On lateral compression of circular aluminum, CFRP and GFRP tubes

Journal Pre-proofs On lateral compression of circular aluminum, CFRP and GFRP tubes Shunfeng Li, Xiao Guo, Qing Li, Dong Ruan, Guangyong Sun PII: DOI: Reference:

S0263-8223(19)32971-X https://doi.org/10.1016/j.compstruct.2019.111534 COST 111534

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Composite Structures

Received Date: Revised Date: Accepted Date:

5 August 2019 21 September 2019 4 October 2019

Please cite this article as: Li, S., Guo, X., Li, Q., Ruan, D., Sun, G., On lateral compression of circular aluminum, CFRP and GFRP tubes, Composite Structures (2019), doi: https://doi.org/10.1016/j.compstruct.2019.111534

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On lateral compression of circular aluminum, CFRP and GFRP tubes Shunfeng Li1, 2, Xiao Guo1, Qing Li3, Dong Ruan4, Guangyong Sun1, 3* 1State

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

2School

of Mechanical, Aerospace and Civil Engineering, The University of Manchester, Manchester, M13 9PL, UK

3School

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

4Faculty

of Science, Engineering and Technology, Swinburne University of Technology, Hawthorn, VIC 3122, Australia Abstract

Thin-walled structures made of lightweight materials, e.g. aluminum, fiber reinforced composites, have been increasingly used as energy absorption structures in vehicles. This work aimed to characterize the lateral crushing behaviors of circular aluminum, glass fiber reinforced plastics (GFRP) and carbon fiber reinforced plastics (CFRP) tubes with different geometric configuration such as diameter-to-thickness (D/T) ratio or thickness. In the experimental investigation, four different D/T ratios varied from 10.78 to 48.02 was considered here for the aluminum, GFRP and CFRP tubes. Crashing behaviors such as force-displacement curves, deformation histories, and crushing force was quantified. The experimental results revealed that the load carrying capacities, energy absorption (EA) and specific energy absorption (SEA) of the circular tubes decline with the increase in the D/T ratio. It was found that better crashworthy characteristics of thicker composites, with a smaller D/T ratio, is due to the more favorable failure modes occurring throughout lateral compression. The lateral crashworthy performance of the GFRP tubes was marginally better than the CFRP counterparts. Due to ductile behavior of aluminum tubes and brittle behavior of composites, aluminum tubes showed much better lateral crashworthiness than that of the composite counterparts. Moreover, with the increase in the D/T ratio, aluminum tubes * Corresponding Author: Tel: +86-13786196408; Email: [email protected]

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exhibited greater advantage on crashworthiness than the composite tubes. On the basis of the experimental data, explicit finite element analysis was further conducted for modeling the lateral crushing behavior of aluminum, GFRP and CFRP tubes. The numerical results were in good agreement with the experimental data, demonstrating the validity of these finite element (FE) models in predicting lateral crushing responses of aluminum, GFRP and CFRP tubes. The proposed FE models can be exploited to further study similar thin-walled metal and composite structures for design optimization.

Keywords: Thin-walled structures; Lateral compression; Failure mode; Energy absorption; Numerical simulation.

1. Introduction Our highly motorized society has drawn growing concerns in environmental sustainability and transportation safety. On the one hand, substantial effort has been devoted to utilization of lightweight materials in automotive, railway, naval and aerospace industries for reducing fuel consumption [1, 2]. On the other hand, safety and legislative requirements have been continuously intensified as this issue becomes more and more critical socioeconomically. For this reason, thin-walled structures made of lightweight composites, e.g. CFRP and GFRP, have been increasingly adopted as energy-absorbing structures in modern vehicles attributable to their particularly lower density and higher specific mechanical properties in comparison with conventional metal materials [3-6]. To date, the conventional metallic structures have been exhaustively investigated. However, the crushing behavior of composites are much more difficult to predict due to their complicated failure mechanisms including fiber fracture, matrix crack, delamination between layers, etc. [7]. Thus, there is urgent need to understand the crashworthiness of composites, which is of great interest to meet safety and energy efficiency requirements of transportation industry. Over the past decades, numerous researchers have investigated the collapse behaviors of thin-walled metallic structures with a particular focus on axial loading conditions. Aluminum alloy, as a relatively newer class of lightweight metallic material, are increasingly adopted to 2

replace traditional steel in modern vehicles. There have been substantial studies on crashworthiness analysis and design of aluminum structures in literatures [8-11]. In this regard, Andrews et al. [12] performed the experimental study on the crushing behavior of aluminum tubes; and they classified four different failure modes, namely concertina, diamond, mixed and bulking failure modes. Guillow et al. [13] further graphed a classification chart of failure mode for aluminum tubes, which was used to understand the effects of geometric sizes on crushing behavior and failure mechanism. It is noted that to date, conventional metallic structures have been exhaustively investigated, whilst fiber reinforced plastic (FRP) composites (such as, CFRP and GFRP) remain under-studied on their crashworthiness. Fiber reinforced composites impose great challenge yet opportunities for new applications in automotive engineering, in which crashworthiness signifies a critical criterion to maintain structural integrity and safety in crushing event. In this aspect, Mamalis et al. [14] studied the crashworthy characteristics of composites tubes by static and dynamic axial crushing tests; and three distinct failure modes were identified, namely progressive end-crushing, unstable local tube wall buckling and mid-length collapse mode. Zhu et al. [15] explored crushing behavior of various CFRP structures with different cross-sectional geometries through the validated finite element models; it was found that the circular tube showed superior capacity of energy absorption to the other cross-sectional counterparts. Liu et al. [16, 17] investigated the crushing behavior of square or double-hat shaped CFRP tubes through the experiments; and they also developed a multiscale approach for predicting 3D elastic model of carbon twill weave fabric composites. While there has been a continuous interest in crashworthiness of thin-walled metal or composite tubes, the existing studies have been largely restricted on axial loading conditions. In automotive practice, tubular structures are prone to subject to oblique or lateral (transverse) crash [10]. For this reason, it is vitally important to understand the crushing behavior of thin-walled tubes under oblique or lateral loading conditions. Concerning oblique loading conditions, for example, Reyes et al. [18] studied the crashworthiness of aluminum tubes through off-axial crushing. Song and Du [19] explored the energy absorption capacities of GFRP tubes under five different oblique loading angles (i.e. 5°, 10°, 15° and 20°, 25°). Recently, the author’s team [20] experimentally investigated the crushing behavior of CFRP 3

and aluminum tubes under axial and four oblique loading angles, varying from 0° to 30°; we found that the loading angle ranged from 0° to 10° has little influence on energy absorption for both CFRP and aluminum tubes; however, the capacity of CFRP tube declines dramatically when the angle increases from 10° to 30°, while that of aluminum tube decreases gradually. With regard to the studies of thin-walled structures subjected to lateral loading conditions. Metallic structures have attracted substantial interest for lateral loading conditions [21-25]. Lu and Yu [26] reviewed the analytical and experimental studies on thin-walled metal tubes under in-plane loads, in which they suggested two deformation modes under lateral compression, namely Burton and Craig’s mode that involves straightening of the tube at the moving contact points; and de Runtz and Hodge’s mode that is of four stationary plastic hinges during compression. Baroutaji and Olabi [27, 28] explored the deformation behavior and the energy absorption of empty and sandwich circular metal tubes under lateral loading; and they further established the optimal configuration of these tubes by performing both parametric and multi-objective optimization design. Relatively speaking, studies on composite structures for lateral crushing are much limited. In this regard, Gupta and Abbas [29] experimentally investigated the GFRP tubes subjected to quasi-static lateral crushing in terms of the peak force, post collapse force, energy absorption and deformation modes. Abdewi et al. [30] carried out the lateral crushing experiments to explore the effects of corrugation geometry on the crashworthiness of woven GFRP tubes; and the results showed that the load bearing capacity was not influenced much by corrugation geometry. More recently, Yan et al. [31] experimentally studied the lateral crushing characteristics of empty and polyurethane-foam natural flax fabric reinforced epoxy composite tubes by comparing with the metal tubes. Almeida Jr. et al. [32] developed a computational model to predict the behavior of the CFRP tubes under radial compression. Although there is an urgent need and particular importance to furthering the crashworthiness study on FRP composite and other lightweight materials for lateral loading conditions, the related reports are limited. This study aimed to investigate the effects of D/T ratio on crashworthy characteristics of thin-walled circular composite tubes through lateral crushing experiments in comparison with the aluminum counterparts. Four different D/T ratios of aluminum, GFRP and CFRP tubes 4

were selected to explore its influence on the lateral crushing behavior. Finite element modeling was used by correlating with the experimental results for these aluminum and composite tubes. Finally, the lateral crushing functional performance and cost efficiency of aluminum, GFRP and CFRP tubes were compared for design guide. 2. Experimental setup 2.1 Materials and specimens This study investigated the lateral crushing behaviors of aluminum, GFRP and CFRP tubes with different geometric configurations. As shown in Fig. 1, several specimens in the three groups of aluminum, GFRP and CFRP tubes were prepared. For ensuring the repeatability of experiments, the same tests were repeated three times. Totally, 36 tubular specimens were used for quasi-static lateral compression.

Fig. 1. Aluminum, GFRP and CFRP tubes with four different D/T ratios.

2.1.1

Geometrical description

Four different thicknesses (T = 1.0, 2.0, 3.0 and 4.0 mm) of aluminum tubes were prepared and tested with the same length and outer diameter (Douter) of 50 mm (Fig. 1). As for the composites, the GFRP and CFRP specimens were made from 0°/90° woven fabric glass-epoxy prepregs and carbon-epoxy prepregs, respectively. Hand lay-up technique by experienced technician was adopted with the same cylindrical mandrel. For investigating the effect of ply numbers of composites on the lateral crushing characteristics; and comparing

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with the aluminum tubes with four thicknesses, both the GFRP and CFRP tubes were fabricated using four ply numbers of 4-ply, 8-ply, 12-ply and 16-ply. Both the length and inner diameter (Dinner) of these two groups of composite specimens are 50 mm (Fig. 1). For facilitating the analysis, the average dimeter of circular tubes (i.e. D = (Douter + Dinner)/2) was employed; and the D/T ratio of specimens were presented in Fig. 1 with the details to be depicted. 2.1.2

Material properties

As shown in Fig. 2, the 2D-Digital Image Correlation (DIC) system was employed to obtain the full-field strain data from the testing samples. Aluminum tubes were made in commercial aluminum alloy AA6063-T6. To determine the material properties of aluminum specimens, the standard in-house tensile tests were conducted by closely following test standard ASTM E8M at a crosshead speed of 2 mm/min. Three tensile testing samples were directly sectioned from the aluminum tubes along the axial direction for avoiding the inconsistence induced by the extrusion process. The typical engineering stress–strain curve of aluminum sample were plotted in Fig. 3 (a). To determine the material properties of the composite tubes, the tensile tests at a crosshead speed of 2 mm/min were carried out according to test standard ASTM D3039. Three testing coupons of GFRP or CFRP materials were cut from the GFRP or CFRP plates made of the same lay-up configuration as the composite tubes. The tensile testing direction of the GFRP/CFRP coupons was aligned with the 0° direction of composite layup, which was also following the axial direction of composite tubes. The typical stress-strain curves of GFRP and CFRP specimens are plotted in Fig. 3 (b).

Aluminum GFRP

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CFRP

Fig. 2. Samples for tensile test.

(a) Stress-strain curve of aluminum alloy

(b) Stress-strain curves of composite materials

Fig. 3. The typical stress-strain curves of three different materials.

2.2 Quasi-static lateral compression As shown in Fig. 4, the quasi-static lateral compression was performed in an INSTRON 5985 testing machine, where the circular tube was compressed by the two rigid and flat platens. During the test, the top platen moved downwards to crush the specimen at a constant crosshead speed of 4 mm/min, whereas the bottom platen was fixed. The final displacement for the lateral compression was set to be the same value of 40 mm. The deformation details of all tubes were photographed and the force-displacement curves were recorded during quasi-static lateral compression.

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Fig. 4. Quasi-static lateral compression test.

2.3 Crashworthiness criteria To investigate the lateral crushing behavior of aluminum, GFRP and CFRP tubes, the crashworthiness indicators, such as the initial collapse force (𝐹0), energy absorption (EA) and specific energy absorption (SEA) were used in this study. The energy absorption (EA) is defined as the energy absorbed by the tube during the lateral crushing, which can be calculated as 𝑑

(1)

EA = ∫0𝐹(𝑥)𝑑𝑥

where 𝑑 is the same crushing distance of 40 mm. Specific energy absorption (SEA), which is defined as the ratio of energy absorption to the mass (𝑚) of the specimen, as SEA =

EA 𝑚

(2)

3. Experimental results The snapshots of the crushing process and the corresponding force-displacement curves of aluminum, GFRP and CFRP tubes were obtained through the quasi-static lateral compression. The same tests were repeated three times to ensure the repeatability of experiments, the average and standard deviation of the experimental results were also reported in this section. 3.1 Aluminum tube 3.1.1

Failure mode

As shown in Fig. 5, four crushing tests for the aluminum tubes with four different D/T ratios (i.e. D/T= 47.55, 23.83, 15.56, 11.48) were presented, from the begin of crushing to the end with every 8 mm compressive distance, i.e. 0, 8, 16 mm, etc. It is observed that deformation mode of the aluminum tubes with the same outer diameter was not affected by the D/T ratio much. Initially, the contact between the tube and two flat platens was

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concentrated in the two lines and the tube wall behaved as a circular arc to deform elastically. Then, the first pair of plastic hinges appeared in the position of the two points (purple points on the Fig. 5(d)) and the second pair of plastic hinges occurred at the horizontal sites of the tubes (red points on Fig. 5(d)), meaning that the aluminum tubes experienced substantial plastic deformation. All aluminum tubes exhibited two pairs of symmetric stationary plastic hinges (see Fig. 5 (d)) during the lateral compression.

(a) AL-T1 (D/T = 47.55)

(b) AL-T2 (D/T = 23.83)

(c) AL-T3 (D/T = 15.56)

(d) AL-T4 (D/T = 11.48) Fig. 5. Deformation histories of the four different aluminum tubes under quasi-static lateral compression.

3.1.2

Force-displacement curve

The force-displacement curves of the aluminum tubes (specimen AL-T2, D/T = 23.83 with a wall thickness of 2 mm) was plotted in Fig. 6 from the three repeated tests. Clearly these three curves of aluminum tubes almost coincided with each other, demonstrating excellent repetition of lateral compression tests.

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Fig. 6. Force-displacement curves of aluminum tubes (AL-T2, D/T = 23.83, wall thickness: 2 mm) under quasi-static lateral compression.

Fig. 7 plots the four force-displacement curves of aluminum tubes with four different thicknesses. Also, the details of aluminum tubes and all experimental results were summarized in Table 1. From the deformation profiles (Fig. 5) and force-displacement curves (Fig. 7), the progressive collapse of aluminum tubes can be divided into three stages, namely, elastic deformation stage, plastic collapse stage and self-contact and densification stage. In the beginning, the crushing force increased linearly in the elastic stage and then experienced an almost plateau where the force increased marginally and most energy was absorbed in the plastic stage (Fig. 7). Finally, the internal self-contact and densification started from the central symmetric lines, in which the top and bottom inner surfaces began to contact (see Fig. 5 (c) and 5 (d)), which resulted in a rapid increase in the lateral crushing force.

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densification stage plastic collapse stage

F0

elastic stage

Fig. 7. Force-displacement curves for four different aluminum tubes under quasi-static lateral compression.

Table 1 The test results of aluminum tubes Specimen

T / mm

D / mm

D/T

Mass / g

F0 / kN

EA / J

SEA / J·g-1

AL-T1-1

1.02

48.98

48.02

21.00

0.50

26.23

1.25

AL-T1-2

1.04

48.96

47.08

21.00

0.50

26.34

1.25

AL-T1-3

1.03

48.97

47.54

21.00

0.50

25.70

1.22

Average

1.03

48.97

47.55

21.00

0.50

26.09

1.24

S.D.

0.01

0.01

0.38

0.00

0.00

0.28

0.01

AL-T2-1

2.01

47.99

23.88

42.00

2.11

116.86

2.78

AL-T2-2

2.02

47.98

23.75

42.00

2.08

116.30

2.77

AL-T2-3

2.01

47.99

23.88

42.00

2.15

118.32

2.82

Average

2.01

47.99

23.83

42.00

2.11

117.16

2.79

S.D.

0.00

0.00

0.06

0.00

0.03

0.85

0.02

AL-T3-1

3.04

46.96

15.45

60.00

5.25

292.29

4.87

AL-T3-2

2.99

47.01

15.72

60.00

5.25

294.71

4.91

AL-T3-3

3.03

46.97

15.50

60.00

5.21

294.38

4.91

Average

3.02

46.98

15.56

60.00

5.24

293.79

4.90

S.D.

0.02

0.02

0.12

0.00

0.02

1.07

0.02

AL-T4-1

4.02

45.98

11.44

79.00

7.08

446.87

5.66

AL-T4-2

4.00

46.00

11.50

78.00

7.10

447.70

5.74

AL-T4-3

4.00

46.00

11.50

79.00

7.03

460.16

5.82

Average

4.01

45.99

11.48

78.67

7.07

451.58

5.74

11

S.D.

0.01

0.01

0.03

0.47

0.03

6.08

0.07

Note: the specimen label, e.g. AL-T1-1, denotes the aluminum tube (AL) with a thickness (i.e. T1) of 1.0 mm of specimen #1 of this group. S.D. means the standard deviation.

3.2 GFRP tube 3.2.1

Failure mode

For investigating the lateral crushing characteristics of GFRP tubes, four different D/T ratios (i.e. D/T = 45.79, 23.22, 14.68, 11.01), corresponding to four ply laminates, were considered for the GFRP tubes. Fig. 8 shows the snapshots of the GFRP tubes under lateral compression, from the beginning to the end of loading with each 8 mm crushing distance, i.e. 0, 8, 16 mm, etc. Different failure modes can be observed during the compression. As shown in Fig. 8 (a), the deformation mode of 4-ply GFRP tube involves straightening of the tube wall at the moving contact lines in between the outer tube and platens; two longitudinal fracture lines were formed at the left and right horizontal ends of the tube. When the ply number of the tube increased to 8 (Fig. 8 (b)), four longitudinal fracture lines were formed at about 90° phase angle as usually observed in the aluminum tubes. As for the failure mode of 12-ply or 16-ply GFRP tubes (Fig. 8 (c) and 8 (d)), delamination occurred before the formation of fracture lines, then propagated along the circumferential direction of tube wall. It is observed that the delamination more likely initiated from the two contact lines. With same inner diameter, the D/T ratio had significant effect on deformation mode. The deformation mode of GFRP tubes became more complex and the longitudinal fracture lines formed earlier with decreasing D/T ratio.

(a) GF-P4 (D/T = 45.79)

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(b) GF-P8 (D/T = 23.22)

(c) GF-P12 (D/T = 14.68)

(d) GF-P16 (D/T = 11.01) Fig. 8. Deformation histories of the four different GFRP tubes under quasi-static lateral compression.

To observe the failure modes of GFRP tubes at a micro level, four microscopic photographs of each tubal thickness are presented in Fig. 9. After the lateral compression, four crushed GFRP tubes (as in Fig. 8) were further examined. It is evident that the fiber breakage and matrix crack are the most common damage modes for a smaller thickness (or a higher D/T ratio, Fig. 9 (a) and (b)); while the delamination is also one of the predominately failure modes in the thicker GFRP tubes (Fig. 9 (c) and (d)), which indicates that GFRP tubes can absorbed more energy through the delamination failure.

(a) GF-P4

(b) GF-P8

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(c) GF-P12

(d) GF-P16

Fig. 9. Microscopic photos to show the failure modes of GFRP tubes in different thicknesses.

3.2.2

Force-displacement curve

Fig. 10 plotted the three force-displacement curves for the 8-ply GFRP tubes under lateral compression. Clearly these curves follow the same trend. Specifically, after a rapid increase and several drops, a low plateau stage was achieved in the crushing force; then increased dramatically at the end of compression. The force level was also very close throughout these three experimental tests, indicating good repeatability for GFRP specimens. For the sake of convenience, only one testing result of each group of GFRP tubes was reported in the following section.

Fig. 10. Force-displacement curves of the three 8-ply GFRP specimens for quasi-static lateral crushing.

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Fig. 11 shows the force-displacement curves of GFRP tubes under lateral loading and Table 2 summarizes the parameters of GFRP tubes as well as the experimental results. Progressive failure mode was seen in all these tests. It is similar to the force-displacement curves of aluminum tubes, the curves of GFRP tubes can be also divided into three stages: namely elastic stage, progressive crushing stage and internal self-contact stage. At the first elastic stage, the crushing force increased to its initial peak approximately elastically (see the dash line in Fig. 11). It should be noted that with decrease of D/T ratio the force increased much more quickly and arrived its peak earlier. After reaching the peak, the force dropped dramatically due to fracture in the ply layers and rose again by engaging residual intact layers. Then the crushing force dropped significantly again due to formation of the second longitudinal fracture line. Note that only two longitudinal fracture lines were formed at the left and right horizontal ends of the tube GF-P4 (Fig. 8 (a)). After formation of two fracture lines, the crushing force reached an almost plateau phase till the end of crushing.

F0

elastic stage progressive crushing stage self-contact stage

Fig. 11. Force-displacement curves of four different GFRP tubes under quasi-static lateral compression.

Table 2 The test results of GFRP tubes Specimen

T / mm

D / mm

D/T

Mass / g

F0 / kN

EA / J

SEA / J·g-1

GF-P4-1

1.13

51.13

45.25

16.00

0.82

14.86

0.93

GF-P4-2

1.13

51.13

45.25

16.00

0.79

15.98

1.00

GF-P4-3

1.09

51.09

46.87

16.00

0.73

15.68

0.98

15

Average

1.12

51.12

45.79

16.00

0.78

15.51

0.97

S.D.

0.02

0.02

0.77

0.00

0.04

0.47

0.03

GF-P8-1

2.26

52.26

23.12

33.00

2.81

50.32

1.52

GF-P8-2

2.25

52.25

23.22

33.00

2.72

47.58

1.44

GF-P8-3

2.24

52.24

23.32

33.00

2.67

48.45

1.47

Average

2.25

52.25

23.22

33.00

2.73

48.79

1.48

S.D.

0.01

0.01

0.08

0.00

0.06

1.14

0.03

GF-P12-1

3.66

53.66

14.66

55.00

6.09

104.87

1.91

GF-P12-2

3.76

53.76

14.30

55.00

6.71

104.60

1.90

GF-P12-3

3.55

53.55

15.08

55.00

5.77

106.62

1.94

Average

3.66

53.66

14.68

55.00

6.19

105.37

1.92

S.D.

0.09

0.09

0.32

0.00

0.39

0.90

0.02

GF-P16-1

5.05

55.05

10.90

75.00

9.17

205.68

2.74

GF-P16-2

4.96

54.96

11.08

76.00

10.23

222.60

2.93

GF-P16-3

4.98

54.98

11.04

76.00

9.98

215.48

2.84

Average

5.00

55.00

11.01

75.67

9.79

214.58

2.84

S.D.

0.04

0.04

0.08

0.47

0.45

6.94

0.08

Note: the specimen label, e.g. GF-P4-1, denotes the 4-ply (P4) laminated GFRP tube (GF) of specimen #1 in this group.

As for the other three specimens (GF-P8, GF-P12 and GF-P16, see Fig. 8 (b), 8 (c) and 8 (d)), four distinct longitudinal fracture lines can be seen throughout the tests, in which the last two fracture lines occurred at the left and right horizontal ends of the tubes. Correspondingly, the crushing forces of 8-ply, 12-ply and 16-ply GFRP tubes underwent a third increase and forth modest ascent. After the formation of the last two longitudinal fracture lines, the force of specimens GF-P8 dropped then reached a long plateau phase. The forces of specimens GF-P12 and GF-P16 declined step by step as a result of the progressive delamination. Finally, the force increased rapidly when the inner tube was contact internally, then bent, delamination and fracture at the self-contact stage. 3.3 CFRP tube 3.3.1

Failure mode

For CFRP tubes with four D/T ratios (i.e. D/T= 45.01, 22.25, 14.53, 10.89) corresponding

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to four ply laminates were considered for the compression tests. Fig. 12 shows the deformation processes of CFRP specimens with different D/T ratios, from the begin to the end of crushing with every 8 mm loading distance, i.e. 0, 8, 16 mm, etc. During crushing, delamination can be seen in the tubes except for specimen CF-P4 (D/T = 45.01). Comparing with the GFRP tubes, all the CFRP specimens exhibited four longitudinal fracture lines regardless of the D/T ratios, in which the first two fracture lines were originated around the two contact lines in between the tube wall and rigid platens. Interestingly, the four fracture lines in specimens CF-P4 and CF-P8 (D/T = 22.25) were located at a 90° phase angle; while those of specimen CF-P12 (D/T = 22.25) and CF-P16 (D/T = 10.89) were located in a smaller phase angle, indicating the effect of D/T ratio.

(a) CF-P4 (D/T = 45.01)

(b) GF-P8 (D/T = 22.25)

(c) CF-P12 (D/T = 14.53)

(d) CF-P16 (D/T = 10.89) Fig. 12. Deformation histories of the four different CFRP tubes under quasi-static lateral compression.

To explore the damage modes of CFRP tubes, an optical microscope was used, and the failure details of these four different CFRP specimens are shown in Fig. 13. Clearly, with 17

increase in ply number, the failure modes become more complex and exhibit more delamination in the microscopic pictures, implying that the specimen can absorbed more energy through the damage of delamination.

(a) CF-P4

(b) CF-P8

(c) CF-P12

(d) CF-P16

Fig. 13. Microscopic photos to show the failure modes of CFRP tubes in different thicknesses.

3.3.2

Force-displacement curve

Three force-displacement curves of 8-ply CFRP tubes were plotted in Fig. 14. The force-displacement trend of these three CFRP tubes was fairly similar to that of the GFRP tubes. After a dramatic increase and several drops, the crushing force entered a low but long plateau stage; after which the force increased significantly. In spite of some difference in these three curves, the trend and the key turning points were fairly close, indicating sufficient

18

repeatability for such composite materials.

Fig. 14. Force-displacement curves of 8-ply CFRP tubes under quasi-static lateral compression.

Fig. 15 compares the force-displacement histories for all these four CFRP groups. Table 3 provides the parametric details of the CFRP tubes and the experimental results. The same pattern of force-displacement curves between the GFRP and CFRP tubes can be seen from Fig. 11 and Fig. 15, respectively. Three different crushing phases, i.e. elastic stage, progressive crushing stage and self-contact stage, can be identified in these plots. After the elastic stage, the crushing force experienced a progressive crushing stage and several sudden drops in the progressive crushing stage. As for the 4-ply CFRP tubes (specimen CF-4), the crushing force underwent four sudden drops due to the formation of four clear longitudinal fracture lines; after that the tube almost lost its load carrying capacity (from 18 mm to 34 mm compression). With regards to the thicker CFRP tubes (i.e. specimens CF-8, CF-12 and CF-16), the crushing force achieved a relatively lower plateau phase after rising and falling in the progressive crushing stage. Finally, all the CFRP specimens ended up with an internal self-contact stage with a dramatic increase in the crushing force.

19

F0

elastic stage progressive crushing stage self-contact stage

Fig. 15. Force-displacement curves of four different CFRP tubes under quasi-static lateral compression.

Table 3 The test results of CFRP tubes Specimen

T / mm

D / mm

D/T

Mass / g

F0 / kN

EA / J

SEA / J·g-1

CF-P4-1

1.17

51.17

43.74

13.00

0.88

12.52

0.96

CF-P4-2

1.12

51.12

45.64

13.00

0.89

12.22

0.94

CF-P4-3

1.12

51.12

45.64

13.00

0.84

12.12

0.93

Average

1.14

51.14

45.01

13.00

0.87

12.29

0.95

S.D.

0.02

0.02

0.90

0.00

0.02

0.17

0.01

CF-P8-1

2.34

52.34

22.37

28.00

3.66

40.42

1.44

CF-P8-2

2.36

52.36

22.19

28.00

3.18

37.55

1.34

CF-P8-3

2.36

52.36

22.19

28.00

3.39

39.60

1.41

Average

2.35

52.35

22.25

28.00

3.41

39.19

1.40

S.D.

0.01

0.01

0.09

0.00

0.20

1.58

0.06

CF-P12-1

3.65

53.65

14.70

44.00

6.74

77.25

1.76

CF-P12-2

3.72

53.72

14.44

44.00

7.21

83.17

1.89

CF-P12-3

3.72

53.72

14.44

44.00

5.94

72.14

1.64

Average

3.70

53.70

14.53

44.00

6.63

77.52

1.76

S.D.

0.03

0.03

0.12

0.00

0.52

4.51

0.10

CF-P16-1

5.11

55.11

10.78

60.00

9.05

128.36

2.14

CF-P16-2

5.01

55.01

10.98

60.00

9.66

116.30

1.94

CF-P16-3

5.05

55.05

10.90

60.00

9.01

133.29

2.22

Average

5.06

55.06

10.89

60.00

9.24

125.98

2.10

S.D.

0.04

0.04

0.08

0.00

0.30

7.14

0.12

Note: the specimen label, e.g. CF-P4-1, denotes the 4-ply (P4) laminated CFRP tube (CF) of specimen #1 20

in this group.

3.4 Comparison between aluminum and composite tubes 3.4.1

Crushing force

For facilitating comparison of the crushing characteristics of aluminum, GFRP and CFRP tubes, the force-displacement curves of different tubes with similar D/T ratios were plotted together in Fig. 16. It can be seen that aluminum tubes exhibited certain load carrying advantage over the composite counterparts, in which the crushing force of aluminum specimens remained to increase with the displacement, whilst the forces of composite tubes increased in a much lower level after sharp drop. Specifically, at the elastic stage, the rising rate of the force of aluminum tubes is fastest than those of the composite tubes; and the rate of CFRP tubes is always greater than that of GFRP counterparts. In addition, the aluminum tubes entered a plastic collapse stage earlier than the CFRP and GFRP tubes. The force of aluminum tubes increased steadily at the plastic collapse stage due to their ductile behaviors, whilst the forces of composite tubes stayed a much lower level after several rises and falls at the progressive crushing stage due to their brittle behaviors. Overall, it was found that the loading capacity of GFRP tubes was higher than that of CFRP tubes during progressive crushing stage. In the end, all the crushing forces increased rapidly in the self-contact or densification stage.

(a) D/T = 45.01 ~ 47.55

(b) D/T = 22.25 ~ 23.83

21

(c) D/T = 14.53 ~ 15.56

(d) D/T = 10.89 ~ 11.48

Fig. 16. Comparison of force-displacement curves of aluminum, GFRP and CFRP tubes under quasi-static lateral compression.

For characterizing the crushing behavior, the initial collapse force (F0) was obtained from the force-displacement curves (see Figs. 7, 11 and 15) and analyzed here. Fig. 17 compares the average initial collapse forces for the aluminum, GFRP and CFRP tubes. The initial collapse force of aluminum tubes was ranged from 0.5 to 7.07 kN (Table 1), whilst the initial collapse forces of the GFRP and CFRP tubes ranged from 0.78 to 9.79 kN (Table 2) and from 0.87 to 9.24 kN (Table 3), respectively. The standard deviations of the aluminum specimens vary from 0 to 0.03, indicating excellent repeatability of the crushing tests; whilst the GFRP/CFRP tubes with a smaller D/T ratio (or thicker tube wall) can lead to more discrepancy of the initial collapse force. This is because the failure modes of thicker composites are much more complicated than the thinner composites (see Figs. 5, 8 and 12).

22

Fig. 17. Initial collapse force (F0) of aluminum, GFRP and CFRP tubes.

Fig. 17 exhibits that all the initial collapse forces show a distinct downward trend with the increase in the D/T ratio. It is noted that the initial collapse force drops faster from D/T = 11 to D/T = 23 than from D/T = 23 to the other lager ratios. This means that when the diameter is almost a constant, the D/T ratio (or thickness) of tubes have significant effect on the initial collapse force, the thicker of the tube wall, the greater the effect. Comparing with the similar D/T values, the initial collapse force of aluminum tubes is always the lowest, while the force of the CFRP tubes is greater than that of the GFRP tubes from D/T = 15 to D/T = 45, but the force of GFRP tube of D/T = 11 is the largest. 3.4.2 Energy absorption capacity Fig. 18 compares the curves of energy absorption (EA) to study the energy absorption capacity of the aluminum, GFRP and CFRP tubes. The energy absorption curves of the aluminum tubes showed a stable upward trend; and the increment was faster and faster due to continuously increasing crushing force during the crushing test (Fig. 16). The curves of composite tubes increased quickly with the displacement due to the higher stiffness (or Young’s modulus) of the composite specimens (Fig. 16); then the curves entered a very low plateau stage where the energy absorption slowly increased. After that, the curves rose rapidly attributable to significant increase in the crushing force at the self-contact stage. Furthermore, although CFRP tubes dissipated more energy than GFRP tubes at the early crushing process, the GFRP tubes absorbed more energy throughout the test due to the fact that the crushing force of GFRP tubes was much higher than that of the CFRP counterparts at the progressive crushing stage.

23

(a) D/T = 45.01 ~ 47.55

(b) D/T = 22.25 ~ 23.83

(c) D/T = 14.53 ~ 15.56

(d) D/T = 10.89 ~ 11.48

Fig. 18. Comparison of energy absorption curves of aluminum, GFRP and CFRP tubes under quasi-static lateral compression.

Fig. 19 graphs the average energy absorption (EA) and specific energy absorption (SEA) of each type of the aluminum, GFRP and CFRP tubes. Evidently, the energy absorption capacity of these three material groups decreased with increasing D/T ratio. The composite tubes were much lower than the aluminum tubes due to the relatively lower crushing force obtained from the composite tubes in the progressive crushing stage; whereas the force of the aluminum tubes were much higher and gradually increased at this stage (Fig. 16). In other words, both EA and SEA of the composite tubes were apparently lower than those of aluminum tubes. The highest EA of CFRP and GFRP tubes was 125.98 J and 214.58 J, respectively; while the aluminum tubes was 451.58 J, more than 3.5 times of the CFRP tubes and 2 times of the GFRP tubes. Also, the highest SEA of GFRP (or CFRP) tubes was 2.84 J/g 24

(2.10 J/g), while the highest for the aluminum tubes was 5.74 J/g, around 2 times of the composite counterparts. Moreover, the gap of EA (or SEA) between the aluminum tubes and composite tubes for a lower D/T ratio is much wider than that for a higher D/T ratio, meaning that with increase in tube thickness (decrease of the D/T ratio). Due to the ductile behaviors of aluminum, the aluminum tubes exhibit a greater advantage on energy absorption capacity than composite counterparts (Fig. 19). Comparing the composite tubes, the GFRP tubes were of a higher EA and SEA than those of CFRP tubes when D/T was in between 11 and 45. The difference between the GFRP and CFRP tubes became smaller at a higher D/T ratio, for example, providing higher EA and SEA (15.51 J and 0.97 J/g) for GFRP tube and lower EA and SEA (12.29 J and 0.95 J/g) for the CFRP counterpart when D/T = 45.

(a) Energy absorption (EA)

(b) Specific energy absorption (SEA)

Fig. 19. Energy absorption and Specific energy absorption of aluminum, GFRP and CFRP tubes.

4. Numerical simulation 4.1 Finite element (FE) models The FE models of the aluminum, GFRP and CFRP tubes were developed and validated in this section to explore more details of lateral crushing behavior. The commercial code ABAQUS/Explicit was employed here to simulate the crushing tests. The material subroutine VUMAT was employed to model the composite tubes. Fig. 20 illustrates the FE model, in which the top and bottom platens were modeled using discrete rigid surfaces. The lateral 25

crushing load was applied through moving the top platen downwards with a constant velocity of 1 m/s, whilst the bottom platen was fixed. It is noted that this loading rate was much higher than the aforementioned quasi-static testing rate (4 mm/min) here to improve computational efficiency for the explicit analysis, however this increase had no much effect on the simulation results through kinetic energy analysis such as Ref [33].

Fig. 20. The schematic of the FE models.

The aluminum tube was modeled using conventional shell elements (S4R, Fig. 20) with four integration points across the thickness. A general contact with friction coefficient of 0.08 was adopted here [34]. The convergence test was carried out to balance computational cost and the numerical accuracy; and the shell element size of 1 × 1 mm was sufficient in this case. As for the FE models of composite tubes, GFRP and CFRP tubes were modeled using continuum shell elements (SC8R) for simulating potential damage of delamination. Following a convergence analysis, the elements size of 1.5 × 1.5 mm for each ply of the CFRP layers was chosen. For modeling the delamination of laminated composites, cohesive contact was set up in between each ply of composite specimen. A general contact with friction coefficient was set to be 0.3 [35]. 4.2 Material Constitutive models 4.2.1 Material model for aluminum The aluminum tube was modeled through an elastoplastic constitutive model with the von 26

Mises isotropic plasticity criterion. The mechanical properties of aluminum tube were adopted as follows: density = 2700 kg/m3, Poisson's ratio = 0.33, Young's modulus = 80 GPa and initial yielding stress = 210 MPa. To define the hardening characteristics in the FE model properly, the stress-strain curve obtained from the in-house test was used (Fig. 3 (a)). 4.2.2 Material model for composites During the progressive lateral crushing process, the failure modes of laminated composites can be classified as intra-laminar damage (fiber damage and matrix cracking) and inter-laminar damage (delamination); thus, both the intra-laminar and inter-laminar constitutive models of composites were adopted in this study. 4.2.2.1 Intra-laminar damage model The fabric-reinforced laminate was modeled as a homogeneous orthotropic material, and the in-plane elastic stress-strain relations are defined as [36]:

{}

𝜀11 𝜀22 = 𝜀𝑒𝑙 12

[

1 (1 ― 𝑑1)𝐸1 𝑣12

―

𝐸1

0

―

𝑣12 𝐸1

1 (1 ― 𝑑2)𝐸2

0

0 0

]{ }

1 (1 ― 𝑑12)2𝐺12

𝜎11 𝜎22 𝜎12

(3)

where the subscripts 1 and 2 denote the warp and weft directions respectively; 𝜺 =

{𝜀11

𝜀22 𝜀𝑒𝑙 12}

𝑇

𝑇 and 𝝈 = {𝜎11 𝜎22 𝜎12} are the elastic strain vector and stress vector

respectively; 𝐸1 and 𝐸2 are the Young’s modulus in the principal orthotropic directions (along the warp and weft directions in this paper); 𝐺12 is the in-plane shear modulus; 𝑣12 is the Poisson’s ratio; 𝑑1 and 𝑑2, are the damage variable associated with fiber breakage in the principal orthotropic directions; and 𝑑12 is the damage variable reflecting the matrix micro-crack due to the in-plane shear deformation. These damage parameters vary between 0 and 1 and represent the degree of stiffness degradation caused by micro-damage in the material. To differentiate between the tensile and compressive fiber failure mode, the subscripts + and – are used in the damage variables, e.g. 𝑑𝑖 + for tensile damage parameter in the i-th direction (i = 1, 2). To describe the initiation and degradation of damage, the effective stresses are introduced, defined as [36]:

27

〈𝜎𝑖𝑖〉

〈 ― 𝜎𝑖𝑖〉

𝜎𝑖 + = 1 ― 𝑑𝑖 +

〈𝜎12〉

𝜎𝑖 ― = 1 ― 𝑑𝑖 ―

(4)

𝜎12 = 1 ― 𝑑12

where 〈𝑥〉 is the Macauley operator defined as 〈𝑥〉 : = (𝑥 + |𝑥|)/2. The damage activation functions 𝐹𝑖 + , 𝐹𝑖 ― and 𝐹12 are used to define the elastic domain as [36]: 𝜎𝑖 +

𝜎𝑖 ―

𝐹𝑖 + = 𝑋𝑖 + ― 𝑟𝑖 + ≤ 0

𝐹𝑖 ― = 𝑋𝑖 ― ― 𝑟𝑖 ― ≤ 0

𝐹12 =

𝜎12 𝑆

(5)

― 𝑟12 ≤ 0

where 𝑋𝑖 + and 𝑋𝑖 ― are the tensile and compressive strength along the i-th direction, and 𝑆 denote the shear strength; 𝑟𝑖 + ，𝑟𝑖 ― and 𝑟12 are the damage thresholds, and they are initially set to be 1. After damage initiation, i.e. when 𝐹𝑖

±

= 0 or/and 𝐹12 = 0, the damage

threshold increased with damage progression according to: 𝑟𝑖 + = max

(

𝜎𝑖 +

)

𝑟𝑖 ― = max

𝑋𝑖 + ,1

(

𝜎𝑖 ―

)

𝑋𝑖 ― ,1

𝑟12 = max

(𝜎𝑆 ,1) 12

(6)

The degradation of material properties after the damage initiation, the damage evolution law is defined as (𝛼 = 1 ± ,2 ± ) [36]: 1

[

2𝑔𝛼0 𝐿𝑐

]

(7)

𝑑𝛼 = 1 ― 𝑟𝛼𝑒𝑥𝑝 ― 𝐺𝛼 ― 𝑔𝛼𝐿 (𝑟𝛼 ― 1) 𝑓

0 𝑐

where 𝐿𝑐 is the characteristic length of the unit element; 𝐺𝛼𝑓 is the fracture energy per unit area; and 𝑔𝛼0 is the elastic energy per unit volume at the point of damage initiation, can be expressed in a form of: 𝑔𝛼0 =

(𝑋𝛼)2

(8)

2𝐸𝛼

Furthermore, according to the Ref [36], the damage evolution law for in-plane shear damage mode can be described as: 𝑑12 = min (𝛽12ln (𝑟12),𝑑𝑚𝑎𝑥 12 )

(9)

where 𝛽12 > 0 and 𝑑𝑚𝑎𝑥 12 ≤ 1 are the material properties, which can be determined experimentally. According to the definition in Eq. (6), the damage thresholds 𝑟𝛼 are non-decreasing quantities, which can ensure that the damage variables are monotonically increasing quantities by the formulation of the damage evolution law in Eq. (7), the schematic diagram of damage variables is plotted in Fig. 21.

28

𝑑𝛼 complete damage

1 𝑜𝑟 𝑑𝑚𝑎𝑥

damage evolution 0

𝑟𝛼 = 1 (damage

𝑟𝛼

initiation) Fig. 21. The sketch of damage variables.

In accordance with Refs. [37, 38], the in-plane shear response of fabric composites dominated by the non-linear behavior of the matrix due to matrix micro-cracking and plasticity. To account for the inelastic behavior, a classical plasticity model with an elastic domain function and a hardening law is implemented. The elastic domain function, 𝐹, is given by: 𝐹 = |𝜎12| ― 𝜎0(𝜀𝑝𝑙) ≤ 0

(10)

and the hardening law is expressed as: 𝑝

𝜎0(𝜀𝑝𝑙) = 𝜎𝑦0 +𝐶(𝜀𝑝𝑙)

(11)

where 𝜎𝑦0 is the initial effective shear yield stress; C and p are the coefficients; and 𝜀𝑝𝑙 is the equivalent plastic strain. Note that the values of 𝜎𝑦0, C and p can be obtained by using the calibration procedure as presented in Ref. [39]. The intra-laminar damage evolution process is illustrated in Fig.22, where point A, B and C represent the damage initiation, partial damage and complete failure respectively, and the area of OABC denote the fracture energy 𝐺𝑓. The linear part of the stress-strain curve from original point O to point A represents the undamaged constitutive behavior. The curve declines gradually after reaching peak point A due to the damage initiation and propagation, which leads to degradation of the elastic modulus. The stress-strain curve follows the path A-B-O upon unloading, while it along a linear path O-B on reloading.

29

𝜎 A (damage initiation)

𝜎0

B (unload) gradual reduction

O

C (complete damage)

reload 𝜀0

𝜀

Fig. 22. Damage propagation process.

To develop the constitutive model for the intra-laminar failure, the in-house experimental data (Fig. 3 (b)) were also employed to determine the elastic properties of composites, see the following Table 4. The material properties for the CFRP and GFRP tubes were selected from the reported literatures [15, 40, 41]. All the material properties for developing the intra-laminar failure model were summarized in Table 4.

Table 4 Material properties for the intra-laminar damage model of composites. Description

Value

Variable GFRP

CFRP

Density (kg/m3)

ρ

1800

1560

Elastic properties (GPa)

E1

26 (Exp)

60 (Exp)

E2

26 (Exp)

60 (Exp)

G12

3.4

4.5

ν12

0.15 (Exp)

0.037 (Exp)

X1+

460

776

X1-

400

704

X2+

460

704

X2-

400

698

S

76

95

𝐺1𝑓𝑐+

40

125

𝐺1𝑓𝑐―

60

250

𝐺2𝑓𝑐+

40

95

𝐺2𝑓𝑐―

60

245

Coefficient in the shear damage variable

𝛽12

0.15

0.18

Maximum value of shear damage variable

𝑑𝑚𝑎𝑥 12

0.99

0.99

Damage initiation (MPa)

Fracture energies (kJ/m2)

30

Initial effective shear yield stress (MPa)

𝜎y0

160

185

Power term in the hardening equation

p

0.41

0.41

Coefficient in the hardening equation

C

1053

1053

4.2.2.2 Inter-laminar damage model For the Inter-laminar damage model, a cohesive contact method in ABAQUS/Explicit was used to simulate the delamination of laminates by using the traction-separation law, which includes damage initiation criterion and damage evolution. The interfacial behavior is described as linear elastic before the damage initiation of delamination, which can be defined as: 𝑡𝑛 𝐾 𝑛 0 0 𝜀𝑛 𝑡 𝒕 = 𝑠 = 0 𝐾𝑠 0 𝜀𝑠 = 𝑲𝜺 𝑡𝑡 0 0 𝐾 𝑡 𝜀𝑡

{} [

]{ }

(12)

where 𝒕 is traction stress vector and 𝜺 is the strain vector, and 𝑲 represents the stiffness matrix of interaction. The subscripts n, 𝑠 and t denote the normal and two shear directions respectively. The damage initiation is governed by a quadratic nominal stress criterion, defined as: 〈𝑡𝑛〉 2

𝑡𝑠 2

𝑡𝑡 2

(13)

( 𝑡0 ) + (𝑡0) + (𝑡0) = 1 𝑛

𝑠

𝑡

where 𝑡0𝑛, 𝑡0𝑠 and 𝑡0𝑡 are maximum admissible value of traction stresses. The function denotes that damage initiates from the moment the left of equal sign reaches a value of 1. After damage onset, damage evolution based on the Benzeggagh-Kenane (BK) law [42], defined as: 𝐺𝑠 + 𝐺𝑡

𝐺𝐶𝑛 + (𝐺𝐶𝑠 ― 𝐺𝐶𝑛)(𝐺𝑛 + 𝐺

𝑠

𝜂

𝐶 + 𝐺𝑡) = 𝐺

(14)

where 𝐺 is the fracture energy and superscript C represents the critical fracture energy; 𝜂 is a cohesive property parameter. The materials parameters for inter-laminar damage model were obtained from Ref. [15, 36] and summarized in Table 5.

Table 5 Material properties describing the inter-laminar damage model of composites Description

Variable

31

Value

Damage initiation (MPa)

Fracture energies (kJ/m2)

BK

GFRP

CFRP

𝑡0𝑛

54

54

𝑡0𝑠

70

70

𝑡0𝑡

70

70

𝐺𝐶𝑛

0.504

0.504

𝐺𝐶𝑠

1.566

1.566

𝐺𝐶𝑡

1.566

1.566

𝜂

2.284

2.284

4.3 Validation and analysis of FE models 4.3.1

Aluminum tube

The numerical simulation was compared with the experimental results for the aluminum tube AL-T2 (D/T = 28.83) for validation purpose. Fig. 23 shows the experimental and numerical deformation of aluminum tube during the lateral compression. It can be observed that both the numerical and experimental results generated two pairs of stationary plastic hinges appeared at the vertical and horizontal ends of the deformed tube, where high stress was concentrated throughout the loading process.

Fig. 23. Comparison of the experimental and numerical deformation and failure modes for aluminum tube.

Fig. 24 compares the experimental and numerical force-displacement curves. Clearly, the numerical results are in very good agreement with the experimental data, indicating that the numerical model developed for the aluminum tubes is validate to predict the crushing responses under the lateral compression.

32

Fig. 24. Comparison of the experimental and numerical force-displacement curves for aluminum tube.

4.3.2 GFRP tubes Fig. 25 compares the experimental and numerical results for GFRP tube GF-P8 (D/T = 23.22) under lateral compression. It can be seen that some stress concentration first appeared in the two contacting areas between the tube and two platens, and at the left/right horizontal ends of the deformed tube (see the second picture in Fig. 25). With increase of loading distance, two pairs of longitudinal fracture lines were taken a shape at the vertical and horizontal positions of the tube, leading to the stress relief (see the third, fourth and fifth pictures in Fig. 25); thus, the tube was fractured into four parts. From both the experimental and numerical results, it is found that the main failure mode were fiber fracture and matrix cracking, and no evident delamination was observed during lateral crushing. Whilst in the aluminum tubes, the stress concentration occurred at vertical and horizontal ends of aluminum tube, and then continued to the end of the test (see Fig. 23).

33

Fig. 25. Comparison of the experimental and numerical deformation and failure modes for GFRP tube.

Fig. 26 compares the force-displacement curves obtained from the numerical analysis and experimental tests. At both the elastic and self-contact stages, the crushing force increased sharply and no much fiber or matrix damage was observed. After the elastic stage, both the experiment and simulation entered a progressive crushing stage with several significant drops, then the crushing force stayed in a low plateau level prior to the self-contact stage. Clearly, the numerical results were in a good agreement with experimental counterparts, validating the FE model for GFRP tubes.

Fig. 26. Comparison of the experimental and numerical force-displacement curves for the GFRP tube.

4.3.3 CFRP tube Fig. 27 compares the numerical and experimental deformation processes for CFRP tube CF-P8 (D/T = 22.25) subjected to lateral compression. It is observed again that the stress concentration was released after the formation of longitudinal fracture lines, also leading the CFRP tube to separate into four parts. Fig. 28 compares the force-displacement curves of the

34

experiment and simulation of CFRP specimens. Apparently, the deformation of CFRP tube can also be also divided into three stages. Comparing the GFRP (Fig. 25) and CFRP tube (Fig. 27), both the numerical and experiment exhibited the same failure modes, including fiber fracture, matrix crack and delamination.

Fig. 27. Comparison of the experimental and numerical deformation and failure modes for the CFRP tube.

Fig. 28. Comparison of the experimental and numerical force-displacement curves for CFRP tube.

On summary, the numerical deformation histories, failure modes and force-displacement curves of the aluminum, GFRP and CFRP tubes showed good agreement with those of the experimental tests, validating the corresponding FE models for further predicting the crushing behaviors of various tubes.

5.

Performance and cost efficiency 35

The crashworthy performance and cost efficiency of aluminum, GFRP and CFRP tubes under the lateral compression was discussed further here for providing a more sensible design guide for engineering applications. To attain a high crashworthy and low-cost lightweight structure, a number of indicators including cost, weight, initial collapse force, energy absorption (EA) and specific energy absorption (SEA), were taken into account here. The piece prices of GFRP and CFRP prepreg are £9.74/m2 and £17.05/m2, respectively (Provided by Shenzhen Hangyu Carbon Fiber Technology Co., Ltd, Shenzhen, China), while the price of aluminum tube is £3.1/kg (Provided by Sichuan Yuantaida Non-ferrous Metal Material Co. LTD, Sichuan, China). All the required data for the performance to cost analysis was summarized in Table 6, the cost of energy absorption (EA/Cost) was used here as an indicator for the cost-performance analysis. It should be pointed out that the processing cost of specimens were not considered from the calculation.

Table 6 Crashworthy performance to cost ratio of aluminum and composite tubes Specimen

D/T

Mass / g

F0 / kN

EA / J

SEA / J·g-1

Cost / £

EA/Cost / J·£-1

AL-T1

47.55

21

0.50

26.09

1.24

0.065

400.77

AL-T2

23.83

42

2.11

117.16

2.79

0.130

899.85

AL-T3

15.56

60

5.24

293.79

4.90

0.186

1579.52

AL-T4

11.48

78.67

7.07

451.58

5.74

0.244

1851.67

GF-P4

45.79

16

0.78

15.51

0.97

0.313

49.58

GF-P8

23.22

33

2.73

48.79

1.48

0.640

76.29

GF-P12

14.68

55

6.19

105.37

1.92

0.985

106.96

GF-P16

11.01

75.67

9.79

214.58

2.84

1.346

159.38

CF-P4

45.01

13

0.87

12.29

0.95

0.548

22.43

CF-P8

22.25

28

3.41

39.19

1.40

1.122

33.55

CF-P12

14.53

44

6.63

77.52

1.76

1.726

44.92

CF-P16

10.89

60

9.24

125.98

2.10

2.359

53.40

Note: all data of specimens are the average.

To more intuitively compare the different crashworthy performance and cost efficiency of aluminum and composite tubes, the radar maps were plotted in Fig. 29, all data in these maps are the average. It was clear that the costs of CFRP tubes were always higher than the

36

aluminum and GFRP tubes, thus its values of energy absorption (EA), specific energy absorption (SEA) and the cost of energy absorption (EA/Cost) were lower than the aluminum and GFRP counterparts with a similar D/T ratio. Moreover, aluminum tubes with a similar D/T ratio can dissipate much more energy (EA) with much lower cost (EA/Cost) than the composite counterparts. With the D/T ratio decreasing from 47.55 to 10.89, the aluminum tubes exhibited more advantage on the crashworthy performance and cost efficiency. For example, the EA/Cost and SEA of aluminum tube were 400.77 J/£ and 1.24 J/g in D/T =47.55 (Fig. 27 (a)), and 1851.67 J/£ and 5.74 J/g in D/T = 11.48 (Fig. 27 (d)).

(a) D/T = 45.01 ~ 47.55

(b) D/T = 22.25 ~ 23.83

(c) D/T = 14.53 ~ 15.56

(d) D/T = 10.89 ~ 11.48

Fig. 29. Radar maps for performance to cost ratio and crashworthy characteristics of circular tubes

37

6.

Conclusions

This paper presented a systematic study on the lateral crushing behaviors of circular aluminum, GFRP and CFRP tubes through experimental and numerical analyses. Within its limitation, the following conclusions can be drawn: (1) Aluminum tubes, due to their ductile behavior, show a much better crashworthy characteristics than the brittle composite tubes; and in particular a thicker aluminum tube had much greater advantages than composite tubes. (2) The D/T ratio exhibited significant effect on the lateral crushing behaviors of all the tubes. Specifically, the tubes performed higher lateral crashworthy capacities with decreasing D/T ratio, such as the higher energy absorption and higher load carrying capacities. Higher energy absorption capacity for composites can be explained by the different major failure modes (fiber breakage and matrix crack). With decrease in the D/T ratio, the delamination became a most common damage mode, enhancing energy absorption. (3) The FE models were developed and validated against the experimental tests. The FE models predicted the deformation behaviors and force-displacement curves accurately, which provided a basis for further design optimization. (4) Aluminum tubes also exhibited much more advantages on overall crashworthy performance and cost efficiency over the GFRP and CFRP tubes. Specifically, with decreasing D/T ratio (or increasing thickness), the tube is of better lateral crushing behavior and higher cost efficiency.

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) and the Open Fund of the State Key Laboratory of Advanced Design and Manufacture for Vehicle Body of Hunan University (31815001). The first author would like to gratefully acknowledge the financial support from China Scholarship Council/University of Manchester Joint PhD Scholarship (201706130138). Dr Guangyong Sun is a recipient of Australian Research Council (ARC) Discovery Early Career Researcher Award (DECRA) in the University of Sydney. 38

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Conflict of interest

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

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