aluminium hybrid structures for transverse loading

Accepted Manuscript

Energy Absorption Mechanics and Design Optimization of CFRP/Alumimun Hybrid Structures for Transverse Loading Guangyong Sun , Hang Yu , Zhen Wang , Zhi Xiao , Qing Li PII: DOI: Reference:

S0020-7403(18)32160-X https://doi.org/10.1016/j.ijmecsci.2018.10.043 MS 4599

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

5 July 2018 15 October 2018 20 October 2018

Please cite this article as: Guangyong Sun , Hang Yu , Zhen Wang , Zhi Xiao , Qing Li , Energy Absorption Mechanics and Design Optimization of CFRP/Alumimun Hybrid Structures for Transverse Loading, International Journal of Mechanical Sciences (2018), doi: https://doi.org/10.1016/j.ijmecsci.2018.10.043

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Highlight 

Investigated energy absorption mechanics and design optimization of CFRP/Alumimun hybrid Tubes under transverse loading.



Found the energy absorption of hybrid tube was affected largely by the number of ply layers (thickness) but marginally by the ply angles. The 𝑆𝐸𝐴 of the optimized CF-AL tube design is increased by 42.96% in comparison

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

with the intial baseline structures.

Energy Absorption Mechanics and Design Optimization of

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CFRP/Alumimun Hybrid Structures for Transverse Loading Guangyong Sun1, 2, 3, Hang Yu1, Zhen Wang1, Zhi Xiao1, *, Qing Li2 1

State Key Laboratory of Advanced Design and Manufacture for Vehicle Body, Hunan University, Changsha, 410082, China

2

School of Aerospace, Mechanical and Mechatronic Engineering, The University of 3

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Sydney, Sydney, NSW 2006, Australia

State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an

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Jiaotong University, Xi’an, 710049, China

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ABSTRACT

This study aimed to explore bending collapse behavior and energy absorption capacity of

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net aluminum (Al), net carbon fiber reinforced plastic (CFRP) and Al/CFRP hybrid tubes, respectively. Based upon the experimental tests, the transverse energy absorption of the

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Al/CFRP hybrid tubes was found to be even higher than the sum of corresponding net Al tube and net CFRP tube. Specifcially, the CF-AL tube (the CFRP tube being placed outside the Al tube) increased the peak force by 6.7% and energy absorption by 20.6%. The AL-CF tube (the CFRP tube being placed inside the Al tube) improved the peak force by 14.1% and energy absorption by 19.1%. The experimental study indicated that overall, the CF-AL tube was of better crashworthiness characteristics. Subsequently, finite element (FE) analyses were carried

* Corresponding Author: Tel: +86-731-88821445; Email: [email protected]. 1

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out by correlating with the experimental results. Based upon this validatedd FE models, a parametric study and design optimization on the CF-AL tube (with respect to length, thickness and ply angle) were further performed. It was found that the optimization increased specific energy absorption (SEA) by 42.96% and mean crushing force by 37.75%; meanwhile the mass of optimum design decreased by 5.02%, exhibiting significant enhancement of

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carashworthoness characteristics.

Keywords: Aluminum/CFRP hybrid structure; Crashworthiness; Three-point bending;

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Structural optimization.

1. Introduction

Nowadays, reduction of fuel consumption and greenhouse gases emission has become a primary goal in automotive industry. It is estimated that 10% reduction of vehicle weight

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could lead to up to 7% saving of fuel; in other words, every 1 kg reduction of vehicle weight will result in about 20 kg reduction in carbon dioxide emission per 100 km [1-3]. Lured by

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growing environmental and economic concerns, lighweighting signifies an important topic of research and development currently. A series of lightweight materials, such as aluminum and

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carbon fiber reinforced plastic (CFRP) composites, have been gradually replacing traditional steel materials in vehicle sturctures.

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As a most typical energy absorber used in vehicle body, thin-walled tubes [4, 5] have been extensively studied under different crash conditions, in which exhuastive attention has

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been paid to axial impact scenarios. Nevertheless, Kallina et al. [6] revealed that up to 90% of the real world vehicle crashes involved collapse failure of structural members in transeverse direction. It is thus particularly important to investigate the crash characteristics and energy absorption of thin-walled structures under lateral conditions [7]. As a relatively light metal, aluminum has been increasingly used in automobile systems, such as powertrain, chassis, and body structures [8]. Substantial studies have been undertaken for a range of aluminum structures and components under transverse loading to improve the energy absorption and crashworthiness [9, 10]. Bending behavior of aluminum tubes was 2

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demonstrated to be rather critical for both static and dynamic loading [11-14]. To further improve the energy absorption characteristics of aluminum tubes, filling and/or mixing with other materials have drawn intensifying attention. For example, Duarte et al. [15] investigated dynamic and quasi-static bending characteristics of aluminum tubes filled with aluminum foam. They demonstrated that the filled tube is of higher load carrying capacity and higher energy absorption efficiency in comparison with the empty tube and foam filler alone.

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Hanssen et al. [16] experimently studied the bending performance of foam filled aluminum tubes under quasi-static three-point bending. They found that aluminum foam could appreciably influence the local deformation modes of the aluminum beams. Guo et al. [17] investigated bending responses of sandwiched double tube filled with aluminum foam; and

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they found that the double tube sandwich structure deformed more uniformly and stably, making it more efficient in load bearing and energy absorption.

As other class of lightweight materials, composites have undergone rapid growth of applications in automotive engineering recently, in which glass fiber reinforced plastic (GFRP)

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and carbon fiber reinforced plastic (CFRP) are two most representative material candidates [18-20]. A properly designed composite structure can have outstanding advantages of

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lightweight, high specific strength and high energy absorption capacity [21]. For example, Liu et al. [22] investigated double hat shaped CFRP tubes under both axial crushing and

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transverse bending conditions; and they found that the peak load and specific energy absorption (SEA) under transverse bending were 10% and 1% of those in axial crushing.

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CFRP structures have a superior capacity of energy absorption through complex failure mechanism in a form of matrix cracking, delamination, fiber breakage, debonding between

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fiber and matrix. However, it was reported that CFRP structures often experience an elastic strain within only 1-2%; and then tend to fail in a brittle fracture mode under a transverse loading. Because of such failure modes, net CFRP tubes could be of relatively low energy absorption capibility, in particular under a bending condition [23, 24]. To enhance energy absorption of CFRP tubes, incorporation of relatively stable plastic deformation in metal materials can be a feasible approach to guide the CFRP to deform and fail in a more effective way. In other words, CFRP could be changed from the catastrophic brittle failure to a more progressive and uniform failure so that more energy can be absorbed 3

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by the CFRP materials per se [25, 26]. For this purpose, Al/CFRP hybrid tubes have been explored recently. In this regard, Sun et al. [27-29] investigated the crushing behavior of Al/CFRP hybrid tubes under quasi-static axial and oblique crushing loads, in which the crushing characteristics of different configuration of Al/CFRP hybrid tubes were explored. Hussein et al. [30] studied the energy absorption of square CFRP tubes wrapped externally with aluminium sheets under quasi-static crushing. Eksi et al. [31, 32] examined the effects of

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inner and outer reinforcements on bending and buckling behaviors of aluminum tube through a series of hybrid configurations. Kim et al. [33] explored the bending behavior and energy absorption of short Al/CFRP hybrid square beam under transverse loading; and they demonstrated sizeable improvement of energy absorption without sacrificing lightweighting

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features.

In spite of these abovementioned studies on the crash characteristics of hybrid tubes, few have compared the crashworthiness characteristics of different hybrid metal/composites configurations and none has explored the effects of structural parameters systematically. This

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study aimed to compare two different hybrid configurations, namely CF-AL and AL-CF tubes, through the experimental tests first. Then, finite element (FE) analyses were performed by

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correlating with the experimental results. Based upon the validated numerical simulation, a parametric study was further carried out to systematically investigate the effects of length,

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thickness and ply angle of CFRP. Finally, a design optimization was conducted for further

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improving the transverse crashworthiness of structures.

2. Materials and Methods for Experiments

2.1 Specimen preparation

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In this study, net aluminum (Al), net CFRP and two different hybrid tubes were tested

under the transverse loading condition. The circular CFRP tubes were made in prepregs with T300 0/90° woven carbon fabric and epoxy. The prepregs of CFRP were wrapped outside a mold and then cured in an autoclave supported by vacuum bag to ensure sufficient pressure. The circular aluminum (Al) tubes were prepared by wire cutting electrical discharge (WCED) process from the rod-shaped aluminum ingots of AA6061O. Figs. 1(a) and (b) illustrate the experimental specimens of circular CFRP and aluminum tubes. The length of the circular CFRP tube was 300 mm, and the outer diameter was 60 mm. 4

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All the CFRP tubes have 9 plies with a total thickness around 2.25 mm. For the aluminum tubes, two different diameters of 55mm and 65mm (namely Al-55 and Al-65) were adopted, respectively. Both the aluminum tubes have a thickness of 2.25 mm, which is the same as the CFRP tube. For the aluminum/CFRP hybrid tubes, the adheisive was evenly smeared on the outer surface of the smaller tube first; then the smaller tube was inserted into the larger tube, as

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illustrated in Figs. 1(c) and (d). According to the different arrangements of inner and outer materials, the hybrid tubes were named as CF-AL (outer CFRP and inner aluminum) and AL-CF (inner CFRP and outer aluminum) tubes, respectively. The details of all the specimens

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are summarized in Table 1.

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(a) (b) (d) (c) Fig. 1 CFRP/Al hybrid configurations: (a) schematic, (b) Aluminum and CFRP tubes, (c) CFRP tube placed outside aluminum tube (CF-AL). (d) CFRP tube placed inside aluminum tube (AL-CF).

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Table 1 Structural parameters of the different tubes. D (Outer diameter)

L (Length)

T (Thickness)

Mass

(mm)

(mm)

(mm)

(g)

CFRP

60.00

300.00

2.25

181.14

Al-55

55.00

300.00

2.25

294.96

Al-65

65.00

300.00

2.25

356.23

CF-AL

60.00

300.00

4.75

498.02

AL-CF

65.00

300.00

4.75

545.77

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Specimen type

A-55: Al tube whose diameter is 55mm, A-65: Al tube whose diameter is 65 mm. CF-AL: hybrid tube where the CFRP tube placed outside the Al tube.

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2.2 Three-point bending tests The quasi-static three-point bending tests have been commonly used to characterize transverse loading behavior [34]. In this study, the specimens of net CFRP tube, net aluminum tube and hybrid tubes were all loaded in a consistent quasi-static three-point bending condition. INSTRON5894 universal testing machine with a loading capacity of 150 kN was

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used to carry out the experiments. The diameter and length of loading nose tip were 25 mm and 80 mm, respectively. The diameter and length of supporter tips were 20 mm and 80 mm, respectively. The loading rate was 2mm/min to avoid a creep effect in the CFRP and adhensive layer [35]. To faciliate the comparison, the maximum displacement of the loading

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nose which can be measured accurately from the test system was set to be 60 mm (i.e. the same as the diameter of net CFRP tube) in all the tests so that drastic failure of tubes can be observed. The specimen was placed on the two cylindric supports as seen in Fig. 2.

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Loading rate: 2 mm/min

Span: 210 mm

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Diameter: 25 mm

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Fig. 2 Experimental setup for three-point bending test.

2.3 Evaluation criteria of crashworthiness Different criteria are commonly used to quantify the crashworthiness performance. In

this study, peak force 𝐹𝑚𝑎𝑥 , energy absorption (𝐸𝐴), specific energy absorption (𝑆𝐸𝐴), mean crushing force ( 𝐹𝑎𝑣𝑔 ), and crush force efficiency ( 𝐶𝐹𝐸 ) were adopted to assess the crashworthiness characheristics. The maximum displacement of the loading nose (60mm) was used to calculate the crashworthiness indicators of all the specimens. The peak force (𝐹𝑚𝑎𝑥 ) can be obtained directly from the load-displacement curve. The

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energy absorption (𝐸𝐴) is obtained by integrating the load-displacement curve, as, 𝑑

𝐸𝐴 = ∫0 𝐹(𝑥)𝑑𝑥

(1)

where 𝑑 is the displacement of loading nose, 𝐹(𝑥) is the instantenous crushing force at displacement 𝑥 . To compare energy absorption capacity of different specimens in consideration of mass (𝑚) effect, the specific energy absorption (𝑆𝐸𝐴) is commonly used as, 𝐸𝐴 𝑚

The mean crushing force (𝐹𝑎𝑣𝑔 ) is calculated as, 𝐹𝑎𝑣𝑔 =

𝐸𝐴 𝑑

(2)

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𝑆𝐸𝐴 =

(3)

To measure the uniformity of crushing force, the crush force efficiency (𝐶𝐹𝐸) is adopted

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in terms of the ratio of the mean crushing force (𝐹𝑎𝑣𝑔 ) to the peak force (𝐹𝑚𝑎𝑥 ), as 𝐹𝑎𝑣𝑔

𝐶𝐹𝐸 = 𝐹

𝑚𝑎𝑥

(4)

3. Experimental Results and Analysis

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3.1 Net aluminum tubes

The load-displacement curves of the net aluminum tubes are shown in Fig. 3 together

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with different failure modes at different loading displacements. In the transverse loading process, the aluminum tube mainly experienced local deformation underneath the loading

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nose. It can be seen that the tube underwent severe plastic deformation during the entire loading process and no crack was observed. The indentation occurred around the compression

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region with local collapse, forming plastic hinges. The main deformation mode was the distortion (ovalization) in the cross-section of net aluminum tubes due to the inward stress

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components [36]. The indentation also occurred near the cylindrical supports. By comparing these two aluminum tubes having different diameters (diameter 55mm for Al-55 and diameter 65mm for Al-65), the collapse modes were rather similar under the three-point bending tests. The load-displacement curves of aluminum tubes with different diameters (55 and 65mm) also followed the similar trends, where the load increased first and then decreased when it reached the peak value. The differences of these two curves are the initial stiffnesss, peak load, and the displacement corrresponding to the peak load. In Fig. 3(a), the load-displacement

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curves can be divided into three regions, namely linear elastic, plastic buckling, and bending collapse with a plastic hinge. In the elastic region, the load increased almost linearly and the slope of the curve thus defines the stiffness of tube. After this process, plastic buckling took place, which was confirmed by the plastic failure mode on the front and rear sides of the specimen underneath the loading nose before reaching the peak load. Due to the plastic deformation, the stiffness reduced, lowering the buckling strength. After reaching the peak

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load, the bending collapse was observed and the load started declining. Fig. 4 displays the finally deformed specimens. During the collapse process, the specific energy absorptions (𝑆𝐸𝐴𝑠) of the aluminum tubes Al-55 and Al-65 were calculated to be 1.161J/g and 1.045J/g,

III

II

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I

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respectively.

Region I

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Region II

Region III Al-65

Region I

(a)

Al-55

Region II

Region III

(b)

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Fig. 3 Load-displacement curves and deformation modes of aluminum tubes under three-point bending tests.

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Indentation Al-55

Plastic hinge

Identation Plastic hinge

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Al-65

Fig. 4 Final deformations of the two aluminum tubes (diameters of 55 and 65mm) after tests.

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3.2 Net CFRP tube

The load-displacement curve and deformation/failure mode of the CFRP tube are shown in Fig. 5. In the beginning, small circumferential crack appeared right underneath the loading nose. With increase in the displacement, cracks propagated along the circumferential direction.

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Buckling was seen around the circumference of the tube, which was similar to the aluminum tubes. It was observed that the region with buckling crushed more catastrophically than the

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other regions; thus the fracture first appeared in the circumferential direction. From the load-displacement curve, it can be seen that the loading capacity dropped sharply at the

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displacement of 30mm, where the circumferential fractures became severe. Due to decline of

nose.

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loading capacity, the high stress was concentrated around the contact region with the loading

With the loading nose pressed onto the upper surface, the axial crack initiated and

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propagated along the axial direction due to the stress concentration. Correspondingly, significant loading drops took place twice after reaching the peak. From these specimen photos at the different displacements shown in Fig. 5, the two major drops of loading capacity were mainly due to cracking in the circumferential and axial directions in sequence. Hence, it can be concluded that the main failure modes of the CFRP tube were the development of cracks in the circumferential and axial directions, both initiated from the loading site. Fig. 6 shows the finally deformed and failed CFRP tube, which exhibits that all the

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cracks on the upper surface followed the circumferential and axial directions. Meanwhile, the symmetric cracks were identified on the other side of the upper surface, indicating that the upper surface of the CRFP tube had almost lost the loading capacity after fractured locally. Neveretheless, the failure on the lower surface was not as severe as that on the upper surface. Specifically, the axial cracks were initiated in the resin and some fractured carbon fibers were observed at the outermost plies of the lower surface without seeing other failure modes. Also,

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some small cracks appeared in the region near the supports. For this reason, the CFRP tube still remained certain capacity of further loading due to the minor damage/cracking on the lower surface. In addition, the phenomenon of rebounding was observed after testing, in which the CFRP tube nearly restored to the initial curvature as soon as unloading, mainly due

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to the fact that there was no major cracking appeared on the lower surface. From the fracture direction, it was expected that the circumferential cracks would keep propagating with increasing displacement of loading nose; and finally could meet the axial cracks on the lower

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surface somewhere, until then the CFRP tube would lose the loading capacity completely.

Fracture in axial direction Fracture in circumferential direction

Slight circumferential crack and initial region of fractures

Fig. 5 Load-displacement curve and failure modes of CFRP tube under the three point bending test.

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Symmetric fractures

(a) Circumferential fracture

Axial fracture

(b)

Two cracks on the region of supports

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Fracture propagate in circumferential direction

Crack in bottom

Fig. 6 Failure modes of CFRP tube after testing : (a) upper surface. (b) lower surface.

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3.3 CF-AL hybrid tube

The displacement-force curve and deformation/failure modes for the CF-AL tube (the CFRP tube being placed outside the Al tube) are shown together in Fig. 7. It was found that the failure modes of the outer CFRP tube are somewhat similar to those of the net CFRP tube. The main failure modes are the circumferential and axial cracking in the loading region of the

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tube which fairly resembled to those of the sole CFRP configuration. It is noted that the

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cracks were initiated from the outer CFRP tube in the circumferential and axial directions underneath the loading nose. Note that the severe fractures on the lower CFRP surface can be also observed, which appeared very different to those on the lower surface of the net CFRP

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tube. This is due to the interaction between the inner aluminum tube and the outer CFRP tube.

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The plastic deformation in the inner aluminum tube engaged severe axial cracking on the lower surface of the outer CFRP tube. As a result, the outer CFRP tube crushed more

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substantially than the sole CFRP tube due to the strong interaction with the inner aluminum tube. Nevertheless, due to the support of the inner aluminum tube, the crack propagation in the outer CFRP tube was smoother and more stable than the sole CFRP tube, indicating the reinforcement role played by the inner aluminum tube.

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Fracture in axial and circumferential direction

Slight circumferential crack and initial region of fractures

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Fracture in circumferential direction

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Fracture in lower face

Fig. 7 Load-displacement curve and deformation modes of CF-AL tube under the transverse loading.

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Fig. 8 shows the finally-deformed specimen after the test. Due to the constraint of the inner aluminum tube and complete failure of the outer CFRP tube, no obvious rebounding

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was observed after unloading. The deformation modes of the inner aluminum tube appeared to be the same as those of the sole aluminum tube and no fracture observed. For the outer

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CFRP tube, indentation of the loading nose on the upper surface appeared to be different with that of the sole CFRP tube. Interestingly, the circumferential fracture was not appearred on the

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upper surface. For the outer CFRP tube, especially on the lower surface, it can be found that the failure region was more extensive than that of the sole CFRP tube, which implies better

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capacity of energy absorption for failure. The load-displacement curve of the CF-AL tube (as plotted in Fig. 7) is smoother than that of the sole CFRP tube, particularly in the high displacement region due to more stable collapse of the outer CFRP tube as discussed above. As a result, the CF-AL tube has a better capacity of energy absorption than the CFRP tube.

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Identation

Fracture in axial direction

(a)

Fracture in

(b) circumferential direction

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circumferential

Fracture in axial direction

Fig. 8 Failure modes of CF-AL tube after testing: (a) Upper surface. (b) Lower surface.

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3.4 AL-CF hybrid tube

The load-displacement curve and deformation modes of the AL-CF hybrid tube are shown in Fig. 9. It was found that the deformation modes of the aluminum tube surface are almost identical to those of the sole aluminum tube. Interestingly, the plastic deformation of

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the outer aluminum tube is not as symmetric as that in the sole aluminum tube, mainly due to the asymmetric failure modes induced by the inner CFRP tube, which had interacted with the

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outer aluminum tube in the different regions. By comparing the curves of the AL-CF tube with that of the sole CFRP tube, a similar pattern up to the 30mm displacement was observed,

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in which the transverse loading dropped sharply and then gradually increased. At this moment, severe fracture occurred in the inner CFRP tube, i.e. the circumferential cracking in the

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loading region of the inner CFRP tube. After the inner CFRP tube fractured, the interaction between the outer aluminum tube and inner CFRP tube became more substantial, leading the

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transverse force to rising again. The smooth load-displacement curve at large displacement indicates that no catastrophic cracking appeared in the CFRP tube. The deformed specimen is shown in Fig. 10 with a fairly similar deformation pattern to

the sole aluminum tube. Neveretheless, the indentation of loading nose on the outer aluminum tube appeared deeper than that of the net aluminum tube. A short crack along the axial direction occurred right underneath the rear side of the loading nose. Due to the internal reinforcement from the CFRP tube, the rigidity of the AL-CF tube was improved in comparison with the net aluminum tube. At the same deflection, the loading on the AL-CF 13

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tube became higher, making the local indentation on the outer surface of the aluminum tube

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deeper.

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Asymmetrical plastic deformation

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Fig. 9 Load-displacement curve and failure modes of AL-CF tube under three-point bending test. Convex region

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Indentation

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Short fracture

Convex region

Short fracture in alumimum

Short fracture in CFRP

Fig. 10 Failure modes and local deformation/failure of AL-CF tube after test.

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By close observation of the short fracture on the upper surface of the outer aluminum tube, it can be seen that the inner CFRP tube experienced a similar circumferential crack right

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underneath the cracking site of the outer aluminum tube. It was estimated that the cracks in the inner CFRP tube initiated before the outer aluminum tube cracked due to the brittleness of CFRP. The cracks in the inner CFRP tube changed the interaction with the outer aluminum tube in this region. Comparing with the sole aluminum tube, the hybrid tube had different plastic deformation modes in the convex region of the outer aluminum tube. Since the failure modes of the inner CFRP tube in this region were rather asymmetric, making the interaction between the inner CFRP tube and outer aluminum tube unbalanced.

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(5)

𝐹(𝑥)(𝐶𝐹𝑅𝑃+𝐴𝑙−65) = 𝐹(𝑥)𝐶𝐹𝑅𝑃 + 𝐹(𝑥)𝐴𝑙−65

(6)

where subscript (𝐶𝐹𝑅𝑃 + 𝐴𝑙 − 55) indicates the force summation of sole CFRP tube and

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sole aluminum tube with the diameter of 55mm; (𝐶𝐹𝑅𝑃 + 𝐴𝑙 − 65) denotes the sum of sole CFRP tube and sole aluminum tube with the diameter of 65mm.

From the displacement-force and displacement-energy absorption curves plotted in Fig. 11, the interactive effects can be observed, which increased the total enregy absorption by 118

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J and 114 J for the CF-AL tube and AL-CF tube, respectively. Moreover, It can be found that the peak force of the sole CFRP tube was much higher than both the sole aluminum tubes with the two different diameters. However, the loading of the sole CFRP tube dropped sharply and became much lower than those of both the sole aluminum tubes (55 and 65mm, respectively) in the large displacement region due to catastrophic fracture in the CFRP wall. A

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stepped drop pattern can be seen in the sum of the force-displacement curves, which followed

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that of the sole CFRP tube. By comparing the curves of the hybrid tubes with the sums of the corresponding sole tubes, it can be found that these two curves are fairly similar at the early stage before the displacement reached 20mm. Overall, the curves of the hybrid tubes are

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smoother than that calculated through the force superpositions, in particular over the large

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displacement region. More interestingly, the loading level of the hybrid tubes is substantially higher than the sum when the displacment is over 20mm. The increase in the peak force is

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mainly due to the sizable interaction between the outer and inner tubes.

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Fig. 11 The load-displacement and energy absorption-displacment curves and the interative

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effects between the CFRP and aluminum tubes: (a) CF-AL. (b) AL-CF.

By observing the region with evident discrepancy between these two curves (i.e. grayed

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area), the enhancement of loading capacity can be identified from the displacement 20mm onwards. At this critical displacement (20mm), the load of the sole CFRP tube reached the

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peak value first; and then the cracks were initiated and propageted on the surface of the sole CFRP tube as analyzed in the previous section. However, due to the support and restriction of the aluminum tube, the hybrid tube still remained high loading capacity and the fracture of the CFRP tube in the hybrid structure developed more stably. This also means that the hybrid tube has better energy absorption capacity than the sum of the corresponding constituent single tubes. For this reason, the hybrid tubes with the two different configurations are both able to largely enhance the loading capacity and energy absorption in comparison with the simple

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superposition of their corresponding constituent tubes. 3.6 Crashworthiness indicators The crashworthiness indicators of such different configurations were summarized in Table 2. It can be seen that the 𝑆𝐸𝐴 of sole CFRP tube is the highest due to its low material density, high specific stiffness and high specific strength. The 𝑆𝐸𝐴 of the hybrid tubes are

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lower than the sole CFRP tube but higher than the sole aluminum tubes; while the 𝐶𝐹𝐸 of the hybrid tubes are higher than the sole CFRP tube but lower than the sole aluminum tubes. From Table 2, it is evident that the energy absorption and peak force were improved in the hybrid tubes. Compared with the sum of sole constituent CFRP and aluminum tubes, the peak force of the CF-AL tube increased by 6.7% and energy absorption by 20.6%; the peak force of

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AL-CF tube increased 14.1% and energy absorption by 19.1%. This indicates that improvement of energy absorption is roughly equivalent for these two different hybrid configurtaions.

In terms of the increase in peak force for the hybrid tubes, it can be seen that the AL-CF

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tube is a little higher than that of the CF-AL tube, which directly lowers the 𝐶𝐹𝐸 of the AL-CF tube compared with the CF-AL tube. In view of overall crashworthiness, it appears

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that the CF-AL tube is better than the AL-CF tube. So, in the following sections, special

optimzation.

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attention will be paid on the CF-AL configuration for numerical analysis and design

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Table 2 Comparison of the crashworthiness indicators for different configurations 𝐹𝑚𝑎𝑥 (N)

𝐹𝑎𝑣𝑔 (N)

𝐸𝐴 (J)

𝐶𝐹𝐸 (%)

𝑆𝐸𝐴 (J/g)

Al-55

5620.03

4645.83

278.75

82.67

0.945

Al-65

6059.24

5011.33

300.68

82.71

0.845

CFRP

8574.37

4906.67

294.40

57.22

1.627

CF-AL

14957.50

11520.33

691.22

77.02

1.388

AL-CF

16382.72

11810.5

708.63

72.09

1.298

CFRP+Al-55

14014.50

9552.67

573.16

68.16

1.204

CFRP+Al-65

14357.63

9918.17

595.09

69.08

1.108

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Specimen type

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4. Numerical Simulation 4.1 Simulation of net aluminum tube and net CFRP tube Three-point bending simulations of net aluminum tube and net CFRP tube were conducted using ABAQUS/Explicit in this section [37]. To capture the main failure modes, four-node thin shell element (S4R) was adopted here. For the CFRP tube, the maximum

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degradation was set for deleting the failed elements from the sturcture. In view of both efficiency and accuracy, the mesh size in the center area of tube was refined. The length of refined mesh area was 100 mm, in which the size of element was set to be 2 mm × 2 mm. General contact was defined in between the loading nose or supporters and the specimen surface, as shown in Fig. 2, with a friction coefficient of 𝑓=0.15 [38]. And the loading nose

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and supporters were modeled to be the rigid bodies, respectively.

For the aluminum tube, the elasto-plastic constitutive model was adopted [31]. The Young’s modulus, yield strength and Poisson’s ratio were 67 GPa, 85 MPa and 0.33, respectively.

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To describe damage initiation and damage process for CFRP, the Hashin’s damage initiation criterion and damage evolution law were applied, in which five failure modes,

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specifically fiber tension, fiber compression, matrix tension, matrix compression and in-plane shear, were considered here to model the onset of the damage [39]. Failure of the CFRP

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laminate initiated when one or more failure modes occurred. The material properties of CFRP were obtained from Ref. [38].

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The simulated load-displacement and energy absorption (𝐸𝐴)-displacement curves of net aluminum tube (Al-55) are shown in Fig. 12 together with the experimental results. It can be

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seen that the experiment and simulation agree well as a whole. The comparison of simulated FEA and experimental deformation processes is exhibited in Fig. 13. Comparing each stage of the experimental and FEA deformation, excellent agreement can be observed.

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Fig. 12 Comparison of experiment and FEA load-displacement and energy absorption-displacement curves for the Al-55 tube.

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Fig. 13 Comparison of experiment and FEA deformation and failure mode for the Al-55 tube.

Fig. 14 Comparison of the experiment and FEA load-displacement and energy absorption-displacement curves for the CFRP tube.

Comparisons of load-displacement and energy absorption (𝐸𝐴)-displacement curves of CFRP tube between the experimental and numerical rusults are plotted in Fig. 14, which exhibited fairly good agreement. According to the 𝐸𝐴-displacement curve, nevertheless, the

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agreement was found to be worse at the large displacement range though the trends remain to be consistent. This was mainly due to catastrophic failure that occurred in the CFRP tube, which was hard to be predicted accurately. Moreover, the CFRP tube was modeled in the single-layer shell elements, where some minor failure modes, such as matrix cracking, delamination etc, had not be able to be modeled in the simulation, making the modeling accuracy relatively lower. Fig. 15 exhibits the experimental and numerical deformation

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processes for the CFRP tube. The major failure modes, namely cracks in circumferential and axial directions, were modeled properly in the FE simulation, which agreed well with the experimental results.

To better understand the failure mechanism of the CFRP tube, which could not be

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observed directly in the crushing experiments, Fig. 16 illustrates the four different failure patterns of the sole CFRP tube, namely tensile failure in the warp direction (SDV1), compressive failure in the wrap direction (SDV2), tensile failure in the fill direction (SDV3) and compressive failure in the fill direction (SDV4), respectively. It is worth noting that the

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fill direction follows the axial directin of the tube in this case of study. It can be observed that the tensile and compressive failure in the warp drection mainly took place in the bottom of

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loading nose, whereas the tensile failure in the fill direction mainly occurred in the upper and

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lower surfaces in the middle of the CFRP tube.

Fig. 15 Comparison of experiment and FEA deformation and failure modes of the CFRP tube.

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4.2 Simulation of the CF-AL tube

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Fig. 16 Simulated failure process of the CFRP tube under different failure criteria: (a) Tensilein failure in the wrap direction (b) Compression failure in the wrap direction (c) Tensile failre in the fill direction (d) Compression failure in the fill direction.

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The CF-AL tube (CFRP placed outside of the inner aluminum) was modeled by combining the abovementioned FE models of the net CFRP tube and net aluminum tube. In

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this hybrid model, the cohesive contact [37] was defined between the outer CFRP tube and inner aluminum tube.

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The experimental and numerical load-displacement curves as well as energy absorption (𝐸𝐴)-displacement curves of the CF-AL tube are plotted in Fig. 17. And again, fairly good

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agreement can be observed between the experimental and FEA results. It is noted that the stepped drops did not occur in the simulation curve which well agreed with the experiment observation. Fig. 18 exhibited the deformation processes obtained from the experiment and simulation. It can be found that better agreement appeared at the early stage (small deformation) but worse at the late stage (large deformation). This was mainly due to the elemental deletion process which made failure of the outer CFRP tube slightly slower in the simulation than in the experiment. Nevertheless, the simulated failure mode and trend were quite consistent with the experiment overall. 21

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The experiment and FEA results of crashworthiness indicators were summarized in Table 3. It can be seen that the errors were fairly small for the CFRP, Al-55 and CF-AL tubes. Note that the CFRP tube had the highest errors of energy absorption, average force and crush force efficiency, counting around 7.00%. Overall, the FE models have exhibited sufficient accuracy

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to predict the crashworthiness indicators properly.

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Fig. 17 Comparison of the experiment and FEA load-displacement and energy absorption (𝐸𝐴)-displacement curves for the CF-AL hybrid tube.

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Fig. 18 Comparison of the experiment and FEA deformation and failure modes for the CF-AL hybrid tube.

Table 3 Comparison of experiment and FEA crashworthiness indicators for the Al-55, CFRP net and CF-AL bybrid tubes.

Experiment

Specimens

𝐸𝐴 (J)

𝑃𝐶𝐹 (kN)

𝐹𝑎𝑣𝑔 (kN)

𝐶𝐹𝐸

Al-55

278.74

5.62

4.65

0.83

CFRP

294.40

8.57

4.91

0.57

CF-AL

691.22

14.96

11.52

0.77

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Error (%)

Al-55

287.79

5.74

4.80

0.84

CFRP

315.03

8.66

5.25

0.61

CF-AL

697.56

15.35

11.63

0.76

Al-55

3.24

2.13

3.23

1.20

CFRP

7.01

1.05

6.92

7.02

CF-AL

0.92

2.6

0.95

1.30

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4.3. Parametric study on the CF-AL tube 4.3.1 Comparative analysis of outer CFRP’s length

In order to improve the crashworthiness performance and reduce the material cost, the outer CFRP tube can be modified to have different lengths in consideration of the localized collapse, which likely occur in the center of the CF-AL tubes. For this reason, the outer CFRP

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tube was only placed at the collapse zone, especially in the area of fracture. In this study, six different lengths of the outer CFRP tube were simulated to explore the effects of the shortened CFRP outer tube on crashworthiness, specifically 𝐿𝑓 = 50, 100, 150, 210 (i.e. up to the

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distance between the two supports), 250, 300 mm, as shown in Fig. 19.

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Lf

Al

300 mm

60 mm

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CFRP

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Fig. 19 Schematic of CF-AL tubes with various lengths of CFRP tube.

The load-displacement curves and energy absorption (𝐸𝐴)-displacement curves are

compared in Fig. 20, in which the similar trends can be observed. The key crashworthiness indicators were listed in Table 4. Interestingly, it is seen that the peak crushing forces are fairly similar when 𝐿𝑓 is longer than or equal to the distance of supports (i.e. 210 mm). This is due to free of restraint at the both ends of specimens. As the outer CFRP tube was shortened (𝐿𝑓 < 210 mm), the loading capacity, in terms of 𝑃𝐶𝐹 , becomes worse due to reduction of

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overall stiffness of the whole specimen. As shown in Fig. 21(b) the trends of 𝑆𝐸𝐴 with different values of 𝐿𝑓 can be observed, in which the 𝑆𝐸𝐴 increased first and then decreased with the increase in the outer CFRP tube length. The 𝑆𝐸𝐴 reached the maximum value when the length was 150mm (i.e. half of the entire hybrid tube length). This means that a longer or shorter CFRP tube may not generate most favorable crashworthiness performance. Moreover, a shorter CFRP tube could indicate

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lower cost on the materials in consideration of high price for the CFRP materials. Considering energy absorption capacity and cost of materials, the CF-AL tube with the outer CFRP tube

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length between 100mm and 150mm would likely have better crashworthiness charateristics.

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Fig. 20 Load-displacement and energy absorption (𝐸𝐴)-displacement curves for different 𝐿𝑓 .

Fig. 21 The 𝑆𝐸𝐴 and 𝐶𝐹𝐸 of the CF-AL tube with different 𝐿𝑓

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ACCEPTED MANUSCRIPT Table 4 Comparison of crashworthiness indicators for the CF-AL tube with different 𝐿𝑓 . 𝑚 (g)

𝑃𝐶𝐹 (kN)

𝐸𝐴 (J)

𝐶𝐹𝐸

𝑆𝐸𝐴 (J/g)

50

330.2

9.14

436.31

0.80

1.32

100

362.5

11.53

560.88

0.81

1.55

150

394.7

13.52

622.30

0.77

1.58

210

433

15.35

681.43

0.74

1.57

250

459

15.32

684.38

0.74

1.49

300

491

15.35

697.56

0.74

1.42

4.3.2 The bending deformation of hybrid tubes

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𝐿𝑓 （mm）

During bending process, the outer CFRP tubes with different lengths were found to have

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similar failure modes which all occurred at the central area. Note that the CF-AL tube with 𝐿𝑓 = 50mm behaved marginal collapse in the outer CFRP tube due to relatively lower rigidity of whole structure, resulting in less energy absorption, and accordingly, the lowest 𝑆𝐸𝐴. It is found that the 𝑆𝐸𝐴 increased with the decrease in 𝐿𝑓 when 𝐿𝑓 ≥ 210mm, due to the fact

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that the area of CFRP than the support span was almost not involved in much deformation during the three-point bending tests. In other words, these areas hardly contributed on energy

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absorption; whereas a shorter CFRP outer tube has a smaller mass, resulting in a higher 𝑆𝐸𝐴. In order to understand the 𝑆𝐸𝐴 and 𝐶𝐹𝐸 for the different CFRP tube lengths, the

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deformation of inner aluminum tube was analyzed further below. To more intuitively describe the deformation severity of the inner aluminum tube, the

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radial deformation profiles were plotted in Fig. 22. The radial deformation profiles of the inner aluminum tube were plotted in Fig. 23 for the different CFRP lengths, in which all the

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profiles are symmetric to the vertical axis, and where 𝐿 is the distance to the center line (left to the loading nose is negative, and to the right is positive), 𝑟𝑖𝑛 is the inner radius of the aluminum tube, 𝑡𝑖𝑛 is the shell thickness of inner aluminum tube, 𝐷1−2 denotes the radial deformation of the tubes in terms of diameter difference, i.e. 𝐷1−2 = 𝐷1 − 𝐷2. The curves can be divided into two parts based upon the difference of the number of peak force of the curve and each part has a similar shape. This means that all the aluminum tubes have two typical deformation modes with the boundary of 𝐿𝑓 = 210 mm; and more severe

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ACCEPTED MANUSCRIPT deformation of aluminum tubes occurred at the region of supports when 𝐿𝑓 < 210 mm. Unsurprisingly, it was observed that the peak deflection occurred in the center of tubes right underneath the loading nose. Note that the peak deflection increased with increasing 𝐿𝑓 when 𝐿𝑓 ≥ 210 mm; while decreased with increasing 𝐿𝑓 when 𝐿𝑓 < 210 mm. Moreover, it is evident to observe the other two deflection peaks in the curves when 𝐿𝑓 < 210 mm. The two deflection peaks are corresponding to the two supporters, respectively. Thus, more

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deformations are participated in the area of supporters according to the other two deflection peaks in the curves when 𝐿𝑓 < 210 mm. In spite of the less deformation in the local loading region when 𝐿𝑓 < 210 mm, more radial deformation was involved in the whole tube. In this case, the total deformation of the inner aluminum tube was greater than those whose 𝐿𝑓 is

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longer than 210mm, thereby obtaining a higher 𝑆𝐸𝐴.

According to radial defomation profiles of the inner aluminum tubes (Fig. 23), the broader region of deformation was found in the tubes when CFRP was placed partly. Although CFRP had a higher 𝑆𝐸𝐴, more severe deformation of the inner aluminum tube

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could compensate the negative effect of less CFRP, thus enhancing the 𝑆𝐸𝐴 of the whole hybrid tube. Note again that less use of CFRP implies lower cost of materials. So a proper

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length of CRFP section could make the whole hybrid tube be of a well-balanced

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crashworthiness capacity and cost efficiency.

Fig. 22 Schematic of deformation mode in three-point bending test.

26

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Fig. 23 The radial deflection profiles of inner aluminum tube for different CFRP lengths. 4.3.3 Analysis of outer CFRP’s thickness and lay-up orientation

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The CFRP parameters, such as thickness and lay-up orientation, are further studied here to understand their influence on the crashworthiness of hybrid tube. For this purpose, five different thicknesses and four ply angles were considered in this study, specifically, 3, 6, 9, 12, 15 layers of plies with the corresponding wall thicknesses of 𝑇𝑐 = 0.75, 1.50, 2.25, 3.00,

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3.75mm (with the ply angles 𝜃=0°), as well as ply angles 𝜃= 0°, -15°, -30°, -45° (with the wall thicknesses of 𝑇𝑐 =2.25mm). The schematic of the hybrid tubes with different thicknesses

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and ply angles of CFRP tube are shown in Fig. 24.

θ=0° θ=90° θ=0° θ=90° θ=0° θ=90°

Ply-9 Ply-8 Ply-7 Ply-6 Ply-5 Ply-4 Ply-3 Ply-2 Ply-1

θ=0° θ=90° θ=0° (b)

(a)

Fig. 24 Schematic of the CF-AL tubes with the different CFRP parameters: (a) Thickness. (b) Ply angles.

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Fig. 25 The simulated load-displacement and energy absorption (𝐸𝐴)-displacement curves of hybrid tubes with different layers of outer CFRP tube.

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Fig. 26 The simulated load-displacement and energy absorption (𝐸𝐴)-displacement curves of hybrid tubes with different ply angles.

(b)

(a)

Fig. 27 𝑆𝐸𝐴 and 𝐶𝐹𝐸 of the hybrid tube with different CFRP parameters: (a) number of stacking layers (thickness); (b) ply angles of each layer.

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ACCEPTED MANUSCRIPT Table 5 Comparison of crashworthiness indicators of the hybrid tubes with different ply layers of the outer CFRP tube (with the ply angles 𝜃=0/90). m (g)

𝑃𝐶𝐹 (kN)

𝐸𝐴 (J)

𝐶𝐹𝐸

𝑆𝐸𝐴 (J/g)

3

362

7.06

343.74

0.81

0.95

6

426

10.50

472.83

0.75

1.11

9

491

15.35

697.56

0.74

1.42

12

560

21.51

966.37

0.75

1.73

15

630

29.72

1252.39

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Layers

0.70

1.99

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Table 6 Comparison of crashworthiness indicators of the hybrid tubes with different ply angles of outer CFRP tube (with the wall thicknesses of 𝑇𝑐 =2.25mm). m (g)

𝑃𝐶𝐹 (kN)

𝐸𝐴 (J)

𝐶𝐹𝐸

𝑆𝐸𝐴 (J/g)

[0/90]

491

15.35

697.56

0.76

1.42

[-15/75]

491

15.57

680.92

0.73

1.39

[-30/60]

491

15.50

695.82

0.77

1.42

[-45/45]

491

16.26

709.09

0.73

1.44

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𝜃 (°)

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The simulated load-displacement curves and energy absorption (𝐸𝐴) -displacement curves for the hybrid tubes are shown in Fig. 25 and Fig. 26, with the different stacking layers

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and ply angles, respectively. It can be seen that the loading capacity and energy absorption increased with increasing layers of the outer CFRP tube. Nevertheless, two sharp drops were

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seen in the load-displacement curves (Fig. 25) in the large displacement when layer number was 15. It means that severe collapes occurred in the outer CFRP tube at large deformation. In

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other words, too thick CFRP could have negative effect on loading stability, loading capacity and energy absorption of whole structures when experiencing large deformation. Compared with the stacking number, the different ply angles resulted in relatively smaller difference on loading capacity and energy absortpion as shwon in Fig. 26. The trends of 𝑆𝐸𝐴 and 𝐶𝐹𝐸 for the hybrid tubes with different stacking layers and ply angles are illustrated in Fig. 27; and the crashworthiness indicators are summarized on Table 5 and Table 6, respectively. It can be seen that 𝑆𝐸𝐴 increased while 𝐶𝐹𝐸 decreased with

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ACCEPTED MANUSCRIPT increasing CFRP’s layers. Further, 𝑆𝐸𝐴 decreased first and then increased with increasing ply angles; whilst the 𝐶𝐹𝐸 changed mariginally with increasing ply angles. Overall, in comparison with the layer number (wall thickness), the ply angles had relatively less effect on 𝑆𝐸𝐴 and 𝐶𝐹𝐸; and it appeared that -45°/45° ply angle has the highest energy absorption.

5. Discrete optimization for hybrid tube CF-AL

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5.1 Discrete optimization methodology

According to the above parametric study, it can be found that the length and thickness of hybrid tubes have stronger effects on the crashworthiness indicators. However, it is difficult to obtain an optimal configuration for the hybrid tubes with various length, thickness and ply

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angles of outer CFRP through the three point bending experimemtal tests. Structural optimization approaches have been considered to be a useful tool to address this issue [40-46]. While the optimization procedure involving continuous design variables has been extensively implemented through conventional mathematical programming algorithms, it may not be able

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to address the issues in real life. In many cases, discrete design variables need to be considered in optimization [47-49], which will be addressed in this study as follows.

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Note that a full factorial design of experiment (DoE) allows generating a best possible solution for sampling the design space with discrete variables, however it can be very costly

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to handle too many expensive highly nonlinear FEA runs. For this reason, an iterative optimization algorithm with arrays was adopted in this study [50]. Due to use of orthogonal

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arrays, the optimization can reduce the number of experiments greatly. A flowchart of such a discrete optimization procedure is shown in Fig. 28 for sake of clarity [51,52].

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Step 1: Definition of problem. In order to formulate a discrete optimization problem, the objective of optimzation,

discrete design variables and design constraints need to be defined. Step 2: Selection of a standard orthogonal array. The minimum orthogonal array was used to reduce computational time. Based upon the number of design variables, the L9(33) orthogonal array was selected as shown in Table 7. Specifically, the L9(33) orthogonal array comprises three levels and nine rows for three discrete design variables. 30

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Step 3: Arrangement of the levels for design variables The discrete design variables were placed into the columns of the L9(33) orthogonal array. a discrete value that is assigned to the second level should be first selected arbitrarily in the initial design. For the neighboring values, the discrete value with one step greater than the second level is assigned to the first level, which can be more likely to consider the first level. On the other hand, the one step smaller value is assigned to the third level.

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Step 4: Calculation of the penalized objective function vector 𝑅𝑛𝑒𝑤 [53]

According to each row of orthogonal array, the characteristic function can be calculated and the matrix experiment can be expressed mathematically, as follows:

(7)

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Find: 𝐱 = (𝑥1 , 𝑥2 , 𝑥3 , … , 𝑥𝑛 )𝑻 {Maxmize: 𝑅(𝐱) Subject to: 𝑔𝑗 (𝐱) ≤ 0, 𝑗 = 1, … , 𝑙

where 𝐱 is the design variable vector, 𝑅(𝐱) is the objective function and 𝑔𝑗 (𝐱) is the j-th constraint function, and 𝑙 is the number of constriants.

To take into account the constraints in the design optmization, the penalty function

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method was utilized to include the constraints into the objective. So the new characteristic objective function 𝑅𝑛𝑒𝑤 is expressed as follows: 𝑅𝑛𝑒𝑤 = 𝑅 − 𝑃(𝐱) 𝑃(𝐱) = 𝑠 × max,0, 𝑣𝑗 -

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{

(8)

where 𝑃(𝐱) is the penalty function, 𝑣𝑖 denotes the violation of the j-th constraint. The

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scaling factor 𝑠 is imposed to set up a value which is at least one-order larger than the objective function.

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Step 5: Determination of the optimum design When characteristic function value 𝑅𝑛𝑒𝑤 was obtained, analysis of mean (ANOM)

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[54,55] is performed to determine the optimum level of each design variable, as shown in Table 8. The new levels should be verified by the confirmation analysis, because the design feasibility or statistical validity of additivity are not always guaranteed. Moreover, if interaction between the different design variables exists, it can emerge an unreasonable determination of the new levels. Therefore, the new levels from ANOM with the best combination from the matrix experiments should be compared. And one can select the best levels as the new levels for the next iteration.

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Step 6: Convergence criteria. There are two convergence criteria in the above iterative algorithm: (1) The procedure is terminated if the number of iteration, in which the responses of the new levels are no longer improved, is more than five. (2) The algorithm is terminated if the number of iteration, in which the responses of the new levels are consecutively not feasible, is more than five.

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If both of the abovementioned convergence criteria are unsatisfied, the optimization process goes back to Step 3. In this step, the former optimum levels are assigned to be the second levels until at least one of convergence criteria is satisfied.

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Start

Select design variables (candidates) for design optimization Select a standard orthogonal array

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Starting n-th iteration Select level values among candidates Level 2 = the initial design value Level=1, 3 = neighboring candidate values of Level 2

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Conduct the matrix experiment with FEA and Evaluate 𝑅𝑛𝑒𝑤 Analysis of means (ANOM)

n=n+1 initial design= optimum level

Select the optimum level of design variables

Termination criteria is satisfied? Yes Stop

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No

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Fig. 28 Flowchart of iterative optimization by using orthogonal arrays

5.2 Discrete optimization for the CF-AL tubes To consider the mass effect on energy absorption, maximization of 𝑆𝐸𝐴 was selected to be the design objective. However, if the mass is too small, some structures may not have

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sufficient capacity of energy absorption to meet the safety requirement even though they have a high 𝑆𝐸𝐴. For this reason, the mean force 𝐹𝑎𝑣𝑔 was adopted here to be a constraint to ensure sufficient level of enregy absorption. Based upon the energy absorption of the net CFRP and net Al tubes mentioned in Section 3, 𝐹𝑎𝑣𝑔 should be no less than 10 kN.

Further, the cost can be another critical criterion. Considering that the cost have nearly

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linear relationship with mass of the CFRP [29], so the mass of CFRP was set to be another constraint here. In order to make the cost of the optimized CF-AL tube less than that of the baseline CF-AL tube, thus the total mass of structure should not be over 498 g. To sum up, a single objective optimization problem with two constraints is formulated

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mathematically as follows: 𝑅 = max{𝑆𝐸𝐴(𝐿𝑓 ，𝑇𝑓 ，𝜃)}

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𝐹𝑎𝑣𝑔 (𝐿𝑓 ，𝑇𝑓 ，𝜃) ≥ 10

(9)

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𝑚(𝐿𝑓 ，𝑇𝑓 ，𝜃) ≤ 498 𝑠. 𝑡. 𝐿𝑓 ∈ *25,50,75,100,125,150,175,200,225,250,275,300+ 𝑇𝑓 ∈ *3,4,5,6,7,8,9,10,11,12,13,14+ { {𝜃 ∈ *0°, 5°, 10°, 15°, 20°, 25°, 30°, 35°, 40°, 45°+

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To take into account the effects of constraints, the penalty function was adopted herein to transform this constrained optimization problem into a non-constrained problem. Specificially,

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the penalty functions for the mean force and CFRP mass were formulated as: {

𝑃1 (𝐿𝑓 ，𝑇𝑓 ，𝜃) = 𝑠1 × max,0,10 − 𝐹𝑎𝑣𝑔 (𝐿𝑓 ，𝑇𝑓 ，𝜃)𝑃2 (𝐿𝑓 ，𝑇𝑓 ，𝜃) = 𝑠2 × max,0, 𝑚(𝐿𝑓 ，𝑇𝑓 ，𝜃) − 498-

(10)

So, this optimization problem can be finally expressed as follows: 𝑅𝑛𝑒𝑤 = max{𝑆𝐸𝐴(𝐿𝑓 ，𝑇𝑓 ，𝜃) + 𝑃1 (𝐿𝑓 ，𝑇𝑓 ，𝜃) + 𝑃2 (𝐿𝑓 ，𝑇𝑓 ，𝜃)} 𝐿𝑓 ∈ *25,50,75,100,125,150,175,200,225,250,275,300+ 𝑠. 𝑡. {𝑇𝑓 ∈ *3,4,5,6,7,8,9,10,11,12,13,14+ 𝜃 ∈ *0°, 5°, 10°, 15°, 20°, 25°, 30°, 35°, 40°, 45°+ {

33

(11)

ACCEPTED MANUSCRIPT where 𝐿𝑓 , 𝑇𝑓 , 𝜃 are the design variables to control the configuration of CFRPs. Length 𝐿𝑓 was ranged from 25 to 300 mm, thickness 𝑇𝑓 was from 3 to 14 plies, and orientation angle θ was from 0° to 45°. All these design variables were considered to be discrete from manufacturing persepctive as summarized in Table 9. In order to explore the dicrete design space, the intermediate values of the design variables were selected to be the initial design for optimization. For the first iteration, the

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parameters of CFRP and characteristic function 𝑅𝑛𝑒𝑤 were listed in Table 10. Then, ANOM was utilized to select the levels of optimum and the calculation results were summarized in Table 11. The highest average level was selected to be the optimal level, in which the best levels (𝐿𝑓 = 200mm, 𝑇𝑓 = 10 plies, and 𝜃 = 5° ) were adopted to be the new levels and

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assigned into the second level for the next iteraiton.

Table 7 The L9(33) orthogonal array Simulation Number

Column number and factor assigned

3 C 1

Rnew1

2

1

2

2

Rnew2

1

3

3

Rnew3

2

1

2

Rnew4

2

2

3

Rnew5

2

3

1

Rnew6

7

3

1

3

Rnew7

8

3

2

1

Rnew8

3

3

2

Rnew9

3

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6

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4 5

9

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2 B 1

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1

1 A 1

Results (Rnew)

Table 8 Mean of Rnew corresponding to each level Design

Levels

variables

1

2

3

A

(Rnew1 + Rnew2 + Rnew3)/3

(Rnew4 + Rnew5 + Rnew6)/3

(Rnew7 + Rnew8 + Rnew9)/3

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ACCEPTED MANUSCRIPT B

(Rnew1 + Rnew4 + Rnew7)/3

(Rnew2 + Rnew5 + Rnew8)/3

(Rnew3 + Rnew6 + Rnew9)/3

C

(Rnew1 + Rnew6 + Rnew8)/3

(Rnew2 + Rnew4 + Rnew9)/3

(Rnew3 + Rnew5 + Rnew7)/3

Table 9 The dicrete values of design variables to the CFRPs: length, thickness and orientation angles. 3

4

5

6

7

8

𝐿𝑓

50

75

100

125

150

175

200

𝑇𝑓

4

5

6

7

8

9

10

𝜃

0

5

10

15

20

25

30

9

10

11

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2

225

250

275

300

11

12

13

14

35

40

45

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1

Tf

θ

Rnew

8

20

-10.63

9

25

-0.51

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Table 10 The discrete values of design variables in the first iteration

10

30

1.61

8

25

-7.64

175

9

30

1.48

175

10

20

1.60

200

8

30

-0.58

200

9

20

1.52

200

10

25

1.70

Lf

1

150

2

150

3

150

4

175

5

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Number serial

6 7

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8

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9

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Table 11 Mean of 𝑅𝑛𝑒𝑤 corresponding to each level for CF-AL tubes Levels

Design variable 1

2

3

A

-3.18

-1.52

0.88

B

-6.28

0.83

1.64

C

-2.50

-2.14

0.84

5.3. Results of optimization The iteration process of the optimization is shown in Fig. 29. It can be seen that the

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design objective of the new level was not enhanced since the 6th iteration, and the optimization procedure was thus terminated at the 9th iteration. Total nine iteration steps over the whole optimization procedure were run before the convergence. An optimum design was reached with 𝑅𝑛𝑒𝑤 = 2.03 (𝑅𝑛𝑒𝑤 = 1.70 in the initial design) at the 6th iteration. The results of the optimized CF-AL tubes are summarized in Table 12, in which the 𝑆𝐸𝐴𝑠 of the structure before and after the optimization are compared. The optimum design is by 42.96% in

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𝐿𝑓 = 225 mm, 𝑇𝑓 = 12 plies, 𝜃 = 45° , which improved the 𝑆𝐸𝐴

comparison with the initial baseline design. Meanwhile, the CFRP mass of optimum design reduced 5.02% and mean crushing force enhanced 37.75%. Note that the CFRP mass and mean force are porportional to the material cost and enregy absorption, respectively.

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Therefore, both the cost and energy absorption were improved substantially through the

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design optimization.

Fig. 29 History of 𝑅𝑛𝑒𝑤 for the CF-AL tube design

To more intuitively describe the improvement of comprehensive crashworthiness

performance after the optimization, the radar maps for each crashworthiness criteria of intial and optimum designs are shown in Fig. 30. It can be seen clearly that the initial design is entirely surrounded by the optimized design in the radar maps, expect for the 𝐶𝐹𝐸 . Considering nearly the same values of 𝐶𝐹𝐸 of these two structures, it can be concluded that

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ACCEPTED MANUSCRIPT the overall crashworthiness performance of the optimized structure (𝐿𝑓 = 225mm, 𝑇𝑓 = 12 plies, 𝜃 = 45° ) has substantially improved. Table 12 The results of optimization design 𝐿𝑓

𝑇𝑓

𝜃

𝐹𝑎𝑣𝑔

m

𝑆𝐸𝐴

300

9

0

498

11.63

1.42

Optimum design

225

12

45

473

16.02

2.03

Increasing ratio

-

-

-

-5.02%

37.75%

42.96%

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Initial design

6. Conclusion

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Fig. 30 The radar map for each crashworthiness criteria of intial design and optimum design

The bending behavior and energy absorption characteristics of the two different hybrid

tubes made of aluminum and CFRP were investigated under a transverse loading condition. The experimental test, numerical simualtion and discrete design optimization were conducted to explore the effects of design parameters on bending behavior and energy absorption. Within its limitation, the following conclusion can be drawn: (1) The experimental tests were carried out to study the interactive effect of hybrid CFRP and aluminum tube as per improvements of loading capacity and energy absorption. The

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strengthening effects on the loading capacity and energy absorption were due to the fact of interactive forces and more severe failure of the outer CFRP tubes. The CF-AL tube was then chosen to be a better hybrid configuration for crashworthiness design. (2) The finite element (FE) simulations of net aluminum and net CFRP tubes as well as CF-AL structure were performed; and good agreement with the experimental results was obtained. Thus, the parametric study and design optimization on CF-AL were then conducted

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using the validated FE models.

(3) The numerical simulations of CF-AL with different structural parameters provided insights into the design of such a hybrid structure. The CF-AL tube whose length of outer CFRP tube was in between 100 and 210mm has better crashworthiness. The deformation

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modes of the inner aluminum tube were analyzed to explain this phenomenon. It was revealed that the energy absorption was affected largely by the number of ply layers (thickness) but marginally by the ply angles.

(4) The discrete optimization was carried out by iteratively using orthogonal array. The

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𝑆𝐸𝐴 of the optimized CF-AL design is 2.03 J/g, which increased by 42.96% in comparison with the initial baseline design. Moreover, the optimized structure has overall better

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crashworthiness performance.

While excellent correlation between the FE and experimental results were obtained, it

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should be pointed out that only one sample was tested experimentally for the consideration of cost. The future study is expected to have substantially larger sample size for statistically

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meaningful results.

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Acknowledgements This work is supported by the Foundation for Innovative Research Groups of the

National Natural Science Foundation of China (51621004), National Natural Science Foundation of China (51575172, 51475154) and the Open Fund of the State Key Laboratory for Strength and Vibration of Mechanical Structures of Xi’an Jiaotong University (SV2017-KF-24). Dr Guangyong Sun is a recipient of Australian Research Council (ARC) Discovery Early Career Researcher Award (DECRA) in the University of Sydney.

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Graphical Abstract

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Specimens

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Experiment And Simulation

T

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0 θ=0 θ=9 0 θ=0

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Parametric Study

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f

Ply-9 Ply-8 Ply-7 θ=0 Ply-6 θ=9 Ply-5 0 θ=0 Ply-4 θ=9 Ply-3 0 θ=0 Ply-2 θ=9 Ply-1

Discrete Optimization

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