J. Cent. South Univ. Technol. (2008) 15: 581−587 DOI: 10.1007/s11771−008−0109−5

Numerical simulation and experimental investigation of

incremental sheet forming process

HAN Fei(韩 飞), MO Jian-hua(莫健华)

(State Key Laboratory of Material Processing and Die & Mould Technology, Huazhong University of Science and Technology, Wuhan 430074, China)

Abstract: In order to investigate the process of incremental sheet forming (ISF) through both experimental and numerical approaches, a three-dimensional elasto-plastic finite element model (FEM) was developed to simulate the process and the simulated results were compared with those of experiment. The results of numerical simulations, such as the strain history and distribution, the stress state and distribution, sheet thickness distribution, etc, were discussed in details, and the influences of process parameters on these results were also analyzed. The simulated results of the radial strain and the thickness distribution are in good agreement with experimental results. The simulations reveal that the deformation is localized around the tool and constantly remains close to a plane strain state. With decreasing depth step, increasing tool diameter and wall inclination angle, the axial stress reduces, leading to less thinning and more homogeneous plastic strain and thickness distribution. During ISF, the plastic strain increases stepwise under the action of the tool. Each increase in plastic strain is accompanied by hydrostatic pressure, which explains why obtainable deformation using ISF exceeds the forming limits of conventional sheet forming. Key words: incremental sheet forming (ISF); sheet metal forming; numerical simulation; finite element method 1 Introduction

Incremental sheet forming (ISF) is an innovative, flexible sheet forming process that uses principles of layered manufacturing for the production of complex-shaped sheet metal parts. This process resolves the complicated geometry information into a series of two-dimensional layers, and then the plastic deformation is carried out layer-by-layer through the computerized numerical controlled movements of a simple spherical forming tool to get the desired part[1−5]. Generally, two main variants of the incremental sheet forming process are known, i.e. the “negative forming process” and the “positive forming” process[6]. In negative incremental forming, a spherical tool moves on a sheet metal, according to a programmed tool path. The sheet is clamped at the periphery by bolts on a support frame. In positive forming (Fig.1), the blank is put on the general mandrel and fastened at its edges by a clamping plate, which can move along the guide posts. The tool deforms the blank into the mandrel and moves along the contour lines until the required shape is formed. In the current study, the latter technique was investigated.

ISF has been investigated mainly by simplified analytical deformation models and by full scale finite

element analysis(FEA). The most prominent analytical model is based on the sine law[7]. Unfortunately, analytical models are limited to the approximate prediction of strains. For further studies, the most commonly used tool was the finite element method. SHIM and PARK[8] performed a numerical simulation of the single layer in the forming of truncated pyramid to find the deformation characters along the tool path. ISEKI[9] modeled the incremental forming of a shell of the frustum of a quadrangular pyramid based on the shell elements without considering material anisotropy and Baushinger effects of shell material, and the formed height of the part was just 5 mm. AMBROGIO et al[10−11] showed that the single point incremental sheet forming process mainly depended on geometrical and process conditions. Particularly, the accuracy of the final geometry was mainly influenced by the tool depth step.

In this paper, a three-dimensional elasto-plastic finite element model was adopted to simulate the incremental sheet forming process. The truncated cone, as the benchmark part, was simulated for different process parameters. The comparison between experimental and FEM results was carried out. Additionally, the influence of the process parameters on the strain history and distribution, the stress state and distribution, sheet thickness distribution during forming

Foundation item: Project(50175034) supported by the National Natural Science Foundation of China Received date: 2007−12−14; Accepted date: 2008−01−21 Corresponding author: HAN Fei, Doctoral candidate; Tel: +86−13691060870; E-mail: hanfei.hust@yahoo.com.cn

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Fig.1 Positive incremental sheet forming process were carefully analyzed. Moreover, the reason why ISF gives higher forming limits than conventional sheet forming processes was explained. 2 Experiment and simulation model 2.1 Experimental setup

For the experiments described in this paper, a special 3-axis numerical controlled machine was used as incremental sheet forming set-up. The blank was fastened by a properly designed fixture. According to the fixture design, the blank with 330 mm×330 mm×1 mm dimensions was 08Al in the annealed state. The circular die used for the experiments had a diameter of 80 mm and a edge radius of 3 mm. A spherical nosed tool (tool steel, Cr13) was used for all applied strategies. The code for tool path was calculated using the computer automated manufacturing module of Unigraphics NX. The tool speed was set to 200 mm/s. Oil was applied to further minimizing the friction. 2.2 Simulation setup

A three-dimensional, elasto-plastic finite element model was set up for the simulation of the ISF process. The explicit finite element package Abaqus/Explicit was used in this work. Because of the nature of the process, there are several nonlinearities involved in the simulation of incremental forming. In addition, for the considered applications, incremental forming processes are typically fully 3D without any symmetry plane. Usually, a large number of elements have to be used and the tool moves along a relatively long trajectory, which will cause finite element analysis to be complicated and time to be consumed. In order to validate the model, similar parameters and material property values have been used in both the simulation and the experimental study. 2.2.1 Material and process parameters

A series of uniaxial tension tests at room temperature with the tensile rate of 2.5 mm/min were conducted to experimentally determine the material

properties for input into the simulation. The main material parameters are shown in Table 1. Fig.2 shows the true stress—strain curve for the blank material. Table 1 Properties of blank for ISF simulations

Item Value

Material 08Al

Yield strength/MPa 175

Ultimate tensile strength/MPa 303

Ultimate elongation/% 44

Elastic modulus/MPa 2.1×105

Density/(g·cm−2) 7.83

Poisson ratio 0.30

Strain-hardening exponent 0.227

Anisotropy coefficients, r r0＝1.71, r45＝1.14, r90＝1.83

Fig.2 Stress—strain curve of 08Al blank

ISF is a complex cold-forming process. Many factors influence the forming process, such as vertical step down (∆z) between the consecutive contours, the wall inclination angle(α) and the tool diameter(D). In order to investigate the effects of control parameters on the output values, a series of experiments and simulations were conducted. The final desired height of the cone was 20 mm. Fig.3 shows the ISF process parameters, and the processing parameters on the investigated simulation are listed in Table 2. 2.2.2 Finite element model

In the model, the 08Al-blank was meshed with 14 400 4-node shell elements (Abaqus type S4R) with five integration points through the thickness. The material was assumed to be planar anisotropic following Hill’s 1948 yield criterion with kinematic hardening. The tool, partial die, clamping plate and backing plate were modeled as rigid surfaces. Coulomb’s friction law was applied with a friction coefficient of 0.05 between the blank and the tool and of 0.15 between the blank and the

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partial die. The contact condition was implemented through a pure Master-Slave contact algorithm. Fig.4 shows the utilized FEM model for the process.

Fig.3 ISF process parameters Table 2 Process parameters utilized for simulations

Process parameter Value

Tool depth step/mm 0.5, 1.0, 1.5, 2.0

Tool diameter/mm 10, 20, 30

Wall inclination angle/(˚) 45, 55, 65

Fig.4 FEM model for ISF process

In order to synchronize the finite element simulation with the experiment, the NC file from the actual experiment was imposed into the simulation to move the forming tool. The tool movement was controlled using predefined displacement constraints in several load steps. Using an artificially high tool velocity to substitute for the real process is considered a potentially good method to shorten the simulation time. So an artificially increased tool feed rate of 2 m/s was utilized. The simulation time is reduced significantly with the increase of tool velocity by 10 times. 2.3 Validation of simulation model 2.3.1 Radial strain

At the process parameters of D=10 mm, α=55˚ and ∆z=1 mm, the experiments and numerical simulations of the truncated cone were performed. For the strain measurements, a grid of 2.5 mm diameter circles, spaced 2.5 mm between centre-points, was used. The results of experiment and simulation are shown in Fig.5. The length of circles is increased in the radial direction and remains nearly constant in the tangential direction. At a height of 16 mm from the top of the truncated cone, the average radial strain εr was 0.325. At the same height, the

average value of εr from simulation was about 0.302. The error between them is about 7.6%.

Fig.5 Strain distribution comparison of numerical results with experimental results: (a) Strain distribution in FEM simulation; (b) Strain got by coordinate grid method 2.3.2 Blank thickness

KITAZAWA et al[12−13] researched the aluminum blank thickness variation in ISF by experiments and based on the shear-dominant deformation model. The thickness variation was expressed as t=t0sin α without considering the material thickness anisotropy, where t was the current wall thickness, t0 was the initial blank thickness and α was the inclination angle of the formed part.

The experimental parts were formed under the process parameters of D=20 mm, α=65˚ and ∆z=1 mm, thickness profiles of workpiece along the section cuts were measured and compared with those of numerical results (shown in Figs.6 and 7). It can be seen that a quite satisfactory agreement is obtained. Values of thinnest points in the two results are almost the same. The absolute error is small and the biggest value is only 0.018 mm, which confirms the effectiveness of the utilized numerical model. The experimental result is also compared with the average theoretical value provided by the sine law, showing that the sine law represents only a crude approximation. 3 Numerical simulation results and discussion 3.1 Strain history and distribution

In the final deformed mesh model (shown in Fig.8),

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five selected nodes on the surface of the cone were highlighted to examine the strain paths during the process.

Fig.6 Numerical simulation of thickness distribution

Fig.7 Thickness distribution comparison of numerical results with experimental results

Fig.8 Selection of nodes

At process parameters of D=10 mm, α=55˚ and ∆z= 1 mm, the three principal strains for node C are presented in Fig.9. The second principal strain is very close to zero, which confirms the plane strain assumption for the process as mentioned by ISEKI[9]. The strain component in the circumferential direction can be negligible.

In order to investigate the strain history and distribution during ISF, the strain path was analyzed for

nodes A, B, D and E indicated in Fig.8. During the simulation, four selected nodes are consecutively affected by the tool movement. The strain paths of nodes A, B, D and E are shown in Fig.10. As can be seen in Fig.10, the strain paths are characterized by steps: each strain increment is directly due to the action of the tool when it passes the particular node. However, no strain occurs when the tool continues its path along the same contour and deviates from the node. As a result, the typically small and localized strain can enhance the formability of the blank.

Fig.9 Principal strains for node C during ISF process

Fig.10 Strain history of nodes A, B, D and E during ISF process

Moreover, node A that is close to the center of the blank, is the first to undergo deformation, but it reaches just a limited strain since it undergoes the tool action just for a few contours. Nodes B and D undergo the tool action for the maximum number of contours and consequently the strains also reach the maximum there. And node E that is close to the blank edge is not affected by the tool in the former contours and the deformation takes place later in the process.

The selected node C on the blank was examined to determine the effect of process parameters on the

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equivalent plastic strain. At process parameters of D= 10 mm and α=45˚, as the step down increases (shown in Fig.11(a)), the strain increments imposed at each loop increase, and the total strain remains the same, since it only depends on the wall angle and height of the cone. Thus, with the decrease of the step down, the deformation will be more uniform.

At process parameters of α=45˚ and ∆z=1 mm, with increasing tool diameter (shown in Fig.11(b)), the strain of one step is smaller and more uniform and the final

Fig.11 Effect of process parameters on equivalent plastic strain: (a) Strain history for different step downs; (b) Strain history for different tool diameters; (c) Strain history for different wall angles

accumulated strain decreases. So, lower equivalent plastic strain arises and greater deformation can be achieved using a bigger tool. At process parameters of D=10 mm and ∆z=1 mm, as the wall angle decreases (shown in Fig.11(c)), the final accumulated strain increases. So, if the wall angle is too small, the strain will exceed the forming limit of the material, and then, the blank will be broken. 3.2 Stress state and distribution

In order to investigate the stress conditions in further detail, Fig.12 shows the equivalent plastic strain and the stress triaxiality ratio for the highlighted node C (with process parameters of D=30 mm, α=45˚ and ∆z= 1 mm). The stress triaxiality ratio q is the ratio of the hydrostatic pressure (σm) to the equivalent von Mises stress ( σ ), where σm=(σ1+σ2+σ3)/3, σi (i=1, 2, 3) is principal stress in the integration points, and

2

)()()( 213

232

221 σσσσσσ

σ−+−+−

= .

It is generally recognized that damage behavior of

sheet metal depends strongly on stress triaxiality σm/σ and equivalent plastic strain ε [14−15]. It can be seen that positive peaks in the course of the stress triaxiality ratio coincide with an increase in plastic strain. Thus, compressive stresses are superimposed whenever the tool deforms the selected nodes, indicating that hydrostatic pressure plays a major role during the process. Compressive hydrostatic stress is thought to be the reason for the high forming limits obtained experimentally.

Node C on the blank was chosen to investigate the effects of process parameters on the maximum principle stress. With the tool diameter increasing (α=45˚, ∆z=1 mm) or the wall angle increasing (D=10 mm, ∆z=1 mm), the maximum principal stress decreases (shown in Fig.13).

Fig.12 Equivalent plastic strain ε and stress triaxiality ratio q

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Fig.13 Effect of process parameters on maximum principal stress: (a) Stress history for different tool diameters; (b) Stress history for different wall angle It is well known that the maximum principal stress is mainly responsible for necking and fracture in incremental sheet forming. Therefore, the blank material experiences a smaller axial stress and lower mechanical damage with a larger tool diameter and larger wall angle. 3.3 Thickness distribution

It is worth pointing out that thinning plays a basic role in incremental sheet forming process. “Safe” thinning in incremental forming (such as maximum thinning before fracture) is much larger than that in conventional stamping due to the stress and strain conditions which characterize the process[16]. It is quite obvious that an effective design and control of incremental forming operations requires an accurate and reliable modeling of thinning.

In order to investigate the thickness distribution in further detail, with varying the process parameters (i.e. tool depth step, tool diameter and wall inclination angle), the thickness distributions are plotted respectively (shown in Fig.14). It can be seen that thickness increases with the decrease of tool depth step (D=10 mm, α=45˚), the increase of tool diameter (α=45˚, ∆z=1 mm) or wall

inclination angle (D=10 mm, ∆z=1 mm). So the lower depth step, larger tool diameter and wall inclination angle will lead to more homogeneous thinning reduction and thickness distribution.

Fig.14 Effect of process parameters on thickness distribution: (a) Thickness distribution for different step downs; (b) Thick- ness distribution for different tool diameters; (c) Thickness distribution for different wall angles 4 Conclusions

1) The radial strain and the thickness distribution of the simulation results are in good agreement with experimental results, so the FEM in this work is proved

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to be effective. 2) The second principal strain is very close to zero

through the ISF process, and the deformation pattern of ISF can be simplified as plane-strain deformation.

3) During incremental sheet forming, the plastic strain increases stepwise under the action of the tool, and the typically small and localized strain can enhance the formability of the blank.

4) Each increase in plastic strain is accompanied by hydrostatic pressure, which explains why ISF exceeds the forming limits of conventional sheet forming.

5) With decreasing depth step, increasing tool diameter and wall inclination angle, the axial stress reduces and leads to thinning reduction and more homogeneous plastic strain and thickness distribution. Moreover, the deformation is more uniform and failure is less likely to occur. References [1] KITAZAWA K. Incremental sheet metal stretch-expanding with

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(Edited by ZHAO Jun)