On crashworthiness design of hybrid metal-composite structures

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On crashworthiness design of hybrid metal-composite structures Zhen Wang , Xihong Jin , Qing Li , Guangyong Sun PII: DOI: Reference:

S0020-7403(19)33480-0 https://doi.org/10.1016/j.ijmecsci.2019.105380 MS 105380

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

13 September 2019 3 December 2019 16 December 2019

Please cite this article as: Zhen Wang , Xihong Jin , Qing Li , Guangyong Sun , On crashworthiness design of hybrid metal-composite structures, International Journal of Mechanical Sciences (2019), doi: https://doi.org/10.1016/j.ijmecsci.2019.105380

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Highlight 1.

Develop a finite element (FE) models for the aluminum/CFRP hybrid structures.

2.

Reveal the interactive effect and energy absorption mechanisms of hybrid structures.

3.

Investigate the effects of wall thickness, sectional dimension and sectional shape on the energy absorption capacity and performance-cost of the hybrid structures.

4.

Design the hybrid structure adopting single/multi-objective discrete optimization method.

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On crashworthiness design of hybrid metal-composite structures Zhen Wang1, Xihong Jin2, Qing Li3, Guangyong Sun1, 3, * 1

State Key Laboratory of Advanced Design and Manufacture for Vehicle Body, Hunan University, Changsha, 410082, China 2

3

CRRC Zhuzhou Locomotive Co., Ltd., Zhuzhou, 412001, China

School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australia Abstract

As a class of promising cost-effective lightweight structures, metal-composite hybrid structures has rapidly emerged in automotive industry largely attributable to their outstanding multifunctional and crashworthy characteristics. Recently, continuous efforts have been devoted to the studies on the crashworthiness of various hybrid tubes, which commonly present two typical configurational schemes, namely metal-composite (i.e. a metal outer tube internally filled with an inner carbon fiber reinforced plastic (CFRP) tube) and composite-metal (i.e. an outer composite tube internally filled with an inner metal tube). Nevertheless, rather limited studies have focused on revealing energy absorption mechanisms of hybrid structures; and how to optimize the performance to cost characteristics of hybrid structures still remains an open question in literature to date. This study aimed to maximize the energy absorption of different configurational aluminum/CFRP (carbon fiber reinforced plastic) hybrid tubes. First, the finite element (FE) models were developed and validated by comparing the damage modes and crashworthiness indictors with the dedicated experimental study. Second, the interactive effects of the hybrid tubes were investigated by analyzing the discrepancies in the deformation pattern and internal energy absorption of each material through the validated FE models. For the AL-CF configuration (i.e. CFRP inner tube with aluminum outer tube), changes of deformation mode increased the internal energies of aluminum and CFRP tubes by 43.6% and 17.8% compared to the net aluminum tube and CFRP tube, respectively; and increased the frictional dissipation energy by 45.6% compared to the sum of that of net aluminum and CFRP tubes, largely enhancing energy absorption of

* Corresponding Author: Tel: +86-13786196408; Email: [email protected]. 2

AL-CF. For the CF-AL configuration (i.e. aluminum inner tube with CFRP outer tube), the internal energy increased by 27.6% for the aluminum tube but decreased 31.9% for the CFRP tube compared to the net aluminum tube and CFRP tube, respectively; whereas the frictional dissipation energy decreased by 47.6% compared to the sum of that of net aluminum and CFRP tubes, indicating the vital importance of hybrid configuration to energy absorption. Third, the effects of wall thickness, sectional dimension and sectional shape on the energy absorption capacity as well as the performance-cost characteristics of the hybrid tubes were further studied. It was found that from a net performance perspective, the hybrid tube with a thicker CFRP tube had higher capacity of energy absorption; whilst from a performance to cost perspective, the hybrid tube with a thinner aluminum tube offered better cost-effective energy absorption characteristics. Moreover, with the same weight, the hybrid tube with a circular sectional shape and a smaller sectional size exhibit a better performance. Finally, a multiobjective discrete optimization was conducted to optimize the AL-CF hybrid tube with various sectional shapes, sizes and wall thicknesses. As a result, the weight, peak crush force (PCF) and cost were finally reduced by 41.3%, 18.0% and 11.2% respectively, while the energy absorption (EA) was enhanced by 48.0% in comparison with the baseline design. Keywords: Aluminum/CFRP hybrid, Multiobjective optimization, Energy absorption, Cost and lightweight. 1. Introduction To meet the ever-growing requirements of fuel efficiency and gas emission reduction, lightweight structures made of fiber reinforced plastic (FRP) composites, aluminum alloy and cellular/foam materials have been more and more extensively used in aerospace, nautical, automotive industries nowadays [1-4]. While the thin-walled FRP structures exhibit superior advantages in energy absorption and weight reduction, their failure mechanisms were much more complicated than metallic counterparts, which involve fiber breakage/buckling, matrix cracking, fiber-matrix de-bonding [5-8], etc. Moreover, the material and manufacture costs of composites are still much more expensive than that of metal by now. Hence, it is unrealistic to use FRP composites to completely replace metals in real life. To reduce the high cost of using net CFRP materials and further improve the structural

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crashworthiness, metal/composite hybrid configurations are introduced to vehicle engineering by combining progressive plastic deformation of metallic materials with high strength and high modulus of lightweight composites [9-15]. However, crushing responses of metal/composite hybrid energy-absorbing devices can be determined by a number of factors such as wall thickness [16], stacking sequence of CFRP [17], etc. For instance, Babbage et al. [18] experimentally investigated the influence of fiber orientation, FRP thickness and epoxy foam on the crushing performance for both circular and squared hybrid aluminum/GFRP tubes subjected to quasi-static compression. Their study revealed that hybrid aluminum/GFRP tubes provided superior crushing characteristics to single aluminum tubes; moreover, the hybrid tubes filled with epoxy foam exhibited even higher crushing performance. Zhu et al. [19] experimentally and numerically studied the compressive behaviors of aluminum/CFRP structures; and their results indicated that the hybrid tubes offered better load bearing capacity than individual aluminum or individual CFRP tubes under complex impacting angles. Sun et al. [20] experimentally investigated the crushing behaviors of aluminum/CFRP structures for the quasi-static loading conditions. They demonstrated that all hybrid structures showed superior load carrying capacity than individual single components. It is noted that the plastic deformation yielded by the metallic hollow component could lead the composite component to a more stable progressive deformation; whilst composite component increases the bending rigidity of the metallic hollow component thanks to its high stiffness, thereby enhancing overall energy absorption of hybrid structure. Moreover, metal/composite hybrid configuration provides a means to balancing the performance and cost of structures, potentially more suitable for extensive applications in automotive engineering. As abovementioned, there have been some reports available in literature concerning the crashworthiness of hybrid composite-metal hollow structures. However, plastic folding developed by metallic component could easily fracture or split the external composite reinforcements, thereby reducing the loading capacity of hybrid structure, as shown in Fig. 1 [19, 21-23]. For this reason, metal-composite hybrid structures configured by inserting/wrapping composite components into metallic components are suggested, as shown in Fig. 2 [23, 24-26]. For example, crashworthiness performance of steel/GFRP hybrid tube composed of inner composite tube externally reinforced with two separate steel tubes was 4

studied by Dlugosch et al. [24]. Energy absorption of the aluminum/CFRP tube composed of inner composite tube reinforced externally with an aluminum tube was tested by Hussein et al. [25]. Crushing behaviors of metal/FRP hybrid structures by inserting the inner FRP components into the outer steel tube were also investigated under both static and dynamic conditions [26].

Fig. 1. Composite-metal hybrid structures studied in literature: (a) Ref [19]; (b) Ref [21]; (c) Ref [22]; (d) Ref [23].

Fig. 2. Metal-composite hybrid structures studied in literature: (a) Ref [24]; (b) Ref [25]; (c) Ref [26]; (d) Ref [23].

Substantial studies have consistently pointed out that energy absorption of a hybrid structure is not equal to the summation of those of individual components due to the

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interactive effect between different material components. It is noted that deformation of hybrid structures with different materials and geometries can be fairly complex, making it difficult to elucidate sophisticated interactive mechanisms of hybrid components merely through experimental tests. Finite element (FE) analysis (FEA) is often considered to be as an effective alternative to exploring sophisticated mechanical responses. In this respect, considerable attention has been paid to build perfect material models consisted of elastic, inelastic and failure responses for simulating the crushing behaviors of laminated composites over years [27-36]. Reuter et al. [37] simulated compressive behaviors of crashworthy aluminum/CFRP structures with axial impacting, in which the simulation was found to well agree with the experiment, providing an efficiency approach for design of hybrid metal/composite components. Sun et al. [38] adopted different

constitutive

models

to

simulate

axial

compressive

responses

of

metal-foam-composite sandwich structures; and their numerical results correlated with the experimental results closely. Due to anisotropic characteristics of composite, nevertheless, it is still fairly difficult to precisely capture the damaged modes of composite components and accurately depict the interaction of hybrid structure subject to large crushing deformation. For promoting wide application of metal/composite structures in industry, the goal of this study is to develop suitable numerical models for exploring energy absorption and deformation mechanisms of aluminum/CFRP tubes on the basis of the recent experimental study [23]. First, the finite element (FE) models of individual components (AL-in, AL-out and CFRP) and hybrid tubes (CFRP-AL and AL-CFRP) were developed in commercial code Abaqus/Explicit together with different constitutive models established from our in-house experimental studies. Second, the interactive crushing effect of CF-AL and AL-CF were identified in detail through the validated models. Third, a parametric study on the influence of wall thickness, sectional shape and sectional dimension on the performance and cost characteristics of these hybrid structures was conducted. Finally, single objective and multiobjective discrete optimizations were performed to obtain desirable hybrid configurations. 2. Experimental and numerical analysis

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2.1 Experimental specimens and tests In this study, aluminum tubes are prepared from commercial aluminum alloy (AA6061-O) rod-shaped ingots through wire cutting electrical discharge (WCED) technique. The CFRP tubes were fabricated using CFRP plain woven prepregs with (0/90)7 through bladder molding process, as described in Ref. [23]. The circular CFRP tubes used here were exactly the same as those reported in [38]. Fig. 3 shows the schematic and specimens for the in-house experimental tests; geometry size and material price information of different specimens were presented in Table 1. Representative crushing history and load-displacement response curves of CFRP, aluminum and hybrid tubes are depicted in Fig. 4.

Fig. 3. Al/CFRP hybrid tubes: (a) dimension, (b) individual aluminum and CFRP tubes, (c) aluminum tube inserted externally in CFRP tube (AL-CF). (d) CFRP tube inserted internally in aluminum tube (CF-AL). Table 1 Geometry size and material price information of all specimens. Specimens AL-in AL-out CF AL-CF CF-AL

Thickness Weight (mm) (g) 1.7 92 1.7 105 1.7 55 3.4 160 3.4 148

Dexternal (mm) 57.1 63.79 60.91 63.79 60.91

Length (mm) 120 120 120 120 120

Material cost ($/kg) [36, 37] 4.56 4.56 88.00 / /

Materials Aluminum Aluminum T300/epoxy AL-out+CFRP CFRP+ AL-in

2.2. Modeling method The FE models for the hybrid tubes CF-AL (CFRP outer tube + aluminum inner tube) and AL-CF (aluminum outer tube + CFRP inner tube) were respectively illustrated in Fig. 5. The rigid body crossheads were modeled by using single layer shell elements (S4R). The CFRP tube was modeled with continuum shell elements (SC8R) for each layer. The aluminum tube was modeled with shell elements (S4R). Following a series of convergence tests, we finally meshed the aluminum tube by shell elements using 2.0×2.0mm; and each ply of the

7

CFRP layers was meshed by continuum shell elements of 1.0×2.0mm [11,19]. The general contact method was used to simulate the contact effects between different materials, and the friction coefficient was set to be 0.15 [35]. The velocity of crosshead was set to be 1m/s to reduce computational time in the explicit FE analysis [38].

Fig. 4. Axial compressive history of CFRP tube in the test.

Fig. 5. FE models of two different hybrid specimens named CF-AL (left) and AL-CF (right).

2.3. Material models 2.3.1. Aluminum material model Sun et al. [39] adopted the ductile failure criterion for aluminum facesheets and honeycomb to simulate the indentation behaviors of sandwich panels with fracture simulation; and they found that the numerical models agreed well with the experimental results. Thus, an elastoplastic material model based upon the von Mise’s isotropic plasticity algorithm was adopted for aluminum as reported in [11] and the material mechanical properties (Young's modulus 69 GPa, density 2.64 g/cm3, Poisson's ratio 0.3) was the same as in [23]. The ductile

8

criterion was adopted herein for modeling the cracking failure of aluminum tubes, and the fracture strains were set to be 0.6 [39]. 2.3.2. CFRP material models The plain woven CFRP laminates could experience various failure modes under crushing loading, such as delamination between adjacent plies, fiber cracks and matrix cracking within the single ply, etc. To capture such different failure modes of CFRP tube, a user-defined material subroutine VUMAT was developed as reported in our previous studies [38,40]. Here, we used the same material properties and constitutive models to investigate the crashworthiness of CFRP and hybrid tubes. The VUMAT flowchart is depicted as follows and in Fig. 6. A maximum stress damage criterion [40] was adopted here to calculate damage activation functions 𝑊𝑘± and 𝑊12 (𝑘 = 1, 2): (i) Tension/compression damage initiation in the kth direction: ̂ 𝜎

𝑊𝑘± = 𝑚𝑎𝑥(𝑋𝑘± ≥ 1)

(1)

𝑘±

(ii) In-plane shear damage initiation： ̂12 𝜎

𝑊12 = 𝑚𝑎𝑥(|

𝑆

≥ 1|)

(2)

where 𝜎̂𝑘± represents the tensile/compressive stresses in the kth direction; and 𝜎̂12 the shear stress; and correspondingly, 𝑋𝑘± denote the ultimate strengths in tensile/compressive tests along the kth direction, respectively. 𝑆 represents in-plane shear strength of a single layer. The damage variable 𝑑𝛼 is calculated as follows (𝛼 = 1±, 2 ±) [41], 2𝑔𝛼 𝐿

1

0 𝑐 𝑑𝛼 = 1 − 𝑟 𝑒𝑥𝑝 [− 𝐺𝛼 −𝑔 𝛼 𝐿 (𝑟𝛼 − 1)] 𝛼

𝑓𝑐

0 𝑐

(3)

𝛼 where 𝐿𝑐 is the characteristic length of the element; 𝐺𝑓𝑐 is the fracture energy; 𝑔0𝛼 is the

elastic energy per unit volume. As for the in-plane shear damage evolution, damage variable 𝑑12 is given as: 𝑚𝑎𝑥 𝑑12 = 𝑚𝑖𝑛,𝛽12 𝑙𝑛(𝑟12 ) , 𝑑12 -

(4)

𝑚𝑎𝑥 where 𝛽12 and 𝑑12 are the material properties determined by experimental tests. ̂ 𝜎

𝑟𝛼 = max(𝑋𝛼 )

(5)

𝛼

where 𝑟𝛼 is the damage threshold to prevent the material damage recovery.

9

Start

End Yes

Initialize finite element model No

Parameter declaration

time≟total time

Read properties and SDVi

Update stress and SDVi

Update strain εij (tn+∆tn+1)

Update stiffness matrix

Calculate stress σij (tn+∆tn+1)

Element deletion

Effective stress 𝜎̂ij (tn+∆tn+1)

dij (tn+∆tn+1) ≥ 0.99

No

Failure check

Update damage variables dij Damage evolution

Tensile/ compressive damage along fill/warp direction

Yes In-plane shear damage

Failure modes VUMAT

Fig. 6. Flowchart for damage calculation of CFRP composites.

3. Experimental validation and energy absorption mechanism 3.1. Experimental validation Figs. 7-9 illustrate load-displacement curves and compressive deformations obtained from the experimental and numerical analyses of these different individual tubes, respectively. For the aluminum tubes, both the numerical and experimental deformation modes exhibited typical concertina modes; and the simulated load-displacement curves are close to those of the experimental results.

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Fig. 7. Deformation modes and load-displacement curves of the AL-in tube.

Fig. 8. Deformation modes and load-displacement curves of the AL-out tube.

Fig. 9. Deformation modes and load-displacement curves of the CFRP tube.

The CFRP tube turned into the typical splaying deformation modes in the end, while the instantaneous crushing force in the simulation fluctuated more sharply than that of the physical data, which was possibly caused by the contact algorithm and mass scaling technique [11]. Nevertheless, the trend of the experimental and numerical load-displacement curves agreed well with each other. Figs. 10-11 compare the experimental and numerical compressive behaviors and

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load-displacement curves of those two different hybrid configurations, namely CF-AL (CFRP outer tube - aluminum inner tube) and AL-CF (aluminum outer tube - CFRP inner tube) respectively. For the CF-AL hybrid tube (outer CFRP component) as shown in Fig. 10 (a), the longitudinal cracks in the CFRP tube wall were observed in both the experimental and numerical analyses, which were caused by restrictions of the internal aluminum tube under large deformation. As seen in Fig. 10(b), the simulated mean crushing force was marginally higher than that of the experimental counterpart, which might be caused by the difference of fiber volume between the experimental specimens and FE models. Even though, the numerical predictions have proven to be fairly accurate. For the AL-CF hybrid tube (outer AL component) as shown in Fig. 11 (a), the cracks on the outer aluminum tubal wall were observed in both the experimental and numerical analyses, which split the aluminum tube into several fronds in the end; whereas the CFRP inner tube completely curled up inside and exhibited a progressive damage process. Clearly, the numerical predictions were also fairly close to the experimental tests.

Fig. 10. Deformation modes and load-displacement response curves of the hybrid CF-AL tube.

Fig. 11. Deformation modes and load-displacement response curves of the hybrid AL-CF tube. 12

Table 2 presents compressive indicators of AL-in, AL-out, CFRP, as well as the hybrid CF-AL and AL-CF tubes. The results confirmed that the numerical models provided fairly accurate prediction to the crashworthiness indicators; thereby the models can be used for the further study. Table 2. Crashworthiness indicators of different configurational hybrid specimens. Specimens

Method

𝐸𝐴 (kJ)

𝑃𝐶𝐹 (kN)

𝐹𝑚𝑒𝑎𝑛 (kN)

𝑆𝐸𝐴 (kJ/kg)

AL-in

Experiment Simulation Experiment Simulation Experiment Simulation Experiment Simulation Experiment Simulation

1.49 1.61 1.67 1.69 3.00 2.86 6.05 5.86 3.98 3.96

36.4 33.0 40.5 37.1 79.6 86.9 123.1 123.9 98.2 94.99

18.6 20.2 20.7 21.1 37.5 35.8 75.6 73.2 49.7 49.5

16.18 17.56 15.80 16.07 54.51 52.07 37.82 36.60 26.87 26.73

AL-out CF AL-CF CF-AL

As defined in Eqs. (1)-(4), five failure modes for CFRP laminates were considered in the FE model, which include the tensile/compressive failures along the longitudinal (warp) direction (i.e. SDV1/SDV2), tensile/compressive failures along the transverse (fill) direction (i.e. SDV3/SDV4), and in-plane shear failure (SDV5). It should be pointed out that it is difficult to observe such failure modes through the experimental tests; whilst the FE models provided an effective way to explore each of such failure modes in the course of the crushing process. Fig. 12 shows these five different damage modes of the net CF tube, as well as the inner CF tube in the AL-CF and outer CF tube in the CF-AL hybrid

structures, in which red color

denotes more severe failure and the blue color presents the less severe failure, respectively. It can be seen that the failures mainly occurred in the warp direction. Since the stacking sequence of laminates used here were set as (0°/90°)7, the in-plane shear failure rarely occurred in the CFRP laminate. Besides, it is found that both the tensile and compressive failures took place in the net CF tube. The compressive failure mainly appeared in the inner CF tube of the AL-CF hybrid configuration; in contrast, the tensile failures mainly occurred in the outer CF tube of the CF-AL hybrid configuration. It is noted that the discrepancy in the

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deformation modes resulted in different failure modes.

Fig. 12. Damaged modes of the CFRP tube.

3.2. Energy absorption mechanisms analysis In the quasi-static crushing process, the total energy absorption (EA) of aluminum/CFRP hybrid tube comprised mainly internal energy (ALLIE), kinetic energy (ALLKE), frictional dissipation energy (ALLFD), artificial strain energy (ALLAE) and others, in which the internal energy contributes a significant portion of the total energy absorption [41-42]. Since the kinetic energy and artificial strain energy did not exceed 5% and 3% of the total energy in all the simulations, the analysis of these two energies can be ignored here. Thus, energy absorption mechanisms of the two hybrid specimens (AL-CF, CF-AL) were conducted through comparing the internal energy and frictional dissipation energy as well as the 14

deformation modes of aluminum and CF tubal components below. Fig. 13 compared the load-displacement curves, internal energy, frictional dissipation energy and deformation modes of the hybrid specimen AL-CF and each of the tubal components. The deformation behaviors of the single CF component in the hybrid AL-CF tube were changed from a typical splaying mode to an internal curling mode, where the inner CF component was split into several internal fronds and these internal fronds gradually curled with the increase of crushing displacement; finally, the inner CF component in the AL-CF tube developed a stable progressive deformation mode. As for the outer aluminum component in the AL-CF hybrid tube, the deformation was changed from a concertina mode of typical sole aluminum tube to a splitting mode. The outer aluminum component generated several longitudinal cracks and the tube wall was split into several curled fronds, as shown in Fig. 13(a). It is found that the damage modes of the outer AL and inner CF components in the hybrid AL-CF tube were different from those of single net tubes, which partly enhanced crushing force of AL-CF in comparison with the sum of those of the individual components, as shown in Fig. 13(b). In addition, the internal energies of both the aluminum and CF components in the hybrid tube were respectively higher than those of the corresponding single net tubes thanks to the change in the deformation modes induced by positive interaction between these two different components. Specifically, the internal energies ALLIEs of the outer aluminum component and inner CF component in the hybrid tube were 43.6% and 17.8% higher than those of respective single net aluminum tube and CF tube, as shown in Figs. 13(c)-(d). Further, substantial frictional dissipation energy ALLFD was generated in the AL-CF hybrid tube, which was 45.6% higher than the sum of the single net AL tube and net CF tube, as shown in Figs. 13(e)-(f).

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Fig. 13. Comparisons in the (a) deformation modes; (b) load-displacement response curves; (c) and (d) ALLIE; (e) and (f) ALLFD of the hybrid AL-CF tube and individual tubes.

Fig. 14 compared the load-displacement curves, internal energies and deformation modes between the hybrid specimen CF-AL and single net tubes. Deformation modes of the outer CF component in CF-AL was changed from a splaying mode to an external curling and splitting

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mode, where the outer CF component was split into several external fronds initially; and these fronds curled externally due to restriction of the inside aluminum tube. With the increase in the crushing displacement, some longitudinal cracks were generated along the compression direction; and the cracks propagated quickly; finally, the CF component wall was completely split from top to bottom. As for the inner aluminum tube in the hybrid configuration, the deformation mode was changed from a concertina mode in a typical individual aluminum tube to an asymmetric mode due to the restriction of the external CF tube, as shown in Figs. 14(a). Both the damage modes of the aluminum and the CFRP components in the CF-AL hybrid tube were different from those of their corresponding single net counterparts and even those in the AL-CF hybrid tube. It is noted that the splitting mode of the CF component in the CF-AL tube directly led to decrease in the crushing force of hybrid tube in comparison with the sum of those of the individual components, as shown in Fig. 14(b). Further, the changes in deformation modes caused 31.9% reduction of internal energy ALLIE in the outer CF component in CF-AL in comparison with that in single net CF tube. While the ALLIE of the hybrid AL component was improved by 27.6% compared to the corresponding net tube thanks to the changes in deformation modes and interactive effect, as shown in Figs. 14(c)-(d). For the ALLFD, it was worth noting that the hybrid AL-CF tube was 47.6% lower than that of the sum of a single net AL tube and single net CF tube, as shown in Figs. 14(e)-(f). The above studies compared the differences in the crushing behavior and energy absorption between net components and hybrid specimens. Importantly, the interactive effect in hybrid specimens was also explored by comparing the energy absorption of each component in the hybrid specimen and their individual net counterparts. It was found that configurational schemes can remarkably affect the energy absorption characteristics and crushing behaviors of both aluminum and CFRP components in a hybrid structure with the same materials. Since the deformation modes largely affected the loading capacity of aluminum and CFRP components, changing the deformation modes of each tubal component might be an effective way to enhance the energy absorption capacity of entire hybrid structures. In addition, not only does automotive industry continuously seek high crashing performance, but 17

also low cost. For this reason, the following study aims to improve the crashworthiness and reduce the cost of hybrid tubes by analyzing the contributions of each component.

Fig. 14. Comparisons in the (a) deformation modes; (b) load-displacement curves; (c) and (d) ALLIE; (e) and (f) ALLFD of the CF-AL and individual tubes.

4. Performance-cost analysis and parametric studies While hybrid structure can be a cost-effective alternative, it is still not easy to obtain a proper hybrid configuration with a range of geometric parameters and material types. In view 18

of lightweight and crashworthiness, the wall thickness, sectional size and sectional shape are some vital structural parameters to influence the deformation modes of crashworthy structures. This section aims to explore the influence of wall thickness of AL (specifically 0.57, 0.85, 1.7, 3.4, and 5.1 mm) and CFRP (4-, 5-, 7-, 9-, and 14-ply) components, sectional size (namely, small, middle, large), and sectional shape (square, hexagon, circle) on the crashworthy performance and cost of hybrid tubes. In this study, the length of all specimens was 120mm, and the stacking sequence of each single ply of the CFRP tube was (0/90). To compare the effects of AL or CFRP tubal components with different wall thicknesses on the crashworthiness of hybrid tubes, the ratios of 𝑇𝐶𝐹 /𝑇𝐶𝐹+𝐴𝐿 (𝑇𝐶𝐹 is the thickness of CFRP tube; 𝑇𝐶𝐹+𝐴𝐿 is the total thickness of the hybrid tube) was introduced to these two different aluminum/CFRP hybrid tubes (i.e. CF-AL and AL-CF).

4.1. Effect of 𝑇𝐶𝐹 /𝑇𝐶𝐹+𝐴𝐿 on crashworthiness of hybrid tube AL-CF Two different design cases were considered here to identify the effects of 𝑇𝐶𝐹 /𝑇𝐶𝐹+𝐴𝐿 ratios on crashworthiness and damage patterns of hybrid tube AL-CF (outer AL and inner CFRP). Their 𝑆𝐸𝐴 (special energy absorption), ratio of 𝑆𝐸𝐴 to cost (performance-cost) and 𝐸𝐴 (energy absorption) are illustrated for the following cases (1) and (2) in Fig. 15. Case (1): 𝑇𝐶𝐹 = 1.7 mm (medium thickness with 7-ply for CFRP), 𝑇𝐴𝐿 = 0.57, 0.85, 1.7, 3.4, and 5.1 mm (correspondingly, 𝑇𝐶𝐹 /𝑇𝐶𝐹+𝐴𝐿 equals to 0.75, 0.66, 0.5, 0.33, 0.25, respectively); Case (2): 𝑇𝐴𝐿 = 1.7 mm (medium thickness for aluminum), 𝑇𝐶𝐹 = 0.972 mm (4-ply), 1.215 mm (5-ply), 1.7 mm (7-ply), 2.187 mm (9-ply), 3.402 mm (14-ply) (correspondingly 𝑇𝐶𝐹 /𝑇𝐶𝐹+𝐴𝐿 equals to 0.36, 0.42, 0.5, 0.56, 0.67, respectively). For case (1), it is found that the 𝑆𝐸𝐴 and ratio of 𝑆𝐸𝐴 to cost of the AL-CF hybrid tube increased with increase in the ratio of 𝑇𝐶𝐹 /𝑇𝐶𝐹+𝐴𝐿 , as shown in Figs. 15 (a) and (b); whilst 𝐸𝐴 decreased with increase in the ratio of 𝑇𝐶𝐹 /𝑇𝐶𝐹+𝐴𝐿 . It is worth noting that during the crushing process, the tubal walls of the outer AL component with thicknesses of 0.57mm and 0.85mm totally flipped outward due to restriction of the inner CFRP tubes; while the CFRP tubal wall were split into several external/internal fronds; and finally, the external fronds spreading outwards beyond the aluminum tube. While the outer AL component with 19

thicknesses of 1.7, 3.4 and 5.1mm deformed plastically with several external fronds, and the inner CFRP tube were split into several fronds and completely fractured inwards. After crushing, all the outer aluminum tubal components experienced plastic deformation with development of several folds and generated the cracks on the tube walls; while the inner CFRP developed a progressive crushing mode (as seen in Fig. 15). The results indicated that the hybrid AL-CF tube with a thinner aluminum tube would be of certain advantages in reducing weight and improving 𝑆𝐸𝐴. For case (2), it was found that both 𝑆𝐸𝐴 and 𝐸𝐴 of hybrid AL-CF tube increased with increasing ratio of 𝑇𝐶𝐹 /𝑇𝐶𝐹+𝐴𝐿 , as shown in Figs. 15 (a) and (b); the corresponding ratio of 𝑆𝐸𝐴 to cost decreased with increasing ratio of 𝑇𝐶𝐹 /𝑇𝐶𝐹+𝐴𝐿 . It was evident that during the crushing process, the walls of AL component in the hybrid AL-CF tube with 9-ply and 14-ply CF components reversed externally due to the restrictionof the inner thicker CFRP tube, which were different from those with the 4-ply, 5-ply and 7-ply CF components. Specifically, the wall of AL tube in the AL-CF structure with 14-ply CF tube completely reversed from the top to bottom end; and the CF tube were split into several fronds and spread beyond the AL tube owing to the high stiffness induced by the thicker inner CF component. However, the AL component with a thinner CF component turned into plastic deformation with development of several plastic folds, in which the inner CFRP component completely spread inwards and turned into a stable progressive damage mode. After crushing, all the external aluminum tubes exhibited a number of plastic folds and cracks on the aluminum tubal walls; while the CF tube experienced a progressive damage mode. The results demonstrated that the hybrid tube AL-CF with a thinner CF tube would be of certain advantages in weight reduction and performance-cost characteristics, but a thicker CF tube exhibited better energy absorption capacity. In comparison with the hybrid AL-CF tube, the net CF tube and net AL tube consistently had the highest and lowest values of 𝑆𝐸𝐴, respectively. The 𝑆𝐸𝐴 of the AL-CF hybrid tube with different ratios of 𝑇𝐶𝐹 /𝑇𝐶𝐹+𝐴𝐿 was always fallen in-between those of the net AL and CF tubes. Increasing the wall thickness of CF tube was an effective way to improve the energy absorption of hybrid tube. Decreasing the wall thickness of AL tube however led to a lower 𝐸𝐴 of AL-CF. Overall, the hybrid tube showed advantages in higher 𝑆𝐸𝐴 and the ratio of 20

𝑆𝐸𝐴 to cost.

21

Fig. 15. Comparisons of (a) 𝑆𝐸𝐴, (b) the ratio of 𝑆𝐸𝐴 to cost and (c) 𝐸𝐴 between these AL-CF hybrid tubes with different ratios of 𝑇𝐶𝐹 /𝑇𝐶𝐹+𝐴𝐿 .

4.2. Effect of 𝑇𝐶𝐹 /𝑇𝐶𝐹+𝐴𝐿 on crashworthiness of hybrid tube CF-AL In this section, two design cases were considered to illustrate the relationship between the ratio of 𝑇𝐶𝐹 /𝑇𝐶𝐹+𝐴𝐿 and crashworthiness and damage patterns of the hybrid AL-CF-AL tube. The 𝑆𝐸𝐴, 𝑆𝐸𝐴 to cost ratio (performance-cost), and 𝐸𝐴 (energy absorption) of the AL-CF-AL specimen are illustrated in Fig. 16 for cases (1) and (2) as. Case (1): 𝑇𝐶𝐹 =1.7mm (7-ply), TAL =0.57, 0.85, 1.7, 3.4, 5.1mm (i.e. 𝑇𝐶𝐹 /𝑇𝐶𝐹+𝐴𝐿 equals to 0.75, 0.66, 0.5, 0.33, 0.25, respectively); Case (2): 𝑇𝐴𝐿 =1.7mm, 𝑇𝐶𝐹 =0.972 (4-ply), 1.215 (5-ply), 1.7 (7-ply), 2.187 (9-ply), 3.402 mm (14-ply) (i.e. 𝑇𝐶𝐹 /𝑇𝐶𝐹+𝐴𝐿 equals to 0.36, 0.42, 0.5, 0.56, 0.67, respectively). For case (1), the 𝑆𝐸𝐴 and ratio of 𝑆𝐸𝐴 to the cost of the hybrid tube CF-AL increased, but the 𝐸𝐴 decreased with increasing ratio of 𝑇𝐶𝐹 /𝑇𝐶𝐹+𝐴𝐿 . It was interesting to note that when the wall thickness of the inner AL component in the hybrid tube was equal to or exceeded 1.7mm; the splitting deformation mode would occur in the outer CF tube due to the restriction of inner AL tube. On the contrary, when the thickness of inner AL tube is less than 1.7mm, the outer CF tube was still split; while the longitudinal cracks propagated with the increase in crushing displacement and the separate parts of CF wall remained mainly complete after crushing. Finally, all the internal AL components developed typical plastic folds and generated a number of cracks on the tubal walls; whilst the CF tubes were split into several external fronds. The results indicated that the AL-CF-AL with a thinner inner AL tubal wall would be of the ability to protect the outer CF tube from being split from top to bottom throughout, thereby enhancing the loading capacity of such a hybrid onfiguration. For CF-AL case (2), it was found that both 𝑆𝐸𝐴 and 𝐸𝐴 increased; while the ratio of 𝑆𝐸𝐴 to cost decreased with the increase in the ratio of 𝑇𝐶𝐹 /𝑇𝐶𝐹+𝐴𝐿 . When the wall thickness of the CF component in the hybrid tube was equal to or exceeded 2.19mm (9-ply), the outer CF component was still split into several outward fronds, but the catastrophic longitudinal crack did not actually appear. In contrast, the CF-AL tubes with 4-ply, 5-ply and 7-ply CF component were split completely by these cracks due to the weak load-bearing capacity of a

22

thinner CF tube. Finally, all the inner AL components turned into a conventional deformation mode with several plastic folds and a few cracks appeared on the tubal walls; whereas the CF components were completely spread outwards. The results indicated that a thicker CFRP wall led to a higher capacity of energy absorption; but the lower performance-cost characteristic of the hybrid CF-AL tube. To sum up, 𝑆𝐸𝐴 of the hybrid CF-AL tubes with various thicknesses was always falling in-between those of the single AL and CF tubes, which is similar to the hybrid AL-CF tube. The net CF tube and net AL tube always kept the highest and lowest values of 𝑆𝐸𝐴, respectively, as shown in Fig. 16. The energy absorption capacity of CF-AL increased with increase in thickness of the CF component, but so did the cost. Decreasing the thickness of AL component could improve the crushing behaviors of CF tubes and lead to a higher ratio of 𝑆𝐸𝐴 to cost for the CF-AL structure. Thus, the hybrid CF-AL tube with a thinner AL component exhibited certain advantages in both weight reduction and performance-cost characteristics.

4.3 Effect of sectional size on crashworthiness of hybrid tube AL-CF The sectional dimension might vary in a certain range with the equal material cost. In this section, crashworthiness and performance-cost of three different sectional sizes (small, middle, large) circular hybrid AL-CF tubes with equal material weight are explored through the numerical methods. All the hybrid tubes were set to be 120mm in length to ensure that the inner CF or the outer AL component with different sectional sizes would remain the same material weight. Table 3 summarizes the geometric details and crashworthiness criteria of these three designs. Fig. 17 illustrates the energy absorption, performance-cost characteristics as well as deformation modes of these three different hybrid tubes. It was found that with the equal material weight, the small hybrid AL-CF component (𝐷𝐴𝐿 = 53.43 and 9 plies CF layup) showed the highest energy absorption capacity and performance to cost ratio; while the larger one showed the worst performance. The reason is that the bending rigidity of the smaller hollow is much greater than the larger one thanks to its smaller diameter and thicker wall thickness. Moreover, a smaller dimensional space made the CF component difficult to change 23

deformation mode, thereby intensifying the interaction between the outer AL and inner CF components.

Fig. 16. Comparisons of (a) 𝑆𝐸𝐴, (b) the ratio of 𝑆𝐸𝐴 to cost and (c) 𝐸𝐴 between these hybrid CF-AL tubes with different ratios of 𝑇𝐶𝐹 /𝑇𝐶𝐹+𝐴𝐿

24

Table 3 Summary of the hybrid tube dimensions and crashworthiness indicators. Specim ens

𝑇𝐴𝐿

𝑃𝐶𝐹

(mm)

(plies)

Small Middle Large

2.09 1.70 1.23

9 7 5

Length 𝑊𝐴𝐿 (g) (mm) (mm)

𝑊𝐶𝐹

𝐷𝐴𝐿

53.43 63.79 86.58

120 120 120

105 105 105

(g)

𝐸𝐴 (J)

55 55 55

6153 5790 4194

𝑆𝐸𝐴 (J/g) 38.5 36.2 26.2

𝐸𝐴/𝐶𝑜𝑠𝑡 (J/$)

𝑆𝐸𝐴/𝐶𝑜𝑠𝑡 (J/g·$)

1089 1025 742

6.81 6.40 4.64

Fig. 17. Comparisons of 𝑆𝐸𝐴, the ratio of 𝑆𝐸𝐴 to cost and 𝐸𝐴, the ratio of 𝐸𝐴 to cost between CF-AL tubes with different section sizes

4.4 Effect of sectional shape on crashworthiness of hybrid tube AL-CF With the same material weight and sectional size (perimeter), different sectional shapes might result in different crushing behaviors and energy absorption capacities. In this section, the influences of different sectional shapes, namely square, hexagon, and circle, on the crashworthiness and performance-cost ratio of the AL-CF hybrid tubes are explored subject to the same material weight to help determine the best tubal profiles. The length of these three hybrid tubes was still kept to be 120mm, the perimeter of the AL tubal section was the same to ensure that the inner CF tube or the outer AL tube had the same material weight. Table 4 summarizes the geometric details and crashworthiness indictors of these three specimens. Fig. 18 illustrates the energy absorption characteristics together with crushing behaviors of these hybrid specimens. With the equal material weight, the circular hybrid AL-CF tubes showed the best crashworthiness and performance-cost features; while the square one showed the worst. For the both square and hexagon hybrid tubes, the outer AL tubes were split into

25

several parts along their edges due to a high stress concentration, which led to considerable decrease in loading carrying and energy absorption capacities.

Table 4 Summary of the different sectional shapes and crashworthiness indicators Sectional shapes Square Hexagon Circle

PCF TAL (mm) (plies) 1.70 7 1.70 7 1.70 7

PerimeterAL (mm) 200.3 200.3 200.3

Length (mm) 120 120 120

MAL (g)

MCF (g)

EA (J)

105 105 105

55 55 55

4109 4570 5970

SEA (J/g) 25.68 28.56 36.2

EA/Cost (J/$)

SEA/Cost (J/g·$)

727 809 1057

4.55 5.05 6.40

Fig. 18. Comparisons of 𝑆𝐸𝐴, the ratio of 𝑆𝐸𝐴 to cost and 𝐸𝐴, the ratio of 𝐸𝐴 to cost between CF-AL tubes with different sectional shapes

5. Discrete optimization for AL-CF tube 5.1 Discrete optimization for AL-CF tubes From the above sections, the AL-CF exhibits overall better performance to cost characteristics which will be further optimized in this section. While the parametric study demonstrated that wall thickness, sectional size and sectional shape have significant effects on the crashworthiness and performance-cost ratio of the AL-CF hybrid tubes, it is inefficient to determine an optimal configuration for such a hybrid tube with a broad range of discrete design variables. In this respect, structural optimization has proven to be effective to address this issue [43-51]. It is noted that for real-life engineering structures, the multiobjective discrete optimization [52] was efficient to search for the best possible hybrid configuration and parameters in this study. For a comparative purpose, the single objective discrete

26

optimization was also performed here. The cross-sectional perimeters of all the inner CF tubes were set to be the same (equal to that of the 7-ply CF tube in the circular AL-CF hybrid tube as tested before); and the diameter of CF tube changes with the thickness of CF tube; while the diameter of the outer AL tube was determined according to the diameter of the CF tube. It was worth noting that sectional size of the hybrid tube can be changed by adjusting the number of layers of the CF tube. Nine different sectional shapes (namely 𝐺𝐻𝑌 : triangle, square, hexagon, heptagon, octagon, enneagon, decagon, circle), twelve different ply numbers (𝑃𝐶𝐹 ) of the CF tube (i.e. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 and 14 layers of plies) and thirteen different wall thicknesses (𝑇𝐴𝐿 ) of the AL tube (i.e. 2.8, 2.6, 2.4, 2.2, 2.0, 1.8, 1.6, 1.4, 1.2, 1.0, 0.8, 0.6, and 0.4 mm) were considered in the design optimization here, respectively. In this study, the square hybrid AL-CF tube as mentioned above (𝑃𝐶𝐹 =7-ply, 𝑇𝐴𝐿 =1.7mm, PerimeterAL=200.3mm) was selected to be the baseline design. The algorithmic details of the single objective discrete optimization and corresponding steps are outlined as follows. Step 1: Defining the discrete optimization problem; A mathematical model of discrete optimization problems is established first, which should include design variables, objectives and constraints. Step 2: Determining the orthogonal array according to the optimization problem; The appropriate orthogonal array is selected according to the number of design variables and design levels. In this study, 𝐿9 (33 ) orthogonal arrays were selected, where “9” indicates the nine rows comprised of nine schemes; superscript “3” represents three design variables; “3” in the base stands for three levels with respect to each design variable, as detailed in [5]. Step 3: Determining the level of variables; The interaction can be ignored in the initial iteration; and arbitrary discrete values are selected from the design space as the middle level. The adjacent smaller values are assigned as the first level; while the adjacent greater values are assigned to the third level. Step 4: Calculating penalty function objective (𝑅𝑛𝑒𝑤 ), as follows; Since an optimization problem can commonly have different constraints, the penalty function was formulated to transform a constrained problem into a non-constrained problem 27

through a penalty strategy, as [52-53]: 𝑅𝑛𝑒𝑤 (𝑥) = 𝑅(𝑥) + 𝜑(𝑥) { 𝜑(𝑥) = 𝑠 × ∑𝑃𝑖=1 max,0, |𝑣𝑖 |-

(6)

where 𝑅𝑛𝑒𝑤 (𝑥) and 𝑅(𝑥) are the new and original objective functions, respectively. 𝜑(𝑥) denotes the penalty function, and 𝑣𝑖 represents the violation of constraint and 𝑠 provides a scaling factor. Step 5: Determining the new design for the next iteration; After obtaining the new function 𝑅𝑛𝑒𝑤 value in step 4, use the analysis of mean (ANOM) as reported in [5] to determine the optimum level. Step 6: Convergence analysis; As long as one of the following conditions is satisfied, the algorithm will be terminated: (1) there is no newly-generated 𝑅𝑛𝑒𝑤 better than the existing 𝑅𝑛𝑒𝑤 over the last five consecutive iterations; or (2) the maximum number of iteration is reached. Otherwise, the design process will return to step 3 until one of the convergence criteria is satisfied. Fig. 19 provides a flowchart of the single objective discrete optimization procedure,

28

Start Select design variable candidates for optimum design Select a standard orthogonal array Starting n-th iteration Select level values among candidates Level 2 = the initial design value Level=1, 3 = neighboring candidate values of Level 2

n=n+1 initial design= optimum level

Conduct the marix experiment with FEA and Evaluate Rnew Analysis of means (ANOM) Select the optimum level of design variables

Termination criteria is satisfied? Yes Stop

Fig. 19 Flowchart of single objective optimization procedure

5.2 Single objective optimization of energy absorption In the optimization, geometry (𝐺𝐻𝑌 ), ply number (𝑃𝐶𝐹 ) of CFRP and wall thickness (𝑇𝐴𝐿 ) of aluminum were considered to be the design variables 𝐱 = (𝐺𝐻𝑌 , 𝑃𝐶𝐹 , 𝑇𝐴𝐿 )𝑇 . To ensure the results of optimization are superior to the baseline design, peaking crushing force (𝑃𝐶𝐹), weight (𝑊 ) and material cost (𝐶𝑜𝑠𝑡) parameters should not be more than 109 kN, 160 g and $5.65, respectively For the safety requirement, energy absorption (𝐸𝐴) should be maximized in the design. Thus, a single objective optimization problem with the three design constraints is formulated mathematically as:

29

𝑅 = max*𝐸𝐴(𝐺𝐻𝑌 , 𝑃𝐶𝐹 , 𝑇𝐴𝐿 )+ 𝑃𝐶𝐹(𝐺𝐻𝑌 , 𝑃𝐶𝐹 , 𝑇𝐴𝐿 ) ≤ 109𝑘𝑁 𝐶𝑜𝑠𝑡(𝐺𝐻𝑌 , 𝑃𝐶𝐹 , 𝑇𝐴𝐿 ) ≤ 5.65$ 𝑊(𝐺𝐻𝑌 , 𝑃𝐶𝐹 , 𝑇𝐴𝐿 ) ≤ 160𝑔 𝑠. 𝑡 𝐺𝐻𝑌 ∈ ( ) (3,4,5,6,7,8,9,10,11,12,13,14) 𝑃𝐶𝐹 ∈ { {𝑇𝐴𝐿 ∈ (2.8,2.6,2.4,2.2,2.0,1.8,1.6,1.4,1.2,1.0,0.8,0.6,0.4)

(7)

The penalty functions for peak crushing force (𝑃𝐶𝐹), material cost (𝐶𝑜𝑠𝑡) and total mass (𝑊 ) as well as the new objective function can be calculated according to Eq. (6) and Eq. (7). Table 5 presents the result after the first iteration. And after seven iterations, the optimization was finally convergent and terminated, as shown in Fig. 20. The comparison in crashworthiness characteristics and geometric details between the baseline design and the optimized hybrid AL-CF tube are summarized in Table 6, in which the constraints of weight, 𝑃𝐶𝐹 and cost of the optimum design (specifically, 𝐺𝐻𝑌 = Circle , 𝑃𝐶𝐹 = 14 -ply, and 𝑇𝐴𝐿 = 1.4𝑚𝑚) were reduced by 37.2%, 11% and 10.6%, while the 𝐸𝐴 was improved by

53.4%.

Table 5 The result of the first iteration. Design variables 𝐺𝐻𝑌 𝑃𝐶𝐹 𝑇𝐴𝐿

Level 2 Enneagon 12 1.4

1 Octagon 11 1.6

3 Decagon 13 1.2

Fig. 20 Convergence history of objective function 𝑅𝑛𝑒𝑤 for the hybrid -AL-CF tube design

30

Table 6 Comparison between the baseline design and optimization design

𝐺𝐻𝑌

𝑃𝐶𝐹

𝑇𝐴𝐿

𝑀

𝐸𝐴

𝑃𝐶𝐹

𝐶𝑜𝑠𝑡

(plies)

(mm)

(g)

(J)

(kN)

(J/g)

Baseline design

Square

7

1.7

160

4109

109

5.65

Optimum design

Circle

14

1.4

100.5

6304

97

5.05

Increasing ratio

-

-

-

-37.2%

53.4%

-11%

-10.6%

5.3. Single objective optimization of cost Similar to the problem in the single objective optimization for energy absorption as presented in section 5.2, a single objective optimization of cost with the three constraints of 𝑃𝐶𝐹, 𝐸𝐴, and 𝑊 is also formulated mathematically as: 𝑅 = max*−𝐶𝑜𝑠𝑡(𝐺𝐻𝑌 , 𝑃𝐶𝐹 , 𝑇𝐴𝐿 )+ 𝑃𝐶𝐹(𝐺𝐻𝑌 , 𝑃𝐶𝐹 , 𝑇𝐴𝐿 ) ≤ 109𝑘𝑁 𝐸𝐴(𝐺𝐻𝑌 , 𝑃𝐶𝐹 , 𝑇𝐴𝐿 ) ≥ 4109𝐽 𝑊(𝐺𝐻𝑌 , 𝑃𝐶𝐹 , 𝑇𝐴𝐿 ) ≤ 160𝑔 𝑠. 𝑡 𝐺𝐻𝑌 ∈ ( ) 𝑃𝐶𝐹 ∈ (3,4,5,6,7,8,9,10,11,12,13,14) { {𝑇𝐴𝐿 ∈ (2.8,2.6,2.4,2.2,2.0,1.8,1.6,1.4,1.2,1.0,0.8,0.6,0.4)

(8)

According to Eq. (6) and Eq. (8), the penalty functions for 𝑃𝐶𝐹, 𝐶𝑜𝑠𝑡 and 𝑊 can be calculated, and the new objective function was updated. Accordingly, Table 7 presents the results for the first iteration. After eight iterations, the optimization procedure was convergent and terminated, as illustrated in Fig. 21. The comparison in the crashworthiness characteristics and geometric details of the baseline design and the optimized design of hybrid AL-CF tubes are summarized in Table 8. It can be seen that the weight, 𝑃𝐶𝐹 and cost of the optimum design (i.e. 𝐺𝐻𝑌 = Circle, 𝑃𝐶𝐹 = 14-ply, and 𝑇𝐴𝐿 = 0.8𝑚𝑚) were reduced by 49.3%, 23.9% and 12.2%, respectively; while the 𝐸𝐴 was improved by 25.7% in comparison with the baseline design.

Design variables 𝐺𝐻𝑌 𝑃𝐶𝐹 𝑇𝐴𝐿

Table 7 The result of the first iteration. Level 1 2 Octagon Enneagon 11 12 1.6 1.4

31

3 Decagon 13 1.2

Fig. 21 Convergence history of objective function 𝑅𝑛𝑒𝑤 for the CF-AL-CF tube design Table 8 Compressive performance indicators of baseline design and optimum design.

𝐺𝐻𝑌

𝑃𝐶𝐹

𝑇𝐴𝐿

𝑊

𝐸𝐴

𝑃𝐶𝐹

𝐶𝑜𝑠𝑡

(plies)

(mm)

(g)

(J)

(kN)

(J/g)

Baseline design

Square

7

1.7

160

4109

109

5.65

Optimum design

Circle

14

0.8

81

5166

83

4.96

Increasing ratio

-

-

-

-49.3%

25.7%

-23.9%

-12.2%

5.4. Multiobjective optimization of energy absorption and cost For a real-life problem, pursuits of better performance and lower product cost should be considered simultaneously. The procedure of multiobjective discrete optimization is very similar to the single objective. Steps 1-4 of the proposed multiobjective discrete optimization are similar to those mentioned in section 5.1. Then, a new step should be considered here to conduct the gray relational analysis which was used to transform multiobjective problems into an equivalent single function as follows [54-55]: Step 5: Conducting grey relational analysis; The grey relevance coefficient (𝛾𝑖𝑗 ) and grey relevance degree (𝛤𝑖 ) can be calculated according to equations blow; 𝜂 −𝑚𝑖𝑛∀𝑗 𝜂𝑖𝑗 ∀𝑗 𝑖𝑗 −𝑚𝑖𝑛∀𝑗 𝜂𝑖𝑗

𝑥𝑖𝑗 = 𝑚𝑎𝑥𝑖𝑗 𝜂

𝑖 = 1,2, … , 𝑀; 𝑗 = 1,2, … , 𝑁

(9)

where 𝜂𝑖𝑗 , 𝑥𝑖𝑗 , 𝑀 and 𝑁 are the value for the j-th criterion, the corresponding normalized value, the number of experiment and the quality characteristics, respectively. 32

Then, the grey relational coefficient (𝛾𝑖𝑗 ) is calculated as: 𝛾𝑖𝑗 =

𝑚𝑖𝑛𝑖 𝑚𝑖𝑛𝑗 |𝑥𝑗∗ −𝑥𝑖𝑗 |+𝜉𝑚𝑎𝑥𝑖 𝑚𝑎𝑥𝑗 |𝑥𝑗∗ −𝑥𝑖𝑗 | |𝑥𝑗∗ −𝑥𝑖𝑗 |+𝜉𝑚𝑎𝑥𝑖 𝑚𝑎𝑥𝑗 |𝑥𝑗∗ −𝑥𝑖𝑗 |

(10)

where 𝑥𝑗∗ is the most ideal normalized response for the j-th performance indicator, and 𝜉 is the identification coefficient (0 ≤ 𝜉 ≤ 1), which is set to be 0.5 [54-55]. Finally, the grey relational grade (𝛤𝑖 ) is calculated by using the grey relational coefficients (𝛾𝑖𝑗 ) and the weighting coefficient 𝑤𝑗 [54] according to Eq. (11). 𝛤𝑖 = ∑𝑁 𝑗=1 𝑤𝑗 𝛾𝑖𝑗

(11)

Step 6: Performing an ANOM for grey relational grade (GRD) [54]; Step 7: Comparing the GRD and determine a new design for the next iteration; Step 8: Conducting a convergence analysis; Fig. 22 provides the flowchart of the multiobjective discrete optimization procedure. Thus, a multiobjective optimization problem with peak force (𝑃𝐶𝐹) and mass (𝑊) constraints is formulated mathematically as follows: 𝑅 = max*𝑓1 (𝐱) = 𝐸𝐴(𝐺𝐻𝑌 , 𝑃𝐶𝐹 , 𝑇𝐴𝐿 ), 𝑓2 (𝐱) = −𝐶𝑜𝑠𝑡(𝐺𝐻𝑌 , 𝑃𝐶𝐹 , 𝑇𝐴𝐿 )+ 𝑃𝐶𝐹(𝐺𝐻𝑌 , 𝑃𝐶𝐹 , 𝑇𝐴𝐿 ) ≤ 109𝑘𝑁 𝑊(𝐺𝐻𝑌 , 𝑃𝐶𝐹 , 𝑇𝐴𝐿 ) ≤ 160𝑔 (12) 𝑠. 𝑡 𝐺𝐻𝑌 ∈ ( ) 𝑃𝐶𝐹 ∈ (3,4,5,6,7,8,9,10,11,12,13,14) { {𝑇𝐴𝐿 ∈ (2.8,2.6,2.4,2.2,2.0,1.8,1.6,1.4,1.2,1.0,0.8,0.6,0.4) After normalizing the sequence (𝑥𝑖𝑗 ), the grey relational coefficient (𝛾𝑖𝑗 ) and grey relational grade (GRD) (𝛤𝑖 ) in the first iteration can be calculated according to Eqs. (9)-(11), and the results were summarized in Table 9. According to Eq. (6) and Eqs. (9)-(12), the penalty functions for 𝑃𝐶𝐹 and 𝑊 can be calculated; then the new objective function 𝑓1 (𝐱) and 𝑓2 (𝐱) can be updated, after that, the GRD can be determined according to the principal component analysis (PCA) as descripted in Ref. [54]. Finally, the levels of the design variables in the first iteration were determined according to the analysis of means (ANOM) for GRD as presented in Table 10. After seven iterations, the optimization procedure was convergent and terminated, as illustrated in Fig. 23. Table 11 shows the comparison in the crashworthiness criteria and geometric details between the baseline design and the optimum design, in which the weight, 𝑃𝐶𝐹 and cost of the optimum design (𝐺𝐻𝑌 = Circle, 𝑃𝐶𝐹 = 14-ply, and 𝑇𝐴𝐿 = 1.2𝑚𝑚) were 33

reduced by 41.3%, 18.0% and 11.2%, respectively; while the 𝐸𝐴 was improved by 48.0%.

Table 9 The sequence after normalizing, the grey relational coefficient and grey relational grade (GRD) The sequence after

NO .

Grey relational coefficient

normalizing

GRD

Order

𝑓1 (𝐱)

𝑓2 (𝐱)

𝑓1 (𝐱)

𝑓2 (𝐱)

𝑓1 (𝐱)

𝑓2 (𝐱)

1

5469

-5.1

0.62

0

0.56

0.33

0.45

9

2

5111

-5.04

0.14

0.6

0.37

0.56

0.46

8

3

5147

-5

0.18

1

0.38

1

0.69

3

4

5355

-5.06

0.46

0.4

0.48

0.45

0.47

7

5

5036

-5.01

0.03

0.9

0.34

0.83

0.59

5

6

5754

-5.06

1

0.4

1

0.45

0.73

2

7

5011

-5.02

0

0.8

0.33

0.71

0.52

6

8

5687

-5.08

0.91

0.2

0.85

0.38

0.62

4

9

5690

-5.03

0.91

0.7

0.85

0.63

0.74

1

Table 10 The levels of design variables in the first iteration obtained from ANOM Level Design variables x 1 2 3 Octagon Enneagon Decagon 𝐺𝐻𝑌 11 12 𝑃𝐶𝐹 13 1.4 1.0 𝑇𝐴𝐿 1.2

Table 11 Comparison between initial baseline design and optimum design.

𝐺𝐻𝑌

𝑃𝐶𝐹

𝑇𝐴𝐿

𝑊

𝐸𝐴

𝑃𝐶𝐹

𝐶𝑜𝑠𝑡

(plies)

(mm)

(g)

(J)

(kN)

(J/g)

Baseline design

Square

7

1.7

160

4109

109

5.65

Optimum design

Circle

14

1.2

94

6082

89.4

5.02

Increasing ratio

-

-

-

-41.3%

48.0%

-18.0%

-11.2%

34

Taguchi approach Start Select design variable candidates for optimum design Select a standard orthogonal array Starting n-th iteration Select level values among candidates Level 2 = the initial design value Level=1, 3 = neighboring candidate values of Level 2

n= n+1 Initial design= Optimum level

Conduct the marix experiment with FEA and Evaluate Rnew

Perform PCA and calculate grey relational grades Conduct ANOM to select the optimum level of the design variables

Grey relational analysis

Calculate grey relational coefficients

Compare the optimum level and rows of orthogonal arrays

Termination criteria is satisfied？

No

Yes Obtain optimal design

Fig. 22 Flowchart of discrete multiobjective optimization procedure

35

Fig. 23 Convergence history of grey relational grade (GRD) for CF-AL-CF design.

6. Conclusion This study investigated the crushing behaviors and energy absorption mechanisms of aluminum/CFRP structures. To explore the crashing characteristics of such metal/composite hybrid structures, a parametric study followed by a multiobjective optimization was conducted to understand the effects of 𝑇𝐶𝐹 /𝑇𝐶𝐹+𝐴𝐿 on the crashworthiness and cost performance. Compared with the previous studies on metal-composite hybrid crashworthiness structures in literature [19-26], this study filled a knowledge gap on analysis of interactive effect of metal-composite hybrid structures on energy absorption and crashworthiness. It is demonstrated that the hybrid configuration did provide an efficient way to enhance structural crashworthiness and reducing the materials cost. The structural optimization of such metal-composite hybrid structure allows to achieve further improvement to balance performance and cost. The following specific conclusion can be drawn from this study. (1) For the AL-CF hybrid (outer AL and inner CF) tube, changes in deformation modes of both AL and CF tubal components contributed to the enhancements of energy absorption for the AL (43.6%) and CF (17.8%) components in the hybrid tube. The frictional energy dissipation of the AL-CF specimen was 45.6% higher than the sum of ebergy absorptions of the individual AL tube and CF tube, largely improving energy-absorbing capacity of the hybrid AL-CF (AL-1.7mm, CF-7ply) tube. (2) For the CF-AL hybrid (inner AL and outer CF) tube, although the energy absorption of

36

the AL tube in the hybrid structure was 27.6% higher than that of the corresponding single net AL tube, the energy absorption of the outer CF tube was 31.9% lower than that of the single net CF tube due to the change in the deformation and failure from splaying mode to splitting mode. Moreover, the friction dissipation energy of the CF-AL hybrid tube was 47.6% lower than the sum of energy absorptions of the corresponding single net AL tube and single net CF tube, thereby decreasing overall energy absorption of the CF-AL configuration. (3) The 𝐸𝐴 of both the AL-CF and CF-AL hybrid tubes increased with the increase in the CF wall thickness and also increase in the AL wall thickness; whilst the 𝑆𝐸𝐴 of both the AL-CF and CF-AL hybrid tubes increased with the increase in the CF wall thickness or decrease in the AL wall thickness. From the performance-cost analysis, it was found that the hybrid tube with a thinner AL tube had greater potential to be a cost-effective energy absorber. With the same usage of material, the hybrid tube with a circular sectional shape or a smaller sectional size exhibited better crushing performance. (4) After the multiobjective optimization, the optimized AL-CF specimen with circle section shape, 14-ply CFRP prepregs and 1.2mm thick aluminum tube exhibited overall better crashworthiness than the baseline design. The weight, peak crushing force 𝑃𝐶𝐹 and material cost of the optimum design were reduced by 41.3%, 18.0% and 11.2%, respectively, while the 𝐸𝐴 was improved by 48.0%.

Author Contribution Statement

Guangyong Sun and Zhen Wang designed the analysis scheme and wrote the manuscript, Xihong Jin and Zhen Wang conducted the simulations, Zhen Wang and Qing Li analyzed the numerical results, and Qing Li revised the manuscript.

37

Conflict of interest The authors declared that they have no conflicts of interest to this work.

Acknowledgements The project was supported by National Natural Science Foundation of China (51575172, 11602161), Australian Research Council (ARC) Discovery Early Career Researcher Award (DE160101633) and Discovery project (DP190103752), Hunan Provincial Innovation Foundation for Postgraduate (CX20190281), and the Open Fund of State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body (31615008).

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