 is the characteristic flow stress which is taken as the average of the initial yield stress 𝜎 y and ultimate strength 𝜎 u , t′ = t − d is the tube thickness at the grooves. The two angles 𝛽 and 𝛾 represents the rotations of the two adjacent lobes whose normal points downwards and upwards, respectively. 𝛽 and 𝛾 are dependent on each other and can be represented by any one of them. Fig. 7(a) shows a special moment when one of the angles reaches its maximum 𝜋/2. At an arbitrary moment, the crushing force is 𝐹horiz = 2𝜋𝑅𝑀p′
d(𝛽 + 𝛾) d𝑢
(7)
Two scenarios are involved in a complete folding period. One scenario is when the two rotating lobes are connected at the triangle base, as indicated in the left wall of Fig. 7(b). The other scenario is when the two rotating lobes are connected at the triangle apex, as indicated in the
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International Journal of Mechanical Sciences 166 (2020) 105239
right wall of Fig. 7(b). It is noted that there are at least two horizontal hinges rotating at the same time, e.g., the hinges a, b, and c with rotation angles 𝛽+𝛾, 𝛾’, and 𝛽’ in Fig. 7(b). The total crushing force contributed by the horizontal hinges is ( ) ⎧[𝑓 (𝛽) + 𝑓 (𝛾 ′ )], 0 ≤ 𝑢 < 𝜆∕2 1 + cos 𝛾0 − cos 𝛽0 2 ( )] ( ) ( ) ⎪[ 1 𝐹horiz = 2𝜋𝑅𝑀p′ ⎨ 𝑓1 (𝛽) + 𝑓1 𝛽 ′ , 𝜆∕2 1 + cos 𝛾0 − cos 𝛽0 ≤ 𝑢 < 𝜆∕2 1 + cos 𝛽0 − cos 𝛾0 [ ( )] ( ) ⎪ 𝑓2 (𝛾) + 𝑓1 𝛽 ′ , 𝜆∕2 1 ≤ + cos 𝛽 − cos 𝛾 𝑢 ≤ 𝜆 0 0 ⎩
(8) where, the two functions f1 (𝛽) and f2 (𝛾) are used to indicate the derivative d(𝛽 + 𝛾)∕d𝑢 for the two folding scenarios. Assuming the oblique hinge lines are rotated simultaneously with the horizontal lines, the crushing force contributed by the oblique lines Fobliq is similar as Eq. (8) by replacing 2𝜋RM′p with 2NLob [cM′p + (1 − c)Mp ] where, Lob is the length of one oblique hinge line, N is the number of lobes along circumference, 𝑐 = 2𝑤∕𝜆 is the ratio of the groove width to the lobe height. The total instantaneous crushing force is Fig. 10. The crushing force versus displacement for the original tube, showing global bending.
Fig. 11. The crushing force versus displacement for the uniformly grooved tube.
𝐹Fold = 𝐹horiz + 𝐹obliq + 𝐹decur + 𝐹comp
(9)
The detailed expressions can be found in Appendix B. For the calculation of the progressive folding process, the grooved thickness t′ and hence the bending moment M′p were updated from fold to fold to account for the effect of the non-uniform groove depth. For the uniformly grooved tube, the crushing process can be divided into two stages. First, global bending took place near the middle point of the tube, causing the upper part of the tube to be rotated quickly. Second, after the fully collapse of the upper part, the deformation switched to progressive folding in the lower part. Therefore, the Wang and Qiu’s model and Singace’s model were used to quantitate the energy absorption during the two stages, respectively. The position where global bending occurred is determined by ( ) 𝑀0 𝑡′ 𝐿′ = (10) 𝐿 𝑀 0 (𝑡 ) where, M0 = 4𝜎 p R2 t is the critical bending moment of the cross-section, L′ is the length of tube above the bending position. It is noted that for all tubes, a partial fold is formed during the initial stage because of the oblique loading condition. For simplicity, the crushing force at this stage is assumed to be cpart FFold , where cpart is the ratio of the perimeter that plays part in the partial fold. In the √ calculations cpart is taken as 0.3, and the fold length is taken as 1.347 𝑅𝑡′ 5. Results 5.1. Comparison between the experimental and the FE numerical results
Fig. 12. The crushing force versus crushing displacement for the gradiently grooved tube, showing progressive folding deformation mode.
Fig. 8 shows the deformed configurations of the three tube specimens after the compression test. Considerable difference can be seen among the three tubes. For the original tube, the deformation is highly localized to the section near the fixed bottom end of the tube due to the large bending moment at that location. The uniformly grooved tube deformed partly in progressive folding mode and partly in global bending mode. On the other hand, the gradiently grooved tube showed no global bending, with all the sections deformed in progressive folding mode. Fig. 9 shows the simulated tube deformations. The FE simulations used exactly the same geometry, the same material properties and the same loading angle as the experiments. Comparing Fig. 9 with Fig. 8, the deformation modes obtained from the FE simulations agree very well with those from the experiments. Figs. 10–12 compare the load-displacement curves from the FE simulations with those from the experiments for the original, the uniformly grooved and the gradiently grooved tube, respectively. Satisfactory agreement was achieved between the simulations and the experiments for all three specimens, especially for the tendency of the crushing curve. As shown in Fig. 10, the force of the original tube took
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International Journal of Mechanical Sciences 166 (2020) 105239
Fig. 13. Chronological snapshots of the deformed configuration during the crushing test at the same loading angle of 15°.
an abrupt drop after reaching its maximum value at the crushing distance of 25 mm, indicating quick loss of its load-bearing capacity after the global bending. It is observed from the experiment that the deformation is highly localized to the fixed end of the original tube. During the final loading stage, a crack appeared near the fixed end of the tube along the circumferential direction. As the loading platen went down, the crack propagated. Therefore, we terminated the loading ahead of time to prevent the specimen from breaking into two pieces. That is why the ultimate experimental displacement of the simple tube is noticeable less than the FE simulation value. From Fig. 11, the force of the uniformly grooved tube first decreased after the first peak, but then increased with further crushing. From Fig. 12, the crushing force of
the gradiently grooved tube shows a steady increasing trend during the crushing process, with its magnitude undulating due to the progressive folding. In the above figures, the apparent bold parts of the experimental curves were actually the small high-frequency fluctuations that were caused by the intermittent slippage between the tube and the loading platen. In the tests, a repeating thumping noise could be heard at the time when the small high-frequency fluctuations on the loading curves were generated. The difference between the deformation modes of the three tubes can be discerned more clearly from the chronological snapshots of the deformed configurations as shown in Fig. 13. For the original tube, global bending buckling took place near the fixed bottom end at the very initial
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International Journal of Mechanical Sciences 166 (2020) 105239
Fig. 14. Absorbed energy versus crushed displacement for the three tubes investigated, from the experiments, the FE simulations and the theoretical models.
Fig. 18. Load-displacement curves of tubes with different sequence of groove depths as in Table. 2 at the loading angle of 15°.
Fig. 15. Deformed configuration of the original tube for loading angles (a) 8° and (b) 10°.
Fig. 16. Deformed configuration of the uniformly grooved tube for loading angles (a) 12° and (b) 14°.
Fig. 19. Absorbed energy of tubes with different sequence of groove depths as in Table. 2 at the loading angle of 15°.
Fig. 17. Deformed configuration of the gradiently grooved tube at the loading angle of 25°. Table 3 Friction coefficients of investigated contact conditions.
Condition 1 Condition 2 Condition 3 Frictionless
Kinetic friction coefficient
Static friction coefficient
0.10 0.15 0.2 0
0.25 0.30 0.2 0
stage. Most of the tube sections reserved integrity during the crushing process, leading to inefficient material usage. For the uniformly grooved tube, global bending can be seen at the location of the sixth groove due to the reduced stiffness of the grooved section. However, the remained part of tube can still bear the crushing load through the progressive folding deformation. For the gradiently grooved tube, the deeper grooves near the loading end would facilitate partial local bulking under the oblique loading, causing the tube axis to rotate towards the loading di-
Fig. 20. Deformed configuration of (a) Tube 3, (b) Tube 4, (c) Tube 5 and (d) Tube 6 in Table. 2 for loading angles 15°.
rection. In this way, the global bending can be effectively prevented, leading to the significant energy absorption. Fig. 14 shows the curves of the dissipated energy during the crushing process, which was obtained by integrating the crushing force over the crushing distance. The solid lines correspond to the experimental results and the dashed lines correspond to the FE results. It can be seen that good consistency was achieved between the experimental curves
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International Journal of Mechanical Sciences 166 (2020) 105239
Fig. 23. Load-displacement curves of the original tube under different contact conditions at the loading angle of 8°, with inserted graphs showing the deformed configurations.
Fig. 21. Load-displacement curves of the original tube under different contact conditions at the loading angle of 15°.
Fig. 22. Load-displacement curves of the uniformly grooved tube under different contact conditions at the loading angle of 15°.
and the simulated curves for all three specimens. For example, taking the crushing distance of 80 mm for the original tube (the maximum crushing distance for this tube) and the crushing distance of 120 mm for the other two, the relative difference between the energy absorption from the simulations and the experiments are 4.2%, 5.7%, and 5.9%, respectively for the three tubes. In addition, the original tube absorbed more energy at a crushing distance smaller than 100 mm, while the gradiently grooved tube was the one that absorbed the most energy when the crushing distance was larger than 100 mm. The energy absorption predicted by the theoretical models are also compared in Fig. 14. Excellent correlation is achieved between the theoretical curves and the curves from the experiments and FE simulations for all three tubes investigated, indicating that the deformation mechanisms are well captured by our theoretical models. The experimental, numerical and theoretical results unanimously show that the gradiently grooved tube is preferable to the uniformly
grooved tube and the original tube. The quantitative agreement between the FE numerical results and the experimental results provides a validation for the FE models. Next, a parametric FE study was carried out to investigate the influences of various parameters such as the groove configuration, the loading angle and the friction coefficients on the energy absorption performance. 5.2. Critical loading angle The deformation mode of the tube is closely related to the loading angle. As the loading angle increases, the deformation mode may change from the progressive folding to the global buckling. The critical loading angle is defined as the loading angle at which the tube changes its deformation mode. The exact value of the critical loading angle is difficult to determine since the transition of deformation mode usually occurs gradually. Therefore, we switched our attention to the determination of
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International Journal of Mechanical Sciences 166 (2020) 105239
Fig. 24. Load-displacement curves of (a) Tube 2, (b) Tube 3, (c) Tube 4 and (d) Tube 5 under different contact conditions at the loading angle of 15°.
the range of the critical loading angle in this paper. For this purpose, a series of FE simulations were carried out with different loading angles increasing from 0° to 25° at an increment of 2°. As shown in Fig. 15, for the original tube, the deformation mode was progressive buckling at the loading angle of 8°, but changed to global bending at the loading angle 10°, indicating that the critical loading angle lies in the range between 8° and 10°. In the same way, the critical loading angle for the uniformly grooved tube was determined to be within the range of 12° to 14°, as shown in Fig. 16, which was obviously larger than the nongrooved counterpart. As for the gradiently grooved tube, the situation was more complex. This was because the deformation always involved obvious progressive deformation mode even at a relative large loading angle. Global bending, on the contrary, was not obvious for this kind of tube. Fig. 9 shows that the progressive deformation mode was well kept at the loading angle of 15°. From Fig. 17, the tube progressive folding played an important role even under a loading angle of 25°. The results indicate that the gradient-groove design could significantly increase the critical loading angle, and thus the energy performance of the circular tube.
5.3. Effect of the different groove depth designs on the energy absorption To seek for the best design of the non-uniformly grooved tube, FE simulations were carried out for a series of tubes with different groove designs. Table 2 shows the investigated tubes with different sequence of groove depths starting from the top end (loading end) of the tube. Each data in the sequence represents the depth of a pair of grooves on the inner and outer surfaces. All the tubes were loaded with the same loading angle of 15°. Fig. 18 shows the crushing force of the investigated tubes. It is obvious to see that global bending took place in the original tube and the uniformly grooved tube (Tube 1), as the crushing force shows an abrupt decrease. On the other hand, Tube 2 and Tube 3 mainly took on a progressive folding deformation mode, as the crushing force increased gradually with periodic undulation. Tube 4 and Tube 5 experienced a combined deformation mode, with first progressive folding and then global bending. From the energy-displacement curves shown in Fig. 19, we can see that the tubes with the groove depth deceasing from top to bottom all showed improved energy absorption compared with the original tube and the uniformly grooved tube when the crushing distance is
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large enough. On the contrary, the tube with the groove depth increasing from top to bottom (Tube 6) showed poor performance in terms of both the deformation mode and the energy absorption. Global bending can be easily initiated at the deepest bottom groove for this kind of tube. The deformation modes of the original, No. 1 and No. 2 tube are shown in Fig. 9 and the other tubes in Table 2 can be seen in Fig. 20. The above results indicate that the depth sequence of the groove design plays an important role in the tube’s deformation behaviour and the energy absorption performance under oblique loading condition. Groove depth in the non-uniformly grooved tube should be decreased from the loading end to the fixed end to achieve the enhanced energy absorption.
International Journal of Mechanical Sciences 166 (2020) 105239
(iii) The sequence of groove depths plays an important role in the tube’s energy absorption performance under oblique loading, with the groove depth decreasing from the loading end to the fixed end corresponding to the best performance, and the reversed depth sequence corresponding to the worst performance, (iv) The deformation mode tends to transit from global bending to progressive folding as the contact friction is increased for the original and the uniformly grooved tubes, and (v) The gradiently grooved tube with a suitable groove sequence is not as sensitive to contact condition as the original and the uniformly grooved tubes.
5.4. Effect of frictional contact condition
Declaration of Competing Interest
We further investigated the effect of contact friction condition on the energy absorption performance. The static-kinetic friction algorithm with exponential decay was used. Four contact conditions were investigated, with the friction parameters shown in Table. 3. Figs. 21 and 22 show the effect of contact condition on the load-displacement curves for the original tube and the uniformly grooved tube, respectively, under the loading angle of 15°. The results indicate that the contact condition had an evident effect on the crushing force and the absorbed energy. For both tubes investigated, the larger friction coefficients resulted in obviously larger crushing force, and thus larger energy absorption. The contact condition may also influence the deformation mode. For an example, Fig. 23 shows load-displacement curves of the original tube with and without friction for contact interaction under the loading angle of 8°. The inset pictures show the deformed configurations for the two contact conditions. The tube with frictional contact property (Condition 1) took on a progressive deformation mode, while an evident global bending can be seen for the frictionless case. It indicates that the deformation mode would transform from global bending to progressive folding as the contact friction is increased for the original tube. Similar phenomenon was also observed for the uniformly grooved tube. In Fig. 24, the load-displacement curves of Tube 2(a), 3(b), 4(c) and 5(d) (in Table 2)are shown for different friction conditions. Contrary to Figs. 21 and 22, Fig. 24(a) shows that the influence of friction coefficients was not significant for Tube 2. Even under frictionless contact condition, Tube 2 still took on the progressive folding deformation mode. As for the other three tubes, the crushing force decreased rapidly after global bending. The results indicate that the tube with a suitable groove sequence is not as sensitive to contact condition as the original and the uniformly grooved tubes, which also highlights the advantage of the gradiently grooved tube. But when global bending occurs, the crushing force decreases a lot.
The authors declare that there is no conflict of interests regarding the publication of the article titled “Enhancing energy absorption of circular tubes under oblique loads through introducing grooves of non-uniform depths” in Int J Mech Sci.
6. Conclusions In this study, dynamic crashing behavior of grooved circular tubes with non-uniform groove depth subjected to oblique impact loading was explored. A series of FE numerical simulations and experimental tests were carried out to investigate the performance of different designs with different sequence of groove depths. Theoretical models were developed to demonstrate the energy absorption mechanisms. The influences of key parameters such as the groove depth, the loading angle and the contact condition were analyzed. The work in this paper can provide a guide for the design of effective energy absorbers under the oblique loading condition. Several conclusions can be drawn as follows. (i) The gradiently grooved tube whose groove depth decreases from loading end to fixed end is advantageous to the original and the uniformly grooved tubes for energy absorption under the oblique loading condition, (ii) The critical loading angle at which global bending occurs can be significantly increased with the introduction of grooves,
Acknowledgements This work was supported by National Natural Science Foundation of China (11772231), the Students Innovation Training Program (SITP) of Tongji University, and Shanghai Supercomputer Center. We also thank Profs Liming Zhu and Jinlong Zhu for their help in the experiments. Appendix A For global bending mode, the crushing forces contributed by the energy dissipation mechanisms within the bending region are 𝑀horiz 𝐿(sin (𝛼 + 𝜃) − 𝜇 cos (𝛼 + 𝜃))
𝐹horiz = 𝐹obliq =
𝑀obliq 𝐿(sin (𝛼 + 𝜃) − 𝜇 cos (𝛼 + 𝜃))
𝐹crush =
𝑀horiz =
𝑀obliq =
𝑀crush =
𝑀crush 𝐿(sin (𝛼 + 𝜃) − 𝜇 cos (𝛼 + 𝜃)) ( ) 0.45 (−𝑅 + 𝑟)2 ∕(𝑅2 𝐻)𝑀0 𝜙3 𝜃∕2
(A1)
(A2)
(A3)
(A4)
(√ ) 1∕32 2∕ 𝑅 2 ( 𝑅 − 𝑟 ) 𝑀 0 𝜙 𝜃∕2 √ 0.25((𝑅 + 𝑟)∕𝑟)𝑀0 (𝜋∕𝑅) 2𝜙 𝜃∕2
where, 𝜙 =
2(𝑅𝐻𝜃∕2)1∕4 √ 𝑅−𝑟
(A5)
(A6)
is the angle characterizing the section ovalisa-
tion, r = 0.6R, H = 1.31R are the geometric parameters, 𝜇 is the friction coefficient. The crushing force contributed by the compression of the tube wall is 𝐹comp =
𝑀comp
𝑀comp 𝐿(sin (𝛼 + 𝜃) − 𝜇 cos (𝛼 + 𝜃))
⎧ 4𝜎 𝑅2 𝑡(𝜃 −sin 𝜃 ) 4 ⎪ p 24 , 𝜃3 ≥ 𝜃4 (𝜃∕2) ⎪ 4𝜎 𝑅cos 2 𝑡(𝜃 −sin 𝜃 ) 3 3 =⎨ p , 𝜃3 ≤ 𝜃4 cos2 (𝜃∕2) ⎪ ⎪ 0, 𝜃3 = 0 ⎩
where, 𝜃3 = arccos
(
) 𝑢 −1 , 4𝑅
(A7)
(A8)
(A9)
K. Tian, Y. Zhang and F. Yang et al.
and
(
𝜃4 = 𝜋 − sin
𝜉 𝑅
International Journal of Mechanical Sciences 166 (2020) 105239
The corresponding force is
) (A10)
are the angular parameters on the loading section and the bending section, respectively. 𝜉 = 𝜋𝑅 − 𝑟𝜙 − (𝜋 − 𝜙)𝑅
(A11)
𝐹decur =
8𝑀p d𝐸decur = d𝑢 sin 𝛽 + tan 𝛾 cos 𝛽
(B16)
The energy dissipated by the compression of the tube wall is d𝐸comp = 2
𝜋
∫0
( ) 𝜎𝑦 𝜀max 1 − cos 𝜃1 𝑡𝑅d𝜃1 d𝑢 = 4𝜋𝜎𝑦 𝑅2 𝑡 tan 𝛼d𝑢∕𝐿
is the length of central hinge AC
where,
Appendix B
𝜀max = 2𝑅 tan 𝛼∕𝐿
(B17)
(B18)
The corresponding force is For the multi-lobe progressive folding mode, the following relations can be obtained from Fig. 7(a) sin 𝛾0 = 𝑚
(B1)
sin 𝛽0 = 1 − 𝑚
(B2)
sin 𝛾 = sin 𝛽 − (1 − 𝑚) ) 𝜆 𝜆( cos 𝛽0 − cos 𝛽 + (1 − cos 𝛾) 2 2
(B3) (B4)
d𝛾 =
cos 𝛽 d𝛽 cos 𝛾
(B5)
d𝑢 =
𝜆 𝜆 (sin 𝛽d𝛽 + sin 𝛾d𝛾) = (sin 𝛽 + tan 𝛾 cos 𝛽)d𝛽 2 2
(B6)
d(𝛽 + 𝛾) d𝛽 2(cos 𝛾 + cos 𝛽) = d𝛽 d𝑢 𝜆 cos 𝛾(sin 𝛽 + tan 𝛾 cos 𝛽)
(B7)
𝑓1 (𝛽) =
For the second scenario, sin 𝛽 ′ = sin 𝛾 ′ − 𝑚 𝑢=
(B8)
) 𝜆 𝜆( cos 𝛾0 − cos 𝛾 (1 − cos 𝛽) + 2 2 cos 𝛾 ′ ′ d𝛾 cos 𝛽 ′
d𝛽 ′ = d𝑢 =
(B9)
(B10)
) 𝜆( ) 𝜆( sin 𝛽 ′ d𝛽 ′ + sin 𝛾 ′ d𝛾 ′ = sin 𝛾 ′ + tan 𝛽 ′ cos 𝛾 ′ d𝛾 ′ 2 2
( ) ( ) 2 cos 𝛽 ′ + cos 𝛾 ′ ( ) d 𝛽 ′ + 𝛾 ′ d𝛾 ′ 𝑓2 𝛾 ′ = = d𝛾 ′ d𝑢 𝜆 cos 𝛽 ′ (sin 𝛾 ′ + tan 𝛽 ′ cos 𝛾 ′ )
(B11)
(B12)
The crushing force contributed by the oblique lines is [ ] 𝐹obliq = 2𝑁 𝐿ob 𝑐 𝑀 ′ p + (1 − 𝑐 )𝑀p ( ) ⎧[𝑓 (𝛽) + 𝑓 (𝛾 ′ )], 0 ≤ 𝑢 < 𝜆∕2) 1 + cos 𝛾0 − 2 ( )] ( ( cos 𝛽0 ) ⎪[ 1 ′ ⎨[𝑓1 (𝛽) + 𝑓1 (𝛽 )], 𝜆∕2 1 + cos 𝛾0 − ( cos 𝛽0 ≤ 𝑢 < 𝜆∕2) 1 + cos 𝛽0 − cos 𝛾0 ⎪ 𝑓2 (𝛾) + 𝑓1 𝛽 ′ , 𝜆∕2 1 + cos 𝛽0 − cos 𝛾0 ≤ 𝑢 ≤ 𝜆 ⎩
(B13) where,
√
𝐿ob =
(
𝜋𝑅 𝑁
)2 +
( )2 𝜆 2
(B14)
The energy dissipated by the decurving of the cylindrical wall Edecur is d𝐸decur = 𝜆
d𝐸comp d𝑢
= 4𝜋𝜎𝑦 𝑅2 𝑡 tan 𝛼∕𝐿
(B19)
References
where, m is the folding parameter indicating the inward ratio of the lobe length. For the first scenario,
𝑢=
𝐹comp =
2𝜋𝑅
∫0
𝑀p 𝜅d𝐿2∕𝜋d𝛽 = 4𝑀p 𝜆d𝛽
(B15)
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K. Tian, Y. Zhang and F. Yang et al. [27] Li G, Xu F, Sun G, Li Q. A comparative study on thin-walled structures with functionally graded thickness (FGT) and tapered tubes withstanding oblique impact loading. Int J Impact Eng 2015;77:68–83. [28] Zhang X, Zhang H. Relative merits of conical tubes with graded thickness subjected to oblique impact loads. Int J Mech Sci 2015;98:111–25. [29] Daneshi GH, Hosseinipour SJ. Experimental study on thin-walled grooved tubes as an energy absorption device. Struct Mat 2002;11:289–98. [30] Abedi MM, Niknejad A, Liaghat GH, Zamani Nejad M. Foam-Filled grooved tubes with circular cross section under axial compression: an experimental study. Iran J Sci Technol 2018;42:401–13. [31] Isaac CW, Oluwole O. Structural response and performance of hexagonal thin-walled grooved tubes under dynamic impact loading conditions. Eng Struct 2018;167:459–70.
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Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Enhancing energy absorption of circular tubes under oblique loads through introducing grooves of non-uniform depths Kun Tian a, Yuan Zhang a, Fan Yang a,∗, Qi Zhao a, Hualin Fan b,∗ a
School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, China Research Center of Lightweight Structures and Intelligent Manufacturing, State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
b
a r t i c l e
i n f o
Keywords: Circular tubes Oblique loading Circumferential grooves Non-uniform depths
a b s t r a c t Compared with the excellent energy absorption performance under axial loading, thin-walled tubes are vulnerable to instable global bending under oblique loading condition. In this paper, a novel design was developed to improve the energy absorption performance of the thin-walled circular tubes under oblique loading by introducing multiple circumferential grooves with non-uniform depths. Quasi-static experiments and finite element simulations were carried out for tube specimens with different groove configurations. Theoretical models were developed to explain the different energy absorption performance of different tubes. The effects of the loading angle and the friction condition on the energy absorption were also investigated. The results highlight the advantages of the gradiently grooved tubes with groove depth decreasing from loading end to fixed end over the uniformly grooved tubes and the original tubes under the oblique loading condition. The work in this paper can provide a guide for the design of advanced energy absorbing devices for arbitrary loading condition.
1. Introduction With the fast development of electric and hydrogen vehicles that are more vulnerable to impact damage, and the light-weight trend of transport facility, increasing attention has been paid to the performance of energy absorbing devices that play an essential role in protecting the occupants in a collision event. The metallic thin-walled tubes have been proved to be efficient energy absorbers, and thus were widely used in automobile, highspeed railway and aerospace industries during the past decades. These tubes can undergo extensive plastic deformation to effectively absorb the kinetic energy. Regarding to various working conditions and emphases, different design criteria are proposed, leading to different configuration of thin-walled tube for optimal performance [1]. In general, a preferred energy absorber should be able to meet these demands: (i) high specific energy absorption (SEA), (ii) tolerable peak crushing force usually taking place at initial stage, and (iii) repeatable collapse mode [2]. A significant number of researches have been conducted to experimentally, numerically and theoretically investigate the energy absorbing behavior of thin-walled tubes, with the configuration being circular tube [3-5], triangular tube [6-8] and square tube [8], etc. Most of the existing efforts were dedicated to the axial loading condition. However, an energy absorber is rarely subjected to pure axial loading in actual crushing events, but rather to oblique loading. It is therefore important to in∗
vestigate the energy absorption and the crushing behavior of thin-walled tubes under the oblique loading condition. Han and Park [9] made a first effort to investigate the static and dynamic collapse behavior of square thin-walled tubes subjected to oblique loading. They found that when the loading angle is large enough, the tubes tend to collapse in global bending mode, which is less stable with a prominent reduction in energy absorption compared with the progressive folding collapse mode. In addition, they discussed the critical angle at which the progressive folding converts to the global bending. Børvik et al. [10] studied the deformation behavior of the circular thin-walled tubes under oblique loading. They also observed that both the energy absorption and the peak force decrease as the loading angle increases. Reyes et al. [11,12] analyzed the energy absorption ability of the aluminum square thin-walled tubes under oblique loading and found that the collapse mode relies on not only the loading angle but also the tube thickness. For conventional thin-walled tubes, the design parameters are very few. Only three geometric parameters can be tuned, namely, the thickness, the diameter and the length. Recently, a number of innovative configurational designs were proposed to improve the energy absorption behavior. The introduction of taper angle is one of the effective designs, and appeals to increasing number of researchers [13-16]. Nagel and Thambiratnam [13] investigated the energy absorption of tapered thin-walled tubes through dynamic simulations, and found the tapered tubes less apt to global bending under oblique loading, compared with
Corresponding authors. E-mail addresses: [email protected] (F. Yang), [email protected] (H. Fan).
https://doi.org/10.1016/j.ijmecsci.2019.105239 Received 5 June 2019; Received in revised form 4 October 2019; Accepted 12 October 2019 Available online 13 October 2019 0020-7403/© 2019 Elsevier Ltd. All rights reserved.
K. Tian, Y. Zhang and F. Yang et al.
the conventional tubes. However, the higher space requirement of the tapered tubes raised an issue for its application as the space for the crushing boxes is limited inside the vehicle. In some work [17-20], the multi-cell tubes of varying cross sections were investigated, and the related studies revealed that they were highly weight-effective. Foamfilling was also proposed [21-23]. Unfortunately, the foam filled tubes did not show any significant improvement in energy absorption under the oblique impact loading condition [10]. Functionally graded thickness tubes with varying thickness were proposed in recent years [2426]. Li et al. [27] compared energy absorption of functionally graded thickness (FGT), tapered uniform thickness (TUT) and straight uniform tubes (SUT) under oblique impact loading and confirmed that FGT is superior to TUT and SUT. Zhang et al. [28] investigated the energy absorption characters of conical tubes with gradient thickness subjected to the oblique loading. In 2002, thin-walled tubes with circumferential grooves were brought forward as a new energy absorber design [29]. An appreciable merit of the grooved thin-walled tubes is that the folding scenario and the energy absorption can be controlled by changing the distance between the grooves or the geometry of each groove. Therefore, the design space of the energy absorbers is tremendously enlarged. Abedi et al. [30] compared the SEA and the initial peak force of the foamfilled simple and grooved specimens under axial compression. Isaac and Oluwole [31] numerically investigated the crashworthiness of hexagonal grooved tubes subjected to dynamic impact. Yao et al. [32] built theoretic and numerical models of thin-walled circular tubes with nonuniform grooves considering both height and thickness variation under axial loading. They found that introducing the thickness gradient grooves can bring a significant improvement in the energy absorption. In 2014, Wei et al. [33] proposed a novel design of energy absorber consisting of an array of thin-walled circular tubes with grooves of gradient depths, which is able to adjust itself for arbitrary loading conditions. It was believed that the non-uniform groove design could coordinate the deformation of the whole structure and lead to the fully utilization of the material. Existing work in the literature provides insights into the energy absorption performance of grooved tubes under the axial loading. However, researches of the grooved tubes under oblique impact loading are still scarce. Motivated by this gap, we carried out a comprehensive investigation in this paper for the crushing behavior of the grooved thin-walled tubes under oblique loading condition. This paper focuses on the thin-walled circular tubes with grooves of varying depths subjected to oblique loading. Finite element (FE) numerical simulations and compression experiments were carried out for the simple tubes, uniformly grooved and non-uniformly grooved tubes to investigate their energy absorption performances and highlight the advantage of the gradiently grooved tubes under oblique loading condition. This paper is organized as follows. Following this introduction, a brief introduction is given for the FE model in Section 2. The details of the experiments are presented and the test results are shown to validate the FE model in Section 3. The influences of various design parameters are investigated through a parametric FE study in Section 4. Some conclusions are drawn in Section 5. 2. Finite element simulations Systematic FE simulations were carried out to investigate the crushing behaviour of the circular tubes with different groove configurations under the oblique loading condition using the general-purpose FE code ABAQUS. The details of the FE model are given below. 2.1. Geometry The geometry of the gradiently grooved thin-walled tube proposed in this study is schematically shown in Fig. 1. The grooves are equally spaced with non-uniform depth, and located alternatively on the inside and outside surfaces of the tube shell. The geometrical parameters of the
International Journal of Mechanical Sciences 166 (2020) 105239
Fig. 1. Schematic configuration of the loading scenario of the specimen of thinwalled tube with non-uniform grooves.
Fig. 2. Stress-strain curve of the specimen material obtained from the uniaxial test. Table 1 Mechanical property parameters of the tube material. Density 𝜌
Modulus E
Poisson’s ratio 𝜈
Yield stress 𝜎 y
2.7 g/mm3
70 GPa
0.334
35.5 MPa
tube include the length L, the outer radius R, and the thickness t. The geometrical parameters of the grooves include the width w, the depth d, and the inter-groove space 𝜆. In this work, attention was focused on the effect of the groove depth d. Other geometric parameters were set to L = 180 mm, R = 30 mm, t = 2 mm, w = 3 mm, 𝜆 = 20 mm, unless otherwise specified. The distance between the first groove and the loading end is 10 mm.
2.2. Material model The tube was made of Aluminium alloy A6061-T6. To obtain the mechanical properties, a uniaxial tensile test was conducted for the dogbone specimen according to Standard GB228. The stress-strain curve obtained from the test is shown in Fig. 2, with the typical material parameters summarized in Table 1. The obtained stress-strain data were incorporated into the FE model to accurately capture the strain hardening properties of the tube material, using the von-Mises plasticity law with isotropic hardening.
K. Tian, Y. Zhang and F. Yang et al.
Fig. 3. Discretized configuration of the FE model.
The 8-noded linear brick element (Type C3D8R) with reduced integration and enhanced hourglass control was selected to discretize the tubes in this work. Fig. 3 shows the meshed configuration, with average element size of 0.5 mm. This mesh size was chosen to account for both computational accuracy and computational cost based on several trial simulations. 2.3. Loading, boundary and contact conditions The tube was loaded by a rigid analytical platen that was moving at a speed of 1 m/s along an oblique direction forming an angle of 𝛼 with the tube axis. All the degrees of freedom (DOFs) of the nodes at the bottom end of the tube were constraint to model the fixed boundary condition. The rigid body movement of the tube is also avoided in this
International Journal of Mechanical Sciences 166 (2020) 105239
way. In addition, a rigid support platen was added just below the tube to prevent the tube shell from penetrating into the support base due to large deformation during the simulation. Several contact interactions may take place between the tube and the loading platen, the tube and the support platen, and the tube surface and itself. The last contact interaction was enforced to prevent selfpenetration of the developed folds. In the simulations, all the contact interactions were properly modeled using the all-inclusive general contact algorithm in ABAQUS. For axial loading condition, previous studies showed that the energy dissipation due to friction is negligible [34,35]. However, for the oblique loading condition, the friction effect cannot be ignored as shown later in this paper. The static-kinetic friction formulation was applied. In order to determine the friction coefficients, we built a number of FE models with different friction coefficients, and it turned out that the results of the numerical model with static coefficient of 0.25, kinetic coefficient of 0.10, and an exponential decay coefficient of 1 showed the closest match with the experimental results. Therefore, unless specified otherwise, the above friction coefficients were adopted for all contact interactions. 3. Experiment To validate the results of the numerical simulations, quasi-static experimental tests were carried out using the universal testing machine (Model CSS-44200, 20 tones) from Changchun Testing Machine Institute. For this purpose, we designed the grooved tube specimens and the test platform that is suitable for the oblique loading condition. The test setup is shown in Fig. 4. The platform included a support base with an oblique platen and a pressing block that could fix the specimen, and a large loading platen that could apply the compression load. The equipment was easy to assemble and could be used multiple times. The platform and the tube specimens were made of carbon steel Q235 and aluminum alloy AL6061-T6, respectively. All the components were manufactured by Shanghai Shili Machineries Co., Ltd. The tube specimens were lathed by a CNC lathe machine with a polycrystalline diamond Fig. 4. Experimental setup, (a) loading platform with the specimen, and (b) half of 3D drawing showing the details of the fixed end.
K. Tian, Y. Zhang and F. Yang et al.
International Journal of Mechanical Sciences 166 (2020) 105239
energy absorption performance is believed to be not much influenced. This is because that the plastic deformation is always the dominant part of energy dissipation/transformation compared with the inertia and the elastic parts, and also aluminum can be taken as a strain-rate insensitive material. 4. Theoretical modelling
Fig. 5. Three specimens tested, (a) original, (b) uniformly grooved, (c) gradiently grooved with groove depth decreasing from top to bottom. Table 2 Investigated tubes with different sequence of groove depths. Tube No.
Sequence of groove depths (mm)
1 2 3 4 5 6
1.0, 1.0, 1.0, 1.0, 1.0, 1.0 1.75, 1.50, 1.25, 1.0, 0.75, 0.50, 0.25 1.50, 1.50, 1.50, 0.75, 0.75, 0.75 1.50, 1.50, 1.50, 0.75, 0.75, 0.75, 0.25, 0.25 1.50, 1.50,1.50,1.00, 1.00, 1.00, 0.50, 0.50 0.50,0.50,0.75,0.75,1.0,1.0
To further demonstrate the mechanisms behind the energy absorption enhancement of the gradiently grooved tube under oblique loading, we developed theoretical models to calculate the crushing force and the absorbed energy of different tubes during the crushing process. The models were established from the energy viewpoint based on the kinematically admissible deformation mechanisms observed from the experiments. For the original tube, global bending is the dominant mechanism. Therefore, we followed the deductions by Wang and Qiu [36] for the energy absorption during the global buckling of a circular tube. The deformation mechanism is shown in Fig. 6. The deformation is localized to the region between the cross-sections BU and DS, where two flattened triangles ABC and ADC were formed and the ovalisation took place in the cross-sections. As per Wang and Qiu’s model, the dissipated energy E is composed of the part contributed by the bending section Ebend and the part contributed by the compression of the tube wall Ecomp . The energy by the bending section can be further decomposed into the energy dissipated at bending of the central horizontal hinge line Ehoriz , bending of the oblique hinge lines Eobliq , and lateral crushing of the wall near the bending section Ecrush . Therefore, 𝐸Bend = 𝐸horiz + 𝐸obliq + 𝐸crush + 𝐸comp The instantaneous crushing force is 𝐹Bend = 𝐹horiz + 𝐹obliq + 𝐹crush + 𝐹comp where, 𝐹horiz = 𝐹obliq =
(PCD) tool. No welding was used for the tubes. Three specimens, i.e., original, uniformly grooved and gradiently grooved tubes were manufactured for comparison purpose, as shown in Fig. 5. The groove depth of the uniformly grooved tube was 1 mm, and the groove depths of the gradiently grooved tube followed the sequence of No.2 in Table 2. Before testing, the tube specimens were annealed at a temperature of 500 Celsius degrees for 1 h followed by a gradual cooling to room temperature at a rate of approximately 100°/h. During the test, the specimen was fixed on the support base with a 15° oblique angle, and compressed by the loading platen that moved at a speed of 4 mm/min. Although the loading speed is much smaller than that in the FE simulation, the
(1)
𝐹crush = 𝐹comp =
d𝐸horiz d𝑢 d𝐸obliq d𝑢 d𝐸crush d𝑢 d𝐸comp d𝑢
(2)
(3)
(4)
(5)
In the above equations, 𝜃 is the rotation angle of the global bending, u is the displacement of the loading platen, M is the bending moment caused by different energy dissipations. The detailed expressions can be found in Appendix A. For the gradiently grooved tube, the deformation takes on a nonaxisymmetric progressive folding mode. Therefore, we calculated the Fig. 6. Deformation mechanism of the global bending mode.
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International Journal of Mechanical Sciences 166 (2020) 105239
Fig. 7. Deformation mechanism of the non-axisymmetric progressive folding mode.
Fig. 8. Deformed tube specimens in the compression experiments, (a) original, (b) uniformly grooved, and (c) gradiently grooved with groove depth decreasing from top to bottom.
Fig. 9. Deformed tube specimens in the FE simulations, (a) original, (b) uniformly grooved, and (c) gradiently grooved with groove depth decreasing from top to bottom.
energy absorption based on the modified version of the model of Singace [37] for the multi-lobe progressive collapse mode. The deformation mechanism is shown in Fig. 7. Each fold layer is composed of multiple triangular lobes, with the triangle apex rotating outward and the triangle base rotating inward. The energy dissipation E is composed of the part by bending the horizontal hinges Ehoriz , the part by bending the oblique hinges Eobliq , the part by decurving of the cylindrical wall Edecur , and the part by compression of the wall Ecomp . The horizontal hinges are assumed to coincide with the grooves, which was indeed the case in the experiments. Neglecting the change of the cross-section peripheral length (which is reasonable according to Ref. [21]), the energy dissipated at bending the horizontal hinges are
d𝐸horiz = 2𝜋𝑅𝑀p′ d(𝛽 + 𝛾)
(6)
where, 𝑀p′ =
𝜎p 𝑡 ′ 2 √ 2 3
is the plane strain plastic bending moment per unit
length of the hinge line, 𝜎p = (𝜎y + 𝜎u )∕2 is the characteristic flow stress which is taken as the average of the initial yield stress 𝜎 y and ultimate strength 𝜎 u , t′ = t − d is the tube thickness at the grooves. The two angles 𝛽 and 𝛾 represents the rotations of the two adjacent lobes whose normal points downwards and upwards, respectively. 𝛽 and 𝛾 are dependent on each other and can be represented by any one of them. Fig. 7(a) shows a special moment when one of the angles reaches its maximum 𝜋/2. At an arbitrary moment, the crushing force is 𝐹horiz = 2𝜋𝑅𝑀p′
d(𝛽 + 𝛾) d𝑢
(7)
Two scenarios are involved in a complete folding period. One scenario is when the two rotating lobes are connected at the triangle base, as indicated in the left wall of Fig. 7(b). The other scenario is when the two rotating lobes are connected at the triangle apex, as indicated in the
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International Journal of Mechanical Sciences 166 (2020) 105239
right wall of Fig. 7(b). It is noted that there are at least two horizontal hinges rotating at the same time, e.g., the hinges a, b, and c with rotation angles 𝛽+𝛾, 𝛾’, and 𝛽’ in Fig. 7(b). The total crushing force contributed by the horizontal hinges is ( ) ⎧[𝑓 (𝛽) + 𝑓 (𝛾 ′ )], 0 ≤ 𝑢 < 𝜆∕2 1 + cos 𝛾0 − cos 𝛽0 2 ( )] ( ) ( ) ⎪[ 1 𝐹horiz = 2𝜋𝑅𝑀p′ ⎨ 𝑓1 (𝛽) + 𝑓1 𝛽 ′ , 𝜆∕2 1 + cos 𝛾0 − cos 𝛽0 ≤ 𝑢 < 𝜆∕2 1 + cos 𝛽0 − cos 𝛾0 [ ( )] ( ) ⎪ 𝑓2 (𝛾) + 𝑓1 𝛽 ′ , 𝜆∕2 1 ≤ + cos 𝛽 − cos 𝛾 𝑢 ≤ 𝜆 0 0 ⎩
(8) where, the two functions f1 (𝛽) and f2 (𝛾) are used to indicate the derivative d(𝛽 + 𝛾)∕d𝑢 for the two folding scenarios. Assuming the oblique hinge lines are rotated simultaneously with the horizontal lines, the crushing force contributed by the oblique lines Fobliq is similar as Eq. (8) by replacing 2𝜋RM′p with 2NLob [cM′p + (1 − c)Mp ] where, Lob is the length of one oblique hinge line, N is the number of lobes along circumference, 𝑐 = 2𝑤∕𝜆 is the ratio of the groove width to the lobe height. The total instantaneous crushing force is Fig. 10. The crushing force versus displacement for the original tube, showing global bending.
Fig. 11. The crushing force versus displacement for the uniformly grooved tube.
𝐹Fold = 𝐹horiz + 𝐹obliq + 𝐹decur + 𝐹comp
(9)
The detailed expressions can be found in Appendix B. For the calculation of the progressive folding process, the grooved thickness t′ and hence the bending moment M′p were updated from fold to fold to account for the effect of the non-uniform groove depth. For the uniformly grooved tube, the crushing process can be divided into two stages. First, global bending took place near the middle point of the tube, causing the upper part of the tube to be rotated quickly. Second, after the fully collapse of the upper part, the deformation switched to progressive folding in the lower part. Therefore, the Wang and Qiu’s model and Singace’s model were used to quantitate the energy absorption during the two stages, respectively. The position where global bending occurred is determined by ( ) 𝑀0 𝑡′ 𝐿′ = (10) 𝐿 𝑀 0 (𝑡 ) where, M0 = 4𝜎 p R2 t is the critical bending moment of the cross-section, L′ is the length of tube above the bending position. It is noted that for all tubes, a partial fold is formed during the initial stage because of the oblique loading condition. For simplicity, the crushing force at this stage is assumed to be cpart FFold , where cpart is the ratio of the perimeter that plays part in the partial fold. In the √ calculations cpart is taken as 0.3, and the fold length is taken as 1.347 𝑅𝑡′ 5. Results 5.1. Comparison between the experimental and the FE numerical results
Fig. 12. The crushing force versus crushing displacement for the gradiently grooved tube, showing progressive folding deformation mode.
Fig. 8 shows the deformed configurations of the three tube specimens after the compression test. Considerable difference can be seen among the three tubes. For the original tube, the deformation is highly localized to the section near the fixed bottom end of the tube due to the large bending moment at that location. The uniformly grooved tube deformed partly in progressive folding mode and partly in global bending mode. On the other hand, the gradiently grooved tube showed no global bending, with all the sections deformed in progressive folding mode. Fig. 9 shows the simulated tube deformations. The FE simulations used exactly the same geometry, the same material properties and the same loading angle as the experiments. Comparing Fig. 9 with Fig. 8, the deformation modes obtained from the FE simulations agree very well with those from the experiments. Figs. 10–12 compare the load-displacement curves from the FE simulations with those from the experiments for the original, the uniformly grooved and the gradiently grooved tube, respectively. Satisfactory agreement was achieved between the simulations and the experiments for all three specimens, especially for the tendency of the crushing curve. As shown in Fig. 10, the force of the original tube took
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Fig. 13. Chronological snapshots of the deformed configuration during the crushing test at the same loading angle of 15°.
an abrupt drop after reaching its maximum value at the crushing distance of 25 mm, indicating quick loss of its load-bearing capacity after the global bending. It is observed from the experiment that the deformation is highly localized to the fixed end of the original tube. During the final loading stage, a crack appeared near the fixed end of the tube along the circumferential direction. As the loading platen went down, the crack propagated. Therefore, we terminated the loading ahead of time to prevent the specimen from breaking into two pieces. That is why the ultimate experimental displacement of the simple tube is noticeable less than the FE simulation value. From Fig. 11, the force of the uniformly grooved tube first decreased after the first peak, but then increased with further crushing. From Fig. 12, the crushing force of
the gradiently grooved tube shows a steady increasing trend during the crushing process, with its magnitude undulating due to the progressive folding. In the above figures, the apparent bold parts of the experimental curves were actually the small high-frequency fluctuations that were caused by the intermittent slippage between the tube and the loading platen. In the tests, a repeating thumping noise could be heard at the time when the small high-frequency fluctuations on the loading curves were generated. The difference between the deformation modes of the three tubes can be discerned more clearly from the chronological snapshots of the deformed configurations as shown in Fig. 13. For the original tube, global bending buckling took place near the fixed bottom end at the very initial
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International Journal of Mechanical Sciences 166 (2020) 105239
Fig. 14. Absorbed energy versus crushed displacement for the three tubes investigated, from the experiments, the FE simulations and the theoretical models.
Fig. 18. Load-displacement curves of tubes with different sequence of groove depths as in Table. 2 at the loading angle of 15°.
Fig. 15. Deformed configuration of the original tube for loading angles (a) 8° and (b) 10°.
Fig. 16. Deformed configuration of the uniformly grooved tube for loading angles (a) 12° and (b) 14°.
Fig. 19. Absorbed energy of tubes with different sequence of groove depths as in Table. 2 at the loading angle of 15°.
Fig. 17. Deformed configuration of the gradiently grooved tube at the loading angle of 25°. Table 3 Friction coefficients of investigated contact conditions.
Condition 1 Condition 2 Condition 3 Frictionless
Kinetic friction coefficient
Static friction coefficient
0.10 0.15 0.2 0
0.25 0.30 0.2 0
stage. Most of the tube sections reserved integrity during the crushing process, leading to inefficient material usage. For the uniformly grooved tube, global bending can be seen at the location of the sixth groove due to the reduced stiffness of the grooved section. However, the remained part of tube can still bear the crushing load through the progressive folding deformation. For the gradiently grooved tube, the deeper grooves near the loading end would facilitate partial local bulking under the oblique loading, causing the tube axis to rotate towards the loading di-
Fig. 20. Deformed configuration of (a) Tube 3, (b) Tube 4, (c) Tube 5 and (d) Tube 6 in Table. 2 for loading angles 15°.
rection. In this way, the global bending can be effectively prevented, leading to the significant energy absorption. Fig. 14 shows the curves of the dissipated energy during the crushing process, which was obtained by integrating the crushing force over the crushing distance. The solid lines correspond to the experimental results and the dashed lines correspond to the FE results. It can be seen that good consistency was achieved between the experimental curves
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International Journal of Mechanical Sciences 166 (2020) 105239
Fig. 23. Load-displacement curves of the original tube under different contact conditions at the loading angle of 8°, with inserted graphs showing the deformed configurations.
Fig. 21. Load-displacement curves of the original tube under different contact conditions at the loading angle of 15°.
Fig. 22. Load-displacement curves of the uniformly grooved tube under different contact conditions at the loading angle of 15°.
and the simulated curves for all three specimens. For example, taking the crushing distance of 80 mm for the original tube (the maximum crushing distance for this tube) and the crushing distance of 120 mm for the other two, the relative difference between the energy absorption from the simulations and the experiments are 4.2%, 5.7%, and 5.9%, respectively for the three tubes. In addition, the original tube absorbed more energy at a crushing distance smaller than 100 mm, while the gradiently grooved tube was the one that absorbed the most energy when the crushing distance was larger than 100 mm. The energy absorption predicted by the theoretical models are also compared in Fig. 14. Excellent correlation is achieved between the theoretical curves and the curves from the experiments and FE simulations for all three tubes investigated, indicating that the deformation mechanisms are well captured by our theoretical models. The experimental, numerical and theoretical results unanimously show that the gradiently grooved tube is preferable to the uniformly
grooved tube and the original tube. The quantitative agreement between the FE numerical results and the experimental results provides a validation for the FE models. Next, a parametric FE study was carried out to investigate the influences of various parameters such as the groove configuration, the loading angle and the friction coefficients on the energy absorption performance. 5.2. Critical loading angle The deformation mode of the tube is closely related to the loading angle. As the loading angle increases, the deformation mode may change from the progressive folding to the global buckling. The critical loading angle is defined as the loading angle at which the tube changes its deformation mode. The exact value of the critical loading angle is difficult to determine since the transition of deformation mode usually occurs gradually. Therefore, we switched our attention to the determination of
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International Journal of Mechanical Sciences 166 (2020) 105239
Fig. 24. Load-displacement curves of (a) Tube 2, (b) Tube 3, (c) Tube 4 and (d) Tube 5 under different contact conditions at the loading angle of 15°.
the range of the critical loading angle in this paper. For this purpose, a series of FE simulations were carried out with different loading angles increasing from 0° to 25° at an increment of 2°. As shown in Fig. 15, for the original tube, the deformation mode was progressive buckling at the loading angle of 8°, but changed to global bending at the loading angle 10°, indicating that the critical loading angle lies in the range between 8° and 10°. In the same way, the critical loading angle for the uniformly grooved tube was determined to be within the range of 12° to 14°, as shown in Fig. 16, which was obviously larger than the nongrooved counterpart. As for the gradiently grooved tube, the situation was more complex. This was because the deformation always involved obvious progressive deformation mode even at a relative large loading angle. Global bending, on the contrary, was not obvious for this kind of tube. Fig. 9 shows that the progressive deformation mode was well kept at the loading angle of 15°. From Fig. 17, the tube progressive folding played an important role even under a loading angle of 25°. The results indicate that the gradient-groove design could significantly increase the critical loading angle, and thus the energy performance of the circular tube.
5.3. Effect of the different groove depth designs on the energy absorption To seek for the best design of the non-uniformly grooved tube, FE simulations were carried out for a series of tubes with different groove designs. Table 2 shows the investigated tubes with different sequence of groove depths starting from the top end (loading end) of the tube. Each data in the sequence represents the depth of a pair of grooves on the inner and outer surfaces. All the tubes were loaded with the same loading angle of 15°. Fig. 18 shows the crushing force of the investigated tubes. It is obvious to see that global bending took place in the original tube and the uniformly grooved tube (Tube 1), as the crushing force shows an abrupt decrease. On the other hand, Tube 2 and Tube 3 mainly took on a progressive folding deformation mode, as the crushing force increased gradually with periodic undulation. Tube 4 and Tube 5 experienced a combined deformation mode, with first progressive folding and then global bending. From the energy-displacement curves shown in Fig. 19, we can see that the tubes with the groove depth deceasing from top to bottom all showed improved energy absorption compared with the original tube and the uniformly grooved tube when the crushing distance is
K. Tian, Y. Zhang and F. Yang et al.
large enough. On the contrary, the tube with the groove depth increasing from top to bottom (Tube 6) showed poor performance in terms of both the deformation mode and the energy absorption. Global bending can be easily initiated at the deepest bottom groove for this kind of tube. The deformation modes of the original, No. 1 and No. 2 tube are shown in Fig. 9 and the other tubes in Table 2 can be seen in Fig. 20. The above results indicate that the depth sequence of the groove design plays an important role in the tube’s deformation behaviour and the energy absorption performance under oblique loading condition. Groove depth in the non-uniformly grooved tube should be decreased from the loading end to the fixed end to achieve the enhanced energy absorption.
International Journal of Mechanical Sciences 166 (2020) 105239
(iii) The sequence of groove depths plays an important role in the tube’s energy absorption performance under oblique loading, with the groove depth decreasing from the loading end to the fixed end corresponding to the best performance, and the reversed depth sequence corresponding to the worst performance, (iv) The deformation mode tends to transit from global bending to progressive folding as the contact friction is increased for the original and the uniformly grooved tubes, and (v) The gradiently grooved tube with a suitable groove sequence is not as sensitive to contact condition as the original and the uniformly grooved tubes.
5.4. Effect of frictional contact condition
Declaration of Competing Interest
We further investigated the effect of contact friction condition on the energy absorption performance. The static-kinetic friction algorithm with exponential decay was used. Four contact conditions were investigated, with the friction parameters shown in Table. 3. Figs. 21 and 22 show the effect of contact condition on the load-displacement curves for the original tube and the uniformly grooved tube, respectively, under the loading angle of 15°. The results indicate that the contact condition had an evident effect on the crushing force and the absorbed energy. For both tubes investigated, the larger friction coefficients resulted in obviously larger crushing force, and thus larger energy absorption. The contact condition may also influence the deformation mode. For an example, Fig. 23 shows load-displacement curves of the original tube with and without friction for contact interaction under the loading angle of 8°. The inset pictures show the deformed configurations for the two contact conditions. The tube with frictional contact property (Condition 1) took on a progressive deformation mode, while an evident global bending can be seen for the frictionless case. It indicates that the deformation mode would transform from global bending to progressive folding as the contact friction is increased for the original tube. Similar phenomenon was also observed for the uniformly grooved tube. In Fig. 24, the load-displacement curves of Tube 2(a), 3(b), 4(c) and 5(d) (in Table 2)are shown for different friction conditions. Contrary to Figs. 21 and 22, Fig. 24(a) shows that the influence of friction coefficients was not significant for Tube 2. Even under frictionless contact condition, Tube 2 still took on the progressive folding deformation mode. As for the other three tubes, the crushing force decreased rapidly after global bending. The results indicate that the tube with a suitable groove sequence is not as sensitive to contact condition as the original and the uniformly grooved tubes, which also highlights the advantage of the gradiently grooved tube. But when global bending occurs, the crushing force decreases a lot.
The authors declare that there is no conflict of interests regarding the publication of the article titled “Enhancing energy absorption of circular tubes under oblique loads through introducing grooves of non-uniform depths” in Int J Mech Sci.
6. Conclusions In this study, dynamic crashing behavior of grooved circular tubes with non-uniform groove depth subjected to oblique impact loading was explored. A series of FE numerical simulations and experimental tests were carried out to investigate the performance of different designs with different sequence of groove depths. Theoretical models were developed to demonstrate the energy absorption mechanisms. The influences of key parameters such as the groove depth, the loading angle and the contact condition were analyzed. The work in this paper can provide a guide for the design of effective energy absorbers under the oblique loading condition. Several conclusions can be drawn as follows. (i) The gradiently grooved tube whose groove depth decreases from loading end to fixed end is advantageous to the original and the uniformly grooved tubes for energy absorption under the oblique loading condition, (ii) The critical loading angle at which global bending occurs can be significantly increased with the introduction of grooves,
Acknowledgements This work was supported by National Natural Science Foundation of China (11772231), the Students Innovation Training Program (SITP) of Tongji University, and Shanghai Supercomputer Center. We also thank Profs Liming Zhu and Jinlong Zhu for their help in the experiments. Appendix A For global bending mode, the crushing forces contributed by the energy dissipation mechanisms within the bending region are 𝑀horiz 𝐿(sin (𝛼 + 𝜃) − 𝜇 cos (𝛼 + 𝜃))
𝐹horiz = 𝐹obliq =
𝑀obliq 𝐿(sin (𝛼 + 𝜃) − 𝜇 cos (𝛼 + 𝜃))
𝐹crush =
𝑀horiz =
𝑀obliq =
𝑀crush =
𝑀crush 𝐿(sin (𝛼 + 𝜃) − 𝜇 cos (𝛼 + 𝜃)) ( ) 0.45 (−𝑅 + 𝑟)2 ∕(𝑅2 𝐻)𝑀0 𝜙3 𝜃∕2
(A1)
(A2)
(A3)
(A4)
(√ ) 1∕32 2∕ 𝑅 2 ( 𝑅 − 𝑟 ) 𝑀 0 𝜙 𝜃∕2 √ 0.25((𝑅 + 𝑟)∕𝑟)𝑀0 (𝜋∕𝑅) 2𝜙 𝜃∕2
where, 𝜙 =
2(𝑅𝐻𝜃∕2)1∕4 √ 𝑅−𝑟
(A5)
(A6)
is the angle characterizing the section ovalisa-
tion, r = 0.6R, H = 1.31R are the geometric parameters, 𝜇 is the friction coefficient. The crushing force contributed by the compression of the tube wall is 𝐹comp =
𝑀comp
𝑀comp 𝐿(sin (𝛼 + 𝜃) − 𝜇 cos (𝛼 + 𝜃))
⎧ 4𝜎 𝑅2 𝑡(𝜃 −sin 𝜃 ) 4 ⎪ p 24 , 𝜃3 ≥ 𝜃4 (𝜃∕2) ⎪ 4𝜎 𝑅cos 2 𝑡(𝜃 −sin 𝜃 ) 3 3 =⎨ p , 𝜃3 ≤ 𝜃4 cos2 (𝜃∕2) ⎪ ⎪ 0, 𝜃3 = 0 ⎩
where, 𝜃3 = arccos
(
) 𝑢 −1 , 4𝑅
(A7)
(A8)
(A9)
K. Tian, Y. Zhang and F. Yang et al.
and
(
𝜃4 = 𝜋 − sin
𝜉 𝑅
International Journal of Mechanical Sciences 166 (2020) 105239
The corresponding force is
) (A10)
are the angular parameters on the loading section and the bending section, respectively. 𝜉 = 𝜋𝑅 − 𝑟𝜙 − (𝜋 − 𝜙)𝑅
(A11)
𝐹decur =
8𝑀p d𝐸decur = d𝑢 sin 𝛽 + tan 𝛾 cos 𝛽
(B16)
The energy dissipated by the compression of the tube wall is d𝐸comp = 2
𝜋
∫0
( ) 𝜎𝑦 𝜀max 1 − cos 𝜃1 𝑡𝑅d𝜃1 d𝑢 = 4𝜋𝜎𝑦 𝑅2 𝑡 tan 𝛼d𝑢∕𝐿
is the length of central hinge AC
where,
Appendix B
𝜀max = 2𝑅 tan 𝛼∕𝐿
(B17)
(B18)
The corresponding force is For the multi-lobe progressive folding mode, the following relations can be obtained from Fig. 7(a) sin 𝛾0 = 𝑚
(B1)
sin 𝛽0 = 1 − 𝑚
(B2)
sin 𝛾 = sin 𝛽 − (1 − 𝑚) ) 𝜆 𝜆( cos 𝛽0 − cos 𝛽 + (1 − cos 𝛾) 2 2
(B3) (B4)
d𝛾 =
cos 𝛽 d𝛽 cos 𝛾
(B5)
d𝑢 =
𝜆 𝜆 (sin 𝛽d𝛽 + sin 𝛾d𝛾) = (sin 𝛽 + tan 𝛾 cos 𝛽)d𝛽 2 2
(B6)
d(𝛽 + 𝛾) d𝛽 2(cos 𝛾 + cos 𝛽) = d𝛽 d𝑢 𝜆 cos 𝛾(sin 𝛽 + tan 𝛾 cos 𝛽)
(B7)
𝑓1 (𝛽) =
For the second scenario, sin 𝛽 ′ = sin 𝛾 ′ − 𝑚 𝑢=
(B8)
) 𝜆 𝜆( cos 𝛾0 − cos 𝛾 (1 − cos 𝛽) + 2 2 cos 𝛾 ′ ′ d𝛾 cos 𝛽 ′
d𝛽 ′ = d𝑢 =
(B9)
(B10)
) 𝜆( ) 𝜆( sin 𝛽 ′ d𝛽 ′ + sin 𝛾 ′ d𝛾 ′ = sin 𝛾 ′ + tan 𝛽 ′ cos 𝛾 ′ d𝛾 ′ 2 2
( ) ( ) 2 cos 𝛽 ′ + cos 𝛾 ′ ( ) d 𝛽 ′ + 𝛾 ′ d𝛾 ′ 𝑓2 𝛾 ′ = = d𝛾 ′ d𝑢 𝜆 cos 𝛽 ′ (sin 𝛾 ′ + tan 𝛽 ′ cos 𝛾 ′ )
(B11)
(B12)
The crushing force contributed by the oblique lines is [ ] 𝐹obliq = 2𝑁 𝐿ob 𝑐 𝑀 ′ p + (1 − 𝑐 )𝑀p ( ) ⎧[𝑓 (𝛽) + 𝑓 (𝛾 ′ )], 0 ≤ 𝑢 < 𝜆∕2) 1 + cos 𝛾0 − 2 ( )] ( ( cos 𝛽0 ) ⎪[ 1 ′ ⎨[𝑓1 (𝛽) + 𝑓1 (𝛽 )], 𝜆∕2 1 + cos 𝛾0 − ( cos 𝛽0 ≤ 𝑢 < 𝜆∕2) 1 + cos 𝛽0 − cos 𝛾0 ⎪ 𝑓2 (𝛾) + 𝑓1 𝛽 ′ , 𝜆∕2 1 + cos 𝛽0 − cos 𝛾0 ≤ 𝑢 ≤ 𝜆 ⎩
(B13) where,
√
𝐿ob =
(
𝜋𝑅 𝑁
)2 +
( )2 𝜆 2
(B14)
The energy dissipated by the decurving of the cylindrical wall Edecur is d𝐸decur = 𝜆
d𝐸comp d𝑢
= 4𝜋𝜎𝑦 𝑅2 𝑡 tan 𝛼∕𝐿
(B19)
References
where, m is the folding parameter indicating the inward ratio of the lobe length. For the first scenario,
𝑢=
𝐹comp =
2𝜋𝑅
∫0
𝑀p 𝜅d𝐿2∕𝜋d𝛽 = 4𝑀p 𝜆d𝛽
(B15)
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