Response of round-hole tubes submitted to pure bending creep and pure bending relaxation

Abstract

This paper presents experimental study on the response of 6061-T6 aluminum alloy round-hole tubes with five different hole diameters of 2, 4, 6, 8, and 10 mm and four different diameter-to-thickness ratios of 30, 40, 50, and 60 submitted to pure bending creep and pure bending relaxation. Pure bending creep or relaxation is defined as bending the tube to the required moment or curvature and maintaining that moment or curvature for a period of time. The experimental results of pure bending creep show that the curvature increases with time. In addition, larger holding moment, diameter-to-thickness ratio, or hole diameter results in larger creep curvature. As the curvature continues to increase, the round-hole tube eventually breaks. The experimental results of pure bending relaxation show that the relaxation moment decreases sharply with time and tends to a stable value. In addition, larger holding curvature, diameter-to-thickness ratio, or hole diameter results in larger drop of the relaxation moment. Due to fixed curvature, the round-hole tube does not break. Finally, formulas proposed by the research team of Pan et al. were respectively improved to simulate the creep curvature-time relationship for pure bending creep in the initial and the secondary stages and the relaxation moment-time for pure bending relaxation. After comparing with the experimental results, it is found that theoretical analysis can reproduce the experimental results reasonably.

Introduction

Round-hole tube (abbreviation: RHT) refers to a tube with a round hole, as shown in Figure 1. RHT is usually used for connecting components of bicycles, motorcycles, or automobiles. When RHT submits to a bending load, the bending rigidity of RHT will progressively reduce as the amount of bending increases, which is called a phenomenon of deterioration. RHT will encounter fracture when the curvature reaches a certain critical value.

Figure 1.

Schematic diagram of an RHT.

The study of smooth tube under bending has achieved plentiful results. In 1987, Kyriakides et al. set up a mechanical device that can perform cyclic bending tests on tubes of various materials (1020 steel, 1018 steel, 304 stainless steel, 6061-T6 aluminum alloy, and NiTi). They have carried out many experimental and theoretical studies on bending with or without internal or external pressure.^1–9

Several other researchers have also published related results. Elchalakani et al. ¹⁰ determined the slenderness limits of cold-formed CHS for fully ductile section by conducting cyclic bending tests with variable amplitudes. Mathon and Liman ¹¹ experimentally studied the pure bending collapse of cylindrical thin shells. Elchalakani and Zhao ¹² investigated the cyclic bending behavior of cold-formed steel tubes with concrete filled. Yazdani and Nayebi ¹³ examined the damage of pipes that undergo periodic bending with a stable internal pressure. Shariati et al. ¹⁴ experimentally discussed the cyclic bending behavior of SS316L cantilevered cylindrical shells. Elchalakani et al. ¹⁵ used the strain detected in bending test to find a new ductility slender limit for the CFT structural plastic designs. Li and Wang ¹⁶ investigated the stability of single-layer cable-stiffened latticed shells submitted to earthquake motions. Chegeni et al. ¹⁷ studied the influence of the shape and depth of corrosion on the performance of a pipe that is bent under a certain internal pressure.

In 1998, Pan et al. started a series of experimental and theoretical investigations on behavior of tubes under monotonic or cyclic bending with various loading paths. Pan et al. ¹⁸ invented and constructed a new measurement equipment to measure the ovalization (change of the outer diameter divides by the outer diameter, ΔD_o/D_o) and curvature (κ) of tubes submitted to bending. They confirmed the function of the equipment by conducting tubes under cyclic bending. Lee et al. ¹⁹ tested the stability of 304 stainless steel tubes with four diameter-to-thickness ratios (D_o/t ratios) submitted to cyclic bending. They found that the log κ-log N_b (number of cycles required to initiate buckling) relationships demonstrate four parallel lines. Lee and Pan ²⁰ examined the pure bending creep behavior of 304 stainless steel tubes with four D_o/t ratios. By modifying the Bailey-Norton law for uniaxial creep, the creep curvature-time relationships of the first two stages were simulated. Pan and Lee ²¹ explored the mean curvature effect on the behavior of tubes undertaken cyclic bending. Different curvature ratios of −1.0, −0.5, and 0 were considered in their experimental investigation. Chang and Pan ²² experimentally explored cyclic bending instability of tubes with four D_o/t ratios. They introduced a new formula for estimating tube’s buckling life.

In 2010, Pan et al. started to experimentally and analytically investigate the behavior of tubes with a circumferential notch submitted to bending. Lee et al. ²³ examined the changes of ΔD_o/D_o for circumferential sharp-notched tubes submitted to cyclic bending. The ΔD_o/D_o-N (number of cycles) curve was identified into three different growing stages. Later, Lee ²⁴ investigated cyclic bending response of circumferentially sharp grooved tubes with different depths. A theoretical formula for estimating N_b was proposed. Thereafter, Lee et al. ²⁵ examined the stability of circumferential sharp-notched 304 stainless steel tubes submitted to cyclic bending at different curvature rates (viscoplastic response). They employed three curvature rates of 0.35, 0.035, and 0.0035 m⁻¹/s to demonstrate the viscoplastic response. Lee et al. ²⁶ inspected the pure bending creep behavior and pure bending relaxation behavior of tubes with a circumferential sharp notch. The formula proposed by Lee and Pan ²⁰ was improved for describing the creep curvature-time and relaxation moment-time relationships. Chung et al. ²⁷ investigated the cyclic bending collapse of circumferential sharp-notched 6061-T6 aluminum alloy tubes. The κ-N_b relationship exhibited considerable differences from that discovered by Lee ²⁴ in circumferential sharp-notched 304 stainless steel tubes subjected to cyclic bending.

In 2016, Lee et al. ²⁸ explored the mechanical behavior of local sharp-cut 6061-T6 aluminum alloy tubes subjected to cyclic bending. In their study, the ANSYS was employed to simulate the ΔD_o/D_o-κ and M (moment)-κ relationships. After that, Lee et al. ²⁹ researched the failure of local sharp-grooved 304 stainless steel tubes submitted to cyclic bending. The M-κ relationships were very similar to that obtained by Lee et al. ²⁵ ; but, the contours of the ΔD_o/D_o-κ relationships were completely different. Lee et al. ³⁰ inspected the deterioration and failure of local sharp-dented 6061-T6 aluminum alloys tubes submitted to cyclic bending. The dent was created by contacting the mold on the tube’s surface. In addition, a theoretical formula was put forward to express the κ-N_f (number of cycles required to initiate failure) relationship.

In this study, the response of 6061-T6 aluminum alloy round-hole tubes (abbreviation: Al-RHTs) with different hole diameters of 2, 4, 6, 8, and 10 mm, and different diameter-to-thickness ratios of 30, 40, 50, and 60 submitted to pure bending creep and pure bending relaxation are examined. In the experimental part, a bending machine was employed to perform relative experiments on Al-RHTs. The magnitudes of M and κ were obtained by the measuring devices on the machine. Additionally, time ( $\overline{t}$ ) was also recorded. In the theoretical part, the formulas proposed by Lee and Pan ²⁰ and Lee et al. ²⁶ were respectively improved to simulate the creep curvature (κ_c)– $\overline{t}$ relationships for pure bending creep in the initial and secondary stages and the relaxation moment (M_r)– $\overline{t}$ relationships for pure bending relaxation.

Experiment

Experimental equipment

The experiments executed by a tube-bending machine. Figure 2(a) and (b) respectively show a schematic diagram and a picture of the machine. It was designed to perform a pure bending test (four-point bending test), which can apply monotonic and cyclic bending. The machine consists of two rotating sprockets, placed symmetrically on two support beams. Two sprockets support two rollers, which apply point loads at each end of the tube. The chains run around these sprockets and are connected to two hydraulic cylinders and load cells to form a closed loop. Once the top or bottom cylinder is contracted, the sprocket will rotate to achieve pure bending of the tube. The load transferred to the tube is the moment formed by the concentrated load of the two rollers. For a detailed description of the tube-bending machine, please refer to the following papers (Lee et al., ¹⁹ Lee and Pan, ²⁰ and Pan and Lee ²¹ ).

Figure 2.

2(a) Schematic diagram of the tube-bending machine and 2(b) picture of the tube-bending machine.

Figure 3(a) and (b) respectively depict a schematic diagram and a picture of an apparatus to measure the ΔD_o/D_o and κ of the tube invented by Pan et al. ¹⁸ When the tube is bent, the two side-inclinometers in the device is able to detect the tube’s angle changes. The κ can be easily calculated according to angle changes. In addition, the central part of the equipment can be used to monitor the change in the diameter of the tube cross section using a magnetic detector. Thus, the ΔD_o/D_o can be measured. For a detailed explanation of the apparatus, please refer to the Pan et al. ¹⁸ study.

Figure 3.

3(a) Schematic diagram of the measurement apparatus and 3(b) picture of the measurement apparatus.

Materials and specimens

The chemical composition by weight percentage of the 6061-T6 aluminum alloy is 0.916% Mg, 0.733% Si, 0.293% Cu, 0.268% Ti, 0.256% Fe, 0.132% Mn, 0.0983% Zn, 0.0682% Cr, and the remaining part Al. The mechanical properties are ultimate tensile stress (σ_u) = 320 MPa, yield stress (σ_o) = 283 MPa, and percent elongation (%ε_f) = 13%.

Smooth 6061-T6 aluminum alloy tubes with D_o = 36.0 mm and t = 3.0 mm were machined on the outer surface to acquire the desired D_o/t ratios of 30, 40, 50, and 60, as shown in Figure 4. However, the inner radius (29.0 mm) of all tested Al-RHTs was unchanged. Next, the tube with every D_o/t ratio was drilled again to obtain the Al-RHT with the required hole diameter d. In this study, five d values were considered, 2, 4, 6, 8, and 10 mm.

Figure 4.

Schematic diagram of tubes with D_o/t ratios of 30, 40, 50, and 60.

Because the hole is local, the hole direction ϕ (Figure 5) may also affect the RHT’s behavior. Since the degradation and failure of the RHT undergoing bending is the most severe when the bending moment direction (z direction) is perpendicular to the ϕ. Therefore, this study only considers ϕ = 0° (y direction).

Figure 5.

Schematic diagram of the cross-section of the RHT with a round-hole direction at ϕ.

Test procedures

The experiment includes pure bending creep (bend the tube to the required M and hold that M for a period of time) and pure bending relaxation (bend the tube to the required κ and hold that κ for a period of time). For each Al-RHT with certain D_o/t and d, different M and κ were respectively held for pure bending creep and pure bending relaxation. The M and κ were measured and controlled by the related equipment. Furthermore, the $\overline{t}$ was also recorded.

Experimental results, simulated results, and discussion

Experimental response of Al-RHTs under pure bending

Figure 6 shows the experimental M-κ curves for Al-RHTs with different D_o/t and different d subjected to pure bending. For each D_o/t ratio, five d values, 2, 4, 6, 8, and 10 mm, were tested. It can be seen that the linear M-κ relationship is observed when the deformation is in the elastic range. Once the deformation is in the plastic range, the M-κ relationship becomes nonlinear. It can also be observed that the RHT with a larger D_o/t ratio has a smaller t. Therefore, the M-κ curve shows a lower M value. In addition, for each D_o/t ratio, the M-κ curves are similar for different d. However, d = 2 mm has the largest breaking curvature and d = 10 mm has shortest breaking curvature. The breaking points are denoted as “*,”“×□”“▴,”“♦” and “■” for d = 2, 4, 6, 8 and 10 mm, respectively. Subsequently, the held moments of Al-RHTs with different D_o/t and different d for pure bending creep and the held curvatures of Al-RHTs with different D_o/t and different d for pure bending relaxation were according to the M-κ curves in Figure 6.

Figure 6.

Experimental M-κ curves for Al-RHTs with different D_o/t and different d subjected to pure bending.

Experimental response of Al-RHTs under pure bending creep

Figure 7 demonstrates the experimental κ– $\overline{t}$ curves of Al-RHTs with D_o/t = 50 and d = 2 mm for the overall bending process (monotonic bending and pure bending creep). The creep starting and creep rupture points are respectively marked by “•” and “×.” It can be found that the curvature of the Al-RHT increases rapidly when the creep begins. Larger held moment results in larger creep curvature. As the curvature continues to increase, the Al-RHT eventually ruptures. Note that the time of rupture for held moment of 120 N m is 4258 s.

Figure 7.

Experimental κ– $\overline{t}$ curves of Al-RHTs with D_o/t = 50 and d = 2 mm for the overall bending process.

Next, We focus on the pure bending creep part from the creep starting point “•” to the creep rupture point “×” in Figure 7. Figure 8 depicts the κ_c– $\overline{t}$ relationship when the held moment is 150 N m. It can be seen that the κ_c– $\overline{t}$ curve can be divided into three stages, which are called in this paper to be the initial stage, secondary stage, and tertiary stage. Once pure bending creep begins, the amount of κ increases, which is called the initial stage. When $\stackrel{·}{\kappa }$ _c gradually decreases, pure bending creep enters the second stage, which is called the secondary stage. The κ_c increases steadily with $\overline{t}$ and $\stackrel{·}{\kappa }$ _c is approximate a constant in this stage. The third stage is called the tertiary stage. The $\stackrel{·}{\kappa }$ _c increases sharply and the Al-RHT ruptures in this stage. In addition, most of pure bending creep time is spent in the initial and secondary stages.

Figure 8.

Experimental κ_c– $\overline{t}$ curve of Al-RHTs with D_o/t = 50 and d = 2 mm for held moment of 150 N m under pure bending creep.

Theoretical simulation for Al-RHTs under pure bending creep

In 2002, Lee and Pan ²⁰ tested smooth 304 stainless steel tubes with different D_o/t ratios submitted to pure bending creep. They proposed a theoretical formulation to describe the κ_c– $\overline{t}$ relationships in the initial and secondary stages as

$$k_{\mathrm{c}}=C{M}^{m_1}{\overline{t}}^{n_1},$$

(1)

in which M is the held moment, C, m₁, and n₁ are material parameters. In their study, m₁ and n₁ were respectively determined to be 5.70 and 0.53 and C was a function of the D_o/t ratio. In 2014, Lee et al. ²⁶ studied the κ_c– $\overline{t}$ relationships for circumferential sharp-notched 304 stainless steel tubes submitted to pure bending creep. Due to the same material and similar trend of the κ_c– $\overline{t}$ relationships, equation (1) was also used as the theoretical formulation. In their study, m₁ and n₁ were also used the same values, and C was proposed to be a function of circumferential sharp notch depth.

In this study, the κ_c– $\overline{t}$ relationships for Al-RHTs with different D_o/t ratios and different d under pure bending creep are similar to that of smooth 304 stainless steel tubes with different D_o/t ratios submitted to pure bending creep tested by Lee and Pan. ²⁰ Therefore, equation (1) is used in this investigation. However, the material parameter C is proposed to relate to D_o/t ratios and d. As for material parameters m₁ and n₁, they can be determined from smooth 6061-T6 aluminum alloy tubes subjected to pure bending creep. Figure 9 demonstrates the experimental κ_c– $\overline{t}$ in the initial and secondary stages for smooth 6061-T6 aluminum alloy tubes subjected to pure bending creep. Note that the smooth 6061-T6 aluminum alloy tubes are with D_o = 36.0 mm and t = 3.0 mm. By curve fitting with equation (1), the values of m₁ and n₁ are respectively obtained to be 1.086 and 0.046.

Figure 9.

Experimental and simulated κ_c– $\overline{t}$ in the initial and secondary stages for smooth 6061-T6 aluminum alloy tubes submitted to pure bending creep.

Figure 10(a) to (e) respectively indicate the experimental κ_c– $\overline{t}$ curves (dotted lines) of Al-RHTs with D_o/t = 30 and d = 2, 4, 6, 8, and 10 mm under pure bending creep in the initial and secondary stages, Figure 11(a) to (e) respectively indicate the experimental κ_c– $\overline{t}$ curves (dotted lines) of Al-RHTs with D_o/t = 40 and d = 2, 4, 6, 8, and 10 mm under pure bending creep in the initial and secondary stages, Figure 12(a) to (e) respectively indicate the experimental κ_c– $\overline{t}$ curves (dotted lines) of Al-RHTs with D_o/t = 50 and d = 2, 4, 6, 8, and 10 mm under pure bending creep in the initial and secondary stages, and Figure 13(a) to (e) respectively indicate the experimental κ_c– $\overline{t}$ curves (dotted lines) of Al-RHTs with D_o/t = 60 and d = 2, 4, 6, 8, and 10 mm under pure bending creep in the initial and secondary stages. It can be seen that for a certain D_o/t ratio under a constant held moment for pure bending creep, a smaller d leads to a faster increasing and larger κ_c. Due to increasing of the curvature, the Al-RHT breaks eventually. Using equation (1) with m₁ = 1.086 and n₁ = 0.0458, the amounts of C for different D_o/t ratios and different d/t are determined by curve fitting from the experimental data in Figures 10 to 13 which are shown in Table 1.

Figure 10.

Experimental and simulated κ_c– $\overline{t}$ curves of Al-RHTs with D_o/t = 30 and d = (a) 2, (b) 4, (c) 6, (d) 8, and (e) 10 mm under pure bending in the initial and secondary stages.

Figure 11.

Experimental and simulated κ_c– $\overline{t}$ curves of Al-RHTs with D_o/t = 40 and d = (a) 2, (b) 4, (c) 6, (d) 8, and (e) 10 mm under pure bending in the initial and secondary stages.

Figure 12.

Experimental and simulated κ_c– $\overline{t}$ curves of Al-RHTs with D_o/t = 50 and d = (a) 2, (b) 4, (c) 6, (d) 8, and (e) 10 mm under pure bending in the initial and secondary stages.

Figure 13.

Experimental and simulated κ_c– $\overline{t}$ curves of Al-RHTs with D_o/t = 60 and d = (a) 2, (b) 4, (c) 6, (d) 8, and (e) 10 mm under pure bending in the initial and secondary stages.

Table 1.

Amounts of C for different D_o/t ratios and different d/t.

D o/t
d/t	30	40	50	60
0.667	0.00187	0.00287	0.00395	0.00567
1.333	0.00193	0.00314	0.00449	0.00596
2.000	0.00205	0.00339	0.00463	0.00669
2.667	0.00221	0.00366	0.00535	0.00724
3.333	0.00243	0.00395	0.00549	0.00845

When we consider the lnC and d/t relationship, four straight lines for D_o/t = 30, 40, 50, and 60 are observed (see Figure 14). Thus, the parameter C can be proposed as

$$\ln C=c_1(d/t)+c_2$$

(2)

where c₁ and c₂ are material parameters which are related to D_o/t ratio. The amounts of c₁ and c₂ can be determined in Figure 14.

Figure 14.

Relationship between lnC and d/t.

From Figure 15(a) and (b), linear relationships of lnc₁-lnD_o/t and lnc₂-lnD_o/t are respectively observed. Therefore, parameters c₁ and c₂ are respectively proposed to be

$$\ln c_1={\alpha }_1\ln D_{\mathrm{o}}/t+{\alpha }_2$$

(3)

Figure 15.

(a) Relationship between lnc₁ and lnD_o/t, and (b) relationship between lnc₂ and lnD_o/t.

and

$$\ln c_2={\alpha }_3\ln D_{\mathrm{o}}/t+{\alpha }_4$$

(4)

where α₁, α₂, α_3, and α₄ are material parameters which are respectively determined from Figure 15(a) and (b) to be 0.679, −1.172, −0.263, and 2.749. The solid lines in Figure 10(a) to (e) respectively indicate the simulated κ_c– $\overline{t}$ curves of Al-RHTs with D_o/t = 30 and d = 2, 4, 6, 8, and 10 mm under pure bending creep in the initial and secondary stages, the solid lines in Figure 11(a) to (e) respectively indicate the simulated κ_c– $\overline{t}$ curves of Al-RHTs with D_o/t = 40 and d = 2, 4, 6, 8, and 10 mm under pure bending creep in the initial and secondary stages, the solid lines in Figure 12(a) to (e) respectively indicate the simulated κ_c– $\overline{t}$ curves of Al-RHTs with D_o/t = 50 and d = 2, 4, 6, 8, and 10 mm under pure bending creep in the initial and secondary stages, and the solid lines in Figure 13(a) to (e) respectively indicate the simulated κ_c– $\overline{t}$ curves of Al-RHTs with D_o/t = 60 and d = 2, 4, 6, 8, and 10 mm under pure bending creep in the initial and secondary stages. It can be seen that the simulation and experimental data have a good correlation.

Experimental response of Al-RHTs under pure bending relaxation

Figure 16 demonstrates a typical M– $\overline{t}$ curves of Al-RHTs with D_o/t = 40 and d = 10 mm under monotonic pure bending followed by pure bending relaxation. The relaxation starting points are marked by “•” It can be found that the moment of the tube drops sharply when the relaxation begins and approaches to a stable value. In addition, a larger held curvature results in a larger drop of the relaxation moment. Due to fixed curvature, the Al-RHT does not break.

Figure 16.

Experimental M– $\overline{t}$ curves of Al-RHTs with D_o/t = 40 and d = 10 mm under monotonic pure bending followed by pure bending relaxation.

Theoretical simulation for Al-RHTs under pure bending relaxation

In 2014, Lee et al. ²⁶ tested circumferential sharp-notched 304 stainless steel tubes submitted to pure bending relaxation. They proposed a theoretical formula to describe the relationship between relaxation moment (M_r) and $\overline{t}$ to be

$$M_{\mathrm{r}}=R{k}^{m_2}{\overline{t}}^{n_2},$$

(5)

where κ is the held curvature, R, m₂, and n₂ are material parameters. In their study, m₂ and n₂ were respectively determined to be 0.15 and −0.07 and the parameter R was proposed as a function of notch depths for circumferential sharp-notched 304 stainless steel tubes.

In this study, the M_r– $\overline{t}$ relationships for Al-RHTs with different D_o/t ratios and different d under pure bending relaxation are similar to that of circumferential sharp-notched 304 stainless steel tubes subjected to pure bending relaxation tested by Lee et al. ²⁶ Therefore, equation (5) is used in this investigation. However, the material parameter R is proposed to relate to the D_o/t and d. As for material parameters m₂ and n₂, they can be determined from smooth 6061-T6 aluminum alloy tubes subjected to pure bending relaxation. Figure 17 depicts experimental M– $\overline{t}$ curves (dotted lines) for smooth 6061-T6 aluminum alloy tubes under monotonic pure bending followed by pure bending relaxation. Note that the relaxation starting points are marked by “•” In addition, the smooth 6061-T6 aluminum alloy tubes are with D_o = 36.0 mm and t = 3.0 mm. By curve fitting of equation (5) (solid lines) with the experimental data (dotted lines) for pure bending relaxation, the values of m₂ and n₂ are respectively obtained to be 0.405 and −0.012.

Figure 17.

Experimental M– $\overline{t}$ (dotted lines) and simulated M_r– $\overline{t}$ (solid lines) curves for smooth 6061-T6 aluminum alloy tubes under monotonic pure bending followed by pure bending relaxation.

Figure 18(a) to (e) respectively indicate the experimental M– $\overline{t}$ curves (dotted lines) of Al-RHTs with D_o/t = 30 and d = 2, 4, 6, 8, and 10 mm under monotonic bending followed by pure bending relaxation, Figure 19(a) to (e) respectively indicate the experimental M– $\overline{t}$ curves (dotted lines) of Al-RHTs with D_o/t = 40 and d = 2, 4, 6, 8, and 10 mm under monotonic bending followed by pure bending relaxation, Figure 20(a) to (e) respectively indicate the experimental M– $\overline{t}$ curves (dotted lines) of Al-RHTs with D_o/t = 50 and d = 2, 4, 6, 8, and 10 mm under monotonic bending followed by pure bending relaxation, and Figure 21(a) to (e) respectively indicate the experimental M– $\overline{t}$ curves (dotted lines) of Al-RHTs with D_o/t = 60 and d = 2, 4, 6, 8, and 10 mm under monotonic bending followed by pure bending relaxation. It can be seen that the M_r- $\overline{t}$ relationship is similar to the reverse κ_c- $\overline{t}$ relationship. For a certain D_o/t ratio under a constant curvature for pure bending relaxation, a smaller d leads to a larger drop of M_r. Due to the constant curvature, the Al-RHT does not break. Using equation (5) with m₂ = 0.405 and n₂ = −0.012, the amounts of R for different D_o/t ratios and different d/t are determined by curve fitting from the experimental data in Figures 18 to 21 which are shown in Table 2.

Figure 18.

Experimental M– $\overline{t}$ (dotted lines) and simulated M_r– $\overline{t}$ (solid lines) curves of Al-RHTs with D_o/t = 30 and d = (a) 2, (b) 4, (c) 6, (d) 8, and (e) 10 mm under monotonic pure bending followed by pure bending relaxation.

Figure 19.

Experimental M– $\overline{t}$ (dotted lines) and simulated M_r– $\overline{t}$ (solid lines) curves of Al-RHTs with D_o/t = 40 and d = (a) 2, (b) 4, (c) 6, (d) 8, and (e) 10 mm under monotonic pure bending followed by pure bending relaxation.

Figure 20.

Experimental M– $\overline{t}$ (dotted lines) and simulated M_r– $\overline{t}$ (solid lines) curves of Al-RHTs with D_o/t = 50 and d = (a) 2, (b) 4,(c) 6, (d) 8, and (e) 10 mm under monotonic pure bending followed by pure bending relaxation.

Figure 21.

Experimental M– $\overline{t}$ (dotted lines) and simulated M_r– $\overline{t}$ (solid lines) curves of Al-RHTs with D_o/t = 60 and d = (a) 2, (b) 4,(c) 6, (d) 8, and (e) 10 mm under monotonic pure bending followed by pure bending relaxation.

Table 2.

Amounts of R for different D_o/t rations and different d/t.

D o/t
d/t	30	40	50	60
0.667	494.15	407.98	350.69	323.07
1.333	477.12	389.93	328.53	302.60
2.000	460.62	371.72	311.48	279.78
2.667	438.96	352.46	290.37	264.28
3.333	418.73	333.14	277.52	246.54

When we consider the R^1/3-d/t relationship, four straight lines for D_o/t = 30, 40, 50, and 60 are observed (see Figure 22). Therefore, parameter R is proposed to be

$$R^{1/3}=r_1(d/t)+r_2$$

(6)

Figure 22.

Relationship between R^1/3 and d/t.

where r₁ and r₂ are material parameters which are related to D_o/t ratio.

From Figure 23(a) and (b), linear relationships of lnr₁-lnD_o/t and lnr₂-lnD_o/t are observed. Therefore, parameters r₁ and r₂ are respectively proposed to be

$$\ln r_1={\beta }_1\ln D_{\mathrm{o}}/t+{\beta }_2$$

(7)

Figure 23.

(a) Relationship between lnr₁ and lnD_o/t, and (b) relationship between lnr₂ and lnD_o/t.

and

$$\ln r_2={\beta }_3\ln D_{\mathrm{o}}/t+{\beta }_4$$

(8)

where β₁, β₂, β₃, and β₄ are material parameters which are respectively determined from Figure 23(a) and (b) to be −0.750, −0.459, −0.199, and 2.759. Figure 18(a) to (e) respectively indicate the simulated M_r– $\overline{t}$ curves (solid lines) of Al-RHTs with D_o/t = 30 and d = 2, 4, 6, 8, and 10 mm under pure bending relaxation, Figure 19(a) to (e) respectively indicate the simulated M_r– $\overline{t}$ curves (solid lines) of Al-RHTs with D_o/t = 40 and d = 2, 4, 6, 8, and 10 mm under pure bending relaxation, Figure 20(a) to (e) respectively indicate the simulated M_r– $\overline{t}$ curves (solid lines) of Al-RHTs with D_o/t = 50 and d = 2, 4, 6, 8, and 10 mm under pure bending relaxation, and Figure 21(a) to (e) respectively indicate the simulated M_r– $\overline{t}$ curves (solid lines) of Al-RHTs with D_o/t = 60 and d = 2, 4, 6, 8, and 10 mm under pure bending relaxation. Again, it can be seen that the simulation and experimental data have a good correlation.

Conclusions

This paper presents the experimental and theoretical research on the response of Al-RHTs with different D_o/t ratios and different d submitted to pure bending creep and pure bending relaxation. Based on experimental and theoretical results, this paper draws the following important conclusions:

1. It is seen from the M-κ curves for monotonic bending that once the deformation is in the plastic range, the M-κ relationship becomes nonlinear. It can also be observed that the larger the D_o/t ratio, the smaller the tube wall thickness. Therefore, a lower M value is found. In addition, for each D_o/t ratio, the M-κ curves are similar for different d. However, a smaller d leads to a larger breaking curvature.

2. It is found from pure bending creep that once the pure bending creep begins, the curvature increases quickly. In addition, the κ_c– $\overline{t}$ relationship can divide into three stages. Most of the time for pure bending creep is spent in the initial and secondary stages. For a certain D_o/t ratio under a constant moment for pure bending creep, a smaller d leads to a faster increasing and larger κ_c. As the curvature continues to increase, Al-RHT eventually breaks.

3. It is observed from pure bending relaxation that once the relaxation begins, the moment drops sharply and gradually becomes a stable value. In addition, the M_r– $\overline{t}$ relationship is similar to the reverse κ_c– $\overline{t}$ relationship. For a certain D_o/t ratio under a constant curvature for pure bending relaxation, a smaller d leads to a larger drop of the M_r. Due to the fixed curvature, the Al-RHT does not break.

4. The theoretical formula proposed by Lee and Pan ²⁰ was employed for simulating the κ_c– $\overline{t}$ relationships for Al-RHTs with different D_o/t ratios and different d submitted to pure bending creep in the initial and secondary stages. In addition, a new formulation of C in equations (2)–(4) was proposed. Moreover, the theoretical formula proposed by Lee et al. ²⁶ was employed for simulating the M_r– $\overline{t}$ relationships for Al-RHTs with different D_o/t ratios and different d submitted to pure bending relaxation. In addition, a new formulation of R in equations (6)–(8) was proposed. By comparing with experimental data, the theoretical formulas can accurately simulate the experimental results, as shown in Figures 10 to 13 and 18 to 21.