Stress—strain curves for different loading paths and yield loci of aluminum alloy sheets

WU Xiang-dong(吴向东)^{1, 2}, WAN Min(万  敏)¹, HAN Fei(韩  非)¹, WANG Hai-bo(王海波)¹

1. School of Mechanical Engineering and Automation, Beijing University of Astronautics and Aeronautics,

Beijing 100083, China;

2. State Key Laboratory of Plastic Forming Simulation and Die and Mould Technology, Wuhan 430074, China

Received 28 July 2006; accepted 15 September 2006

Abstract: To carry out biaxial tensile test in sheet metal, the biaxial tensile testing system was established. True stress—true strain curves of three kinds of aluminum alloy sheets for loading ratios of 4:1, 4:2, 4:3, 4:4, 3:4, 2:4 and 1:4 were obtained by conducting biaxial tensile test in the established testing systems. It shows that the loading path has a significant influence on the stress—strain curves and as the loading ratio increases from 4:1 to 4:4, the stress—strain curve becomes higher and n-value becomes larger. Experimental yield points for three aluminum alloy sheets from 0.2% to 2% plastic strain were determined based on the equivalent plastic work. And the geometry of the experimental yield loci were compared with the yield loci calculated from several existing yield criteria. The analytical result shows that the Barlat89 and Hosford yield criterion describe the general trends of the experimental yield loci of aluminum alloy sheets well, whereas the Mises yield criterion overestimates the yield stress in all the contours.

Key words: aluminum alloy sheet; biaxial tensile test; stress—strain curve; yield locus

1 Introduction

As one of the key problems urgent to be resolved in the fields of mechanics, material science and sheet metal forming, constructing reasonable yield criterion, work-hardening law and stress-strain relationship is very important for predicting the distribution of stress and strain in sheet metal forming, as well as for improving FEA simulation precision and resolving engineering problems.

Also, in the practical sheet metal forming, the loading path usually deviates from the linear paths due to the effects of the geometry boundary and the friction condition, which is seen in complicated part forming and multistage forming. Therefore, the accurate description of the plastic deformation behavior under different loading paths and the establishment of the reasonable yield model of sheet metals are very important [1-2].

In literatures some experimental methods such as uniaxial tensile test, hydraulic bulge test with round plate, the compression test along thickness of sheet and notched or grooved strip tension test have been used to study sheet metals with anisotropic properties. The disadvantage of these experiments is that only some yield locus points can be determined, for example, the uniaxial tension stress state, equi-biaxial stress state, plane strain stress state, etc. Therefore, the experimental method of realizing variable biaxial loading paths has been a major topic of sheet metal forming research.

Biaxial tensile test with different types of cruciform specimens has become the most promised method to realize various stress state of biaxial tension by changing proportion of load or displacement of two axes. MAKINDE et al [3], LI et al [4] have designed different testing apparatus to carry out biaxial tensile test.

In biaxial tensile test, one of the key problems is to determine the stress of center area in cruciform specimen. To minimize the calculation error of stress, many researchers have optimized the shape of cruciform specimen use photoelastic model, FEM and different optimization algorithm (MONTGOMERY [5], DEMMERLE and BOEHLER [6], KUWABARA et al [7], YU et al [8]).

On the base of biaxial tensile system and optimized cruciform specimen, biaxial tensile test has been widely used in investigating mechanical behavior of sheet metals under biaxial tensile states.

KUWABARA et al[9-11] studied the experimental yield loci of different work hardening of cold-rolled steel sheet and an aluminum alloy sheet under biaxial tension. They found that Gotoh’s yield criterion describes the yield behavior of steel sheet with good accuracy whereas the Taylor model fit the 6××× aluminum alloy sheet very well.

GREEN et al[12] used a type of cruciform specimen with thinned center and slots in the arms to study the biaxial tensile behavior of an 1145 aluminum alloy sheet. In addition, finite element analysis was carried out using different phenomenological models of anisotropic plasticity. Instead of evaluating the applicability of different yield criteria, GREEN et al designed a procedure to determine the anisotropic parameters of various yield functions and obtain best fits to these functions.

In this paper, in order to study the plastic deformation behavior of the anisotropic sheet metal, the biaxial tensile testing system is established and three types of aluminum alloy sheets that have different Lankford coefficient are studied using a biaxial tensile test system and optimized cruciform specimens. And the experimental result is given and analyzed.

2 Test system

The biaxial tensile testing system used in this paper includes a loading test machine, a control unit and an application software. The loading test machine, which consists of six independent controlled axes and each axis is actuated by a hydraulic cylinder, was designed and assembled in Beihang University. The Structure of hardware system is illustrated in Fig.1.

Fig.1 Block diagram of control system

The machine frame consists of a top beam, two vertical poles and a worktable. The four jaws, which are connected with different horizontal hydraulic cylinders, were used to clamp the specimen. Some of the corresponding specifications of the test system are as follows.

1) The outlook size is length 2 m, width 2 m and height 2 m.

2) The displacement scope of the four axes is 300-700 mm.

3) The maximum load capability of each axis is  100 kN.

4) The velocity of the four axes is 0-200 mm/min.

The control system of machine contains a computer, A/D and D/A converters, serial interface boards, amplifiers and other necessary boards. During the experiment, the control system performed the measurements and calculations, sent the control signals to the process, and regulated how the other devices function.

This PC-based system adopted an improved digital PID controller and the accuracy of system under the proportional and non-proportional loading paths is very good[13].

3 Experimental

The shape and dimension of cruciform specimen were optimized by means of combining FEM with orthogonal design, which is illustrated in Fig.2.

Fig.2 Schematic diagram of cruciform specimen (mm)

Three kinds of aluminum alloy sheets are used in the test and the properties of them are listed in Table 1.

4 Results and discussion

4.1 Stress—strain curves under different loading paths

Fig.3 shows the true stress—true strain curves of three kinds of aluminum alloy sheets at loading ratios of 4:1, 4:2, 4:3, 4:4, 3:4, 2:4 and 1:4. In Fig.3, the loading ratio is the ratio of load along rolling direction to load along transverse direction. And the left part is the stress—strain curve of rolling direction, i.e. the loading direction is along the rolling direction of sheet metal in test procedure, and the right part is the curve of transverse direction. It can be seen that the stress—strain curves under different loading paths are different from each other, the stress—strain curve for loading ratio 4:4 is high and n-value is large, whereas the other curves are low and n-value are small.

Table 1 Mechanical properties of aluminum alloy sheet metals used in test

Fig.3 True stress—true strain curves of aluminum alloy sheets for different loading ratios: (a), (a′) 2024-O; (b), (b′) 5754-M; (c), (c′) LY12-M

4.2 Comparison between experimental yield points and theoretical yield loci

According to the principle of constant amount of total plastic work per unit volume, the relationship between stress—strain along two directions in biaxial tensile test and equivalent stress-equivalent strain can be written as

                     (1)

where  σ₁ and ε₁ are the stress and strain of rolling direction whereas σ₂ and ε₂ are transverse directions. The symbol  and  designate the equivalent stress and equivalent strain.

Based on Eqn.(1) and the experimental stress—strain curves, points of aluminum alloy sheet metals under different loading paths for different equivalent plastic strains from 0.2% to 2% were determined. Also, the theoretical yield loci calculated based on Hill 48, Hill 90, Hosford, Barlat 89 for different plastic works are drawn to compare with yield points, as shown in Figs.4, 5 and 6.

Fig.4 Comparison of experimental yield points of 2024-O aluminum alloy sheet and existing yield loci for different equivalent plastic strains: (a) ε=0.002; (b) ε=0.01; (c) ε=0.02

Fig.5 Comparison of experimental yield points of 5754-M aluminum alloy sheet and existing yield loci for different equivalent plastic strains: (a) ε=0.002; (b) ε=0.01; (c) ε=0.02

Fig.6 Comparison of experimental yield points of LY12-M aluminum alloy sheet and existing yield loci for different equivalent plastic strains: (a) ε=0.002; (b) ε=0.01; (c) ε=0.015

The analytical result shows that the Barlat 89 and Hosford yield criterion describe the general trends of experimental yield loci of aluminum alloy sheets better, and then Hosford, Hill 90 and Hill 48 criterion. Mises yield criterion overestimated the yield stress in all the contours, especially for LY12-M.

5 Conclusions

1) The biaxial tensile testing system is established which includes a test machine and cruciform specimen. Test result shows that the system meets the requirement of biaxial tensile test and three kinds of aluminum alloy sheets were used to carry out the experiment.

2) According to the experimental results, the true stress—true strain curves of aluminum alloy sheets vary under different loading paths and the loading ratio higher, the stress value and n-value higher.

3) Comparison between experimental yield points and theoretical yield loci shows that the Barlat 89 and Hosford yield criteria describe the experimental yield loci of aluminum alloy sheets studied in this paper better than Hill 90, Hill 48 and Mises criteria.

References

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(Edited by LONG Huai-zhong)

Foundation item: Project (50475004) supported by the National Natural Science Foundation of China; Project (05-2) supported by the Foundation of the State Key Laboratory of Plastic Forming Simulation and Die & Mould Technology of HUST; Project (2004036197) supported by the Postdoctoral Science Foundation of China

Corresponding author: WU Xiang-dong; Tel: +86-10-60877704; E-mail: xdwu@buaa.edu.cn