摘要
本文主要針對五種不同深度之凹痕6061-T6鋁合金管作循環彎曲負載的實驗，以探討其相關的力學行為與裂縫損壞關係。透過彎管實驗機與曲度-橢圓化量測器來進行實驗數據的控制、量測及蒐集。從實驗之彎矩-曲度曲線中顯示，在對稱循環彎曲時，沒有凹痕的圓管呈現對稱且穩定的迴圈，而有凹痕的圓管因凹痕接觸的程度不同，呈現不對稱且不穩定的迴圈。其次，由實驗之橢圓化-曲度曲線中發現，在對稱循環彎曲時，沒有凹痕的圓管呈現對稱且棘齒狀增加的變化，而有凹痕的圓管因凹痕接觸的程度不同呈現不對稱且棘齒狀增加的變化。而且凹痕深度越深，橢圓化-曲度曲線就越不對稱。當橢圓化值增加到某一臨界值時，圓管便會發生皺曲或破裂而導致管件損壞。此外，由實驗之控制曲度-循環至損壞圈數曲線中可看出，控制曲度越大則循環至損壞圈數就越少，且由其雙對數座標關係圖中可發現，控制曲度-循環至損壞圈數呈現五條不平行的直線。最後，本文提出一相關的理論方程式來描述不同深度凹痕圓管在循環彎曲負載時控制曲度與循環至損壞圈數關係，在與實驗值做比較後發現，理論分析能充分的描述實驗結果。

關鍵字 : 凹痕圓管、循環彎曲、力矩、曲度、橢圓化、循環至損壞圈數

Experimental Study on the Response of Dented Circular Tubes under Cyclic Bending

WU,CHENG-CHANG
PAN,WEN-FUNG
Department of Engineering Science
National Cheng Kung University

SUMMARY
This thesis investigates the behavior and failure of 6061-T6 aluminum alloy tubes with five different dent depths under cycling bending. The tube bending machine and curvature-ovalization measurement apparatus were used to control, measure and collect the experimental data. During the symmetrical cycling bending, it can be observed from the experimental moment-curvature curves that the curve becomes symmetrical and stable for tubes without a dent. Due to the contact of the dent’s sides, the curves show different degrees of unsymmetry for tubes with different dent depths. Next, from the experimental ovalization-curvature curve, the ovalization of the tube’s cross-section increases in a symmetrical and ratcheting manner for tubes without a dent. However, due to the contact of the dent’s sides, the ovalizations show different degrees of unsymmetry, ratcheting and increasing for tubes with different dent depths. A higher dent depth leads to a more unsymmetrical ovalization-curvature curve. The tube buckles or fractures when the ovalization of the tube’s cross-section reaches a critical amount. In addition, it is shown from the experimental controlled curvature-number of cycles to failure relationship that the larger controlled curvature leads to a fewer amount of the number of cycles to failure. It can be observed from the aforementioned relationship in a log-log scale that the relationship shows five nonparallel straight lines. Finally, a theoretical formulation was proposed in this thesis to simulate the relationship between the controlled curvature and the number of cycles to failure. By comparing the theoretical analysis with the experimental data, it is shown that the theoretical formulation can properly simulate the experimental results.

Keywords : Dented Circular Tubes, Cycling Bending, Moment, Curvature, Ovalization, Number of Cycles of Failure

INTRODUCTION

Since the tubes are with the characteristic of high bending strength, light weight and isotropy, the tubes are widely used in the industrial applications. Tube components are often subjected to cycling bending load. This load will cause the tube’s cross-section to ovalize. The tube buckles or fractures when the ovalization of the tube’s cross-section reaches to a critical amount. The damaged tube will lead to inestimable loss. Therefore, it is necessary to study tubes in the industrial applications.

MATERIALS AND METHODS

Material
The material of tube used was 6061-T6 aluminum alloy. Table 1 shows its proportion of the chemical composition. The ultimate stress, yield stress, the Young’s modulus and Poisson's ratio were 310 MPa, 286 MPa, 69 GPa and 0.33, respectively.
Table 1. Chemical composition of 6061-T6 aluminum alloy
Element Cr Mn Fe Si Cu Zn Al
proportion (%) 0.09 0.15 0.7 0.5 0.3 0.25 97.4
The size of tube was 500 mm in length, 30 mm in outside diameter and 0.5 mm in wall thickness. The type of local dent is semicylinderical and with five different dented depths (a) of 0.0, 0.3 , 0.6, 0.9 and 1.2 mm.

Method
A four-point bending machine was used to conduct the local dented circular tubes under cyclic bending. A curvature-ovalization measurement apparatus (COMA) was used to control the curvature. The curvature-rate was 0.035 m-1s-1 in this study. The magnitude of the bending moment was measured by two load cells mounted in the bending device, and the magnitudes of the curvature and ovalization of the tube’s cross-section were measured by COMA. In addition, the number of cycles to failure was also recorded.

RESULTS AND DISCUSSION

The experimental results
Figures 1(a)-(e) show the moment-curvature relationships for dented 6060-T6 aluminum alloy circular tubes under cyclic bending with a = 0.0, 0.3, 0.6, 0.9 and 1.2 mm, respectively. For a = 0.0 mm, the magnitudes of the maximum and minimum stresses were almost the same. For a  0.0 mm, the maximum stress was larger than the minimum stress. Due to the contact between the dent’s sides in the reverse bending, the curves exhibited increasing first and gradually decreasing manner. However, the loops become steady eventually.

Figure 1. Moment-curvature relationships for dented 6060-T6 aluminum alloy circular tubes under cyclic bending with a = (a) 0.0, (b) 0.3, (c) 0.6, (d) 0.9 and (e)1.2 mm

Figures 2(a)-(e) show the ovalization-curvature relationships for dented 6060-T6 aluminum alloy circular tubes under cyclic bending with a = 0.0, 0.3, 0.6, 0.9 and 1.2 mm, respectively. For a = 0.0 mm, the relationship was symmetrical. For a  0.0 mm, the relationships become unsymmetrical. Higher dent depth leads to sever symmetrical and fast increasing manner.

Figure 2. Ovalization-curvature relationships for local-dented 6060-T6 aluminum alloy circular tubes under cyclic bending with
a = (a) 0.0, (b) 0.3, (c) 0.6, (d) 0.9 and (e)1.2 mm

Figure 3 shows the relationship between controlled curvature and number of cycles to failure for dented 6061-T6 aluminum alloy tube with different dent depths. It can be seen that the larger controlled curvature or the deeper dented tube leads to a fewer amount of the number of cycles to failure. Figure 4 shows the relationship between controlled curvature and number of cycles to failure for dented 6061-T6 aluminum alloy tube with different dent depths in a log-log scale. Five straight lines were determined by least square method. It can be seen that these five straight lines are with different slopes and intercepts.

Figure 3. Relationship between controlled curvature and number of cycles to failure for dented 6061-T6 aluminum alloy tubes with different dent depths

Figure 4. Relationship between controlled curvature and number of cycles to failure for dented 6061-T6 aluminum alloy tubes with different dent depths
in a log-log scale

Theoretical analysis
Kyriakides and Shaw【2】have proposed a formulation of the relationship between controlled curvature (c/o) and number of cycles to failure (Nf) for the material they tested as
c/o = C (Nf)- (1)
where C and  are the material parameters, which are related to the material properties. The material parameter C is the controlled cyclic curvature magnitude at Nf = 1, and  is the slope in the log-log plot. By referring Lee at al.【5】and Fan【15】, C and  were proposed to be
　 C=C_0 〖(e)〗^(-β(a/t)) (2)
and
α=α_0 〖(e)〗^(-γ(a/t)) (3)
where C0, , 0 and  are material parameters. From Figure 5 and 6 the values of C0, , 0 and  can be determined to be 0.94、0.171、0.106 and -0.1753, respectively. Figure 7 shows the experimental and simulated results of cyclic curvature versus the number of cycles to failure for the five different dented depths of 0, 0.3, 0.6, 0.9, 1.2 mm in a log-log scale. Good agreement between the experimental and simulated results has been achieved.

Figure 5. The relationship between lnC and a/t ratio

Figure 6. The relationship between lnα and a/t

Figure 7. Experimental and simulated relationship between controlled curvature and number of cycles to failure for 6061-T6 aluminum alloy
tubes with different dent depths in a log-log scale

CONCLUSIONS
From the relationship between moment and curvature, for tubes without a dent, the magnitudes of the maximum and minimum stresses were almost the same. For tubes with a dent, the maximum stress was larger than the minimum stress. Due to the contact between the dent’s sides in the reverse bending, the curves exhibited increasing first and gradually decreasing manner. However, the loops become steady eventually.
From the relationship between ovalization and curvature, for tubes without a dent, the relationship was symmetrical. For tubes with a dent, the relationships become unsymmetrical. Higher dent depth leads to sever symmetrical and fast increasing manner.
From the relationship between controlled curvature and number of cycles to failure in a log-log scale, five straight lines were determined by least square method. It can be seen that these five straight lines are with different slopes and intercepts.
The formulations (2) and (3) proposed in this study were used with formulation (1) for simulating the relationship between the controlled curvature and number of cycles to failure of dented 6061-T6 aluminum alloy tubes for different dented depths under cyclic bending. By comparing with the experimental data, the theoretical formulation can properly describe the experimental result.

1.P. K, Shaw. and Kyriakides, S., “Inelastic Analysis of Thin-Walled Tubes under Cyclic Bending”, International Journal of Solids and Structures, Vol. 21, No. 11, pp.1073-1110(1985).

2.Kyriakides, S. and P. K, Shaw.,”Inelastic Buckling of Tubes under Cyclic Loads”,ASME Journal of Pressure Vessel and Technology, Vol. 109, pp. 169-178 (1987).

3.Corona. E. and Kyriakides, S.,”On the collapse of Inelastic Tubes under Combined Bending and Pressure”,International Journal of Solids and Structures, Vol. 24, pp. 505-535(1988).

4.Corona. E. and Kyriakides, S.,”An Experimental Investigation of the Degradation and Buckling of Circular Tubes under Cyclic Bending and External Pressure”,Thin-Walled Structures, Vol. 12, pp. 229-263 (1991).

5.W. F. Pan, T. R. Wang and C. M. Hsu,”A curvature-ovalization measurement apparatus for circular tubes under cyclic bending”,Experimental Mechanics, Vol. 38, No. 2, pp.99-102 (1998).

6.W. F. Pan andY. S. Her, “Viscoplastic collapse of thin-walled tubes under cyclic bending”,Journal Engineering Materials and Technology, Vol. 120, No. 4, pp. 287-290 (1998)

7.W. F. Pan andC. H. Fan, “An experimental study on the effect of curvature-rate at preloading stage on subsequent creep or relaxation of thin-walled tubes under pure bending”, JSME InternationalJournal, Series A, Vol. 41, No. 4, pp. 525-531 (1998).

8.K. L. Lee, W. F.Pan andJ. N. Kuo,” The influence of the diameter-to-thickness ratio on the stability of circular tubes under cyclic bending”,International Journalof Solids and Structures, Vol. 38, No. 14, pp. 2401-2413(2001).

9.W. F. Pan andK. L. Lee,”The effect of mean curvature on the response and collapse of thin-walled tubes under cyclic bending”,JSME International Journal, Series A, Vol. 45, No. 2, pp. 309-318 (2002).

10.K. L. Lee and W. F. Pan, “Pure bending creep of SUS304 stainless steel tubes”, Steel and Composite Structures, Vol. 2, No. 6, pp.461-474 (2002).

11.徐建民、李國龍和潘文峰，「圓管在對稱與不對稱循環彎曲負載下力學行為之實驗研究」，技術學刊，第十八卷，第二期，257-262頁(2003)。

12.K. L. Lee, W. F. Pan andC. M. Hsu,” Experimental and theoretical evaluations of the effect between diameter-to-thickness ratio and curvature-rate on the stability of circular tubes under cyclic bending”,JSME International Journal, Series A, Vol. 47, No. 2, pp. 212-222 (2004).

13.徐日豐、徐建民和潘文峰，「圓管在循環彎曲負載下力學行為之理論研究」，技術學刊，第二十卷，第三期，305-313頁(2005)。

14.K. H. Chang, C. M. Hsu, S. R. Sheu and W. F. Pan,“Viscoplastic response and collapse of 316L stainless steel under cyclic bending”,Steel and Composite Structures, Vol. 5, No. 5, pp. 359-374 (2005).

15.范揚東，「尖銳凹槽薄壁管在循環彎曲負載下力學行為和皺曲損壞之研究」，國立成功大學工程科學研究所碩士論文，(2008)。

16.張高華、李國龍和潘文峰，「圓管承受循環彎曲負載截面變形量測器之設計」，技術學刊，第二十三卷，第一期，21-28頁(2008)。

17.K. H. Chang, W. F. Pan andK. L.,“Mean moment effect on circular thin-walled tubes under cyclic bending”,Structural Engineering and Mechanics, Vol. 28, No. 5, pp. 495-514 (2008).

18.K. H. Chang andW. F. Pan,“Buckling life estimation of circular tubes under cyclic bending”,International Journal of Solids and Structures, Vol. 46, No. 2, pp. 254-270 (2009).

19.K. L. Lee, C. Y.Hung, H. Y. Chang andW. F. Pan,”Buckling life estimation of circular tubes of different materials under cyclic bending of each cross-section of circular tube under cyclic bending”,Journalof Chinese Institute Engineers, Vol. 33, No. 2, pp. 177-189 (2010).

20.K. L. Lee, C. Y. Hung and W. F. Pan, “Variation of ovalization for sharp-notched circular tubes under cyclic bending”, Journal of Mechanics, Vol. 26, No. 3, pp. 403- 411 (2010).

21.K. L. Lee, C. Y.Hung andW. F. Pan,“CCD digital camera system for measuring curvature and ovalization of each cross-section of circular tube under cyclic bending”,Journalof Chinese Institute Engineers, Vol. 34, No. 1, pp. 75-86 (2011).

22.K. L. Lee, C. M. Hsu and W. F. Pan, “The influence of diameter-to-thickness ratios on the response and collapse of sharp-notched circular tubes under cyclic bending”,Journal of Mechanics, Vol. 28, No. 3, pp. 461-468 (2012).

23.李國龍、洪兆宇和潘文峰，「循環彎曲負載下橢圓形凹槽圓管皺曲損壞之實驗分析」，中正嶺學報，第四十一卷，第二期，183-190頁(2012)。

24.K. L. Lee, C. M. Hsu and W. F. Pan, “The influence of mean curvatures on the collapse of sharp-notched circular tubes under cyclic bending”,Journal of Chinese Society of Mechanical Engineering, Vol. 34, No. 5, pp. 461-468 (2013).

25.K. L. Lee, C. M. Hsu and W. F. Pan,” Viscoplastic collapse of sharp-notched circular tubes under cyclic bending”, Acta Mechanics Solida Sinica, Vol. 26, No. 6, pp. 629- 641 (2013).

26.K. L. Lee, C. H. Mengand W. F. Pan, “The influence of notch depth on the response of local sharp-notched circular tubes under cyclic bending”,Journal of Applied Mathematics and Physics, Vol. 2, No. 6, pp. 335-341 (2014).

27.沈睿騰，「不同深度切痕圓管在循環彎曲負載下行為之實驗研究」，國立成功大學工程科學研究所碩士論文，(2014)。